Logarithmic modules for chiral differential operators of nilmanifolds
aa r X i v : . [ m a t h . QA ] J un LOGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORSOF NILMANIFOLDS
BELY RODRIGUEZ MORALESA bstract . We describe explicitly the vertex algebra of (twisted) chiral di ff eren-tial operators on certain nilmanifolds and construct their logarithmic modules.This is achieved by generalizing the construction of vertex operators in terms ofexponentiated scalar fields to Jacobi theta functions naturally appearing in thesenilmanifolds. This provides with a non-trivial example of logarithmic vertex alge-bra modules, a theory recently developed by Bakalov.
1. I ntroduction
To any smooth manifold M (and a choice of ω ∈ H ( M , Z )) satisfying some mildtopological properties (the first Pontryagin class vanishes), Malikov, Schechtmannand Vaintrob [1] and independently Beilinson and Drinfeld [2] attach a sheaf ofvertex algebras O chM called the sheaf of chiral di ff erential operators . In the simplestcase when ω =
0, locally on a coordinate patch U with coordinates n x i o i = ,..., dim M ,the sections O chM ( U ) form an dim M -dimensional βγ -system, i.e., the vertex algebragenerated by fields n β i , γ i o i = ,..., dim M satisfying the OPE β i ( z ) · γ j ( w ) ∼ δ i , j z − w , β i ( z ) · β j ( w ) ∼ γ i ( z ) · γ j ( w ) ∼ . On intersections of coordinate patches, the fields γ i change as coordinates do whilethe fields β i change as vector fields.This construction works in the algebraic, holomorphic, real-analytic or C ∞ -setting.In this work we will be mainly concerned with the C ∞ -setting. Little is known aboutthe structure of the global sections V M = Γ ( M , O chM ) of this sheaf. Only recently inthe context of supermanifolds, Bailin Song proved that, in the holomorphic setting,the vertex algebra V M coincides with the simple small N = c = M = T ∗ [1] N is the shifted cotangent bundle to a K N [3]. In this work we will describe explicitly V M when M is the Heisenberg3-dimensional nilmanifold.The vertex algebra V M (or rather its super-extension) is expected to play a centralrole in Mirror-Symmetry. In particular, for M and N a mirror pair of Calabi-Yaumanifolds, one expects a natural isomorphism V M ≃ V N of vertex algebras. Theircharacters are known to be equal by work of Borisov and Libgober [4].If M is non-simply-connected, a subtle phenomenon arises as one needs to con-sider non-trivial windings . Aldi and Heluani showed in [5], based on ideas of C.Hull [6], that when ( M , ω ) is the three torus T with its generator of H ( T , Z ) ≃ Z , Mathematics Subject Classification.
Key words and phrases. chiral di ff erential operators, vertex algebra, logarithmic module, logarith-mic quantum field. or if M is its mirror dual: the Heisenberg 3-dimensional nilmanifold N with van-ishing ω , the vertex algebra V M can be naturally represented in a Hilbert space.This Hilbert space is associated to a 6-dimensional nilmanifold Y which fibers overboth T and N .For certain M , we can describe explicitly V M in terms of a larger manifold fiberingover M . Suppose that the ω -twisted Courant algebroid TM ⊕ T ∗ M of M is paral-lellizable. That is, there exists a global frame of vector fields (cid:8) β i (cid:9) and dual basis ofdi ff erential forms n α i o such that [ β i , β j ] Lie + ι β i ι β j ω is a constant combination of β i ’sand α i ’s, and Lie β i α j is a constant linear combination of the α i ’s. In other words,there exists a Lie algebra g , dim g = M , with a symmetric invariant bilinearpairing of signature (dim M , dim M ), and a trivialization TM ⊕ T ∗ M ≃ g × M . TheCourant-Do ff man bracket of the frame n β i , α i o is given by the bracket in g .The approach, following ideas of C. Hull [6] and exploited for example in [7] isthat one may try to find a manifold N with the property that dim N = M . Itfibers over M , N ։ M and its parallelizable, such that TN ≃ g × N , that is, the Liebracket of vectors in a frame is identified with the Lie bracket of g . In this situationwe consider the g -module C ∞ ( N ), the Kac-Moody a ffi nization ˆ g of g and its inducedmodule from C ∞ ( N ). We have an embedding C ∞ ( M ) ֒ → C ∞ ( N ) given by pullback,inducing the sequence of embeddings: V ( g ) ⊂ Ind ˆ g ˆ g + C ∞ ( M ) ⊂ H = Ind ˆ g ˆ g + C ∞ ( N ) , where the first module is induced from the constant function 1, coincides with thevacuum module for the algebra ˆ g and is known to be a vertex algebra. The secondmodule coincides with the vertex algebra V M and is here represented as a subspaceof H .Let G be the unipotent Lie group with Lie algebra g . It is also an extension of R by R . Let Γ ⊂ G be the subgroup generated by a basis of the quotient R of G . Itis a cocompact subgroup, the quotient Y = G / Γ is a six-dimensional nilmanifoldwhich is a non-trivial T -fibration over T . In fact we have the central extensions:(1.1) 0 → R → G → R → → Z → Γ → Z → Y as a T = R / Z fibration over T .The Lie group G acts on L ( Y ) and its Lie algebra g acts on C ∞ ( Y ). We extend the g = g ⊗ t ⊂ ˆ g = g [ t , t − ] ⊕ C K action on C ∞ ( Y ) into a representation of ˆ g + = g [ t ] ⊕ C K by letting K act by 1 and a n act by 0 if n ≥
0. The vector space H is the correspondingˆ g -induced module H = Ind ˆ g ˆ g + C ∞ ( Y ).Properly speaking H is its L completion, but we will not care about unitarityproperties in this work.Notice that the constant function 1 defines an embedding V ( g ) ֒ → H of thevacuum representation of ˆ g into H . As it is well known V ( g ) is a vertex algebraand this embedding makes H into a V ( g )-module. Consider now the three Torus T = R / Z , the morphism Y ։ T provides an embedding C ∞ ( T ) ֒ → C ∞ ( Y ).It is easy to see that this is an embedding of g -modules. The induced ˆ g -module OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 3 coincides with V T , that is V ( g ) ⊂ V T ≃ Ind ˆ g ˆ g + C ∞ ( T ) ⊂ H . However a little work is required to check that H is a vertex algebra module over V T . The fields associated to vectors f ∈ C ∞ ( T ) involve explicitly logarithms of theformal variable z . The situation is very similar to that of the lattice vertex algebrawhere the logarithms only appear exponentiated, hence appealing to the identityexp(log( z )) = z one can get rid of them.The situation with the Heisenberg nilmanifold is a quite di ff erent. Any line L ⊂ R determines a central character χ L : Z ( G ) ≃ R → R of G . We can view L as a onedimensional subgroup of G (in the quotient R ). The subgroup K = ker χ L ⊕ L ⊂ G is normal and its cokernel 0 → K → G → Heis ( R ) → , is the 3-dimensional real Heisenberg group. If the line L is generated by an elementof Γ , this sequence is compatible with Γ in the sense that there exists an analogoussequence 0 → K Γ → Γ → Heis ( Z ) → , whose quotient is now the integer Heisenberg group. This construction shows Y as a fibration over the Heisenberg nilmanifold N = Heis ( R ) / Heis ( Z ) (it is not hardto see that the fiber is also a three torus T ).We obtain thus an embedding C ∞ ( N ) ֒ → C ∞ ( Y ). As before it is easy to see thatthis is an embedding of g -modules. It turns out that the induced ˆ g -module is alsoisomorphic to the vertex algebra V N : V ( g ) ⊂ V N ≃ Ind ˆ g ˆ g + C ∞ ( N ) ⊂ H . This time however, logarithms are unavoidable. In fact, the fields associated tovectors of V N have explicit logarithms of z on them when acting on H . It isonly by restricting to V N ⊂ H that they disappear by use of the same identityexp(log( z )) = z . However, when analyzing the action of V N on H these logarithmsremain, making H a logarithmic module over V N .In order to describe explicitly the fields of V N and their action on H , we need to usecertain results from harmonic analysis. In particular, since the representation of G in L ( Y ) is unitary, it decomposes into direct sum of irreducible representations. Theserepresentations turn out to be induced from unitary irreducible representations ofthe real Heisenberg group, and by the Stone-von Neumann theorem they areunique once we choose a central character. One can choose explicit cyclic vectorsfor these representations: they are given by appropriate constant (the vacuumvector), exponential (functions from T ), or Jacobi theta functions. We constructvertex operators associated to these Jacobi theta functions in complete analogy ashow one constructs vertex operators associated to exponential functions. Theseoperators however, carry an explicit dependency on the logarithm of the formalvariable. We show by explicit computation the locality and translation invariantproperty as well as the axioms for a logarithmic module as in [8]. The main resultsof this work are: Theorem 1.1. H has the structure of V T -module. Theorem 1.2. H has the structure of logarithmic V N -module. MORALES B. R.
2. Q uantum F ields and V ertex A lgebras Let V be a vector space over C , the quantum fields on V are defined as Field ( V ) = Hom ( V , V (( z ))) where V (( z )) = V [[ z ]][ z − ] denotes the space of Laurent series on V ;i.e. a field on V is a formal series a ( z ) = P n ∈ Z a ( n ) z − − n where a ( n ) ∈ End ( V ) and foreach v ∈ V , a ( n ) v = n large enough.Two quantum fields a ( z ) , b ( z ) are called local if there is N ∈ N such that(2.1) ( z − z ) N [ a ( z ) , b ( z )] = . Lemma . [9] Let a ( z ) , b ( z ) , c ( z ) be pairwise local fields on V then a ( z ) ( n ) b ( z ) , ∂ z a ( z ) , b ( z ) , c ( z ) n ∈ Z are also pairwise local fields.The n -product of two local fields is defined as(2.2) (cid:16) a ( z ) ( n ) b ( z ) (cid:17) ( z ) v = ∂ ( N − − n ) z (cid:16) ( z − z ) N a ( z ) b ( z ) v (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) z = z = z for v ∈ V , n < N , and (cid:16) a ( z ) ( n ) b ( z ) (cid:17) ( z ) v = n ≥ N .Given and operator A we will use the notation A ( k ) = A k k ! .Given a field a ( z ) = P n ∈ Z a ( n ) z − − n the annihilation and creation parts of a ( z ) aredefined respectively as: a ( z ) − = X n ≥ a ( n ) z − − n , a ( z ) + = X n ≤− a ( n ) z − − n ;The normally ordered product of two fields a ( z ) , b ( z ) is defined by: a ( z ) b ( z ) : = a ( z ) + b ( z ) + b ( z ) a ( z ) − . Definition . A vertex algebra is a vector space V , a distinguished vector ∈ V and linear map Y z : V → Field ( V ) , v Y ( v , z ) , such that the following axioms are satisfied:(vacuum axiom) Y ( , z ) = id , Y ( v , z ) ∈ V [[ z ]], Y ( v , z ) | z = = v ;(translation invariance) [ T , Y ( v , z )] = ∂ z Y ( v , z );(locality axiom) For every v , v ∈ V there is N large enough such that( z − z ) N (cid:2) Y z ( v ) , Y z ( v ) (cid:3) = . Where the translation endomorphism T ∈ End ( V ) is defined as Tv = ∂ z Y ( v , z ) | z = . Remark:
There are several equivalent approach to define vertex algebra [10].Following [11] let g be a Lie algebra with a non degenerate symmetric invariantbilinear form h· , ·i : g × g → C , for instance, every finite dimensional semisimpleLie algebra has such bilinear form. The Kac-Moody a ffi ne Lie algebra b g is definedas vector space by b g = g [ t , t − ] ⊕ C K with the commutator[ at m , bt n ] = [ a , b ] t m + n + m h a , b i δ m , − n K , where K is central. Let us introduce the notation a n = at n .Consider the subalgebra of the Kac-Moody a ffi ne algebra given by g [ t ] ⊕ C K andits one dimensional representation C1 , where K acts by multiplication by a givenscalar k and the elements of g [ t ] acts by zero. OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 5
Proposition . [9] The b g module V k ( g ) = Ind b gg [ t ] ⊕ C K ≃ U (cid:16)b g (cid:17) ⊗ U ( g [ t ] ⊕ C K ) C1 has a vertex algebra structure.The map Y z : V k ( g ) → Field (cid:16) V k ( g ) (cid:17) is defined as Y ( a − , z ) = X n ∈ Z a n z − − n , a ∈ g , and in general for the generators of V k ( g )(2.3) Y ( a − n k · · · a r − n kr , z ) = : ∂ ( n k − z Y ( a − , z ) · · · ∂ ( n kr − z Y ( a r − , z ) : . This vertex algebra V k ( g ) is known as the universal a ffi ne vertex algebra of level k oras the Kac-Moody vertex algebra of level k . Definition . A module over a vertex algebra V is a vector space W equipped witha linear map Y z : V → Field ( W ) such that: • Y ( ) = id • Y ( a ( n ) b ) = Y ( a ) ( n ) Y ( b ) for all n ∈ Z .3. L ogarithmic Q uantum F ields and L ogarithmic M odules Logarithmic Fields.
It is convenient to extend the notion of quantum fieldsdefined before to include logarithms, i.e., it is often needed to have the notion oflogarithm in the formal theory of fields, in this section the basic results of logarith-mic quantum fields will be stated following the ideas developed by Bojko Bakalovin [8]. Let us start by introducing the formal variable log( z ) which intuitively canbe thought as the logarithm of z . Since there are now two formal variables we havetwo possible derivations D z = ∂ z + z − ∂ log( z ) , D log( z ) = z ∂ z + ∂ log( z ) . Notice that here we are using the derivatives D z and D log( z ) instead of ∂ z and ∂ log( z ) because the former derivations carry formally the data coded in the analyticequation " ∂ z log( z ) = z " while the latter derivations do not.Let W be a vector space over C , let α ∈ C / Z and define LField α ( W ) = Hom (cid:0) W , W [log( z )][[ z ]] z − α (cid:1) , the space of logarithmic quantum fields on W is defined to be LField ( W ) = M α ∈ C / Z LField α ( W ) . The logarithmic fields will be denoted as a ( z ) instead of a (log( z ) , z ) when no confu-sion arise. Definition . Two logarithmic fields a ( z ), b ( z ) are local if for N >> z − z ) N [ a ( z ) , b ( z )] = . MORALES B. R.
Definition . The n -product of two local logarithmic fields a ( z ) and b ( z ) is definedas(3.2) (cid:16) a ( z ) ( n ) b ( z ) (cid:17) ( z ) w = D ( N − n − z (cid:16) ( z − z ) N a ( z ) b ( z ) w (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) z = z = z for w ∈ W and n < N . For n ≥ N the n -product is defined by (cid:16) a ( z ) ( n ) b ( z ) (cid:17) = Remark:
Even when the expression log( z ) is a formal variable we can define formallylog (cid:0) xy (cid:1) = log ( x ) + log (cid:0) y (cid:1) , (3.3) log xy ! = log ( x ) − log (cid:0) y (cid:1) , (3.4) log (1 − x ) = − X n > x n n , (3.5)therefore the expression log ( z − z ) may be interpreted in the following way:log ( z − z ) = log ( z ) + log (cid:18) − z z (cid:19) = log ( z ) − X n > z − n z n n . It is easy to derive the following properties from the Leibniz rule( D z a ) ( n ) b = − na ( n − b , (3.6) D z (cid:16) a ( n ) b (cid:17) = ( D z a ) ( n ) b + a ( n ) ( D z b ) , (3.7) (cid:16) ∂ log( z ) a ( n ) b (cid:17) = (cid:16) ∂ log( z ) a (cid:17) ( n ) b + a ( n ) (cid:16) ∂ log( z ) b (cid:17) . (3.8)Once again there is a Dong’s Lemma for logarithmic fields: Lemma . Let a ( z ) , b ( z ) , c ( z ) be pairwise local logarithmic fields then a. a ( z ) ( n ) b ( z ) and c ( z ) are local fields for all n ∈ Z , b. D z a ( z ), b ( z ) and D log( z ) a ( z ) are pairwise local. Proof.
