Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds
UUUITP-24/11NSF-KITP-11-195
Sheaves of N = 2 supersymmetric vertexalgebras on Poisson manifolds Joel Ekstrand a , Reimundo Heluani b and Maxim Zabzine a,c a Department of Physics and Astronomy, Uppsala university,Box 516, SE-751 20 Uppsala, Sweden b IMPA, Rio de Janeiro, RJ 22460-320, Brazil c Kavli Institute for Theoretical Physics, University of California,Santa Barbara, CA 93106 USA
Abstract
We construct a sheaf of N = 2 vertex algebras naturally associatedto any Poisson manifold. The relation of this sheaf to the chiral deRham complex is discussed. We reprove the result about the existenceof two commuting N = 2 superconformal structures on the space ofsections of the chiral de Rham complex of a Calabi-Yau manifold, butnow calculated in a manifest N = 2 formalism. We discuss how thesemi-classical limit of this sheaf of N = 2 vertex algebras is related tothe classical supersymmetric non-linear sigma model. To any smooth manifold M one can associate a sheaf of vertex algebras [1],which is called the chiral de Rham complex (CDR) of M . Locally on a d -dimensional manifold M one attaches d copies of the free bosonic βγ -systemtensored with d copies of the free fermionic bc -system. These local modelsare then glued along intersections of the corresponding patches on M usingappropriate automorphisms of these free field systems. One can combinethese 4 d fields into 2 d N = 1 superfields to obtain a sheaf of N = 1 SUSY a r X i v : . [ h e p - t h ] S e p ertex algebras [2]. More generally, one can construct a sheaf of N = 1SUSY vertex algebras associated to any Courant algebroid E over M , andthis can be done in a coordinate free fashion [3]. Geometric properties of M are reflected in algebraic properties of CDR. For example, if E admitsa generalized Calabi-Yau structure, then there exists an embedding of the N = 2 superconformal vertex algebra into global sections of this sheaf [4].The reader may find more results along these lines in [2, 3, 5].There is a quasiclassical version of CDR as a sheaf of Poisson vertexalgebras [6]. This can be naturally related to the Hamiltonian treatmentof supersymmetric classical non-linear sigma models [7]. Indeed, CDR canbe interpreted as a formal quantization of the non-linear sigma model. By“formal” we mean here that instead of working with the actual loop space of M , one deals with the space of formal loops into M [8]. Nevertheless, therelation to sigma models is quite inspiring, e.g. see [5].In this note, we present a very simple construction of a sheaf of N = 2SUSY vertex algebras on any Poisson manifold M . We discuss the relationof this N = 2 sheaf to CDR as a sheaf of N = 1 SUSY vertex algebras.We recover the main result from [3], about the existence of two commuting N = 2 superconformal structures on the space of sections of CDR in theCalabi-Yau case, but now calculated in a manifest N = 2 formalism. Wealso briefly discuss the semiclassical limit of this N = 2 sheaf and its relationto the Hamiltonian treatment of N = (2 ,
2) supersymmetric sigma modelswith a K¨ahler target.The paper is organized as follows. In Sect. 2 the definition of an N K = 2SUSY Vertex algebra is given. In Sect. 3 we construct the sheaf of N=2 SUSYvertex algebras on any Poisson manifold. In Sect. 4 we discuss the case of asymplectic manifold. Sect. 5 deals with the case of a Calabi-Yau manifold. InSect. 6 we consider the semiclassical limit of the N = 2 sheaf and discuss therelation to the N = (2 ,
2) supersymmetric sigma model. Section 7 containsa summary of the results in this article and a discussion of open questions.All technical calculations are collected in the appendices. For the reader’sconvenience, we collect the rules for Λ-brackets in Appendix A. Appendix Bcontains the calculation for the symplectic case. Appendix C presents theproof of the existence of an embedding of the N = (2 ,
2) superconformalalgebra in the Calabi-Yau case, in manifest N = 2 formalism. Appendix Dcontains the details of the Hamiltonian treatment of the N = (2 ,
2) sigmamodel with a K¨ahler target. 2 N = 2 SUSY Vertex algebra
In this section we briefly review the definitions of vertex algebras and their N = 2 supersymmetric counterparts. The number of supersymmetriesintroduced are in general arbitrary, but since we are mainly interested inthe case of two supersymmetries in this work, we choose to be concrete. Formore details, the reader is referred to [9] and [10].Given a vector space V , a field is defined as an End( V )-valued distributionin a formal parameter z : A ( z ) = (cid:88) j ∈ Z z j +1 A ( j ) , where A ( j ) ∈ End( V ) , (2.1)and, for all B ∈ V , A ( z ) B contains only finitely many negative powers of z .A vertex algebra is a vector space V (the space of states ), with a vector | (cid:105) ∈ V (the vacuum ), a map Y from a given state A ∈ V to a field Y ( A, z )(called the state-field correspondence ), and an endomorphism ∂ : V → V (the translation operator ). The field Y ( A, z ) will also be denoted by A ( z ).These structures must fulfill a set of axioms. The vacuum should beinvariant under translations: ∂ | (cid:105) = 0. Acting with ∂ on a field should be thesame as differentiation of the field with respect to the formal parameter z :[ ∂, Y ( A, z )] = ∂ z Y ( A, z ) . (2.2)We will use ∂ to denote both the endomorphism and ∂ z , and it should beclear from the context what we mean by ∂ . The field Y ( A, z ) correspondingto a given state A creates the same state from the vacuum in the limit z → Y ( A, z ) | (cid:105)| z =0 = A ( − | (cid:105) = A . (2.3)The construction easily extends to the case when V is a super vector space.The state-field correspondence Y should respect this grading, ∂ should bean even endomorphism, and the vacuum should be even.From the endomorphisms A ( j ) of Y ( A, z ) (called the
Fourier modes ), wecan define the λ -bracket :[ A λ B ] = (cid:88) j ≥ λ j j ! ( A ( j ) B ) , (2.4)where λ is an even formal parameter. The λ -bracket can be viewed as aformal Fourier transformation of Y ( A, z ) B :[ A λ B ] = Res z e λz Y ( A, z ) B , (2.5)3here Res z picks the z − -part of the expression. The locality axiom of thevertex algebra says that the sum (2.4) is finite for all A and B , in otherwords, all fields in a vertex algebra are mutually local.The λ -bracket captures the operator product expansion of the correspond-ing (chiral) fields in a two dimensional quantum field theory. Taking theresidue in (2.5) picks out the pole in z , which can be considered to be aformal δ -function. The parameter λ then keeps track of how many derivativesact on the δ -function. In other words, (2.4) is equivalent, in the familiarnotation of OPEs, to A ( z ) · B ( w ) ∼ (cid:88) j ≥ (cid:0) A ( j ) B (cid:1) ( w )( z − w ) j . (2.6) N K = 2 supersymmetric vertex algebra. A vertex algebra endowed with extra supersymmetries can conveniently bedescribed by the formalism of SUSY vertex algebras. By introducing twoadditional formal parameters, θ and θ , that are odd, and promoting thefields A ( z ) to superfields A ( z, θ , θ ), we obtain the notion of N K = 2 SUSYvertex algebra of [10]. In the following, we will often drop the subscript K .We let Z = ( z, θ , θ ) and consider N = 2 superfields of the form A ( Z ) = (cid:88) j ∈ Z z j +1 (cid:0) A ( j | + θ A ( j | + θ A ( j | + θ θ A ( j | (cid:1) , (2.7)where A ( j |∗∗ ) ∈ End( V ) and, as before, for all B ∈ V , A ( Z ) B contains onlyfinitely many negative powers of z . The state-field correspondence Y ( A, Z )maps a state A , to a superfield A ( Z ). We have two odd endomorphisms: D and D satisfying [ D i , D j ] = δ ij ∂ and [ D i , ∂ ] = 0. The vacuum is translationinvariant: D i | (cid:105) = 0. We require translation invariance,[ D i , Y ( A, Z )] = ( ∂∂θ i − θ i ∂ z ) Y ( A, Z ) . (2.8)In addition to the even formal parameter λ , we introduce two odd formalparameters, χ and χ , with the relations [ χ i , χ j ] = − δ ij λ and [ χ i , λ ] = 0.We can then define the N=2 SUSY Λ -bracket :[ A Λ B ] = res Z e ( zλ + θ χ + θ χ ) Y ( A, Z ) B = (cid:88) j ≥ λ j j ! (cid:0) A ( j | − χ A ( j | + χ A ( j | − χ χ A ( j | (cid:1) B , (2.9)4here res Z is