The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions
We show that the Morita equivalences Cliff(4)≃H, Cliff(7)≃Cliff(−1), and Cliff(8)≃R arise from quantizing the Hamiltonian reductions R⁰|4//Spin(3), R⁰|⁷//G₂, and R⁰|⁸//Spin(7), respectively.
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nasplib_isofts_kiev_ua-123456789-1485532025-02-09T21:54:49Z The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions Johnson-Freyd, T. We show that the Morita equivalences Cliff(4)≃H, Cliff(7)≃Cliff(−1), and Cliff(8)≃R arise from quantizing the Hamiltonian reductions R⁰|4//Spin(3), R⁰|⁷//G₂, and R⁰|⁸//Spin(7), respectively. We show that the Morita equivalences Cliff(4)≃H, Cliff(7)≃Cliff(−1), and Cliff(8)≃R arise from quantizing the Hamiltonian reductions R⁰|⁴//Spin(3), R⁰|⁷//G₂, and R⁰|⁸//Spin(7), respectively. This paper is a contribution to the Special Issue “Gone Fishing”. The full collection is available at http://www.emis.de/journals/SIGMA/gone-fishing2016.html. I would like to thank the referees for their comments and improvements to this paper. This work was completed during the “Gone Fishing 2016” conference at University of Colorado, Boulder, which was supported by the NSF grant DMS-1543812. This research was also supported by the NSF grant DMS-1304054. The Perimeter Institute for Theoretical Physics is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. 2016 Article The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions / T. Johnson-Freyd // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 8 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 15A66; 53D20; 16D90; 81Q60 DOI:10.3842/SIGMA.2016.116 https://nasplib.isofts.kiev.ua/handle/123456789/148553 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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We show that the Morita equivalences Cliff(4)≃H, Cliff(7)≃Cliff(−1), and Cliff(8)≃R arise from quantizing the Hamiltonian reductions R⁰|4//Spin(3), R⁰|⁷//G₂, and R⁰|⁸//Spin(7), respectively. |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions |
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quaternions and bott periodicity are quantum hamiltonian reductions |
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The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions / T. Johnson-Freyd // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 8 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 116, 6 pages
The Quaternions and Bott Periodicity
Are Quantum Hamiltonian Reductions?
Theo JOHNSON-FREYD
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
E-mail: theojf@perimeterinstitute.ca
URL: https://perimeterinstitute.ca/personal/tjohnsonfreyd/
Received August 30, 2016, in final form December 09, 2016; Published online December 11, 2016
http://dx.doi.org/10.3842/SIGMA.2016.116
Abstract. We show that the Morita equivalences Cliff(4) ' H, Cliff(7) ' Cliff(−1), and
Cliff(8) ' R arise from quantizing the Hamiltonian reductions R0|4//Spin(3), R0|7//G2,
and R0|8//Spin(7), respectively.
Key words: Clifford algebras; quaternions; Bott periodicity; Morita equivalence; quantum
Hamiltonian reduction; super symplectic geometry
2010 Mathematics Subject Classification: 15A66; 53D20; 16D90; 81Q60
This note provides (super) symplectic origins for the quaternion algebra H and for the eight-
fold “Bott periodicity” of Clifford algebras (due originally to Cartan [3]) in terms of quantum
Hamiltonian reduction. Clifford algebras arise in symplectic supergeometry as the Weyl (aka
canonical commutation) algebras of purely-odd symplectic supermanifolds R0|n. As we explain,
Hamiltonian reductions quantize to bimodules, which are often Morita equivalences. In particu-
lar, we will show that the well-known Morita equivalence H ' Cliff(4) is the quantization of the
Hamiltonian reduction R0|4//Spin(3), where Spin(3) = SU(2) acts on R0|4 as the underlying real
module of the defining action of SU(2) on C2, and that the reduction R0|8//Spin(7) coming from
the spin representation quantizes to the “Bott periodicity” Morita equivalence Cliff(8) ' R. We
also show that the Morita equivalence Cliff(7) ' Cliff(−1) arises from the Hamiltonian reduc-
tion R0|7//G2, where G2 ⊆ SO(7) is the exceptional Lie group of automorphisms of the octonion
algebra O.
