A question on generalization of partition functions of CY 3-folds in String Theory
aa r X i v : . [ phy s i c s . g e n - ph ] D ec A question on generalization of partition functions of CY3-folds in String Theory
Mohammad Reza RahmatiDecember 29, 2020
Abstract
This is an expositoray article on the topological string partition function promoting anextension of the partition function of open Gromov-Witten theory of CY 3-folds definedby the trace of vertex operators. We also give a brief survey of their connection to thetheory of Hilbert scheme of points on surface. Specifically; we apply infinitely manyCasimir operators twisted to the vertex operator computing the amplitude. The case offinite number of twists has been well discussed in the mathematics and Physics literature.
The vertex operators provide a representation theoretic and the combinatorial framework forthe generating series in string theory. To a CY two generating series can be associated; one thetopological vertex partition function and the other, the Guage index generating series that canbe otained as the elliptic genus of certain Hilbert scheme of points on surface. The topologicalvertex partition function is obtained from a framed gluing of topological vertex associated todifferent C patches of the CY 3-fold. The partition function has various interpretations whichare some how related to each other . One can mention; BPS content of the Guage actions,Fock space formalism of Hilbert scheme of points, Knot invariants, GW correlation functions,Generating series of quiver Guage theory, etc.The vertex operators act on the Fock space of infinite dimension and their trace producesgenerating series which are crucial in quantum Physics. The vertex operator provide a powerfultool to compute string theory amplitudes. We follow a computation of the character of theinfinite wedge representation (Fock space) [8] where a product formula is established for thecharacter. Bloch-Okounkov prove that the associated character is quasimodular.In various cases the partition function can be formulated through the Fock space structureon L n H ( n ) where H ( n ) stands for the cohomology of the Hilbert scheme of n-points on C ,[7]. The generating series appears as a correlation functions involving chern characters of theHilbert scheme associated to cohomology classes on the CY 3-fold X . The GW-theory of CY3-folds can be explained by the χ y -genus of the Hilbert scheme of points on surface.In general the successive application of commutation rule of the boson fields interacting tothe vertex operator produces product factors [see Appendix in [2]]. The product formulas forthe string partition functions are specially of interests. The GupaKumar-Vafa invariants canbe obtained from these product formulas.We consider the partition functions as characters of representations and specifically as traceof vertex operators, twisted by one or more Casimir operators. We give a natural extension ofthe trace to the case when infinitely many Casimirs. We ask of a representation and Physicstheoretic interpretations for the trace of the extended vertex operator.1 Hilbert Scheme of points on Surface