the coefficient of θ θ z − . The locality axiom of the SUSYvertex algebra requires that the sum (2.9) is finite for all A and B , i.e. , allfields in a SUSY vertex algebra are mutually local.Let us define the normal ordered product :: between two states by V ⊗ V → V, A ⊗ B (cid:55)→ : AB : ≡ A ( − | B . (2.10)In the following, we will often omit the symbol ::, and use parenthesis toindicate when the ordering is important. Properties of the normal orderingproduct and the relations between the Λ-bracket and the normal orderingare given in Appendix A. We note however that the normal ordered productis not associative nor commutative. The Λ-bracket and the normally orderedproduct satisfy a Leibniz-like rule (A.7) known as the non-commutative Wickformula. In fact, one can define an N K = 2 SUSY vertex algebra as a tuple( V, | (cid:105) , :: , [ Λ ] , D , D , ∂ ) satisfying the axioms of Appendix A.If one drops the integral terms in the axioms, one arrives to the notion ofa Poisson N = 2 SUSY vertex algebra [10, § { Λ } for the Λ-bracket, and we note that V with its operation · becomes aunital super-commutative associative algebra since the quantum correctionsin (A.5) and (A.6) vanish. Moreover, the Poisson Λ-bracket { Λ } now isdistributive with respect to :: (i.e. the Leibniz rule holds) since the quantumcorrection in (A.7) vanish.Let us consider the situation when one has a family V (cid:126) of N = 2 SUSYvertex algebras parametrized by (cid:126) , that is, an N = 2 SUSY vertex algebraover C [[ (cid:126) ]], such that the fiber at (cid:126) = 0 is a Poisson vertex algebra V withthe operations defined as: AB : := lim (cid:126) → : AB : (cid:126) , { A Λ B } := lim (cid:126) → (cid:126) [ A Λ B ] (cid:126) . We then say that the family is a quantization of V , or that V is the quasiclassical limit of V (cid:126) . This happens for example when V is the universalenveloping SUSY vertex algebra of a conformal Lie algebra, namely, when V is generated by some fields A i such that their OPE only involves the fields A i and their derivatives. In this situation, one may consider the algebra V (cid:126) generated by the same { A i } with the Λ-bracket[ A i Λ A j ] (cid:126) := (cid:126) [ A i Λ A j ] , We easily see that the quantum corrections of (A.5) and (A.6) are of order (cid:126) ,and therefore they vanish on V . We thus obtain a quasiclassical limit of V (cid:126) .5 .2 Example: The λ -brackets of an N = 2 superconformalvertex algebra. The N = 2 superconformal vertex algebra is generated by a Virasoro field L , two odd fields G + and G − , an even field J , and a central element c (thecentral charge) [9], with[ L λ L ] = ( ∂ + 2 λ ) L + λ c , [ L λ G i ] = ( ∂ + 32 λ ) G i , (2.11)[ L λ J ] = ( ∂ + λ ) J , (2.12)[ G + λ G − ] = L + ( λ + 12 ∂ ) J + λ c , [ G ± λ G ± ] = 0 , (2.13)[ J λ G ± ] = ± G ± , [ J λ J ] = λ c . (2.14)In an N K = 2 SUSY vertex algebra, the same algebra is generated by asingle field G , with the Λ-bracket [10][ G Λ G ] = (2 λ + 2 ∂ + χ D + χ D ) G + λχ χ c , (2.15)where the superfield G ( Z ) is expanded as G ( Z ) = − iJ ( z ) + iθ (cid:0) G + ( z ) − G − ( z ) (cid:1) − θ (cid:0) G + ( z ) + G − ( z ) (cid:1) + 2 θ θ L ( z ) . (2.16) In this section, we recall the concepts of gradings by conformal weights andcharge in the supersymmetric case. As always, we restrict to the case of 2supersymmetries, and we will omit the terms N K = 2 below.Recall from [10, Def. 5.6] that a SUSY vertex algebra V is called conformalif it admits a vector τ ∈ V such that defining G ( Z ) = Y ( τ, Z ) this fieldsatisfies (2.15), and moreover • τ (0 | = 2 ∂ , τ (0 | = − D , τ (0 | = D . • The operator H := τ (1 | acts diagonally with eigenvalues boundedbelow and with finite dimensional eigenspaces.In this case the eigenvalues of H are called the conformal weights . More-over, it follows from [10, Thm. 4.16 (4)] that, ∀ a ∈ V ,[ H, Y ( a, Z )] = (cid:18) z∂ z + 12 (cid:0) θ ∂ θ + θ ∂ θ (cid:1)(cid:19) Y ( a, Z ) + Y ( Ha, Z ) . (2.17)6 SUSY vertex algebra will be called graded if there exists a diagonaloperator H satisfying (2.17). If Ha = ∆ a for ∆ ∈ C we say that a hasconformal weight, or dimension, ∆. It is easy to see that in this case:∆( ∂a ) = ∆( a ) + 1 , ∆( D i a ) = ∆( a ) + 12 , ∆(: ab :) = ∆( a ) + ∆( b ) , (2.18)and if we let ∆( λ ) = 1 and ∆( χ i ) = 1 / a Λ b ] have conformal weight ∆( a ) + ∆( b ), so that the OPE (or the Λ bracket)becomes a graded operation of degree zero. This is a special property of the N = 2 case, in general the OPE is of degree N/ − N = 2 supersymmetric. In this case, the Λ-bracket has to be anotherfield of conformal weight zero. In particular, the Λ-bracket of functions is anoperation on functions.In fact, the following is a simple exercise in SUSY vertex algebras: Theorem 1.
Let V be a graded N K = 2 SUSY vertex algebra such that theconformal weights are bounded by . Let V be the space of conformal weight vectors, then V is naturally a Poisson algebra, with multiplication beingthe normally ordered product, and the Poisson bracket being the Λ -bracket. Immediately we see that if we want the dimension zero sector of ourtheory to consists of functions on the target manifold M then M has to be aPoisson manifold. This is the content of the next section.The theorem above can be generalized as in the non-SUSY case. Indeed,given a SUSY vertex algebra V , it is easy to see that P ( V ) := V : V D V : + : V D V : , (2.19)is naturally a Poisson algebra, the associative commutative product is inducedfrom the normally ordered product and the Poisson bracket is induced fromthe (0 | V is graded, then P ( V ) inherits the grading, andtherefore the zero-th weight space is a Poisson subalgebra. N = 2 VA from a Poisson structure
In this section, we construct a sheaf of SUSY vertex algebras on any Poissonmanifold ( M, Π). The heuristic is simple; we first attach a local model to an7ffine space and then we need to prescribe how these local fields change underthe allowed local automorphisms (depending on whether we work in thealgebraic, real-analytic or smooth setting). In [1], the authors attach to R n (in the smooth setting) and coordinates { x ν } a free βγ - bc -system. That is avertex algebra generated by 2 n fermionic fields { b ν , c ν } , and 2 n bosonic fields { γ ν , β ν } . What the authors noticed is that under changes of coordinates, thefields γ ν transform as the coordinates { x ν } do, the fields b ν (respectively c ν )transform as the vector fields ∂/∂ x ν do (respectively the differential forms dx ν ). The fields β ν , however, do not transform as tensorial objects, but in arather complicated way. In fact, one may think of the generating fields γ µ , β µ , c µ and b µ as coordinates on the graded supermanifold ˜ M := T ∗ [2] T [1] M and CDR may be thought of as a formal quantization of loops into thismanifold.It was noticed in [2] that if we instead of looking at 4 n -fields as generators,we study 2 n -superfields as generators, these objects transform as tensors.This corresponds to trading supersymmetry in the target by supersymmetryin the worldsheet, namely, instead of loops into ˜ M as above, we are lookingat N = 1 superloops (maps from S | ) into the supermanifold M (cid:48) := T ∗ [1] M .In terms of the previous generators (in the non-SUSY case), the superfieldsare given by φ ν = γ ν + θc ν , S ν = b ν + θβ ν , where the superfields φ ν are even and transform as the coordinates { x ν } do,while the superfields S ν are odd and transform as the vector fields ∂/∂ x ν do.In this article we exploit further this mechanism by which we trade thecomplexity of each generator (they are superfields with more components),by simplicity of the transformation formula under changes of coordinates.For this we will look at N = 2 superloops into M . Locally, to R n we willattach a SUSY vertex algebra generated by n superfields ( N = 2) Φ ν (whichin components account for the 4 n generators in the classical sense) such thatthey transform as coordinates do. It follows from Theorem 1 that the OPEof these fields has to be of the form:[ Φ µ Λ Φ ν ] = Π(Φ) µν , (3.1)where Π µν are the components of a Poisson bivector. In fact, we have thefollowing Theorem 2.
