aa r X i v : . [ m a t h . AG ] D ec Rationality of capped descendent vertex in K -theory Andrey Smirnov
Abstract
In this paper we analyze the fundamental solution of the quan-tum difference equation (qde) for the moduli space of instantons ontwo-dimensional projective space. The qde is a K -theoretic general-ization of the quantum differential equation in quantum cohomology.As in the quantum cohomology case, the fundamental solution of qdeprovides the capping operator in K -theory (the rubber part of thecapped vertex). We study the dependence of the capping operatoron the equivariant parameters a i of the torus acting on the instantonmoduli space by changing the framing. We prove that the cappingoperator factorizes at a i →
0. The rationality of the K -theoretic 1-legcapped descendent vertex follows from factorization of the cappingoperator as a simple corollary. Contents
Quantum toroidal algebra gl Y ( z ) . . . . . . . . . . . . . . . . . . . . . . 223.6 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . 24 The rationality of partition functions in the presence of descendents is animportant and long standing conjecture in enumerative geometry of 3-folds.The deepest insight into this conjecture was achieved in the theory of stablepairs where the conjecture was proved for nonsingular toric 3-folds and localcurve 3-folds [10, 9] (the theory of stable pairs is conjecturally equivalentto Gromov-Witten theory and cohomological Donaldson-Thomas theory of3-folds [2]). The central result here is the rationality of the cohomologicalcapped 1-leg descendent vertex, see Theorem 3 in [10], which was obtainedby detailed analysis of the poles of the descendent vertex. The rationality2f the partition functions for local curve 3-folds and nonsingular toric 3-folds can be derived from this result by applying standard arguments such asdegeneration and geometric reduction of 3-leg descendent vertex to the caseof 1-leg.In this paper we prove the rationality of the capped K -theoretic 1-legdescendent vertex - Theorem 2. This function is defined in the theory ofstable quasimaps to the Hilbert scheme of points on the complex plane. Theenumerative geometry of stable quasimaps developed in [5] is a K -theoreticgeneralization of quantum cohomology. We should stress here that our proofof rationality is completely different (and, we believe, simpler) than one in[10]. Our main idea is to analyze the capped K -theory vertex in broadercontext: instead of restricting ourself to quasimaps to the Hilbert schemeof points Hilb n ( C ) we consider the quasimaps to the moduli of instantons M ( n, r ) of arbitrary rank r and topological charge n . The Hilbert schemesare covered by a special case r = 1 when M ( n,
1) = Hilb n ( C ). For a general r the coefficients of power series for both the bare vertex and the cappingoperator are rational functions of additional equivariant parameters a · · · a r corresponding to the action of framing torus A = ( C ∗ ) r on M ( n, r ). Weanalyze the asymptotic behaviour of the capping operator and vertex in thelimit a i → quantum difference equation (qde), see equation (7). As explained in [8], in K -theory this equation playsthe role similar to one of the quantum differential equation in quantum co-homology of instanton moduli space M ( n, r ). The capping operator in K -theory is given by the fundamental solution of qde. We note here, that in thespecial case r = 1, the cohomological limit of the operator M O (1) ( z ) in qde(7) coincides with the operator of quantum multiplication by the first Chernclass of tautological bundle in the quantum cohomology of Hilb n ( C ) givenby formula (6) in [6]. Therefore, in this limit the qde turns to the quantumdifferential equation for Hilb n ( C ) described in [7]. As a consequence, ratio-nality of the descendent vertex in the quantum cohomology follows from ourmain Theorem 2 through the cohomological limit.The paper is organized as follows. First we recall some necessary factsabout M ( n, r ) and various tori acting on this moduli space. Following [5]we define the bare and capped vertex functions in Section 1.3. We then3ormulate the factorization Theorems 3 and 4 and prove the main Theorem2 as a corollary.In Section 2 we outline the theory of the quantum toroidal algebra U ~ ( bb gl ).Using the results of [8] we then describe the action of this algebra on theequivariant K -theory of the instanton moduli spaces. In Section 3 we give auniversal formula for qde for instanton moduli space of arbitrary rank r interms of Heisenberg subalgebras of U ~ ( bb gl ). We then prove the factorizationTheorem 3 for the capping operator. In Section 4 we give an explicit formulafor the bare 1-leg K -theoretic vertex. We use it to prove a factorizationtheorem for the vertex. Let M ( n, r ) be the moduli space of framed rank r torsion-free sheaves F on P with fixed second Chern class c ( F ) = n . A framing of a sheaf F is achoice of an isomorphism: φ : F | L ∞ → O ⊕ rL ∞ (1)where L ∞ is the line at infinity of C ⊂ P . This moduli space is usuallyreferred to as rank r instanton moduli space. Note that in the special case r = 1 this moduli space is isomorphic to the Hilbert scheme of n -points onthe complex plane M ( n,