The part a is proven in [8], for the part b just notice that D z a ( z ) = a ( z ) ( − id , D log( z ) = zD z then use part a . (cid:3) In order to define the normally ordered product for logarithmic fields some extrastep is required, for α ∈ C / Z select a representative α such that − < Re ( α ) ≤ a ( z ) ∈ LField α ( W ) can be uniquely expressed as a ( z ) = X n ∈ Z a n (cid:0) log( z ) (cid:1) z − n − α , where for every w ∈ W holds a n (cid:0) log( z ) (cid:1) w = n >>
0. The annihilation andcreation parts of a ( z ) are defined: a ( z ) − = X n ≥ a n (cid:0) log( z ) (cid:1) z − n − α , a ( z ) + = X n ≤ a n (cid:0) log( z ) (cid:1) z − n − α , and this concepts can be extended linearly to LField ( W ); then the normally orderedproduct of logarithmic fields is defined by the usual formula: a ( z ) b ( z ) : = a ( z ) + b ( z ) + b ( z ) a ( z ) − . OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 7
The propagator of two logarithmic fields a ( z ), b ( z ) is defined as P ( a , b ; z , z ) = [ a ( z ) − , b ( z )] = a ( z ) b ( z ) − : a ( z ) b ( z ) : . The propagator can be used to compute the n -products [8]: Proposition . If a ( z ), b ( z ) are local logarithmic fields then the n -product for n ≥ (cid:16) a ( z ) ( n ) b ( z ) (cid:17) ( z ) w = D ( N − n − z (cid:16) ( z − z ) N P ( a , b ; z , z ) w (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) z = z = z where N is large enough such that the equation 3.1 holds and n < N .3.2. Logarithmic Modules.
Logarithmic modules over vertex algebras are a gen-eralization of the usual notion of modules over vertex algebras but allowing loga-rithmic quantum fields, formally we define:
Definition . A logarithmic module over a vertex algebra V is a vector space W equipped with a linear map Y z : V → LField ( W ) such that: • Y ( ) = id • Y ( a ( n ) b ) = Y ( a ) ( n ) Y ( b ) for all n ∈ Z .Moreover, if V is a vertex algebra equipped with an automorphism ϕ and W is alogarithmic module over V such that Y ( ϕ a ) = e π iD log( z ) Y ( a ) holds for every a ∈ V ,then W is called a ϕ -twisted logarithmic module [8].Let W be a vector space and let W ⊆
LField ( W ) be a collection of logarithmicfields which are pairwise local, denote by W the smallest C [ D log( z ) ] submoduleof LField ( W ) containing W ∪ { id } and closed under n -products; then, because ofProposition 3.1, W is again a collection of pairwise local logarithmic fields. Theorem 3.3 (Bakalov) . Let W be a vector space and W be defined as above. Then W with the n-product of logarithmic fields has the structure of vertex algebra where thevacuum vector is id and the translation operator is D z . From this it becomes clear that W is a logarithmic module over W , just take themap Y : W →
LField ( W ) to be the inclusion map; moreover, it is e π iD log( z ) -twistedmodule. Corollary . Let V be a vertex algebra and W a vector space, then giving a loga-rithmic V -module structure on W is equivalent to give a vertex algebra morphism V → W for a local collection of logarithmic fields W ⊆
LField ( W ), moreover, if V is equipped with an automorphism ϕ then the module will be twisted if and onlythe associated vertex algebra morphism transforms ϕ into e π iD log( z ) .4. F unctions on the D ouble T wisted T orus Double twisted torus.
Let V be a 3-dimensional real vector space and consider G the extension 0 / / ∧ V / / G / / V / / v , ζ ) ( v ′ , ζ ′ ) = ( v + v ′ , ζ + ζ ′ + v ∧ v ′ ) , v , v ′ ∈ V , ζ, ζ ′ ∈ ∧ V MORALES B. R. making G into a group. Using a coordinate system n x i , x ∗ i o , i = , ,
3, where n x i o are the coordinates on the canonical basis n e i o of V and n x ∗ i o are coordinates on thebasis n e ∗ i = ǫ ijk e j ∧ e k o of ∧ V , this product translates as (cid:16) x i , x ∗ i (cid:17) (cid:16) y i , y ∗ i (cid:17) = (cid:18) x i + y i , x ∗ i + y ∗ i + ǫ ijk x j y k (cid:19) , where ǫ ijk denotes the totally antisymmetric tensor. The double twisted torus Y isdefined as the quotient of G modulo the subgroup Γ generated by e i , i = , , V ≃ R . The tangent bundle TY is trivialized by the left invariantvector fields of G : α i = ∂ x ∗ i , β i = ∂ x i − ǫ ijk x j ∂ x ∗ k , being h β i , β j i = ǫ ijk α k the only non trivial commutators, therefore they span a Liealgebra g ; moreover this Lie algebra is equipped with a non degenerate symmetricinvariant bilinear form D β i , α j E = δ i , j . Now consider the space of polynomials C [ x i , x ∗ i ] which is a g -module via therestriction of the action on C ∞ ( G ), let b g = g [ t , t − ] ⊕ C K be the a ffi ne Kac-Moody Liealgebra associated to g and extend the action for the elements a n = at n , a ∈ g , n ≥ K act as the identity, then define the b g -module(4.1) H = Ind b gg [ t ] ⊕ C K C [ x i , x ∗ i ] . Notice that H has naturally the structure of V ( g )-module.Let’s start by defining some operators on H that will be useful later, particularlywhen we try to fit H into an algebraic structure:Define the operators x in : = − n α in for n ,
0, note that those operators commutewith each other, define x i acting on an element of C [ x i , x ∗ i ] as f x i f , impose that h x jn , x i i = h β j , n , x i i = δ i , j δ n , K ; therefore x i can be extended to H .Define the operators W i : = α i on H for i = , , P i : = β i , + ǫ ijk x j W k − ǫ ijk X m mx j − m x km , note that since all the α in commute with each other then all the x in and W j commutewith each other. Also define the operators x ∗ i , n : = − n β i , n + ǫ ijk x jn W k − ǫ ijk X m mx jn − m x km , n , , the operators x ∗ i , will be defined acting on functions as f x ∗ i f with the commu-tation relations h x ∗ i , , x jn i = h x ∗ i , , x ∗ j , i = h x ∗ i , , β j , n i = ǫ ijk x kn , h W j , x ∗ i , i = δ i , j K . Remark:
The operators P i and x ∗ i , n are well defined because even when the sumappearing in the last term runs over the integers it is actually finite since x im acts by OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 9 zero for m big enough.It would be convenient to compute explicitly for later reuse all the commutatorsof the previously defined operators. It is obvious that h α jm , x in i = , (4.2) h β j , m , x in i = δ i , j δ n , − m K , (4.3) h α im , W j i = , (4.4) h β j , m , W i i = , (4.5) h α jm , x ∗ i , n i = δ i , j δ n , − m K (4.6) h α jm , P i i = h P i , W j i = . (4.8)For β jm and x ∗ i , n with n , h β j , m , x ∗ i , n i = − n h β j , m , β i , n i − ǫ ipq n h β j , m , x pn W q i + ǫ ipq n X s s h β j , m , x pn − s x qs i = ǫ ijk n α kn + m − ǫ ipq n δ j , p δ m , − n W q + ǫ ipq n X s s h β j , m , x pn − s i x qs + ǫ ipq n X s sx qs h β j , m , x pn − s i = ǫ ijk n α kn + m − ǫ ijk n δ m , − n W k + ǫ ipq n X s s δ j , p δ m , s − n x qs + ǫ ipq n X s s δ j , q δ m , − s x pn − s = ǫ ijk n α kn + m − ǫ ijk n δ m , − n W k + ǫ ijk n ( m + n ) x km + n + ǫ ijk n mx km + n = (1 − δ m , − n ) ǫ ijk n α kn + m + δ m , − n ǫ ijk n α kn + m − ǫ ijk n δ m , − n W k + (1 − δ m , − n ) ǫ ijk n ( m + n ) x km + n + (1 − δ m , − n ) ǫ ijk n mx km + n + δ m , − n ǫ ijk n mx km + n = (1 − δ m , − n ) (cid:16) ǫ ijk n α kn + m + ǫ ijk n ( m + n ) x km + n + ǫ ijk n mx km + n (cid:17) + δ m , − n (cid:16) ǫ ijk n α kn + m − ǫ ijk n W k + ǫ ijk n mx km + n (cid:17) = (1 − δ m , − n ) (cid:16) − ǫ ijk n ( m + n ) x kn + m + ǫ ijk n ( m + n ) x km + n + ǫ ijk n mx km + n (cid:17) + δ m , − n (cid:16) ǫ ijk n α k − ǫ ijk n W k − ǫ ijk x km + n (cid:17) = − (1 − δ m , − n ) ǫ ijk x km + n − δ m , − n ǫ ijk x km + n = − ǫ ijk x km + n , notice that x ∗ i , was defined in a way such that the previous formula is also satisfied.For β jm and P i the bracket is computed as follows h β j , m , P i i = h β j , m , β i i + ǫ ipq h β j , m , x p W q i − ǫ ipq X n n h β j , m , x p − n x qn i = − ǫ ijk α km + ǫ ipq h β j , m , x p i W q − ǫ ipq X n n h β j , m , x p − n i x qn − ǫ ipq X n nx p − n h β j , m , x qn i = − ǫ ijk α km + ǫ ipq δ j , p δ m , W q − ǫ ipq X n n δ j , p δ m , n x qn − ǫ ipq X n n δ j , q δ m , − n x p − n = − ǫ ijk α km + ǫ ijk δ m , W k − ǫ ijk m x km + ǫ ikj m δ m , − n x km = − ǫ ijk α km + ǫ ijk δ m , W k − ǫ ijk mx km = ǫ ijk (1 − δ m , )( − α km − mx km ) + ǫ ipk z δ m , ( − α k + W k ) = ǫ ijk (1 − δ m , )( mx km − mx km ) + ǫ ipk z δ m , ( − W k + W k ) = . Similarly, the remaining commutators can be computed obtaining: h β j , m , x ∗ i , n i = − ǫ ijk x km + n , (4.9) h β j , m , P i i = , (4.10) h P i , P j i = − ǫ ijk W k , (4.11) h x ∗ i , n , x ∗ j , i = ǫ ijk n x kn , (4.12) h x i , m , x ∗ j , n i = m + n mn ǫ ijk + m ǫ ijk W k δ m , − n . (4.13)Let us define the fields α i ( z ) = X n α in z − − n ,β i ( z ) = X n β i , n z − − n , and the logarithmic fields x i ( z ) = W i log( z ) + X n ∈ Z x in z − n , x ∗ i ( z ) = P i log( z ) + X n ∈ Z x ∗ in z − n + log( z )2 ǫ ijk W j x k ( z ) , Note that following di ff erential equation holds(4.14) D z x i ( z ) = α i ( z ) , it will be useful to find a similar equation for the derivative of x ∗ i ( z ). The followingequation can be obtained by straightforward computation:(4.15) D z x ∗ i ( z ) = β i ( z ) + ǫ ijk x j ( z ) D z x k ( z ) . OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 11
5. M odules and the D ouble T wisted T orus Fibration over the 3 Torus.
As it was noticed before, the group G acts on L ( G / Γ ) as left translations and therefore the Lie algebra g acts on smooth functionson G / Γ as left invariant vector fields, i.e., g acts on a dense subspace of L ( G / Γ );so similarly to the case of polynomials it is possible to obtain a b g -module out of itinducing H = Ind b gg [ t ] ⊕ C K C ∞ ( G / Γ ) , where as usual α in and β i , n act as zero when n > K acts as the identity.Every function f ∈ C ∞ ( G / Γ ) can be interpreted as a rapidly decreasing smoothfunction in six variables f = f ( x i , x ∗ i ) invariant under the action of Γ on the right,i.e. for every ( γ i , γ ∗ i ) ∈ Γ holds f ( x i , x ∗ i ) = f ( x i + γ i , x ∗ i + γ ∗ i + ǫ ijk x j γ k ). Such functionscan be decomposed in a Fourier series with respect to the orthonormal system n e π i ω i x ∗ i o ω ∈ Z as: f ( x i , x ∗ i ) = X ω ∈ Z e π i ω i x ∗ i f ω ( x i ) , where f ω satisfies f ω ( x i + γ i ) = e − π i ǫ ijk ω i x j γ k f ω ( x i ).Define C ω = n e π i ω i x ∗ i f ; f : R → C , f ( x i + γ i ) = e − π i ǫ ijk ω i x j γ k f ( x i ) o , then L ( G / Γ ) ≃ M ω ∈ Z C ω , specifically for ω = C = n f : f ( x i + γ i ) = f ( x i ) o = M ρ ∈ Z C e π i ρ i x i . Define also V T = Ind b gg [ t ] ⊕ C K C = Ind b gg [ t ] ⊕ C K M ρ ∈ Z C e π i ρ i x i = M ρ ∈ Z Ind b gg [ t ] ⊕ C K C e π i ρ i x i . Geometrically we have a T fibration over the T T / / G / Γ / / T , and embeddings C ⊂ C ∞ ( T ) ⊂ C ∞ ( G / Γ ) , which leads to V ( g ) ⊂ V T = Ind b gg [ t ] ⊕ C K C ∞ ( T ) ⊂ H = Ind b gg [ t ] ⊕ C K C ∞ ( G / Γ ) . We know V ( g ) is a vertex algebra, i.e., it is the Kac-Moody vertex algebra of level1, it turns out V T is a vertex algebra too. Theorem 5.1. V T is a vertex algebra. The vertex algebra V T is called the vertex algebra of chiral di ff erential operators associated to T whose existence is proven abstractly in [1], however it would beconvenient to give a prove of the theorem computing explicitly the quantum fields. Proof.
Define the vacuum vector as the constant 1 function, and consider the statefield correspondence map Y : V T → Field ( V T ) e π i ρ i x i : e π i ρ i x i ( z ) : = e π i ρ i x i z π i ρ i W i exp π i ρ i X n < x in z − n exp π i ρ i X n > x in z − n ,α i − α i ( z ) ,β i , − β i ( z ) . Let us quickly check the vertex algebra axioms : Y ( e π i ρ i x i , z ) (cid:12)(cid:12)(cid:12) z = = e π i ρ i x i z π i ρ i W i exp π i ρ i X n < x in z − n exp π i ρ i X n > x in z − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = = e π i ρ i x i . The fields of the form Y ( e π i ρ i x i , z ) commute with each other because all the x in and W i commute, therefore they are local. The fields of the form α i ( z ), β i ( z ) arepairwise local. Now because h α jm , x i ( z ) i = α j ( z ) and Y ( e π i ρ i x i , z )commute.The locality for the fields β j ( z ) and e π i ρ i x i ( z ) is checked as follows: h β j , n , x i ( z ) i = δ i , j Kz n , then h β j , n , e π i ρ i x i ( z ) i = δ i , j π i ρ i e π i ρ i x i ( z ) z n , from this follows h β j ( z ) , e π i ρ i x i ( z ) i = X n h β j , n , e π i ρ i x i ( z ) i z − − n = δ i , j X n π i ρ i e π i ρ i x i ( z ) z − − n z n = δ i , j π i ρ i e π i ρ i x i ( z ) δ ( z , z ) , therefore the fields β j ( z ) and e π i ρ i x i ( z ) are local. The locality for any other pair offields follows from Dong’s Lemma 2.1.The last condition remaining to be proved is the translation invariance of the fields,let us define the translation endomorphism T in V T . Initially it is convenient todefine T acting on x i , T ( x i ) should be a vector such that Y ( T ( x i ) , z ) = ∂ z Y ( x i , z ),but this equation is satisfied by α i ( z ) because of the equation (4.14), so it becomesnatural to define T ( x i ) = α i − .Now it easy to define T on any function as(5.1) T (cid:16) e π i ρ i x i (cid:17) = π i ρ i e π i ρ i x i T ( x i ) = π i ρ i e π i ρ i x i α i − . We define T ( ) = T recursively by the formula [ T , a n ] = − na n − and impose the commutationrelation h T , x i i = α − , finally T extends to the whole V T as a derivation of the nor-mally ordered product. Note that T was defined in a way so it satisfies translationinvariance for the fields α i ( z ) and β i ( z ), and for x i ( z ) holds h T , x i ( z ) i = h T , W i i log( z ) + X n h T , x in i z − n = h T , α i i log( z ) + X n − n h T , α in i z − n OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 13 = X n − n ( − n ) α in − z − n = X n α in z − − n = α i ( z ) = ∂ z x i ( z ) , so now we can compute h T , e π i ρ i x i ( z ) i = π i ρ i e π i ρ i x i ( z ) h T , x i ( z ) i = π i ρ i e π i ρ i x i ( z ) ∂ z x i ( z ) = ∂ z e π i ρ i x i ( z ) . (cid:3) Theorem 5.2.
