Supergeometry in mathematics and physics
Mikhail Kapranov
February 21, 2016

Contents
0 Introduction

1

1 Supergeometry as understood by mathematicians
1.1 Commutative superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The symmetric monoidal category of super-vector spaces . . . . . . . . . . .
1.3 Superschemes and supermanifolds . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Lie supergroups and superalgebras . . . . . . . . . . . . . . . . . . . . . . .

3
3
4
5
7

2 Supergeometry as understood by physicists
2.1 Idea of non-observable square roots . . . . . . . . . . . . . . . . . . . . . . .
2.2 Square root of d/dt and theta-functions . . . . . . . . . . . . . . . . . . . . .
2.3 Square roots of spacetime translations . . . . . . . . . . . . . . . . . . . . .
2.4 Quadratic spaces and intersections of quadrics . . . . . . . . . . . . . . . . .
2.5 Supersymmetry, superspace and constraints . . . . . . . . . . . . . . . . . .
2.6 Constraints and complete intersection slices . . . . . . . . . . . . . . . . . .

8
9
10
12
13
15
17

3 Homotopy-theoretic underpinnings of supergeometry
3.1 The skeleton of the Koszul sign rule . . . . . . . . . . . . . . . . . . . . . . .
3.2 (Higher) Picard groupoids and spectra . . . . . . . . . . . . . . . . . . . . .
3.3 The sphere spectrum and the free Picard n-groupoid . . . . . . . . . . . . .
3.4 Towards higher supergeometry . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
21
22
24

0

Introduction

Supergeometry is a geometric tool which is supposed to describe supersymmetry (“symmetry
between bosons and fermions”) in physics. In the impressive arsenal of physico-geometric
tools, it occupies a special, somewhat mysterious place.
To an outside observer, it projects a false sense of accessibility, presenting itself as asimple modification of the familiar formalism. However, this is only an appearance. Like a
1

bewitched place, it is protected not by barriers but by something less tangible and therefore
much more powerful. It is not even so clear where this place is localized. A naive attempt
to “forge ahead” may encounter little resistance but may also end up missing the real point.
Indeed, the physical origin of supergeometry lies in the comparison of the Bose and Fermi
statistics for identical particles. So mathematically, supergeometry zooms in on our mental
habits related to questions of commutativity and identity. These habits predate quantum
physics and therefore are deeply ingrained, but they do not correspond to the ultimate
physical reality. The new way of thinking, offered by supergeometry in mathematics and by
supersymmetry in physics, requires changes that cannot be reduced to simple recipes (suchas “introducing ± signs here and there” which is, in the popular mind, the best known featureof supergeometry).
Put diferently, if introducing ± signs can take us so far, then something is happeningthat we truly don’t understand. In addition, one can almost say that mathematicians andphysicsts mean different things when they speak about supergeometry. This contributes tothe enduring feeling of wonder and mystery surrounding this subject.
The present article (chapter) is subdivided into three parts. Following the specifications
for the volume, §1 presents a very short but self-contained exposition of supergeometry as
it is generally understood by mathematicians. Not surprisingly, the presentation revolves
around the Koszul sign rule (1.1.1) which appears first at the level of elements of a ring, andlater again at the level of categories. More systematic expositions can be found in the books[33] [16] and in many articles, esp. [9].
In §2 we discuss the aspects of supergeometry that are used by physicists in relation tosupersymmetry. From the mathematical point of view, this amounts to much more than the
study of super-manifolds or of the Koszul rule. Some of the basic references are [49] [33] [16][10]. The entry point for a mathematician here could be found in the idea of taking natural“square roots” of familiar mathematical and physical quantities.
In our presentation, we introduce the abstract concept of a quadratic space (data of Γmatrices) of which various situations involving spinors form particular cases. This has theadvantage of simplifying the general discussion and also of relating the subject to the mathematical theory of intersections of quadrics. In particular, the familiar dichotomy of complete
intersections of quadrics vs. non-complete ones has a direct significance for understanding
the “constraints" which are usually imposed on super-fields.
Finally, §3 is devoted to an attempt to uncover deeper roots for the mysterious power ofthe super-geometric formalism, of which the remarkable consistency of its sign rules is justone manifestation. It seems that the right language to speak about such things is given byhomotopy theory. Indeed, this theory provides a systematic modern way to talk about theissues of identity: instead of saying that two things are “the same”, we say that they are“homotopic”, and specify the homotopy (the precise reason why they should be considered
the same). We can also consider homotopies between homotopies and so on. From thispoint of view, the group {±1} of signs howering over supergeometry, is nothing but π1st ,2

the first homotopy group of the sphere spectrum S. As emphasized by Grothendieck in amore categorical language, S can be seen as the most fundamental homotopy commutative
object. This suggests that “mining the sphere spectrum” beyond the first level should leadto generalizations ofpossibly, supersymmetry) involving not just signs√ supergeometry (and24
stbut, for instance, 1 to account for π3 = Z/24.
My understanding of supergeometry owes a lot to lectures and writings of Y. I. Manin.
In particular, the idea of square roots provided by the super formalism, was learned fromhim long time ago. The homotopy-theoretic considerations of §3 were stimulated by thejoint work with N. Ganter [17]. I would also like to thank N. Ganter, N. Gurski and Y.
I. Manin for remarks and suggestions on the preliminary versions of this text. This work
was supported by World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan.

1Supergeometry as understood by mathematicians

1.1

Commutative superalgebras

For a mathematician, supergeometry is the study of supermanifolds and superschemes: objects whose rings of functions are commutative superalgebras.
We fix a field k of characteristic 6= 2. By an associative superalgebra over k one means
simply a Z/2-graded associative algebra A = A0̄ ⊕ A1̄ . Elements of A0̄ are called even (orbosonic), elements of A1̄ are called odd (or fermionic).
An associative superalgebra A is called commutative, if it satisfies the Koszul sign rulefor commutation of homogeneous elements:(1.1.1)

ab = (−1)deg(a)·deg(b) ba.

The terms “commutative superalgebra” and “supercommutative algebra” are used withut
distinction.
L
Often, one considers Z-graded supercommutative algebras A = i∈Z Ai , with the samecommutation rule (1.1.1) imposed on homogeneous elements.
Examples 1.1.2. (a) Any usual commutative algebra A becomes supercommutative, if put
in degree 0̄. The most important example A = k[x1 , · · · , xm ] (the polynomial algebra).
(b) The exterior (Grassmann) algebra Λ[ξ1 , · · · ξn ] over k, generated by the symbols ξi ofdegree 1̄, subject only to the relations
ξi2 = 0,ξi ξj = −ξj ξj , i 6= j,

is supercommutative. It can be seen as the free supercommutative algebra on the ξi : therelations imposed are the minimal ones to ensure supercommutativity.

3

(c) If A, B are the commutative superalgebras, then the tensor product A ⊗k B with thestandard product grading deg(a ⊗ b) = deg(a) + deg(b) and multiplication(a1 ⊗ b1 ) · (a2 ⊗ b2 ) = (−1)deg(b1 )·deg(a2 ) (a1 · b1 ) ⊗ (a2 · b2 )(ai , bi homegeneous), is again a supercommutative algebra. For example,
k[x1 , · · · , xm ] ⊗ Λ[ξ1 , · · · , ξn ],deg(xi ) = 0̄, deg(ξj ) = 1̄,is a commutative superalgebra. This is a general form of a free commutative superalgebra
on a finite set of even and odd generators.
(d) The de Rham algebra Ω•X of differential forms on a C ∞ -maniflold X, is a Z-gradedsupercommutative algebra over R.
(e) The cohomology algebra H • (X, k) of any CW-complex X, is a Z-graded supercommutative algebra over k.
The importance and appeal of such studies are based on the following heuristic
1.1.3. Principle of Naturality of Supers. All constructions and features that make commutative algebras special among all algebras, can be extended to supercommutative algebras,
and make them just as special.
The most important of such features is the relation to geometry: a commutative algebra A can be seen as the algebra of functions on a geometric object Spec(A) which can beconstructed from A and used to build more complicated geometric objects by gluing. Supergeometry (as understood by mathematicians) is the study of similar geometric objects forsupercommutative algebras.

