AGT relations for sheaves on surfaces

We consider a natural generalization of the Carlsson–Okounkov Ext operator on the K –theory groups of the moduli spaces of stable sheaves on a smooth projective surface. We compute the commutation relations between the Ext operator and the action of the deformed W –algebra on K –theory, which was developed by the author in previous work. The conclusion is that the Ext operator is closely related to a vertex operator, thus giving a mathematical incarnation of the Alday–Gaiotto–Tachikawa correspondence for a general algebraic surface.

Geometry & Topology Assumption A implies that M is proper and there exists a universal sheaf 1 (1-2)

U M S
Assumption S implies that M is smooth.

1.2
The enumerative geometry of the moduli space of stable sheaves is quite rich, as evidenced by Donaldson invariants arising as certain integrals of cohomology classes on M. In the present paper, we will consider algebraic K-theory instead of cohomology, a process which accounts for the adjective "deformed" in the representation-theoretic structures explained in Section 1.6.Explicitly, we consider the following algebraic K-theory groups with Q coefficients: (1-3) Let m 2 Pic.S/, and consider two copies M and M 0 of the moduli space (1-1).These two copies may be defined with respect to a different c 1 and stability condition H , but we assume that the rank r of the sheaves parametrized by M and M 0 is the same.In this paper, we will mostly be concerned with the virtual vector bundle (1-4) (a straightforward generalization of the construction of Carlsson and Okounkov [7]) given by (1-5) E m D R.m/ R .RHom.U 0 ; U ˝m//: The RHom is computed on M M 0 S: the notation U, U 0 and m stands for the pullback of the universal sheaf from M S and M 0 S, respectively, as well as the pullback of the line bundle m from S. Similarly, W M M 0 S !M M 0 is the standard projection, so E m is a complex of coherent sheaves on M M 0 .
1.3 Any Schur functor applied to E m gives rise to a K-theory class on M M 0 , which in turn induces an operator from K M 0 to K M via the usual formalism of correspondences.With this in mind, let us consider the following immediate generalization of Carlsson and Okounkov [7,Equation (3)] and Carlsson, Nekrasov and Okounkov [6,Equation (19)].
Definition 1.4 Consider the so-called Ext operator with 1 and 2 as in (1)(2)(3)(4).The pushforward and pullback maps are well-defined due to the properness and smoothness of M and M 0 , respectively.
In (1)(2)(3)(4)(5)(6), the symbol ^ E m denotes the total exterior power of E m ; as E m is in general a complex of coherent sheaves, some explanation is in order.Specifically, consider where the right-hand side is the power series expansion of a rational function in t; see Section 3.1 for details.Then the quantity ^ E m in (1-6) denotes the t D 1 specialization of (1)(2)(3)(4)(5)(6)(7).If this specialization is not well-defined, then all the results in Sections 1.6 and 1.9 hold with m replaced by m=t, and with all formulas being equalities of rational functions in t; see Section 3.1 for details.
Example 1.5 Let M D M 0 and m D O S =t, with t being a formal parameter.Then Assumption S implies that E O S is locally free (up to a constant sheaf) and that where M M 0 denotes the diagonal.By a simple computation involving correspondences, the isomorphism above implies that Tr.A O S =t / D X k 0 . t/ k .M; ^k Tan M / (up to a constant rational function in t).The right-hand side is the t -genus of the moduli space M, as considered for example in Göttsche and Kool [10].
1. 6 In the present paper, we will seek to determine the Ext operator A m using the representation-theoretic properties of the vector space K M .To this end, we need to make K M into a representation of an appropriate algebra which is "big" enough, in order to constrain the operator A m as much as possible.A candidate for such an algebra is A r , namely a particular integral form of the deformed W -algebra of type gl r (initially defined in Awata, Kubo, Odake and Shiraishi [1] and Feigin and Frenkel [8]).
The main purpose of our work in [15; 17; 16] is to construct an action A r Õ K M ; we will recall the construction in Section 2, but let us summarize the main idea here.In [17, Section 6.7], we construct certain geometric operators Under Assumptions A and S, we show in [16,Theorem 4.15] that the operators W n;k satisfy the quadratic commutation relations developed in [1] and [8]; see  for the specific form of these relations in our language.In [17, Theorem 6.9], we further show that W n;k D 0 for all n 2 Z and k > r , which tautologically implies that the operators (1-8) yield an action Given two copies M and M 0 of the moduli space of stable sheaves, each with its own universal sheaf U and U 0 , respectively, we may write for the determinant line bundles on M S and M 0 S, respectively.We set which is the class of a line bundle on M M 0 S (it is implicit that m and q are pulled back from S).Our main result, which will be proved in Section 3, is: We have the following interaction between the Ext operator (1-6) and the generators (1-8) of the W -algebra action: where W k .x/D P n2Z W n;k =x n .The series coefficients of the two sides of (1-12) are maps K M 0 !K M S which arise from certain correspondences in K M M 0 S .
