The Derived Pure Spinor Formalism as an Equivalence of Categories
We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duali...
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| description | We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duality for the supertranslation algebra. After introducing and describing the category of supermultiplets, we define the derived pure spinor construction explicitly as a dg-functor. We then show that the functor that takes the derived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor construction, using an explicit calculation. Finally, we illustrate our findings with examples and use insights from the derived formalism to answer some questions regarding the ordinary (underived) pure spinor superfield formalism.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 022, 37 pages
The Derived Pure Spinor Formalism
as an Equivalence of Categories
Chris ELLIOTT a, Fabian HAHNER b and Ingmar SABERI c
a) Department of Mathematics and Statistics, Amherst College,
220 South Pleasant Street, Amherst, MA 01002, USA
E-mail: celliott@amherst.edu
b) Mathematisches Institut der Universität Heidelberg,
Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
E-mail: fhahner@mathi.uni-heidelberg.de
c) Ludwig-Maximilians-Universität München,
Theresienstraße 37, 80333 München, Germany
E-mail: i.saberi@physik.uni-muenchen.de
Received July 12, 2022, in final form April 04, 2023; Published online April 18, 2023
https://doi.org/10.3842/SIGMA.2023.022
Abstract. We construct a derived generalization of the pure spinor superfield formalism
and prove that it exhibits an equivalence of dg-categories between multiplets for a super-
translation algebra and equivariant modules over its Chevalley–Eilenberg cochains. This
equivalence is closely linked to Koszul duality for the supertranslation algebra. After in-
troducing and describing the category of supermultiplets, we define the derived pure spinor
construction explicitly as a dg-functor. We then show that the functor that takes the de-
rived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor
construction, using an explicit calculation. Finally, we illustrate our findings with examples
and use insights from the derived formalism to answer some questions regarding the ordinary
(underived) pure spinor superfield formalism.
Key words: pure spinor superfields; equivalence of categories; supersymmetry; field theory;
BV formalism
2020 Mathematics Subject Classification: 81R25; 17B55; 17B81
1 Introduction
Much effort has been expended over the last fifty years to develop efficient techniques for con-
structions of, and computations in, supersymmetric field theories. The essential difficulty is
visible even in the simplest models: because one asks that the supersymmetry transformations
square to ordinary translations, the supersymmetry transformations on the space of fields typ-
ically only define a representation of the supersymmetry algebra after imposing equations of
motion for fermionic fields. Similarly, if gauge fields are present, the algebra is only represented
up to gauge transformations. For the purposes of quantization, it is desirable for the symmetry
to act on the full space of fields, without regard to the dynamics; to attain this, one must often
pass to a more complicated model, which tends to involve additional “auxiliary fields” that do
not change the physics of the theory. Such auxiliary fields may or may not exist, depending on
the example in which one is interested.
Relatedly, since the action of the Poincaré group on spacetime is geometric in nature and the
super Poincaré group is (under certain assumptions) the only interesting nontrivial possibility
for extended spacetime symmetry that is available for consideration [33], it is pleasing to think of
supersymmetry as arising from the action of particular geometric symmetries on an appropriate
mailto:celliott@amherst.edu
mailto:fhahner@mathi.uni-heidelberg.de
mailto:i.saberi@physik.uni-muenchen.de
https://doi.org/10.3842/SIGMA.2023.022
2 C. Elliott, F. Hahner and I. Saberi
supermanifold, which amounts to giving “superfield” formulations of supersymmetric theories.
Since supersymmetry necessarily acts on the space of superfields, giving such a formulation
implicitly requires extending the field content of the theory, in one fashion or another, by some
set of auxiliary fields as described above. (The literature on supersymmetry is immense, and
we cannot hope to give an overview here; for foundational work in the subject, the reader is
referred to the collection of reprints [26], or the early review [54] and references therein.)
Finding systematic techniques to construct superfield formulations and extend the range
of theories for which they are available has thus been a subject of great interest. Numerous
approaches have been developed, including harmonic superspace [29] and the “rheonomy” ap-
proach to supergravity theories [7]. There have also been approaches in which the space of
so-called pure spinors is used to construct appropriate sets of auxiliary fields, beginning more
than thirty years ago in papers by Nilsson [50] and Howe [35, 36]. Pure spinors were used to
great effect in Berkovits’ formulation of the superstring [4], and their applications to superfield
formulations have been developed in a wide variety of examples, notably in work of Cederwall
and collaborators. See [6, 13, 14] for early papers, and [11] for a review with references to further
literature.
There have been numerous related studies of ideas connected to the space of pure spinors,
for example by Krotov, Losev and collaborators [1, 40] and, notably, in the work of Movshev
and Schwarz on ten-dimensional supersymmetric Yang–Mills theory [44, 45, 46, 47, 48]. See also
the related work of Kapranov [39]. This work was further developed in a mathematical context
in [32], in [31], and in [30], where connections to the theory of Koszul duality were emphasized.
(In particular, the graded Lie algebra Koszul dual to functions on the nilpotence variety was
studied, and used to compute the cohomology of the pure spinor superfield.) Relatedly, work by
Movshev, Schwarz, and Xu on the Lie algebra cohomology of supersymmetry algebras [49, 56]
made the appearance of the space of square-zero elements in that context clear. The cohomology
of the supertranslation algebra appears in various guises in the physics literature, notably in
the context of the “brane scan” [17, 27, 37], where certain cohomology classes are identified as
WZW terms in worldvolume action functionals.
Remark 1.1. It is worth making an orienting remark on terminology at this point. The term
“pure spinor” is often used in the physics literature to refer to points in (or coordinates on)
the space of odd square-zero elements in the supertranslation algebra. This space is sometimes
also called the nilpotence variety [20]. However, the relevant condition here is that of being
a Maurer–Cartan element (having zero self-bracket), which has nothing at all to do with the
spin representation, at least a priori. On the other hand, pure spinors, in the sense of Cartan and
Chevalley [15], are defined to be elements of the spin representation for which the dimension of
the annihilator under Clifford multiplication is maximal; the pure spinors form the minimal orbit
in the action of the spin group on the (projectivized) spin representation. In ten-dimensional
minimal supersymmetry, which was the first example studied [50], the odd elements consist of
a single copy of the spin representation, the bracket is defined using Clifford multiplication, and
the spaces of square-zero elements and pure spinors coincide. This coincidence is responsible
for the confusing terminology; our usage of “pure spinor formalism” in this paper is historically,
rather than logically, motivated.
The pure spinor superfield formalism was reinterpreted in terms of sheaves over the nilpotence
variety in [20]; a detailed analysis of the formalism in the general setting of appropriate gener-
alizations of the supertranslation algebra—odd central extensions of abelian Lie algebras—was
given in [19]. In this version, the pure spinor superfield formalism constructs a supersymmetric
multiplet (a collection of classical fields equipped with an action of the supersymmetry algebra)
out of the datum of a graded module over the ring of functions on the nilpotence variety of the
supertranslation algebra. The output of the pure spinor superfield formalism is a locally free
The Derived Pure Spinor Formalism as an Equivalence of Categories 3
sheaf on superspace, equipped with a differential and a module structure making it a multiplet.
One can think of it as resolving the more standard component field formulations freely over
superspace (and thus a type of superfield formalism). There is a filtration that allows one to
pass to a minimal quasi-isomorphic presentation that is a locally free sheaf on the spacetime;
this minimal presentation agrees in examples with the typical component-field descriptions of
supermultiplets in the literature. Depending on the example being considered, a set of auxiliary
fields may be generated automatically; more generally, though, the output includes differen-
tials that resolve the equations of motion of the free multiplet, thus giving a description in
the Batalin–Vilkovisky formalism. In this formalism, auxiliary fields are not essential, and an
on-shell action of supersymmetry is formalized as a homotopy action of supersymmetry on the
derived critical locus of the action.
It is natural to ask for a characterization of all multiplets which arise in this manner. In [19],
an example of a multiplet which cannot be constructed using the technique was described.1 The
failure in this particular case was linked to the geometry of the underlying nilpotence variety:
it is not Cohen–Macaulay, so that the dualizing complex of its ring of functions has cohomology
in multiple degrees and is not quasi-isomorphic to a single homogeneous module. At the level
of multiplets, the structure sheaf gives rise to the four-dimensional N = 1 vector multiplet. On
general grounds, one expects that the dualizing module gives rise to the dual (or “antifield”)
multiplet. However, due to the failure of the Cohen–Macaulay property, this does not work on
the nose. Instead, the appearance of the dualizing complex suggests that a generalization of the
pure spinor superfield formalism to the world of derived algebraic geometry is necessary.
In this paper, we tackle these questions systematically, and show that a derived generalization
of the pure spinor formalism can be used to produce every supermultiplet in a very general
setting. Let n2 and n1 be vector spaces. We regard n2 and Πn1 (where Π denotes the parity
shift functor) as abelian super Lie algebras, and consider an odd central extension of the form
0 → n2 → n → Πn1 → 0. (1.1)
Since the Lie algebra cohomology group H2(Πn1) with trivial coefficients is, here, just given
as Sym2 n∗1, such a central extension is equivalent to a choice of map Γ: Sym2(n1) → n2 that
defines the bracket. We can extend by a chosen subalgebra g0 of the even automorphisms of n,
getting another extension sequence of the form
0 → n → g → g0 → 0.
(n clearly has no even inner automorphisms.) The super Lie algebra g will act on the supergroup
N = exp(n), and its even subalgebra g+ will act on the body V of N . The automorphism Lie
algebra aut(n) will include an abelian summand than can be thought of as “scale transforma-
tions,” giving rise to an internal weight grading which places n1 in degree one, n2 in degree two,
and g0 → aut(n) in degree zero; this will either be internal or not, depending on whether the
summand is included in g0.
Our motivating example will be the case where V = Rn is a Riemannian vector space, g+
is the Poincaré algebra of infinitesimal isometries of Rn, and g is the super Poincaré algebra.
However, other examples are relevant to physics as well, for example when considering twisted
theories in the pure spinor formalism along the lines of [52].
Let Multg denote the dg-category of g-multiplets: roughly speaking, these will be differential
graded vector bundles on V , equipped with a g-action for which the action of g+ is “geometric”,
i.e., inherited from the action on spacetime. We will denote multiplets by triples (E,D, ρ), con-
sisting of a vector bundle, differential and g-action; precise definitions will be given in Section 2.
Our main result, stated somewhat informally, says the following.
1The obstruction to constructing this multiplet using the pure spinor formalism was previously observed by
Martin Cederwall.
4 C. Elliott, F. Hahner and I. Saberi
Theorem (Theorem 4.3). There is an equivalence of dg-categories
Multg ⇆ Modg0C•(n) (1.2)
between g-multiplets and g0-equivariant modules over the Chevalley–Eilenberg algebra C•(n).
We will view the functor from left to right as a natural derived enhancement of the pure
spinor construction. Indeed, C•(n) can be regarded as a bigraded commutative differential
algebra, since the action of rescalings in g0 has integral eigenvalues. Taking Lie algebra cochains
of the exact sequence (1.1), we obtain the sequence
C → C•(Πn1) → C•(n) → C•(n2) → C
of bigraded cdgas, witnessing C•(n) as an R-algebra, where R := C•(Πn1) is the free commu-
tative algebra on n∗1. If we totalize the bigrading, the complex is concentrated in non-positive
degrees. Its degree-zero cohomology then takes the form R/I, where I is the quadratic ideal
that arises as the image of the dual Γ∗ to the bracket map. The (affine) nilpotence variety is
Spec(R/I), so that modules over this ring correspond to quasi-coherent sheaves on the nilpotence
variety. Furthermore, there is a natural equivariant map
C•(n) → R/I,
so that any R/I-module is a C•(n)-module in a natural way. When applied to modules of this
sort, our derived enhancement agrees on the nose with the standard pure spinor formalism.
From this point of view, we can view C•(n) as a derived enhancement of the nilpotence
variety: its spectrum is an affine derived scheme whose underlying classical affine scheme is
exactly the affine nilpotence variety, but whose derived data remembers the higher cohomology
of n with respect to the totalized grading.
Remark 1.2. While we will not need any technology from the theory of derived algebraic
geometry in this paper, we refer the interested reader to the article [55] of Toën for a survey.
The functor in (1.2) from Modg0C•(n) to multiplets is defined by extending the construction
in [19]. It is closely connected to more standard versions of Koszul duality in at least two ways.
First of all, it can be thought of as arising from the dual pair (C•(n), U(n)). The standard
Koszul duality functor is generated by the kernel
(K′,D′) =
(
C•(n)⊗ U(n), Xi ⊗ ni
)
,
where
{
Xi
}
is a basis for n∗, therefore a set of generators for C•(n), and ni denotes the corre-
sponding basis of n (thus set of generators for U(n)).
If we now view C∞(N) as a right module for U(n), where n acts by right-invariant vector
fields—in particular, ni by the vector field Di := ρ(ni)—we can modify this kernel to obtain
a (C•(n), C∞(N))-bimodule of the form
(K,D) =
(
C•(n)⊗ C∞(N), Xi ⊗Di
)
.