1.2

The symmetric monoidal category of super-vector spaces

The main guiding principle for extending properties of usual commutative algebras to supercommutative ones is also sometimes called the Koszul sign rule. It goes like this:1.2.1. When we move any quantity (vector, tensor, operation) of parity p ∈ Z/2 past anyother quantity of parity q, this move should be accompanied by multiplication with (−1)pq .
An instance is given by the first formula in Example 1.1.2(c). This rule can be formalized
as follows.
Let SVectk be the category of super-vector spaces over k, i.e., Z/2-graded vector spaces
V = V 0̄ ⊕ V 1̄ . This category has a monoidal structure ⊗, the usual graded tensor product.
The operation ⊗ is associative up to natural isomorphisms, and has a unit object 1 = k (putin degree 0̄). Define the symmetry isomorphisms(1.2.2)

RV,W : V ⊗ W −→ W ⊗ V,v ⊗ w 7→ (−1)deg(v)·deg(w) w ⊗ v.
4

Proposition 1.2.3. The family R = (RV,W ) makes SVectk , ⊗, 1, R) into a symmetric monoidal
category.
For background on symmetric monoidal categories we refer to [32, 7]. A basic example ofa symmetric monoidal category is given by the usual category of vector spaces Vectk , withthe usual tensor product, the unit object 1 = k, and the symmetry given by v ⊗ w 7→ w ⊗ v.
The meaning of the proposition is that SVectk with symmetry (1.2.2) satisfies all the sameformal properties as the “familiar" category Vectk .
If dim(V 0̄ ) = m and dim(V 1̄ ) = n, we write dim(V ) = (m|n). In particular, we have thestandard coordinate superspaces km|n . We denote by Π the parity change functor on SVectk
given by multiplication with k0|1 on the left.
It is well known that one can develop linear algebra formalism (tensor, symmetric, exterior
powers etc.) in any symmetric monoidal k-linear abelian category (V, ⊗, 1, R). This gives a
way to define commutativity. That is, a commutative algebra in V is an object A ∈ V together
with a morphism µA : A ⊗ A → A satisfying associativity and such that the composition
RA,A

µ

A
A
A ⊗ A −→ A ⊗ A −→is equal to µA . Given two commutative algebras (A, µA ) and (B, µB ), the object A ⊗ B isagain a commutative algebra with respect to µA⊗B given by the composition
A⊗B⊗A⊗B

Id⊗RB,A ⊗Id

−→

µ ⊗µ

A
B
A ⊗ A ⊗ B ⊗ B −→A ⊗ B.

For V = SVectk , this gives Example 1.1.2(c).
Remark 1.2.4. Symmetric monoidal categories can be seen as categorical analogs of commutative algebras: instead of the equality ab = ba, we now have canonical isomorphisms
V ⊗W 'W ⊗V.

1.3

Superschemes and supermanifolds

Given a supercommutative algebra A, the even part A0̄ is commutative, and so has theassociated affine scheme Spec(A0̄ ). Explicitly, it is a ringed space Spec(A0̄ ), OSpec(A0̄ ) ,where Spec(A0̄ ) is the set of prime ideals in A0̄ with the Zariski topology and OSpec(A0̄ ) isa sheaf of local rings on this space obtained by localization of A0̄ . That is, the value of
OSpec(A0̄ ) on a “principal open set”
Uf = {p ∈ Spec(A0̄ ) : f ∈ p},

f ∈ A0̄ ,is given by OSpec(A0̄ ) ()Uf ) = A0̄ [f −1 ].
Further, A0̄ is not only commutative but lies in the center of A as an associative algebra. Therefore, associating to Uf the commutative superalgebra A[f −1 ], we get a sheaf ofcommutative superalgebras0̄1̄
OSpec(A) = OSpec(A)⊕ OSpec(A),50̄OSpec(A)= OSpec(A0̄ )on the same topological space Spec(A0̄ ) as before. The pair (ringed space)
Spec(A) :=Spec(A0̄ ), OSpec(A)is the fundamental geometric object associated to A, see [30, 33].
The stalks of OX are commutative superalgebras which, considered as ordinary associative
algebras, are local rings. Indeed, we have:
Proposition 1.3.1. If B is a commutative superalgebra such that B 0̄ is a local ring withmaximal ldeal m0̄ , then B itself is a local ring with maximal ideal m = m0̄ ⊕ B 1̄ .
Working with sheaves of local rings is a fundamental technical feature of Grothendieck’s
theory of schemes. So we introduce the following
Definition 1.3.2. A super-locally ringed space over k is a pair X = (X, OX ), where X is a0̄1̄
topological space and OX = OX
⊕ OX
is a sheaf of commutative k-superalgebras on X, withstalks being local rings.
A morphism of super-locally ringed spaces f : (X, OX ) → (Y , OY ) is a pair f = (f] , f [ ),where:• f] : X → Y is a continuous map of topological spaces.
• f [ : f]−1 (OY ) → OX is a morphism of sheaves of commutative superalgebras, which, in
addition, takes the maximal ideal of each local ring (f]−1 OY )x := OY,f] (x) , x ∈ X into
the maximal ideal of the local ring OX,x .
The resulting category of super-locally ringed spaces over k will be denoted SLRS k
Proposition 1.3.3. For any two commutative superalgebras A, B we have an isomorphism
HomSAk (A, B) ' HomSLRS k (Spec(B), Spec(A)).
Definition 1.3.4. A superscheme over k is a super-locally ringed space over k locally isomorphic to Spec(A) for a commutative superalgebra A. The category SSchk of superschemes
over k is defined to be the full subcategory of SLRS k formed by superschemes.
Examples 1.3.5. (a) The affine superspace of dimension (m|n) is the superschemeAm|n = Spec k[x1 , · · · , xm ] ⊗ Λ[ξ1 , · · · , ξn ] .
(b) The next class of examples is provided by algebraic supermanifolds of dimension
(m|n). By definition, they are superschemes locally of the form Spec A ⊗ Λ[ξ1 , · · · , ξn ] ,where A is the ring of functions on a smooth m-dimensional affine algebraic variety over k.

6

0̄
1̄ 2
Each superscheme X = (X, OX ) gives rise to an ordinary scheme X 0̄ = (X, OX/(OX))
called the even part of X. If X is an algebraic supermanifold of dimension (m|n), then X 0̄
is a smooth algebraic variety of dimension m.
As with usual schemes, a superscheme X is defined by its functor of points on the categoryof supercommutative algebras

A 7→ X(A) = HomSSchk (Spec(A), X).
Differentiable or analytic supermanifolds are similarly defined as locally ringed spaces
X = (X 0 , OX ) where X 0 is an ordinary differentiable or analytic manifold and OX is a sheafof commutative superalgebras locally isomorphic to OX ⊗k Λ[ξ1 , · · · , ξn ]. Here OX is thesheaf of C ∞ or analytic functions on X and k = R for differentiable or real analytic and C
for complex analytic manifolds.

1.4

Lie supergroups and superalgebras

Here is an illustration of the Principle of Naturality of Supers: extension of the formalismof differential geometry to the super-case looks totally straightforward.
Given a k-linear symmetric monoidal category A, one can speak about algebras in A of
any given type, for example Lie algebras. In the case A = SVect, this gives the familiar
concept.
Definition 1.4.1. A Lie superalgebra over k is a k-super-vector space g = g0̄ ⊕ g1̄ equippedwith a homogeneous (degree 0̄) operation (x, y) 7→ [x, y] satisfying[x, y] = −(−1)deg(x)·deg(y) [y, x];[x, [y, z]] = [[x, y], z] + (−1)deg(x)·deg(y) [y, [x, z]].
The following examples can also be given in any symmetric monoidal category, we use
the case V = SVectk .
Examples 1.4.2. (a) If R is any associative superalgebra, we can make it into a Lie superalgebra using the supercommutator[x, y] = x · y − (−1)deg(x)·deg(y) y · x.
(b) If A is an associative superalgebra, then we have the Lie superalgebra Der(A) = Der0̄ (A)⊕Der1̄ (A) of super-derivations of A. Here Deri (A) is the space of k-linear maps D : A → A
satisfying the super-Leibniz ruleD(a · b) = D(a) · b + (−1)i·deg(a) a · D(b).
The bracket is given by the supercommutator as in (a).