1 Symplectic supermanifolds and Clifford algebras
A superalgebra is a Z/2-graded associative algebra (meaning, in particular, that the multiplica-
tion adds degree modulo 2); morphisms are grading-preserving. A supermodule is a Z/2-graded
module. If M is a left A-supermodule, the algebra EndA(M) of all A-linear endomorphisms
of M is naturally a superalgebra acting on M from the right (with multiplication fg = g ◦ f).
Two superalgebras A and B are super Morita equivalent if there are Z/2-graded bimodules AMB
and BNA with grading-preserving bimodule isomorphisms M ⊗B N ∼= A and N ⊗AM ∼= B. We
will generally suppress the word “super”: for example, “module” and “Morita equivalence” will
henceforth always be meant in the super sense.
A superalgebra A is commutative if for homogeneous elements x and y (with degrees |x|
and |y|), yx = (−1)|x|·|y|xy. Note in particular that for an odd element x in a commutative
superalgebra, x2 = −x2, and so x2 = 0. By definition, odd n-dimensional space R0|n is the
“spectrum” of the commutative superalgebra O(R0|n) = R[x1, . . . , xn] where the coordinate
functions x1, . . . , xn are odd, and R[. . . ] denotes the free commutative superalgebra on “. . . ”.
?This paper is a contribution to the Special Issue “Gone Fishing”. The full collection is available at
http://www.emis.de/journals/SIGMA/gone-fishing2016.html
mailto:theojf@perimeterinstitute.ca
https://perimeterinstitute.ca/personal/tjohnsonfreyd/
http://dx.doi.org/10.3842/SIGMA.2016.116
http://www.emis.de/journals/SIGMA/gone-fishing2016.html
2 T. Johnson-Freyd
Thus O(R0|n) ∼=
∧•Rn. In general, a supermanifold is a “space” that looks locally like Rm|n =
Rm × R0|n. See [5] for details on superalgebras and supermanifolds.
A symplectic structure on a supermanifold is an even nondegenerate closed de Rham 2-form.
De Rham forms can be defined for commutative superalgebras just like for commutative algebras,
but behave differently in one important way: if x is an odd coordinate, then dx is even, and so
dx∧dx 6= 0, and if x and y are both odd, then dx∧dy = dy∧dx with no sign. A side effect of this
is that symplectic structures on odd manifolds behave somewhat like metrics on even manifolds.
Equip R0|n with the positive-definite symplectic form ω =
∑
i
(dxi)
2
2 . The corresponding Pois-
son structure on R0|n is given by the Poisson brackets {xi, xj} = −2δij . (The sign depends on
an essentially-arbitrary choice of convention for inverse matrices in superalgebra.) The symplec-
tic form ω on R0|n is translation-invariant and so admits a canonical quantization to the Weyl
algebra W(R0|n) = R〈x1, . . . , xn〉/([xi, xj ] = {xi, xj}), where by definition in a superalgebra the
commutator is defined on homogeneous elements by [x, y] = xy − (−1)|x|·|y|yx. Thus W(R0|n)
is the Clifford algebra Cliff(n) = R〈x1, . . . , xn〉/(x2i = −1, xixj = −xjxi for i 6= j) with its
usual Z/2-grading in which all generators xi are odd. The Weyl algebra of R0|n equipped with
symplectic form −ω is Cliff(−n) = R〈x1, . . . , xn〉/(x2i = 1, xixj = −xjxi).
2 Quantum Hamiltonian reduction
A moment map for the action of a super Lie group G on a symplectic supermanifold M is a map
µ : M → g∗ = Lie(G)∗ of Poisson supermanifolds such that the infinitesimal action of an element
a ∈ g is given by the Hamiltonian vector field for the function m 7→ 〈µ(m), a〉, where 〈 , 〉 denotes
the pairing of the vector space g with its dual; such data is equivalent to a Lie algebra map
µ∗ : g → O(M), where the latter is treated as a super Lie algebra with its Poisson bracket.
When certain cohomology groups of M and g vanish, µ exists and is unique. The Hamiltonian
reduction M//G of this data is the quotient space µ−1(0)/G. This can be defined in the super
case via its algebra of functions (O(M)/〈µ∗g〉)G, where 〈µ∗g〉 denotes the ideal generated by
the image of µ∗. As Marsden and Weinstein explained in the even case [7], when 0 is a regular
value of µ and the action of G on µ−1(0) is free and proper, the manifold M//G is naturally
symplectic. The natural maps µ−1(0) ↪→ M and µ−1(0) � M//G are together a Lagrangian
correspondence between M and M//G. Super Hamiltonian reduction can be cleanly expressed
as an example of coisotropic reduction of super Poisson algebras [4, 8].