We briefly follow [7] on basics of Hilbert scheme of points on surface. Our purpose is topresent the Fock space stricture on the cohomology of Hlbert scheme of points on surface, as analternative way to produce partition functions of CY 3-folds. Fix a quiver (
I, H ), with vertexset I and edges H . We shall consider the quiver variety M ζ ( v, w ) = µ − ( ζ ) ss //GL v C (2.1)where v ∈ Z I + , w ∈ Z I ≥ are dimension and framing vectors and ζ = ( ζ i ) ∈ C I ! M i ∈ I ζ i Id V i ∈ Z ( gl v ) (2.2)( GL v = Q i GL ( V i )), and Z ( . ) is the center. The map µ is defined as µ : M ( v, w ) −! gl v µ ( B, a, b ) = X in ( h )= i ǫ ( h ) B h B ¯ h + a i b i i ∈ I ∈ ⊕ i gl ( V i ) = gl v (2.3)where M ( v, w ) = M h ∈ H Hom ( V out ( h ) , V in ( h ) ) M ( Hom i ∈ H ( W i , V i ) ⊕ Hom ( V i , W i )) (2.4)The variety M ζ ( v, w ) can be interpreted as the moduli of representations of the quiver ( I, H ).If A = ( a ij ) be the adjacency matrix of the quiver ( I, H ), then C = 2 Id − A is a Cartan matrix,and R + = { β = ( β i ) ∈ Z I ≥ | β T Cβ ≤ } (2.5)is a set of positive roots of a Lie algebra. Lets for simplicity ( I, H ) be a quiver of ˜ A l − -typewith cyclic orientation, That is the associated lie algebra is e sl l = sl l [ t, t − ] ⊕ C c ⊕ C d (2.6)Set w = (1 , , ..., ∈ Z I ,ζ = (0 , ζ R ) ,ζ R ∈ { η = ( η i ) ∈ R I | η i < , ∀ i } (2.7)Let e i be the i -th coordinate vector of Z I . Define L i ( v ) = { ( ̺ , ̺ ) | ̺ ⊂ ̺ , ̺ ∈ M ζ ( v, w ) , ̺ ∈ M ζ ( v + e i , w ) } Lagrangian ⊂ M ζ ( v, w ) × M ζ ( v + e i , w ) (2.8)Consider the two projections p , p from M ζ ( v, w ) × M ζ ( v + e i , w ) onto the first and secondfactors. For α ∈ H ∗ ( M ζ ( v, w )) and β ∈ H ∗ ( M ζ ( v + e i , w )) we can define a representation e i E i : (cid:0) β ( − v i − + v i p ∗ ( p ∗ βL i ( v )) (cid:1) ,f i F i : (cid:0) α ( − v i +1 + v i p ∗ ( p ∗ αL i ( v )) (cid:1) (2.9)2t is possible to choose ζ such that M ζ ( v, w ) ∼ = X [ n ]0 (2.10)Then one gets naturally the action on X [ n ]0 . Set H n,iX = H i ( X [ n ] , C ) , H nX = n M i =0 H n,iX , H X = M n H nX (2.11)and | i = 1 ∈ H ( X [0] ) = C (2.12)Define the cycles Q [ m + n,m ] = { ( ξ, x, η ) ∈ X [ m + n ] × X × X [ m ] | ξ ⊃ η, Supp ( I η /I ξ ) = { x }} (2.13)where m ≥ , Q [ m,m ] = 0. One has dim Q [ m + n,m ] = 2 m + n + 1. If α ∈ H ∗ ( X ) , for each n ≥ a − n ( α ) : a p ∗ ( Q [ m + n,m ] .p ∗ α.p ∗ a ) a n ( α ) : b p ∗ ( Q [ m + n,m ] .p ∗ α.p ∗ b ) (2.14)For n = 0, the correspondences a − n ( α ) have bidegree ( n, n − | α | ). We have a − n ( α ) † =( − n a n ( α ) with respect to the cup product. They satisfy the commutaion relation[ a m ( α ) , a n ( α )] = − mδ m, − n ( α, β ) (2.15)of Heisenberg type. H X is an irreducible module over Heisenberg algebra generated by a n ( α )with highest weight | i = 1 ∈ H ( X [0] ). It is linearly generated by a − n ( α ) ... a − n k ( α k ) | i , n , ..., n k > , α , ..., α n ∈ H ∗ ( X ) (2.16)One can consider the above elements factored by the image of the map τ k ∗ : H ∗ ( X ) ! H ∗ ( X k )induced from the diagonal embedding τ k : X ! X k . We set a n ... a n k ( τ k ∗ α ) = X a n ( α j, ) ... a n k ( α j,k ) , τ k ∗ α = X j α j, ⊗ ... ⊗ α j,n (2.17)which is well defined cf. loc. cit. We define the normal ordering product: a m a m := ( a m a m , m ≤ m a m a m , m ≥ m (2.18)Set L n = − / X m ∈ Z : a m a n − m : (2.19)These operators satisfy the Virasoro relations, and also act on the former operators as shifts[ L m ( α ) , L n ( β )] = ( m − n ) L m + n ( αβ ) + m − m δ m, − n ( Z X e X αβ ) Id H X [ L m ( α ) , a n ( β )] = − n a m + n ( αβ ) (2.20)where e X is the Euler class. 