1) = Hilb n ( C ) . Let A ≃ ( C × ) r be the framing torus acting on M ( n, r ) by scaling the i -thsummand in isomorphism (1) with a character which we denote by a i . Thistorus acts on the instanton moduli space preserving the symplectic form. Letus denote by T = A × ( C × ) where the second factor acts on C ⊂ P byscaling the coordinates on the plane with characters which we denote by t and t . This induces an action of T on M ( n, r ). The action of this torusscales the symplectic form with a character ~ = t t .Let C ⊂ A be a one-dimensional subtorus acting on the r = r + r -dimensional framing space with the character r + ar . In this situationwe say that subtorus C splits the framing r to r + ar . We note that thecomponents of C -fixed point set are of the form: M ( n, r ) C = a n + n = n M ( n , r ) × M ( n , r ) . (2)4f we set M ( r ) = ∞ ` n =0 M ( n, r ) then M ( r ) C = M ( r ) × M ( r ) . (3) Let QM d ( n, r ) be the moduli space of stable quasimaps from P to M ( n, r ) .Let us consider an action of a one-dimensional torus on P which comes fromscaling the standard coordinate on P . The fixed point set of this actionconsist of two points { p , p } = { , ∞} ⊂ P . We denote the character of T p P by q and the torus by C ∗ q . This action induces an action of C ∗ q on QM d ( n, r ). We denote the total torus acting on QM d ( n, r ) by G = T × C ∗ q .For a point p ∈ P let QM dp ( n, r ) ⊂ QM d ( n, r ) be the open subset ofquasimaps non-singular at p . This subset comes together with the evaluationmap: ev p : QM dp ( n, r ) −→ M ( n, r )sending a quasimap to its value at p .The moduli space of relative quasimaps [ QM dp ( n, r ) is a resolution of themap ev meaning that we have a commutative diagram: d QM dp ( n, r ) b ev p % % ❑❑❑❑❑❑❑❑❑❑❑ QM dp ( n, r ) +(cid:11) rrrrrrrrrrr ev p / / M ( n, r )with a proper evaluation map b ev p . The explicit construction of the modulispace of relative quasimaps is given in Section 6.4 of [5].Let L = C n and τ ∈ K GL ( L ) = Λ[ x ± , · · · , x ± n ] be a symmetric Laurentpolynomial. Every such polynomial corresponds to a virtual representation τ ( L ) of GL ( L ) which is a tensorial polynomial in L . For example, elementarysymmetric function τ = e k ( x i ) corresponds to τ ( L ) = Λ k L . The bare vertex with a descendent τ is defined by the following formal power series: V ( τ ) ( z ) = ∞ X d =0 ev p , ∗ (cid:16) QM dp ( n, r ) , b O vir ⊗ τ ( V | p ) (cid:17) z d ∈ K G (cid:16) M ( n, r ) (cid:17) loc ⊗ Q [[ z ]] (4) In this paper we follow the terminology and notations of [5]. A good introduction tothe stable quasimaps is Section 4.3 of [5]. V is the rank n degree d bundle on P defining the quasimap and b O vir is the virtual structure of QM dp ( n, r ). The pushforward ev p , ∗ is not proper;however, the fixed locus of G is (in fact, the fixed locus is a set of finitelymany isolated points in this case). Therefore, the pushforward is well definedin localized K -theory.The capped vertex with a descendent τ is a similar object defined forquasimap moduli space relative to p :ˆ V ( τ ) ( z ) = ∞ X d =0 b ev p , ∗ (cid:16)d QM dp ( n, r ) , b O vir ⊗ τ ( V | p ) (cid:17) z d ∈ K G (cid:16) M ( n, r ) (cid:17) ⊗ Q [[ z ]] (5)Now, b ev p is proper and the result lives in non-localized K -theory.By definition, the degree zero quasimaps d QM p ( n, r ) = M ( n, r ) corre-spond to the constant maps from P to the instanton moduli space. In thiscase we have V | p = V where V is a tautological bundle on M ( n, r ). Bydefinition of the capped vertex :ˆ V ( τ ) ( z ) = τ ( V ) K / + O ( z )where K is the canonical bundle on M ( n, r ).As an element of non-localized K -theory the capped vertex is a simplerobject. For example at large r all quantum corrections vanish: Theorem 1. (Theorem 7.5.23 in [5]) For every τ there exist r ≫ suchthat ˆ V ( τ ) ( z ) = τ ( V ) K / . In fact, numerical computations shows that one can give the precise boundfor r in this theorem. Assume that the descendent is given by a Schurpolynomial τ = s λ ( x , · · · , x n ) for a partition λ = ( λ ≥ λ ≥ · · · ≥ λ k ). Conjecture 1.
For every τ = s λ ( x , · · · , x n ) the capped vertex is classical ˆ V ( τ ) ( z ) = τ ( V ) K / if and only if r > λ . In particular, this conjecture implies that for the Hilbert schemes of pointson a plane Hilb n ( C ) corresponding to r = 1 the capped vertex is classicalonly in the absence of descendents, i.e, for τ = 1. In general, all terms inthe power series (5) are non-vanishing. However, our main theorem says thatthis power series is in fact a rational function: Theorem 2.
The power series ˆ V ( τ ) ( z ) is the Taylor expansion in z of arational function in Q ( u , · · · , u r , t , t , q, z ) . .4 Capping operator In computing the capped vertex one can separate the contributions of the QM dp ( n, r ) (which corresponds to the bare vertex) and the “rubber” part ofthe relative moduli space, see Section 7 of [5]:ˆ V ( τ ) ( z ) = Ψ( z ) V ( τ ) ( z ) . (6)Here Ψ( z ) = K G (cid:16) M ( n, r ) (cid:17) ⊗ loc ⊗ Q [[ z ]]is the so called capping operator corresponding to the contribution of therubber part. The matrix Ψ( z ) can be computed explicitly as a matrix of the fundamental solution of the quantum difference equation :Ψ( z ) O (1) = M O (1) ( z )Ψ( z ) (7)where O (1) is an operator of multiplication by the corresponding line bun-dle in K T ( M ( n, r )). The operator M O (1) ( z ) acts in K T ( M ( n, r )) and hasrational in z matrix coefficients. It is constructed explicitly in Section 3.1. Assume that the torus C splits the framing r to r + ar , so that the set of C -fixed points is given by (3). By definition, the coefficients of power series Ψ( z )and V ( τ ) ( z ) are given by classes of localized K -theory and thus are rationalfunctions of a . We are interested in a → h on K T (cid:16) M ( r ) (cid:17) . Let α k , k ∈ Z \ { } and K be the standard generators of h . InSection 3.5 we prove the following result: Theorem 3. If Ψ ( r ) ( z ) is the solution of (7) for M ( r ) then: lim a → Ψ ( r ) ( z ) = Y ( r ) , ( r ) ( z ) Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ) (8) where Y ( r ) , ( r ) ( z ) is the evaluation of the following universal element Y ( z ) = exp (cid:16) − ∞ X k =1 ( ~ − ~ − ) K k ⊗ K − k − z − k K k ⊗ K − k α − k ⊗ α k (cid:17) ∈ h ⊗ ( z ) (9) in the h ⊗ - representation K T (cid:16) M ( r ) (cid:17) ⊗ K T (cid:16) M ( r ) (cid:17) . α − k ⊗ α k act on the corresponding K -theory as locally nilpotentoperators. Thus, the coefficients of Y ( r ) , ( r ) ( z ) are rational functions of z forall r and r . In Section 4 we also prove a similar result for the bare vertex: Theorem 4.