Let M be a Poisson manifold and let O be its sheaf of smoothfunctions. There exists a sheaf V of SUSY vertex algebras on M generated y O , together with an embedding ι : O → V , such that ι ( f g ) =: ι ( f ) ι ( g ) : , ι { f, g } = [ ι ( f ) Λ ι ( g )] , (3.2) for all local sections f, g of O . This sheaf satisfies a universal propertysuch that for any other sheaf V (cid:48) satisfying (3.2) , then there exists a uniquesurjective morphism j : V → V (cid:48) .Proof.
The proof of this statement is straightforward just as in the con-struction of the chiral de Rham complex [1] (see also [3, Prop. 4.6]). Sincethe construction in the N = 2 supersymmetric case is simpler than in thenon-SUSY case of [1] and the N = 1 case of [3] we sketch here the proof.Locally, one can proceed as follows. For a Poisson algebra O we consider thefree H -module generated by O (see Appendix A for notation). This modulehas a structure of SUSY Lie conformal algebra with the operation[ f Λ g ] := { f, g } , (3.3)extended by Sesquilinearity. We can consider its universal enveloping SUSYvertex algebra V (cid:48) [10]. We now consider its quotient V by the ideal generatedby the relations f g =: f g : , D i ( f g ) :=: ( D i f ) g : + : f D i g : , O = | (cid:105) , (3.4) ∀ f, g ∈ O , i = 1 ,
2. Since the operations are defined locally, this ideal iscompatible with localization and in fact we obtain a sheaf locally describedby this quotient V .Notice that V (cid:48) is naturally graded (declaring O to be of degree zero).Since the ideal (3.4) is homogeneous it follows that V is also graded. In fact,we see that locally O is just the degree zero part of V . Remark 1.
There is a subtlety when we say that this sheaf is generatedby n -superfields satisfying (3.1) . If we are in the algebraic setting and thebivector Π is algebraic then we can use arguments of formal geometry tomake sense of the RHS of (3.1) . In the smooth setting we may construct thesheaf as in the proof of the Theorem, or argue as in [11]. This sheaf of N K = 2 SUSY vertex algebras can also be viewed as asheaf of vertex algebras, generated by the components of Φ. Naming thecomponents of Φ as Φ µ = γ µ + θ c µ + θ d µ + θ θ δ µ , (3.5)9he bracket (3.1) is equivalent to the λ -brackets[ γ µ λ δ ν ] = Π µν , [ c µ λ d ν ] = Π µν , (3.6)[ c µ λ δ ν ] = Π µν,τ c τ , [ d µ λ δ ν ] = Π µν,τ d τ , (3.7)[ δ µ λ δ ν ] = Π µν,τρ
12 ( d τ c ρ − c τ d ρ ) , (3.8)where Π is evaluated at γ and the rest of the brackets are zero. Note that, fora linear Poisson-structure the δ ’s commute. Here γ is even, and transformsas a coordinate. The odd fields c and d transforms as vectors, end the evenfield δ transforms in an in-homogenous way.Alternatively, we can generate the sheaf by N = 1 superfields. Expand Φas Φ µ = φ µ ( z, θ ) − θ S µ ( z, θ ). We then have[ φ µ Λ S ν ] N K =1 = Π µν , [ S µ Λ S ν ] N K =1 = Π µν,τ S τ . (3.9)This shows that the Poisson calculus, in the sense of [12], can be mappedto the N K = 1 vertex algebra corresponding to (3.1). For any Poissonmanifold, the cotangent bundle is equipped with the non-trivial structure ofLie algebroid. Namely, in local coordinates we have { dx µ , dx ν } = Π µν,τ dx τ , { f ( x ) , dx µ } = Π µν ∂ ν f , (3.10)where f ( x ) ∈ C ∞ ( M ) and dx is the local basis for differential forms. Thus,on a Poisson manifold one can construct the Courant algebroid (bi-algebroid T M ⊕ T ∗ M with the above bracket on T M and the trivial bracket on T ∗ M )and the corresponding sheaf of N = 1 SUSY vertex algebras is generated bythe relations (3.9). The N = 2 sheaf can be related to the Chiral de Rham complex (the sheaf of N = 1 SUSY vertex algebras associated to the standard Courant algebroidon T M ⊕ T ∗ M ). It is instructive to expand the superfield Φ in such way sowe make contact with previous [3, 2, 7] calculations.Let φ µ ( z, θ ) be an even N = 1 superfield, and S ν ( z, θ ) an odd N = 1superfield with the expansions φ µ ( z, θ ) = γ µ ( z ) + θ c µ ( z ) , (3.11)and S µ ( z, θ ) = b µ ( z ) + θ β µ ( z ) . (3.12)10he field φ µ ( z, θ ) transforms as a coordinate, and S ν ( z, θ ) as a one-form.Recall that the defining Λ-bracket of the Chiral de Rham complex is[ φ µ Λ S ν ] N K =1 = δ µν , (3.13)with [ φ µ Λ φ ν ] N K =1 and [ S µ Λ S ν ] N K =1 being zero. Written as λ -brackets, e.g. , with no manifest supersymmetry, this is[ β ν λ γ µ ] = δ µν , [ c µ λ b ν ] = δ µν , (3.14)and the rest of the brackets are zero.From these brackets and fields, we can construct an N = 2 superfield Φ µ ,that will fulfill (3.1), byΦ µ ( z, θ , θ ) = φ µ ( z, θ ) − θ Π µν ( φ ( z, θ )) S ν ( z, θ ) . (3.15)In components, this isΦ µ = γ µ + θ c µ − θ Π µν b ν + θ θ (Π µν β ν + (Π µν,τ c τ ) b ν ) . (3.16)If the Poisson structure is degenerate, this Φ may differ from the most generalΦ fulfilling (3.1), and it is only on a symplectic manifold where the sheafgenerated by (3.1) is the same as the CDR. The labeling of the two θ ’s in the definition of the SUSY vertex algebrais arbitrary, and when we have more then one supersymmetry, we alsohave an R -symmetry. In particular, the bracket (3.1) is invariant under thetransformations θ → − θ , θ → θ . (3.17)If we also let D → − D and D → D , then axiom (2.8) is still satisfied. Thisautomorphism may induce non trivial transformations on the components ofthe superfields. The sheaf V constructed above admits a quasi-classical limit P as a sheafof SUSY Poisson vertex algebras. It is generated by O just as in (3.2) withthe normally ordered product :: replaced by the associative commutativeproduct of the Poisson vertex algebra and its Λ-bracket [ Λ ] replaced by thePoisson Λ-bracket { Λ } . 11 N = 2 algebra on a symplectic manifold In this section, we discuss the case of a symplectic structure. If the Poissonbivector Π is invertible, then M is symplectic and we will use a differentnotation for this case: Π µν = ω µν . The symplectic structure ω µν is a closednon-degenerate two form, such that ω µν ω νρ = δ µρ . We can then associate asheaf of N = 2 vertex algebras to the manifold, generated by[ Φ µ Λ Φ ν ] = ω (Φ) µν . (4.1)The symplectic case is interesting since we have a canonical two form ω µν .From the Φ’s, we can construct objects that transforms as vectors, D i Φ µ , or ∂ Φ µ . To construct target space diffeomorphism invariant operators, currents,out of these objects, we need tensors with covariant indicies that we cancontract with, e.g. , forms. The most apparent example to study is the caseof a symplectic manifold.As noted above, this sheaf is essentially the same as the Chiral de Rhamcomplex. If we expand Φ as in (3.15), we can use ω to project out S ν . Thebrackets (3.13) and (4.1) are then equivalent.The automorphism (3.17) induces an automorphism on the componentsof Φ, given by γ µ → γ µ , β µ → β µ + ( ω τν,µ ω νσ )( c τ b σ ) + ω µσ,ν ∂ω νσ , (4.2) c µ → − ω µν b ν , b µ → ω µν c ν . (4.3)This automorphism of the βγ − bc -system was discovered, in the case of aCalabi-Yau target manifold, in [3, Theorem 6.4]. N = 2 superconformal algebra On the symplectic manifold, the sheaf carries the structure of an N = 2superconformal algebra. We can construct a generator G ω by G ω = 12 ω µν ( D Φ µ D Φ ν + D Φ µ D Φ ν ) . (4.4)There are no order ambiguities in this expression. The operator G ω is a welldefined section of the sheaf, and there is no need for any quantum corrections.The operator fulfill the N = 2 superconformal algebra[ G ω Λ G ω ] = (2 λ + 2 ∂ + χ D + χ D ) G ω + λχ χ c , (4.5)with central charge c = 3 dim M . The proof is given in Appendix B.12 N = (2 , vertex algebra on a Calabi-Yau mani-fold Let us consider a K¨ahler manifold M , with K¨ahler form ω . Consider the N K = 2 SUSY vertex algebra generated by[ Φ α Λ Φ ¯ β ] = ω α ¯ β . (5.1)Here the fields Φ α and Φ ¯ β correspond to holomorphic and anti-holomorphiccoordinates. Let us define an operator H by H = ( g α ¯ β D Φ α ) D Φ ¯ β − ( g α ¯ β D Φ α ) D Φ ¯ β . (5.2)As it stands, this operator is not a well-defined section of the sheaf of vertexalgebras for a general K¨ahler manifold. It may need a ”quantum correction”,as we will see soon. At this stage, the operator might seem rather ad-hoc,but we will motivate it by the discussion of sigma model in section 6.In order to construct a well defined section of the sheaf of vertex algebras,we need to investigate how H transforms under coordinate changes. Let { z α } be a holomorphic coordinate system, and let ˜ z α = F α ( z β ) be an invertible,holomorphic change of coordinates. We have ˜ g δ ¯ ε = g α ¯ β Φ α,δ Φ ¯ β, ¯ ε and˜ g δ ¯ ε D ˜Φ δ = ( g α ¯ β Φ α,δ Φ ¯ β, ¯ ε )( ˜Φ δ,γ D Φ γ ) = g α ¯ β Φ ¯ β, ¯ ε D Φ α , (5.3)and, using quasi-associativity (A.6), (cid:16) ˜ g δ ¯ ε D ˜Φ δ (cid:17) D ˜Φ ¯ ε = (cid:16) g α ¯ β Φ ¯ β, ¯ ε D Φ α (cid:17) (cid:16) ˜Φ ¯ ε, ¯ ρ D Φ ¯ ρ (cid:17) = (cid:16)(cid:16) g α ¯ β Φ ¯ β, ¯ ε D Φ α (cid:17) ˜Φ ¯ ε, ¯ ρ (cid:17) D Φ ¯ ρ − (cid:18)(cid:90) ∇ d Λ ˜Φ ¯ ε, ¯ ρ (cid:19) [ g α ¯ β Φ ¯ β, ¯ ε D Φ α Λ D Φ ¯ ρ ]= (cid:0) g α ¯ β D Φ α (cid:1) D Φ ¯ β − i (cid:16) ∂ ˜Φ ¯ ε, ¯ ρ (cid:17) Φ ¯ ρ, ¯ ε . (5.4)Therefore, under the inverse change of coordinates ˜ z → z , H transforms as H → H − i (cid:16) ∂ ˜Φ ¯ α, ¯ ρ (cid:17) Φ ¯ ρ, ¯ α = H − i ∂ det ¯ A det ¯ A , (5.5)where A αβ = ∂ ˜ z α /∂z β is the Jacobian of the change of coordinates and ¯ A is its complex conjugate. We see immediately that H will define a globalsection of our sheaf if M is Calabi-Yau. In that case, this section looks like135.2) in the holomorphic coordinate system where the holomorphic volumeform is constant.To find the expression for this section in a general holomorphic coordinatesystem we must add a quantum correction to H that cancels the inhomoge-neous transformations. On a Calabi Yau manifold, we can write the volumeform as Ω ∧ ¯Ω, where Ω is a holomorphic volume form, Ω = e f ( z ) dz ∧ . . . ∧ dz d/ .Under the change of coordinates ˜ z → z , f transforms as a density:˜ f = f + log det Φ α,β = f − log det A . (5.6)We can use this to cancel the inhomogenious transformation of H . Thus, ingeneral holomorphic coordinates of a Calabi-Yau manifold, H = H − i∂ ¯ f = ( g α ¯ β D φ α ) D φ ¯ β − ( g α ¯ β D φ α ) D φ ¯ β − i∂ ¯ f (5.7)is a well defined section.Let us now define G ± by G ± = G ω ∓ H , (5.8)where G ω is the operator constructed in (4.4). Introducing new odd deriva-tives, D ± , that are linear combinations of the derivatives D and D , by D ± ≡ √ D ∓ iD ) , (5.9)we can write (5.8) in a general holomorphic coordinate system as G ± = ( ω α ¯ β D ± Φ α ) D ∓ Φ ¯ β ± i∂ ¯ f . (5.10)The following is the main result of [3] now stated in manifest N = 2formalism. The proof can by found in Appendix C. Theorem 3.
Let M be a Calabi-Yau manifold and G ± be defined by (5.10) .The sections G ± generate two commuting N = 2 superconformal algebras, [ G ± Λ G ± ] = (2 λ + 2 ∂ + χ D + χ D ) G ± + λχ χ c , [ G ± Λ G ∓ ] = 0 , (5.11) each with a central charge c = dim M . The N = 2 Hamiltonian of an N = (2 , super-symmetric sigma model We now want to relate the above discussion to the Hamiltonian treatmentof the supersymmetric sigma model, and thereby motivate the expression(5.2). To do this, we consider the classical supersymmetric sigma model, andwe derive a Hamiltonian formulation thereof. A similar treatment of the N = 1 sigma model was initiated in [13, 14] and its relation to CDR wassuggested in [7]. Here, we suggest the similar relation between the N = (2 , N = 2 supersymmetric vertex algebras on the same Calabi-Yau.We restrict ourself to the N = (2 ,
2) supersymmetric sigma model withthe target manifold M being a K¨ahler manifold, which is not the most generalsigma model with this amount of supersymmetry. The action functional fora classical N = (2 ,
2) supersymmetric sigma model is given by S = (cid:90) dσdτ dθ dθ − dθ dθ − K (Φ , ¯Φ) , (6.1)where the integral performed over Σ | with even coordinates t, σ and fourodd θ coordinates. For the sake of simplicity, we assume that Σ = R × S .Φ and ¯Φ are maps from Σ | to M which satisfy some first order differentialequation (see Appendix D). In physics, Φ = { Φ α } is called a chiral superfield,and ¯Φ = { Φ ¯ α } is an anti-chiral superfield. K is the K¨ahler potential, which isdefined only locally, but nevertheless the action functional (6.1) is well-defined.Upon integration of the odd θ -coordinates, the functional (6.1) reduces tothe more familiar form of the non-linear sigma model and its critical pointsare the generalizations of harmonic maps from Σ to M . In Appendix D weset the notation and present some properties of this N = (2 ,
2) model whichare needed for the derivation. For more on supersymmetric sigma modelsand their applications, the reader may consult the book [15].We would like to consider the Hamiltonian formulation of (6.1). By doinga change of the odd variables, and integrating out two of them, the action(6.1) can be written as S = (cid:90) dσdτ dθ dθ (cid:18) iK ,α ∂ φ α − H (cid:19) , (6.2)with H = g α ¯ β D φ α D φ ¯ β − g α ¯ β D φ α D φ ¯ β (6.3)15eing the Hamiltonian and ∂ being the derivative along τ (time). Here, θ and θ , with corresponding odd derivatives, D i = ∂∂θ i + θ i ∂ σ , are theremaining two odd coordinates. Also, K ,α ¯ β = g α ¯ β . See Appendix D for amore detailed derivation.From (6.2), we see that the Poisson bracket is given by { φ α , φ ¯ β } = ω α ¯ β , (6.4)and that the Hamiltonian density of the sigma model is given by (6.3).The bracket (6.4) is the same as the bracket of the Poisson vertex algebracorresponding to the vertex algebra generated by (5.1). The expression (6.3)is the classical version of the operator H considered in (5.7). Thus, followingthe logic presented in [7], we can think of the sheaf of N = 2 supersymmetricvertex algebras on a Calabi-Yau as a formal quantization of the N = (2 , In this note, we construct a sheaf of N = 2 supersymmetric vertex algebras fora Poisson manifold. We also study the properties of this sheaf on symplecticand Calabi-Yau manifolds. We relate the corresponding semiclassical limit tothe N = (2 ,
2) non-linear sigma model. Let us conclude with a few remarks. • As mentioned above, given an N K = 2 SUSY vertex algebra V , thequotient P ( V ) defined by (2.19) is a Poisson algebra. Just as in thenon-SUSY case, there exist an analogous construction of the Zhualgebra associated to V , this is a one parameter family of associativesuperalgebras P (cid:126) ( V ) such that the special fiber (cid:126) = 0 coincides with P ( V ) and all other fibers are isomorphic. In general it is not truethat this family is flat, or that P (cid:126) ( V ) is a deformation of the Poissonalgebra P ( V ). However, given the construction in this article, startingfrom a Poisson manifold M with its sheaf of Poisson algebras O , weconstructed a sheaf of SUSY vertex algebras V and we obtain a oneparameter family of associative algebras P (cid:126) ( V ). We easily see that P ( V ) = O .This immediately leads one to question whether this family is indeeda deformation in this particular case, obtaining thus a natural wayof quantizing Poisson manifolds. We plan to return to this topic in afuture publication. 16 The most general N = (2 ,
2) non-linear sigma models are related togeneralized K¨ahler geometry [16]. Thus, there should be an analogousHamiltonian treatment of these general models, and it should suggesthow to define sheaves of N = 2 Poisson vertex algebras for a widerclass of manifolds. However, it may require a bigger set of fields thanconsidered in this article. This problem remains to be studied. Acknowledgement
M.Z. thanks KITP, Santa Barbara where part of this work was carriedout. The research of M.Z. is supported by VR-grant 621-2008-4273 and wassupported in part by DARPA under Grant No. HR0011-09-1-0015 and bythe National Science Foundation under Grant No. PHY05-51164.