7Given a supermanifold X = (X, OX ) of dimension (m|n) in the smooth, analytic or
algebraic category, super-derivations of OX form a sheaf TX of Lie superalgebras on X
called the tangent sheaf. It is also a sheaf of OX -modules, locally free of rank (m|n). Its
fiber at a point x ∈ X 0 will be denoted Tx X and called the tangent space to X at x. It is asuper-vector space of dimension (m|n).
A Lie supergroup, resp. an algebraic supergroup over k, is a group object G in the categoryof smooth supermanifods, resp. algebraic supermanifolds over k. In particular, it has a unit
element e ∈ G0 . The tangent space Te G is a Lie superalgebra denoted by g = Lie(G).
Note that g0̄ = Lie(G0̄ ) is the usual Lie algebra corresponding to a Lie or algebraic group.
Conversely, given a finite-dimensional Lie algebra g and a Lie or algebraic group G0̄ suchthat Lie(G0̄ ) = g0̄ , one can integrate it to a Lie or algebraic supergroup G with Lie(G) = g.
We will specially use the case when g is nilpotent, that is, the iterated commutators[x1 , [x2 , · · · , xn ]...] vanish for n greater than some fixed N (degree of nilpotency). We assumechar(k) = 0. In this case we can associate to g a canonical algebraic group G = eg . by meansof the classical Hausdorff series(1.4.3)

1
x · y = x + y + [x, y] + · · ·2This means the following. Suppose we want to find the set-theoretic group of Λ-pointseg (A) = Hom(Spec(A), eg ), where A is a super-commutative algebra. As a set, this groupis defined to be (g ⊗k A)0̄ , which has a structure of an ordinary (purely even) nilpotent Liealgebra and therefore can be made into a group by means of the ordinary Hausdorff series
above. This is the group eg (A).
Informally, one says that eg “is” the Lie algebra g considered as a manifold with themultiplication given by (1.4.3) but the formal meaning is that the arguments in commutatorsin (1.4.3) always belong to some ordinary Lie algebra.
1.4.4. Pfaff systems and Frobenius theorem. Let X be a supermanifold. A Pfaff system
in X is a subbundle (i.e., a subsheaf of OX -modules which is locally a direct summand)
C ⊂ TX . For a Pfaff system C the Lie algebra structure in sections of TX induces the
OX -linear map known as the Frobenius pairingzC : Λ2OX C −→ TX /C.
A Pfaff system C is called integrable, if zC = 0, i.e., C is closed under the bracket of vector
fields. In this case we a super-analog of the Frobenius theorem: in the smooth or analytic caseC can be locally represented as the relative tangent bundle to a submersion of supermanifolds
X →Y.

2

Supergeometry as understood by physicists

For a physicist, the really important concept is supersymmetry, and supermanifolds per seare of interest only tangentially. For example, they arise as infinite-dimensional spaces of8bosonic and fermionic classical fields, over which Feynman integrals are taken. One way(not the only one!) to construct supersymmetric field theories is by using superspace, aconcept not synonymous with “supermanifold” of mathematicians. In fact, superspace is asupermanifold with a rather special “spinor-conformal" structure.

2.1

Idea of non-observable square roots

To understand the idea of supersymmetry, it is useful to include it into the following moregeneral heuristic principle.
2.1.1. Principle of square roots. It is useful to represent observable quantities of immediate physical interest (real, positive, bosonic) as bilinear combinations of more fundamental
quantities which can be complex, fermionic and not even observable by themselves.
In other words, it is useful to take “square roots” of familiar objects. Let us give several
examples.
Example 2.1.2 (Wave functions and probability density). In elementary quantum
mechanics, the wave function ψ(x) of a particle (say, electron), is a complex quantity which
can not be measured. But the expression
P (x) = |ψ(x)|2 = ψ(x) · ψ(x) ≥ 0
represents the probability density of the electron which is real, non-negative and measurable.
Example 2.1.3 (Laplace operator on forms). Let X be a C ∞ Riemannian manifold.
The space Ω• (X) of all differential forms on X is Z-graded and so can be considered as asuper-vector space. The Laplace operator on forms is defined as∆ = d ◦ d∗ + d∗ ◦ d = [d, d∗ ],where d is the exterior derivative and d∗ is its adjoint with respect to the Riemannian metric.
Thus ∆ is non-negative definite, real (self-adjoint) and bosonic, while d and d∗ are fermionic.
Example 2.1.4 (Spinors as square roots of vectors). Let V be a d-dimensional vector
space over k = R or C, equipped with a non-degenerate quadratic form q. We refer to V as
the spacetime, or the Minkowski space. Elements of V (vectors in the physical sense of theword) can be represented as bilinear combinations of more fundamental quantities, spinors.
By definitions, spinors are vectors in minimal (spinor) representations of Spin(V ), a double
covering of the group SO(V ). There is one spinor representation S, if d = dim(V ) is odd,
and two representations, S+ and S− , if d is even. Further properties of these representations
depends on the residues modulo 8 of d and of the signature of q, if k = R (Bott periodicity).
We refer to [3, 8] for a detailed treatment.
The expression of vectors as bilinear combinations of spinors comes from Spin(V )-equivariant
maps γ (Dirac’s gamma-matrices) whose nature we indicate in three cases d = 2, 4, 10 with
Minkowski signature.
9d = 2: S+ , S− can be defined over R and have real dimension 2. They are self-dual: S±∗ = S± .
The gamma maps are γ : Sym2 (S± ) → V .
d = 4: S+ , S− are complex and have complex dimension 2. They are hermitian conjugate of∗each other: S ± = S∓ . The gamma map is γ”S+ ⊗ S− → V .
d = 10: S+ , S− are real, of dimension 16 and self-dual. The gamma maps are γ : Sym2 (S± ) →V.
Needless to say, forming the double covering Spin(V ) → SO(V ) itself can be seen as taking
square roots of rotations. The appearance of spinors is a reflection of this procedure at thelevel of representations of groups.
Example 2.1.5 (Weil conjecture over finite fields). It is tempting to add to this list of“square roots” the following classical example. Let X be a smooth projective curve over afinite field Fq . The étale cohomology group H 1 (X ⊗ Fq , Ql ) is acted upon by the Frobenius
element Fr, generating Gal(Fq /Fq ). It was proved by A. Weil (and motivated the moregeneral Weil conjectures) that each eigenvalue λ of Fr is an algebraic integer whose image√in each complex embedding satisfies |λ| = q, i.e., λ · λ = q. So the first cohomology, a
fermionic structure, gives rise to factorizations q = λ · λ.
Further, it was suggested by Y. I. Manin that the motive of a supersingular elliptic curve
over Fq can be seen as a “spinorial square root" of the Tate motive. See [37] for developmentsin this direction.

2.2

Square root of d/dt and theta-functions

The simplest example of supersymmetry can be obtained by looking at the differential operator (super-derivation)


∂
∂
∂
0
1
+ξ
=
, Q2 =(2.2.1)
Q =
∂/∂t
0
∂ξ
∂t
∂t
in the algebra O(A1|1 ) = C[t] ⊗ Λ[ξ]. One can replace C[t] by the algebras of smooth oranalytic functions of t. The second equality sign in (2.2.1) is an instance of component
analysis: we view C[t] ⊗ Λ[ξ] as a free C[t]-module with basis 1, ξ and write Q as a 2 × 2
matrix differential operator in t alone. One verifies immediately that Q2 = ∂/∂t, so Q gives
a square root of the Hamiltonian (time translation).
Already this example is quite non-trivial: it provides a natural explanation of Sato’s
approach to theta functions [39, 40]. We do it in two stages. First, consider the exponential(2.2.2)eQ = 1 + Q +

Q2 Q3
+
+ ···
2!
3!

as series of operators acting in complex analytic functions in t, ξ.

10

Proposition 2.2.3. eQ converges to a local operator, i.e., to an endomorphism of the sheaf
OC1|1 of analytic functions on C1|1 .
A standard example of a local operator in this sense is a linear differential operator P on
the line. An operator with constant coefficients is just a polynomial in d/dt:
P = h(d/dt),h(z) =NX

an z n .

n=0

PnReplacing polynomials by entire analytic functions h(z) = ∞n=0 an z , we get expressions
which, even when they converge, need not give local operators. For instance, h(z) = ez gives
the shift operator(ed/dt f )(t) = f (t + 1).
P
However, if h(z) is sub-exponential, i.e., the series k(z) = n!an z n still represents an entire
function, then h(d/dt) acts on analytic functions in a local way. Indeed, by the Cauchy
formula,


I
∞
X

1
−1
0
(n)
f (t )kh(d/dt)f (t) =an f (t) =
dt0 ,
0
2πi
t
−
t
0
|t −t|=εn=0(2.2.4)

where ε can be arbitrarily√small. So if f is analytic in an open U ⊂ C, then so is h(d/dt)f .
For
P∞ example, h(z)n = cos z gives a local operator. One can similarly make sense of series
n=0 an (t)(d/dt) , where an (t) are analytic functions in t of sub-exponential growth in n.
These series define local operators on analytic functions on C known simply as differential
operators of infinite order [40]. They form a sheaf of rings on C, denoted DC∞ . Similarly for
any complex analytic (super-)manifold such as C1|1 .
Returning to the situation of Proposition 2.2.3, we see that Q, being a square root of ∂/∂t,
has exponential eQ which is a local operator: a global section of DC∞1|1 or, after component
analysis, of Mat2 (DC∞ ).
Remark 2.2.5. Although Q is a super-derivation, it is not a derivation in the usual sense and
eQ is not a ring automorphism. In particular, forming eQ is not an instance of exponentiatinga Lie superalgebra to a Lie supergroup.
We now consider the simplest theta-function
X 2
θ(t, x) =en t+nx , <(t) < 0, x ∈ C.
n∈Z