The space H has the structure of V T -module.Proof. We must define a field for each vector of V T . Set Y ( e π i ρ i x i , z ) = e π i ρ i x i ( z ) , Y ( α i − , z ) = α i ( z ) , Y ( β i , − , z ) = β i ( z ) , and for the rest of the elements define the associated field by the normally orderedproduct, exactly as in the Kac-Moody algebra, for example Y ( α i − e π i ρ j x j ) = : α i ( z ) e π i ρ j x j ( z ) : . Note that we are defining the fields exactly as in the proof of theorem 5.1 andproving the locality condition for those fields we never used the fact that thelogarithmic terms disappeared, i.e., what we actually prove there was that thoselogarithmic fields are pairwise local.Now in this way because of 3.6 and because the normally ordered product oftwo fields is the − Y : V T → Field ( H ) commutes with the n -products when n <
0. For any two pairs of fields of the form α i ( z ) and β j ( z ), it istrivial to see that the n -product condition holds. So it is only left to check it for pairof fields (cid:16) α i ( z ) , e π i ρ j x j ( z ) (cid:17) , (cid:16) β i ( z ) , e π i ρ j x j ( z ) (cid:17) , (cid:16) e π i ρ i x i ( z ) , e π i ρ i x i ( z ) (cid:17) and for n ≥ e π i ρ i x i ( z ) and e π i ρ j x j ( z ) commute we know that e π i ρ i x i ( z )( n ) e π i ρ j x j ( z ) = n ≥ e π i ρ i x i ( n ) e π i ρ j x j = (cid:16) α i ( z ) , e π i ρ j x j ( z ) (cid:17) .For the last pair of fields (cid:16) β i ( z ) , e π i ρ j x j ( z ) (cid:17) we only need to compute for n = β i ( z ) (0) e π i ρ j x j ( z ) = ( z − z ) h β i ( z ) − , e π i ρ j x j ( z ) i(cid:12)(cid:12)(cid:12)(cid:12) z = z = z = π i δ i , j ρ j e π i ρ j x j ( z ) = Y (cid:16) π i ρ j δ i , j e π i ρ j x j , z (cid:17) = Y (cid:16) β i , − (0) e π i ρ j x j , z (cid:17) . (cid:3) Fibration over the Heisenberg Nilmanifold.
As it was explained before thereis a T fibration over the Heisenberg nilmanifold T / / G / Γ / / N = Heis ( R ) / Heis ( Z ) , the goal of this subsection is proving that H is a logarithmic module over the ver-tex algebra of chiral di ff erential operators over the Heisenberg nilmanifold ( V N ),we will do this by carefully restricting to a subspace of C ∞ ( G / Γ ) such the vectorfields identify with the Heisenberg Lie algebra, i.e., we will describe explicitly thestructure of V N and compute the quantum fields. In order to achieve that it will be required to use some techniques from harmonic analysis on the Heisenberg groupto deduce the structure of the space and therefore define the quantum vectors (log-arithmic) fields for the algebra and for the module H . The reader interested in adeeper study of harmonic analysis of the Heisenberg group may consult [12].Define the symbol ξ ijk as ξ = ξ = ξ = ξ ijk = ξ ijk − ξ ikj = ǫ ijk .It is convenient to change the coordinates x i x i , x ∗ i x ∗ i + ξ ijk x j x k , so that the group law turns into( x i , x ∗ i )( y i , y ∗ i ) = ( x i + y i , x ∗ i + y ∗ i + ξ ijk x j y k ) , and the action of α i and β j in these coordinates looks like α i = ∂ x ∗ i β i = ∂ x i + ξ ijk x k ∂ x ∗ j . We still have the Fourier type decomposition L ( G / Γ ) ≃ L ω ∈ Z C ω but this time C ω is given by C ω = n e π i ω i x ∗ i f ; f : R → C , f ( x i + γ i ) = e − π i ω i ξ ijk x j γ k f ( x i ) o . Consider vectors ω ∈ Z of the form ω = (0 , , n ) C (0 , , n ) = n e π i nx ∗ f , , n : f , , n ( x i + γ i ) = e − π i nx γ f , , n ( x i ) o , now those functions f , , n can be decomposed into Fourier series once again f , , n ( x , x , x ) = X m ∈ Z e π i mx f , , n , m ( x , x ) , from this it follows the decomposition of C (0 , , n ) = L m ∈ Z C (0 , , n , m ) . Let us take m = C (0 , , n , for all n ∈ Z , i.e. the functions f n : R → C such that f n ( x + γ , x + γ ) = e − π i nx γ f n ( x , x ). Define C = M n ∈ Z C (0 , , n , = X n e π i nx ∗ f n ( x , x ) ⊂ L ( G / Γ ) . Note that restricting to elements of G of the form ( x , x , , , , x ∗ ) is the same asworking in Heis ( R ), the (polarized) Heisenberg group x x ∗ x , so Heis ( R ) acts on C , , n , by right translations, i.e., looking at the functions on Y depending only on the variables x , x , x ∗ is the same as looking at the Heisenbergnilmanifold N = Heis ( R ) / Heis ( Z ). OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 15
Remark:
The space C constructed above using Fourier analysis can be describedintrinsically as follows: since the Heisenberg group is a central extension1 → R → Heis ( R ) → R → , and if we take a one dimensional complex representation of the center, i.e., a centralcharacter χ n : R → C ∗ , χ n ( x ∗ ) = e π i nx ∗ then C (0 , , n , = CoInd
Heis ( R ) R C χ n and C can beexpressed as C = M n ∈ Z C (0 , , n , = M n ∈ Z CoInd
Heis ( R ) R C χ n . Define for n , m ∈ { , , . . . | n | − } the k-linear map Θ m : L ( R ) → C (0 , , n , as(5.2) Θ m ( g )( x , x , , , , x ∗ ) = e π i nx ∗ X k ∈ Z e π i ( nk + m ) x g ( x + k ) , and for γ , γ ∈ Z it holds Θ m ( g )( x + γ , x + γ , , , , x ∗ ) = e π i nx ∗ X k ∈ Z e π i ( nk + m )( x + γ ) g ( x + γ + k ) = e π i nx ∗ X k ∈ Z e π i ( nk + m ) x g ( x + γ + k ) , making s = k + γ = e − π i nx γ e π i nx ∗ X s ∈ Z e π i ( ns + m ) x g ( x + s ) = e − π i nx γ Θ k ( g )( x , x , , , , x ∗ ) , then Θ m ( g ) ∈ C , , n , , so Θ m is a well defined linear, because of the orthogonalityrelations between the exponentials the maps Θ m are monomorphisms so then theydefine a unique monomorphism Θ : L ( R ) ⊗ C | n | → C (0 , , n , . Let e π i nx ∗ f be an element in C (0 , , n , then because f ( x + , x ) = f ( x , x ) it can bedecomposed into Fourier series e π i nx ∗ f ( x , x ) = e π i nx ∗ X k ∈ Z e π i kx f k ( x ) = | n |− X m = e π i nx ∗ X k ∈ Z e π i ( nk + m ) x f kn + m ( x ) , from the property f ( x , x + γ ) = e − π i nx γ f ( x , x ) it follows f nk + m ( x ) = f m ( x + k ) so e π i nx ∗ f ( x , x ) = | n |− X m = e π i nx ∗ X k ∈ Z e π i ( nk + m ) x f m ( x + k ) , this means that any function in C , , n , is uniquely determined by f , f , . . . , f | n |− ,i.e., there is a linear injection Φ : C (0 , , n , → L ( R ) ⊗ C | n | f (cid:0) f , . . . , f | n |− (cid:1) , moreover Φ and Θ m are inverse functions, so we can make L ( R ) ⊗ C | n | a Heis ( R )-module. Now because of the Stone-von Neumann theorem[13], L ( R ) is the only irreducibleunitary representation of Heis ( R ) with the central character χ n ( t ) = e π i nt and thereis a unique decomposition C (0 , , n , ≃ L ( R ) ⊗ C p n being Θ an isomorphism and | n | = p n . A fully detailed proof of this can be found in [12]. Proposition . If n , C (0 , , n , ≃ L ( R ) ⊗ C | n | and for n = C (0 , , n , ≃ L ( T ), i.e., C ≃ L ( T ) ⊕ M n , L ( R ) ⊗ C | n | ≃ M ρ ∈ Z C e π i ρ i x i ⊕ M n , L ( R ) ⊗ C | n | . Proposition 5.3 means that the G -module C identifies with the g -module M ρ ∈ Z C e π i ρ i x i ⊕ M n , S ( R ) ⊗ C | n | , where S ( R ) denotes the Schwartz space of rapidly decreasing smooth functions in R . We will also denote this space by C . Since one space is the completion of theother one and it will always be clear to distinguish which one we are using.Define the action of elements of at n ∈ t g [ t ] on C by zero and the action of K as theidentity so it is possible now to induce V N = Ind b gg [ t ] ⊕ C K C = Ind b gg [ t ] ⊕ C K M ρ ∈ Z C e π i ρ i x i ⊕ M n , S ( R ) ⊗ C | n | = M ρ ∈ Z Ind b gg [ t ] ⊕ C K C e π i ρ i x i ⊕ M n , Ind b gg [ t ] ⊕ C K S ( R ) ⊗ C | n | Theorem 5.4.