Our functor is defined as the integral transform associated to this kernel. From this perspective,
we can interpret the functor heuristically in two steps:
1. First form the Koszul dual U(n)-module of a module over the Chevalley–Eilenberg complex.
2. Then apply the associated bundle construction to obtain a dg-vector bundle over the
supergroup N , with a residual right N -action. In particular, we can forget down to a dg-
vector bundle on the even part V of N , with a geometric action of the supersymmetry
algebra n.
The Derived Pure Spinor Formalism as an Equivalence of Categories 5
It is another version of Koszul duality which is perhaps most closely connected. Kapranov [38]
established a version of Koszul duality that provides a Quillen equivalence between the model
categories of D-modules on a space and Ω•-modules over the same space. Our construction
can be understood as a version of this equivalence in the case where that space is the nilpotent
super Lie group N ; it relates translation-invariant natural vector bundles to modules over the
translation-invariant differential forms, which are precisely the Lie algebra cochains. The con-
nection to the notion of “multiplet” in the physics literature has, as far as we know, not been
appreciated before.
Remark 1.3. While Koszul duality provides a natural language in which to understand the pure
spinor construction, our proof of Theorem 4.3, and all of our discussion of examples, will proceed
by direct computation using the kernel (K,D). This is meant to emphasize that the technique
provides not just an abstract equivalence, but a set of efficient and practical computational
techniques. We also emphasize that the version of Koszul duality encapsulated in the pure
spinor formalism naturally produces dg vector bundles whose sections are sheaves on the site of
manifolds equipped with appropriate structure: in the case of standard super-Poincaré algebras,
this means that one knows how to place the resulting multiplet on any Riemannian manifold.
We see this as being connected to ideas in Cartan geometry for the model space N ∼= G/G0,
and hope to return to versions of the formalism on non-flat spacetimes (for nontrivial or more
general Cartan geometries) in future work.
Finally, it is worth remarking that the functor from right to left in (1.2) is also easy to
understand: it is just taking the derived n-invariants of a multiplet. This is intuitively satisfying
in various respects. For example, the twist of a multiplet consists of the derived invariants of
some abelian odd subalgebra of n, whereas its dimensional reduction consists of the invariants
of an abelian even subalgebra of n. The fact that a multiplet can be recovered from the datum
of its derived n-invariants thus says, in a sense, that it is equivalent to all of its possible twists,
considered as a natural family over the derived classifying space Bn.
The upshot of this result is that, after an appropriate derived upgrade, every supermultiplet
has a pure spinor superfield description. We can use the equivalence of categories to shed some
light on the motivating questions concerning the underived pure spinor formalism discussed
above. For example, the following is an immediate consequence:
Theorem (Corollary 4.6). A given multiplet (E,D, ρ) lies in the image of the underived pure
spinor formalism if and only if the Chevalley–Eilenberg cohomology H•(n, E) is concentrated in
a single degree.
We illustrate the derived formalism with some applications and examples; we also show that
the derived formalism gives rise to a construction of certain maps relating the multiplets asso-
ciated to the different Chevalley–Eilenberg cohomology groups of the supertranslation algebra,
associated to a filtration on C•(n). We give explicit calculations for examples with minimal
supersymmetry in dimensions three and four.
Further directions
The realization of the pure spinor formalism as an equivalence of dg-categories offers potential
insight to numerous more involved applications. We give a few indications of possible directions
here:
1. Given any multipletM = (E,D, ρ), associated to any supersymmetry algebra g with asso-
ciated supertranslation algebra n, it is possible to realize a C•(n)-module Γ generating M
via the pure spinor formalism: one can simply set Γ = C•(n, E). These modules have
6 C. Elliott, F. Hahner and I. Saberi
cohomology consisting of a graded sum of finitely generated modules over the ring of func-
tions on the nilpotence variety, together with additional information encoding the action
of the generators of C•(n) in nonzero total degree. By the results of [52], they efficiently
and fully encode the information of the various twists of the original multiplet M : one
can compute the stalk of the module at a classical point Q : C•(n, E) → C, obtaining de-
scriptions of different twists for different orbits under the action of Spin(n) and the group
of R-symmetries. We refer to the discussion in [20, 22, 23] for details of this classification.
It would, for example, be interesting to explicitly understand the modules associated to
multiplets such as the N = (1, 0) tensor multiplet in six dimensions.
2. One can use the Batalin–Vilkovisky (BV) formalism to construct interacting supersym-
metric classical field theories using the pure spinor formalism. In the BV formalism,
an interacting supersymmetric field theory is encoded by a cyclic L∞ structure on a mul-
tiplet M , that is, an L∞ algebra structure together with a pairing of degree −3. This
additional data can be carried along the pure spinor functor; it is enough to define a Lie
algebra structure and shifted symplectic pairing internal to equivariant C•(n)-modules.
This will require a bit more input; it is not straightforward to define a natural tensor
product on the category of multiplets, so one would need to work instead with the D-
modules obtained by taking the sheaf of sections of the multiplet with its natural tensor
product. One would then aim use the equivalence to establish a “convolution” monoidal
structure ∗ making the functor monoidal.
For example, as computed in [14] and discussed in [52, Section 7.3] and [18], if n is the
eleven-dimensional N = 1 supertranslation algebra, the eleven-dimensional supergravity
multiplet arises from the zeroth cohomology of C•(n), placed in degree −3 with respect
to the natural weight grading, as a C•(n)-module. In order to obtain an interaction,
it would suffice to define, up to homotopy, a Lie structure on this module. It would be
interesting to understand the action functionals of [8, 10] in this language, and to use them
to connect to component actions for perturbative supergravity, either twisted or untwisted.
An interacting BV theory conjecturally describing the minimal twist of eleven-dimensional
supergravity was studied in [51].
3. There are several sheaves over the (derived) nilpotence variety that automatically carry
Lie structures. For example, if h is a finite-dimensional semisimple Lie algebra, one can
form the tensor product H0(n)⊗h.2 It has been known for some time that this Lie algebra
describes interacting ten-dimensional super Yang–Mills theory in the BV formalism; the
correct gauge algebras for lower-dimensional super Yang–Mills theories are also obtained
in this fashion.
In dimensions higher than eleven we expect C•(n) to have no higher cohomology, so
Spec(C•(n)) will be purely classical. Equivalently, the nilpotence variety is expected to
be a complete intersection according to Hartshorne’s conjecture: roughly speaking, the
nilpotence variety is determined by a system of n equations, while the number of variables
is of order 2n/2. Hartshorne’s conjecture states that any smooth projective variety in Pn
with codimension < n/3 is a complete intersection. The codimension condition applies
to nilpotence varieties with minimal supersymmetry in dimension ≥ 12, although these
varieties are not smooth (but see for instance the recent article [24] for version of this
result that would be applicable to the example of nilpotence varieties in high dimensions).
However, it is often possible to obtain non-trivial cohomology by taking—rather than the
2When we discuss Lie algebra cohomology H•(n), we will view n as being a Z-graded object with respect to
the weights of the natural rescaling action. As such, the complex C•(n) is naturally bigraded; we refer here to the
zeroth cohomology with respect to the total grading. We will discuss these degree conditions further in Section 3.1.
The Derived Pure Spinor Formalism as an Equivalence of Categories 7
entire Chevalley–Eilenberg complex—a non-trivial quotient associated to an orbit closure
within the nilpotence variety.
To give a further example, the tangent sheaf Der(C•(n)) carries a Lie bracket for all super-
translation algebras n. While the associated multiplet is not generally associated to a BV
theory—it does not typically carry a −1-shifted symplectic pairing—one can always build
a BV theory by applying the cotangent theory construction and considering the multiplet
M ⊗M ![−1], where M ! =M∗⊗Ωtop(V ) is the density-valued dual. This procedure allows
for the construction of a very general family of interacting classical supersymmetric field
theories.
4. The equivalence of categories allows for a program to classify families of multiplets starting
from the algebraic geometry of the (derived) nilpotence variety. This direction is explored
in [34], where (among other things) a description of all six-dimensional N = (1, 0) multi-
plets whose derived invariants form a single line bundle on the projective nilpotence variety
is given.
2 The category of multiplets
In [19], the notion of a g-multiplet for a super Lie algebra g was formalized. Let us quickly review
the relevant definitions, and then move on to define morphisms between g-multiplets and study
the resulting category Multg. We start with the case of strict multiplets and strict morphisms
and then discuss the homotopy-theoretic generalization.
In brief, by a multiplet on a smooth manifold X we will mean a graded vector bundle E
on X together with a differential D and an action ρ of an appropriate super Lie algebra g,
compatible with an action of the even part of g on the base manifold X. The reader should
consider the prototypical situation where X = Rn, and g is a super Lie algebra whose even
part is the Poincaré algebra of infinitesimal isometries of Rn. A multiplet can be thought of as
modelling the perturbations of a classical field in the BV formalism.
Remark 2.1. We would like to clearly emphasize here the distinction between multiplets and
theories in this context. In general, the data of a multiplet is less data than the data of a classical
field theory: it does not include enough data to specify dynamics. The data of a perturbative
classical supersymmetric field theory in the BV formalism includes the data of a multiplet,
together with some additional data. If a multiplet is equipped with the additional data of a BV
bracket (an odd shifted symplectic structure), it can be viewed as modelling a free theory in
the BV formalism. Such structures can easily be constructed from multiplets by taking the
sum of a multiplet with a shift of its dual. If one wishes to include non-trivial interactions, one
must additionally equip the sections of the bundle E with an L∞ structure, compatibly with
the supersymmetry action.
So, while the results of this paper do apply to BV field theories, when we speak of multiplets
we are referring to a less constrained notion, and to obtain a genuine field theory from a multiplet
one needs to specify strictly more data.
2.1 Gradings
Many objects appearing throughout this work (e.g., vector spaces, vector bundles, Lie algebras
etc.) will carry a grading by Z×Z/2Z as well as a differential of bidegree (1,+). We refer to such
objects with the prefix “dgs” standing for “differential graded super”. Typically these gradings
have a clear physical meaning: the integer grading refers to the ghost number or cohomological
degree, while the Z/2Z-grading corresponds to the intrinsic parity (fermion number modulo two).
8 C. Elliott, F. Hahner and I. Saberi
The total parity denotes the Z/2Z-grading which arises by forgetting the Z-grading to Z/2Z and
then totalizing with the intrinsic parity. It is the total parity which governs all signs.
It will often be convenient to introduce an additional piece of data: an auxiliary action of
a one-dimensional abelian Lie algebra R on a dgs vector space. Many of our examples will be
built from the following simple observation.
Example 2.2. Let n be a dg Lie algebra concentrated in degrees 1 and 2. For all non-negative
integers k, there is a natural action of R on Symk(n∗), where R acts on duals to the degree 1
and 2 summands of n∗ with weight −1 and −2 respectively.
The results in this paper will make sense for arbitrary R-equivariant dgs vector spaces, but
our most common examples will have the following behavior.
Definition 2.3. A lifted dgs vector space is an R-equivariant dgs vector space where the R-
action has integral weights, and the Z/2Z-grading coincides with the R-weight modulo two.
Let us now introduce some notation that we will use when we discuss lifted dgs vector bundles
on a smooth manifold X. Let us write
E =
⊕
(w,d)∈Z2
Ew,d
for a lifted dgs vector bundle, where the first index indicates the decomposition of E by R-weight,
and the second index indicates internal Z-degree on E. In some of our calculations, particular
in Section 5, we will place the summands in a two-dimensional array, where the two coordinates
are given by w − d and d. For example,
E =
· · · E−1,0 E0,0 E1,0 E2,0 · · ·
· · · E0,1 E1,1 E2,1 E3,1 · · ·
· · · E1,2 E2,2 E3,2 E4,2 · · ·
.
So we recover w as the total degree with respect to this sheared bigrading. The overall parity is
then determined by the column.
If the differential on our dgs vector bundle decomposes as a sum of homogeneous differential
operators of order k, say D =
∑
k≥0Dk, the summand Dk acts—with respect to the sheared
grading—with bidegree (2k − 1, 1). So the homogeneous summands all increase the vertical
degree by 1, but they may modify the horizontal degree by any odd integer ≥ −1.
2.2 Strict multiplets
For a dgs vector bundle3 (E,D) over a base smooth manifold X, we denote the space of global
smooth sections by E = Γ(X,E). The endomorphisms End(E) form a dgs Lie algebra, where the
bracket is given by the commutator and the differential is [D,−]. Inside (End(E), [D,−]) there
is a sub dgs Lie algebra consisting of all endomorphisms which act on sections via differential
operators. We denote this subalgebra by (D(E), [D,−]).
Definition 2.4. A strict local dgs g-module is a triple (E,D, ρ) where (E,D) is a dgs vector
bundle and
ρ : g −→ (D(E), [D,−])
is a map of dgs Lie algebras. Here the super Lie algebra g is viewed as a dgs Lie algebra in
cohomological degree zero with trivial differential.
3Note that each graded piece Ek is a finite rank vector bundle, but the total rank of E may still be infinite.