Its value at x = 0 (the Thetanullwert)
θ(x, 0) =

X

2

qn ,

n∈Z

11

q = et , |q| < 1,

is a modular form, so there is a relation (modular transformation) relating its values at t
and at 1/t (in our normalization). The main result of [40] is a characterization of θ(t, 0) by
two differential equations of infinite order in t alone (i.e., by local conditions in t). They arethen used to deduce the modular transformation because the system of equations is invariant
under it.
This can be done as follows. Note that θ(t, x), as a function of two variables, satisfies theheat equation
∂ 2θ
∂θ
=
.
∂t
∂x2
This means that ∂/∂x acts on θ as a square root of ∂/∂t. On the other hand, θ has periodicity
properties(
θ(t, x + 2πi) = θ(t, x)θ(t, x + 2t) = e−x−t θ(t, x).
Using (2.2.4) in the x-variable and the heat equation, we can write this formally as( √e2πi ∂/∂t θ(t, x) = θ(t, x)√
e2t ∂/∂t θ(t, x) = e−t−x θ(t, x),pafter which we can specialize to x = 0. Now, replacing ∂/∂t with Q, we get a system of
two differential equations of infiite order in t alone, satisfied by θ(t, 0).

2.3

Square roots of spacetime translations

Representing just the Hamiltonian (the operator of energy, or time translation) as a bilinear
combination of fermionic operators (§2.2, or Example 2.1.3), is a non-relativistic procedure.
Relativistically, we cannot separate ∂/∂t from any other constant vector field ∂v (momentum
operator) on the Minkowski space V and so should represent them all. This leads to thefollowing concept.
Definition 2.3.1. A quadratic space over a field k is a datum of finite-dimensional k-vector
spaces V, B and a surjective linear map Γ : Sym2 (B) → V . We will often view Γ as a
V -valued scalar product (b1 , b2 ) 7→ Γ(b1 , b2 ) ∈ V on B
A quadratic space Γ gives rise to the Lie superalgebra(2.3.2)t = tΓ ,t0̄ = V, t1̄ = B,with the only non-zero component of the bracket being Γ. We call t the supersymmetry
algebra associated to Γ. The abelian central subalgebra t0̄ = V is the usual Lie algebra of0̄
infinitesimal spacetime translations, and we denote by T 0̄ = et the corresponding algebraic
group (i.e., V considered as an algebraic variety and equipped with vector addition). Since t
is nilpotent: [x, [y, z]] = 0 for any homogeneous x, y, z, it is easily integrated to an algebraic
supergroup T = et called the supersymmetry group using the truncated Hausdorff series(1.4.3).
12

Examples 2.3.3. In physical applications (see [10, 16] for a detailed exposition), V is the
Minkowski space (i.e., a vector space over R or C with a non-degenerate quadratic form)and B is a direct sum of (possible several copies of) the spinor bundle(s). Different possible
choices of such B are known as different types of extended supersymmetry. More precisely:
N = p supersymmetry (SUSY) means that d = dim(V ) is odd and B = S ⊕p .
N = (p, q) supersymmetry (SUSY) means that d is even and B = S+⊕p ⊕ S−⊕q .
The map Γ of the quadratic space is constructed out of the gamma-maps γ for spinors.
Quadratic spaces obtained in this way will be called spinorial. For example:(a) d = 10, N = (1, 0) SUSY: Here B = S+ and Γ = γ : Sym2 (S+ ) → V . This is themost fundamental example in many respects.
(b) d = 4, N = (1, 1) SUSY: Here B = S+ ⊕ S− and Γ is the compositionγ

Sym2 (S+ ⊕ S− ) −→ S+ ⊗ S− −→ V.
(c) d = 2, N = (1, 1) SUSY is often written explicitly using two operators H (energy= ∂/∂t) and P (momentum = ∂/∂x). The space S+ is spanned by two vectors Q+ , Q∗+ andS− by Q− , Q∗− , and the supersymmetric algebra (i.e., the commutation rule in t) is writtenas[Q+ , Q+ ] = H + P, [Q− , Q− ] = H − P, [Q+ , Q− ] = 0.
For a spinorial quadratic space Γ the group Spin(V ) acts on t and on T = et . The
corresponding semidirect product P = Spin(V ) n T is known as the super-Poincaré group(corresponding to the type of extended supersymmetry represented by Γ).

2.4

Quadratic spaces and intersections of quadrics

In algebraic geometry, the term intersection of quadrics [38, 47] means one of two closely
related objects:(1) A homogeneous intersection of quadrics: a subscheme Y in a vector space B, given byhomogeneous quadratic equations. That is, the ideal I(Y ) ⊂ Sym• (B ∗ ) is generated
by I2 (Y ) ⊂ Sym2 (B ∗ ), its degree 2 homogeneous part.
(2) A projective intersection of quadrics: the projectivization Z = P(Y ) ⊂ P(B) of Y asabove. In this way, projective intersections of quadrics Z ⊂ P(B) are in bijection withthose homogeneous intersections of quadrics Y ⊂ B which are not entirely supported
at 0.
Each quadratic space Γ : Sym2 (B) → V gives a homogeneous intersection of quadrics
YΓ = {b | Γ(b, b) = 0} ⊂ B,
13and we denote by ZΓ ⊂ P(B) its projectivization. Alternatively,YΓ = {b ∈ t1̄Γ | [b, b] = 0}
is the scheme of Maurer-Cartan elements of the supersymmetry algebra tΓ .
Proposition 2.4.1. Each homogeneous intersection of quadrics Z ⊂ B can be obtained inthis way from some quadratic space Γ : Sym2 (B) → V , defined uniquely up to an isomorphism.
Proof: Take V = I2 (Y )∗ , the space dual to the space of quadratic equations of Y and takefor Γ the canonical projection.
So quadratic spaces are simply data encoding intersections of quadrics. Classically, thesimplest intersections of quadrics are as follows.
Definition 2.4.2. (1) A projective intersection of quadrics Z ⊂ B is called a complete
intersection, if dim I2 (Z), the number of quadratic equations of Z, is equal to the codimensionof Z in P(B).
(2) A quadratic space Γ : Sym2 (B) → V is called of complete intersection type, if ZΓ is acomplete intersection, i.e., dim(V ) = codim(ZΓ ).
Examples 2.4.3. (a) For the spinorial quadratic space Γ : Sym2 (S+ ) → V of d = 10,
n = (1, 0) SUSY (Example 2.3.3(a)), XΓ is the 10-dimensional space of pure spinors Σ10 ⊂P(S+ ) = P15 . It can be identified with one component of the variety of isotropic 5-planes in
V . It is not a complete intersection: it has codimension 5 but is given by d = 10 equations.
(b) For the quadratic space Γ : Sym2 (S+ ⊕S− ) → V of d = 4, N = (1, 1) SUSY (Example
2.3.3(b)), XΓ ⊂ P(S+ ⊕ S− ) = P3 is the disjoint union of two skew lines P(S+ ) t P(S− ) 'P1 t P1 . It is not a complete intersection: it has codimension 2 but is given by d = 4
equations.
(c) The variety Σ10 ⊂ P15 is a particular case of the following: a partial flag variety G/P(G reductive algebraic group, P ⊂ G parabolic), equivariantly embedded into P(E), where
E is an irreducible highest weight representation of G. All such varieties are known to beintersections of quadrics.
So our physical spacetime is really the space of equations of an auxiliary intersection(typically, not a complete intersection!) of quadrics.
We will also need families of quadratic spaces parametrized by superschemes.
Definition 2.4.4. Let X be a superscheme. A quadratic module over X is a datum of two
locally free sheaves B, V of OX -modules, both of purely even rank, and of an OX -linear map
Γ : Sym2OX (B) → V with two properties:(1) q is surjective, and therefore the dual map Γ∨ : B ∨ → Sym2OX (B ∨ ) is an embedding of
a locally direct summand.
14

(2) The OX -algebra Sym•OX (B ∨ )/(Γ∨ (V ∨ )) is flat over OX .
In other words, a quadratic module gives a family of intersections of quadrics, parametrized
by X, and the condition (2) means that this family is flat, in particular, its fibers have thesame dimension. This is important for non-complete intersections.