Appendices
A Rules for Λ -brackets in N K = 2 SUSY vertexalgebras
In this appendix we collect some properties of Λ-bracket calculus. For furtherexplanations and details, the reader may consult [10]. • The operators D i , ∂ and the parameters χ j , λ , where i, j = 1 ,
2, havethe commutator relations [ ∂, χ i ] = [ D i , λ ] = [ ∂, λ ] = 0, and[ D i , D j ] = 2 δ ij ∂ , [ χ i , χ j ] = − δ ij λ , [ D i , χ j ] = 2 δ ij λ . (A.1)We will denote by H the super-algebra with two odd generators D , D and one even generator ∂ = [ D , D ] commuting with both D and D . • Sesquilinearity:[ D i a Λ b ] = − χ i [ a Λ b ] , [ a Λ D i b ] = ( − a ( D i + χ i ) [ a Λ b ] , (A.2a)[ ∂a Λ b ] = − λ [ a Λ b ] , [ a Λ ∂b ] = ( ∂ + λ ) [ a Λ b ] . (A.2b) • Skew-symmetry: [ a Λ b ] = − ( − ab [ b − Λ −∇ a ] . (A.3)17he bracket on the right hand side is computed as follows: first compute[ b Γ a ], where Γ = ( γ, η ), then replace Γ by ( − λ − ∂, − χ − D ). • Jacobi identity:[ a Λ [ b Γ c ] ] = [ [ a Λ b ] Γ+Λ c ] + ( − ab [ b Γ [ a Λ c ] ] . (A.4)where the first bracket on the right hand side is computed as in (A.3).An H -module with an operation [ Λ ] satisfying sesquilinearity, skew-symmetry, and the Jacobi identity is called a SUSY Lie conformalalgebra. • Quasi-commutativity: ab − ( − ab ba = (cid:90) −∇ [ a Λ b ] d Λ , (A.5)where the integral (cid:82) −∇ d Λ is defined as ∂∂χ ∂∂χ (cid:82) − ∂ dλ . • Quasi-associativity:( ab ) c − a ( bc ) = (cid:18)(cid:90) ∇ d Λ a (cid:19) [ b Λ c ] + ( − ab (cid:18)(cid:90) ∇ d Λ b (cid:19) [ a Λ c ] . (A.6) • Quasi-Leibniz (non-commutative Wick formula):[ a Λ bc ] = [ a Λ b ] c + ( − ab b [ a Λ c ] + (cid:90) Λ0 [ [ a Λ b ] Γ c ] d Γ . (A.7) B N = 2 algebra on a symplectic manifold We want to show that G ω = 12 ω µν ( D Φ µ D Φ ν + D Φ µ D Φ ν ) (B.1)fulfill [ G ω Λ G ω ] = (2 λ + 2 ∂ + χ D + χ D ) G + λχ χ dim M , (B.2)using the bracket [ Φ µ Λ Φ ν ] = ω (Φ) µν , (B.3)18here ω µν ω ντ = δ µτ . Note that there are no ambiguities in the order of thenormal ordering in (B.1). Since each term only contains one type of D , therecan be no χ χ -terms when the brackets of the constituents are calculated.Thus, no terms survive the integration in (A.6).We are free to choose any coordinates we want. Since we are on asymplectic manifold, we can choose Darboux coordinates, where ω is constant.This simplifies the calculations considerably. Let G i ≡ ω µν D i Φ µ D i Φ ν . (B.4)We first want to calculate [ G i Λ G i ]. We have [ D i Φ µ Λ D i Φ ν ] = λ ω µν , so[ D i Φ µ Λ G i ] = λ D i Φ µ . Skew-symmetry then gives[ G i Λ D i Φ µ ] = ( λ + ∂ ) D i Φ µ . (B.5)From this, we see that [ G i Λ G i ] = (2 λ + ∂ ) G i . (B.6)We now want to calculate [ G G ]. We have [ D Φ µ Λ G ] = − χ χ D Φ µ .Using skew-symmetry, we then get[ G D Φ µ ] = ( ∂ + χ D + χ D ) D Φ µ + χ χ D Φ µ . (B.7)From this we see that[ G D Φ µ D Φ ν ] = ( ∂ + χ D + χ D ) ( D Φ µ D Φ ν )+ χ χ ( D Φ µ D Φ ν + D Φ µ D Φ ν ) + (cid:90) , (B.8)where the integral term is given by (cid:90) Λ0 [ ( ∂ + χ D + χ D ) D Φ µ + χ χ D Φ µ Γ D Φ ν ] d Γ = − (cid:90) Λ0 χ χ [ D Φ µ Γ D Φ ν ] d Γ = − λχ χ ω µν . (B.9)From (B.8), it is now easy to see that[ G G ] = ( ∂ + χ D + χ D ) G + χ χ ω µν D Φ µ D Φ ν + λχ χ dim M , (B.10)and, finally,[ G ω Λ G ω ] = [ G G ] + [ G G ] + [ G G ] + [ G G ]= (2 λ + 2 ∂ + χ D + χ D ) G + λχ χ dim M . (B.11)19 N = (2 , algebra on a Calabi-Yau manifold We want to calculate the algebra generated by G + and G − , defined in (5.10),under the bracket (5.1). We are free to work in any coordinates we want.A convenient choice is to choose the coordinates where the holomorphicvolume form is constant. On a Calabi-Yau, we can always choose suchcoordinates locally. In this coordinates, the quantum correction ± i∂ ¯ f ( z )vanishes. Also note that Γ ααβ = 0 in these coordinates. To the metric, we havea corresponding K¨ahler potential K . Let subscripts of K denote derivatives: K µ ...µ k ≡ ∂ µ . . . ∂ µ k K , so K α ¯ β = g α ¯ β = iω α ¯ β , with g being the metric and ω the K¨ahler form of the manifold.Let us define p α ≡ iK α , and G ± = D ∓ p α D ± φ α , M = iK αβ D + φ α D − φ β . (C.1)We then have G ± = G ± ± M . (C.2)Note that M vanishes for a flat manifold . The definition of p implies thebrackets [ φ α Λ p β ] = δ αβ , [ φ ¯ α Λ p β ] = iω ¯ αα K αβ , (C.3a)in addition to [ φ α Λ φ ¯ β ] = ω α ¯ β . (C.3b)In light of the derivation of the Hamiltonian density in section 6, p α can beunderstood as the conjugate momenta to Φ α , and the brackets (C.3) is thecorresponding Dirac brackets, see (D.25).Let us define linear combinations of χ and χ , to better suit the base(5.9): χ ± = 1 √ χ ± iχ ) . (C.4)The relations between D ± and χ ± are[ D ± , D ∓ ] = 2 ∂ , [ χ ± , χ ∓ ] = − λ , [ D ± , χ ± ] = 2 λ , (C.5a)[ D ± , D ± ] = 0 , [ χ ± , χ ± ] = 0 , [ D ± , χ ∓ ] = 0 . (C.5b)Note that the rules of sesquilinearity give[ D ± a Λ b ] = − χ ∓ [ a Λ b ] , [ a Λ D ± b ] = ( − a ( D ± + χ ∓ ) [ a Λ b ] . (C.6)20e want to prove that G + and G − gives two commuting N = 2 supercon-formal algebras, i. e.[ G ± Λ G ± ] = (2 λ + 2 ∂ + χ + D + + χ − D − ) G ± + λχ χ d , [ G ± Λ G ∓ ] = 0 . (C.7)We first prove that G and G − fulfill the algebra (2.15). In terms of the split(C.2), we then need to prove that[ G ± Λ M ] + [ M Λ G ± ] ± [ M Λ M ] = (2 λ + 2 ∂ + χ + D + + χ − D − ) M , (C.8a)[ G ∓ Λ M ] − [ M Λ G ± ] ∓ [ M Λ M ] = 0 . (C.8b) C.1 Algebra of G ± . The calculation of [ G ± Λ G ± ] is straightforward. We do the calculation for G , the calculation for G − can be deduced by exchanging + and − . We have[ p α Λ G ] = χ − D − p α , [ G
