Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence
The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on the relevant Grassmannian, and give a mirror symmetry interp...
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| description | The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on the relevant Grassmannian, and give a mirror symmetry interpretation, following a physical analysis of Eager, Hori, Knapp, and Romo.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 028, 22 pages
Stringy Kähler Moduli for the Pfaffian–Grassmannian
Correspondence
Will DONOVAN
Yau Mathematical Sciences Center, Tsinghua University,
Haidian District, Beijing 100084, China
E-mail: donovan@mail.tsinghua.edu.cn
Received September 29, 2020, in final form March 10, 2021; Published online March 24, 2021
https://doi.org/10.3842/SIGMA.2021.028
Abstract. The Pfaffian–Grassmannian correspondence relates certain pairs of derived equi-
valent non-birational Calabi–Yau 3-folds. Given such a pair, I construct a set of deri-
ved equivalences corresponding to mutations of an exceptional collection on the relevant
Grassmannian, and give a mirror symmetry interpretation, following a physical analysis
of Eager, Hori, Knapp, and Romo.
Key words: Calabi–Yau threefolds; stringy Kähler moduli; derived category; derived equi-
valence; matrix factorizations; Landau–Ginzburg model; Pfaffian; Grassmannian
2020 Mathematics Subject Classification: 14F08; 14J32; 14M15; 18G80; 81T30
1 Introduction
Birational Calabi–Yau 3-folds are known to have equivalent derived categories [4]. There also
exist pairs of Calabi–Yau 3-folds which are not birational but may be proved to be derived
equivalent. A much-studied class of examples comes from the “Pfaffian–Grassmannian” cor-
respondence, concerning pairs of 3-folds arising as linear sections of the Grassmannian G(2, 7)
and its projective dual Pfaffian. Rødland conjectured that such pairs share a mirror [15], leading
to an expectation that they are derived equivalent: this was proved by Borisov and Căldăraru [3],
and Kuznetsov [13].
Meanwhile, Hori and Tong [10] gave a physical explanation of how such pairs of 3-folds arise
from the same gauged linear σ-model. In work of Addington, Segal, and the author [1], a partial
mathematical interpretation of this was given by constructing a particular derived equivalence
for each pair using categories of matrix factorizations.
According to mirror symmetry, the derived symmetries of a variety may be determined
by monodromy on a stringy Kähler moduli space (SKMS). Hori and Tong described this space
for such 3-folds: implicit in this was a prediction of further equivalences, corresponding to “grade
restriction windows”, see [8, 16]. Later physics work made these windows explicit, and argued
that differences between equivalences are given by spherical twists: see Eager, Hori, Knapp, and
Romo [6], and Hori [9, end of Section 5].
In this paper, I interpret this physics work by constructing, for each pair of 3-folds coming
from the Pfaffian–Grassmannian correspondence, a set of equivalences corresponding to the
“windows” above by extending the methods of [1]. I then show that these equivalences, along
with appropriate spherical twists, may be organized into an action of the fundamental group
of the relevant SKMS.
This paper is a contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji
Saito for his 77th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Saito.html
mailto:donovan@mail.tsinghua.edu.cn
https://doi.org/10.3842/SIGMA.2021.028
https://www.emis.de/journals/SIGMA/Saito.html
2 W. Donovan
1.1 Calabi–Yau pairs
I recall the construction of the 3-folds YG and YP. Start with a 7-dimensional vector space V ,
and consider the following.
� The Grassmannian of 2-planes in its Plücker embedding
G(2, V ) ⊂ P
(
∧2V
)
.
� The Pfaffian of 2-forms on V of rank at most 4, denoted
P(4, V ) ⊂ P
(
∧2V ∨
)
.
These varieties are projectively dual. The Pfaffian P(4, V ) is singular along the locus of forms
of rank at most 2, but taking sufficiently generic hyperplane sections yields a smooth Calabi–Yau
3-fold YP. Taking a dimension 7 subspace Π ⊂ ∧2V ∨ and its annihilator Π◦ ⊂ ∧2V , we then
obtain smooth Calabi–Yau 3-folds as follows:1
YG = G(2, V ) ∩ PΠ◦,
YP = P(4, V ) ∩ PΠ.
1.2 Equivalences
For each pair YG and YP, I construct a set of derived equivalences depending on a discrete
parameter given as follows:
m = (m0,m1,m2) ∈ Z3 such that ml ≤ ml+1 ≤ ml + 1.
Each choice ofm gives an exceptional collection on G(2, V ) by successive mutations of a collection
due to Kuznetsov (Proposition 2.2). This collection is determined by a Lefschetz block
OG(m0), S(m1), Sym2 S(m2),
where S is the rank 2 tautological subspace bundle on G(2, V ).
Now, by extending the construction of [1], I prove the following.
Theorem 1.1 (Theorem 5.8). For each m ∈ Z3 as above, an equivalence
Ψm : Db(YG)
∼−→ Db(YP)
may be constructed using the exceptional collection on G(2, V ) given by m.
Section 3 outlines how Ψm is constructed: it is a “window equivalence”, where generators
are obtained from the collection given by m. The equivalence obtained in [1] corresponds to
m = (6, 7, 8) (Remark 4.3).
1.3 Physics
The SKMS in our case is described in the physics literature as a sphere with five punctures:
see [10, Fig. 1] and [6, Section 4]. Our 3-folds correspond to “large radius limits” near two of
these punctures, shown below as the poles. In this picture, homotopy classes of paths between
large radius limits are expected to correspond to derived equivalences, given by “grade restriction
windows”. Such equivalences are supplied by Theorem 1.1. Monodromy around the other three
punctures are then expected to correspond to spherical twists [9, end of Section 5]. Theorems 1.2
and 1.4 below confirm these expectations from the physics literature.
1For details, see [3]. In fact, if Π is chosen such that YP is a smooth 3-fold, then the same is true of YG [3,
Corollary 2.3].
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 3
1.4 Groupoid action
I show that the fundamental groupoid of the SKMS acts on the derived categories Db(YG)
and Db(YP).
Let M = S2 − {5 points} where two of the punctures are the poles, and basepoints mG
and mP are chosen near them. Following the physics analysis [6, Section 4], I choose a finite
subset Ψk, 0 ≤ k ≤ 3, of the equivalences Ψm of Theorem 1.1 as follows:
Ψk = Ψm, where ml =
{
−1, l < k,
0, l ≥ k.
The corresponding exceptional collections are illustrated in Section 2 as Collections 0–3. Com-
bining the construction of Theorem 1.1 with standard techniques for manipulating window
equivalences, I obtain the following.