Suppose that G acts instead on an associative superalgebra A. A comoment map is a Lie alge-
bra map µ∗ : g→ A, where A is treated as a super Lie algebra with its commutator bracket, such
that the infinitesimal action of a ∈ g is given by the inner derivation [µ∗(a),−]. Corresponding
to the zero section µ−1(0) is the quotient module A/〈µ∗g〉, where 〈µ∗g〉 denotes the left ideal
generated by the image of µ∗. Corresponding to the quotient M//G = µ−1(0)/G is the quantum
Hamiltonian reduction A//G = (A/〈µ∗g〉)G. This is naturally an algebra because it is isomor-
phic to EndA(A/〈µ∗g〉). When G is compact, A//G ∼= AG/
(
AG ∩ 〈µ∗g〉
)
, where AG denotes
the G-invariant subalgebra of A. From this perspective, the algebra structure on A//G arises
because, although 〈µ∗g〉 is merely a left ideal in A, its intersection with AG is a two-sided ideal,
as µ∗g is central in AG. The module A/〈µ∗g〉 is by construction a bimodule between A and A//G.
Example 1. Suppose that M is a linear symplectic supermanifold and C ⊆M is a coisotropic
submanifold cut out by linear equations r1 = · · · = rp+q = 0, where r1, . . . , rp are even and
rp+1, . . . , rp+q are odd. The Hamiltonian flows for r1, . . . , rp+q define an action on M of the
abelian Lie supergroup Rp|q. Let C⊥ ⊆ C denote the symplectic orthogonal to C. The Hamil-
tonian reduction M//Rp|q is then canonically linearly symplectomorphic to C/C⊥.
SinceM is linear, it admits a canonical quantization to the Weyl algebraW(M)=T (M∗)/([a,b]
= {a, b}, a, b ∈ M∗). The quotient W(C) = W(M)/〈r1, . . . , rc〉 is the canonical quantization
The Quaternions and Bott Periodicity are Quantum Hamiltonian Reductions 3
of C, andW(M)//Rp|q ∼=W(C/C⊥). In the purely-odd case, which is the only case of concern in
this paper, W(C) is a Morita equivalence betweenW(M) andW(C/C⊥): it suffices to consider
the case M = R0|2 with “split” symplectic form (dx)2
2 − (dy)2
2 and Lagrangian C ∼= R0|1 spanned
by the vector (1, 1); then W(M) ∼= Mat(1|1) is the algebra of 2× 2 matrices in which
(
1 0
0 0
)
and(
0 0
0 1
)
are even and
(
0 1
0 0
)
and
(
0 0
1 0
)
are odd, andW(C) is the defining (1|1)-dimensional module.
(When there are even coordinates, W(C) is not a Morita equivalence. The Stone–von Neu-
mann theorem can be understood as saying that for purely even M , W(C) becomes a Morita
equivalence after appropriate functional analytic completions. The mixed case can be handled
by decomposing M and C into even and odd parts.)
In particular, linear Lagrangians provide Morita equivalences W(M) ' R. This does not
explain why Cliff(8) = W(R0|8) ' R, because the positive-definiteness of the symplectic form
prevents R0|n from admitting Lagrangian sub-supermanifolds, linear or not.
Lemma 1. If the Hamiltonian reduction Cliff(n)//G is not the zero algebra, then Cliff(n)/〈µ∗g〉
is a Morita equivalence between Cliff(n) and Cliff(n)//G.
Proof. For any superalgebra A, an A-module X is a Morita equivalence between A and
EndA(X) if and only if X is a finitely-generated projective generator of the supercategory
of A-modules. The holomorphic symplectic supermanifold C0|n = R0|n ⊗ C admits a linear
Lagrangian L if n is even and an (n + 1)/2-dimensional linear coisotropic C if n is odd. Via
Example 1, these linear coisotropics provide Morita equivalences Cliff(n)⊗C ' C =W(L/L⊥)
or Cliff(1) ⊗ C = W(C/C⊥). For C and Cliff(1) ⊗ C, any non-zero finitely-generated module
is a projective generator. But “non-zero”, “finitely-generated”, and “projective” are Morita-
invariant notions, so these properties hold also for Cliff(n)⊗ C and hence for Cliff(n). �
3 Cliff(4) and H
Corresponding to the exceptional isomorphism SO(4) ∼= Spin(3)×Z/2Spin(3) are two commuting
actions of Spin(3) on R0|4 by linear symplectic automorphisms. (Odd symplectic groups are
even orthogonal groups; metaplectic groups correspond to spin groups.) Denote the coordinates
on R0|4 by {w, x, y, z} and the bases for two copies of so(3) by {a+, b+, c+} and {a−, b−, c−},
normalized so that their brackets are [a±, b±] = ±2c±, [b±, c±] = ±2a±, [c±, a±] = ±2b±. The
comoment maps for the actions are:
a± 7→ 1
2(wx± yz), b± 7→ 1
2(wy ± zx), c± 7→ 1
2(wz ± xy).