The Derived Pure Spinor Formalism as an Equivalence of Categories 9
Remark 2.5. This definition (as well as many of the following) has a natural generalization for g
a dgs Lie algebra. Since we are ultimately interested in the pure spinor superfield formalism, we
restrict our attention to super Lie algebras with no cohomological grading.
Note that, since a super Lie algebra g has no differential, ρ commutes with the differential
on the dgs vector bundle,
[D, ρ(x)] = 0 ∀x ∈ g.
It is standard to encode a g-module structure ρ on (E,D) as a dgs Lie algebra structure on the
direct sum g⊕ E . Concretely we set for the unary and binary operations
µ1(x1, σ1) = (0, Dσ1),
µ2((x1, σ1), (x2, σ2)) =
(
[x1, x2], ρ(x1)σ2 − (−1)|σ1||x2|ρ(x2)σ1
)
, (2.1)
where x1, x2 ∈ g and σ1, σ2 ∈ E .
There is an obvious notion of morphisms between strict local g-modules.
Definition 2.6. A strict morphism of strict local g-modules (E,D, ρ) and (E′, D′, ρ′) is a map
of cochain complexes
ψ : E −→ E ′
realized by differential operators such that
ψ ◦ ρ(x) = ρ′(x) ◦ ψ
for all x ∈ g.
A strict morphism ψ : (E,D, ρ) −→ (E′, D′, ρ′) gives rise to a strict morphism of the associ-
ated dgs Lie algebras by setting ψ̃ = idg×ψ : g ⊕ E −→ g ⊕ E ′. Conversely it is easy to check
that every strict morphism of dgs Lie algebras of that form gives rise to a strict morphism of
g-modules. We call ψ a quasi-isomorphism if it is a quasi-isomorphism of dgs vector bundles;
equivalently ψ̃ is a quasi-isomorphism of dgs Lie algebras.
Now, suppose that (E,D) is a dgs vector bundle over an affine space V (so, in the notation
used above, V = X). Let us additionally assume that g is equipped with a map of super Lie
algebras
ϕ : aff(V ) −→ g.
In essence, when we refer to strict g-multiplets, we are referring to strict local g-modules for
which the affine transformations acts in a geometric way.
Definition 2.7. A strict g-multiplet (E,D, ρ) on V is an affine dgs vector bundle over V
equipped with a strict local g-module structure
ρ : g −→ D(E)
such that the pullback of the module structure along ϕ agrees with the natural action on sections
of the affine vector bundle. Concretely, this means that the following diagram commutes:
g D(E).
aff(V )
ρ
aff
ϕ (2.2)
10 C. Elliott, F. Hahner and I. Saberi
Here aff denotes the natural action of the affine algebra on E. We define the category of
strict multiplets with strict morphisms to be the full subcategory of the category of strict local
g-modules with objects strict g-multiplets. We denote this category by Multstrictg .
Remark 2.8. The condition (2.2) stating that the translations act geometrically heavily depends
on ϕ. In particular, since the action aff is effective, the corresponding category of g-multiplets
is trivial if ϕ is not injective.
Remark 2.9. Note that a morphism of strict local g-modules is automatically compatible with
the action of aff(V ). For example let ψ : (E,D, ρ) −→ (E′, D′, ρ′) be a strict morphism. Then
we have
ψ ◦ aff(x) = ψ ◦ ρ(ϕ(x)) = ρ′(ϕ′(x)) ◦ ψ = aff ′(x) ◦ ψ.
Therefore it is sensible to define Multstrictg as a full subcategory of strict local g-modules.
2.3 Homotopy theory of multiplets
It is well known that various strict algebraic structures, such as associative algebras or Lie
algebras, admit homotopy-theoretic generalizations (A∞ or L∞ algebras in these cases). The
same is true for strict g-module structures, which generalize to L∞ g-modules. In the context
of this work, we will refer to such homotopy module structures simply as module structures
and choose to emphasize whenever a module is strict instead. In this spirit we can easily define
(not necessarily strict) local g-modules and g-multiplets by replacing Lie maps in the above
definitions with L∞ maps.
Thus, a local g-module is just a dgs vector bundle (E,D) together with an L∞ map of L∞
algebras
ρ : g⇝ D(E).
Recall that this means there are component maps
ρ(k) : g⊗k −→ D(E)[1− k], k ≥ 1
satisfying a series of compatibility relations the lowest of which reads[
ρ(1)(x), ρ(1)(y)
]
− ρ(1)([x, y]) =
[
D, ρ(2)(x, y)
]
∀x, y ∈ g.
Clearly a local g-module is strict if and only if ρ(k) = 0 for all k ≥ 2.
Remark 2.10. As we discussed in the introduction, considering this homotopical version of
a module for a super Lie algebra is very natural in supersymmetric field theory. In most super-
symmetric classical field theories, when one describes the most natural multiplet of fields, there
is only a strict action of the supersymmetry algebra on-shell, i.e., after imposing the equations
of motion. If one, however, allows L∞ actions, it is possible to describe an L∞ action of the
supersymmetry algebra on the space of fields in the BV formalism directly, without introducing
auxiliary fields. This idea goes back to the beginnings of the BV formalism; applications to
global symmetries are discussed, for example, in [3]. The example of super Yang–Mills theory
is discussed rigorously and systematically in all dimensions in [23].
Similarly to the strict case, we can conveniently encode a g-module structure ρ as an L∞
structure on g ⊕ E . To this end we supplement the operations (2.1) by the following brackets
for k ≥ 3 (for details we refer to [2, 42])
µk((x1, v1), . . . , (xk, vk)) =
(
0,
k∑
i=1
±ρ(k−1)
(
x1, . . . , x̂i, . . . , xk
)
vi
)
.
The Derived Pure Spinor Formalism as an Equivalence of Categories 11
One can define a morphism ψ : (E,D, ρ) −→ (E′, D′, ρ′) of local g-modules by component maps
ψn : g⊗n−1 ⊗ E −→ E ′,
which are given by differential operators and satisfy a series of compatibility relations (see [2]
for details). We recover strict morphisms by restricting to those where the only component map
is ψ1. Again, we can describe such morphisms by morphisms of the associated L∞ algebras
ψ̃ : g⊕ E ⇝ g⊕ E ′
by supplementing ψ̃1 = idg×ψ1 with
ψ̃k((x1, σ1), . . . , (xk, σk)) =
(
0,
k∑
i=1
±ψk
(
x1, . . . , x̂i, . . . , xk, σi
))
for k ≥ 2. ψ is a quasi-isomorphism of local g-modules if ψ1 is a quasi-isomorphism of cochain
complexes, or equivalently if ψ̃ is a quasi-isomorphism of L∞ algebras. In general, encoding
module structures as L∞ structures is very convenient because it allows the use of many known
tools from the theory of L∞ algebras, like homotopy transfer for instance.
Remark 2.11. An equivalent realization of the notion of a non-strict morphism between a pair
of local L∞ algebras g, g′ is provided by considering the Chevalley–Eilenberg chain complex C•(g)
of an L∞ algebra g. This is a cocommutative dg coalgebra whose underlying graded coalgebra
is Sym•(g[−1]), with differential induced from the L∞ structure on g as a sum of terms given
by the L∞ brackets on g. The data of a morphism ψ : g → g′ of local L∞ algebras is equivalent
to that of a morphism of cocommutative dg coalgebras
ψ′ : C•(g) → C•(g
′)
given by differential operators. This interpretation allows us to view local L∞ algebras as
objects of a dg-category, so that we can discuss (for example) homotopies between L∞ algebra
morphisms. When the source g is finite-dimensional, we can dually consider morphisms of
commutative dg algebras C•(g′) → C•(g). This is for example the case for the local g-module
structures ρ : g⇝ D(E) appearing in the definition of multiplets.
Let us now give the definition of a (not necessarily strict) multiplet.
Definition 2.12. A g-multiplet (E,D, ρ) is an affine dgs vector bundle over V equipped with
a local g-module structure
ρ : g⇝ D(E)
such that the pullback of the module structure along ϕ agrees strictly with the natural action
on sections of the affine vector bundle. Concretely, this means that the following diagram
commutes:
g D(E).
aff(V )
ρ(1)
aff
ϕ
We define the dg-category of g-multiplets to be the full subcategory of local g-modules with
objects being g-multiplets and denote it by Multg.
12 C. Elliott, F. Hahner and I. Saberi
Remark 2.13. Note that, in particular, the action ρ ◦ ϕ obtained by restricting the L∞ action
to affine transformations is necessarily strict.
We will sometimes wish to refer to the full subcategory of strict multiplets, but allowing
arbitrary morphisms.
Definition 2.14. Denote by Multstrict-obg the full sub dg-category of Multg generated by strict
multiplets.
We can obtain homotopy categories from the dg-categories Multg as well as Multstrictg by
replacing the hom space HomMultg(E,E
′) by its zeroth cohomology H0(HomMultg(E,E
′)). We
denoted the resulting categories by Ho(Multg) and Ho(Multstrictg ). Speaking physically, quasi-
isomorphisms of g-multiplets correspond to perturbative equivalences of multiplets, where by
“perturbative” here we mean equivalences of the derived formal neighborhoods of a point in the
classical moduli field space. Therefore isomorphism classes in the homotopy category correspond
to perturbatively distinct multiplets.
Remark 2.15. Because every L∞ algebra can be strictified [41, Part II, Corollary 1.6], the
natural inclusion
Ho
(
Multstrict-obg
)
→ Ho(Multg)
is an equivalence of categories, where Multstrict-obg denotes the full subcategory of multiplets
spanned by strict objects.
Example 2.16. Let us give one example of a non-strict multiplet for three dimensional N = 1
supersymmetry. Recall that Spin(3) ∼= SU(2); we denote the two dimensional spinor represen-
tation by S and the three-dimensional vector representation by V . We fix g to be the super
Poincaré algebra, whose underlying Z/2Z-graded vector space is of the form
g = (so(3)⊕ V )⊕ΠS
where Π indicates the shift into odd parity. The symmetric bracket is induced from the iso-
morphism Γ: Sym2(S) ∼= V of SU(2)-representations. Let us define E to be the trivial vector
bundle over V = R3 with fibers
E−1
+ = C, E0
+ = V, E0
− = S,
where we have used the subscript ± to indicate Z/2Z-degree. The field content consists of
a zero-form, a one-form and a fermion field. The differential D operates on sections as the de
Rham differential d : Ω0 −→ Ω1, and vanishes elsewhere. This dgs vector bundle can be lifted
to a Z× Z-graded vector bundle where E0
+ has weight one, and E0
− has weight two.
We summarize this field content with the below array, using the conventions of Section 2.1: Ω0
Ω1 S
d
.
The even part of g acts in the standard geometric fashion. For Q ∈ g− we set
ρ(1)(Q) : S −→ Ω1, ψ 7→ Γ(Q,ψ),
Ω1 −→ S, A 7→ Q ∧ /∂A,
ρ(2)(Q,Q) : Ω1 −→ Ω0, A 7→ ι[Q,Q]A.
Here ι denotes the contraction of a differential form by a vector field and we view [Q,Q] as
a constant vector field on X.
The Derived Pure Spinor Formalism as an Equivalence of Categories 13
2.4 Linear structure
We can define the direct sum of two g-multiplets as follows.
Definition 2.17. The direct sum of two g-multiplets (E,D, ρ) and (E′, D′, ρ′) is defined to be
the multiplet
(E,D, ρ)⊕ (E′, D′, ρ′) = (E ⊕ E′, D ⊕D′, ρ⊕ ρ′).
Remark 2.18. In contrast, defining a tensor product on the category of g-multiplets is not
straightforward. Considering the D-modules of global sections, one can take the tensor product
in the category of D-modules, but this does not take the additional structures on a multiplet
into account. As alluded to in the introduction, one can might also try using the equivalence
established in Section 4 to define a product on the category which makes the functor monoidal.
We will not discuss this issue further in this work.
For further reference, we also define the dual of a g-multiplet. In physics context, these are
usually called antifield multiplets
Definition 2.19. The dual of a g-multiplet (E,D, ρ), also referred to as the associated antifield
multiplet is the g-multiplet
(
E!, D∗, ρ∗
)
. Here E! is the linear dual vector bundle to E twisted
by the canonical bundle. The action ρ∗ denotes the map
ρ∗ : g⇝ D
(
E!
)
given by ρ∗(k)(Q1, . . . , Qk) = ρ(k)(Q1, . . . , Qk)
∗.
Note that we will typically be working over the flat space Rn, so we may choose a trivialization
of the canonical bundle if we wish to identify E! with E∗.
3 Derived pure spinor superfields
In this section, we define the derived generalization of the pure spinor superfield construction,
which will provide one of the two functors that witness the equivalence of categories. We begin
by setting up the general context in which we want to work.