Theorem 1.2 (Theorem 6.3). There is an action of the fundamental groupoid π1(M, {mG,mP})
on Db(YG) and Db(YP), given by the following diagram:
M
mG
mP
⊗OYG
(1)
Ψ0Ψ1Ψ2
Ψ3
⊗OYP
(1)
1.5 Mutations and group action
The exceptional collections for the Ψk are related by mutations of exceptional objects as follows:
Collection 0 1 2 3.
OG S Sym2 S
The restrictions of these three exceptional objects to YG are the three spherical objects below
(Proposition 7.6):
OYG , SYG , Sym2 SYG . (1.1)
Remark 1.3. This phenomenon is analogous to exceptional objects restricting to spherical
objects on an anticanonical divisor [18, Example 3.14(c)], though here the codimension of YG
is 7.
I then show (Proposition 7.7) that the “differences” between equivalences(
Ψj+1
)−1 ◦Ψj
are spherical twists around the objects (1.1). Combining this with Theorem 1.2, I deduce the
following.
4 W. Donovan
Theorem 1.4 (Theorem 7.8). There is an action of the fundamental group π1(M,mG) on
Db(YG) given by the following diagram:
M
Sym2S
S
O
mG
⊗OYG
(1)
Ψ−1(⊗OYP
(1))Ψ
Here we let Ψ = Ψ3, and the loops around equatorial holes indicate spherical twists around the
objects (1.1).
It would be interesting to recover M, and the above actions, from an analysis of Bridgeland
stability conditions.
1.6 Contents
Section 2 explains the exceptional collections in Theorem 1.1, Section 3 outlines the structure
of the proof, and then Sections 4 and 5 give the details. Section 6 proves Theorem 1.2, and
Section 7 proves Theorem 1.4.
2 Exceptional collections
In this section, I explain the exceptional collections in Theorem 1.1, and convenient ways to
visualize them.
The construction in [1] used the following full exceptional collection on the Grassmannian
G(2, V ) for a 7-dimensional vector space V [14, Theorem 4.1], where S denotes the rank 2
tautological subspace bundle.{
Syml S∨ ⊗ (detS∨)m : l ∈ [0, 3) , m ∈ [0, 7)
}
. (2.1)
I depict the collection (2.1) as follows, where as usual we write O(1) for detS∨, and for conve-
nience in this section put S2 for Sym2 S:
O
S∨
S2∨
· · ·
· · ·
O(6)
S∨(6)
S2∨(6)
Remark 2.1. The diagonal display format matches with [9] and is used later to visualize the
direction of Homs in the collection: see Remark 4.10.
It will be helpful instead to take a dual exceptional collection as follows.2
2Note that this is an exceptional collection because, for locally free sheaves E and F , we have Ext•(E ,F) =
Ext•
(
F∨, E∨
)
.
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 5
Collection 0.
O(−6)
S(−6)
S2(−6)
· · ·
· · ·
O
S
S2
From this collection, we deduce others by mutation. Namely, fixing an order for the collection
〈E1, . . . , En〉, it is standard that mutating En from right to left gives
L1 · · ·Ln−1(En) = S(En),
where the Li denote left mutation functors, and S is the Serre functor
S = −⊗ ωG[dim G] = −⊗O(−7)[dim G].
Mutating in this way at O, and noting that the property of being an exceptional collection is
invariant under the shift [dim G], we obtain the following:
Collection 1.
O(−7)
S(−6)
S2(−6)
· · ·
· · ·
O(−1)
S
S2
Mutating at S then gives:
Collection 2.
O(−7)
S(−7)
S2(−6)
· · ·
· · ·
O(−1)
S(−1)
S2
We now have two choices for how to mutate, at O(−1) or S2.
Mutating at S2 gives a twist by O(−1) of Collection 0:
Collection 3.
O(−7)
S(−7)
S2(−7)
· · ·
· · ·
O(−1)
S(−1)
S2(−1)
Mutating at O(−1) gives:
Collection 4.
O(−8)
S(−7)
S2(−6)
· · ·
· · ·
O(−2)
S(−1)
S2
Noting then that being a full exceptional collection is invariant under twisting by O(k), we
obtain further collections summarized in the following.
Proposition 2.2. Let m = (m0,m1,m2) be a non-decreasing sequence of integers such that
ml+1 ≤ ml + 1. Then{
Syml S(m) : l ∈ [0, 2] , m ∈ (ml − 7,ml]
}
(2.2)
gives a full exceptional collection on G(2, V ), for dimV = 7.
6 W. Donovan
3 Structure of equivalence proof
I outline the proof of the derived equivalences Ψm between YG and YP in Theorem 1.1, before
giving the proof in the following Sections 4 and 5.
These derived equivalences are obtained by showing that YG and YP are derived equivalent,
in an appropriate sense, to Landau–Ginzburg models
(XG, f) and (XP, f).
Here the space XG is a bundle over G(2, V ), the space XP is an Artin stack which is described
in the next section, and f is a function defined on both these spaces: f arises by restriction
of a function from an Artin stack X, of which XG and XP are open substacks. This follows
a physics construction of Hori–Tong [10], where this function is a “superpotential”.
Each equivalence Ψm is a composition of three equivalences as below, where Db(XG, f) is
a category of matrix factorizations (see for instance [1, Section 2]), and Brm(XP, f) is a closely
related “B-brane category” (Definition 4.5):
Db(XG, f) Brm(XP, f)
Db(YG) Db(YP)
Ψ
m
W
∼
ΨG
∼
Ψ
m
P
∼
Ψm
The vertical arrows are constructed as in [1], using that the spaces Y may be recovered
from the spaces X. In particular, YG is the critical locus of f in XG, giving the left-hand
equivalence ΨG via Knörrer periodicity [19]. The right-hand equivalence Ψ
m
P is a generalized
Knörrer periodicity, established in [1, Section 5].
The horizontal equivalence Ψ
m
W is a “window equivalence”, which factors via a certain sub-
category in Db(X, f), determined by one of the exceptional collections from Proposition 2.2.
This is explained and proved in the following Section 4. The rest of the proof is then given
in Section 5.
Remark 3.1. There is a general theory of window equivalences [2, 7], however it does not yet
apply in this case. In particular, it gives exceptional collections related to that of Kapranov [11]
rather than those in Proposition 2.2: this seems to make it unsuitable for giving an equivalence
with the Calabi–Yau YP. For more discussion, see [1, Remark 4.12].
Remark 3.2. In the physics literature [6, Section 7] a grade restriction window for m = (7, 7, 7)
is associated with the derived equivalence of Borisov and Căldăraru [3]. It would be interesting
to compare the latter with the Ψm obtained in this paper.
4 Equivalences for Landau–Ginzburg models
In this section, I construct the window equivalences Ψ
m
W used to prove Theorem 1.1, after
reviewing the construction of XG and XP, the underlying spaces of the Landau–Ginzburg models
discussed above.
Let S be a 2-dimensional vector space, and consider the quotient stack
G =
[
Hom(S, V )
/
GL(S)
]
.
The open substack of G of rank 2 homomorphisms is equivalent to the variety G(2, V ). Fur-
thermore, the vector bundle on G induced by the representation S of GL(S) restricts to the
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 7
tautological subspace bundle on G(2, V ). The representation detS∨ similarly corresponds to
the bundle O(1) on G(2, V ).
Take the quotient stack
X =
[
Hom(S, V )⊕Hom
(
V,∧2S
)/
GL(S)
]
.
Representations of GL(S) also give vector bundles on X, and its substacks. In particular, the
representation S gives a bundle on X, and the notation S will also be used to denote this bundle.
For brevity, we write the line bundle corresponding to the representation
(
detS∨
)⊗k
as O(k),
by analogy with the notation on G.
Now, with x ∈ Hom(S, V ) and p ∈ Hom
(
V,∧2S
)
, let
XG = {rkx = 2} and XP = {rk p = 1}
be open substacks of X. These have the structure of a vector bundle over G(2, V ) and an Artin
stack P, respectively, where
P =
[
Homrk=1
(
V,∧2S
)/
GL(S)
]
.
We let f denote a function on the stack X, and use the same notation for its restriction
to substacks. As the particular form of f is not used in this section, we defer its definition to
the following Section 5.
The following categories correspond to the collections of Proposition 2.2.
Definition 4.1 (window subcategories Wm). Let m = (m0,m1,m2) be a non-decreasing sequ-
ence of integers such that ml+1 ≤ ml + 1. Then{
Syml S(m) : l ∈ [0, 2] , m ∈ (ml − 7,ml]
}
(4.1)
gives a set of bundles on X, and we write Wm for the full subcategories of Db(X) and Db(X, f)
generated by this set.
For brevity, we notate bundles as follows.
Notation 4.2. Let Sl,m denote the bundle Syml S(m) on X, or its restriction to a substack.
Remark 4.3. In [1] a set of generators Tl,m = Syml S∨(m) was used. Note that Tl,m ∼= Sl,m+l,
using that S∨ ∼= S(1) because S∨ ∼= S ⊗ detS∨. The window subcategory used in [1], with
generators given in (2.1) above, is therefore the window Wm with m = (6, 7, 8) in our notation
here.
We write embeddings of stacks as follows:
XG
iG−→ X
iP←− XP.
Proposition 4.4. For Wm ⊂ Db(X, f) the derived functor
i∗G : Wm → Db(XG, f)
is an equivalence, and the derived functor
i∗P : Wm → Db(XP, f)
is fully faithful.
8 W. Donovan
Proof. We first prove the analogous statement with Db(XG, f) and Db(XP, f) replaced with
Db(XG) and Db(XP), before explaining how to deduce the proposition.
To prove fully faithfulness, we generalize the argument given in [1, Lemma 4.3]. It suffices
to check fully faithfulness on generators: take then two of the generators Sl,m and Sl′,m′ of Wm.
The Homs between these are equal to the Homs between their restrictions to XG and XP,
respectively, because the complements of the latter have codimensions at least 2. Furthermore,
there are no higher Exts between them because they are vector bundles on an affine stack, so it
suffices to check that they do not acquire any higher Exts after applying the functors i∗. Namely,
we require the following:
Ext>0
XG
(
i∗GSl,m, i
∗
GSl′,m′
)
= 0, (4.2)
Ext>0
XP
(
i∗PSl,m, i
∗
PSl′,m′
)
= 0. (4.3)
These vanishings are proved in Sections 4.1 and 4.2, respectively.
We then claim that i∗G is essentially surjective. For this, note that the generators Sl,m, when
considered as sheaves on G(2, V ) give generators for the derived category by Proposition 2.2.
Essential surjectivity then follows by a general argument, exactly as in [1, Lemma 4.6].
To complete the proof of the proposition, we repeat the arguments in [1, Proposition 4.9,
Lemma 4.10]. The important point here is that morphisms in the categories Db(XG, f) and
Db(XP, f) are related to those in the categories Db(XG) and Db(XP) by certain spectral sequ-
ences. �
Using the above proposition we have the following.
Definition 4.5 (B-brane categories). We define a subcategory
Brm(XP, f) ⊂ Db(XP, f)
to be the image of Wm ⊂ Db(X, f) under i∗P.
Remark 4.6. In other words, Brm(XP, f) is the full subcategory of Db(XP, f) generated by
the bundles given by (4.1).
Definition 4.7 (window equivalences). We write
Ψ
m
W = i∗P ◦ (i∗G)−1 : Db(XG, f)→Wm → Brm(XP, f).
4.1 Grassmannian side
I start with some vanishing results on G(2, V ), before proving the vanishing (4.2) on XG. I recall
the following, again writing Tl,m = Syml S∨(m).
Lemma 4.8 ([1, Lemma 4.5]). Let G = G(2, V ), with dimV = n odd. If 0 ≤ l, l′ ≤ 1
2n − 1
and m′ ≥ m then we have
Ext>0
G
(
Tl,m, Tl′,m′
)
= 0.
We quickly obtain the following, where again Sl,m = Syml S(m).
Corollary 4.9. In the setting of Lemma 4.8 above, so that in particular we have m′ ≥ m,
Ext>0
G
(
Sl,m, Sl′,m′
)
= 0 and Ext>0
G
(
Sl,m, Sl′,m′+l′−l
)
= 0.
Proof. For the first, we exchange l and l′, and note that the result is isomorphic to the Ext-
group in Lemma 4.8 above, using local freeness of the Syms. For the second, we use that
Tl,m ∼= Sl,m+l, and the local freeness of O(1). �
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 9
Remark 4.10. In terms of the following picture, this corollary says that there are no higher
Exts on G(2, V ) from an object on the lines pictured, say L, to any object on a parallel line L′
lying to the right of L:
· · ·
· · ·
S0,−1 S0,0
S1,0
S2,0
S0,1
S1,1
S2,1
S0,2
S1,2
S2,2 S2,3
· · ·
· · ·
I now show the vanishing (4.2). By a standard calculation using the structure of XG as
a vector bundle over G, we have
RHomXG
(
i∗GSl,m, i
∗
GSl′,m′
) ∼= ⊕
n≥0
RHomG
(
Sl,m, Sl′,m′ ⊗ SymnO(1)⊕7
)
.