Together these six elements are a basis for the space of homogeneous-quadratic functions on R0|4.
The quadratic Casimir for both so(3)s is θ = ±2a2± = ±2b2± = ±2c2± = wxyz. Completing the
basis for O(R0|4) are the unit 1 and xyz, wyz, wzx, and wxy. Basis vectors 1, a±, b±, c±, and θ
are even, and x, y, z, w, xyz, wyz, wzx, and wxy are odd.
We now consider the quantization Cliff(4) = W(R0|4), for which we can use the same basis
{1, w, x, y, z, a+, b+, c+, a−, b−, c−, xyz, wyz, wzx,wxy, θ} (with the same grading). Note that,
whereas inO(R0|4) we had a2± = b2± = c2± = ±θ/2, in Cliff(4) we have a2± = b2± = c2± = 1
2(±θ−1).
Since the actions of Spin(3) on R0|4 are linear, they lift to Cliff(4), and the same assignments
a+, . . . , c− provide the quantum comoment maps.
Theorem 1. The quantum Hamiltonian reduction of either of the so(3)-actions on R0|4 produces
a Morita equivalence Cliff(4) ' H = R〈i, j, k〉/(i2 = j2 = k2 = ijk = −1) (where the quaternion
algebra H is purely even).
Proof. There is a manifest symmetry interchanging the two so(3)-actions; we will work with
the action of the a−, b−, and c−. It is not hard to see that [a−,−], [b−,−], and [c−,−] preserve
4 T. Johnson-Freyd
polynomial degree. We have observed already that the quadratic elements a+, b+, c+ commute
with the generators a−, b−, c− of the action, as well as with θ = wxyz. The subspace of Cliff(4)
spanned by {a−, b−, c−} is a submodule for the action of {a−, b−, c−} isomorphic to the adjoint
action. The subspaces spanned by {w, x, y, z} and {xyz, wyz, wzx,wxy} are each isomorphic to
the underlying real module of the defining module of su(2). Thus a basis for the so(3)-fixed
subalgebra Cliff(4)Spin(3) is given by the five even elements {1, a+, b+, c+, θ}.
The left ideal 〈µ∗so(3)〉 in Cliff(4) generated by {a−, b−, c−} is eight-dimensional with basis
{a−, b−, c−, w− xyz, x+wyz, y +wzx, z +wxy, θ+ 1}. This ideal intersects Cliff(4)Spin(3) only
in the one-dimensional space spanned by θ + 1. It follows that in the quotient Cliff(4)Spin(3)/(
Cliff(4)Spin(3)∩〈µ∗so(3)〉
)
we have a2+ = 1
2(θ−1) ≡ 1
2(−2) = −1, and so the map (a+, b+, c+) 7→
(i, j, k) identifies the quantum Hamiltonian reduction Cliff(4)//Spin(3) with the quaternion
algebra H. Lemma 1 completes the proof. �
Theorem 1 suggests that the “classical limit” of H is the Hamiltonian reduction R0|4//Spin(3).
Since 0 is not a regular value of the classical moment map, R0|4//Spin(3) is not a supermanifold.
It does make sense as an affine super scheme: its algebra of functions is the purely even Poisson
algebra R[a, b, c]/(a2 = b2 = c2 = ab = bc = ca = 0) with Poisson brackets {a, b} = 2c, {b, c} =
2a, and {c, a} = 2b. Thus the “classical limit” of H is the first-order neighborhood of 0 in so(3)∗.