2.5

Supersymmetry, superspace and constraints

We start with a quick explanation of some physical terms.
2.5.1. Supersymmetry is a feature of a field theory (say, a collection of fields ϕ plus a
Lagrangian action S[ϕ]) defined, a priori, on the usual (non-super!) Minkowski space V .
It means that the action of the usual Poincaré group SO(V ) n V on fields by changes ofvariables (which leaves any relativistic Lagrangian invariant) is extended, in some way, toan action of the super-Poincaré group P so that S[ϕ] is still invariant. Here P is constructed
out of one of the spinorial quadratic spaces Γ : Sym2 (B) → V (Example 2.3.3).
Thus the new datum in supersymmetry is the extension of the action of V to an actionof tΓ . This means that we need to represent all the momentum operators ∂v , v ∈ V , asbilinear combinations of fermionic “supercharges" Db , b ∈ B so that we have the commutation
relations[Db , Db0 ] = ∂Γ(b,b0 ) , [Db , ∂v ] = [∂v , ∂v00 ] = 0.
For this to be possible, there should be about equally many bosonic and fermionic fields
in the theory. This explains why supersymmetry is sometimes called “symmetry between
bosons and fermions”.
2.5.2. Superspace is a tool to construct supersymmetric theories by replacing the mysterious “in some way" above by a natural construction. More precisely, a superspace is asupermanifold S extending the spacetime V , (so that V = S 0̄ is its even part) and which
admits a natural action of t.
The simplest choice (flat superspace) is S = T , the underlying manifold of the supersymmetry group, on which t = Π(B) ⊕ V acts by left-invariant vector fields Db , ∂v , see[10, 16]. Any field on S (referred to as superfield) gives an entire multiplet of usual fields on
V by component analysis: writing OS = OV ⊗ Λ• (B ∗ ) as a free OV -module of rank 2dim(B) .
The Lie algebra t acts naturally on superfields, so working only with such fields, we get
supersymmetry seemingly “for free”
This construction can, of course, be done for an arbitrary quadratic space Γ : Sym2 (B) →V . If Γ is spinorial, then the action of t on superfields extends to an action of the superPoincaré group P.
Remark 2.5.3. Similarly to §2.2, the exponentials eDb of the supercharges are local operatorson analytic superfields, while the shifts e∂v are not. It would be interesting to understandthe consequences of this phenomenon. The situation of §2.2 corresponds to the simplest
example of a quadratic space, when B is 1-dimensional, V = Sym2 (B) is also 1-dimensional,and Γ = Id.
15

2.5.4. The difficulty with supersymmetry is that it tends to require too many fields (onV ) for all of them to make physical sense. The following result [35] is usually intepreted bysaying that “supersymmetry in > 11 dimensions is not sensible”.
2.5.5. Nahm’s theorem. Any supersymmetric theory with d > 11 contains fields of spin
≥ 2.
For the superspace construction this is easy to understand. Already the simplest kind of
a superfield, a function on S, is a section of OV ⊗ Λ• (B ∗ ), where B is the direct sum of one
or several spinor spaces. As d grows, the decomposition of Λ• (B ∗ ) into Spin(V )-irreducibles,
quickly begins to contain higher spin representations such as Symj (V ), j ≥ 2.
However, even in the remaining dimensions d ≤ 11, the superspace construction typicallygives too many component fields. To eliminate some of the components, one usually imposes(in a seemingly ad hoc way) some additional restrictions on superfields known as constraints.
In the next subsection we discuss a conceptual point of view on such constraints.
Supergeometry as understood by physicists, is the study of various versions of (not necessarily flat) superspaces. All the examples that have been considered, fit into the following
concept.
Definition 2.5.6. An abstract superspace is a supermanifold S (smooth, analytic or algebraic) of dimension (m|n) together with a Pfaff system C ⊂ TS of rank (0|n) satisfying thefollowing properties:(1) The restriction C|S 0̄ coincides with the odd part of (TS )|S 0̄ .
(2) Denote B = Π(C) and V = (T S)|C . Then, the Frobenius pairing zC : Λ2 C → (TS )/C,
written as an OS -linear map Γ : Sym2OS (B) → V , is a quadratic module over S, see
Definition 2.4.4.
Example 2.5.7 (Supercurves, as understood by physicists). For a mathematician,an (algebraic) supercurve is an algebraic supermanifold of dimension 1|n for some n. Fora physicist, a supercurve is a superspace of dimension (1|n), so the Pfaff system C is anecessary part of the structure. See [33, 11, 12]. The geometry of a supercurve of dimension(1|1) is locally modeled on the setting of §2.2.
Example 2.5.8 (Spinorial curved superspaces). Definition 2.5.6 is quite general. It does
not require that the fibers of the quadratic module Γ be spinorial quadratic spaces. However,
the intersections of quadrics related to spinorial spaces (such as the space of pure spinors) are
rigid both as abstract algebraic varieties and as intersections of quadrics. This means thata quadratic module whose one fiber is a spinorial quadratic space, has all neighboring fibers
spinorial of the same type. Therefore if S is a superspace (in our sense) with the commutator
pairing Γ spinorial at one even point, then we have a similar isomorphic “spinorial structure"in each neighboring tangent space. This amounts to a differential-geometric structure on S
including a conformal structure in the quotient bundle (TS )/C (in particular, on the ordinary
manifold S 0̄ ) and a “choice of spinors” for this conformal structure. Cf. [18, 31]
Cobordism categories of such curved spinorial superspaces provide a language for Atiyahstyle approach to supersymmetric quantum field theories [45, 46]
16

2.6

Constraints and complete intersection slices

Various recipes of imposing constrains on superfields can be understood using the idea ofsimple plane slices of complicated intersections of quadrics.
Let Z ⊂ P(B) be an intersection of quadrics. A plane slice of Z is a scheme of the form
Z ∩ M , where M ⊂ P(B) is a projective subspace. It is an intersection of quadrics in M .
The two simplest possibilities are as follows:(0) Z ∩ M = Z, i.e., M is contained in Z entirely.
(1) Z ∩ M is a complete intersection of quadtrics in M .
Let Γ : Sym2 (B) → V be the quadratic space corresponding to Z. A projective subspace
M corresponds to a linear subspace B 0 ⊂ B, and Z ∩ M corresponds to the quadratic
space Γ0 : Sym2 (B 0 ) → V 0 , where V 0 = Γ(Sym2 (B 0 )) and Γ0 is the restriction of Γ. The
supersymmetry algebra tΓ0 is the Lie sub(super) algebra in t generated by B 0 ⊂ t1̄Γ . We willcall such quadratic spaces slices of Γ.
The case (0) above means that tΓ0 = B 0 is abelian and purely even. In the case (1) (which
includes the case (0)) we will say that tΓ0 is a null-subalgebra.
Examples 2.6.1. (a) Let Z = C tC 0 is the union of two skew lines in P3 (Example 2.4.3(b)).
Clearly, each of the two lines gives an instance of Case (0) above. Other then that, we havea P1 × P1 worth of complete intersection slices Z ∩ M , with M being a chord passing throughone point on C and one point on C 0 .
(b) Let Γ : Sym2 (S+ ) → V be the d = 10 N = (1, 0) SUSY quadratic space. Each vectorv ∈ V, q(v) = 0, gives a linear operator Γv : S+ → S+ given by the Clifford multiplication
by v (transpose of the quadratic space map Γ). If v is a non-zero null vector, i.e., q(v) = 0,then Γ2v = 0 and Ker(Γv ) = Im(Γv ) is an 8-dimensional subspace in S+ which we denote Lv .
The intersection P(Lv ) ∩ Σ10 is a quadric hypersurface in P(Lv ), and the space of equationsof this hypersurface is spanned by v. In other words, Γ(Sym2 (Lv )) = C · v, andh = Lv ⊕ C · v ⊂ S+ ⊕ V = t
is a (maximal) null subalgebra. The (1|8)-dimensional linear subspaces Lv ⊕C·v ⊂ t = C10|16
are known as super-null-geodesics [51, 28, 34, 10].
It seems that the P(Lv ) ∩ Σ10 are precisely the maximal complete intersection slices ofΣ10 .
Assuming this, we can formulate the constraints on superfields as follows.
2.6.2. Spin 0 constraints are imposed on scalar superfields which are functions on S or,
more generally maps Φ : S → X where X is a given target manifold. They have the form
Db Φ = 0,b ∈ B0,where B 0 is one fixed subspace of B on which Γ vanishes (case (0) above), maximal with thisproperty. Maps Φ satisfying the constraints are known as chiral superfields.
172.6.3. Spin 1 constraints are imposed on gauge superfields (connections ∇ in principal
bundles on S). They have the form of integrability (F∇ )|g·h = 0, where h = tΓ0 runs over all
null-subalgebras in tΓ and g · h is the left translation of h in S. In other words,[∇b1 , ∇b2 ] = ∇Γ0 (b1 ,b2 ) ,[∇b , ∇a ] = [∇a1 , ∇a2 ] = 0,∀b, b1 , b2 ∈ B 0 , a, a1 , a2 ∈ V 0