For a descendent τ ∈ Λ[ x , · · · , x n ] we have : lim a → V ( r ) , ( τ ) ( z ) = V ( r ) , ( τ ) ( z ~ r ) ⊗ V ( r ) , (1) ( z ~ − r q − r ) (10)Here, as well as in Theorem 3, the additional superscript ( r ) correspondsto the ranks of the instanton moduli. The proof of Theorem 2 is now elementary. Indeed, for arbitrary r and τ and in Theorem 4 let ˆ V ( r ) , ( τ ) ( z ) be the corresponding capped vertex. Weneed to check that ˆ V ( r ) , ( τ ) ( z ) is a rational function of z . First, by Theorem1 we can find r large enough for the corresponding capped descendent vertexto be classical ˆ V ( r ) , ( τ ) ( z ) = τ ( V ) K / , i.e. independent of z .We have: τ ( V ) K / = Ψ ( r ) ( z ) V ( r ) , ( τ ) ( z ) . The bundle K does not depend on a and by our choice of τ we have lim a → τ ( V ) = τ ( V ) ⊗
1. Thus, by the factorization theorems we obtain:( τ ( V ) ⊗ K / = Y ( r ) , ( r ) ( z )Ψ ( r ) ( z ~ r ) V ( r ) , ( τ ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ) V ( r ) , (1) ( z ~ − r q − r )Now, the first factor on the right side gives the capped vertex with shifted pa-rameter Ψ ( r ) ( z ~ r ) V ( r ) , ( τ ) ( z ~ r ) = ˆ V ( r ) , ( τ ) ( z ~ r ). The operator Y ( r ) , ( r ) ( z )from Theorem 3 is explicitly invertible, therefore:ˆ V ( r ) , ( τ ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ) V ( r ) , (1) ( z ~ − r q − r ) = Y ( r ) , ( r ) ( z ) − ( τ ( V ) ⊗ K / In full generality the descendent τ can be a symmetric Laurent polynomial τ ∈ K GL ( n ) ( · ) = Λ[ x ± , · · · , x ± n ], but it will be clear in the proof of this proposition that caseswith inverse powers of x i are treated in the same way when a → ∞ . The result remainsthe same if we change the roles of r and r . Y ( r ) , ( r ) ( z ) has rational coefficients, so Y ( r ) , ( r ) ( z ) − has. Thus,in the right side we have a vector whose components are rational functionsof z . The first component of this vector is ˆ V ( r ) , ( τ ) ( z ~ r ) and thus is also arational function. Of course, the property to be rational does not depend onthe shift, so ˆ V ( r ) , ( τ ) ( z ) is a rational function of z . (cid:3) I would like to thank A. Okounkov for guidance and his interest to thiswork. The author was also supported in part by RFBR grants 15-31-20484mol-a-ved and RFBR 15-02-04175. gl Let us set Z = Z , Z ∗ = Z { (0 , } and: Z + = { ( i, j ) ∈ Z ; i > i = 0 , j > } , Z − = − Z + Set n k = ( t k − t − k )( t k − t − k )( ~ − k − ~ k ) k and for vector a = ( a , a ) ∈ Z denote by deg( a ) the greatest common divisorof a and a . We set ǫ a = ± a ∈ Z ± . For a pair non-collinear vectorswe set ǫ a , b = sign(det( a , b )).The “toroidal” algebra U ~ ( bb gl ) is an associative algebra with 1 generatedby elements e a and K a with a ∈ Z , subject to the following relations [12]: • elements K a are central and K = 1 , K a K b = K a + b • if a , b are two collinear vectors then:[ e a , e b ] = δ a + b K − a − K a n deg( a ) (11)9 if a and b are such that deg( a ) = 1 and the triangle { (0 , , a , b } hasno interior lattice points then[ e a , e b ] = ǫ a , b K α ( a , b ) Ψ a + b n where α ( a , b ) = (cid:26) ǫ a ( ǫ a a + ǫ b b − ǫ a + b ( a + b )) / ǫ a , b = 1 ǫ b ( ǫ b b + ǫ b b − ǫ a + b ( a + b )) / ǫ a , b = − a are defined by: ∞ X k =0 Ψ k a z k = exp (cid:16) ∞ X m =1 n m e m a z m (cid:17) for a ∈ Z such that deg( a ) = 1. e (2 , − e (2 , − e (2 , e (2 , e (2 , e (0 , − e (0 , − e (0 , e (0 , e (1 , − e (1 , − e (1 , e (1 , e (1 , e ( − , − e ( − , − e ( − , e ( − , e ( − , e ( − , − e ( − , − e ( − , e ( − , e ( − , Figure 1: The line with slope w = 2 corresponds to Heisenberg subalgebragenerated by α k = e k, k for k ∈ Z \ { } .10 .2 Slope Heisenberg subalgebras For w ∈ Q ∪ {∞} we denote by d ( w ) and n ( w ) the denominator and numera-tor of rational number. We set d ( ∞ ) = 0, n ( ∞ ) = 1 and n (0) = 0, d (0) = 1.Let us set: α wk = e ( d ( w ) k,n ( w ) k ) , k ∈ Z \ { } From (11) we see that for fixed w ∈ Q ∪ {∞} these elements generate aHeisenberg with the following relations:[ α w − k , α wk ] = K kd ( w )(1 , − K − kd ( w )(1 , n k We will informally refer to this algebra as “Heisenberg subalgebra with aslope w ” and denote it h w ⊂ U ~ ( bb gl ). It is convenient to visualize the algebra U ~ ( bb gl ) as in the Figure 1. Heisenberg subalgebras of h w correspond to lineswith slope w in this picture.The Heisenberg subalgebra with slope w = 0 will play distinguished rolein this paper. In this special case we will often omit the slope superscript: h = h , α k = α k and K = K (1 , . The algebra U ~ ( bb gl ) carried different inequivalent Hopf structures. We willuse the Hopf structure with zero slope defined by a coproduct ∆ : U ~ ( bb gl ) → U ~ ( bb gl ) ⊗ U ~ ( bb gl ), which has the following explicit form on Heisenberg sub-algebra with slope zero:∆( α − k ) = α − k ⊗ K − k ⊗ α − k ∆( α k ) = α k ⊗ K k + 1 ⊗ α k ∆( K ) = K ⊗ K (12)where k >
0. The algebra U ~ ( bb gl ) is a triangular Hopf algebra, which meansthere exist an element R ∈ U ~ ( bb gl ) ⊗ U ~ ( bb gl ) (the universal R -matrix) whichenjoys the following properties. In U ~ ( bb gl ) ⊗ it satisfies the quantum Yang-Baxter equation: R R R = R R R U ~ ( bb gl ) ⊗ the corresponding R -matrix acts. In addition we have:1 ⊗ ∆( R ) = R R , ∆ ⊗ R ) = R R and: R ∆( g ) = ∆ op ( g ) R , ∀ g ∈ U ~ ( bb gl ) . where ∆ op is the opposite coproduct. Explicitly, the universal R -matrix isgiven by the following formula (Khoroshkin-Tolstoy factorization formula) R = ← Y w ∈ Q ∪{∞} exp (cid:16) ∞ X k =0 n k α w − k ⊗ α wk (cid:17) (13)the order of factors in this product is given explicitly as : R = → Y w ∈ Q w< R − w R ∞ ← Y w ∈ Q w ≥ R + w = ← Y w ∈ Q w ≥ (cid:16) R − w (cid:17) − (cid:16) R ∞ (cid:17) − → Y w ∈ Q w< (cid:16) R + w (cid:17) − where the order is the standard order on the set of rational numbers, and: R ± w = ∞ Y k =0 exp( n k α w ± k ⊗ α w ∓ k ) = exp (cid:16) ∞ X k =0 n k α w ± k ⊗ α w ∓ k (cid:17) (14) Set R = Q ( t / , t / , a ). Let F ( a ) = R [ p , p , · · · ] be the set of polynomialsin infinitely many variables which we consider as R vector space. Let ν bea partition and P ν ∈ F ( a ) be the corresponding Macdonald polynomials