aa r X i v : . [ m a t h . AG ] A ug W –ALGEBRAS ASSOCIATED TO SURFACES ANDREI NEGUT , Abstract.
We define an integral form of the deformed W –algebra of type gl r ,and construct its action on the K –theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S , under certain assumptions.Our construction generalizes the action studied by Nakajima, Grojnowski andBaranovsky in cohomology, although the appearance of deformed W –algebrasby generators and relations is a new feature. Physically, this action encodesthe AGT correspondence for 5d supersymmetric gauge theory on S × circle. Introduction
The purpose of the present paper is to study the representation theory of the K –theory groups of moduli spaces of sheaves on a smooth surface S over an alge-braically closed field of characteristic 0 (henceforth denoted by C ). For any r > Z [ q ± , q ± ]-algebra A r and construct a module for the special-ization of A r when q and q are set equal to the Chern roots of Ω S . When S = A ,this module is the C ∗ × C ∗ equivariant K –theory of the moduli space of rank r framed sheaves on P . When S is projective, the module is defined as follows. Fix c ∈ H ( S, Z ) and an ample divisor H ⊂ S such that gcd( r, c · H ) = 1. Let:(1.1) K M = ∞ M c = ⌈ r − r c ⌉ K ( M ( r,c ,c ) )where M ( r,c ,c ) denotes the moduli space of stable sheaves on S with Chernclasses ( r, c , c ) (see [15] for a review of the theory). The fact that stablesheaves have c bounded below by r − r c is called the Bogomolov inequality. Thecoprimality of r and c · H is called Assumption A in [23], and it implies that: • every semistable sheaf is stable, hence M is projective • there exists a universal sheaf U on M × S where (once we have fixed r and c ) we will always write: M = ∞ G c = ⌈ r − r c ⌉ M ( r,c ,c ) As we recall in Section 3, A r is closely related to the deformed W –algebra of type gl r ([1], [8]). It is generated over Z [ q ± , q ± ] by symbols W n,k indexed by a “degree” , n ∈ Z and a “level” k ∈ { , ..., r } . There exists an r → ∞ limit of this construction,which we denote by A ∞ , with generators W n,k indexed by n ∈ Z , k ∈ N . We have:(1.2) A r = A ∞ ( W n,k ) n ∈ Z ,k>r Our construction for the action A r y K M can be summarized logically as follows:(1.3) A y K M A ∞ ֒ → b A ) ⇒ A ∞ y K M (1.4) { W n,k } k>rn ∈ Z act by 0 in K M ⇒ A r y K M where A is the double shuffle algebra of (3.1), and b A is its completion. Conjecture 1.1.
There exists an “action” of the double shuffle algebra: (1.5) A y K M given explicitly in Subsection 5.4. The fact that the action (1.5) extends to the completion b A is a consequence of thefact that K M is a good A -module (see Definition 3.9), which in turn follows fromBogomolov’s inequality. Explicitly, to each generator W n,k ∈ A , we associate inSection 5 an abelian group homomorphism via certain explicit correspondences:(1.6) K M w n,k −−−→ K M× S satisfying the compatibility conditions spelled out in Definition 5.2: this is themeaning of the quotes around the word “action” in the statement of Conjecture1.1. When S = A , the same construction was done in [24] in the context of theequivariant K –theory of the moduli space of framed sheaves, but the arbitrarysurface case poses interesting features: for instance, the parameters q and q ofthe algebra A r are identified with the Chern roots of Ω S , viewed as elements in K S . Theorem 1.2.