As in the introduction, let n be a two-step nilpotent super Lie algebra, defined by a central
extension
0 → n2 → n → Πn1 → 0 (3.1)
of the odd abelian super Lie algebra Πn1. We imagine such objects as generalizations of su-
pertranslation algebras. The automorphisms of n, which are all outer, will contain an abelian
factor generating scale transformations, with respect to which n1 has weight one and n2 weight
two. There is also a natural map
aut(n) → gl(n2).
The kernel of this map can be thought of as the R-symmetry algebra; in physical examples,
aut(n) will be the product of so(n), scale transformations, and the R-symmetry algebra.
All of our constructions will take place in reference to a fixed super Lie algebra g of the
following type:
14 C. Elliott, F. Hahner and I. Saberi
Definition 3.1. A super Lie algebra g is of super Poincaré type if it can be written as an
extension
0 → n → g → g0 → 0,
where n is a two-step nilpotent super Lie algebra of the form (3.1) and g0 is a Lie algebra
equipped with a Lie map g0 → aut(n).
This means that the Z/2Z grading on g can be lifted to a Z-grading concentrated in degrees
zero, one, and two:
g = g0 ⊕ n1 ⊕ n2.
The most important examples are the super Poincaré algebras in various dimensions and with
various amounts of supersymmetry.
It is possible to consider examples where g0 is any subalgebra of aut(n). We will typically wish,
however, to take g0 to be such that any linear transformation of n2 arising from an automorphism
of n can be generated by an element of g0. In the super-Poincaré case, this will mean that so(n)
is contained in g0. In this paper, we will always simply take g0 = aut(n).
Let us choose a basis dα for n1 and eµ for n2 such that we can expand the symmetric bracket
in structure constants4[
dα, dβ
]
= fµαβeµ.
We denote by R = Sym• (n∗1) = C[λα] the ring of polynomial functions on n1. For Q in n1, the
equation [Q,Q] = 0 defines an ideal I in R, explicitly given by
I =
(
λαfµαβλ
β
)
.
As we will discuss shortly, we can identify the quotient algebra R/I with the degree zero Lie
algebra cohomology H0(n). Here, by degree zero, we mean with respect to the totalization of
the canonical Z× Z-grading on Chevalley–Eilenberg cochains.
The pure spinor superfield formalism gives a systematic tool to construct g-multiplets from
the input datum of a graded g0-equivariant R/I-module. Here we generalize the pure spinor
superfield construction to g0-equivariant C
•(n)-modules and show that it defines a functor.
3.1 The category of C•(n)-modules
Recall that the Chevalley–Eilenberg complex of n takes the form
C•(n) =
(
Sym• (n∗[1]),dCE
)
,
where the Chevalley–Eilenberg differential dCE is induced by the dual of the bracket. Here the
notation n∗[1] means that we shift n∗ down in cohomological degree by one. The Chevalley–
Eilenberg complex has a Z× Z/2Z-grading endowing it with the structure of a dgs algebra. In
our present case, since the the Z/2Z grading of n lifts to a Z-grading where ni has weight i, C
•(n)
can be given a Z×Z-grading. Totalizing this grading, the generators of n∗1 sit in degree 0 while
the generators of n∗2 live in degree −1. We denote these generators by λα and vµ respectively.
We can thus identify
C−p(n) = ∧pn∗2 ⊗R
with respect to the totalized grading.
4In the context of super Poincaré algebras, these structure constants are typically expressed in terms of the
matrix elements of the gamma matrices.
The Derived Pure Spinor Formalism as an Equivalence of Categories 15
The Chevalley–Eilenberg differential acts on these generators according to
dCEλ
α = 0,
dCEv
µ = λαfµαβλ
β.
In coordinates we will often write the Chevalley–Eilenberg algebra in the form
(C•(n), dCE) =
(
C[λα, vµ], dCE = λαfµαβλ
β ∂
∂vµ
)
.
Using this description, we immediately see that we indeed recover H0(n) = R/I.
C•(n) is a dgs algebra, therefore we can consider dgs modules over it.
Definition 3.2. A C•(n)-module is a dgs vector space (Γ,dΓ) together with a morphism
(C•(n), dCE) −→ (End(Γ), [dΓ,−]),
of dgs algebras.
Definition 3.3. A morphism of C•(n)-modules (Γ,dΓ) and (Γ′, dΓ′) is a cochain map
f : (Γ,dΓ) −→ (Γ′, dΓ′)
such that
f(x ·Γ γ) = x ·Γ′ f(γ).
In other words, f is just a morphism of dgs modules.
Note that g0 acts on both n1 and n2 and thus also on C•(n).
Definition 3.4. A C•(n) module Γ is called g0-equivariant if Γ is also a representation of g0
and the module structure map is g0-equivariant. We further assume that each degree Γk is
finite dimensional as a complex vector space. We will denote the dg-category of g0-equivariant
C•(n)-modules by Modg0C•(n).
Our main results will take place in this category of g0-equivariant modules.
3.2 The derived pure spinor superfield formalism
There is a canonical strict multiplet associated to the super Lie algebra g: the free superfield.
Recall that n is a two-step nilpotent super Lie algebra and let N = exp(n) be the associated
nilpotent super Lie group. Since all such nilpotent super Lie groups are split, N can be chosen
to be the total space of a bundle with fiber ΠN1 over N2, where N2 is a vector space treated
as an additive abelian group. (As super vector spaces, n and N are isomorphic; the group
operation can be constructed using the Baker–Campbell–Hausdorff formula, which terminates
at quadratic order in this case.)
Left and right translations induce two commuting, aut(n)-equivariant actions of n on N by
vector fields
L , R : n −→ Vect(N).
Let us choose coordinates xµ on the abelian group N2 and θα on N1. In terms of these coordi-
nates, the vector fields given by the action of the odd elements can be described as follows:
R(dα) =
∂
∂θα
− fµαβθ
β ∂
∂xµ
,
L (dα) =
∂
∂θα
+ fµαβθ
β ∂
∂xµ
.
16 C. Elliott, F. Hahner and I. Saberi
In addition, the even elements simply act by derivatives
R(eµ) = L (eµ) =
∂
∂xµ
.
The free superfield is the strict g-multiplet with E = C∞(N), vanishing differential, and module
structure given by the left translations L . Note that by “free” here, we mean in the sense of
the rank one free module over the ring of smooth functions on superspace (“unconstrained”),
rather than “non-interacting”. Recall that all our multiplets are a priori non-interacting: the
inclusion of interactions is additional data with which a multiplet can be equipped.
Let us now generalize the pure spinor superfield formalism to a derived setting.
Definition 3.5. We define the pure spinor functor A• : Modg0C•(n) −→ Multstrictg by setting
A•(Γ) =
(
C∞(N)⊗C Γ,D
)
,
for an object (Γ,dΓ). The differential D is constructed using the right action R and the C•(n)-
module structure on Γ. Explicitly, it takes the form
D = λαR(dα) + vµR(eµ) + dΓ.
For a morphism f : Γ −→ Γ′ we define
A•(f) = idC∞(N)⊗f : A•(Γ) −→ A•(Γ′).
A few comments are in order.
1. The differential D squares to zero precisely since Γ is a C•(n)-module.
2. We can equip A•(Γ) with the structure of a dgs vector bundle over the spacetime V := N2
by placing C∞(N) in cohomological degree zero. Note that in particular the differential is
of bidegree (1,+).
3. Further, g acts on C∞(N) via left translations and on Γ by the trivial extension of the
g0-module structure which was part of the input datum. The tensor product of these two
action makes A•(Γ) into a strict multiplet.
4. It is immediate to check that A•(f) is a strict morphism of strict multiplets. Since f is
a morphism of C•(n)-modules we have
A•(f) ◦ D = D′ ◦A•(f).
In addition,
A•(f) ◦ L = L ◦A•(f)
obviously follows from the definition.
5. A• is additive. The direct sum of two C•(n)-modules is mapped to the direct sum of the
respective multiplets, A•(Γ⊕ Γ′) = A•(Γ)⊕A•(Γ′).
This construction is a direct generalization of the pure spinor superfield formalism as de-
scribed in [19]. To see this, we notice that the category of graded equivariant of R/I-modules
sits as a subcategory inside ModstrictC•(n), namely precisely as those modules concentrated in coho-
mological degree zero. Indeed, every R/I-module is a C•(n)-module by the map
C•(n) = C
[
λα, vµ
]
−→ R/I,
(
λα, vµ
)
7→ λα.
The Derived Pure Spinor Formalism as an Equivalence of Categories 17
The other way round, let (Γ,dΓ) be a C•(n)-module concentrated in cohomological degree zero.
Then the differential dΓ vanishes and vµ acts trivially for degree reasons. Therefore one has,
for γ any element of Γ,
0 = dΓ
(
vµ · γ
)
=
(
dCEv
µ
)
γ =
(
λfµλ
)
γ,
which endows Γ with the structure of an R/I-module.
Restricting the functor A• to graded equivariant R/I-modules, we obtain a functor
A•
R/I : Modg0R/I −→ Multstrictg
where the output simplifies to
A•
R/I(Γ) = (C∞(N)⊗C Γ, λαR(dα))
such that we precisely recover the pure spinor superfield formalism as presented in [19].
Further, this is a derived generalization of the pure spinor superfield construction in the sense
that C•(n) can be viewed as a derived replacement of the ring R/I. We will therefore refer to
this construction as the derived pure spinor superfield formalism.
Remark 3.6. We can alternatively view this geometrically as a derived enhancement of the
affine nilpotence variety Spec(R/I). We can view a multiplet as arising from a quasi-coherent
sheaf over the nilpotence variety, but this point of view requires forgetting some of the data given
by the C•(n) action. The philosophy of derived algebraic geometry suggests instead retaining
this information by viewing a multiplet as arising from a coherent sheaf over the affine derived
scheme Spec(C•(n))—the derived analogue of the nilpotence variety.
Remark 3.7. In many cases the multiplet A•(Γ) can be equipped with additional structures
such as higher brackets yielding an L∞ structure. For example if Γ is not only a C•(n)-module
but in addition an algebra, one can take a Lie algebra h such that the tensor product Γ⊗h forms
a dgs Lie algebra. Then A•(Γ⊗h) = A•(Γ)⊗h is also a dgs Lie algebra. An L∞ structure on the
component fields arises via homotopy transfer. This was studied in the case of ten-dimensional
super Yang–Mills theory in [1, 19].
3.3 The multiplet associated to C•(n)
As a first example we can plug C•(n) itself into the derived pure spinor functor A• and study
the associated multiplet.
Lemma 3.8. There is a natural equivalence
A•(C•(n)
)
≃ Ω•(N).
Proof. First, we can describe
A•(C•(n)
)
= (C∞(N)⊗ C•(n),D = λαR(dα) + vµR(eµ) + dCE) .
Let us write V for the even part N2 of N , viewed as an affine space. We can identify C∞(N) =
C∞(V )⊗ C[θα] and C•(n) = C[λα, vµ]. The differential takes the explicit form
D = λα
∂
∂θ
− λαfµαβθ
β ∂
∂xµ
+ vµ
∂
∂xµ
+
(
λαfµαβλ
β
) ∂
∂vµ
.
On the other hand, N is parallelizable, therefore its de Rham complex takes the form
Ω•(N) ∼= C∞(N)⊗ Sym•(n∗1)⊗ ∧•n∗2.
18 C. Elliott, F. Hahner and I. Saberi
Identifying dθα = λα and dxµ = vµ, the de Rham differential takes the form
ddR = λα
∂
∂θα
+ vµ
∂
∂xµ
.
Further, g acts on the de Rham complex via left translation making it a strict multiplet. The
map defined in coordinates via(
A•(C•(n)
)
,D,L
)
−→ (Ω•(N), ddR,L ) ,
(
xµ, λα, θα, vµ
)
7→
(
xµ, λα, θα, vµ + λfµθ
)
is a quasi-isomorphism of g-multiplets. Therefore, we can identify the multiplet associated to
C•(n) itself as the differential forms on the super Lie group N . ■
Remark 3.9. We can further compute cohomology with respect to D0 = λα ∂
∂θα and obtain
a retraction5 to the de Rham complex on the spacetime manifold V :
(
Ω•(N),D0
) (
Ω•(V ), 0
)
.h
p
i
(3.2)
Here, i is the embedding of the factor of polynomial degree 0 in the λ and θ variables, and p
is the obvious projection. The homotopy h is given by h = θα ∂
∂λα . The induced differential
via homotopy transfer is the de Rham differential on V . The module structure, however, is no
longer strict. In fact the strict part now vanishes, but there are now quadratic pieces appearing.
These take the form
ρ(2)(Q,Q) = ι[Q,Q] : Ωk(V ) −→ Ωk−1(X).
Here we view [Q,Q] as a constant vector field on V and ι denotes the contraction of a differential
form with a vector field.
There is of course a further quasi-isomorphism of g-modules to the trivial g-module C. In this
spirit, each of the multiplets A•(C•(n)
)
, Ω•(N), and Ω•(V ) can be viewed as free resolutions
of the trivial module—either free over spacetime (i.e., as C∞(V )-modules), or even free over
superspace (i.e., as C∞(N)-modules). Note, however, that the trivial module does not form a g-
multiplet since the translations do not act geometrically. Therefore this last quasi-isomorphism
is just a quasi-isomorphism of g-modules.