By expanding the Symn piece into irreducibles we see that this splits into summands
RHomG
(
Sl,m, Sl′,m′+n
)
,
where Sl,m, Sl′,m′ ∈ Wk and n ≥ 0. It therefore suffices to know that there are no higher Exts
from the generators Sl,m of Wm to other generators Sl′,m′ of Wm or to sheaves Sl′,m′+n “to their
right”. But this is clear from the form of the exceptional collections in Section 2, combined with
Corollary 4.9 above.
4.2 Pfaffian side
We now show the vanishing (4.3). Similarly to the above, by a standard calculation we have
RHomXP
(
i∗PSl,m, i
∗
PSl′,m′
) ∼= ⊕
n≥0
RHomP
(
Sl,m, Sl′,m′ ⊗ Symn S⊕7
)
.
We check the following very straightforward bound. This will give the vanishing for the Sym0
piece: the vanishing for the Symn piece with n > 0 will then follow by induction.
Lemma 4.11. For two generators Sl,m and Sl′,m′ of Wm we have
m′ −m < max(l′ − l, 0) + 7.
Proof. By assumption
m ∈ (ml − 7,ml] and m′ ∈ (ml′ − 7,ml′ ].
Notice then that
m′ −m < ml′ − (ml − 7) = (ml′ −ml) + 7.
Recall that the ml form a non-decreasing sequence with ml+1 ≤ ml + 1. If l′ ≥ l then
ml′ −ml ≤ l′ − l,
and otherwise ml′ −ml ≤ 0, hence the claim. �
Remark 4.12. The collections W(0,0,0) and W(−2,−1,0) (given as Collections 0 and 4 in Section 2)
show that, amongst such bounds which are uniform for all the Wm, the bound of Lemma 4.11
is the best possible.
10 W. Donovan
For our induction, we now observe that
Sl,m ⊗ S = Syml S(m)⊗ S
∼=
(
Syml+1 S ⊕ Syml S ⊗ ∧2S
)
(m)
∼= Syml+1 S(m) ⊕ Syml S(m− 1)
= Sl+1,m ⊕ Sl,m−1. (4.4)
Now suppose we use this repeatedly to expand the Symn piece, namely
RHomP
(
Sl,m, Sl′,m′ ⊗ Symn S⊕7
)
.
I claim we get summands
RHomP
(
Sl,m, Sl′′,m′′
)
(4.5)
for which the inequality
m′′ −m < max(l′′ − l, 0) + 7. (4.6)
is satisfied. This follows by induction: the base case n = 0 is shown in Lemma 4.11, and the
inequality still holds when we increment l′′ or decrement m′′ each time we apply (4.4).
Let us now calculate the cohomology of the summands (4.5). First note
RHomP
(
Sl,m, Sl′′,m′′
) ∼= Syml S∨(−m)⊗ Syml′′ S(m′′)
∼= Syml S(l −m)⊗ Syml′′ S(m′′)
∼=
(
Syml S ⊗ Syml′′ S
)
(l −m+m′′).
Applying RΓP to this, we will obtain RHomP ∼= RΓP ◦ RHomP . Observe then that there is
a morphism of stacks δ : P → P6 induced by det : GL(S)→ C∗, and we have RΓP ∼= RΓP6 ◦ δ∗.
The functor δ∗ takes ker(det)-invariants where ker(det) = SL(S). Expanding Syml S ⊗ Syml′′ S
into irreducibles, the only sheaf that survives this is (∧2S)⊗l ∼= O(−l) in the case that l = l′′.
We thence deduce that the only contribution to the cohomology of (4.5) comes from
RΓPO(−m+m′′)
in the case l = l′′ and that, setting n = m′′ −m, this contribution is
RΓP6δ∗O(n) ∼= RΓP6OP6(−n). (4.7)
But (4.6) implies that if l = l′′ then n < 7, so the required vanishing of higher Exts in (4.3)
follows from standard cohomology vanishing on P6.
Remark 4.13. The minus sign in (4.7) appears because the bundle O(1) on
P =
[
Hom
(
V,∧2S
)
− {0}
/
GL(S)
]
corresponds to the representation ∧2S∨ by the definitions at the beginning of this section. The
bundle OP6(1), on the other hand, corresponds to ∧2S.
Remark 4.14. The last part of the argument above, the calculation of the cohomology of the
summands (4.5), parallels the proof of [1, Lemma 4.3]. We include it here to explain some
details, and for convenience.
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 11
5 Equivalences for Calabi–Yau 3-folds
We now show that the categories Db(XG, f) and Brm(XP, f) are equivalent, respectively, to the
derived categories of our Calabi–Yau 3-folds YG and YP, completing the proof of Theorem 1.1.
In this section we require the definition of the function f on X from [1], as follows. Take
a surjective map A : ∧2 V → V such that the dual subspaces Π and Π◦ used to construct the
3-folds in Section 1.1 are given by
Π = kerA∨ and Π◦ = ImA.
Recalling that
X =
[
Hom(S, V )⊕Hom
(
V,∧2S
)/
GL(S)
]
with x ∈ Hom(S, V ) and p ∈ Hom
(
V,∧2S
)
take a function on X defined by
f(x, p) = p ◦A ◦ ∧2x
via the canonical isomorphism Hom
(
∧2S,∧2S
) ∼= C.
In addition to the data (X, f), the Landau–Ginzburg model includes the data of a C∗-action
on X for which f has weight 2. For this, we let C∗ act with weight 0 on x, and weight 2 on p.
Note that this action has been suppressed in our notation so far. Following physics terminology,
it is sometimes know as the “R-charge”. For more details, see [1, Section 2.1].
5.1 Grassmannian side
Recall that XG is by definition the Artin stack
XG =
[{
(x, p) ∈ Hom(S, V )⊕Hom
(
V,∧2S
)
: rk(x) = 2
}/
GL(S)
]
,
which is isomorphic to the total space of the vector bundle O(−1)⊕7 over the Grassman-
nian G(2, V ). Denote the projection morphism for this bundle by π. The Calabi–Yau 3-fold YG
defined in Section 1.1 is the zero locus of a transverse section of O(1)⊕7 given by
s = A ◦ ∧2x.
Then we have f = sp, where p is the tautological section of π∗O(−1)⊕7. The C∗-action preserves
each fibre of the bundle, and acts with weight 2 on those fibres.
Given this geometric situation, an equivalence from Db(YG) to Db(XG, f) follows from a ver-
sion of Knörrer periodicity, as follows. Let XG|YG denote the base change of the bundle XG
over G to the base YG. Then we have
YG
π←− XG|YG
k−→ XG,
where k is the inclusion.