Assuming Conjecture 1.1, we have w n,k = 0 in (1.6) for all n ∈ Z and k > r , hence the “action” (1.5) factors through an “action” A r y K M . The group K M may be interpreted as the Hilbert space of 5d supersymmetricgauge theory on the projective surface S times a circle. Our placing an actionof the deformed W –algebra on K M yields a mathematical generalization of theAlday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory andconformal field theory (along the lines of [17], [24], [31], which treated the S = A case). Historically, this correspondence has usually been studied for toric surfacesusing Nekrasov’s equivariant generalization of partition functions. Therefore, theprojective surface situation treated in the present paper is a new phenomenon,which to the author’s knowledge has not previously been studied for rank r > A r y K M with the Carlsson-Okounkov Ext operator ([5]),thus establishing the equality of partition functions that AGT stipulates forbifundamental matter. However, one should note that, in order to completely –ALGEBRAS ASSOCIATED TO SURFACES 3 set up the AGT correspondence for an arbitrary surface S , the deformed W –algebra studied herein needs to be enlarged. Such an enlargement should prob-ably deform the vertex operator algebras (VOAs) associated to 4-manifolds (see [7]).The action A y K M is defined via the correspondences Z and Z • (Definitions 4.11and 4.16, respectively), which yield resolutions of singularities of the cohomologicalconstructions of Nakajima [20] and Grojnowski [14] in rank 1, and Baranovsky [2]in rank r . In general, the correspondences Z and Z • are defined as dg schemes,but we show in Propositions 4.19 and 4.20 that they are smooth schemes wheneverthe moduli spaces M are smooth (more precisely, under Assumption S of (4.27)).The proof of the smoothness of Z and Z • relies on presenting their tangent spacesin terms of cones of Ext groups, which is an idea that has appeared repeatedly inthe literature (see [13] for a development in the context of virtual degeneration loci).Through an analysis of the relation between the double shuffle algebra and W –algebras, which we perform in Sections 2 and 3, we show that Conjecture 1.1boils down to Conjecture 5.7. We prove both Conjectures under Assumption B ofSubsection 5.10, which we expect to hold for rational surfaces (see [16]). In general,the full statement of Conjecture 1.1 would follow if one knew that the shuffle algebrais a localization of the K –theoretic Hall algebra in the sense of [29], but this fact isopen (see [19] for certain results when S = T ∗ C is the cotangent bundle to a curve).When S = A , Theorem 1.2 was proved in [24] using the fact that the group K M is(generically) an irreducible module for the W –algebra. We do not have this featurefor a general surface S , and we have little control on the size of the abelian group K M . Instead, we give a geometric proof of Theorem 1.2 in Section 6, which admitsan obvious lift to the level of functors between derived categories. Let us mentioncertain possible avenues for generalizing our results.(1) Do the results in the present paper hold without Assumption A, namely thefact that gcd( r, c · H ) = 1? This would involve replacing universal sheavesby quasi-universal or twisted sheaves in all our definitions and computa-tions, and we make no claims about the technical difficulties that may arise.(2) Do the results in the present paper hold for r = 0? What about when onereplaces S by a quasi-projective surface endowed with a torus action withprojective fixed point set? Although the philosophy clearly generalizes,in either case, care must be taken in choosing the moduli space M correctly.(3) Do the results hold if the moduli space M of stable sheaves is replaced bythe moduli stack of all sheaves? It is conceivable that A still acts on K M ,but whether the algebra A ∞ still acts is unclear. The reason for this is thatelements of A ∞ are certain infinite sums inside A , which act correctly on(1.1) due to the fact that the grading by c is bounded below.I would like to thank Sergei Gukov, Tamas Hausel, Davesh Maulik, AlexanderMinets, Georg Oberdieck, Francesco Sala, Olivier Schiffmann, Richard Thomasand Alexander Tsymbaliuk for many interesting discussions on the subject, and forall their help in my understanding the framework of W –algebras and sheaves onsurfaces. I gratefully acknowledge the support of NSF grant DMS–1600375. ANDREI NEGUT , Shuffle algebras
Let q = q q . Throughout this paper, we will often encounter the ring:(2.1) K = Z [ q ± , q ± ] symmetric in q ,q and its field of fractions:(2.2) F = Q ( q , q ) symmetric in q ,q Let us recall the trigonometric version of the Feigin-Odesskii shuffle algebra [11]:
Definition 2.1.
