Superspaces

Definition. The superspace E is Z2-graded

E = E+ ⊕ E−.

A superalgebra is an algebra A whose underlying vector space is a superspace, and whose products respects the Z2-grading; in other words,

Ai · Aj ⊂ Ai+j .

Example 1. An exterior algebra is a supertrace with the Z2-grading

Λ±E = X ΛiE. (−1)i=±1

+ Example 2. An ungraded vector space E is implicitly Z2-graded with E = E and E− =0. • The algebra of End(E) of a superspace is a superalgebra, when graded in the usual way:

End+(E) =Hom(E+,E+) ⊕ Hom(E−,E−) End+(E) =Hom(E+,E+) ⊕ Hom(E−,E−).

Definition. A superbundle on a manifold M is a bundle E = E+ ⊕E−, where E+ and E− are two vector bundles on M such that the fibers of E are superspaces.

• If E is a vector bundle, we will identify it with the superbundle E+ = E and E− =0. • There is a general principle in dealing with elements of a superspace, that whenever a formula involves commuting an element a past another b, one must insert a sign (−1)|a||b|, where |a| is the parity of a, which equals 0 or 1 accrording to whether the degree of a is even or odd. – Thus, the supercommutator of a pair of odd parity elements of a superalgebra is actually their anticommutator, due to the extra minus sign. • We will use the same bracket notation for this supercommutator as is usually used for the commutator:

[a, b]=ab − (−1)|a|·|b|ba.

This bracket satisfies the axioms of a Lie superalgebra: (i) [a, b]+(−1)|a|·|b|[b, a]=0 (ii)[a, [b, c]] = [[a, b],c]+(−1)|a|·|b|[b, [a, c]].

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Definition. A superalgebra is said to be super-commutative if its superbracket vanishes identically.

Example. The exterior algebra ΛE is super-commutative. Definition. A on a superalgebra A is a linear form φn A satisfying φ([a, b]) = 0.

• On a superalgebra End(E), there is a canonical linear form given by the formula

Tr + (a) − Tr − (a), if a is even, Str(a)= E E 0, if a is odd.

Proposition 1. The linear form Str defined above is a supertrace on End(E). Proof. We must verify that Str[a, b]=0. (i) If a and b have opposite parity, then [a, b] is odd in parity and hence Str[a, b]=0. a+ 0 b+ 0 (ii) If a =   and b =   are both even, then 0 a− 0 b−

[a+,b+]0 [a, b]=  0[a−,b−]

has vanishing supertrace, since Tr[a+,b+]=Tr[a−,b−]=0. 0 a− 0 b− (iii) If a =   and b =   are both odd, then a+ 0 b+ 0

a−b+ + b−a+ 0 [a, b]=  0 a+b− + b+a−

has supertrace

− + − + + − + − Str[a, b]=TrE+ (a b + b a ) − TrE+ (a b + b a )=0. 

• If E = E+ ⊕ E− and F = F + ⊕ F − are two supertraces, then their E × F is the superapace with underlying vector space E ⊗ F and grading

(E ⊗ F )+ =(E+ ⊗ F +) ⊕ (E− ⊗ F −), (E ⊗ F )− =(E+ ⊗ F −) ⊕ (E− ⊗ F +).

• If A and B are superalgebras, then their tensor product A⊗B is the superalgebra whose underlying space is Z2-graded tensor product of A and B, and whose product is defined by the rule

|b1|·|a2| (a1 ⊗ b1) · (a2 ⊗ b2)=(−1) a1a2 ⊗ b1b2. 3

Definition. If A is a supercommutative algebra and E is a supertrace, we extend the supertrace on End (E) to a map Str : A ⊗ End(E) → A, by the formula Str(a ⊗ M)=aStr(M) for a ∈ A and M ∈ End(M). • The extension of the supertrace vanishes on supercommtators in A ⊗ End(M), since [a ⊗ M,b ⊗ N]=(−1)|M|·|b|ab ⊗ [M,N] wnere A is supercommutative. • If E1 and E2 are two supertraces, we may identify End(E1 ⊗E2) with End(E1)⊗ End(E2), the action being as follows:

|a2|·|e1| (a1 ⊗ a2)(e1 ⊗ e2)=(−1) (a1e1) ⊗ (a2e2),

where ai ∈ End(Ei) and ei ∈ Ei. ± Definition. If E is a supertrace such that dim (E )=m∓, we define its deter- minant line to be the one-dimensional vector space det(E)=(Λm+ E+)∗ ⊗ Λm− E−. • If E+ = E−, then det(E) is canonically isomorphic to R. Definition. If E = E+ ⊕E− is a superbundle, then its determinant line bundle det(E) is the bundle det(E)=(Λm+ E+)∗ ⊗ Λm− E−, ± where m± =rk(E ). • If M is an n-dimensional manifold, det (TM) is the volume-form bundle ΛnT ∗M. Definition. If E is a Hermitian supertrace, we say that u ∈ End−(E) is odd self-adjoint if it has the form ou− u =   , u+ 0 where u+ : E+ → E−, and u− is the adjoint of u+.

j • Let V be a real vector space with basis ei and dual basis e . Denote k k ιk = ι(ek),ε= ε(e ), k where ε is the exterior product and ιk is the interior product. k ∗ k ∗ I I – We will identify Λ V with (Λ V ) by setting he ,eJ i = δJ , where 1 k I = {1 ≤ i < ···

I i1 ik are multiindices and e = e ∧···∧e , eJ = ej1 ∧···∧ejk . – Using this identification, it is easy to see that ε(α)∗ = ι(α) ∈ End(ΛV ∗) and that ι(ν)∗ = ε(ν) ∈ End(ΛV ). • If A ∈ End(V ), we denote by λ(A) the unique derivation of the superalgebra ΛV which coincides with A on V ⊂ ΛV ; it is given by the explicit formula j k λ(A)=Xhe ,Aekiεj ι . j,k 4

Definition. A non-zero T :ΛV → R which vanishes on ΛkV for k

i i ∗ If ei is a basis for V with dual basis e , the element e ∈ ΛV play the role of coordinate function on V . – From this point of view, the interior multoplication ι(ν):ΛV ∗ → ΛV ∗,(ν ∈ V ), is the operation of differention in the direction ν. – We see that a Berezin integral is an analogue of the Lebesgue integral: a Berezin integral T :ΛV ∗ → R is a linear form on the function space ΛV ∗ which vanishes on “partial derivatives”: T · ι(ν)α =0, ∀ν ∈ V and α ∈ ΛV ∗. • If V is an oriented Euclidean vector space, there is a canonical Berezin integral, defined by projecting α ∈ ΛV onto the component of the monomial e1 ∧···∧en; here n is the dimension of V and ei form an oiented orthonormal basis of V . – We will denote the Berezin integral by T :

 1, |f| = n, T (eI )= 0, otherwise.

If α ∈ ΛV , we will often denote T (α)byα[n], n although strictly speaking α[n] is an element of Λ V and not of R. 2 • If A ∈ Λ V , its exponential in the algebra ΛV will be denoted by expΛ A = P(Ak/k!). Definition. The Pfaffian of an element A ∈ Λ2V is the number

PfΛ(A)=T (expΛ A). The Pfaffian of an element A ∈ Lie(so(V )) is the number

Pf(A)=T (expΛ XhAei,ejiei ∧ ej ). i

2 Example. If V = R with orthonormal basis {e1,e2}, and if 0 −θ A =   θ 0 then the Phaffian of A is θ. • The Phaffian vanishes if the dimension of V is odd. • If the dimension of V is odd, Pf(A) is a polynomial of homogeneous order n/2 in the components of A. If the orientation of V is reversed, it changes sign. 5

Proposition 2. The Pfaffian f an antisymmetric linear map is a square root of the determinant (Pf(A)2)=detA.

Proof. By the spectral theorem, we can choose an oriented basis ej of V such that ≤ ≤ n there are real numbers cj ,1 j 2 , for which

Ae2j−1 =cj e2j

Ae2j = − cj e2j−1.