3et L be a line bundle on X . Define a virtual vector bundle E L on X [ k ] × X [ l ] of rank k + l by E L | ( ξ,η ) = χ ( O , L ) − χ ( I η , I ξ ⊗ L ) (2.21)There is an Ext operator of Carlson W ( L, z ) : H X [[ z, z − ]] ! H Z [[ z, z − ]] h W ( L, z ) η, ξ i = Z X [ k ] × X [ l ] ( η ⊗ ξ ) c k + l ( E L ) , η ∈ H ∗ ( X [ k ] ) , ξ ∈ H ∗ ( X [ l ] ) (2.22)Define the vertex operators Γ ± ( L, z ) = exp (cid:16)P n z ∓ n n a ± n ( L ) (cid:17) . They satisfy the commutationrelation Γ + ( L ) . Γ − ( L ′ ) = (1 − yx ) h L,L ′ i Γ − ( L ′ )Γ + ( L ) (2.23)We can write the Carlson Ext operator in terms of vertex operators W ( L, z ) = Γ − ( L − K X , z )Γ + ( − L, z ) (2.24)where K X is the canonical class.Consider Z n = { ( ξ, x ) ⊂ X [ n ] × X | x ∈ Supp ( ξ ) } (2.25)with two projections p , p . For γ ∈ H ∗ ( X ) set G ( γ ) n = p ∗ ( ch ( O Z n ) p ∗ td ( X ) p ∗ γ ) ∈ H ∗ ( X [ n ] ) , n > G ( γ ) n = ⊕ i G i ( γ ) n , G i ( γ ) n ∈ H s +2 i ( X [ n ] ) (2.27)For a sheaf F on X define F [ n ] = p ∗ p ∗ F (2.28)This operation reduces to K -groups. Then one can show that X n c ( L [ n ] ) = exp X r ≥ ( − r − r a − r ( c ( L )) ! | i (2.29)By setting c ℏ ( L [ n ] = P i c i ( L [ n ] ) ℏ i , We can write this identity as a generating series X n c ℏ ( L [ n ] w n ) = exp X r ≥ ( − r − r a − r ( c ℏ ( L )) w r ! | i = c ℏ ( L ) exp( a − (1 X ) w ) | i (2.30)It satisfies the differential equation ddw Z = C ℏ ( L ) Z, C ℏ ( L ) = c ( L ) a − (1 X ) c ( L ) − (2.31)4et L be a line bundle on X and ch k ( L [ n ] ) be the k -th chern character of L [ n ] . Define thegenerating series h ch L k ...ch L n k n i : = X n ≥ q n (cid:18)Z X [ n ] ch k ( L n ] ) ...ch k n ( L n [ n k ] ) c ( T X [ n ] ) (cid:19) h ch L k ...ch L n k n i = h ch L k ...ch L n k n ih ih i =( q : q ) − χ ( X ) ∞ , ( a : q ) n = n Y i =0 (1 − aq i ) (2.32)These correlation functions are conjecturally multiple q -zeta values, i.e. a q -deformation ofusual multiple zeta values. They are equal to the following generating series up to a factor. Let α , ..., α n ∈ H ∗ ( X ), F α ...α n k ...k n = X n q n Z X [ n ] n Y i =1 G k i ( α i , n ) c ( T X [ n ] ) ! = T r q n W ( L , s ) Y i L k i ( α i ) ! (2.33)They appear as the constant coefficient in a more general generating series, cf. [7]. The materials of this section are well known. The references are [1, 35, 34, 33, 31, 30, 26, 27,28, 25, 24, 23, 22, 6, 5, 4, 3, 2]. We explain the topological string partition function of CY3-folds and the method to write it in terms of topological vertex. We employ the concept oftoric varieties and their fan diagrams. Specifically we shall consider the web diagrams of CY3-folds on the plane.Let T = ( C × ) p be the p -dimensional torus. It is the complexification of U (1) p . Assume Σbe a collection of cones called a fan in N R = R m and M = N ∨ the dual. Associated to a fan Σwe can define a variety M Σ = C m \ Z (Σ) T (3.1)where T is a product of torus with a finite abelian group. The equivalence classes turns outto be ( z , ..., z m ) ∼ ( λ Q a z , ..., λ Q ma z m ) , λ ∈ C × (3.2)and the relation X Q ia v i = 0 , fan condition (3.3)where v i are generators of Σ (called fan condition). If D i be the divisor defined by z i = 0, thecanonical bundle of M Σ can be written as K M = O ( − P D i ). It is trivial if Σ D i ∼