inHaiman’s normalization [1] . Recall that P ν is a basis of the vector space F ( a ).Let us define a homomorphism ev a : U ~ ( bb gl ) → End (cid:16) F ( a ) (cid:17) defined ex-plicitly in the basis of Macdonald polynomials by: These two infinite products give the same element in the completion of U ~ ( bb gl ) ⊗ .There is, however, an important difference - evaluated in the tensor product of two Fockrepresentations F ( a ) ⊗ F ( a ) the fist product converges in the topology of power series in a /a while the second as power series in a /a . In this paper we are dealing with limits a /a → The parameters t , q of Macdonald polynomials are related to t and t by q = t / , t = t / On central elements:ev a ( K (1 , ) P ν = t − t − P ν , ev a ( K (0 , ) P ν = P ν (15) • On Heisenberg subalgebra with slope 0:ev a ( e ( m, ) P ν t m/ − t − m/ )( t m/ − t − m/ ) p − m P ν m < − m ∂P ν ∂p m m > • On Heisenberg subalgebra with slope ∞ :ev a ( e (0 ,m ) ) P ν = a − m sign( k ) − t m ∞ X i =1 t m ( ν i − t m ( i − ! P ν (17)Note that e ,n , e n, , K (0 , and K (1 , generate U ~ ( bb gl ) thus the last set offormulas define a homomorphism ev u .The representation ev a is called Fock representation of U ~ ( bb gl ) evaluatedat a . The parameter a is called the evaluation parameter of Fock representa-tion. Let us note here that by (17) the generators act in F ( a ) as monomialsin the evaluation parameter a :ev a ( α wk ) ∼ a − kn ( w ) (18) Let us denote F ( r ) = F ( a ) ⊗ · · · ⊗ F ( a r ) (19)and define representation of quantum toroidal algebra ev ( r ) : U ~ ( bb gl ) → End (cid:16) F ( r ) (cid:17) by:ev ( r ) ( α ) = ev a ⊗ · · · ⊗ ev a r (cid:16) ∆ ( r ) ( α ) (cid:17) , α ∈ U ~ ( bb gl )13here ∆ is a coproduct from Section 2.3. We set: R ( r ) , ( r ) = ev ( r ) ⊗ ev ( r ) ( R )As shown in [8], in the tensor product of Fock spaces, the infinite product (13)converges in the topology of power series in a i and R ( r ) , ( r ) is a well definedelement of End (cid:16) F ( r ) ⊗ F ( r ) (cid:17) whose matrix coefficients are rational functionsof evaluation parameters a i . Let ev ( r ) a = ev a a ⊗ · · · ⊗ ev a r a be a shiftedevaluation. Denote R ( r ) , ( r ) ( a ) = ev ( r ) ⊗ ev ( r ) a ( R ) and R ± , ( r ) , ( r ) w ( a ) =ev ( r ) ⊗ ev ( r ) a ( R ± w ). Proposition 1. (Section 2 of [8]) • The operator R − , ( r ) , ( r )0 def = R − , ( r ) , ( r )0 ( a ) does not depend on a . • R − , ( r ) , ( r ) w (0) = 1 for w > and R + , ( r ) , ( r ) w (0) = 1 for w < • R ( r ) , ( r ) ∞ (0) = ~ − Ω where ~ Ω acts on F ( r )( n ) ⊗ F ( r )( n ) by multiplication on ~ ( n r + n r ) / . • In particular, we have: R ( r ) , ( r ) (0) = ( R − , ( r ) , ( r )0 ) − ~ Ω . The universal R -matrix R satisfies the quantum Yang-Baxter equation.Thus, its limit R = ~ − Ω R − is also a solution of this equation. It is not difficultto see that R is a universal R -matrix for h . Let us introduce some convenient notations and terminology here. The Fockspace is equipped with the grading: F ( a ) = ∞ M n =0 F ( n ) ( a )corresponding to the degree function deg( p k ) = k . This induces the gradingon the tensor products: F ( r ) = ∞ M n =0 F ( r )( n ) F ( r )( n ) denotes the subspace spanned by elements of degree n . Through-out this paper we use the following terminology: an operator A : F ( r ) ⊗ F ( r ) → F ( r ) ⊗ F ( r ) is lower-triangular (upper-triangular) if A = ∞ L k =0 A ( k ) (respectively A = ∞ L k =0 A ( − k ) ) where A ( k ) : F ( r )( n ) ⊗ F ( r )( n ) → F ( r )( n + k ) ⊗ F ( r ) n − k . Wesay that the operator A is strictly lower or upper-triangular if in addition A (0) = 1. We say the operator is block-diagonal if A ( k ) = 0 for k = 0. Forexample, the operators R − w and R + w defined by (14) are strictly lower andupper-triangular respectively. Finally, for a formal variable z we denote ablock-diagonal operator z d ( i ) acting on a F ( r )( n ) ⊗ F ( r )( n ) as multiplication by z n i . In the special case r = 1 the instanton moduli space coincides with theHilbert scheme of n points on the complex plane M ( n,
1) = Hilb n ( C ). Asa vector space the equivariant K -theory of the Hilbert scheme of points isisomorphic to the Fock space: ∞ M n =0 K T (Hilb n ( C )) = F ( a ) (20)where T is a torus from Section 1.2. The summands here correspond to thegrading from the previous section K T (Hilb n ( C )) = F ( n ) ( a ). The parameters t , t and a of the Fock space are identified with the corresponding equivariantparameters of T .The fixed point set Hilb n ( C ) T is discrete. Its elements are labeled bypartitions ν with | ν | = n . The structure sheaves of the fixed points O ν forma basis of the localized K -theory. The polynomials representing the elementsof this basis under isomorphism (20) are the Macdonald polynomials P ν inHaiman normalization [1].In [12] the geometric action of U ~ ( bb gl ) on the equivariant K -theory ofHilb n ( C ) was constructed. As a representation, this action coincides withthe Fock representation described in Section 2.4. In particular, formulas(15)-(17) describe this geometric action explicitly in the basis of the fixedpoints. 15 .8 Coproducts and stable envelopes In [4, 3] the geometric action of U ~ ( bb gl ) on K T ( M ( r )) was constructed. Asa representation K T ( M ( r )) is isomorphic to a product of r Fock spaces F ( r ) .The evaluation parameters a i of this representation are identified with theequivariant parameters of framing torus A . An alternative construction of U ~ ( bb gl ) action on equivariant K -theory was given in [8] using FRT formalism[11]. The key ingredient of [8] is the notion of the stable map, which is ageometric description of the coproduct. Here we recall basic facts about thestable map, the details can be found in Section 2 of [8].Assume that a one-dimensional subtorus C ⊂ A splits the framing as r to r + ar such that M ( r ) C = M ( r ) × M ( r ). In this case we have a canonicalisomorphism of the corresponding K -theories, called stable envelope : K T ( M ( r ) C ∗ ) = K T ( M ( r )) ⊗ K T ( M ( r )) Stab −→ K T ( M ( r ))The stable map Stab and the antipode ∆ make the following diagram com-mutative: K T ( M ( r )) ⊗ K T ( M ( r )) ∆( α ) (cid:15) (cid:15) Stab / / K T ( M ( r )) α (cid:15) (cid:15) K T ( M ( r )) ⊗ K T ( M ( r )) Stab / / K T ( M ( r ))for every element α ∈ U ~ ( bb gl ).Let us consider a chain of splittings by whole A , such that all factors r i = 1 in the end: r → a r + a r → a r + a r + a r → · · · → a + · · · + a r (21)This gives a canonical isomorphism of U ~ ( bb gl )-modules: K T ( M (1)) ⊗ · · · ⊗ K T ( M (1)) = F ( r ) Φ ( r ) −→ K T ( M ( r )) (22) To be precise, the definition of stable envelope map requires a choice of a chamberin C ⊂ Lie ( C ) and an alcove ∇ ⊂ P ic ( M ( n, r )) ⊗ Q . In this paper the chamber C corresponds to a → ∇ unique alcove lying in the opposite of the ample cone whoseclosure contains 0 ∈ H ( M ( n, r ) , R ). This is the same choice as in Theorem 3 in [8]. Theother choices of alcove ∇ correspond to a freedom in the choice of the Hopf structure, i.e.,the antipode ∆ for U ~ ( bb gl ). F ( r ) is defined by (19), and Φ ( r ) is a composition of the stable envelopescorresponding to splittings. By construction of stable envelope, Φ ( r ) does notdepend on the order of splittings in (21) and thus it is well defined. Thisproperty reflects the coassociativity of coproduct. First let us consider an element B ( z ) ∈ U ~ ( bb gl )[[ z, q ]] given explicitly by: B ( z ) = ← Y w ∈ Q − ≤ w< : exp (cid:16) ∞ X k =0 n k ~ − krd ( w ) / − z − kd ( w ) q kn ( w ) ~ − krd ( w ) / α w − k α wk (cid:17) : . The symbol :: here denotes the normal ordered exponent - all annihilationoperators α wk with k > K T ( M ( r )) withrepresentation F ( r ) of U ~ ( bb gl ) through isomorphism Φ ( r ) . Define the followingoperator acting on K T ( M ( r )) : M ( r ) O (1) ( z ) = O (1) ev a ⊗ · · · ⊗ ev a (cid:16) ∆ ⊗ r ( B ( z )) (cid:17) (23)where O (1) is the operator of multiplication by the corresponding line bundlein K -theory. Note that the coefficients of normally ordered exponentialsact in F ( r ) as a locally nilpotent operators. This means that the operators M ( r ) O (1) ( z ) has rational matrix coefficients for all r . The qde for the rank r instanton moduli space is given by the following equation [8]:Ψ ( r ) ( zq ) O (1) = M ( r ) ( z )Ψ ( r ) ( z ) . (24)The capping operator in quantum K -theory is a fundamental solution of qde,i.e., corresponds to the following boundary condition :Ψ ( r ) (0) = 1 (25)which geometrically means that the classical limit z = 0 of the capped de-scendent vertex coincides with the corresponding K -theory class. Here 1 ∈ K G ( M ( n, r )) stands for the class of the structure sheaf O M ( n,r ) . .2 Cocycle identity Out goal to to compare the capping operator Ψ ( r ) ( z ) for M ( r ) with the(shifted) capping operator for the C -fixed point set given by Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ). By definition, the first operator is the fundamental solution of(24) while the second solvesΨ ( r ) ( z ~ r q ) ⊗ Ψ ( r ) ( z ~ − r q ) O (1) = M ( r ) O (1) ( z ~ r ) ⊗ M ( r ) O (1) ( z ~ − r )Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r )We conclude that the operator: J ( r ) , ( r ) ( z ) = Ψ ( r ) ( z ) (cid:16) Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ) (cid:17) − (26)is the solution of the following difference equation: J ( r ) , ( r ) ( zq ) M ( r ) O (1) ( z ~ r ) ⊗ M ( r ) O (1) ( z ~ − r ) = M ( r ) O (1) ( z ) J ( r ) , ( r ) ( z ) (27) Proposition 2.
The operator J ( r ) , ( r ) ( z ) satisfies the dynamical cocycle iden-tity: J ( r + r ) , ( r ) ( z ) J ( r ) , ( r ) ( z ~ r ) = J ( r ) , ( r + r ) ( z ) J ( r ) , ( r ) ( z ~ − r ) (28) Proof.
Consider the chain of splittings: M ( r + r + r ) → M ( r + r ) × M ( r ) → M ( r ) × M ( r ) × M ( r )or, alternatively: M ( r + r + r ) → M ( r ) × M ( r + r ) → M ( r ) × M ( r ) × M ( r )The result does not depend on the choice of the order and the propositionfollows from definition of J ( r ) , ( r ) ( z ).By definition, Ψ ( r ) ( z ) is a power series in z whose coefficients are rationalfunctions of a . As we discussed in [8] the limit lim a → Ψ ( r ) ( z ) exists and is lower-triangular. The power series Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r ) is independent of a .We conclude: 18 orollary 1. The operator Y ( r ) , ( r ) ( z ) = lim a → J ( r ) , ( r ) ( z ) (29) is a lower-triangular operator satisfying the dynamical cocycle identity (28). The difference operators M ( r ) O (1) ( z ) are given by evaluation of a universalelements from U ~ ( bb gl )[[ z, q ]] in the Fock representation. It means that thesolution J ( r ) , ( r ) ( z ) and Y ( r ) , ( r ) ( z ) are also given by evaluation of someuniversal elements. In particular Y ( r ) , ( r ) ( z ) = ev ( r ) ⊗ ev ( r ) ( Y ( z )) for someuniversal Y ( z ). In the following sections we find Y ( z ) explicitly.Finally, let us note here that by definition (26) we have J ( r ) , (0) ( z ) = J (0) , ( r ) ( z ) = 1 and thus: Y ( r ) , (0) ( z ) = Y (0) , ( r ) ( z ) = 1 , ∀ r. (30) The global dependence of the capping operator Ψ ( r ) ( z ) on the equivariantparameter a is governed by certain q -difference equation dual to qde, the socalled quantum Knizhnik-Zamolodchikov equation, [8] (Section 4.2):Ψ ( r ) ( z, aq ) E ( z, a ) = S ( z, a )Ψ ( r ) ( z, a ) (31)Moreover, under identification (22) the operators are explicitly expressiblethrough the universal R -matrices from Section 2.5: S ( z, a ) = z d (1) R ( r ) , ( r ) ( a ) , E ( z, a ) = z d (1) (cid:16) R ( r ) , ( r ) ∞ ( a ) (cid:17) . Proposition 3.
The operator Ψ ( r ) ( z, is a lower-triangular solution of wallKnizhnik-Zamolodchikov (wKZ) equation: z d (1) ( R − , ( r ) , ( r )0 ) − ~ Ω Ψ ( r ) ( z,
0) = Ψ ( r ) ( z, ~ Ω z d (1) (32) Proof.
The existence of a limit Ψ ( r ) ( z,
0) was shown in [8]. The proof followsfrom Proposition 1.Note that the operator ~ Ω z d (1) commutes with any block diagonal-operator,such as for example Ψ ( r ) ( z ~ r / ) ⊗ Ψ ( r ) ( z ~ − r / ). Thus, we conclude: Corollary 2.
The operator Y ( r ) , ( r ) ( z ) defined by (29) is a lower-triangularsolution of wKZ equation: z d (1) ( R − , ( r ) , ( r )0 ) − ~ Ω Y ( r ) , ( r ) ( z ) = Y ( r ) , ( r ) ( z ) ~ Ω z d (1) (33)19 .4 Uniqueness of solutions of wKZ equation Lemma 1.