1.8 A major motivation for the study of the Ext operator A m stems from mathematical physics: as explained in Carlsson, Nekrasov and Okounkov [6], the operator A m encodes the contribution of bifundamental matter to partition functions of 5d supersymmetric gauge theory on the algebraic surface S times a circle.Moreover, the deformed Walgebra A r encodes symmetries of Toda conformal field theory.In this language, (1-12) becomes a mathematical manifestation of the Alday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory and conformal field theory, by describing the Ext operator A m in terms of its commutation with W -algebra generators.To the author's knowledge, the present paper is the first mathematical treatment of AGT over general algebraic surfaces in rank r > 1 (the reference [6] used different techniques from ours to describe the Ext operator in the r D 1 case).
However, we note that formulas (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) are not enough to completely determine A m for a general smooth projective surface S, and one should instead work with a deformed vertex operator algebra which properly contains several deformed W -algebras A r .In the nondeformed case, a potential candidate for such a larger algebra was studied in Feigin and Gukov [9], where the authors expect that it contains operators which modify sheaves on S along entire curves, on top of our operators W n;k which modify sheaves at individual points.While we give a complete algebrogeometric description of the latter operators, we do not have such a description for the former operators.Once such a description is available, we hope that one can extend Theorem 1.7 to a bigger vertex operator algebra properly containing A r .
There is a situation where formulas (1-12) do indeed determine the Ext operator A m completely: this corresponds to taking S D A 2 , replacing M by the moduli space of framed rank r sheaves on the projective plane, and working with torus equivariant K-theory; see Section 4.1 for details.In this particular case, we showed in [14] that K M is isomorphic to the universal Verma module of A r .Theorem 1.7 holds in the situation at hand, and we will show in Theorem 4.5 that our formulas completely determine the Ext operator A m .This precisely yields the AGT correspondence for 5d supersymmetric gauge theory on A 2 S 1 ; see for instance Braverman, Finkelberg and Nakajima [4], Bruzzo, Pedrini, Sala and Szabo [5], Maulik and Okounkov [12] and Schiffmann and Vasserot [18] for the history of this correspondence in mathematical language.
Theorem 1.10 We have the following interaction between the Ext operator (1-6) and the Heisenberg operators P ˙n for all n > 0: In A r , the series W r .x/matches the normal-ordered exponential of the generating series of the P n ; see Theorem 2.8.With this in mind, it is straightforward to show that the k D r case of Theorem 1.7 follows from Theorem 1.10.
For any ˛2 K S , we will write P n f˛g for the composition where proj 1 and proj 2 are the projections from M S to M and S, respectively.Let q 1 and q 2 denote the Chern roots of the cotangent bundle 1 S .Any symmetric Laurent polynomial in q 1 and q 2 gives rise to a well-defined element of K S , via .q n 1/q nr OEn q 1 OEn q 2 ; where OEn x D 1 C x C C x n 1 .The expression in curly brackets is an element of K S because OEn q 1 OEn q 2 is a unit in the ring K S .
Remark To see that OEn q 1 OEn q 2 is a unit in the ring K S , since the Chern character gives us an isomorphism K S Š A .S; Q/, we have q 1 C q 2 D OE 1 S 2 2 C N and q D OE! S 2 1CN , where N K S denotes the nilradical.Therefore OEn q 1 OEn q 2 2 n 2 CN , and is thus invertible in the ring K S .
Acknowledgements I thank Boris Feigin, Sergei Gukov, Hiraku Nakajima, Nikita Nekrasov, Andrei Okounkov, Francesco Sala and Alexander Tsymbaliuk for many interesting discussions on the subject of Ext operators and W -algebras.I gratefully acknowledge the support of NSF grant DMS-1600375.
2 The moduli space of sheaves 2.1 Throughout the present paper, we will work with smooth projective varieties over the field C. For such varieties X , we let K X D K 0 .X / ˝Z Q be the Grothendieck group of the category of coherent sheaves on X , with scalars extended to Q. Derived tensor product yields a ring structure on K X , and we have pullback and pushforward maps for any proper l.c.i.morphism X !Y .Definition 2.2 Given smooth projective varieties X and Y , any class 2 K X Y (called a "correspondence" in this setup) defines an operator where proj X ; proj Y denote the projection maps from X Y to X and Y , respectively.
The composition of operators (2-1) can also be described as a correspondence where proj X Y , proj Y Z and proj X Z are the standard projections from X Y Z to X Y , Y Z and X Z. Throughout the present paper, all operators K Y !K X arise from explicit correspondences.While we will use the language of composition of operators for convenience, what is really happening behind the scenes is composition of correspondences under the operation .; 0 / 7 ! 00of (2-3).