0. Thiscondition can be interpreted as the existence of a lattice point m ∈ ( R m ) ∨ such that h v i , m i = 1for all i . Thus M Σ is CY iff the vectors v i all lie in a hyperplane. It follows that X i Q ia = 0 , CY condition (3.4)5hus a toric CY is never compact.We will deal with CY 3-folds. In this case the 3-dimensional fan is projected on a 2-dimensional graph, called the web diagram, and T = ( C × ) m − × finite abelian gp. There is amoment map µ : M Σ ! ∆ M (3.5)where ∆ M is the polytope of M , i.e is dual to the fan σ . The moment map is given by m − µ a : C m ! C , a = 1 , , ..., m −
3, and is described by the equations m X i =1 Q ia | z i | = Re ( t a ) , t a ∈ C (3.6)called Witten D -term equations. We have an action of U (1) m − on coordinates z j z j exp (cid:0) iQ ia α a (cid:1) z j , a = 1 , , ..., m − M Σ = T m − i =1 µ − ( Re ( t a )) T (3.8)is a CY 3-fold and t a are complexified Kahler parameters. We look at CY 3-folds as T = ( S ) -fibrations over 3-dimensional base with corners. Introduce new variables( p , θ ) , ..., ( p k , θ k ) , p i = | z i | , z i = | z i | e iθ i (3.9)The 3-fold is then parametrized by the coordinates p i and θ i , where the latter describes T .The CY 3-fold is constructed by gluing many open C -patches to each other along edges. Theweb diagram of CY 3-folds records the information of a framing when gluing C -patches. Theframe is settled by choosing a vector f i perpendicular to the vectors v i of the fan. The choice offraming affects the partition function of the CY, however here we assume the frame is chosento be standard, i.e f i ∧ v i = 1 for all i . Fermionic model: [26] To each C -patch of the CY variety we associate a partition functioncalled topological vertex C R ,R ,R which can be written in the following two equivalent forms X R ,R ,R C f ,f ,f R ,R ,R Y i =1 T r Ri V i = X −! k (1) , −! k (2) , −! k (3) C f ,f ,f −! k (1) , −! k (2) , −! k (3) 3 Y i =1 n −! k ( i ) T r −! k ( i ) V i (3.10)where T r R V denotes the trace of V as appears in the representation R , and χ R ( C ( −! k )) is thecharacter of the symmetric group calculated at the conjugacy class C ( −! k ). The sum in the lefthand side runs over all the representations R , R , R of the symmetric group S N for all N . Thetrace of representations of the symmetric groups can also be stated via the conjugacy classesand we simply have T r −! k V = X R χ R ( C ( −! k )) T r R V (3.11)The two sums are related by the following identity C f ,f ,f R ,R ,R = X −! k (1) , −! k (2) , −! k (3) C f ,f ,f −! k (1) , −! k (2) , −! k (3) 3 Y i =1 χ Ri (cid:16) C ( −! k ( i )) (cid:17) n −! k ( i ) (3.12)6he V i are holonomy variables, given by the determinant of the Wilson line V i = P R Γ i A i . Infact one notes that in counting the holomorphic maps from curves of different genus to