For a given block diagonal operator D ( r ) , ( r ) there exists uniquelower triangular solution J ( r ) , ( r ) of wKZ: z d (1) ( R − , ( r ) , ( r )0 ) − ~ Ω J ( r ) , ( r ) = J ( r ) , ( r ) ~ Ω z d (1) (34) and diagonal part J ( r ) , ( r )(0) = D ( r ) , ( r ) .Proof. Write the last equation in the form: Ad ~ − Ω z − d (1) ( J ( r ) , ( r ) ) = ˜ RJ ( r ) , ( r ) with ˜ R = ~ − Ω ( R − , ( r ) , ( r )0 ) − ~ Ω . The operator ˜ R is strictly lower triangular,thus, taking the n -th component of this equation we obtain: Ad ~ Ω z − d ( J ( r ) , ( r )( n ) ) = J ( r ) , ( r )( n ) + · · · where dots stand for terms J ( r ) , ( r )( k ) with k < n . As the operator 1 − Ad ~ Ω z − d is invertible for general z we can solve this linear equation for J ( r ) , ( r )( n ) recur-sively through the lower terms J ( r ) , ( r )( k ) with k < n . By induction, all termsare thus expressed through the lowest term J ( r ) , ( r )(0) = D ( r ) , ( r ) . Corollary 3.
The solution of wKZ with J ( r ) , ( r )(0) = D ( r ) , ( r ) is given by J ( r ) , ( r ) = E ( r ) , ( r ) D ( r ) , ( r ) where E ( r ) , ( r ) is unique strictly lower triangularsolution.Proof. Indeed, by above Proposition there exist unique strictly lower-triangularsolution E ( r ) , ( r ) . The operator ~ Ω z d (1) commutes with any block diagonal op-erator D ( r ) , ( r ) and thus J ( r ) , ( r ) = E ( r ) , ( r ) D ( r ) , ( r ) is the solution of wKZwith J ( r ) , ( r )(0) = D ( r ) , ( r ) .Finally, we give explicit universal formula for the strictly lower triangularsolution E . Proposition 4.
There exist the universal element E ( z ) ∈ U ~ ( bb gl ) ⊗ [[ z ]] givenexplicitly by: E ( z ) = exp (cid:16) ∞ X k =1 n k − z − k K − k ⊗ K k K − k α − k ⊗ K k α k (cid:17) (35)20 uch that E = ev ( r ) ⊗ ev ( r ) ( E ( z )) is the strictly lower-triangular solution ofwKZ equation in F ( r ) ⊗ F ( r ) .Proof. To show that E ( z ) is a universal solution of wKZ it is enough tocheck that it solves the following equation in U ~ ( bb gl ) ⊗ ( z ): z − d (1) E ( z ) z d (1) = ( R − ) − ~ Ω E ( z ) ~ − Ω (36)The operator R − is given by (14). Note, that the operators z d (1) and ~ Ω areonly defined in the representation F ( r ) ⊗ F ( r ) . However, the operators Ad z d (1) and Ad ~ Ω are well defined in U ~ ( bb gl ) ⊗ : z − d (1) α − k ⊗ α k z d (1) = z − k α − k ⊗ α k , and Lemma 2 below gives: ~ Ω α − k ⊗ α k ~ − Ω = K k α − k ⊗ K − k α k From (14) we see that (36) is equivalent to the set of equations: n k K − k ⊗ K k z − k − z − k K − k ⊗ K k = − n k + n k − z − k K − k ⊗ K k which are identities. Lemma 2.
For all r , r the identity: ~ Ω α − k ⊗ α k ~ − Ω = K − k α − k ⊗ K k α k holds in F ( r ) ⊗ F ( r ) .Proof. Note that α − k ⊗ α k : F ( r )( n ) ⊗ F ( r )( n ) → F ( r )( n + k ) ⊗ F ( r )( n − k ) . Thus, from theexplicit action of ~ Ω on the fock space described in Proposition 1 we obtain: ~ Ω α − k ⊗ α k ~ − Ω = ~ k ( r − r ) / α − k ⊗ α k The central element K acts on F ( r ) by multiplication on a scalar ~ − r/ . Thelemma is proven. Corollary 4.
For all r and r we have: Y ( r ) , ( r ) ( z ) = ev ( r ) ⊗ ev ( r ) ( E ( z ) D ( z )) where D ( z ) universal operator such that D ( r ) , ( r ) ( z ) is block diagonal in everyrepresentation. .5 Universal form of Y ( z ) Our final step is to prove that D = 1 which means: Proposition 5.
For all r and r we have: Y ( r ) , ( r ) ( z ) = ev ( r ) ⊗ ev ( r ) ( E ( z )) Proof.
By Corollary 4 we have: Y ( r ) , ( r ) ( z ) = E ( r ) , ( r ) ( z ) D ( r ) , ( r ) for some block diagonal matrix D ( r ) , ( r ) and we need to show that D ( r ) , ( r ) =1. By Corollary 1 we have: Y ( r + r ) , ( r ) ( z ) Y ( r ) , ( r ) ( z ~ r ) = Y ( r ) , ( r + r ) ( z ) Y ( r ) , ( r ) ( z ~ − r ) (37)Let us consider left and right wKZ operators (they are coproducts of thewKZ operators in the first and second components respectively): A L ( J ) = R R z d (3) J z − d (3) ~ Ω +Ω A R ( J ) = R R z − d (1) J z d (1) ~ Ω +Ω where R as in Section 2.5. It is obvious that the left side of (37) satisfies theequation A L ( J ) = J . Similarly, the right side satisfies A R ( J ) = J . Thereforewe must have: A R ( Y ( r + r ) , ( r ) ( z ) Y ( r ) , ( r ) ( z ~ r )) = Y ( r + r ) , ( r ) ( z ) Y ( r ) , ( r ) ( z ~ r )Taking degree zero part of this equality in the third component we obtain: R − ~ − Ω z − d (1) D ( r + r ) , ( r ) Y ( r ) , ( r ) ( z ~ r ) z d (1) ~ Ω +Ω = D ( r + r ) , ( r ) Y ( r ) , ( r ) ( z ~ r )The block-diagonal operator D ( r + r ) , ( r ) commutes with ~ − Ω and z − d (1) so wecan rewrite the last equation in the form: R D ( r + r ) , ( r ) ~ − Ω z − d (1) Y ( r ) , ( r ) ( z ~ r ) z d (1) ~ Ω +Ω = D ( r + r ) , ( r ) Y ( r ) , ( r ) ( z ~ r )22y Lemma 3 this is equivalent to the equality: R D ( r + r ) , ( r ) ( z ~ r ) − d (1) Y ( r ) , ( r ) ( z ~ r )( z ~ r ) d (1) ~ − Ω = D ( r + r ) , ( r ) Y ( r ) , ( r ) ( z ~ r )By definition the operator Y r ,r ( z ) solves wKZ equation (34) which gives: R D ( r + r ) , ( r ) ( R ) − = D ( r + r ) , ( r ) (38)The same argument for A L gives similar restriction for the second componentof D : R D ( r ) , ( r + r ) ( R ) − = D ( r ) , ( r + r ) (39)The universal element has the form: D ( z ) = P i,j f i,j ( z ) b i ⊗ b j ∈ U ~ ( bb gl ) ⊗ [[ z ]]in a basis of degree zero elements b i . The first equation (38) gives ∆ op ( b i ) =∆( b i ). The only elements of U ~ ( bb gl ) with this properties are b i = K i . Simi-larly, the second (39) equation gives b j = K j . We conclude that the universaloperator has the following form: D ( z ) = P i,j f i,j ( z ) K i ⊗ K j . The diagonalpart of (30) gives D ( r ) , (0) ( z ) = D (0) , ( r ) ( z ) = 1 for all r and r . which meansthat f , ( z ) = 1 and f i,j ( z ) = 0 of i = 0 or j = 0. We conclude that D = 1.To finish the proof we need the following simple lemma. Lemma 3.