In this way we reduce the proof to the case in which V is the vector space R2, and Ae1 = θe2, Ae2 = −θe1. 0 −θ In this case Pf(A)=θ, while the determinant of A =   is θ2.  θ 0 6

Superconnections • If M is a manifold and E = E+ ⊕E− is a superbundle on M, let A(M,E) be the space of E-valued differential forms on M. – This space has a Z-grading given by the degree of a differential form, and we will denote the component of exterior degree i of α ∈A(M,E)byα[i]. – In addition, we have the total Z2-grading, which we will denote by A(M,E)=A+(M,E) ⊕A−(M,E); and which is defined by A(M,E)=A2i(M,E±) ⊕A2i+1(M,E∓); for example, the space of sections of E± is contained in A±(M,E). • If g is a bundle of Lie subalgebras on M, then A(M,g) is a Lie superalgebra with respect to the Lie superbracket defined by

|X1|·|α2| [α1 ⊗ X1,α2 ⊗ X2]=(−1) (α1 ∧ α2) ⊗ [X1,X2]. Likewise, if E is a superbundle of modulus for g with respect to the action ρ, then A(M,E) is a supermodule for A(M,g), with respect to the action ρ(α ⊗ X)(β ⊗ ν)=(−1)|X|·|β|(α ∧ β) ⊗ (ρ(X)ν). In particular, this construction may be applied to the bundle of Lie superalgebra End(E), where E is a superbundle of M, since ΛT ∗M ⊗E is a bundle of modules for the superalgebra ΛT ∗M ⊗ End(E); we see that A(M,End E) is a Lie superalgebra, which has A(M,E) as a super- module. • Any differential operator on A(M,E) which supercommutes with the action of A(M) is given by the action of element of A(M,End(E)); such an operator will be called local. (This is consistent with the “super” point of view, that A(M) is the algebra of functions on a supermanifold fibered over M, and elements of A(M,End(E)) are the zeroth order differential operators on this supermanifold.) Definition. If E is a bundle of supertraces over a manifold M, then a supercon- nection on E is an odd-parity first-order differential operator A : A±(M,E) →A±(M,E) which satisfies Leibniz’s rule in the Z2-graded sense: if α ∈A(M) and θ ∈A(M,E), then A(α ∧ θ)=dα ∧ θ +(−1)|α|α ∧ Aθ. • Let A be a superconnection on E. The operator A can be extended to act on the space A(M,End(E)) in a way consistent with Leibniz’s rule: Aα =[A,α], ∀α ∈A(M,End(E)). – To check that Aα, as defined by this fomula, is an element of A(M,EndE), we need only check that the operator [A,α] commutes with exterior multiplication by any differential form β ∈A(M). 7

Definition. The curvature of a superconnection A is defined to be the operator A2 on A(M,E).

Proposition 3. The curvature is a local operator, and hence is given by the action of a differential form F ∈A(M,End(E)), which has total degree even, and satisfies the Bianchi identity AF =0.

• A superconnection is entirely determined by its restriction to Γ(M,E), which may be any opeator A :Γ(M,E+) →A∓(M,E) that satisfies

A(fs)=df · s + fAs, ∀f ∈ C∞(M),s∈ Γ(M,E).

Indeed, if we define

A(α ⊗ s)=dαs +(−1)|α|αAs, ∀α ∈A(M),s∈ Γ(M,E).

this gives extention of A to A(M,E). • In order to bette understand what a superconnection consists of, we can break 1 it into its homogeneous components A[i], which maps Γ(M,E)toA (M,E):

A = A[0] + A[1] + A[2] + ···

Proposition 4. (1) The operator A[1] is a covariant derivative on the bundle E which preserves the sub-bundles E+ and E−. (2) The operator A[i] for i =16 are given by the action of differential forms

i ω[i] ∈A(M,End(E))

i −  A (M,End (E)) if i is even, on A(M,E), where ωi ∈ Ai(M,End+(E)) if i is odd.

Corollary 5. The space of superconnections on E is an affine space modelled on the vector space A−(M,End(E)). Thus, if As is a smooth one-parameter family of superconnections, then

dA s ∈A−(M,End(E)) ∀s. ds

• We call A[1] the covariant derivative component of the superconnection A. Example.