a CY3-fold, these curves must go to the vertices or edges of the web diagram, and to illustrate them,they wrap arround the edges with different holonomies. The change of the framing affect as C f − n v ,f − n v ,f − n v R ,R ,R = ( − P n i l ( R i ) q P i n i k ( R i ) / C f ,f ,f R ,R ,R (3.13)[See [26] for the definition of indices l ( R i ) , k ( R i ) and n −! k ( i ) ]. The topological vertex C R ,R ,R is invariant under circle change of the representations R i , i.e they have cyclic symmetry. Wehave the rule v ! − v C R ,R ,R ! ( − l ( R ) C R t ,R ,R (3.14)In the gluing of two graphs γ and Γ with partition functions Z (Γ ) and Z Γ we produce termsas X Q Z (Γ )( − l ( Q e l ( Q ) t Z (Γ ) Q t ! X −! k Z (Γ ) exp (cid:16) − l ( −! k ) t (cid:17)Q j k j ! j k j Z (Γ ) (3.15)The rules to write the partition function in terms of topological vertex are as follows,(1) The edges of the graph are labeled by integral vectors v i . To each edge associate arepresentation R i .(2) For smooth CY the graph can be divided into trivalent vertices corresponding to C -patches.(3) To each vertex there associates an ordered triple ( v i , v j , v k ) by reading counterlockwise.(4) If all the edges are incoming associate C R i ,R j ,R k to ( v i , v j , v k ), otherwise replace the cor-responding representation by its transpose times ( − l ( R ) .(5) If the vertex ( v i , v j , v k ) shares the i -th edge with the vertex ( v i , v j ′ , v k ′ ), we glue theamplitudes by summing over the representations on the i -th edge X R i C R j R k R i (cid:0) e − l ( R i ) t i ( − ( n i +1) l ( R i ) q − n i k ( R i ) / (cid:1) C R ti R j ′ R k ′ (3.16)where n i = | v k ∧ v k ′ | .(6) The length of edge can be read from the Witten D-term equation on the Kahler moduliof X . The edges of the graph Γ are straight lines on the plane with rational slope.To the i -th edge in the ( p i , q i )-direction of length x i we associate a Kahler parameter t i = x i / p p + q .(7) For non-compact edges the corresponding representation is trivial, denoted R = 0 , ∅ , • .(8) If there are more than one D -branes say n on the edge, we have contributions of (cid:0) n (cid:1) open strings stretching between D -branes. The effect of integrating out these strings isexp (cid:0) − P m trU m U m (cid:1) = P R ( − l ( R ) tr R U tr R U . It produces contributions of the form X R i ,Q La,i ,Q Ra,i C R j ,R k ,R i ⊗ na =1 Q La,i (cid:0) ( − s ( i ) e − L ( i ) q f ( i ) (cid:1) C R ti ⊗ na =1 Q Ri,a ,R ′ j ,R ′ k n Y a =1 T r Q La,i V a T r Q Ra,i V − a . (3.17)7here the exponents in the middle term is read from the Young diagram of the represen-tations R l , R ′ l , Q La,i , Q
Ra,i , cf. [26].(9) The topological vertex is given by C λµν = q | k ( µ ) / S ν t ( q − ρ ) X η S λ t /η ( q − v − ρ ) S µ/η ( q − ν t − ρ ) (3.18)where q ν − ρ = { q − ν i +(2 i − / | i = 1 , , ... } , q = e πiǫ , and k λ k = P λ i , k ( µ ) = | µ |− k µ k . Bosonic model: [26] We explain another model to express the partition function of CY 3-foldsinvolving Boson operators. We shall consider the Hilbert space H of states to be generated by | −! k i = Y j α k j − j | i , −! k = ( k , k , ..., k m ) (3.19)is a vector involving winding numbers. On H ⊗ we define an