2.3
In Section 1.6, we referred to various operators K M !K M S as defining an action of a certain algebra on K M , and we will now explain the meaning of this notion.Given two arbitrary homomorphisms (of abelian groups) their "product" xyj is defined as the composition where S ! S S is the diagonal.It is easy to check that .xyj/zj D x.yzj /j , hence the aforementioned notion of product is associative, and it makes sense to define x 1 x n j for arbitrarily many operators x 1 ; : : : ; Similarly, given two operators (2-4), we may define their commutator as the difference of the two compositions where swapW S S ! S S is the permutation of the two factors.In all cases studied in the present paper, we will have 2 OEx; y D .z/ for some K M z !K M S which is uniquely determined (the diagonal embedding is injective because it has a left inverse), and which will be denoted by z D OEx; y red .We leave it as an exercise to the interested reader to prove that the commutator satisfies the Leibniz rule in the form OExyj ; z red D xOEy; z red j C OEx; z red yj , and the Jacobi identity in the form OEx; y red ; z red C OEy; z red ; x red C OEz; x red ; y red D 0.
Finally, we consider the ring homomorphism K D ZOEq ˙1 1 ; q ˙1 2 Sym !K S given by sending q 1 and q 2 to the Chern roots of the cotangent bundle of S (therefore, q D q 1 q 2 goes to the class of the canonical line bundle).We will often abuse notation, and write q 1 ; q 2 ; q for the images of the indeterminates in the ring K S .For any 2 K and any operator (2-4), we may define their product as the composition where we identify 2 K with its image in K S .With this in mind, the ring K S can be thought of as the "ring of constants" for the algebra of operators (2-4).