Definition 5.1. We define an equivalence ΨG by the composition
ΨG : Db(YG)
π∗−→ Db(XG|YG)
k∗−→ Db(XG, f).
This is an equivalence by a result of Shipman [19, Theorem 3.4].
12 W. Donovan
5.2 Calculations
Later, in Section 7, I study certain bundles on YG to prove Theorem 1.4: for use there, in this
subsection I calculate the images of these objects under ΨG. I first give the method in a simpler
case, before applying it to YG.
Example 5.2. Consider a line bundle L over a variety B, and a subvariety Y ⊂ B cut out
by a transverse section s of L. Let X be the total space of L∨, with projection π, and p the
tautological section of π∗L∨. We make the setup C∗-equivariant so that s and p have weights 0
and 2, respectively. Take notation as follows, where X|Y is the total space of the base change
of the bundle L over B to Y , and j is the inclusion of the zero section B in X:
XX|Y
BY
k
k
π jπ
Then in Db(X, f), where f = sp, we have an object K given by
π∗L∨[−1] OX ,
s
p
where the shift [−1] denotes a change in C∗-weight. Forgetting the right- and left-moving
morphisms respectively in K gives Koszul resolutions for C∗-equivariant sheaves
π∗k∗OY and j∗OB ⊗ π∗L∨[−1]. (5.1)
Note that these determine objects of Db(X, f) because they are supported on the zero locus
of f , namely X|Y ∪ B. Indeed, they may be written as pushforwards from this locus using
isomorphisms
π∗k∗OY ∼= k∗π
∗OY and j∗OB ⊗ π∗L∨ ∼= j∗
(
OB ⊗ j∗π∗L∨
) ∼= j∗
(
OB ⊗ L∨
)
which follow by flat base change, and the projection formula for j, respectively. Now, by
a standard argument with matrix factorizations, both the objects of (5.1) are isomorphic to K
in Db(X, f): see for instance [19, proof of Lemma 3.2]. Putting all this together, we obtain
an isomorphism in Db(X, f) as follows.
k∗π
∗OY ∼= j∗(OB ⊗ L∨)[−1].
We generalize the method of this example to deduce the following.
Proposition 5.3. We have that
ΨG : OYG 7→ j∗OG(−7)[−7] ∼= j!OG,
where j is the inclusion of the zero section of XG, and similarly with occurences of O replaced
with S or Sym2 S.
Proof. Take F = O(1)⊕7, a bundle on G(2, V ) of rank r = dimV = 7, and set notation for
morphisms as follows:
XGXG|YG
GYG
k
π jπ
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 13
Following the argument of the example above, we take K in Db(XG, f) as follows:
∧rπ∗F∨[−r] . . . π∗F∨[−1] OXG
,
s
p
s
p
s
p
which gives an isomorphism of objects
k∗π
∗OYG ∼= j∗
(
OG ⊗ detF∨
)
[−r].
Recalling that j! = j∗(ωj [dim j] ⊗ −), and noting that ωj = detF∨ and dim j = −r, the first
claim follows. Twisting K by S or Sym2 S then gives the further claim. �
5.3 Pfaffian side
Recall that XP is the Artin stack
XP =
[{
(x, p) ∈ Hom(S, V )⊕Hom
(
V,∧2S
)
: p 6= 0
}/
GL(S)
]
.
Given A : ∧2 V → V , each value of p determines a ∧2S-valued 2-form on V , namely ωp = p ◦A.
We then have that f(x, p) = ωp ◦ ∧2x by the definition of f .
In this subsection, we construct an equivalence
Db(YP)→ Brm(XP, f)
for each choice of m. In [1], such an equivalence was constructed for m = (6, 7, 8) by considering
isotropic subspaces for the ωp as p varies. An outline is given in [1, Section 5.1] with details
in the subsections following. The argument is lengthy and subtle, but to obtain an equivalence for
general m I only need to modify the very last step of it: this subsection explains the modification.
The argument uses the following locus in XP. Note that XP is a bundle over P and also
over P6 by composition with the morphism δ : P → P6 induced by det : GL(S)→ C∗.
Definition 5.4 ([1, Definition 5.5]). Let Γ ⊂ XP be the closed substack with points (x, p),
where p corresponds to a point of YP ⊂ P6, and the composition of x with a quotient morphism
S
x−→ V → V/ kerωp
has rank at most 1.
Remark 5.5. Note that, when rk(x) = 2, the above condition matches the condition defining
the correspondence between YG and YP which appears in Borisov and Căldăraru’s work [3,
Section 0.7]. They show that the ideal sheaf of this correspondence gives a derived equivalence.
Note that Γ is a flat family of stacks over YP. The following proposition gives the construction
of Ψ
m
P . Take notation as below, where j and k denote inclusions:
XPXP|YP
P6YP
Γ
k
k
ππ
j
Proposition 5.6. There exists an equivalence
Ψ
m
P = k∗ ◦ j∗ ◦ j∗ ◦ π∗ : Db(YP)
∼−→ Brm(XP, f) ⊂ Db(XP, f).
14 W. Donovan
Proof. I explain how choices can be made so that the given composition has essential image
in Brm(XP, f). This refines the argument of [1, proof of Proposition 5.9].
I first explain why, for E a sheaf on YP , we may make choices so that Ψ
m
P (E) ∈ Brm(XP, f).
The result then follows by generation. Using that πj is flat, Ψ
m
P (E) is a sheaf on XP. Further-
more, by the projection formula,
Ψ
m
P (E) = k∗j∗j
∗π∗(E) ∼= k∗(j∗OΓ ⊗ π∗E).
Now j∗OΓ has an Eagon–Northcott resolution
∧4Q∨ ⊗ Sym2 S(1)→ ∧3Q∨ ⊗ S(1)→ ∧2Q∨(1)→ O,
where Q = V/ kerωp, by for instance [20, Section 6.1.6]. This may be made C∗-equivariant by
adding appropriate shifts of C∗-weight, and it follows that Ψ
m
P (E) has a C∗-equivariant resolution
π∗F2 ⊗ Sym2 S → π∗F1 ⊗ S → π∗F0 → π∗F ,
where F and F0, . . . ,F2 are sheaves in the image of j∗ : Coh(YP)→ Coh(P6). Using a Beilinson
collection on P6, each of these Fs can be replaced by a quasi-isomorphic complex whose terms
are direct sums of line bundles OP6(−m) with m ∈ (n− 7, n], for any choice of integer n.