The following identity holds in F ( r ) ⊗ F ( r ) ⊗ F ( r ) ~ − Ω ( α − k ⊗ α k ⊗ ~ Ω = ( ~ r / ) − d (1) ( α − k ⊗ α k ⊗ ~ r / ) d (1) (40) Proof.
From explicit action of Heisenberg algebra we know that the operator α − k ⊗ α k ⊗ F ( r ) ⊗ F ( r ) ⊗ F ( r ) : F ( r )( n ) ⊗ F ( r )( n ) ⊗ F ( r )( n ) → F ( r )( n + k ) ⊗ F ( r )( n − k ) ⊗ F ( r )( n ) Thus, from the definition of ~ Ω given in Proposition 1 we obtain: ~ − Ω ( α − k ⊗ α k ⊗ ~ Ω = ~ − kr / ( α − k ⊗ α k ⊗ z d ( i ) from Section 2.6 we obtain the same result( ~ r / ) − d (1) ( α − k ⊗ α k ⊗ ~ r / ) d (1) = ~ − kr / ( α − k ⊗ α k ⊗ .6 Proof of Theorem 3 The theorem 3 in now proved. Indeed, by definition of J ( r ) , ( r ) (26) anddefinition of Y ( r ) , ( r ) ( z ) (9) we havelim a → Ψ ( r ) ( z ) = Y ( r ) , ( r ) ( z )Ψ ( r ) ( z ~ r ) ⊗ Ψ ( r ) ( z ~ − r )By Proposition 5 the operator Y ( r ) , ( r ) ( z ) is an evaluation of the universalelement (35), which is the claim of Theorem 3. (cid:3) In this section we consider the descendent bare vertex defined by (4). Thepush-forward in (4) can be computed explicitly by the equivariant localiza-tion to the locus of the fixed point set of G -action on the moduli space ofnonsingular quasimaps QM dp ( n, r ). This provides a formula for the barevertex as a power series in z with explicit coefficients. Here we sketch itsderivation and use it to prove the factorization Theorem 4.First, recall that the fixed set QM dp ( n, r ) G consist of finitely many isolatedpoints p which parameterize the following data: QM dp ( n, r ) G = { p = ( −→ λ , −→ d ) : |−→ λ | = n, |−→ d | = d } where −→ λ = ( λ (1) , · · · , λ ( r ) ) is an r -tuple of Young diagrams with total numberof |−→ λ | = | λ (1) | + · · · + | λ ( r ) | = n boxes. The degree data −→ d = { d (cid:3) ≥ (cid:3) ∈ −→ λ } assigns a non-negative integer to each box of −→ λ such that |−→ d | = P (cid:3) ∈−→ λ d (cid:3) = d . The degree −→ d is assumed to be stable, which meansthat the data d (cid:3) for (cid:3) ∈ λ ( i ) for i = 1 , · · · , r define r plane partitions . Theequivariant localization formula thus gives the following power series for thebare descendent vertex: V ( τ ) ( z ) = ∞ X d =0 X p ∈ QM dp ( n,r ) G a ( p ) z d ∈ K G ( M ( n, r )) loc ⊗ Q [[ z ]] (41)where a ( p ) is a contribution of a fixed point p .24he fixed set M ( n, r ) T consists of isolated points labeled by r -tuples ofYoung diagrams −→ λ with |−→ λ | = n . In fact, this is a special case of thefixed points on the moduli space of quasimaps (the degree zero or constantquasimaps) M ( n, r ) = QM p ( n, r ). The classes of the fixed points [ −→ λ ] ∈ K G ( M ( n, r )) loc form a basis in the localized K -theory. Assume that in thisbasis the bare vertex has the expansion V ( τ ) ( z ) = P |−→ λ | = n V ( τ ) −→ λ ( z )[ −→ λ ]. In (41)only the fixed points p = ( −→ ν , −→ d ) with −→ ν = −→ λ contribute to the coefficient V ( τ ) −→ λ ( z ). Therefore we have: V ( τ ) −→ λ ( z ) = X d (cid:3) ≥ (cid:3) ∈−→ λ a −→ λ (( −→ λ , −→ d )) z |−→ d | where the sum runs over the stable degrees −→ d . Before we give an explicitformula for a −→ λ (( −→ λ , −→ d )) let us introduce the following auxiliary functions:we set n ( (cid:3) ) = k if (cid:3) ∈ λ ( k ) and denote by x ( (cid:3) ) and y ( (cid:3) ) the standardcoordinates of a box in a partition. Define: ϕ −→ λ ( (cid:3) ) = a n ( (cid:3) ) t x ( (cid:3) )1 t y ( (cid:3) )2 and V ( −→ λ , −→ d ) = X (cid:3) ∈−→ λ ϕ −→ λ ( (cid:3) ) q d (cid:3) , W = a + · · · + a r . (42)For each fixed point we define a Laurent polynomial in equavariant parame-ters S ( −→ λ , −→ d ) ∈ K G ( pt ) by: S ( −→ λ , −→ d ) = W ∗ V ( −→ λ , −→ d ) + W V ( −→ λ , −→ d ) ∗ t − t − − V ( −→ λ , −→ d ) ∗ V ( −→ λ , −→ d )(1 − t − )(1 − t − ) . (43)The symbol ∗ corresponds to taking dual in K -theory and acts by inversionof all weights ∗ : w w − . 25 .2 Virtual tangent space The G -character of the virtual tangent space T vir ( −→ λ ,dd ) QM dp ( n, r ) is given ex-plicitly by [5]: T vir ( −→ λ , −→ d ) QM dp ( n, r ) = S ( −→ λ , −→ S ( −→ λ , −→ d ) − S ( −→ λ , −→ q − −→ { d (cid:3) = 0 : ∀ (cid:3) ∈ −→ λ } . The stability of −→ d implies that T vir ( −→ λ , −→ d ) QM dp ( n, r )is a Laurent polynomial without a constant term and with both positive andnegative coefficients (negative coefficients appear because it is a virtual tan-gent space).As we noted above M ( n, r ) = QM p ( n, r ) and thus the T -character ofthe tangent space to the instanton moduli space at a point −→ λ ∈ M ( n, r ) T isgiven by T −→ λ M ( n, r ) = S ( −→ λ , −→ Let τ ∈ K GL ( n ) = Λ[ x ± , x ± , · · · , x ± n ] a symmetric polynomial representingthe descendent. We define the evaluation of such polynomial at fixed point( −→ λ , −→ d ) by τ ( −→ λ , −→ d ) = τ ( x (cid:3) = ϕ −→ λ ( (cid:3) ) q d ( (cid:3) ) ) (recall that the r -tuple of Youngdiagrams −→ λ corresponding to a fixed point ( −→ λ , −→ d ) has exactly |−→ λ | = n boxes so we may think that the variables x i are labeled by boxes in −→ λ ).The localization in the equivariant K -theory gives the coefficients in (41) : V ( τ ) −→ λ ( z ) = X d (cid:3) ≥ (cid:3) ∈−→ λ ˆ a (cid:16) T vir ( −→ λ , −→ d ) QM dp ( n, r ) (cid:17) τ ( −→ λ , −→ d ) ( − | d | r q − r | d | z | d | (45) In this formula q − r | d | is the contribution of polarization to virtual structure sheaf- see (6.1.9) in [5]. In fact, as we discussed in [8] the capping operator is given by thefundamental solution of qde in which we should substitute z → ( − r z . In this paper,however, we prefer to move this sign to the vertex, in which case all final formulas looksimpler. This corresponds to the appearance of factor the ( − | d | r in (45). a ( x + y ) = ˆ a ( x )ˆ a ( y ) , ˆ a ( x ) = 1 x − x − . The coefficients of the power series for the bare descendent vertex (45) arerational functions of the equivariant parameters a , · · · , a r corresponding toframing torus A . The action of subtorus C ⊂ A corresponds to the substitu-tion of framing characters( a , · · · , a r , a r +1 · · · , a r ) → ( a , · · · , a r , aa r +1 · · · , aa r ) . (46)We are interested in the limit lim a → V ( r ) , ( τ ) ( z ) under this substitution. Let −→ λ = ( λ (1) , · · · , λ ( r ) , λ ( r +1) · · · , λ ( r ) ) be a fixed point on M ( n, r ). Denote by −→ λ = ( λ (1) , · · · , λ ( r ) ) and −→ λ = ( λ ( r +1) · · · , λ ( r ) ) the corresponding classeson M ( n, r ) and M ( n, r ). The theorem 4 is equivalent to the followingfactorization of the coefficients of the bare vertex:lim a → V ( r ) , ( τ ) −→ λ ( z ) = V ( r ) , ( τ ) −→ λ ( z ~ r ) V ( r ) , (1) −→ λ ( z ~ − r q − r ) (47)To prove it, first note that after substitution (46) for the functions (42) weobtain: V ( −→ λ , −→ d ) = V ( −→ λ , −→ d ) + a V ( −→ λ , −→ d ) , W = W + a W where W = a + · · · a r and W = a r +1 + · · · + a r . Let us consider thecontribution to the vertex of the term V ( −→ λ , −→ d ) V ( −→ λ , −→ d ) ∗ in (43). The con-tribution of a weight w ∈ V ( −→ λ , −→ d ) V ( −→ λ , −→ d ) ∗ to ˆ a ( T vir ( −→ λ , −→ d ) QM dp ( n, r )) has thefollowing form:ˆ a (cid:16) − w (1 − t − )(1 − t − ) (cid:17) = ( w − w − )( w t − t − − w − t t )( w t − − w − t )( w t − − w − t )If w ∈ V ( −→ λ i , −→ d i ) V ( −→ λ j , −→ d j ) ∗ with i = j then w ∼ a ± and in the limit a → a → V ( −→ λ , −→ d ) V ( −→ λ , −→ d ) ∗ to T vir −→ λ , −→ d QM dp ( n, r ) splits to asum of contributions of V ( −→ λ , −→ d ) V ( −→ λ , −→ d ) ∗ and V ( −→ λ , −→ d ) V ( −→ λ , −→ d ) ∗ .Thus, the contribution to the vertex, given by roof function ˆ a factors to aproduct of corresponding contributions.Second, we analyse terms V ( −→ λ i , −→ d i ) W j with i = j . We assume i = 1 and j = 2. The corresponding contribution to S ( −→ λ , −→ d ) has the form: W ∗ V ( −→ λ , −→ d ) − V ( −→ λ , −→ q − W ( V ( −→ λ , −→ d ) ∗ − V ( −→ λ , −→ ∗ )( q − t t = r P j =1 P (cid:3) ∈−→ λ (cid:16) ϕ −→ λ ( (cid:3) ) aa j d ( (cid:3) ) − P i =0 q i − aa j ϕ −→ λ ( (cid:3) ) t t q d ( (cid:3) ) − P i =0 q − i (cid:17) Thus the corresponding contribution to the vertex gives: Q (cid:3) ∈−→ λ r Q j =1 d ( (cid:3) ) − Q i =0 (cid:16) aajϕ −→ λ (cid:3) ) qit t q (cid:17) / − (cid:16) aajϕ −→ λ (cid:3) ) qit t q (cid:17) − / (cid:16) ϕ −→ λ (cid:3) ) qiaaj (cid:17) / − (cid:16) ϕ −→ λ (cid:3) ) qiaaj (cid:17) − / a → −→ Q (cid:3) ∈−→ λ r Q j =1 d ( (cid:3) ) − Q i =0 (cid:16) − t / t / q / (cid:17) = ( − ( ~ q ) / ) r |−→ d | Same calculation for i = 2 , j = 1 gives ( − ( ~ q ) / ) − r |−→ d | . The terms V ( −→ λ , −→ d ) W and V ( −→ λ , −→ d ) W do not depend on a . We conclude, that the contributionof V ( −→ λ , −→ d ) W to the vertex in the limit a → V ( −→ λ , −→ d ) W and V ( −→ λ , −→ d ) W shifted by some powers of − ( ~ q ) / . Fi-nally, we have: lim a → ϕ −→ λ ( (cid:3) ) = ( ϕ −→ λ ( (cid:3) ) if (cid:3) ∈ −→ λ (cid:3) ∈ −→ λ which for a symmetric polynomial τ means lim a → τ ( −→ λ , −→ d ) = τ ( −→ λ , −→ d ) Over-28ll, we obtain:lim a → V ( r ) , ( τ ) −→ λ ( z ) = P −→ d , −→ d ˆ a (cid:16) T vir ( −→ λ , −→ d ) QM d p ( n , r ) (cid:17) ˆ a (cid:16) T vir ( −→ λ , −→ d ) QM d p ( n , r ) (cid:17) × ( − ( ~ q ) ) r |−→ d | ( − ( ~ q ) ) − r |−→ d | z |−→ d | + |−→ d | τ ( −→ λ , −→ d )( − | d | r q − | d | r = V ( r ) , ( τ ) −→ λ (cid:16) z ~ r (cid:17) V ( r ) , (1) −→ λ (cid:16) z ~ − r q − r (cid:17) (cid:3) References [1] M. Haiman. Notes on Macdonald polynomials and the geometry ofHilbert schemes. In
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