where B 0 runs over all the maximal complete intersection slices of Y (case (1) above).
Remark 2.6.4. When imposing constraints on superfields, it is obviously desirable not to
end up restricting their dependence in the usual, even directions of spacetime. In the case
2.6.3 this is ensured by the fact that the null-subalgebras h have dim(h0̄ ) = 1 (they are
the usual null-lines). In other words, all the complete intersection slices of Z = Σ10 are
quadric hypersurfaces: Z ∩ M is a hypersurface in M . It would be interesting to study otherintersections of quadrics Z with this property.
2.6.5. Lie algebra meaning of complete intersections. Given a quadratic space Γ :Sym2 (B) → V , we can associate to it another, Z-graded Lie algebra (i.e., a Lie algebra in
VectZk )etΓ = FL(B[−1]) (Ker(Γ))
Here FL(−) means the free graded Lie algebra generated by a graded vector space. In ourcase B[−1] is B put in degree +1, so the degree 2 part of FL(B[−1]) is Sym2 (B). The
Lie algebra etΓ is obtained by quotienting FL(B[−1]) by the graded Lie ideal generated byKer(Γ) ⊂ Sym2 (B). Thus
et1Γ = B, et2Γ = V,
but it is not required that V commutes with the generators and therefore etΓ can be nontrivial in degrees ≥ 3. Cf. [10] §11.3. We note that tΓ can be considered as a Z-graded Liealgebra by lifting the degree 0̄ part to degree 2 and degree 1̄ part to degree 1 (this is possible
is possible since the degree 0̄ part lies in the center). With this understanding, we have asurjective homomorphism of graded Lie algebrasp : etΓ −→ tΓ .
Denote byRΓ = Sym• (B ∗ ) I2 (YΓ )the graded coordinate algebra (commutative in the usual sense) of the homogeneous intersection of quadtics YΓ ⊂ B. Then, the enveloping algebra U (etΓ ) is identified with RΓ! , thequadratic dual of the quadratic algebra RΓ , see [36] for background. In particular, we havea homomorphism of graded algebrasη : U (etΓ ) −→ Ext•RΓ (k, k).
The algebra RΓ is called Koszul, if η is an isomorphism. This is the case in all spinorial
examples. The role of complete intersections from this point of view is as follows.
18

Proposition 2.6.6. The following are equivalent, and if they are true, then RΓ is Koszul:(i) Γ is of complete intersection type, i.e., YΓ is a complete intersection of quadrics.
(ii) We have etiΓ = 0 for i ≥ 3, i.e., the morphism p : etΓ → tΓ is an isomorphism. In
particular, the condition of commutativity of et2Γ = V with et1Γ = B already follows fromthe defining relations of etΓ .
Proof: This is a particular case of the general principle in commutative algebra that locallycomplete intersections are characterized by the cotangent complex being quasi-isomorphic
to a 2-term complex. The special case of intersections of quadrics was studied in [27].
Further, in many cases (including those related to spinors) the algebra etΓ can be identified with the amalgamated free product of all its null-subalgebras etΓ0 = tΓ0 . This relates
the integrability conditions on null-subalgebras with the Koszul duality point of view on
constraints for SYM advocated in [34].

3

Homotopy-theoretic underpinnings of supergeometry

3.1

The skeleton of the Koszul sign rule

To understand the nature of the Koszul sign rule 1.2.1, let us “minimize” the symmetricmonoidal category SVectk incorporating it.
To account just for the signs, we can disregard all morphisms in SVectk which are notisomorphisms. as well as all objects which have total dimension > 1. Restricted to 1dimensional super-vector spaces (dim(V 0̄ ) + dim(V 1̄ ) = 1) and their isomorphisms, we get asymmetric monoidal category 1-SVectk . Similarly, to capture the Z-graded sign rule, we canrestrict to the category 1-VectZk of Z-graded 1-dimensional vector spaces. These categories
are examples of the following concept.
Definition 3.1.1. A Picard groupoid is a symmetric monoidal category (G, ⊗, 1, R) in which
all objects are invertible under ⊗ and all morphisms are invertible under composition.
A Picard groupoid G gives rise to two abelian groups:• The Picard group of G, denoted Pic(G), or π0 (G). It is formed by isomorphism classesof objects, with the operation given by ⊗.
• The group π1 (G) = AutG (1) of automorphisms of the unit object. It is canonically
identified with the group of automorphisms of any other object.
In our case
1-SVectk :π0 = Z/2,

1-VectZk :π0 = Z,19

π1 = k∗ ;π1 = k∗ .

Here, k∗ is still unnecessarily big: to formulate the sign rule. we need only the subgroup{±1} ⊂ k∗ . So we cut these Picard groupoids further.
For this, we replace k with the ring Z, since {±1} = Z∗ is precisely its group of invertible
elements. Accordingly, we replace 1-dimensional k-vector spaces with free abelian groupsof rank 1. This gives Picard groupoids 1-SAb, 1-AbZ . Their objects are Z/2- or Z-gradedabelian groups which are free of rank 1. As before, the morphisms are isomorphisms„ ⊗ isthe graded tensor product over Z and the symmetries are given by the Koszul sign rule. The
πi of these Picard groupoids are now as follows:
1-SAb :

π0 = Z/2,

π1 = {±1} = Z/2,

1-AbZ :

π0 = Z,π1 = {±1} = Z/2.

We can call 1-SAb and 1-AbZ the sign skeleta of the Koszul sign rule (Z/2-graded andZ-graded versions). They contain all the data needed to write the sign rule but nothing
more.
The following simple but remarkable fact can be seen as a mathematical explanation of
the Principle of Naturality of Supers 1.1.3.
Proposition 3.1.2. 1-AbZ is equivalent to FL , the free Picard groupoid generated by one
formal object (symbol) L.
By definition, FL has, as objects, formal tensor powers L⊗n , n ∈ Z. It further has only
those morphisms that are needed to write the symmetry isomorphisms
RL⊗m ,L⊗n : L⊗m ⊗ L⊗n = L⊗(m+n) −→ L⊗n ⊗ L⊗m = L⊗(n+m)
satisfying the axioms of a symmetric monoidal category (as well as composition, tensor
products etc. of such morphisms).
Sketch of proof: L corresponds to the group Z placed in degree 1, so L⊗n is Z in degree n.
Further, RL,L ∈ Aut(L⊗2 ) = Aut(1) corresponds to (−1) ∈ Z∗ (note that RL,l ◦ RL,L = Id bysymmetry. The axioms of a symmetric monoidal category give that RL⊗m ,L⊗n correspondsto (−1)mn , so we recover the Koszul rule.
In other words, the category VectZk which is at the basis of all supergeometry, can beobtained as a kind of k-linear envelope of a free Picard groupoid. More precisely, we havethe following construction.
Definition 3.1.3. Let G be a Picard groupoid, and χ : π1 (G) → k∗ be a homomorphism.
By a (G, χ)-graded k-vector space we will mean a functor V : G → Vectk , whose value on
objects will be denoted A 7→ V A , satisfying the following condition. For each object A, theaction of each λ ∈ π1 (G) ' Aut(A) on A is taken into the multiplication by χ(λ) on V A . We(G,χ)
denote by Vectk
the category of (G, χ)-graded k-vector spaces.
0

Since V is a functor, the spaces V A and V A for isomorphic objects A, A0 are identified,so a (G, χ)-graded k-vector space V can be viewed as a π0 (G)-graded vector space in theusual sense.
20

(G,χ)

Proposition 3.1.4. (a) The category Vectk
the operation given by

has a structure of a monoidal category withk
(V ⊗ W )A = limVect
V B ⊗k W C ,−→{B⊗C→A}

the colimit taken over the category formed by pairs of objects B, C ∈ G together with an(G,χ)(iso)morphism B ⊗ C → A. Further, the symmetry in G makes Vectk
into a symmetricmonoidal category.
(G,χ)(b) If G = FL and χ : π1 (FL ) = Z/2 → k∗ is the embedding of {±1}, then Vectk
identified with the category VectZk with the symmetry given by the Koszul sign rule.

3.2

is

(Higher) Picard groupoids and spectra

One of the insights of Grothendieck in his manuscript “Pursuing stacks" (cf.[6], p. 114) was
the correspondence between Picard groupoids and a particular class of spectra in the sense ofhomotopy topology. See [13] for a discussion and [17] for a slightly more detailed treatment
which we follow here. A systematic account can be found in [25].
The concept of a spectrum arises as a result of stabilizing the homotopy category ofpointed topological spaces (say CW-complexes) under the two operations (adjoint functors)
Σ = reduced suspension, Ω = loop space,
Hom(ΣX, Y ) = Hom(X, ΩY ).
For example, the spheres S n satisfy Σ(S n ) = S n+1 . We always have a canonical map (unitof adjunction)
εX : X −→ ΩΣX.
A spectrum Y can be seen as a topological space Ω∞ Y together with a sequence of deloopings:
spaces Ω∞−j Y equipped with compatible homotopy equivalences Ωj (Ω∞−j Y ) ∼ Ω∞ Y . A
spectrum Y has homotopy groups πi (Y ), i ∈ Z defined by
πi (Y ) = πi+j (Ω∞−j Y ),j  0.

