Super Characteristic Classes and Riemann-Roch Type Formula

The main purpose of this article is to define the super characteristic classes on a super vector bundle over a superspace. As an application, we propose the examples of Riemann-Roch type formula. We also introduce the helicity group and cohomology with respect to coefficient of the helicity group. As an application, we propose the examples of Gauss-Bonnet type formula.


Introduction
In this paper, we define various characteristic classes on a super vector bundle over a superspace, so called super characteristic classes.We also propose the super Riemann-Roch formulas and the super Gauss-Bonnet formulas as its application.In contrast, it is justified the definition of the super characteristic classes by establishing those formulas.In [1], we defined the super Chern classes with values in the super number ( ) a b , , a b ∈  and we succeeded in applying the super ADHM construction of the super Yang-Mills instantons.But essentially the super Chern classes ought to take with values in an integer a ∈  .Meaning like it, we introduce the new defini- tion of the super Chern classes with values in integer.In general, the characteristic classes consider that given the vector bundles it corresponds to some cohomology class of the base manifolds.Hence, we need the cohomology reflecting the properties of superspaces.Therefore, we will define the cohomology with respect to coefficient of the some finitely generated group, which is called the helicity group.
This article is organized as follows.After a brief sketch on the definition and examples of superspaces and its cohomology in Section 2 ([1]- [6]), main result in this paper is that we define the Chern class, Chern character, Todd class, Pontrjagin class, Eular class, Â -genus and L-genus as in the case of super category in Section 3. In Section 4, as an application, we have the Riemann-Roch type formula of super structure sheaf on the complex supercurves of dimension ( ) 1 N with genus g.Moreover, it generalizes the structure sheaf to any super line sheaves.In particular, in the case of dimension ( ) 1 1 , with 1 N = supersymmetric structure, we obtain the Atiyah-Singer index type formula for any super line bundles.In Section 5, we attempt to define the helicity group and cohomology with respect to coefficient of the helicity group.In Section 6, we give the Gauss-Bonnet type formula on the complex supercurves of dimension ( ) 1 N with genus g and the complex super projectve space of dimension ( ) n N .

Supermanifolds
We will summarize the definitions here in order to establish terminology and notation ([1]- [6]).Definition 2.1 A superspace is defined to be a local ringed space ( ) is a local ring.In particular case of a superspace, a supermanifold is defined by the following.

Definition 2.2 A supermanifold of dimension ( )
n N is a ringed space ( ) with the following properties: 1) the structure sheaf ,0 ,1  is locally isomorphic to the exterior algebra ⋅ Λ  .A supermanifold is said to be split if the isomorphism 3) holds globally.A local section ( ) can be expressed as follows: ( ) as a local coordinate of a supermanifold M .

Example 2.1 1)
The typical example is the real (or complex) linear superspace 2) A real super sphere of dimension ( ) , ,  is the sheaf of the ring of differential functions on n S .
3) A complex super projective space of dimensin ( ) We denote by


, that is ( ) The above are examples of the supermanifolds in Definition 2.2.5) We have a new example of superspace in Definition 2.1 as follows.The complex supercurves of dimension ( ) 1 N with genus g is defined by , , , is the canonical line bundle on the classical Riemann surfaces Σ and . In the case of 1 N = , it becomes the super Riemann surfaces with 1 N = SUSY structure (c.f. [7],p.162).In the case of 2 N ≥ , we do not kown whether or not there exists a SUSY structure.We can construct the super Euler sequence as follows ( [1]).

( ) ( )
Tensoring this with ( ) Considering the super determinant ( so called Berezin bundle ) of the super Euler sequence, we obtain The following is given by Penkov ([8]).Theorem 2.1 (Super Serre Duality) Let E be a complex super vector bundle over M .Suppose that * BerT M is the canonical super line bundle of M .Then we have the following.

Super Characteristic Class
In this section, we will give a main result in this paper.Let where ι is the natural injection and exp is defined by

( )
Ker exp =  .This induces the exact sequence of cohomology groups: We can identify ( ) , M H M  with the equivalence classes of ( ) 1 0 or ( ) Then we can define the super first Chern class of ( ) 1 0 -super line bundle L and ( )  .We consider the line sheaf ( ) over the complex super projective space The super first Chern calss and the classical first Chern class denote by 1 c and 1 c  , respectively.Then we have Hence, we see that for the superline bundle L ( ) ( ) ( ) ( ) We will propose the axiomatic definition of super Chern classes (cf.[1] [2] [9]- [15]).We consider the category of complex ( ) r s -super vector bundles over an ( ) Axiom 1 For each complex super vector bundle E over M and for each positive integer i, the i-th super Chern class  is given, and ( ) We set ( ) ( )

Axiom 2 (Naturality)
Let E be a complex super vector bundle over a superspace N and where * f E is the pull-back bundle over M .Axiom 3 (Whitney sum formula) Let 1 2 , , , q L L L  be complex line bundles of rank ( ) 1 0 or ( ) Then it can be axiomatically as follows: In order to explicitly define the super characteristic classes we need the splitting principle ([2] Proposition 3.7) as follows.
Proposition 3.1 (Bartocci, Bruzzo, Hernandez-Ruiperez) Let E be a complex ( ) r s -super vector bundle over an ( ) n N -supermanifold M .Then there exists a supermanifold ( ) F M and a proper fibration ( ) 2) The pull-back bundle * E π splits into a direct sum of even complex line bundles i l of rank ( ) 1 0 and odd complex line bundles j m of rank ( ) We will explicitly give the super characteristic classes.