element | P i = exp − t X j n α − j α j ! | i ⊗h | = X −! k e − l ( −! k ) t ( − h n −! k | −! k i ⊗h −! k | , n −! k = Y j k j ! j k j (3.20)called propagator. The topological vertex is defined as a state in H ⊗ . If we write the partitionfunction Z formally as Z = X −! k i C −! k , −! k , −! k Q i n −! k i T r −! k V T r −! k V T r −! k V (3.21)Thinking of topological vertex C as a state in H ⊗ we write Z by Z = X −! k i T r −! k V T r −! k V T r −! k V Q i n −! k i h −! k | ⊗ h −! k | ⊗ h −! k | C i (3.22)It follows from the last formula that C can be written in the form | C i = exp X −! k i F −! k , −! k , −! k ( g s ) α − −! k α − −! k α − −! k | i ⊗ | i ⊗ i (3.23)We may formally write Z = h V | ⊗ h V | ⊗ h V | C i where V = h | exp (cid:18)X n T rV ⊗ n α n (cid:19) (3.24) Example. • The partition function of the web diagram obtained by gluing three C patchesalong a triangle is Z = X R ,R ,R ( − P i l ( R i ) e − l ( R i ) t q P i k Ri C • R R t C • R R t C • R R t (3.25)8 The partition function of an (
M, N )-web diagram with M vertical and N horizontalhexagons is written as Z ( M, N ) = X α N Y a =1 Q | α a | b a X µ,ν ( − Q m ) P b | µ b |−| ν b | Y b ( − Q b ) | ν b +1 | Y a C µ a ν a α a ( t, q ) Y b C ν ta µ ta β ta ( t, q )(3.26)where α a is a set of M partitions α a , ..., α aM , and β b = α a +1 b , [2]. The material of this section are well known. The references are [14, 29, 30] as well as manyother available texts. We include this section as part of the task to make our terminology moreconcrete and understandable. We begin with the definition of the Fock space. The Fock space F = V ∞ V is the vector space spanned by semi-infinite wedge product of a fixed basis of theinfinite dimensional vector space V = P i ∈ / Z C v i , i.e. monomials v i ∧ v i ∧ ... such that • i > i > ... • i j = i j − − / j ≫ ψ k : v i ∧ v i ∧ ... v k ∧ v i ∧ v i ∧ ...ψ ∗ k : v i ∧ v i ∧ ... ( − l v i ∧ v i ∧ ... ∧ d v i l = k ∧ ... (4.1)The monomials can be parametrized by partitions | λ i = v λ = λ − / ∧ λ − / ∧ ... (4.2)We can also write this using Frobenius coordinates of partitions | λ i = l Y i =1 ψ ∗ a i ψ b i | i , a i = λ i − i + 1 / , b i = λ ti − i + 1 / α n = X k ∈ / Z ψ k + n ψ ∗ k (4.4)They satisfy [ α n , ψ k ] = ψ k + n , [ α n , ψ ∗ k ] = − ψ k − n . The operators of the formΓ + ( x ) = exp X n ≥ x n n α n ! , Γ − ( x ) = exp X n> x n n α − n ! (4.5)are called vertex operators. They are adjoint with respect to the natural inner product. Wehave a commutation relation Γ + ( x )Γ − ( y ) = (1 − xy )Γ − ( y )Γ + ( x ) (4.6)9e have Γ + ( x ) v µ = X λ ⊃ µ s λ/µ ( x ) v λ (4.7)Vertex operators provide powerful tools to express partitions.Γ + (1) | µ i = X λ ⊃ µ | λ i Γ − (1) | µ i = X λ ⊂ µ | λ i (4.8)For example we may write the McMahon function as Z = X q ♮ boxes = h ( ∞ Y t =0 q L Γ + (1)) q L ( − Y t = −∞ Γ − (1) q L ) i = h Y n> Γ + ( q n − / ) Y n> Γ − ( q − n − / ) i (4.9)We may divide a 3-dimensional partitions into slices of two dimensional partitions , for instancealong the diagonals or any other way this could be done. In this way the vertex operator dividesinto multiplication of many vertex operators of the slices, Z ( { x ± m } ) = h ... Y u i 2) can bewritten as Z ′ = h Y