The space V N has a vertex algebra structure. The vertex algebra V N is the vertex algebra of chiral di ff erential operators over theHeisenberg nilmanifold, once again instead of following [1] we will compute ex-plicitly the quantum fields so it become easier to describe its logarithmic module. Remark.
Notice that after the change of coordinates previously done the fields x i ( z )remain invariant but the fields x ∗ i ( z ) don not, specifically x ∗ ( z ) transforms into x ∗ ( z ) = P log( z ) + X i ∈ Z x ∗ , i z − i + log( z ) W X i ∈ Z x i z − i + X i , j ∈ Z x i x j z − i − j + W W (cid:0) log( z ) (cid:1) , Moreover P and W i act trivially on V N so all the terms with logarithms in x ∗ ( z )and x i ( z ) are zero. Proof.
The vacuum vector ∈ V N will be the 1 constant function, let us start definingfields for the basis elements: once again the fields associated to elements of theform a − n k · · · a − n kr ⊗ a − n k ∈ b g will be the same as for the Kac-Moody vertexalgebra, for elements e π i ρ i x i we define Y ( e π i ρ i x i ) = : e ( π i ρ i x i ( z ) ) : = exp (cid:16) π i ρ i x i ( z ) + (cid:17) exp (cid:16) π i ρ i x i ( z ) − (cid:17) , which can also be written as Y ( e π i ρ i x i ) = e π i ρ i x i exp π i ρ i X n < x in z − n exp π i ρ i X n > x in z − n . OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 17
Denote F m ∈ C the image of e − ( x ) by Θ m , i.e. F m = Θ m ( e − ( x ) ) = X k ∈ Z e π i ( nk + m ) x e π i nx ∗ e − ( x + k ) , and define Y ( F m , z ) = X k ∈ Z : exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) : , since h x ∗ ( z ) ± , x i ( z ) ∓ i = h x ∗ ( z ) ± , (cid:0) x ( z ) (cid:1) ∓ i = h x i ( z ) ± , (cid:0) x ( z ) (cid:1) ∓ i = i = , Y ( F m , z ) = X k ∈ Z exp (cid:16) π i nx ∗ ( z ) + (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) + (cid:17) exp (cid:16) − ( x ( z )) + − kx ( z ) + − k (cid:17) · exp (cid:16) π i nx ∗ ( z ) − (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) − (cid:17) exp (cid:16) − ( x ( z )) − − kx ( z ) − (cid:17) , so for every g ∈ C it holds Y ( F m ) g ∈ V N (( z )). Now for a = β , a = β or a = α wewould like to define Y ( f ) for every f ∈ L n , S ( R ) ⊗ C | n | imposing the followingequation(5.3) (cid:2) a ( z ) , Y ( f , z ) (cid:3) = Y ( a f , z ) δ ( z , z ) . Define Y ( aF m ) by the formula Y ( aF m ) = a Y ( F m ) − Y ( F m ) a , clearly if g ∈ C then Y ( aF m ) g ∈ V N (( z )), note that since L ( R ) is an irreducible heis ( R )-mod then every element f ∈ L n , S ( R ) ⊗ C | n | is obtained acting succes-sively on F m with (cid:8) β , β , α (cid:9) and adding, so with the above formula it is proven that Y ( f ) g ∈ V N (( z )) for every functions f , g . It remains to prove that Y ( f ) v ∈ V N (( z )) forevery function f and every v = a n k . . . a n g ∈ V N , this can be done by induction on k , the base case ( k =
0) is already proven, for the recursive case just write v = a q w where Y ( f ) w ∈ V N (( z )), so expanding the equation 5.3 we get Y ( f , z ) a ( z ) w = a ( z ) Y ( f , z ) w − Y ( a f , z ) δ ( z , z ) w , multiplying by z q and taking residues we get res z z q Y ( f , z ) a ( z ) w = res z z q a ( z ) Y ( f , z ) w − res z z q Y ( a f , z ) δ ( z , z ) w , Y ( f , z ) a q w = a q Y ( f , z ) w − z q Y ( a f , z ) δ ( z , z ) w , Y ( f , z ) x = a q Y ( f , z ) w − z q Y ( a f , z ) δ ( z , z ) w . but a q Y ( f , z ) w ∈ V N (( z )) and z q Y ( a f , z ) δ ( z , z ) w ∈ V N (( z )) because the inductionhypothesis and then follows the desired result.From this Y : V N → Field ( V N ) is fully determined since the field for the remainingvectors is determined by taking the normally ordered product of the above fieldsand by linearity.From the previous analysis it is also deduced that Y ( F m ) ∈ V N [[ z ]] and from thisit follows that Y ( v , x ) ∈ V N [[ z ]] for all v ∈ V N , and it is clear that Y ( F m , z ) | z = = F m so it holds for every element in V N .The computations to check the locality condition for the fields α i ( z ) , β i ( z ) , Y ( e π i ρ i x i , z )are analogous to the ones done for the Kac-Moody vertex algebra and the Theorem x ( z ) commute with itself, the fields Y ( F m , z ) and Y ( F m , z )commute, therefore the only remaining pairs to check are n α i ( z ) , Y ( F m , z ) o and n β i ( z ) , Y ( F m , z ) o .From h α ir , x ∗ ( z ) i = δ i , z r it follows that h α ir , e π i nx ∗ ( z ) i = π i nz r e π i nx ∗ ( z ) δ i , , and so h α ir , Y ( F m , z ) i = h α ir , P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1)i = P k ∈ Z h α ir , exp (cid:16) π i nx ∗ ( z ) (cid:17)i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) h α ir , exp (cid:16) π i ( nk + m ) x ( z ) (cid:17)i exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) h α ir , exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1)i = P k ∈ Z h α ir , exp (cid:16) π i nx ∗ ( z ) (cid:17)i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = π i nz r δ i , P k ∈ Z e π i nx ∗ ( z ) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = π i nz r δ i , Y ( F m , z ) , so finally this leads to h α i ( z ) , Y ( F m , z ) i = X r ∈ Z h α ir , Y ( F m , z ) i z − − r = π i n δ i , Y ( F m , z ) X r ∈ Z z r z − − r = π i n δ i , Y ( F m , z ) δ ( z , z ) . So the fields α i ( z ) , Y ( F m , z ) are a local pair.From 4.2 and 4 . β ( z ) and Y ( F m , z ) are a local pair.Let’s prove that β ( z ) and Y ( F m , z ) are local, from the relations proven in section4 we deduce h β , r , x ∗ ( z ) i = W z r log ( z ) , h β , r , x ( z ) i = , h β , r , x ( z ) i = z r K , (cid:20) β , r , (cid:16) x ( z ) (cid:17) (cid:21) = z r x ( z ) , from this follows h β , r , e π i nx ∗ ( z ) i = π i nW e π i nx ∗ ( z ) z r log ( z ) , h β , r , e π i ( nk + m ) x ( z ) i = , h β , r , e − kx ( z ) i = − kz r e − kx ( z ) , (cid:20) β , r , e − ( x ( z ) ) (cid:21) = − z r x ( z ) e − ( x ( z ) ) , OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 19 which finally leads to h β ( z ) , e π i nx ∗ ( z ) i = π i nW e π i nx ∗ ( z ) log ( z ) δ ( z , z ) , h β ( z ) , e π i ( nk + m ) x ( z ) i = , h β ( z ) , e − kx ( z ) i = − ke − kx ( z ) δ ( z , z ) , (cid:20) β ( z ) , e − ( x ( z ) ) (cid:21) = − x ( z ) e − ( x ( z ) ) δ ( z , z ) . Now we can compute the brackets (cid:2) β ( z ) , Y ( F m , z ) (cid:3) = h β ( z ) , P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1)i = P k ∈ Z h β ( z ) , exp (cid:16) π i nx ∗ ( z ) (cid:17)i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) h β ( z ) , exp (cid:16) π i ( nk + m ) x ( z ) (cid:17)i exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) (cid:2) β ( z ) , exp (cid:0) − ( x ( z )) (cid:1)(cid:3) exp (cid:0) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) (cid:1) (cid:2) β ( z ) , exp (cid:0) − kx ( z ) (cid:1)(cid:3) exp( − k ) = π i nW log ( z ) δ ( z , z ) P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + − δ ( z , z ) x ( z ) P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + − δ ( z , z ) P k ∈ Z k exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = π i nW Y ( F m , z ) log ( z ) δ ( z , z ) − Y (cid:16) Θ m ( xe − x ) , z (cid:17) δ ( z , z ) , therefore ( z − z ) (cid:2) β ( z ) , Y ( F m , z ) (cid:3) =
0, so they are local.Let’s prove that β ( z ) and Y ( F m , z ) are local, once again from the relations provenin section 4 we get h β , r , x ∗ ( z ) i = z r ( x ( z ) − W log ( z )) = z r ˜ x ( z ) , here we use the notation ˜ x i ( z ) = x i ( z ) − W i log ( z ) = P n x in z − n , it holds h β , r , x ( z ) i = z r K , h β , r , x ( z ) i = , (cid:20) β , r , (cid:16) x ( z ) (cid:17) (cid:21) = , which means h β , r , e π i nx ∗ ( z ) i = π i nz r ˜ x ( z ) e π i nx ∗ ( z ) , h β , r , e π i ( nk + m ) x ( z ) i = π i ( nk + m ) z r e π i ( nk + m ) x ( z ) , h β , r , e − kx ( z ) i = , (cid:20) β , r , e − ( x ( z ) ) (cid:21) = , and this translates into h β ( z ) , e π i nx ∗ ( z ) i = π i n ˜ x ( z ) δ ( z , z ) , h β ( z ) , e π i ( nk + m ) x ( z ) i = π i ( nk + m ) e π i ( nk + m ) x ( z ) δ ( z , z ) , h β ( z ) , e − kx ( z ) i = , (cid:20) β ( z ) , e − ( x ( z ) ) (cid:21) = . Now the commutator of the fields is computed (cid:2) β ( z ) , Y ( F m , z ) (cid:3) = h β ( z ) , P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1)i = P k ∈ Z h β ( z ) , exp (cid:16) π i nx ∗ ( z ) (cid:17)i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) h β ( z ) , exp (cid:16) π i ( nk + m ) x ( z ) (cid:17)i exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) (cid:2) β ( z ) , exp (cid:0) − ( x ( z )) (cid:1)(cid:3) exp (cid:0) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) (cid:1) (cid:2) β ( z ) , exp (cid:0) − kx ( z ) (cid:1)(cid:3) exp( − k ) = π i ˜ x ( z ) δ ( z , z ) P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + π i ( nk + m ) δ ( z , z ) P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = π i n ˜ x ( z ) Y ( F m , z ) δ ( z , z ) + π i mY ( F m , z ) δ ( z , z ) − nx ( z ) Y ( F m , z ) δ ( z , z ) + n π i ( x ( z ) + k ) δ ( z , z ) P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = − π i nW Y ( F m , z ) log ( z ) δ ( z , z ) + π i mY ( F m , z ) δ ( z , z ) + π i nY ( Θ m ( xe − x ) , z ) δ ( z , z ) . The locality condition for the remaining fields follows from Dong’s Lemma.Now it is only left to prove the translation invariance of the fields, we will proceedsimilarly to the proof of Theorem 5.1, let us define the translation endomorphism T in V N . We already know how to define T ( x i ) and proceeding exactly as in the proofof 5.1 we get that the fields e π i ρ i x i ( z ) satisfy the translation invariance condition.For T ( x ∗ ) the situation is similar but slightly more complicated, once again a vectorsuch that Y ( T ( x ∗ ) , z ) = ∂ z Y ( x ∗ , z ) is needed, but unfortunately the equation 4.15 is alittle bit more complicated. We start noticing that after the change of coordinateswe made the equation 4.15 was transformed into D z x ∗ i ( z ) = β i ( z ) − ξ ijk x k ( z ) D z x j ( z ) , so taking i =
3, acting on the vacuum vector and evaluating z = T ( x ∗ ) should be defined as T ( x ∗ ) = β , − − α − x , and force the commutation relation h T , x ∗ , i = β , − − ǫ jk X m mx j − − m x km . Now it is easy to define T on any function as T ( f ) = ∂ x f T ( x ) + ∂ x f T ( x ) + ∂ x ∗ f T ( x ∗ ) . To make computations easier here we will actually use the fact that W , W and P act by zero so we have no logarithms in the fields. OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 21
Consider the field ˜ x ∗ ( z ) = P n x ∗ z − n , it is convenient to prove that translationinvariance holds for the field ˜ x ∗ ( z ), for n , h T , x ∗ . n i = T , − β , n n + ǫ jk n X m mx jn − m x km = − n (cid:2) T , β , n (cid:3) + ǫ jk n X m m h T , x jn − m x km i = β , n − + ǫ jk n X m m h T , x jn − m i x km + ǫ jk n X m mx jn − m h T , x km i = β , n − + ǫ jk n X m m α jn − m − x km + ǫ jk n X m mx jn − m α km − = β , n − − ǫ jk n X m m ( n − m − x jn − m − x km − ǫ jk n X m m ( m − x jn − m x km − = β , n − − ǫ jk n X m m ( n − m − x jn − m − x km − ǫ jk n X m ( m + mx jn − m − x km = β , n − − ǫ jk X m mx jn − m − x km , then h T , ˜ x ∗ ( z ) i expands as h T , ˜ x ∗ ( z ) i = X n h T , x ∗ , n i z − n = X n β , n − z − n − ǫ jk X n X m mx jn − m − x km z − n = X n β , n z − n − − ǫ jk X n X m mx jn − m x km z − n − = β ( z ) − ǫ jk X n X m x jn − m z − n + m mx km z − m − = β ( z ) + ǫ jk X n x jn z − n X n − nx kn z − n − = β ( z ) + ǫ jk x j ( z ) ∂ z x k ( z ) = ∂ z ˜ x ∗ ( z ) , here the last equality hold because 4.15 and ˜ x ∗ ( z ) coincides with the x ∗ ( z ) as definedin section 4 when setting the formal variable log( z ) =
0, i.e., when deleting all termswith P , W and W . Now the field x ∗ ( z ) after the change of coordinates (withoutthe logarithmic terms) can be written as: x ∗ ( z ) = ˜ x ∗ ( z ) + x ( z ) x ( z ) , but now it becomes easy to prove translation invariance for x ∗ ( z ) as we alreadyknow it holds for x ( z ) and x ( z ) h T , x ∗ ( z ) i = h T , ˜ x ∗ ( z ) i + h T , x ( z ) x ( z ) i = ∂ z ˜ x ∗ ( z ) + h T , x ( z ) i x ( z ) + x ( z ) h T , x ( z ) i = ∂ z ˜ x ∗ ( z ) + ∂ z x ( z ) x ( z ) + x ( z ) ∂ z x ( z ) = ∂ z (cid:18) ˜ x ∗ ( z ) + x ( z ) x ( z ) (cid:19) = ∂ z x ∗ ( z ) . Finally we have the tools for proving the translation invariance condition for thefields Y ( F m , z )[ T , Y ( F m , z )] = T , X k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) = X k ∈ Z h T , exp (cid:16) π i nx ∗ ( z ) (cid:17)i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) + X k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) h T , exp (cid:16) π i ( nk + m ) x ( z ) (cid:17)i exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) + X k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) h T , exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17)i = P k ∈ Z π i n h T , x ∗ ( z ) i exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z π i ( nk + m ) exp (cid:16) π i nx ∗ ( z ) (cid:17) h T , x ( z ) i exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z − x ( z ) + k ) exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) (cid:2) T , x ( z ) (cid:3) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = P k ∈ Z π i n ∂ z (cid:16) x ∗ ( z ) (cid:17) exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z π i ( nk + m ) exp (cid:16) π i nx ∗ ( z ) (cid:17) ∂ z (cid:16) x ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z − x ( z ) + k ) exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) ∂ z (cid:0) x ( z ) (cid:1) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) = P k ∈ Z ∂ z (cid:16) exp (cid:16) π i nx ∗ ( z ) (cid:17)(cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) ∂ z (cid:16) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17)(cid:17) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1) + P k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) ∂ z (cid:0) exp (cid:0) − ( x ( z )) − kx ( z ) − k (cid:1)(cid:1) = ∂ z X k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) = ∂ z Y ( F m , z ) . (cid:3) We have the embeddings C ⊂ C ∞ ( N ) ⊂ C ∞ ( G / Γ ) , which leads to V ( g ) ⊂ V N ⊂ H , OGARITHMIC MODULES FOR CHIRAL DIFFERENTIAL OPERATORS OF NILMANIFOLDS 23 so it is expected that H is a logarithmic module over V N . Theorem 5.5.
The space H has the structure of logarithmic V N -module.Proof. We must define a logarithmic module for each vector of V N , set Y ( e π i ρ i x i , z ) = e π i ρ i x i ( z ) , Y ( α i − , z ) = α i ( z ) , Y ( β i , − , z ) = β i ( z ) , Y ( F m , z ) = X k ∈ Z exp (cid:16) π i nx ∗ ( z ) (cid:17) exp (cid:16) π i ( nk + m ) x ( z ) (cid:17) exp (cid:16) − ( x ( z )) − kx ( z ) − k (cid:17) , Now we extend Y to any function f ∈ L n , S ( R ) ⊗ C | n | exactly as we did in theprevious theorem 5.4, i.e., through the formula 5.3, in particular for every a ∈ g thelogarithmic field Y ( aF m , z ) is defined by the formula Y ( aF m , z ) = [ a , F m ] and finallywe extend Y to the rest of the vectors via the normally ordered product.Notice that during all the analysis made in the proof of 5.4 to show that Y ( f ) wasactually a field was never used the fact that the logarithmic terms acted by zero, sowhat we actually prove back there was that the Y ( f ) for any function was actuallya logarithmic field. Similarly we proceeded, on propose, when proving that thefields were pairwise local, so what we actually prove was that those are pairwiselocal logarithmic fields.Let’s prove that the function Y : V N → LField ( H ) preserves the n -products,because of the way we defined the fields and equation 3.6 it is clear that Y preservesall negative n -products. For positive n -products involving only the fields e π i ρ i x i ( z ) , α i ( z ), β i ( z ) the n -product condition holds, the analysis is complete analogous to theone made in the proof of theorem 5.2. It is also clear that for n ≥ Y ( F m ( n ) F m , z ) = = Y ( F m , z ) ( n ) Y ( F m , z )because Y ( F m , z ) and Y ( F m , z ) commute.For α i ( z ) and Y ( F m , z ) we have α i ( z ) (0) Y ( F m , z ) = ( z − z ) h α i ( z ) − , Y ( F m , z ) i(cid:12)(cid:12)(cid:12)(cid:12) z = z = z = π i n δ i , Y ( F m , z ) = Y ( α i ( o ) F m , z ) . For the fields β ( z ) and Y ( F m , z ) there is nothing to prove since they commute.For the fields β ( z ) and Y ( F m , z ) holds β ( z ) (0) Y ( F m , z ) = ( z − z ) (cid:2) β ( z ) − , Y ( F m , z ) (cid:3)(cid:12)(cid:12)(cid:12) z = z = z = π i nW Y ( F m , z ) log ( z ) − Y (cid:16) Θ m ( xe − x ) , z (cid:17) = (cid:2) β , , Y ( F m , z ) (cid:3) = Y ( β F m , z ) . Finally for β ( z ) and Y ( F m , z ) holds β ( z ) (0) Y ( F m , z ) = ( z − z ) (cid:2) β ( z ) − , Y ( F m , z ) (cid:3)(cid:12)(cid:12)(cid:12) z = z = z = − π i nW Y ( F m , z ) log ( z ) + π i mY ( F m , z ) + π i nY ( Θ m ( xe − x ) , z ) . = (cid:2) β , , Y ( F m , z ) (cid:3) = Y ( β F m , z ) . Therefore H is a logarithmic V N -module. (cid:3) R eferences [1] Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob. Chiral de rham complex. Commu-nications in mathematical physics , 204(2):439–473, 1999.[2] Alexander Beilinson and Vladimir Drinfeld.
Chiral algebras , volume 51. American MathematicalSoc., 2004.[3] Bailin Song. Vector bundles induced from jet schemes. arXiv preprint arXiv:1609.03688 , 2016.[4] Lev A Borisov and Anatoly Libgober. Elliptic genera of toric varieties and applications tomirror symmetry.
Inventiones mathematicae , 140(2):453–485, 2000.[5] Marco Aldi and Reimundo Heluani. Dilogarithms, ope, and twisted t-duality.
InternationalMathematics Research Notices , 2014(6):1528–1575, 2012.[6] Chris Hull and Barton Zwiebach. Double field theory.
Journal of High Energy Physics ,2009(09):099, 2009.[7] Peter Bouwknegt, Jarah Evslin, and Varghese Mathai. T-duality: topology change from h-flux.
Communications in mathematical physics , 249(2):383–415, 2004.[8] Bojko Bakalov. Twisted logarithmic modules of vertex algebras.
Communications in MathematicalPhysics , 345(1):355–383, 2016.[9] Victor G Kac.
Vertex algebras for beginners . Number 10. American Mathematical Soc., 1998.[10] Alberto De Sole and Victor G Kac. Finite vs a ffi ne w-algebras. Japanese Journal of Mathematics ,1(1):137–261, 2006.[11] Roger William Carter.
Lie algebras of finite and a ffi ne type , volume 96. Cambridge UniversityPress, 2005.[12] Louis Auslander and Richard Tolimieri. Abelian Harmonic Analysis, Theta Functions and Func-tional Algebras on a Nilmanifold , volume 436. Springer, 2006.[13] Gerald B Folland.
A course in abstract harmonic analysis . Chapman and Hall / CRC, 2016.
E-mail address ::