Example 3.2.1. A topological space X gives the suspension spectrum Σ∞ X, with Ω∞ Σ∞ X =limn Ωn Σn X and Ω∞−j Σ∞ X = limn Ωn−j Σn X (limits under powers of ). The homotopy−→−→
groups of Σ∞ X are the stable homotopy groups of X:
πi (Σ∞ X) = πist (X) := lim πi+n Σn X.
−→
nSpectra form (after inverting homotopy equivalences), a triangulated category SHo known
as the stable homotopy category. This category has a symmetric monoidal structure (smash

21

product of spectra). Let m ≤ n be integers (m = −∞ or n = ∞ allowed). By an [m, n]spectrum we mean a spectrum Y with πi (Y ) = 0 with i ∈/ [m, n], and we denote by SHo[m,n] ⊂SHo the full subcategory of [m, n]-spectra. There is a canonical “truncation" functor
τ[m,n] : SHo −→ SHo[m,n] .
Grothendieck’s correspondence can be formulated as follows.
Theorem 3.2.2. There is an equivalence of categories

B : Picard groupoids [eq−1 ] −→ SHo[0,1] ,so that πi (B(Gc) = πi (G), i = 0, 1. Here [eq−1 ] means that equivalences of Picard groupoids
are invertex, similarly to inverting homotopy equivalences in forming SHo.
A more precise result is proved in [25]. The spectrum BG corresponding to a Picard
groupoid G, is a version of the classifying space of G. That is, the space Ω∞ BG = BG isthe usual classifying space of G as a category, and the deloopings are constructed using thesymmetric monoidal structure, see [17] (3.1.6) for an explicit construction.
The further point of Grothendieck is that more general [0, n]-spectra should have a description in terms of Picard n-groupoids, an algebraic concept to be defined, meaning “symmetric monoidal n-categories with all the objects and higher morphisms invertible in all
possible senses”. Here we can formally allow the case n = ∞.
Incredible complexity of the stable homotopy category, well known to topologists, prevents us from hoping for a simple algebraic definition of Picard n-groupoids. Nevertheless,for small values of n = 2, 3, this can be accessible and useful. The case n = 2 is being treatedin the paper [21] building on the theory of symmetric monoidal 2-categories [20, 22].

3.3

The sphere spectrum and the free Picard n-groupoid

The fundamental role in homotopy theory is played by the sphere spectrum S = Σ∞ S 0 definedas the suspension spectrum of the 0-sphere. Its homotopy groups are the stable homotopy
groups of spheres
πi (S) = πist := lim πi+n (S n ),−→nwhich vanish for i < 0, so S is a [0, ∞]-spectrum. The spectrum S is the unit object in thesymmetric monoidal structure on SHo and for this reason can be considered as a homotopytheoretic analog of the ring Z of integers.
This motivates the further installment in Grothendieck’s vision of a dictionary between
spectra and Picard n-groupoids:
Conjecture 3.3.1. The Picard n-groupoid corresponding to τ[0,n] S, shoud be identified with(n)
FL the free Picard n-groupoid on one formal object L.

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π0st = Z

π1st = Z/2

π2st = Z/2

π3st = Z/24

Cardinality of sets,dimension of vector
spacesSign of permutation,
determinant, Koszul
ruleSpin, central extensions of symmetricgroups√“String”, 24 1 in transformation of η(q),χ(K3) = 24

K0 (k) = Z

K1 (k) = k∗

K2 (R) → π1 (SLn (R)) Dilogarithms etc.
π1 (SLn (R)) = Z/2

Table 1: The first few πist and their significance.
The concept of a free Picard n-groupoid presumes that we already have a system of(n)axioms for what a Picard n-groupoid is. If we have such axioms, then FL contains asobjects, formal tensor powers L⊗n and only those higher morphisms which are needed towrite the necessary “higher symmetry isomorphisms".
As before, for large n this seems unattainable directly, but for small n ≤ 2 this can bemade into a theorem. In particular, the case n = 1, proved in [25, Prop. 3.1] has enormous
significance for super-geometry. Indeed, combining it with Proposition 3.1.2, we arrive at:Corollary 3.3.2. The sign skeleton 1-SAbZ of the Z-graded Koszul sign rule, is the Picard
groupoid corresponding to the [0, 1]-truncation of S. The groups Z = π0 (1-AbZ ) and Z/2 =π1 (1-AbZ ) are the first two stable homotopy groups of spheres.
In other words, the entire super-mathematics is obtained by unravelling the first two layersof the sphere spectrum.
In Table 1 (which expands, somewhat, a table from the online encyclopedia nLab) wegive the values of the πist for i ≤ 3 and indicate mathematical and physical phenomena thatthese groups govern. We also compare S with another spectrum, the algebraic K-theory
spectrum K(k) of a field k, which has πi (K(k)) = Ki (k), the Quillen K-groups. These
groups are indicated at in the bottom row. We notice that the first two groups are πi of
our intermediate Picard groupoid 1-VectZk which corresponds to the [0, 1]-truncation of K(k),see [6], §4. A philosophy going back to [4] and to Quillen, says that S can be heuristicallyconsidered as the K-theory spectrum of F1 , the (non-existent) field with one element, thesymmetric group Sn being the “limit”, as q → 1, of the general linear groups GLn (Fq ).
The first two columns are self-explanatory. In the third column, the phenomenon ofspin(ors) is based on the fundamental group π1 (SOn ) = π1 (SLn (R)) = Z/2 which is the same
as π1st (SOn ) and is identified with π1st = Z/2 via the map SOn → Ωn+1 (S n+1 ) (this is known
as J-homomorphism). The existence of central extensions of symmetric and alternatinggroups Sn and An (with center Z/2) and of corresponding projective representations [42][29] is a related phenomenon: An embeds into SOn , and taking the preimage in Spin(n), we
get a Z/2-extension.
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One thing is worth noticing. Supergeometry, as understood by mathematicians, tackles
only the first two columns of Table 1. A a similar-sounding concept (supersymmetry) usedby physicists, dips into the third column as well: fermions are always wedded to spinors in
virtue of the Spin-Statistics Theorem. In fact, there is something in the very structure of thesphere spectrum that seems to relate spin (third column) and statistics (the second column).
At the most naive level, this is the coincidence of π1st = Z/2 with π2st = Z/2.
Further, consider the [1, 2]-truncated spectrum τ[1,2] S. Its loop spectrum Ωτ[1,2] S is a[0, 1]-spectrum with π0 = π1st = Z/2 and π1 = π2st = Z/2.
3.3.3. Homotopy-theoretic Spin-Statistics Theorem. The Picard groupoid corresponding to Ωτ[1,2] S, is equivalent to 1-SAb, the sign skeleton of the Z/2-graded Koszul sign rule.
In other words, Ωτ[1,2] S is homotopy equivalent to (τ[0,1] S)/2, the reduction of the spectrum
τ[0,1] S by the element 2 of its π0 .
So there is not one, but two ways in which the same Koszul sign rule appears out of
the sphere spectrum, one through statistics, the other one through spin. Note that the“topological proof” of the usual physical Spin-Statistics Theorem, going back to Feynman[15] (see also [14], §20), is based on the intuitive claim that interchanging two particles
is “equivalent” to tracing a non-trivial loop in the rotation group, and this claim needs
something like 3.3.3 to be consistent.
Proof of 3.3.3: This is an exercise on known facts in homotopy theory. A [0, 1]-spectrum
Y (or a Picard groupoid) is classified by its π0 , π1 and the Postnikov invariant which is agroup homomorphism kY : π0 (Y ) → π1 (Y ) satisfying 2kY = 0. Explicitly, kY is given by
the composition product with the generator η ∈ π1st . For Y = τ[0,1] S we have therefore thatkY : Z → Z/2 is the surjection, and for Y = τ[0,1] S/2 we have kY = IdZ/2 .
If both π0 = π1 = Z/2 and kY 6= 0, then kY = Id, like for Y = (τ[0,1] S)/2 and so
Y ∼ (τ[0,1] S)/2. Now, for Y = Ωτ[1,2] S the Postnikov invariant is the map π1st → π2st given bycomposition with η. It is known [50] that η 2 ∈ π2st is the generator and so kY 6= 0.
The 4th column of Table 1, headed by π3st = Z/24, is related to various “string-theoretic"
mathematics such as the appearance of 24th roots of 1 in Dedekind’s formula for the modulartransformation of his η-function [2], the Euler characteristic of a K3 surface being 24, theimportance of the central charge modulo 24 in conformal field theory and so on, cf. [23].