Now π∗OP6(−m) ∼= O(m), with the sign appearing as in Remark 4.13. Therefore, we may
obtain a resolution of Ψ
m
P (E) by vector bundles in the window subcategory Wm, by letting n = ml
for Fl, and n = m0 for F . Doing likewise for E ranging over a set of generators for Db(YP),
it follows that Ψ
m
P may be constructed to have essential image in Brm(XP, f). The proof of the
equivalence property is exactly as in [1, Theorem 5.12]. �
Remark 5.7. A different construction of such a functor is given in [17, Section 4.2], avoiding
much of the difficult analysis in [1]. However, the discussion in the latter suffices for our purposes.
5.4 Equivalences
We now complete the proof of Theorem 1.1.
As before, let m = (m0,m1,m2) be a non-decreasing sequence of integers such that ml+1 ≤
ml + 1.
Theorem 5.8. For each m as above, there is an equivalence
Ψm : Db(YG)
∼−→ Db(YP),
which factors through the subcategory Wm of Db(X, f).
Proof. Recall that in the previous section we obtained (Definition 4.7) an equivalence as follows.
Ψ
m
W = i∗P ◦ (i∗G)−1 : Db(XG, f)→Wm → Brm(XP, f).
Composing this with the equivalences of this section, from Definition 5.1 and Proposition 5.6,
we get
Ψm = (Ψ
m
P )−1 ◦Ψ
m
W ◦ΨG : Db(YG)→ Db(YP)
and the theorem is proved. �
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 15
6 Groupoid action
In this section I construct the groupoid action of Theorem 1.2. Let us first recall the notation
which was used to state this theorem in Section 1.4.
Notation 6.1. For brevity, we write windows, and equivalences,
W0 = W(0,0,0),
W1 = W(−1,0,0),
W2 = W(−1,−1,0),
W3 = W(−1,−1,−1),
Ψ0 = Ψ(0,0,0),
Ψ1 = Ψ(−1,0,0),
Ψ2 = Ψ(−1,−1,0),
Ψ3 = Ψ(−1,−1,−1),
and similarly write Ψl
W for l = 0, . . . , 3.
Theorem 1.2 will follow from the fact that W3 = W0⊗O(−1), and the following proposition.
Proposition 6.2. There are natural isomorphisms as follows
(−⊗OXG
(1)) ◦ΨG
∼= ΨG ◦ (−⊗OYG(1)),
(−⊗OXP
(1)) ◦ΨP
∼= ΨP ◦ (−⊗OYP(−1)).
Proof. Recall from Definition 5.1 that ΨG is the composition
Db(YG)
π∗−→ Db(XG|YG)
k∗−→ Db(XG, f),
where morphisms are as follows:
XGXG|YG
GYG
k
π
Now assume given sheaves L on XG and M on YG such that k∗L ∼= π∗M. Then, using the
projection formula, we have isomorphisms as follows:
(−⊗ L) ◦ΨG = (−⊗ L) ◦ (k∗π
∗) ∼= k∗ ◦ (−⊗ k∗L) ◦ π∗
∼= k∗ ◦ (−⊗ π∗M) ◦ π∗ ∼= (k∗π
∗) ◦ (−⊗M)
∼= ΨG ◦ (−⊗M).
We immediately deduce the first statement, by taking L = OXG
(1) and M = OYG(1).
From Proposition 5.6 we have that Ψ
m
P is isomorphic to the composition
Db(YP)
(πj)∗−−−→ Db(Γ)
(kj)∗−−−→ Db(XP, f),
where morphisms are as shown below. Here we write Db(Γ) rather than Db(Γ, f) because f
restricts to zero on Γ ⊂ XP by Definition 5.4.
XPXP|YP
P6YP
Γ
k
π
j
Applying a similar argument to ΨP using sheaves L = OXP
(1) and M = OYP(−1), we deduce
the second statement. In this case it suffices that k∗L ∼= π∗M, as this gives (kj)∗L ∼= (πj)∗M
after applying j∗. The minus sign here arises as in Remark 4.13. �
16 W. Donovan
Theorem 6.3. There is an action of the groupoid π1(M, {mG,mP}) for M = S2 − {5 points}
on Db(YG) and Db(YP), given by the following diagram.
M
mG
mP
⊗OYG
(1)
Ψ0Ψ1Ψ2
Ψ3
⊗OYP
(1)
Proof. Removing a point from the far side of the sphere, the claim is that the groupoid acts
by the following diagram, subject to a relation: that monodromy starting at either Db(YG)
or Db(YP) and then going around a large circle gives the identity up to isomorphism:
Db(YG) Db(YP)
Ψ3
Ψ2
Ψ1
Ψ0
⊗OYP
(1)⊗OYG
(1)
Now we have that W3 = W0 ⊗OX(−1), and note that
i∗GOX(m) ∼= OXG
(m) and i∗POX(m) ∼= OXP
(m).
It then follows easily from Definition 4.7 of the window equivalences Ψ
m
W, of which the Ψl
W are
examples, that
Ψ3
W
∼= (−⊗OXP
(−1)) ◦Ψ0
W ◦ (−⊗OXG
(1)),
noting the opposite signs in the line bundle twists. Combining with the above Proposition 6.2
then gives an isomorphism
Ψ3 ∼= (−⊗OYP(1)) ◦Ψ0 ◦ (−⊗OYG(1)),
which yields the required relation, and completes the proof. �
7 Monodromy action
In this section I describe the differences between the equivalences Ψl in Theorem 1.2 (Theo-
rem 6.3 above) as spherical twists on the Calabi–Yau YG, and thereby prove Theorem 1.4.
To describe the difference Ψ′−1 ◦Ψ between such equivalences Ψ and Ψ′, we will use a func-
tor Tr that “transfers” between the two corresponding windows W and W′. The difference is
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 17
then described as a twist functor, denoted Tw. The following proposition gives the formal part
of this procedure.
For convenience, take two functors from W = Wm as follows:
ΦG = (ΨG)−1 ◦ i∗G, ΦP = (Ψ
m
P )−1 ◦ i∗P
so that the factorization of Ψ = Ψm via W takes the form
Ψ ∼= ΦP ◦ (ΦG)−1 : Db(YG)→W→ Db(YP).
Proposition 7.1 ([5, Proposition 2.2]). For windows W and W′, assume given a functor
Tr: W → W′ that intertwines with an autoequivalence Tw of Db(YG) and with the identity
on Db(YP), namely
ΦG ◦ Tr ∼= Tw ◦ΦG and Φ′P ◦ Tr ∼= ΦP. (7.1)
Then there is an isomorphism
Ψ′−1 ◦Ψ ∼= Tw,
where Ψ and Ψ′ are the window equivalences associated to W and W′.