0+ Λ p α ] = ( χ − + D + ) D − p α , (C.9a)[ φ α Λ G ] = χ + D + φ α , [ G
0+ Λ φ α ] = ( χ + + D − ) D + φ α , (C.9b)and ( χ ± + D ∓ )( χ ∓ + D ± ) = − χ ∓ χ ± − D ± D ∓ + 2 ∂ + χ ± D ± − χ ∓ D ∓ , (C.10)so, remembering that ( D ± ) = 0,[ G
0+ Λ G ] = [ G
0+ Λ D − p α ] D + φ α + D − p α [ G
0+ Λ D + φ α ] + (cid:90) = (( χ + + D − )([ G
0+ Λ p α ])) D + φ α + D − p α (( χ − + D + )[ G
0+ Λ φ α ]) + (cid:90) = − χ − χ + G + ((2 ∂ + χ + D + ) D − p α ) D + φ α − χ + χ − G + D − p α ((2 ∂ + χ − D − ) D + φ α ) + (cid:90) = (2 λ + 2 ∂ + χ + D + + χ − D − ) G + (cid:90) . (C.11)21he integral term is given by (cid:90) [ [ G
0+ Λ D − p α ] Γ D + φ α ] d Γ = (cid:90) [ (2 ∂ + χ + D + − χ − χ + ) D − p α Γ D + φ α ] d Γ= − (cid:90) ( χ − χ + + 2 γ ) η + η − δ αα d Γ = i (cid:90) ( χ − χ + + 2 γ ) η η d Γ d − iλ ( χ − χ + + λ ) d λχ χ d . (C.12)To see that G and G − commutes, we note that( χ ± + D ∓ ) = 0 , (C.13)so [ G
0+ Λ D + p α ] = ( χ − + D + )[ G
0+ Λ p α ] = ( χ − + D + ) D − p α = 0 , (C.14a)[ G
0+ Λ D + φ α ] = ( χ + + D − )[ G
0+ Λ φ α ] = ( χ + + D − ) D + φ α = 0 . (C.14b)Thus,[ G
0+ Λ G − ] = [ G
0+ Λ D + p α ] D − φ α + D + p α [ G
0+ Λ D − φ α ] = 0 . (C.15)There is no integral term. C.2 Algebra of G ± and M . Let us define some shorthand notation, and calculate some brackets we aregoing to use later. Let B αβ ≡ D + φ α D − φ β , (C.16)so M can be written M = iK αβ B αβ . (C.17)Let E ± ≡ Γ σαβ K σγ D ± φ γ B αβ . (C.18)Also, note that [ K αβ Λ φ γ ] = i Γ γαβ . (C.19)22 .2.1 The bracket [ M Λ M ] . We want to calculate [ M Λ M ]. Since both the first and second argument ofthe bracket is the same expression, M , we only need to calculate the polesrepresented by an odd number of λ ’s and χ ’s, and from skew-symmetry wecan deduce the full answer. We have[ M Λ M ] = i [ M Λ K αβ ] B αβ + iK αβ [ M Λ B αβ ] + (cid:90) , (C.20)where (cid:82) represents the integral term in the quasi-Lebniz. First term of (C.20) . We start with the first term in (C.20), so we wantto calculate [ M Λ K αβ ]. Now,[ K αβ Λ M ] = i [ K αβ Λ K γδ ] B γδ + iK γδ [ K αβ Λ B γδ ] . (C.21)We then need to calculate [ K αβ Λ B γδ ]:[ K αβ Λ B γδ ] = [ K αβ Λ D + φ γ ] D − φ δ + D + φ γ [ K αβ Λ D − φ δ ] + (cid:90) = i ( D + + χ − )(Γ γαβ ) D − φ δ + iD + φ γ ( D − + χ + )(Γ δαβ ) + (cid:90) . (C.22)The integral term of (C.22) is (cid:90) Λ0 [ [ K αβ Λ D + φ γ ] Γ D − φ δ ] d Γ = (cid:90) Λ0 [ ( D + + χ − )[ K αβ Λ φ γ ] Γ D − φ δ ] d Γ= (cid:90) Λ0 i ( − η − η + )[ Γ γαβ Γ φ δ ] d Γ= − λ [ Γ γαβ Γ φ δ ] . (C.23)From (C.21), using skew-symmetry, we have[ M Λ K αβ ] = χ + Γ δαβ K γδ D + φ γ − χ − Γ γαβ K γδ D − φ δ − λ iK γδ [ Γ γαβ Γ φ δ ] − D + φ γ D − K γδ Γ δαβ − D + K γδ Γ γαβ D − φ δ + . . . , (C.24)where the dots represents terms with no poles, or no odd derivatives, orcontaining only terms where D ± hits holomorphic φ . So, using the notationdefined in (C.18), the first term in (C.20) is χ + iE + − χ − iE − + λ K γδ [ Γ γαβ Γ φ δ ] B αβ + O ( λ ) . (C.25)23 econd term of (C.20) . To calculate the second term of (C.20), we firstcalculate [ B αβ Λ M ] using (C.22):[ B αβ Λ M ] = i [ B αβ Λ K γδ ] B γδ = χ + Γ αγδ D + φ β B γδ − χ − Γ βγδ D − φ α B γδ − λ i [ Γ αγδ Γ φ β ] B γδ + O ( λ ) . (C.26)The second term of (C.20) then is χ + iE + − χ − iE − + λ K γδ [ Γ γαβ Γ φ δ ] B αβ + O ( λ ) . (C.27) Integral term of (C.20) . There will be an integral term in (C.20), givenby i (cid:90) Λ0 [ [ M Λ K αβ ] Γ B αβ ] d Γ . (C.28)Skew-symmetry still guaranties that we only need to calculate the polesrepresented by an odd number of λ ’s and χ ’s. The integral gives at least λ ,and the possible poles then are λ and λχ χ . Higher poles are not possibledue to dimensional arguments. Let M K ≡ [ M Λ K αβ ]. We have[ M K Γ B αβ ] = [ M K Γ D + φ α ] D − φ β + D + φ α [ M K Γ D − φ β ] + (cid:90) . (C.29)The integral term can not be relevant here, since this would give at least a γ ,and the integration in (C.28) would give at least λ , but the highest possiblepower of λ is one. Recall that the only terms surviving the integration is the η + η − -terms.We first calculate the first term in (C.29). We have[ M K Γ D + φ α ] = ( η − + D + )[ M K Γ φ α ] . (C.30)So, we need the η + - and η + η − -part of [ M K Γ φ α ], which can be found bylooking at the corresponding terms of [ φ α Γ M K ]. These, in turn, can befound by using (C.24), and we get[ φ α Γ [ M Λ K αβ ] ] η + ,η + η − = η + [ φ α Γ K γδ ]Γ δαβ D + φ γ = − iη + Γ αγδ Γ δαβ D + φ γ , (C.31)and [ M K Γ D + φ α ] η + η − = − iη − η + Γ αγδ Γ δαβ D + φ γ , (C.32)24o the relevant part of the first term of (C.29) is − iη − η + Γ αγδ Γ δαβ B γβ . (C.33)The second term can be calculated by exchanging + and − , and yields thesame term. In total, the integral terms is (cid:90) Λ0 η − η + Γ αγδ Γ δαβ B γβ d Γ = − iλ Γ αγδ Γ δαβ B γβ . (C.34) In total.