2.4
Recall the universal sheaf (1-2), and consider the derived scheme Since U is isomorphic to a quotient V=W of vector bundles on M S (Proposition 2.2 of [15]), the projectivization in (2-5) is defined as the derived zero locus of a section of a vector bundle on the projective bundle P M S .V/.However, it was shown in [15, Proposition 2.10] that under Assumption S, the derived zero locus is actually a smooth scheme whose connected components are given by and F 0 x F means that F 0 F and the quotient F=F 0 is isomorphic to the length one skyscraper sheaf at the point x 2 S.This scheme comes with projection maps (2-7) More generally, we defined a derived scheme Z 2 in [17, Definition 4.17], which was shown (under Assumption S, in [17,Proposition 4.21]) to be a smooth scheme whose connected components are given by This scheme is equipped with projection maps as in (2-9) below, but we observe that the rhombus is not derived Cartesian (and this is key to our construction): (2-9) Note that all of the maps in the diagram above are proper, l.c.i.morphisms.Define (2-10) whose connected components are given by derived fiber products (2-11) Z cCn;c D Z cCn;cCn 2 Z cCn 1;cCn 2 : : : While Z n is a derived scheme, we note that its closed points are all of the form (2-12) Z cCn;c D f.F cCn ; : : : ; F c / sheaves with F cCn x x F c for some x 2 Sg: Therefore, we have the following projection maps, which only remember the smallest and the largest sheaf in a flag (2-12): (2-13) (the notation generalizes (2-7)).In diagram (2-13), the maps p ˙are l.c.i.morphisms, and the maps p ˙ p S are proper (they inherit these properties from the maps in (2-9)).Finally, we consider the line bundles L 1 ; : : : ; L n on Z n , whose fibers are given by (2-14) L i j .F cCn ;:::;F c / D F cCn i;x =F cCn iC1;x on the connected component Z cCn;c Z n .
Note that (2-20) is an infinite sum, but its action on K M is well-defined because the operators L n;k (resp.U n;k ) increase (resp.decrease) the c 2 of stable sheaves by n, and Bogomolov's inequality ensures that the moduli space of stable sheaves is empty if c 2 is small enough.
The following theorem was stated in [17, Theorem 3.13 and Proposition 3.15] and proved in [16,Theorem 4.15] under Assumption S. Theorem 2.10 We have the following formulas for all n; n 0 2 Z and k; k 0 2 N: (2-29) where the coefficients c m;m 0 ;l;l 0 n;n 0 ;k;k 0 .q 1 ; q 2 / 2 K S were computed algorithmically in [17].They are certain universal symmetric Laurent polynomials in q 1 and q 2 .Indeed, we show in [17,Theorem 3.13] that (2-28) is equivalent to the defining relation in the deformed W -algebra A r (with replaced by .1 q 1 /.1 q 2 /).Similarly, relation In other words, these power series are considered as operators We will also consider the operators Furthermore, we will consider the generating series and also set The definition of the W -algebra generators in (2-20) is equivalent to where D x is the q-difference operator in the variable x, ie D x .f.x//D f .xq/.In formula (2-35), we place all powers of D x to the right (resp.left) of all powers of x when writing down the power series L.x; yD x / (resp.U.xq; yD x /).In terms of generating series, formula (2-30) reads (2-36) OEW k .x/;P ˙n D ˙OEn q 1 OEn q 2 OEk q n q n.r k/ı C ˙ x ˙nW k .x/: 2.12 Given a rational function F.z/, whose set of simple poles is partitioned into two disjoint sets P 1 t P 2 (which may be empty), we will write (2-37) The first equality is a definition, and the second equality is the residue theorem.If F.z 1 ; : : : ; z n / is a rational function with simple poles of the form z i D c and z i D z j for various c 2 P 1 t P 2 and various scalars in some set Q, then we set (2-38) as the result of the n-step process which starts with F.z 1 ; : : : ; z n /=z 1 z n , and at the i th step replaces a rational function in z i ; : : : ; z n by the sum of its residues of the form z i D c 1 i 1 for various c 2 P 1 and 1 ; : : : Just like in (2-37), the residue theorem implies that the answer is the same as .1/ n times the result of the n-step process which starts with F.z 1 ; : : : ; z n /=z 1 z n , and at the i th step replaces a rational function in z 1 ; : : : ; z nC1 i by the sum of its residues of the form z nC1 i D c 1 i 1 for various c 2 P 2 and 1 ; : : : Proposition 2.13 [17, following the proof of Proposition 5.12] We have the following formulas for the maps (2-13): .p C p S / r .L 1 ; : : : ; .p p S / r .L 1 ; : : : ; where x/ and r .z 1 ; : : : ; z n / is a rational function with coefficients in .p˙ p S / .K M S / whose poles are all of the form z i D c, where c 2 f0; 1g t P for some finite set P.
Note that the integrands in (2-39)-(2-40) have poles when z i equals q 1 or 0 times one of the Chern roots of U. Thus, the location of the symbol U in the subscripts of the integrals (2-39)-(2-40) indicates whether these poles are thought to lie in the set P 1 or P 2 for the sake of the notation (2-37).
3 Computing the Ext operator 3.1 To properly define the Ext operator (1-6), note that the complex E m of (1-4) can be written as a difference V 1 V 2 of vector bundles.Then we define and interpret it as a rational function in t, with coefficients in K M M 0 .Strictly speaking, the object ^ E m in (1-6) refers to the specialization of this rational function at t D 1.