3.4 Component fields and homotopy transfer
The multiplet A•(Γ) given by applying the pure spinor functor does not at first glance resemble
the more standard component field formulations known from physics. These component field
multiplets are distinguished by the fact that they are given by dgs vector bundles whose total
rank (as vector bundles over spacetime) is finite. In other words, component field multiplets
are usually defined with the assumption that they only contain a finite number of component
fields. With some care, it is always possible to choose a homotopy representative for A•(Γ)
that satisfies this condition, as long as Γ satisfies a finiteness condition. In other words, we will
establish the following lemma.
Lemma 3.10. Let Γ be a C•(n)-module such that only finitely many cohomology groups H•(Γ)
are non-vanishing and each of these is finitely generated as an R-module. Then A•(Γ) is quasi-
isomorphic to a multiplet of finite rank.
5Recall that a retraction between two cochain complexes is a diagram as in (3.2), such that p◦ i = idΩ•(V ) and
i ◦ p− idΩ•(N) = D0 ◦ h + h ◦ D0.
The Derived Pure Spinor Formalism as an Equivalence of Categories 19
In the rest of this section we will explain an algorithm that provides explicit finitely generated
component field representatives for A•(Γ), thus establishing the lemma. In [19], a general
procedure to extract such a component field multiplet out of the underived model A•
R/I(Γ)
was explained. In essence, this is done by homotopy transfer: one identifies a retraction to
another quasi-isomorphic complex and then applies homotopy transfer to the structures present
for the multiplet. By construction, this yields a quasi-isomorphic and thus physically equivalent
multiplet. Crucially, the component field multiplets obtained in that way are not necessarily
strict anymore. Higher-order terms in the module structure can arise during the transfer.
Let us begin by summarizing the procedure for the underived pure spinor formalism A•
R/I
and then describe the generalization to A•.
3.4.1 Minimal models for A•
R/I
Recall that the differential on
(
A•
R/I(Γ),D
)
admits an obvious splitting
D = D0 +D1 = λα
∂
∂θα
− λαfµαβθ
β ∂
∂xµ
,
which can be viewed by equipping A•
R/I(Γ) with a filtration by polynomial degree on V and
splitting the differential into the terms that preserve and lower filtered degree. (To filter by
polynomial degree on V is imprecise, since that filtration is not complete. A complete filtra-
tion that serves the same purpose can be constructed as explained in [19], roughly speaking
by assigning both λ and θ filtered degree one. Alternatively, under the assumption that the
generator of scalings is included in g0, one can use the weight grading as discussed in the next
subsection.) We can take cohomology with respect to D0 and then perform homotopy transfer
along the diagram
(A•(Γ),D0) (H•(A•(Γ),D0), 0).h
p
i
We observe, by [43, Theorem 10.3.15], that although the transfer data depends on a choice of
section for the projection onto the cohomology, the multiplet obtained by homotopy transfer is
independent of this choice up to isomorphism. In this example, it is always “minimal” in the sense
that its differential does not contain terms of order zero in differential operators. Intuitively, this
means that we cannot take any further cohomology without leaving the category of multiplets.
(In more general examples coming from C•(n)-modules, “minimal” is not related to having no
terms of order zero in the differential; the massive Klein–Gordon field is a simple example.)
Starting from a multiplet of the form A•
R/I(Γ), we will refer to the corresponding minimal
multiplet as µA•
R/I(Γ). One can identify the D0-cohomology with the Koszul cohomology of Γ,
tensored with functions on spacetime,
H•(A•
R/I(Γ)) = C∞(V )⊗H•(K•(Γ)).
The Koszul cohomology is conveniently computed by a minimal free resolution L of Γ in R-
modules. The minimal multiplet takes the form
µA•
R/I(Γ) =
(
C∞(V )⊗ (L⊗R C),D′, ρ′
)
,
where D′ is the differential induced from D1 and ρ
′ the module structure induced from L via ho-
motopy transfer. We refer to [19] for details. For Γ a finitely generated module, Hilbert’s syzygy
theorem states that the minimal free resolution exists, consists of finitely generated modules and
its length is less or equal than dim(n1) (see for example [21, Theorem 1.13]; for a discussion in
the equivariant case we refer to [28, Proposition 2.4.9, Remark 2.4.10]). Therefore, µA•
R/I(Γ) is
indeed of finite rank over spacetime, i.e., it provides a reasonable component field multiplet.
20 C. Elliott, F. Hahner and I. Saberi
3.4.2 Minimal models for A•
Let us now discuss a generalization of this procedure to the functor A•. We will assume in this
section that Γ carries an action of the abelian Lie algebra R compatible with the action of R
on n where ni has R-weight i. This will be used to construct a filtration at the very end of this
section (however, this assumption is not required for Lemma 3.10).
Recall that the differential on A•(Γ) takes the form
D = λα
∂
∂θα
− λαfµαβθ
β ∂
∂xµ
+ dΓ + vµ
∂
∂xµ
.
One can construct a finite-dimensional component field model in two steps. First one takes
cohomology with respect to the internal differential of the module dΓ and performs homotopy
transfer along the diagram
(A•(Γ),dΓ) (C∞(N)⊗H•(Γ)), 0).h
p
i
The differential and the C•(n)-module structure of the resulting multiplet may contain addi-
tional pieces induced from homotopy transfer. Since H•(Γ) is a C•(n)-module with vanishing
differential, each homogeneous summand Hk(Γ) carries the structure of an R/I-module. Thus,
we can proceed by taking cohomology with respect to the Koszul differential D0 = λα ∂
∂θα , and
applying the homotopy transfer along a retraction on to this cohomology. As before, the D0-
cohomology is computed by minimal free resolutions of the individual Hk(Γ) in R-modules, and
as before, it is independent of the choice of transfer datum. The resulting multiplet is thus of
the form
C∞(V )⊗
(⊕
k
L•
k ⊗R C
)
,
where L•
k is the minimal free resolution of Hk(Γ). As long as the cohomology H•(Γ) is bounded,
this is already a finite rank vector bundle over spacetime. The field content of this multiplet is
just the field content of the direct sum of all the minimal multiplets associated to the cohomology
groups, i.e.,⊕
k
µA•(Hk(Γ)).
At this stage we have already obtained a finite-dimensional model, as required by Lemma 3.10.
However, additional acyclic differentials induced by homotopy transfer can still be present. At
this final stage, we will use the filtration on each multiplet µA•(Hk(Γ)) by weight with respect
to the R-action on Γ. We define µA•(Γ) by taking the cohomology with respect to the summand
of the total differential of filtered degree zero (in other words, we pass to the E1 page of the
associated spectral sequence), and apply homotopy transfer. We will illustrate this procedures
using examples in Section 5.
4 An equivalence of categories
We now show that the derived pure spinor superfield formalism provides an equivalence of
categories between the dg categories of g0-equivariant C
•(n)-modules and g-multiplets. This
implies in particular that, up to quasi-isomorphism, every g-multiplet can be constructed using
the derived pure spinor superfield formalism.
The Derived Pure Spinor Formalism as an Equivalence of Categories 21
Remark 4.1. Recall that for a general Lie algebra there is a Koszul duality equivalence be-
tween the dg-categories of C•(g)-comodules and U(g)-modules. If g is, for instance, finite dimen-
sional, we can instead consider C•(g)-modules. For a relevant discussion in a similar context,
see [16, Sections 7 and 8]. Relatedly, Kapranov [38] gives a formulation of Koszul duality that
establishes a Quillen equivalence between the categories of D-modules and Ω•-modules on the
same space. Our results show that the pure spinor formalism admits a natural derived gen-
eralization that is closely related to Kapranov’s construction; however, in our setting, one is
working on a supermanifold, and asks for appropriate equivariance conditions. Alternatively,
our procedure can be viewed in two steps, as an explicit form of commutative/Lie Koszul dual-
ity tailored to the examples in question, combined with an associated bundle construction that
realizes a U(n)-module as an n-equivariant dg vector bundle over V = N2.
4.1 The inverse functor: derived n-invariants
Any g-module is in particular a n-module, and we can thus take derived invariants with respect
to n. For multiplets, this defines a functor in the opposite direction to the functor A•, assigning
a strict C•(n)-module to a multiplet. The resulting C•(n)-module is also g0-equivariant. The
functor takes the form
C•(n,−) : Multstrictg −→ Modg0C•(n),
It maps the multiplet (E,D, ρ) to C•(n, E); this Chevalley–Eilenberg complex can be written
more explicitly as
C•(n, E) =
(
C•(n)⊗ E , dCE +D + λαρ(dα) + vµρ(eµ)
)
.
It is a strict C•(n)-module; the action is on C•(n) via multiplication and on E via the identity.
Morphisms are mapped according to the rule
ψ 7→ idC•(n)⊗ψ.
Remark 4.2. The assignment on the level of objects is also well defined for not necessarily
strict modules. Then the higher-order terms of the g-module structure enter the differential
such that the term λαρ(dα) is replaced by
λ · ρ :=
∞∑
k=1
λα1 · · ·λαkρ(k)(dα1 , . . . , dαk
).
Notice that the output is still a strict C•(n)-module.
There are several ways to intuitively understand the fact that the inverse functor is given by
the derived invariants of supertranslations. One can of course appeal to the general structure
of Koszul duality. On a more down-to-earth level, one can recall that the component fields
of a supermultiplet in the usual pure spinor superfield formalism correspond to the generators
of the minimal free resolution of that module over R = C•(Πn1). The resolution differential
was then proven in [19] to agree with those supersymmetry transformations that are order zero
in spacetime derivatives, verifying a conjecture of Berkovits. Since spacetime derivatives act
by zero on translation-invariant sections of E, it is clear that one can think of the minimal
free resolution of the module as arising from the derived Πn1-invariants of translation-invariant
sections: R consists of the Lie algebra cochains of Πn1, the generators of the free graded R-
module arise from the component fields of the multiplet, and the differential encodes the action
of supersymmetry on translation-invariant sections. It is clear that this story is just a two-step
22 C. Elliott, F. Hahner and I. Saberi
procedure to compute derived n-invariants by first taking n2-invariants, and then accounting for
the action of supersymmetry.
To flesh this story out, we now describe the module C•(n, E) in some more detail and sketch
how its cohomology can be computed. Recall that, since E is a multiplet, the action of n2 is just
given by derivatives along the coordinate directions
ρ(eµ) =
∂
∂xµ
.
Thus taking cohomology with respect to the term vµρ(eµ) in the differential means restricting
to translation invariant sections of E and eliminating vµ. Denoting the fiber of E over 0 by E0
we find a quasi-isomorphism
C•(n, E) ≃
(
R⊗ E•
0 , λ · ρconstants
)
. (4.1)
Note that E•
0 carries a Z× Z/2Z-grading since it comes from a dgs vector bundle. In addition,
there is an integer grading by polynomial degree in λ present. In the following, we will label
cohomology groups by the former degree. Then each cohomology group is an g0-equivariant R/I-
module graded by polynomial degree in λ. If the cohomology is concentrated in a single degree,
the complex on the right hand-side can be viewed as the minimal free resolution of the R/I-
module forming the cohomology. Note that this is indeed a minimal free resolution, since
all terms in the differential are of nonzero polynomial degree in λ. It will follow from the
theorem below that this R/I-module is precisely the algebraic input datum the multiplet can
be constructed from in the pure spinor superfield formalism, i.e., by applying the functor A•
R/I .
4.2 Main theorem and proof
Let us now show that the functors A• and C•(n,−) induce an equivalence of dg-categories
between the homotopy categories of multiplets and equivariant C•(n)-modules.
Theorem 4.3. A• and C•(n,−) provide an equivalence of dg-categories between Multstrict-obg and
Modg0C•(n).
Proof. We first show that there is an equivalence of dg-functors
idMod
g0
C•(n)
→ C• ◦A•.
We will naturally construct quasi-isomorphisms
Γ ≃ C•(n, A•(Γ))
for each equivariant C•(n)-module (Γ, dΓ).
We can explicitly describe C•(n, A•(Γ)) by(
C
[
λ′, v′
]
⊗ C∞(N)⊗ Γ,
dΓ +
(
λ′fµλ′
) ∂
∂v′µ
+ v′µ
∂
∂xµ
+ λ′αL (dα) + λαR(dα) + vµR(eµ)
)
.
Here we made a notational distinction between the generators of the C•(n) in the construction
(denoted by λ′ and v′) and the action of C•(n) on Γ (denoted by λ and v).
The differential contains a piece of the form
ddR = v′µ
∂
∂xµ
+ λ′α
∂
∂θα
.
The Derived Pure Spinor Formalism as an Equivalence of Categories 23
Thus, we can identify(
C•(n, A•(Γ)), v′µ
∂
∂xµ
+ λ′α
∂
∂θα
)
= (Ω•(N),ddR)⊗ Γ.