Definition 3.1 1)
The total super Chern class ( ) The total super Chern character ) Let E  be a real vector bundle of rank ( ) We can consider that it is justified these definitions by the following (cf.[13] [14]).Lemma 3.1 The first few terms of ( ) ( ) td E are given by the following.
Proof.Let E be a complex rank-( ) 2 1 super vector bundle over a complex ( ) 4 4 -dimensional supermanifold M .Then, total super Chern class is written by Hence, we have The total super Chern character is written by ) ( ) It is well-known thtat Hence the total super Todd class is written by ( ) Then, they satisfy that The first few terms of ( ) p E  are given by the following. ( Proof. ( ) p E  similarly form in the classical case.Therefore ( ) L E  are of same argument (cf.[13]).
Let E be a complex rank-( ) 2 2 super vector bundle over a complex ( ) 4 6 -dimensional supermanifold M .The total super Chern class is written by The total super Pontrjagin class is written by Then, they satisfy that , , be the complex supercurves with genus g, where in Example 2.1 (5).Then the canonical super line bundle on Σ is explicitly written by Note that for any object E and F the parity change functor Π satisfies Λ  is a supermanifold, then its tangent bundle can be written by ( ) [16]).Hence we have Theorem 4.1 Let Σ be the complex ( ) 1 N -dimensional supercurves with genus g.Then, we have a Noether type formula as follows.
be the genus on the classical Riemann surfaces and be the number of linear independent Dirac zero modes or harmonic spinors which is not topologically invariant.The structure sheaf of the complex supercurves have decomposition In the case of genus 2 g ≥ , we have (cf.( [17])) In the case of genus for any p.In the case of genus In the case of genus 2 g ≥ , we have the following.


Note that equal of second make use of the classical Serre duality.Hence we obtain In the case of genus 1 g = and 0 g = , we can prove similarly. Corollary 4.1 Let Σ be the complex ( ) 1 N -dimensional supercurves with genus g.Then we have a Riemann-Roch type formula as follows.
From Theorem 4.1, this completes the proof of Corollary 4.1. The following Corollary essentially has been obtained by [18].It needs the 1 N = supersymmetric structure on the 1 N = super Riemann surfaces (cf. [7] [19] [20]).The following rewrite the result of [21] to the super characteristic classes.Corollary 4.2 Let Σ be the complex ( ) where ( ) h Z is a super holomorphic function.We put . Then the inverse element of x , which is unique, is given by the formula ( ) [23]).As an application, we have a main theorem as follows.
Theorem 4.2 Let Σ be the complex ( ) where [ ] ( ) is defined by the transition functions ( )  is called the super skyscraper sheaf.Tensoring this with 1 L , we have The map ψ is defined by ( ) Taking the alternative sum, we have ( ) We also take the exact sequence 0 ˆˆ1 0 0.
Taking also the alternative sum, we have isomorphic to the ( ) n N  -valued cohomology with coefficient in  of the classical manifold M using the uni- versal coefficient theorem.That is to say, we have the following.
This isomrphism is applied in section 6.

Gauss-Bonnet Type Formula
In this section, we will apply the super cohomology with coefficient in helicity group ( ) Theorem 6.1 Let Σ be the complex ( ) 1 N -dimensional supercurves with genus g.Then we have a Gauss-Bonnet type formula as follows.

4 )
The super Eular class ( ) e E is defined by

2 2 r
s .The i-th super Pontrjagin class ( ) i p E  and the total super Pontrjagin class are defined by

1 1 -
dimensional super Riemann surfaces and ˆL super line bundles of rank ( ) 1 0 on Σ .Then we have a Atiyah-Singer index type formula as follows. is the fundamental homology class.Proof.The canonical super line bundle KΣ of a super Riemann surface Σ can be defined by splitting the Berezin bundle * BerT Σ using the super complex structure , ( ) ẑ θ ∈ Σ .Note that the operator D is 1 N = supersymmetric anti-holomorphic vector fields.Tensoring this exact sequence with any super line bundles LΣ , we have as the space of sections s of LΣ satisfying the condition ˆ0

1
N -dimensional supercurves with genus g and ˆL L Σ Σ Σ = ⊗ be any super line bundles of rank ( ) 1 0 on Σ .Then we have a Riemann-Roch type formula as follows.

(
the fundamental homology class.Proof.Let us consider the super divisor .The super Weil divisor can be considered as the super Cartier divisor.Then there is the exact sequence From the super Euler sequence, we can compute the total Chern class of holomorphic tangent bundle |