Summing the contributions, and using skew-symmetry, we have[ M Λ M ] = A + χ + iE + − χ − iE − + λ Q = − A + ( χ + + D − )2 iE + − ( χ − + D + )2 iE − + ( λ + ∂ ) Q , (C.35)where A is the part of the bracket with no λ ’s or χ ’s, and Q ≡ ( K γδ [ Γ γαβ Γ φ δ ] − i Γ γαδ Γ δγβ ) B αβ . (C.36)Thus[ M Λ M ] = (2 χ + + D − ) iE + − (2 χ − + D + ) iE − + (2 λ + ∂ ) Q . (C.37)
C.2.2 The bracket [ M Λ G ] . We want to calculate [ M Λ G ] + [ G
0+ Λ M ], and later [ M Λ G − ] − [ G
0+ Λ M ].We start with [ M Λ G ], and the other terms can then be calculated byusing skew-symmetry and by exchanging + and − . Using Leibniz andsesquilinearity, we see that[ M Λ G ] =[ M Λ D − p γ ] D + φ γ + D − p γ [ M Λ D + φ γ ] + (cid:90) =( χ + + D − )([ M Λ p γ ]) D + φ γ + D − p γ ( χ − + D + )([ M Λ φ γ ]) + (cid:90) . (C.38) First part of (C.38) . We first note that[ p γ Λ M ] = i [ p γ Λ K αβ ] B αβ + iK αβ [ p γ Λ B αβ ] , (C.39)with no integral term, and[ p γ Λ B αβ ] = χ + δ βγ D + φ α − χ − δ αγ D − φ β , (C.40)25o [ p γ Λ M ] = i [ p γ Λ K αβ ] B αβ + iχ + K γα D + φ α − iχ − K γα D − φ α . (C.41)Using skew-symmetry, we have[ M Λ p γ ] = − i [ p γ Λ K αβ ] B αβ + i ( χ + + D − )( K γα D + φ α ) − i ( χ − + D + )( K γα D − φ α ) . (C.42)From (C.5), we see that ( χ ± + D ∓ ) = 0, and we note that( χ + + D − )( χ − + D + ) = χ + D + − χ − D − − χ − χ + + D − D + , (C.43)so, the first part of (C.38) is − i ( χ + + D − )([ p γ Λ K αβ ] B αβ ) D + φ γ − i ( χ + D + − χ − D − − χ − χ + + D − D + )( K γα D − φ α ) D + φ γ = − i ( χ + + D − )([ p γ Λ K αβ ] B αβ ) D + φ γ + χ + D + M− χ − χ + M − iχ − D − K αβ B αβ − iD − D + ( K γα D − φ α ) D + φ γ . (C.44) Second part of (C.38) . We have[ φ γ Λ M ] = i [ φ γ Λ K αβ ] B αβ = Γ γαβ B αβ , (C.45)so, the second part of (C.38) is − D − p γ ( χ − + D + )(Γ γαβ B αβ ) . (C.46)In the coordinates chosen, this is( χ − + D + )(Γ γαβ B αβ ) D − p γ − i∂ ([ p γ Λ Γ γαβ ] B αβ ) . (C.47) Integral term of (C.38) . The integral term in (C.38) is given by (cid:90) Λ0 [ [ M Λ D − p γ ] Γ D + φ γ ] d Γ . (C.48)We neeed[ [ M Λ D − p γ ] Γ D + φ γ ] = − ( η − + D + )[ [ M Λ D − p γ ] Γ φ γ ]= − ( η − + D + )[ ( D − + χ + )[ M Λ p γ ] Γ φ γ ]= ( η − + D + )( η + + χ + )[ [ M Λ p γ ] Γ φ γ ] . (C.49)26sing (C.42) we get[ φ γ Γ [ M Λ p γ ] ] = − i [ φ γ Γ [ p γ Λ K αβ ] ] B αβ + i ( D − + η + + χ + )[ φ γ Γ K γα ] D + φ α − i ( D + + η − + χ − )[ φ γ Γ K γα ] D − φ α . (C.50)We have [ φ γ Γ K γα ] = Γ γγα = 0, so[ [ M Λ p γ ] Γ φ γ ] = i [ φ γ Γ [ p γ Λ K αβ ] ] B αβ , (C.51)and [ [ M Λ D − p γ ] Γ D + φ γ ] = iη − η + [ φ γ Γ [ p γ Λ K αβ ] ] B αβ , (C.52)so the quantum correction is λ [ φ γ Γ [ p γ Λ K αβ ] ] B αβ . (C.53)Using Jacobi, this is λ [ p γ Λ [ φ γ Γ K αβ ] ] B αβ = − iλ [ p γ Λ Γ γαβ ] B αβ . (C.54) In total.
Thus, (C.38) is the sum of (C.44), (C.47) and (C.54): χ + D + M − χ − χ + M + χ − (Γ γαβ B αβ D − p γ − iD − K αβ B αβ ) − iχ + ([ p γ Λ K αβ ] B αβ D + φ γ ) + D + (Γ γαβ B αβ ) D − p γ − iD − D + ( K γα D − φ α ) D + φ γ − iD − ([ p γ Λ K αβ ] B αβ ) D + φ γ − ( λ + ∂ )( i [ p γ Λ Γ γαβ ] B αβ ) . (C.55)We have i [ p γ Λ K αβ ] B αβ D + φ γ = iE + , (C.56)Γ γαβ B αβ D − p γ − iD − K αβ B αβ = iE − . (C.57)Also, − iD − D + ( K γα D − φ α ) D + φ γ = − iD + ( D − K αβ B αβ )+ i ∂K αβ B αβ + i K αβ D + φ α ∂D − φ β , (C.58)so [ M Λ G ] = − χ − χ + M + χ + D + M + i ( χ − + D + ) E − − i ( χ + + D − ) E + + 2 i∂K αβ B αβ + 2 iK αβ D + φ α ∂D − φ β − Γ γαβ B αβ D + D − p γ + i [ p γ Λ K αβ ] B αβ D − D + φ γ − ( λ + ∂ )( i [ p γ Λ Γ γαβ ] B αβ ) . (C.59)27 G
0+ Λ M ] , and taking the sum. Using skew-symmetry, from (C.59), wecalculate [ G
0+ Λ M ]:[ G
0+ Λ M ] = χ − χ + M + 2 λ M + 2 ∂ M + χ − D − M + iχ − E − − iχ + E + − i∂K αβ B αβ − iK αβ D + φ α ∂D − φ β + Γ γαβ B αβ D + D − p γ − i [ p γ Λ K αβ ] B αβ D − D + φ γ − λi [ p γ Λ Γ γαβ ] B αβ . (C.60)Taking the sum of (C.59) and (C.60), we have[ G
0+ Λ M ] + [ M Λ G ] =(2 λ + 2 ∂ + χ − D − + χ + D + ) M + i (2 χ − + D + ) E − − i (2 χ + + D − ) E + − (2 λ + ∂ )( i [ p γ Λ Γ γαβ ] B αβ ) . (C.61) C.2.3 Summing the results from C.2.1 and C.2.2
We want to show that (C.8a) is fulfilled. From (C.37) and (C.61), we get[ G
0+ Λ M ] + [ M Λ G ] + [ M Λ M ] = (2 λ + 2 ∂ + χ + D + + χ − D − ) M− (2 λ + ∂ ) (cid:16)(cid:16) − K γδ [ Γ γαβ Γ φ δ ] + i Γ γαδ Γ δγβ + i [ p γ Λ Γ γαβ ] (cid:17) B αβ (cid:17) . (C.62)The parenthesis of the last line is − K γδ [ Γ γαβ Γ φ δ ] + i Γ γαδ Γ δγβ + i [ p γ Λ Γ γαβ ] = i Γ γαδ Γ δγβ − i∂ γ Γ γαβ = 0 . (C.63)This is zero in the coordinates chosen. So, (C.8a) is fulfilled. The corre-sponding equation for the − -sector comes by exchanging + and − . We havethus shown that[ G ± Λ G ± ] = (2 λ + 2 ∂ + χ + D + + χ − D − ) G ± + λχ χ d . (C.64)We now want to show that G + and G − commute.28 .2.4 [ G − Λ M ] − [ M Λ G ]We want to calculate [ G − Λ M ] − [ M Λ G ]. From (C.60) we see that[ G − Λ M ] is[ G − Λ M ] = χ + χ − M + 2 λ M + 2 ∂ M + χ + D + M + iχ + E + − iχ − E − − i∂K αβ B αβ + 2 iK αβ D − φ α ∂D + φ β + Γ γαβ B αβ D − D + p γ − i [ p γ Λ K αβ ] B αβ D + D − φ γ + λi [ p γ Λ Γ γαβ ] B αβ . (C.65)Note that when we exchange + and − , or equivalently 1 and 2 in thenumbering of the supersymmetries, we keep the integration order in theintegrals fixed. This yields an extra minus sign in the quantum term above.The difference between (C.65) and (C.59) is[ G − Λ M ] − [ M Λ G ] = ( χ + χ − + χ − χ + + 2 λ + 2 ∂ ) M + i (2 χ + + D − ) E + − i (2 χ − + D + ) E − − i∂K αβ B αβ − iK αβ D + φ α ∂D − φ β − iK αβ ∂D + φ α D − φ β + Γ γαβ B αβ ( D − D + + D + D − ) p γ − i [ p γ Λ K αβ ] B αβ ( D − D + + D + D − ) φ γ + i (2 λ + ∂ ) ([ p γ Λ Γ γαβ ] B αβ )= + i (2 χ + + D − ) E + − i (2 χ − + D + ) E − + 2 ∂ M − i∂K αβ B αβ − iK αβ ∂B αβ + 2Γ γαβ B αβ ∂p γ − i [ p γ Λ K αβ ] B αβ ∂φ γ + i (2 λ + ∂ ) ([ p γ Λ Γ γαβ ] B αβ ) . (C.66)The third line of (C.66) can be simplified, notingΓ γαβ B αβ ∂p γ − i [ p γ Λ K αβ ] B αβ ∂φ γ = i Γ γαβ B αβ K γσ ∂φ σ + i Γ γαβ B αβ K γ ¯ σ ∂φ ¯ σ − i Γ σαβ K σγ B αβ ∂φ γ + iK αβγ B αβ ∂φ γ = iK αβ ¯ γ B αβ ∂φ ¯ γ + iK αβγ B αβ ∂φ γ = i∂K αβ B αβ . (C.67)So, finally,[ G − Λ M ] − [ M Λ G ] = i (2 χ + + D − ) E + − i (2 χ − + D + ) E − + i (2 λ + ∂ ) ([ p γ Λ Γ γαβ ] B αβ ) , (C.68)29nd, using (C.37) and (C.63),[ G − Λ M ] − [ M Λ G ] − [ M Λ M ] =+ (2 λ + ∂ ) (cid:16)(cid:16) − K γδ [ Γ γαβ Γ φ δ ] + i Γ γαδ Γ δγβ + i [ p γ Λ Γ γαβ ] (cid:17) B αβ (cid:17) = 0 . (C.69)Thus [ G + Λ G − ] = 0 . (C.70) D Derivation of the Hamiltonian of the N = (2 , supersymmetric sigma model The action for an N = (2 ,
2) supersymmetric sigma model with a K¨ahlertarget manifold is given by S = (cid:90) dσdτ dθ dθ − dθ dθ − K (Φ , ¯Φ) , (D.1)where K is the K¨ahler potential, and Φ = { Φ α } is a chiral superfield, and¯Φ = { Φ ¯ α } is an anti-chiral superfield. We use indicies µ, ν, . . . to denote realcoordinates, and α, β, . . . to denote complex coordinates.We have two copies of the N = (1 ,
1) algebra :( D i ± ) = i∂ ± , { D i + , D j − } = 0 , { D ± , D ± } = 0 , i, j = 1 , , (D.2)where D i ± = ∂∂θ i ± + iθ i ± ∂ ± , ∂ ± = ∂ ± ∂ . (D.3)For chiral and anti-chiral superfields, the two supersymmetries are related: D ± Φ α = iD ± Φ α , D ± Φ ¯ α = − iD ± Φ ¯ α . (D.4)In physics literature D i ± are typically combined into complex operators andalso it is more customary to use the the complex θ ’s (for example, see theconventions in [16]). Here we misuse the spinor notation. For example, the partial derivative ∂ + should beunderstood as ∂ ++ in spinor indices. Since we are after the Hamiltonian treatment, theLorentz covariance is not the issue.
30e now want to go to Hamiltonian formalism. We are going to integrateout two odd coordinates, and write (D.1) in a first order form.Let us define a new set of θ ’s: θ θ θ θ = 1 √ − i i − i i − i − ii − i − θ θ − θ θ − . (D.5)The action then is S = − (cid:82) dσdτ dθ dθ dθ dθ K . We now want to integrateout θ and θ . Introduce new differential operators: D = √ ( D + D − ) , D = √ ( D + D − ) , (D.6) D = √ ( D + iD − ) , D = √ ( D + iD − ) . (D.7)We have D = ∂∂θ + iθ ∂ + 12 ( iθ − θ ) ∂ + 12 ( iθ + θ ) ∂ , (D.8) D = ∂∂θ + iθ ∂ + 12 ( iθ − θ ) ∂ + 12 ( iθ + θ ) ∂ , (D.9)and ( D ) = ( D ) = i∂ , ( D ) = ( D ) = i∂ . (D.10)Under integration, S = − (cid:90) dσdτ dθ dθ D D K | θ = θ =0 . (D.11)Now, D D K = K µν D Φ µ D Φ ν + K µ D D Φ µ . (D.12)Due to (D.4), we have D Φ α = − iD Φ α , D Φ ¯ α = + iD Φ ¯ α , (D.13)and K µ D D Φ µ = − iK α D D Φ α + iK ¯ α D D Φ ¯ α = K α ∂ Φ α − K ¯ α ∂ Φ ¯ α = 2 K α ∂ Φ α + total derivative. (D.14)31lso, K µν D Φ µ D Φ ν = K µα D Φ µ D Φ α + K µ ¯ α D Φ µ D Φ ¯ α = − iK ¯ βα D Φ ¯ β D Φ α + iK β ¯ α D Φ β D Φ ¯ α = − iK ¯ βα D Φ ¯ β D Φ α . (D.15)Using (D.4) again, we note that D Φ α = √ ( D + D − )Φ α = √ ( D + iD − )Φ α = D Φ α , (D.16) D Φ ¯ α = √ ( D + D − )Φ ¯ α = √ ( − iD + D − )Φ ¯ α = − iD Φ ¯ α , (D.17)so, K µν D Φ µ D Φ ν = − K ¯ βα D Φ ¯ β D Φ α , (D.18)and S = (cid:90) d σdθ dθ (cid:16) K ¯ βα D Φ ¯ β D Φ α − K α ∂ Φ α (cid:17) (cid:12)(cid:12)(cid:12) θ = θ =0 . (D.19)Denote θ ≡ √ iθ , θ ≡ √ iθ and ∂ ≡ ∂ . Let D ≡ − i √ iD | θ = θ =0 = ∂∂θ + θ ∂ , (D.20) D ≡ − i √ iD | θ = θ =0 = ∂∂θ + θ ∂ , (D.21)and φ µ ≡ Φ µ | θ = θ =0 . Then S = (cid:90) d σdθ dθ (cid:16) iK α ∂ φ α − K ¯ βα D φ ¯ β D φ α (cid:17) = (cid:90) d σdθ dθ (cid:18) iK α ∂ φ α − H (cid:19) , (D.22)with H = K ¯ βα D φ ¯ β D φ α − K ¯ βα D φ ¯ β D φ α . (D.23)From (D.22) we see that the Hamiltonian density is given by (D.23). Themomenta is p α = iK α , p ¯ α = 0 . (D.24)The definitions of the momentas give the second class constraints p α − iK α = 0and p ¯ α = 0, leading to the Dirac brackets { φ α , φ ¯ β } ∗ = ω α ¯ β , { φ α , p β } ∗ = δ αβ , { φ ¯ α , p β } ∗ = iω ¯ αα K αβ , (D.25)32ith the remaining brackets being zero.Let us define new combinations of the derivatives D and D : D ± ≡ √ D ∓ iD ) . (D.26)We can then write the Hamiltonian (D.23) as H = G c − − G c + , (D.27)with G c ± = D ∓ p α D ± φ α + iK αβ D ± φ α D ∓ φ β . (D.28) References [1] F. Malikov, V. Schechtman, and A. Vaintrob, “Chiral de RhamComplex,”
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