Since the de Rham complex on N is acyclic, this complex is quasi-isomorphic to Γ. We can fix
homotopy data
(Ω•(N)⊗ Γ, ddR) (Γ, 0).h
p
i
It is easy to see that the only induced differential on the right hand side is dΓ, so that we obtain
a quasi-isomorphism uniformly for all choices of Γ:
C•(n, A•(Γ)) ≃ (Γ, dΓ).
Now, for any morphism f : Γ → Γ′ of C•(n)-modules, we can identify
C•(A•(f)) = idC•(n)⊗ idC∞(N)⊗f.
So there is a commutative square
Γ C•(n, A•(Γ))
Γ′ C•(n, A•(Γ′))
f A•(f)
inducing an equivalence of hom complexes Hom(Γ,Γ′) → Hom(C•(n, A•(Γ)), C•(n, A•(Γ′))).
Conversely, we will construct an equivalence of dg-functors
A• ◦ C• → idMultstrict-obg
.
Let (E,D, ρ) be a g-multiplet. We can describe A•(C•(n, E)) explicitly by(
C[λ, v]⊗ E ⊗ C∞(N), D + λfµλ
∂
∂vµ
+ λ · ρ+ vµρ(eµ) + λαR(dα) + vµR(eµ)
)
.
We denote coordinates on the base of the vector bundle E by xµ and on N2 by yµ. In this
notation, we find
vµ (ρ(eµ) + R(eµ)) = vµ
(
∂
∂xµ
+
∂
∂yµ
)
.
Taking cohomology with respect to this piece, we obtain a quasi-isomorphic complex of the form(
C[λ, θ]⊗ E , D + λ · ρ+ λαR(dα)
)
.
The differential contains a piece corresponding to the Koszul differential
dK = D0 = λα
∂
∂θα
.
24 C. Elliott, F. Hahner and I. Saberi
Clearly, taking cohomology with respect to the Koszul differential we arrive at E . Concretely
let us consider homotopy data
(C[λ, θ]⊗ E ,D0) (E , 0),D†
0
p
i
where the homotopy is given by D†
0 = θ ∂
∂λ , while i is the obvious inclusion and p evaluates at
λ = θ = 0. It is easy to see that the induced differential is just D. Thus, we find homotopy data
(C[λ, θ]⊗ E , d) (E , D),D′†
0
p′
i′
(4.2)
providing a quasi-isomorphism of cochain complexes. The induced maps are given by
i′ =
∞∑
n=0
(
D†
0(D +D1 + λ · ρ)
)n ◦ i,
p′ = p ◦
∞∑
n=0
(
D†
0(D +D1 + λ · ρ)
)n
= p,
D′†
0 = D†
0 ◦
∞∑
n=0
(
(D +D1 + λ · ρ)D†
0
)n
. (4.3)
By construction, p ◦ D†
0 = 0 and thus p′ = p. We note that these sums are all finite by degree
reasons, since the left hand side in (4.2) is concentrated in finitely many degrees with respect to
the grading by polynomial degree in the θ-variables, and in each expression in equation (4.3),
the operator being raised to the power n within the sum raises this θ-degree. Therefore for n
sufficiently large, all terms in the sum defining our maps vanish.
We have to check that this not only provides a quasi-isomorphism of cochain complexes, but
of multiplets. Therefore we transfer the module structure induced by L to the right hand side
and check that it agrees with the original module structure ρ of the multiplet.
Explicitly, the transferred module structure can be described by
ρ
(k)
L (Q, . . . , Q) = pL (Q)
(
D′†
0 L (Q)
)k−1
i′
= pL (Q)
(
D†
0
∞∑
n=0
((D +D1 + λ · ρ)D†
0)
nL (Q)
)k−1
×
∞∑
j=0
(
D†
0(D +D1 + λ · ρ)
)j ◦ i.
Since p projects onto λ = θ = 0, only terms of order zero in λ and θ can contribute. It is easy
to see that the only such term is
ρ
(k)
L (Q, . . . , Q) = pϵ
∂
∂θ
(
D†
0ϵ
∂
∂θ
)k−1
D†
0λ · ρ(k)i.
Here we expressed Q in a basis, Q = ϵαdα. Noting that{
D†
0, ϵ
∂
∂θ
}
= ϵ
∂
∂λ
,
The Derived Pure Spinor Formalism as an Equivalence of Categories 25
we deduce
ρ
(k)
L (Q, . . . , Q) = ρ(k)(Q, . . . , Q).
This shows that A•(C•(n, E)) ≃ (E,D, ρ) as multiplets.
As before, for any morphism ψ : E → E ′ we can realize
A•(C•(ψ)) = C•(ψ)⊗ idC∞(N) .
So there is a commutative diagram
A•(C•(n, E)) E
A•(C•(n, E ′)) E ′
C•(ψ) ψ
inducing an equivalence of hom spaces, and hence we have an equivalence of dg-categories. ■
Remark 4.4. The equivalence of dg-categories automatically induces an equivalence of the un-
derlying homotopy categories. Hence, each multiplet is perturbatively equivalent to a multiplet
constructed via the derived pure spinor superfield formalism.
4.3 Some consequences of the theorem
Let us now discuss some consequences of the equivalences of categories.
Corollary 4.5. Let (E,D, ρ) be any multiplet. A•(C•(n, E)) is a strictification.
Note that this does not imply that there exist a strict finitely generated component field
multiplet that is equivalent to E . In particular, E need not admit an auxiliary field formulation
in the usual sense. The strictification A•(C•(n, E)) typically contains infinitely many component
fields.
We can further use the equivalence of categories to derive some statements on the (underived)
pure spinor superfield formalism, i.e., the functor A•
R/I .
Corollary 4.6. The essential image of the functor A•
R/I consists of those multiplets for which
H•(n, E) is concentrated in a single Z-degree after totalization of the natural Z× Z-grading.
This gives a description of all multiplets which can be constructed via the pure spinor su-
perfield formalism. In [19], it was argued that the antifield multiplet of the four-dimensional
vector multiplet cannot be constructed via A•
R/I . We come back to this example in Section 5.2
where we compute the relevant cohomology spaces and explain how the multiplet is built in the
derived formalism.
Corollary 4.7. The functors A•
R/I and H•(n, E) provide an equivalence of categories between
the essential image of A•
R/I and the category of graded g0-equivariant R/I-modules.
Suppose we have an R/I-module Γ with associated minimal multiplet µA•
R/I(Γ). As discussed
earlier, the fields of µA•
R/I(Γ) take values in the minimal free resolution of Γ. In addition, there
is a close link between the supersymmetry module structure and the resolution differential. In
more detail, let us pull back µA•
R/I(Γ) along the inclusion
{0} ↪→ V.
This restricts the multiplet to the fiber µA•(Γ)0. The multiplet µA•
R/I(Γ)0 carries a module
structure for the odd abelian super Lie algebra Πn1. This module structure coincides with the
resolution differential in the following sense.
26 C. Elliott, F. Hahner and I. Saberi
Corollary 4.8. Let Γ be an R/I-module and let (L,dL) be its minimal free resolution in R-
modules. Let us identify the fiber
µA•
R/I(Γ)0 = L⊗R C.
The map generated over R by the Πn1-module structure on µA•
R/I(Γ)0
ρconstants : µA•
R/I(Γ)0 ⊗
(⊕
k
n⊗k1
)
−→ µA•
R/I(Γ)0
coincides with the resolution differential. In coordinates we can express this as
λ · ρconstants =
∑
k
ρ
(k)
constants
(
λα1dα1 , . . . , λ
αkdαk
)
= dL,
where dα is a basis for n1.
Proof. By construction, we have H•(n, µA•(Γ)) = Γ. But by (4.1) we know that there is
a quasi-isomorphism
C•(n, µA•(Γ)) ≃ (µA•(Γ)0 ⊗R, λ · ρconstants) .
Thus, the cochain complex on the right is the minimal free resolution of Γ in R-modules and we
obtain the desired result. ■
In practice this means that the resolution differential contains all the information on the
supersymmetry transformations which are of order zero in the derivatives. This result was
conjectured by Berkovits in [5] and proved by direct computation in [19].
Let us now state some results on the duality operations in the category of multiplets.
Corollary 4.9. Let (E,D, ρ) be a g-multiplet and let (E∗, D∗, ρ∗) be the respective dual (or
antifield) multiplet. If these are both quasi-isomorphic to a multiplet in the image of A•
R/I and
we have
(E,D, ρ) ≃ A•
R/I(Γ),
then there is also a quasi-isomorphism(
E∗, D∗, ρ∗
)
≃ A•
R/I
(
Extn−qR (Γ, R)
)
.
Here, n = dim(n1) and q = dimR(Γ).
Proof. Move to the minimal multiplet µA•
R/I(Γ) ≃ (E,D, ρ). Corollary 4.8 implies that
C•(n, E) ≃ C•(n, µA•
R/I(Γ)) ≃ (L,dL),
where (L, dL) is the minimal free resolution of Γ in R-modules. There is a quasi-isomorphism(
E∗, D∗, ρ∗
)
≃
(
µA•
R/I(Γ)
)∗
.
This in turn implies that
C•(n, E∗) ≃ C•(n, µA•
R/I(Γ)
∗) ≃ (L∗, d∗L
)
,
where (L∗,d∗L) is the dual of the minimal free resolution (L,dL). By assumption the derived
invariants of (E∗, D∗, ρ∗) are concentrated in a single degree. Hence, (L∗,d∗L) again resolves
a single R/I-module, namely Extn−qR (Γ, R). ■
The Derived Pure Spinor Formalism as an Equivalence of Categories 27
In general, however, taking dual multiplets can lead outside of the image of A•
R/I . In fact,
the above proof implies that (A•
R/I(Γ))
∗ is in the essential image of A•
R/I precisely when Γ is
a Cohen–Macaulay R-module. In general this may not be the case—the dual of the vector
multiplet in four-dimensional N = 1 supersymmetry is an example of this type, that does not
occur in the image of the underived pure spinor functor. We will discuss this example in the
generalized formalism in Section 5.2.
If Γ is not Cohen–Macaulay, we can still compute the derived invariants of the associated
multiplet to find
C•(n, A•
R/I(Γ)
∗) ≃ RHomR(Γ, R) ≃ Ext•R(Γ, R).
We can therefore deduce the following natural description for the dual of a multiplet obtained
using the (underived) pure spinor formalism.
Corollary 4.10. Let Γ be an R/I-module and µA•
R/I(Γ) the associated component field multiplet.
Then there is a C•(n)-module structure on the Ext-algebra Ext•R(Γ, R) such that
µA•
R/I(Γ)
∗ ∼= µA•(Ext•R(Γ, R)).
4.4 First examples
Let us discuss the pure spinor formalism as an equivalence of categories in a few simple examples.
Example 4.11. First, let us consider the situation where there is no supersymmetry at all.
Let V be an n-dimensional vector space. Let g denote its Poincaré Lie algebra of infinitesimal
isometries, and let n denote its Lie algebra of translations, viewed as a dg Lie algebra concen-
trated in degree 2. So C•(n) ∼= Sym•(n∗[1]) is an exterior algebra in n generators v1, . . . , vn of
degree −1. We are, therefore, interested in g0 ∼= so(n)-equivariant dg-modules over the exterior
algebra. We restrict attention to those modules with finite-dimensional cohomology in each
degree.
On the other hand, the dg-category Multg of multiplets is nothing but the dg-category of
Poincaré equivariant dg-vector bundles on the affine space V : the g action is completely de-
termined (and all multiplets are automatically strict). Again, let us restrict attention to bun-
dles with finite-dimensional cohomology in each degree. Such multiplets are determined by
their restriction to a formal neighborhood of the origin, which is a Poincaré equivariant dg-
module over the completed Weyl algebra D̂n = C[∂1, . . . , ∂n][[z1, . . . , zn]]/([∂i, zi] − 1). Transla-
tion equivariance guarantees that all such modules are induced from so(n)-equivariant modules
over C[∂1, . . . , ∂n]. Our statement then reduces to ordinary Koszul duality as a relationship
between modules over an exterior and a commutative algebra.
Example 4.12. Let Σ be a finite-dimensional vector space. Let us now briefly discuss the exam-
ple where g = n = R⊕ΠΣ as a graded vector space, with Lie bracket given by a non-degenerate
inner product Σ⊗2 → R. This is the background for supersymmetric classical mechanics. Now
C•(n) ∼=
(
C
[
v, λ1, . . . , λN
]
,dCE
)
,
where v is an odd generator, λi are even generators, dCEλ
i = 0 for all i, and dCEv =
(
λ1
)2
+
· · ·+
(
λN
)2
(so the λi are linearly dual to a choice of orthonormal basis of Σ).
Let (Γ,dΓ) be a C•(n)-module. The derived version of the pure spinor formalism associates
to Γ the multiplet
A•(Γ) =
(
C∞(Rt)⊗ C[θ1, . . . , θN ]⊗ Γ,dΓ +
(
v − λaθa
) ∂
∂t
+ λa
∂
∂θa
)
, (4.4)
28 C. Elliott, F. Hahner and I. Saberi
where θ1, . . . , θN are, again, odd generators, and the expression λaθa implicitly uses the choice
of inner product on Σ.