3.2
The main goal of the present section is to compute the commutation relations between the Ext operator A m W K M 0 !K M of (1-6) and the operators A m P ˙n henceforth refers to and analogously for W n;k instead of P ˙n.As opposed to the operators (3-2), the operator A m acts nontrivially between all components of the moduli space In principle, the moduli spaces of sheaves in the domain and codomain can correspond to different choices of first Chern class and stability condition, but we always require them to have the same rank r .Therefore, there are two universal sheaves of the same rank r , where M (resp.M 0 ) is the union of the moduli spaces that appear in the codomain (resp.domain) of .The determinants of these universal sheaves are denoted by u and u 0 , respectively, as in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10).
studied in the present paper (such as the compositions W n;k A m or P ˙nA m that appear in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (1-15)) arise from K-theory classes on M M 0 S .Then the product qz refers to the operator corresponding to the class proj S .q/, while the product z refers to the operator corresponding to the class where M M 0 S proj M S ; proj M 0 S ; proj S !M S; M 0 S; S are the projections.Equation (3-4) is best restated in the language of correspondences from Section 2.1.In these terms, P ˙n is given by a K-theory class supported on the locus where F 0 nx F means that F 0 F and that F=F 0 is a length n sheaf supported at x. Then (3-4) merely states that the universal sheaves U cCn and U c have isomorphic determinants when restricted to C. This is just the version "in families" of the wellknown statement that a codimension-2 modification of a torsion-free sheaf does not change its determinant.As a consequence of Proposition 3.4, of (1-11) will behave just like a constant in all our computations, ie it will not matter where we insert in any product of operators among P ˙n, H ˙n and W n;k .

3.5
Our main intersection-theoretic computation is the following:

Lemma 3.10
We have the following relations involving the Ext operator A m : The two sides of (3-42) and (3-43) map K M 0 to K M S y 1 .The symbol j applied to any term that involves three of the series L; E; U means that we restrict a certain operator K M 0 !K M S S S y 1 to the small diagonal.
Proof of Theorem 1.7 In terms of the generating series (2-32), formulas (3-42) and (3-43) take the form Now let us replace the variable y by the symbol yD x , where D x denotes the qdifference operator D x .f.x//D f .xq/.However, we make the following prescription: in the first equation above, the D x 's are placed to the right of all x's, while in the second equation, the D x 's are placed to the left of all the x's.We thus obtain Now let us multiply the first equation on the right by U.qx; yD x / (with the D x 's placed to the left of all the x's) and the second equation on the left by L.x ; yD x =m/ (with the D x 's placed to the right of all the x's): .1 x/A m L.x; yD x /E.yD x /U.xq; The two terms in the right-hand sides of the above equations are pairwise equal to each other (this is not manifestly obvious for the second term, because y differs from yq, but this is a consequence of commuting D x past x).We conclude that Recalling the definition (2-35), this implies Taking the coefficient of .yDx / k implies (1-12).In doing so, the right-most factor 1 x changes into 1 x=q k due to the fact that the operators 1=D k x must pass over it.
The action of an arbitrary generator W n;k on the basis vector (4-2) is prescribed by the commutation relations (2-28), together with the relations W 0;k j¿i D e k .u 1 ; : : : ; u r /j¿i for all k; where e k denotes the k th elementary symmetric polynomial.
Theorem 4.3 [14,Theorem 3.12] We have an isomorphism of modules for the deformed W -algebra of type gl r (the action on the left-hand side is given by (4-1)) (4-3) K M Š M u 1 ;:::;u r ; induced by sending the K-theory class of the structure sheaf of M 0 M to j¿i.