4 Cliff(7) and G2
The 14-dimensional exceptional Lie group G2 is the subgroup of SO(7) preserving the alterna-
ting 3-form ε on R7 defined by identifying R7 with the pure-imaginary octonions and setting
ε(a, b, c) = (ab)c− a(bc) ∈ R [2]. Since SO(7) acts by linear symplectic automorphisms of R0|7,
we get an induced symplectic action of G2.
Theorem 2. The quantum Hamiltonian reduction Cliff(7)//G2 provides the Morita equivalence
Cliff(7) ' Cliff(−1).
Proof. By Lemma 1, it suffices to compute Cliff(7)//G2. As in Theorem 1, the Poincaré–
Birkoff–Witt isomorphismO(R0|7) =
∧•(R7) ∼= Cliff(7) is SO(7)-equivariant by the functoriality
of the Weyl algebra construction. The G2-fixed algebra has as its basis a set of the form
{1, ε, ε̄, θ}, where ε is the cubic 3-form defining G2, ε̄ is its dual quartic, and θ is the generator
of
∧7R7. In Cliff(7), ε̄ = θε and θ2 = 1.
An explicit presentation of the action of Lie(G2) = g2 is given in [1] as follows. Denote the
coordinates on R0|7 by x1, x2, . . . , x7. The cubic function ε is
ε = x1x2x3 + x1x4x5 + x1x6x7 + x2x4x6 + x2x7x5 + x3x7x4 + x3x6x5.
Consider the quadratic functions e1, . . . , e7 defined by ei = ∂
∂xi
ε. For example, e1 = x2x3 +
x4x5 + x6x7. Orthogonal to each ei is a two-dimensional vector space of commuting quadratics
given by the differences of the monomials in ei. For example, orthogonal to e1 are x2x3 − x4x5,
x4x5−x6x7, and their sum x2x3−x6x7. A basis for the image of g2 under µ∗ is given by choosing
for each i = 1, . . . , 7 two quadratics orthogonal to ei. For example:
x2x3 − x4x5, x4x5 − x6x7, x3x1 − x4x6, x4x6 − x7x5, x1x2 − x7x4,
x7x4 − x6x5, x5x1 − x6x2, x6x2 − x3x7, x1x4 − x2x7, x2x7 − x3x6,
x7x1 − x2x4, x2x4 − x5x3, x1x6 − x5x2, x5x2 − x4x3.
We wish to compute EndCliff(7)(Cliff(7)/〈µ∗g2〉) = Cliff(7)G2/(〈µ∗g2〉∩Cliff(7)G2), where 〈µ∗g2〉
is the left ideal generated by these 14 elements.
The Quaternions and Bott Periodicity are Quantum Hamiltonian Reductions 5
Note that x1x4x5x6x7(x2x3 − x4x5) = θ + x1x6x7, where θ = x1x2 · · ·x7 ∈ Cliff(7)G2 . The
numerics of the second summand are: x2x3 − x4x5 was orthogonal to e1; x6x7 is the unused
monomial in e1. Similarly, for each monomial µ in the cubic ε one can find θ + µ ∈ 〈µ∗g2〉, and
summing shows that 7θ + ε ∈ 〈µ∗g2〉 ∩ Cliff(7)G2 ; hence also 7 + ε̄ ∈ 〈µ∗g2〉 ∩ Cliff(7)G2 . It
follows that Cliff(7)G2/(〈µ∗g2〉 ∩ Cliff(7)G2) is a quotient of the copy of Cliff(−1) spanned by
the classes of 1 and θ.
Finally, for any basis element α ∈ µ∗g2, we have α(θ − ε) = 0, from which it follows that
1 6∈ 〈µ∗g2〉. The ideal cannot mix even and odd terms without setting both to 0, and so we find
Cliff(7)G2/(〈µ∗g2〉 ∩ Cliff(7)G2) ∼= Cliff(−1). �
5 Spin(7) and Bott periodicity
We conclude by providing a Hamiltonian reduction whose quantization is the famous “Bott
periodicity” equivalence Cliff(8) ' R. The irreducible real spin representations of all four
groups Spin(5), Spin(6), Spin(7), and Spin(8) are eight-real-dimensional. The reduction
Cliff(8)//Spin(8) vanishes since the image of the comoment map consists of all quadratic ele-
ments of Cliff(8), including some which are invertible, and so the Spin(8)-action does not induce
a Morita equivalence. The reader is invited to compute Cliff(8)//Spin(5) and Cliff(8)//Spin(6).
We will show:
Theorem 3. Cliff(8)//Spin(7) ∼= R.