Proof. This is a diagram chase of functors. In the reference Tr was assumed to be a restriction
of an endofunctor of the category containing W, in our case Db(X, f), but the proof proceeds
without this assumption. �
To construct functors Tr we define endofunctors of Db(X, f), and then show that they restrict
to functors between windows. Recall that X is a vector bundle over a stack
G =
[
Hom(S, V )
/
GL(S)
]
and write j : G → X for the inclusion of the zero section. Recall also that
j! = j∗(ωj [dim j]⊗−) ∼= j∗(OG(−7)[−7]⊗−),
where we use that ωj is the determinant of the normal bundle of G.
Definition 7.2. Take endofunctors of Db(X, f)
Tr0 = Tr(j!OG), Tr1 = Tr(j!S), Tr2 = Tr
(
j!S
2
)
,
where
Tr(E) = Cone
(
E ⊗HomX(E ,−)→ id
)
.
Here, and elsewhere in this section, we again put S2 for Sym2 S.
Remark 7.3. The functorial cone above is shorthand for the usual Fourier–Mukai constructions.
I then claim the following.
Proposition 7.4. The Trl restrict to functors Trl : Wl →Wl+1.
18 W. Donovan
Proof. I first show this for Tr0, as the others are similar. I claim that the functor acts as the
identity on all generators of the window W0 except O. First observe that
HomX(j!OG ,−) ∼= HomG(OG , j∗−). (7.2)
This is zero except on O by inspection of Collection 0, noting the vanishing in Corollary 4.9.
It also implies, using that objects in the collection are exceptional, that
HomX(j!OG ,O) = C.
A generator of this Hom may be seen explicitly by writing down the Koszul resolution of j!OG ,
namely
O → O(−1)⊕7 → · · · → O(−6)⊕7 → O(−7). (7.3)
We thus see that Tr0(O) = Cone(j!OG → O) is quasi-isomorphic to a complex given by
O(−1)⊕7 → · · · → O(−6)⊕7 → O(−7)
which, in particular, lies in the window W1 given as Collection 1.
For the other two functors Trl for l = 1, 2, we replace (7.2) with
HomX
(
j!S
l,−
) ∼= HomG
(
Sl, j∗−
)
and repeat the same argument, where we tensor (7.3) by Sl. �
I now complete the proof of the assumptions of Proposition 7.1, in particular property (7.1),
after constructing autoequivalences of Db(YG) as follows.
Definition 7.5. Take endofunctors of Db(YG)
Tw0 = Tw(OYG), Tw1 = Tw(SYG), Tw2 = Tw
(
S2
YG
)
,
where
Tw(F) = Cone
(
F ⊗HomYG(F ,−)→ id
)
.
Proposition 7.6. Each of the Twl is a twist by a spherical object, and therefore is an auto-
equivalence.
Proof. To see this for Tw0, write k : YG ↪→ G and note that
Ext•YG(OYG ,OYG) = Ext•YG(k∗OG, k
∗OG) ∼= Ext•G(OG, k∗k
∗OG).
After taking a Koszul resolution of k∗k
∗OG, this may be calculated by a spectral sequence from
Ext•G(OG,∧•OG(−1)⊕7). By examination of the exceptional collectional given as Collection 0,
the only non-trivial terms occuring are Ext0(OG,OG) ∼= C and
Ext•G(OG,OG(−7)) ∼= H•G(OG(−7)) ∼= C[−dim G]
by duality. Noting that YG is Calabi–Yau, the result follows. A similar argument applies to the
other Twl. �
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 19
Proof of intertwinement (7.1). The isomorphism Φ′P ◦ Tr0 ∼= ΦP follows immediately from
the cone construction of Tr0: recall that ΦP = (Ψ
m
P )−1◦i∗P and note that i∗P j!OG = 0 for support
reasons. The same argument applies to the other Trl.
The other intertwinements
ΦG ◦ Trl ∼= Twl ◦ΦG (7.4)
are more involved, but essentially follow from the exceptional collection property. We give the
argument for Tr0 before explaining how it adapts to the other Trl.
As preparation, recall that ΦG = Ψ−1
G ◦ i∗G, and write i in place of iG for clarity. It will be
more convenient to prove the following, which is equivalent to (7.4).
i∗ ◦ Trl ∼= ΨG ◦ Twl ◦Ψ−1
G ◦ i
∗. (7.5)
By standard facts about spherical twists ΨG ◦ Tw0 ◦Ψ−1
G is given by the spherical twist around
ΨG(OYG). By Proposition 5.3 this is isomorphic to j!OG, and therefore
ΨG ◦ Tw0 ◦Ψ−1
G
∼= Cone(j!OG ⊗HomXG
(j!OG,−)→ id).
Now we have a fibre product diagram
XXG
GG
i
i
jj
and noting that the i are open immersions and therefore flat, we find by base change that
j!OG = j!i
∗OG ∼= i∗j!OG .
We thence have isomorphisms of functors Db(X, f)→ Db(XG, f) as follows:
i∗ ◦ Tr0 ∼= Cone(i∗j!OG ⊗HomX(j!OG ,−)→ i∗),
ΨG ◦ Tw0 ◦Ψ−1
G ◦ i
∗ ∼= Cone(i∗j!OG ⊗HomXG
(i∗j!OG , i∗−)→ i∗).
Comparing the two cones above suggests that the intertwinement may follow from an iso-
morphism
HomX(j!OG ,−) ∼= HomXG
(i∗j!OG , i∗−) on W0. (7.6)
This isomorphism does indeed hold: we follow an argument for this, which furthermore gi-
ves (7.5), in [5, proof of Lemma 3.17, end of Section 3.2.2]. (Note that the setting there is more
general, involving a spherical functor, not just a spherical object.) According to the argument,
it suffices if the natural morphism of functors from Db(G)
τ : RΓG → RΓG ◦ i∗
is an isomorphism on the subcategory j∗W0 of Db(G). To see why we take this subcategory,
note that j∗ : Db(X, f)→ Db(G) because f |G = 0, and the left-hand side of (7.6) is given by
HomX(j!OG ,−) ∼= HomG(OG , j∗−) ∼= RΓG(j∗−). (7.7)
20 W. Donovan
Now we have that j∗− ∼= HomG(O, j∗−). It therefore suffices to check if τ is an isomorphism
on objects
A = HomG(O, j∗B),
where B is a generator of W0.
We determine τA. Firstly, R>0ΓG(A) = 0 becauseA is a vector bundle on an affine stack. Fur-
thermore, R>0ΓG(i∗A) ∼= Ext>0
G (O, i∗j∗B). But then this vanishes by the exceptional collection
property, as O is on the right-hand side of the collection associated to W0, namely Collection 0.