4.4
The Ext (respectively vertex) operator A m (respectively ˆm) for S D A 2 was studied in [14, Section 4], where we obtained an analogue of Theorem 1.7 in the case k D 1 (some coefficients in the formulas of loc.cit.differ from those of the present paper, because their operator A m differs from ours by an equivariant constant).However, having only proved the case k D 1 in loc.cit.led to weaker formulas than (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12).Thus, the present paper strengthens the results of loc.cit.; see Remark 4.8 therein.Specifically, Corollary 1.11 completely determines the operator ˆm (hence also A m ) in the case S D A 2 , due to Theorems 4.3 and 4.5.

1
Fix a smooth projective surface S over an algebraically closed field of characteristic zero (henceforth denoted by C), and invariants .r;c 1 / 2 N H 2 .S; Z/.An important object in algebraic geometry is the moduli space (1-1)M D

( 2 - 1 X 1 X
29) is the defining relation in the deformed Heisenberg algebra.As we explained in [17, Definition 5.2 and formulas (5.20)-(5.21)]and proved in [16, Theorem 4.15], the fact that the operators (2-20), (2-21) and (2-23) satisfy the relations in Theorem 2.10 is precisely what we mean when we say that the deformed W -algebra A r and the deformed Heisenberg algebra act on the groups K M .2.11 Let us consider the operators of Section 2.5 and form the generating series (2-31) L n .y/D kD1 L n;k .y/ k and U n .y/D kD1 U n;k .y/ k : 20), (2-21) and (2-23) for all n 2 Z and n 0 ; k 2 N.One must be careful what one means by "commutation relation".While the operator P ˙nA m unambiguously refers to K M 0 A m !K M P ˙n !K M S ;

Proposition 3 . 4
We have the equality of correspondencesK M c˙n !K M c S (3-4) P ˙n .detU c˙n / D .detU c / P˙n for all c 2 Z. Formula (3-4) also holds with P ˙n replaced by W n;k or H ˙n.

0
Cn;c 0 , where U denotes the universal sheaf on M c S, and (3-14) .p 0Id/ E m D .p 0 C Id/ E m .L 1 C C L n /.p 0 S Id/ .U 0 _ m/ on Z c;c n M c 0 , where U 0 denotes the universal sheaf on M c 0 S. Proof To prove (3-13), consider the diagram (3-15)
homotopic to the identity in dimension 3, II: Branching foliations THOMAS BARTHELMÉ, SÉRGIO R FENLEY, STEVEN FRANKEL and RAFAEL POTRIE 3183 The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores MARTIN BRIDGEMAN, JEFFREY BROCK and KENNETH BROMBERG 3229 Weighted K-stability and coercivity with applications to extremal Kähler and Sasaki metrics VESTISLAV APOSTOLOV, SIMON JUBERT and ABDELLAH LAHDILI 3303 Anosov representations with Lipschitz limit set MARIA BEATRICE POZZETTI, ANDRÉS SAMBARINO and ANNA WIENHARD 3361 The deformation space of geodesic triangulations and generalized Tutte's embedding theorem YANWEN LUO, TIANQI WU and XIAOPING ZHU and (3-12) by ‡ n;y D .Id p S p / yq/U.x; yq/: Change the variables x 7 !xq, y 7 !y=q in the second equation, and multiply the first equation by E.y/ and the second equation by E.y=m/.Thus we obtain The subscription price for 2023 is US $740/year for the electronic version, and $1030/year ( C $70, if shipping outside the US) for print and electronic.Subscriptions, requests for back issues and changes of subscriber address should be sent to MSP.Geometry & Topology is indexed by Mathematical Reviews, Zentralblatt MATH, Current Mathematical Publications and the Science Citation Index.Geometry & Topology (ISSN 1465-3060 printed, 1364-0380 electronic) is published 9 times per year and continuously online, by Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.Periodical rate postage paid at Oakland, CA 94615-9651, and additional mailing offices.POSTMASTER: send address changes to Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.