ZλµνZ ∅∅∅ = C λµν is called refined topological vertex. The aforementionedtechnics also applies to the refined version of topological vertex.To give some sense of computations we calculate the trace of a general vertex operator actionon the Fock space F , cf. [14]. We have the following formula for the trace of a vertex operatoracting on F = V ∞ V ; T r q L exp X n A n α − n ! exp X n B n α n !! = Y n X k k X l =0 n l A ln B ln l ! q nk (cid:18) kl (cid:19) (4.17)where L is the charge operator, q L | λ i = | λ || λ i .To prove the formula, denote the operator in the trace by T . We have the isomorphism ∞ ^ V = ∞ O n =1 ∞ M k =0 α k − n | i (4.18)which implies T r ( T ) = ∞ Y n =1 T r (cid:16) T | L ∞ k =0 α k − n | i (cid:17) = Y n X k h α k − n | i (cid:12)(cid:12)(cid:12) q L e A n α − n e B n α n (cid:12)(cid:12)(cid:12) | α kn | ii = Y n X k,l,m A ln B mn l ! m ! q n ( l − m + k ) h α k − n | i (cid:12)(cid:12)(cid:12) α l − n α mn (cid:12)(cid:12)(cid:12) | α kn | ii = Y n X k,l A ln B ln l ! l ! q nk h α k − n | i (cid:12)(cid:12)(cid:12) α l − n α ln (cid:12)(cid:12)(cid:12) | α kn | ii = Y n X k k X l =0 n l A ln B ln l ! q nk (cid:18) kl (cid:19) (4.19)11 Generalization In this section we extend the partition function of a CY 3-fold as the trace vertex operatorstwisted by certain Casimir operators, and present our main result. Our motivation is a com-putation on the character of the infinite wedge Fock space in [8]. Let F = V ∞ V be the Fockspace on a fixed basis of V = L j ∈ Z C v j . Then the character of F as a representation of gl ∞ isgiven by ch ( gl ∞ , F ) = Y n ≥ (1 + q q n +1 / q n +3 / .... )(1 + q − q n − / q − ( n − / .... ) (5.1)where q j = e πiτ j . The partition function of U (1) theory can be written in the form Z ( τ, m, ǫ ) = T r Q L τ exp X n ≥ Q nm − n ( q n/ − q − n/ ) α n ! exp X n ≥ Q n − m − n ( q n/ − q − n/ ) α − n !! (5.2)Using the commutation relation of α ± it can be written as Z ( τ, m, ǫ ) = Y k (1 − Q kτ ) − Y i,j (1 − Q kτ Q − m q i + j − )(1 − Q kτ Q m q i + j − )(1 − Q kτ q i + j − ) (5.3)cf. [12]. The partition function in (8.4) can be generalized to Z ( τ, m, ǫ, t ) = T r Q L τ e P n t n L n exp X n ≥ Q nm − n ( q n/ − q − n/ ) α n ! exp X n ≥ Q n − m − n ( q n/ − q − n/ ) α − n !! (5.4)In the limit m Z ( τ, m = 0 , ǫ, t ) = T r (cid:0) Q L τ e P n t n L n (cid:1) (5.5)We can write the partition function in Remark 8.2 in terms of the Gromov-Witten potentials Z ( τ, m, ǫ ) = exp X g ≥ ǫ g − F g ! (5.6)where e F = Y k (1 − Q kτ ) − (cid:18) (1 − Q kτ ) Q − m )(1 − Q kτ Q m ) (1 − Q kτ ) (cid:19) / (5.7)One may ask if the above equation is equal to 6.8 with q = q = .... = 1. Problem: Consider the following trace as a twist of the two former ones, T r exp X j ≥ πiL j ! exp X n> A n α − n ! exp X n> B n α n !! (5.8)We pose the following questions; • How to compute the trace in terms of the former traces.12 How the last trace is related to the former two characters in Theorems 8.1 and 8.2. Whatis the representation theory interpretation of that. • In case that the coefficients A n , B n are suitably chosen what is the Physical interpretationof the trace in terms of string theory partition functions. • Is there any product formula for the trace.