By Lemma 1, Theorem 3 establishes that the cyclic module Cliff(8)/〈µ∗so(7)〉 is a Morita
equivalence between Cliff(8) and R.
Proof. The following construction of Spin(7), and its eight-dimensional spin representation,
are developed in [6]. Consider the octonion algebra O and the 4-form φ ∈
∧4O∗ defined
by φ(a, b, c, d) = 〈a, b × c × d〉, where the triple cross product is by definition b × c × d =
1
2
(
b(c̄d)− d(c̄b)
)
and c̄ is the octonionic conjugate of c. Then Spin(7) is precisely the subgroup
of SO(8) fixing φ. In terms of coordinates x1, x2, . . . , x8 on R0|8, φ corresponds to the function
φ = x1234 + x1256 + x1278 + x1357 − x1368 − x1458 − x1467
+ x5678 + x3478 + x3456 + x2468 − x2457 − x2367 − x2358,
where we have abbreviated xij...k=xixj · · ·xk. Let θ=x12345678. The fixed algebra Cliff(8)Spin(7)
has basis {1, φ, θ} and multiplication θ2 = 1, θφ = φθ = φ, and φ2 = 14θ + 14− 12φ.
The image µ∗so(7) ⊆
∧2R8 ⊆ Cliff(8) of the comoment map is spanned by quadratic ele-
ments of the form α = xij±xkl such that −1
2α
2−1 = ∓xijkl is (with the given sign) a monomial
in φ. (There are 3 ·14 such αs; given {i, j}, there are three αs that include the monomial xij and
three that are disjoint from {i, j}, and these six span a three-dimensional space; this counts cor-
rectly the 21-dimensional space so(7).) It follows that φ+14, being a sum of 14 terms of the form
−1
2α
2, is in the ideal 〈µ∗so(7)〉, and so Cliff(8)//Spin(7) is a quotient of R{1, φ, θ}/〈φ+14〉 ∼= R.
Each α = xij ±xkl ∈ µ∗so(7) determines a splitting φ = xijkl(1 + θ) +κ+λ where κ is a sum
of four quartic monomials each of which has indices containing either {i, j} or {k, l} but not
both, and λ is a sum of eight quartic monomials each of which has indices intersecting the sets
{i, j} and {k, l} at one element each. For example, when α = x13 − x57, we have
φ = x1357 + x2468︸ ︷︷ ︸
xijkl(1+θ)
+x1234 − x1368 + x5678 − x2457︸ ︷︷ ︸
κ
+ x1256 + x1278 − x1458 − x1467 + x3478 + x3456 − x2367 − x2358︸ ︷︷ ︸
λ
.
6 T. Johnson-Freyd
We see that [α, xijkl(1 + θ)] = 0 and [α, κ] = 0. Since we know that [α, φ] = 0, we find [α, λ] = 0
as well. Suppose β is a quadratic monomial and ν a quartic monomial such that the indices in β
and ν overlap at one element. Then βν = 1
2 [β, ν], from which it follows that αλ = 1
2 [α, λ] = 0.
It’s also clear that ακ = 0, since κ factors as (xij∓xkl)(. . . ) where α = xij±xkl and (. . . ) is a sum
of two quadratic monomials that have no overlap with α. Finally, αxijkl(1 + θ) = α(−1 − θ).
All together, we find:
α(φ+ 1 + θ) = α
(
xijkl(1 + θ) + κ+ λ+ (1 + θ)
)
= α(−1− θ) + 0 + 0 + α(1 + θ) = 0.
It follows that 1 6∈ 〈µ∗so(7)〉, and so Cliff(8)//Spin(7) 6∼= 0, completing the proof. �
Acknowledgements
I would like to thank the referees for their comments and improvements to this paper. This work
was completed during the “Gone Fishing 2016” conference at University of Colorado, Boulder,
which was supported by the NSF grant DMS-1543812. This research was also supported by the
NSF grant DMS-1304054. The Perimeter Institute for Theoretical Physics is supported by the
Government of Canada through the Department of Innovation, Science and Economic Develop-
ment Canada and by the Province of Ontario through the Ministry of Research, Innovation and
Science.
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1 Symplectic supermanifolds and Clifford algebras
2 Quantum Hamiltonian reduction
3 Cliff(4) and H
4 Cliff(7) and G2
5 Spin(7) and Bott periodicity
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