3.4

Towards higher supergeometry

Conjecture 3.3.1 means that the sphere spectrum S is the ultimate source for meaningful
twists of commutativity, i.e., for designing truly commutative-like structures, flexible enoughto serve as a basis of geometry. The existing super-mathematics uses only the first two levels(0th and 1st) of S, with physical applications exploiting the parallelism between the 1st andthe 2nd levels.
This opens up a fantastic possibility of higher super-mathematics which would use, as its“sign skeleton", the spectrum S in its entirety or, at least, the truncations τ[0,n] S and the free(n)Picard n-groupoids FL for as long as we can make sense of them algebraically. Here we24sketch the first step in this direction, the formalism for n = 2. For convenience, we adopt a
genetic approach.
3.4.1. Idea of supersymmetric monoidal categories. Super-mathematics begins withreplacing commutatibity ab = ba with supercommutativity (1.1.1). Categorical analogs ofcommutative algebras are symmetric monoidal categories, where we have coherent isomorphisms V ⊗ W ' W ⊗ V . So we introduce “categorical minus signs" into these isomorphisms
as well.
More precisely, by a k-superlinear category we mean a module category V (a category
tensored over) the symmetric monoidal category SVectk . In such a category we have theparity change functor Π given by tensoring with the super-vector space k0|1 . We take Π as thecategorical analog of the minus sign. In doing so, we use the identification of π0 (S/2) = Z/2(the Z/2-grading) with π1st = Z/2 (the ±1 signs).
We now consider k-superlinear categories A which are Z- or Z/2-graded, i.e., split into a
categorical direct sum A = i Ai , where i ∈ Z or Z/2. We assume that A is equipped withgraded SVectk -bilinear bifunctors⊗ = ⊗i,j : Ai × Aj −→ Ai+j
subjects to associativity isomorphisms of the usual kind and want to impose twisted commutativity isomorphisms
RV,W : V ⊗ W −→ Πdeg(V )·deg(W ) (W ⊗ V )subjects to natural axioms, in which, further, various numerical minus signs will be introduced.
3.4.2. Definition of supersymmetric monoidal categories. For simplicity consider the(2)
Z-graded version. We use Proposition 3.1.4 as a guideline, and start with F = FL , the free
Picard 2-groupoid on one object L. It corresponds to the truncation τ[0,2] S. Thus π0 (F), thegroup of equivalence classes of objects, is identified with π0st = Z and will account for thegrading.
The category AutF (1) formed by automorphisms of the unit object and 2-morphisms
between them, is a usual Picard groupoid which corresponds to Ωτ[1,2] S and so, by the"Spin-Statistics Theorem” 3.3.3, it is identified with 1-SAb, the sign skeleton of SVectk .
We denote by SCatk the 2-category of k-superlinear categories. It serves as a categorical analog of the category of ordinary vector spaces. We will denote by V SVectk W thecategorical tensor product [19] of two k-superlinear categories V and W
The Picard groupoid 1-SVectk plays the role of the multiplicative group k∗ for SCatk :it acts on each object by equivalences. The monoidal functor (embedding) χ : 1-SAb →1-SVectk is therefore an analog of the homomorphism χ from Definition 3.1.3.
(F ,χ)
We now consider the 2-category SCatk
formed by all 2-functors V : F → SCatk whichtake the action of AutF (1) on each object A into the action on V A given by χ. As before,

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⊗i

a datum of such V is the same as a datum of a family of superlinear categories V i = V L ,i ∈ Z, one for each equivalence class of objects of F. Now, the formula(V  W)A = 2 lim{B⊗C→A} V B SVectk W C−→(F ,χ)

makes SCatk
into a symmetric monoidal 2-category. It can be seen as the categoricalanalog of the category of super-vector spaces.
By definition, a supersymmetric monoidal k-category is a symmetric monoidal object A(F ,χ)in SCatk . An explicit algebraic model for F was proposed in [5], Ex. 5.2, see also [41],Ex. 2.30. Taking this model for F, we can unravel the data involved in A. These datain particular, contain superlinear categories Ai , bifunctors ⊗i,j and isomorphisms RV,W as
outlined in 3.4.1.
The definition of a Z/2-graded supersymmetric monoidal category is similar, using thePicard 2-groupoid F/2.
If V is an object of a symmetric monoidal category, then there is an action of the symmetric group Sn on V ⊗n . If, instead V ∈ A2m+1 is an odd object of a supersymmetric monoidal
category A, then V ⊗n ⊕ Π(V ⊗n ) has an action of the (spin) central extension of Sn , first
discovered by Schur [42].
Examples 3.4.3. (a) The exterior algebra of a superlinear category. Similarly tousual linear algebra (Examples 1.1.2), an example of a supersymmetric monoidal category
can be extracted from the categorical version of the exterior power construction developedin [17]. This construction is based on the categorical sign character which is a functor of
monoidal categories
sgn2 : Sn −→ 1-SAb(Sn is the symmetric group considered as a discrete monoidal category). It combines theusual sign character sgn : Sn → Z/2 at the level of π0 and the “spin-cocycle" c ∈ H 2 (Sn , Z/2)
at the level of π1 . The exterior power Λn V of a superlinear category V is obtained fromthe tensor power V n by considering objects equipped with sgn2 -twisted Sn -equivariancestructure (see [17] for precise context and details). The analog of the wedge product is givenby the functors∧m,n : Λm V × Λn V −→ Λm+n V
given by partial Π-antisymmetrization, as in [17] §4.2.
(b) Superalgebras of types M and Q and the half-tensor product of Sergeev.
Let k be algebraically closed. It is known since C.T.C. Wall [48] that simple finite-dimensional
associative superalgebras over k are of two types: type M, formed by the matrix superalgebras Mp|q = End(kp|q ) and type Q, formed by the so-called queer superalgebras Qn ⊂ Mn|n ,see [26] and [29]. The simplest nontrivial queer algebra is the Clifford algebra Cliff 1 on one
generator

Q1 = Cliff 1 , Cliff n = k[ξ1 , · · · ξn ]/ ξi2 = 1, ξi ξj = −ξj ξi , deg(ξi ) = 1̄.
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Their behavior under tensor multiplication is(3.4.4)

Mp|q ⊗ Mm|n ' Mpm+qn|pn+qm ,Mp|q ⊗ Qn ' Q(p+q)n ,Qm ⊗ Qn ' Mmn|mn .

This means that the super-Brauer group formed by Morita equivalence classes of these algebras, is identified with Z/2, with type M mapping to 0̄ and type Q mapping to 1̄. This Z/2
is nothing but π2st , responsible for spin.
As a consequence, irreducible objects of any semisimple k-superlinear category V also split
into two types M and Q, according to their endomorphism algebras being k or Q1 . Denoting
V 0̄ the subcategory formed by direct sum of objects of type M and V 1̄ the subcategory formedby sums of objects of type Q, we get an intrinsic Z/2-grading on V. By (3.4.4), any exact
monoidal structure ⊗ on V preserves this grading.
Further, if V, W are irreducible objects of type Q, then V ⊗W is acted upon by Q1 ⊗Q1 'End(k1|1 ) and so is identified with the direct sum of some object with its shift:
V ⊗ W ' (2−1 V ⊗ W ) ⊕ Π(2−1 V ⊗ W ),2−1 V ⊗ W := I ⊗Q1 ⊗Q1 (V ⊗ W )where I ' (k1|1 )∗ is an irreducible right module over Q1 ⊗ Q1 . We get in this way a new
monoidal operationV ⊗1̄,1̄ W := 2−1 V ⊗ W.

⊗1̄,1̄ : V 1̄ × V 1̄ −→ V 0̄ ,This operation was introduced by A. Sergeev [43], see also [29], p.163 for more discussion.
If ⊗ is symmetric, then ⊗1̄,1̄ satisfies
V ⊗1̄,1̄ W ' Π(W ⊗1̄,1̄ V ).
Indeed, the interchange of the factors in V ⊗ W corresponds to the interchange of ξ1 andξ2 in Cliff 2 = Q1 ⊗ Q1 , and the pullback of I under this interchange is isomorphic to Π(I)(which, as a right Cliff 2 -module, is not isomorphic to I). The operation ⊗1̄,1̄ can be used asa source of examples of supersymmetric monoidal structures.

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Kavli Institute for Physics and Mathematics of the Universe (WPI), 5-1-5 Kashiwanoha, Kashiwa-shi,
Chiba, 277-8583, Japan. Email: mikhail.kapranov@ipmu.jp

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