On the other hand, we can study multiplets on R for the super Lie algebra n directly. These
are affine dgs-vector bundles on R equipped with N commuting odd symmetries, each of which
squares to the action of translation. One can see this structure more concretely from the ex-
pression of equation (4.4) by applying homotopy transfer to take the cohomology with respect
to the final summand λa ∂
∂θa of the differential.
There is a rich theory underlying the classification of multiplets in supersymmetric mechanics;
see for example [25]. It would be interesting to investigate the connections between this existing
work and the point of view described here.
5 Applications
In this final section we discuss some examples of multiplets in the light of the derived formalism.
This also provides some insight to some of the curiosities of the underived pure spinor superfield
formalism. In particular, we connect the multiplets associated to different Lie algebra cohomol-
ogy groups by curvature maps and construct antifield multiplets (in the sense of Definition 2.19)
for four-dimensional N = 1 supersymmetry. In addition, we show how both the on- and off-shell
version of the chiral multiplet can be constructed.
5.1 Multiplets from Lie algebra cohomology
We noticed in Section 3.3 that the multiplet associated to C•(n) can be identified with the de
Rham complex of the super Lie group N . On the other hand, it was already appreciated in the
previous literature that the individual Lie algebra cohomology groups of n give an interesting
class of R/I-modules which yield multiplets via the underived pure spinor construction A•
R/I .
Some of these multiplets were studied for various super Poincaré algebras in [19, Section 6.3];
see also [49]. The derived formalism gives a new tool to study these multiplets and to highlight
the relations among the multiplets associated to the different Lie algebra cohomology groups.
To start with, recall that when the nilpotence variety Spec(R/I) associated to n is a complete
intersection, the Chevalley–Eilenberg cohomology of n is concentrated in degree zero. The
associated multiplet to H0(n) is then simply quasi-isomorphic to the de Rham complex on the
supertranslation group Ω•(N).
In contrast, for cases with higher Lie algebra cohomology, there is no direct quasi-isomorphism
between the multiplets associated to C•(n) and H•(n). However—as advertised in Section 3.4—
we can use the derived formalism to study the multiplets associated to the individual Lie algebra
cohomology groups and their relationships. As discussed, there is a quasi-isomorphism identify-
ing the multiplet associated to C•(n) with the de Rham complex on spacetime,
A•(C•(n)) ≃ Ω•(N).
On the other hand, we can first take cohomology with respect to the Chevalley–Eilenberg differ-
ential. Let us filter the complex A•(C•(n)) by the total degree on the ring C•(n): the internal
Z-degree plus the R-weight: the Chevalley–Eilenberg cohomology has degree one for this filtra-
tion, and the remaining piece of the differential has degree zero. The homotopy transfer theorem
leads to an equivalence
A•(C•(n)) ≃
(⊕
k
A•(Hk(n)),D′
)
,
The Derived Pure Spinor Formalism as an Equivalence of Categories 29
where the differential D′ on the right hand side is induced by homotopy transfer, and consists
of a sum of terms
D′
j,k : A•(Hk(n))→ A•(Hk−j(n))
for j ≥ 1. An alternative way of understanding these differentials is to consider higher dif-
ferentials in the spectral sequence associated to the filtration discussed above, by total degree
on C•(n).
For example, let us consider the case j = 1. The process we have just discussed gives rise to
differential operators between the associated component field multiplets:
∇ : µA•(Hk(n)) −→ µA•(Hk−1(n)
)
.
As we will see in an example momentarily, these operators can intuitively be thought of in the
case k = 0 by viewing µA•(H−1
)
as the field strength multiplet of µA•(H0
)
, with ∇ acting as
the field strength or curvature map. In particular, as discussed in [19], the multiplet associated
to H0 always contains a summand corresponding to a p-form abelian gauge field for some p.
Since the deformation we describe deforms the sum of the minimal multiplets µA•(Hk(n)) to (an
object quasi-isomorphic to) the de Rham complex on spacetime, one expects that the operator
mapping µA•(H0
)
to µA•(H−1
)
will include a de Rham differential that carries the p-form to
a (p+1)-form “field strength” component field in the multiplet µA•(H−1
)
. In examples, we will
see that this is the case.
We expect that our operators should provide a general coordinate-free description of opera-
tors of the type “Ra”, considered by Cederwall in constructing pure spinor superfield actions;
see [9]. These operators appear, for example, in the discussion of interacting eleven-dimensional
supergravity in [8, 10], and for the Dirac–Born–Infeld action in [12].
Example 5.1. Consider N = 1 supersymmetry in three dimensions. Let p denote the N = 1
super Poincaré algebra in three dimensions, and let g = p⊕R, where R acts on n by rescaling, with
weight one on the odd piece and with weight two on the even piece. We will use the R action to
lift the Z×Z/2Z-grading on our multiplets. We denote the associated super translation algebra
by n = g>0 and the super translation group by N . Recall that the supertranslation algebra is
of the form
n = S[−1]⊕ V [−2],
where S is the two-dimensional spin representation of Spin(3) ∼= SU(2), and V is the three-
dimensional vector representation of Spin(3), with the bracket being induced from the equiv-
ariant isomorphism Sym2(S) ∼= V . We can choose bases for these representations to obtain
generators and relations for the Chevalley–Eilenberg complex as follows:
C•(n) = C
[
λα, vµ
]
, dCEv
1 =
(
λ1
)2
, dCEv
2 = λ1λ2, dCEv
3 =
(
λ2
)2
.
It will sometimes be convenient to write the differential more compactly as
dCEv
(αβ) = λ(αλβ).
The Lie algebra cohomology is easily computed.
Hk =
R/I if k = 0,((
λ2v1 − λ1v2
)
C
[
λ1
]
⊕
(
λ2v2 − λ1v3
)
C
[
λ2
])
/I−1 if k = −1,
0 otherwise.
30 C. Elliott, F. Hahner and I. Saberi
The ideal I−1 appearing in the case where k = −1 is spanned by λ2
(
λ2v1 − λ1v2
)
− λ1
(
λ2v2 −
λ1v3
)
. As explained in [19], µA•(H0
)
is identified with the 3d N = 1 gauge multiplet and
µA•(H−1
)
with the corresponding antifield multiplet. In the array notation introduced in Sec-
tion 2.1, we can denote the direct sum of these multiplets as follows:
µA•(H0
)
⊕ µA•(H−1
)
=
Ω0
Ω1 S
S Ω2
Ω3
d
d
.
This direct sum is not yet isomorphic to the de Rham complex, but we can clearly see how the
differential can be deformed such that this is the case: one has to add an acyclic piece which
cancels the two spin representations and an additional differential of order one forming the de
Rham differential between Ω1 and Ω2.
Let us now see how this deformation arises from the differential on A•(C•(n)) via homotopy
transfer. We start by taking cohomology with respect to the Chevalley–Eilenberg differential dCE
and fix homotopy data
(A•(C•(n)), dCE)
(
C∞(N)⊗H•(n), 0
)
h
p
i
,
such that we find induced differentials of the form D′ = pDi+ pDhDi+ · · · . By degree reasons,
only the first and the second summand can contribute non-zero terms. Taking cohomology with
respect to D0 and transferring to the minimal multiplets we finally obtain the map
∇ : µA•(H0
)
−→ µA•(H−1
)
,
which we view as a field strength map. Let us now study this map explicitly using our basis.
Running the techniques described in [19] (see also [40] for an earlier account), one finds the
following representatives for the component fields in µA•(H0
)
:[
1 − −
− λ(αθβ) λαθ2
]
,
where internal degree is indicated by vertical position—concentrated in degrees zero and one—
and the R×-weight minus the degree is indicated by horizontal position. For µA•(H−1
)
one
finds [
λαv
(αβ) λ(αθβ)vγδ −
− − θ2λαλβv
(αβ)
]
,
where now the rows indicate degree 2 and 3. We can fix a homotopy h for dCE by
h
(
λαλβ
)
= v(αβ).
There will be terms in ∇ of the form
pλ
∂
∂θ
hλ
∂
∂θ
i.
The Derived Pure Spinor Formalism as an Equivalence of Categories 31
Acting on representatives of the fermions in the multiplet µA•(H0
)
, we find
λαθ2 7→ λαλβθβ 7→ v(αβ)θβ 7→ λβv
(αβ).
This is a map between µA•(H0
)
and µA•(H−1
)
, which induces an acyclic deformation on their
direct sum.
In addition, ∇ contains terms of order one given by
pv
∂
∂x
i.
Investigating the representatives, it is clear that this acts as a de Rham differential between
the one-form in µA•(H0
)
and the two-form in µA•(H−1
)
. This maps the one-form gauge field
present in µA•(H0
)
to its field strength, which is part of µA•(H−1
)
.
5.2 Antifield multiplets in four dimensions
Let us discuss an example of a multiplet which is of physical relevance, but is not in the essential
image of the functor A•
R/I . For four-dimensional N = 1 supersymmetry, µA•(R/I) can be
identified with the vector multiplet. The corresponding antifield multiplet, however, cannot be
constructed in the underived pure spinor superfield formalism. This failure is linked to the fact
that the relevant nilpotence variety is not Cohen–Macaulay [19].
Here we describe the antifield multiplet using component fields first and then calulate the
derived n-invariants. As expected we find that the cohomology is not concentrated in a single
degree. We begin by recalling the structure of the supertranslation algebra in four dimensions.
Definition 5.2. The four-dimensional N = 1 supertranslation algebra is the super Lie algebra
with underlying Z/2Z-graded Spin(4)-module
n = R4 ⊕Π(S+ ⊕ S−),
with Lie bracket given by the canonical isomorphism S+ ⊗ S− → R4 of Spin(4)-representations.
Example 5.3. Let (E,D, ρ) denote the antifield multiplet to the vector multiplet on R4. It
is concentrated in R-weight 0 to 4 and cohomological degree 0 and 1, and can be concretely
described in array notation as
µA•(R/I)∗ =
Ω0
(
R4
)
Γ
(
R4, S+ ⊕ S−
)
Ω1
(
R4
)
Ω0
(
R4
)⋆d⋆
. (5.1)
We can also describe the L∞ action of the supertranslation algebra n concretely. We will only
need the action on constant sections. There, odd elements act by the following formula:
ρ
(1)
constants(Q) : S+ ⊕ S− −→ Ω0,
(
ψ∨, ψ̄∨) 7→ Q+ ∧ ψ∨ +Q− ∧ ψ̄∨,
Ω1 −→ S+ ⊕ S−, A∨ 7→ Q+ ∧A∨ +Q− ∧A∨,
ρ
(2)
constants(Q1, Q2) : Ω0 −→ Ω1, c∨ 7→ Γ(Q1, Q2)⊗ c∨,
where we have denoted the fields by
(
ψ∨, ψ̄∨) ∈ Γ
(
R4, S+⊕S−
)
, A∨ ∈ Ω1
(
R4
)
, and c∨ ∈ Ω0
(
R4
)
in degree one. We have similarly denoted the positive and negative helicity summands of Q ∈
S+ ⊕ S− by Q+ and Q− respectively.
32 C. Elliott, F. Hahner and I. Saberi
Let us write E0 for the fiber of E over 0 ∈ R4 (not to be confused with the summand of
degree zero), so concretely
E0 =
R S+ ⊕ S− R4
R
.
We can use this to describe the Chevalley–Eilenberg complex of n with coefficients in our mul-
tiplet, using the L∞ action of n on the sheaf E of sections of E. We find the following:
Proposition 5.4.
C•(n, E) ∼= (E0 ⊗ Sym(S+ ⊕ S−), dρ),
where dρ is generated over Sym(S+ ⊕ S−) by the sum of the terms
ρ(1)|constants : E0 ⊗ (S+ ⊕ S−) → E0,
ρ(2)|constants : E0 ⊗ Sym2(S+ ⊕ S−) → E0
obtained by restricting the L∞ action ρ to constant sections of E.
Proof. Note that
C•(n) ∼=
(
Sym
((
R4
)∗
[1]
)
⊗ Sym(S+ ⊕ S−),dCE
)
,
where we have identified S± with its dual using its canonical inner product. We obtain the
given description by taking the cohomology by the operator dual to the action of the algebra of
translations on E ; the result is quasi-isomorphic to C•(n, E), no additional homotopical correction
terms appear. ■
Let us now discuss the cohomology of the Chevalley–Eilenberg complex.
Proposition 5.5. We have an isomorphism
H•(n, E) ∼= C⊕ (Sym(S+)⊕ Sym(S−))[−1].
Proof. This is a straightforward calculation using the description that we have just given. In
the weight zero term of E0, all elements are dρ-closed, but all such elements other than constants
are also dρ-exact. In weight 1 in E0, the dρ-closed elements are generated over 1⊗Sym(S+⊕S−)
by ∧2S+ ⊕∧2S− ⊆ (S+ ⊕ S−)⊗ (S+ ⊕ S−). When we quotient by dρ-exact elements we are left
with
(
∧2S+ ⊗ Sym(S−)
)
⊕
(
∧2S− ⊗ Sym(S+)
)
. In weight 2 in E0 the closed and exact elements
coincide, and in weight > 2 in E0 there are no dρ-closed elements. ■
From our general results, we know abstractly that A•(C•(n, E)) is dual to the vector multiplet.