Finally, we have that H0τA is an isomorphism by normality of G, as the codimension of G −G
is greater than two. We deduce that τA is an isomorphism, and thence τ is an isomorphism,
thereby proving the intertwinement (7.4) for Tr0.
For the other transfer functors Tr1 and Tr2, it now suffices that τ is an isomorphism on S∨⊗
j∗W1 and S∨2 ⊗ j∗W2, respectively. These categories appear because the role of (7.7) is repla-
ced by
HomX(j!S,−) ∼= RΓG
(
S∨ ⊗ j∗(−)
)
,
HomX
(
j!S
2,−
) ∼= RΓG
(
S∨2 ⊗ j∗(−)
)
.
But then the above argument suffices, using that S is on the right-hand side for Collection 1,
and S2 is on the right-hand side for Collection 2, and that Proposition 5.3 continues to hold
with occurences of O replaced with S or S2. �
Combining the above, and applying Proposition 7.1, we can describe differences between
equivalences as follows.
Proposition 7.7. The twists Tw correspond to “window shifts” as follows:(
Ψ1
)−1 ◦Ψ0 ∼= Tw0 = Tw(OYG),(
Ψ2
)−1 ◦Ψ1 ∼= Tw1 = Tw(SYG),(
Ψ3
)−1 ◦Ψ2 ∼= Tw2 = Tw(S2
YG
).
We may then complete the proof of Theorem 1.4, as follows.
Theorem 7.8. There is an action of the fundamental group π1(M,mG) on Db(YG) given by
the following diagram, with Ψ = Ψ3:
M
Tw2
Tw1
Tw0
mG
⊗OYG
(1)
Ψ−1(⊗OYP
(1))Ψ
Proof. We take the groupoid action of Theorem 1.2 (Theorem 6.3) and forget one of the
basepoints mP, as follows. Restricting that action to a chart containing both basepoints, we
obtain the left-hand picture below. Proposition 7.7 then says that monodromies at Db(YG)
around equatorial holes are given by the right-hand picture:
Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence 21
Db(YG)
Tw2
Tw1
Tw0
Db(YG) Db(YP)
Ψ3
Ψ2
Ψ1
Ψ0
The remaining two monodromies are immediate, giving the result. �
Remark 7.9. It is natural to ask for a Pfaffian analogue of Theorem 7.8 with a basepoint mP
in place of mG. This seems an interesting and approachable problem, but it appears that the
method here would need modification to solve it, as follows.
A first step could be to establish spherical objects on the 3-fold YP and prove an analogue
of Proposition 5.3, giving their images under appropriate equivalences Ψ
m
P . To continue the
strategy above, we would then need manageable resolutions for the images, as in the Koszul
resolution of j!OG in (7.3). These should be obtainable, but can be expected to be more compli-
cated than Koszul resolutions, because of the Eagon–Northcott resolutions in the construction
of Ψ
m
P from the proof of Proposition 5.6.
Remark 7.10. I do not discuss whether the action of Theorem 7.8 is faithful. If it is, then the
spherical twists by the objects in Proposition 7.7 act as a free group: it could be interesting
to try to prove this by some B-side analogue of results of Keating on free group actions by
symplectic Dehn twists [12].
Acknowledgements
In celebration of his 77th birthday, I am pleased to express my gratitude to Kyoji Saito for his
kindness and interest over the years. I also want to thank N. Addington and E. Segal for the
great experience of working on our paper [1], of which this work is a continuation.
I am supported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan. I also
acknowledge the support of WPI Initiative, MEXT, Japan, and of EPSRC Programme Grant
EP/R034826/1.
I am grateful for conversations with K. Hori, who made me aware of the windows used here,
for discussions with R. Eager, A. Kuznetsov, and M. Romo, and for helpful suggestions from
anonymous referees. In the last stage of this project, I was away from my home institution due
to the coronavirus pandemic, so I am especially appreciative of hospitality and support from
Kavli IPMU, in particular Y. Ito and M. Kapranov, and from B. Kim at KIAS, and M. Wemyss
at Glasgow University.
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1 Introduction
1.1 Calabi–Yau pairs
1.2 Equivalences
1.3 Physics
1.4 Groupoid action
1.5 Mutations and group action
1.6 Contents
2 Exceptional collections
3 Structure of equivalence proof
4 Equivalences for Landau–Ginzburg models
4.1 Grassmannian side
4.2 Pfaffian side
5 Equivalences for Calabi–Yau 3-folds
5.1 Grassmannian side
5.2 Calculations
5.3 Pfaffian side
5.4 Equivalences
6 Groupoid action
7 Monodromy action
References
|
| id | nasplib_isofts_kiev_ua-123456789-211321 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:49:20Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Donovan, Will 2025-12-29T11:11:01Z 2021 Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence. Will Donovan. SIGMA 17 (2021), 028, 22 pages 1815-0659 2020 Mathematics Subject Classification: 14F08; 14J32; 14M15; 18G80; 81T30 arXiv:2009.12630 https://nasplib.isofts.kiev.ua/handle/123456789/211321 https://doi.org/10.3842/SIGMA.2021.028 The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on the relevant Grassmannian, and give a mirror symmetry interpretation, following a physical analysis of Eager, Hori, Knapp, and Romo. In celebration of his 77th birthday, I am pleased to express my gratitude to Kyoji Saito for his kindness and interest over the years. I also want to thank N. Addington and E. Segal for the great experience of working on our paper [1], of which this work is a continuation. I amsupported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan. I also acknowledge the support of WPI Initiative, MEXT, Japan, and of EPSRC Programme Grant EP/R034826/1. I am grateful for conversations with K. Hori, who made me aware of the windows used here, for discussions with R. Eager, A. Kuznetsov, and M. Romo, and for helpful suggestions from anonymous referees. In the last stage of this project, I was away from my home institution due to the coronavirus pandemic, so I am especially appreciative of hospitality and support from Kavli IPMU, in particular Y. Ito and M. Kapranov, and from B. Kim at KIAS, and M. Wemyss at Glasgow University. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence Article published earlier |
| spellingShingle | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence Donovan, Will |
| title | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence |
| title_full | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence |
| title_fullStr | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence |
| title_full_unstemmed | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence |
| title_short | Stringy Kähler Moduli for the Pfaffian-Grassmannian Correspondence |
| title_sort | stringy kähler moduli for the pfaffian-grassmannian correspondence |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211321 |
| work_keys_str_mv | AT donovanwill stringykahlermoduliforthepfaffiangrassmanniancorrespondence |