It is instructive to calculate the component field formulation for this multiplet explicitly using
the recipe presented in Section 3.4 in order to see how the is related to the corresponding
calculation in the underived pure spinor superfield formalism.
Our calculation will follow the same outline as the calculation we performed in the previous
section. We will first study the multiplet associated to each of the two cohomology groups
of C•(n, E) in isolation, then we will compute the additional differential in A•(H•(n, E)) relating
these two individual terms, obtained by homotopy transfer.
First, recall that
A•(C) ≃ C∞(N) ≃ Γ
(
R4,Sym(S+[1]⊕ S−[1])
)
.
The Derived Pure Spinor Formalism as an Equivalence of Categories 33
For the non-trivial summand of the cohomology, we can compute that
A•(Sym(S+)) ≃
(
Γ
(
R4,Sym(S+)⊕ Sym(S+[1]⊕ S−[1])
)
,D
)
≃ Γ
(
R4,Sym(S−[1])
)
,
where the differential D is generated by the degree one isomorphism S+[1] → S+. Similarly
A•(Sym(S+)) ≃ Γ
(
R4,Sym(S+[1])
)
.
So altogether, when we compute A•(H•(n, E)), we obtain a multiplet with the following Betti
numbers:[
1 4 6 4 1
− 2 4 2 −
]
.
Now, let us compute the correction terms that allow us to obtain A•(C•(n, E)) in full. As
discussed in the previous section, Section 5.1, there is an additional differential coming from
homotopy transfer. We will compute this differential in coordinates.
Running the procedure described in [19], we find the following local coordinates for our
multiplet:[
1
(
θα, θ̄α̇
) (
θ2, θαθ̄α̇, θ̄
2
) (
θ2θα, θ̄
2θ̄α̇
)
θ2θ̄2
−
(
λs, λ̄s̄
) (
λsθα, λ̄s̄θ̄α̇
) (
λsθ2, λ̄s̄θ̄2
)
−
]
.
Let us unpack the notation. Recall that we identified C•(n, E) with (E0 ⊗ Sym(S+ ⊕ S−),dρ).
We use
{
sα, s̄α̇
}
for a basis of S+ ⊕ S− in the first factor, and {λα, λ̄α̇} for a basis of S+ ⊕ S−
in the second factor, and, e.g., λs is the image of 1 under the canonical map 1 → S+ ⊗ S+
dual to the scalar valued pairing. We then use
{
θα, θ̄α̇
}
for the linear odd coordinate functions
S∗
+ ⊕ S∗
− ⊆ C∞(N).
With this concrete basis in hand, let us now investigate the additional pieces in the differential
coming from homotopy transfer. For this purpose, let us fix a homotopy h for the differential dρ.
We have
dρ(sα) = λα,
and therefore we can set
h(λα) = sα
and similarly for λ̄ and s̄. There are two nontrivial terms contributing to the transferred differ-
ential
D0hD0
(
θ2
)
= λs
and again similarly for the complex conjugates. We see that this induces a differential that
cancels some of the representative basis elements in pairs. The remaining representatives are[
1
(
θ, θ̄
)
θθ̄ −
− − − λsθ2 + λ̄s̄θ̄2
]
.
It is immediate to see that these representatives span the Spin(4) representations occurring
in (5.1). In addition there is a differential of order one described by
D0hD1
(
θαθ̄β̇Aαβ̇
)
=
(
λsθ2 + λ̄s̄θ̄2
)
∂µAµ.
We thus recover the anticipated description of the vector antifield multiplet.
34 C. Elliott, F. Hahner and I. Saberi
5.3 The chiral multiplet revisited
Let us discuss one further example in the derived formalism, which will illustrate the relation be-
tween on- and off-shell multiplets within the formalism, i.e., the appearance of non-trivial L∞ ac-
tions of the supersymmetry algebra. We will again work specifically with four-dimensionalN = 1
supersymmetry.
In [19], it was demonstrated that the minimal multiplet µA•
R/I(Γ) associated to the mod-
ule Γ = Sym(S+) is equivalent to the BRST version of the chiral multiplet. Constructing the
associated cotangent theory yields the standard off-shell BV theory of the chiral multiplet whose
component fields include a scalar field ϕ, a chiral spinor field ψ, and an auxiliary scalar field F ,
as well as their associated complex conjugates and antifields. Of course, one can integrate out
the auxiliary field F and obtain an equivalent BV theory, but the supersymmetry algebra action
is now no longer strict. This is referred to as the on-shell formulation of the chiral multiplet.
This is discussed in the L∞-module language in [53], but the idea is much older, for example,
the related example of an N = 2 hypermultiplet is discussed in the on-shell language in [3].
Let us again start from the component field formulation for these multiplets and compute
their derived n-invariants. Plugging in this module in the derived pure spinor formalism we
find that the minimal multiplet µA•(C•(n, E)) can be explicitly identified with the on-shell
formulation for the chiral multiplet described above. The off-shell formulation including the
auxiliary field is given by a quasi-isomorphic non-minimal multiplet.
The component fields of the chiral multiplet in the on-shell formulation take the following
form.
E =
Ω0 ⊗ C2 Ω0 ⊗ (S+ ⊕ S−)
Ω0 ⊗ (S+ ⊕ S−) Ω0 ⊗ C2
⋆d⋆d
/∂
.
In order to describe the n action, we will denote the component fields by
(
ϕ, ϕ̄
)
in degree zero,
weight zero and
(
ψ, ψ̄
)
in degree zero, weight one and their respective antifields in degree one
by
(
ϕ∨, ϕ̄∨
)
and
(
ψ∨, ψ̄∨). The odd elements of n act in the following way.
ρ(1)(Q) : S+ ⊕ S− −→ Ω0 ⊗ C2,
(
ψ, ψ̄
)
7→ p+(Q) ∧ ψ + p−(Q) ∧ ψ̄,
Ω0 ⊗ C2 −→ S+ ⊕ S−,
(
ϕ, ϕ̄
)
7→ Q ∧ /∂
(
ϕ+ ϕ̄
)
,
ρ(2)(Q,Q) : S+ ⊕ S− −→ S+ ⊕ S−,(
ψ+, ψ̄+
)
7→ p−(Q)⊗ p+(Q) ∧ ψ∨ + p+(Q)⊗ p−(Q) ∧ ψ̄∨.
With this description we can compute the cohomology of the Chevalley–Eilenberg complex. We
will use the following notation. Let (e, ē) and (s, s̄) denote bases of C2 and S+⊕S− respectively.
As in the previous section, let us use λα, λ̄α̇ to denote even basis elements for C•(n), and
write λs, λ̄s̄ to indicate the images of the canonical maps R → S±⊗S∗
±. One finds the following
description for the Chevalley–Eilenberg cohomology.
H•(n, E) ∼= (Sym(S−)e⊕ Sym(S+)ē) +
(
Sym(S+)λs⊕ Sym(S−)λ̄s̄
)
[−1].
Again, we see that the cohomology is not concentrated in a single degree.
We can calculate the multiplet associated to this module following a method similar to the
previous section, beginning with the multiplets associated to the summands of the cohomology
described above, and then computing the correction terms associated to homotopy transfer. We
obtain a quasi-isomorphic multiplet(⊕
k
A•(Hk(n, E)),D′, ρ′
)
,
The Derived Pure Spinor Formalism as an Equivalence of Categories 35
where the new differential D′ contains terms induced via homotopy transfer. Individually, both
µA•(H0(n, E)
)
and µA•(H1(n, E)
)
contain the field content of a chiral and an antichiral BRST
multiplet. We have the following Betti numbers for the sum of the multiplets induced from the
individual cohomology groups:[
2 4 2 −
− 2 4 2
]
.
There are explicit elements representing the cohomology which take the forrm[(
e, ē
) (
θαe, θ̄α̇ē
) (
θ2e, θ̄2, ē
)
−
−
(
λs, λ̄s̄
) (
λsθ̄α̇, λ̄s̄θ
)
α
(
λsθ̄2, λ̄s̄θ2
)] .
Now let’s take a look at the induced differentials under homotopy transfer. For an explicit
homotopy h we can choose
h(λαe) = sα, h
(
λ̄α̇ē
)
= s̄α̇.
Applying this to the representatives, we find
pD0hD0i(F ) = pD0hD0
(
Fθ2e
)
= p((λs)F ) = F,
pD1hD0i(ψ) = pD1hD0(ψθe) = p
(
(λs)θ̄ /∂ψ
)
= /∂ψ,
pD1hD1i(ϕ) = pD1hD1(ϕe) = p
(
(λs)θ̄2∂2ϕ
)
= ∂2ϕ.
Similar results hold for the complex conjugate fields. Thus, we find that there are induced
differentials of order zero, one and two. With respect to the weight grading, these operators
alter the weight grading by minus one, one and three respectively. We can summarize the
multiplet in the following diagram.
Ω0 ⊗ C2 Ω0 ⊗ (S+ ⊕ S−) Ω0 ⊗ C2
Ω0 ⊗ C2 Ω0 ⊗ (S+ ⊕ S−) Ω0 ⊗ C2
⋆d⋆d
/∂ id
.
This is precisely the off-shell BV model for the chiral multiplet. Taking cohomology with respect
to the acyclic differential we obtain the minimal multiplet µA•(C•(n, E)) which is precisely the
original on-shell formulation that we started with.
Acknowledgements
We would like to give special thanks to R. Eager, J. Walcher, and B. R. Williams for conversa-
tions and collaboration on related projects. We also gratefully acknowledge conversations with
I. Brunner, M. Cederwall, I. Contreras, O. Gwilliam, J. Huerta, J. Palmkvist, and S. Noja.
In addition, we would like to thank the anonymous referees for useful suggestions. This work
is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un-
der Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES
Excellence Cluster). I.S. is supported by the Free State of Bavaria.
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1 Introduction
2 The category of multiplets
2.1 Gradings
2.2 Strict multiplets
2.3 Homotopy theory of multiplets
2.4 Linear structure
3 Derived pure spinor superfields
3.1 The category of C^bullet(n)-modules
3.2 The derived pure spinor superfield formalism
3.3 The multiplet associated to C^bullet(n)
3.4 Component fields and homotopy transfer
3.4.1 Minimal models for A^bullet_{R/I}
3.4.2 Minimal models for A^bullet
4 An equivalence of categories
4.1 The inverse functor: derived n-invariants
4.2 Main theorem and proof
4.3 Some consequences of the theorem
4.4 First examples
5 Applications
5.1 Multiplets from Lie algebra cohomology
5.2 Antifield multiplets in four dimensions
5.3 The chiral multiplet revisited
References
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| id | nasplib_isofts_kiev_ua-123456789-211921 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:08:30Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Elliott, Chris Hahner, Fabian Saberi, Ingmar 2026-01-16T11:20:04Z 2023 The Derived Pure Spinor Formalism as an Equivalence of Categories. Chris Elliott, Fabian Hahner and Ingmar Saberi. SIGMA 19 (2023), 022, 37 pages 1815-0659 2020 Mathematics Subject Classification: 81R25; 17B55; 17B81 arXiv:2205.14133 https://nasplib.isofts.kiev.ua/handle/123456789/211921 https://doi.org/10.3842/SIGMA.2023.022 We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duality for the supertranslation algebra. After introducing and describing the category of supermultiplets, we define the derived pure spinor construction explicitly as a dg-functor. We then show that the functor that takes the derived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor construction, using an explicit calculation. Finally, we illustrate our findings with examples and use insights from the derived formalism to answer some questions regarding the ordinary (underived) pure spinor superfield formalism. We would like to give special thanks to R. Eager, J. Walcher, and B. R. Williams for conversations and collaboration on related projects. We also gratefully acknowledge conversations with I. Brunner, M. Cederwall, I. Contreras, O. Gwilliam, J. Huerta, J. Palmkvist, and S. Noja. In addition, we would like to thank the anonymous referees for their useful suggestions. This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1– 390900948 (the Heidelberg STRUCTURES Excellence Cluster). I.S. is supported by the Free State of Bavaria. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Derived Pure Spinor Formalism as an Equivalence of Categories Article published earlier |
| spellingShingle | The Derived Pure Spinor Formalism as an Equivalence of Categories Elliott, Chris Hahner, Fabian Saberi, Ingmar |
| title | The Derived Pure Spinor Formalism as an Equivalence of Categories |
| title_full | The Derived Pure Spinor Formalism as an Equivalence of Categories |
| title_fullStr | The Derived Pure Spinor Formalism as an Equivalence of Categories |
| title_full_unstemmed | The Derived Pure Spinor Formalism as an Equivalence of Categories |
| title_short | The Derived Pure Spinor Formalism as an Equivalence of Categories |
| title_sort | derived pure spinor formalism as an equivalence of categories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211921 |
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