Consider the rational function: (2.3) ζ ( x ) = (1 − xq )(1 − xq )(1 − x )(1 − xq ) and the vector space: S big = ∞ M k =0 F ( z , ..., z k ) Sym where the superscript
Sym refers to rational functions that are symmetric withrespect to z , ..., z k . We endow the vector space S big with the shuffle product: (2.4) R ( z , ..., z k ) ∗ R ′ ( z , ..., z k ′ ) == Sym R ( z , ..., z k ) R ′ ( z k +1 , ..., z k + k ′ ) ≤ i ≤ k Y k +1 ≤ j ≤ k + k ′ ζ (cid:18) z i z j (cid:19) Define the shuffle algebra : (2.5) S ⊂ S big to be the K -subalgebra generated by the elements: (2.6) E d • = Sym z d ...z d k k (cid:16) − z qz (cid:17) ... (cid:16) − z k qz k − (cid:17) Y ≤ i S ⊗ K F It was shown in [27] that the localized shuffle algebra (2.7) is actually generated by { E ( d ) = z d } d ∈ Z . For example, the identity: E (0) ∗ E (1) − E (1) ∗ E (0) = (1 − q )(1 − q )(1 − q ) z z ( z + z )( z − z q )( z − z q ) = (1 − q )(1 − q ) E (1 , suggests that one can obtain the generator E (1 , from shuffle products of E (1) and E (0) iff one inverts the element (1 − q )(1 − q ) in the ground ring. We will not doso in the present paper, and therefore emphasize the fact that the shuffle algebra S of (2.5) has a more complicated description than the algebra (2.7). For example,the latter admits a straightforward description in terms of Feigin-Odesskii type –ALGEBRAS ASSOCIATED TO SURFACES 5 wheel conditions, but we do not know an analogous description of the former. Proposition 2.2. The algebra S is generated over K by the elements: (2.8) E k,n = q gcd( k,n ) − E ( d ,...,d k ) where d i = l nik m − l n ( i − k m + δ ki − δ i , as k ranges over N , and n ranges over Z .Proof. Since S is generated by E d • for arbitrary d • = ( d , ..., d k ) ∈ Z k , it is enoughto show that any given E d • can be written as a K -linear combination of productsof (2.8). We will prove this statement by induction on k . The case k = 1 is trivial,and for the induction step, consider arbitrary k , d • and let n = d + ... + d k . Then: z d ...z d k k − q a k Y i =1 z ⌈ nik ⌉ − ⌈ n ( i − k ⌉ + δ ki − δ i i =(2.9) = k − X i =1 (cid:18) − z i +1 qz i (cid:19) · (Laurent polynomial in q, z , ..., z k )for some suitably chosen a ∈ Z . Plugging formula (2.9) in (2.6) gives us:(2.10) E d • − q a E k,n = k − X i =1 X d ′• ,d ′′• E d ′• ∗ E d ′′• where in the right-hand side, d ′• = ( d ′ , ..., d ′ i ) and d ′′• = ( d ′′ , ..., d ′′ k − i − ) are certainvectors of integers modeled after the monomials that appear in the right-hand sideof (2.9). By the induction hypothesis, the right-hand side of (2.10) can be writtenas a K -linear combination of products of the elements (2.8), hence so can E d • . (cid:3) E k,n . The idea for doing so is inspired by [3], and combiningtheir construction with the results of [27], allows us to conclude that, as F -modules:(2.11) S ⊗ K F = k i ∈ N , n i ∈ Z M n k ≤ ... ≤ ntkt F · E k ,n ...E k t ,n t In other words, any element of S can be written as a linear combination of ordered monomials E k ,n ...E k t ,n t if one allows the coefficients to be rational functions in q and q . Our main purpose for the remainder of this section is to show that theequality (2.11) still holds without localization, i.e. the fact that, as K -modules:(2.12) S = k i ∈ N , n i ∈ Z M n k ≤ ... ≤ ntkt K · E k ,n ...E k t ,n t By Proposition 2.2, any element of S can be written as a K -linear combination ofproducts of E k,n . Our task thus reduces to establishing an integral version of the“straightening lemma” of [3]: we must show that an arbitrary product of E k,n ’s isequal to a sum of ordered products of E k,n ’s, in non-decreasing order of nk . Thisstatement follows by repeated applications of the following result: ANDREI NEGUT , Theorem 2.4. For any k, k ′ ∈ N and n, n ′ ∈ Z such that nk > n ′ k ′ , we have: (2.13) [ E k,n , E k ′ ,n ′ ] = ∆ k i ∈ N , P k i = k + k ′ n i ∈ Z , P n i = n + n ′ X n ′ k ′ ≤ n k ≤ ... ≤ ntkt ≤ nk p k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) · E k ,n ...E k t ,n t where p k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) ∈ K , and ∆ = (1 − q )(1 − q ) . nk > n ′ k ′ are implied by the specialcase when the triangle spanned by the vectors ( k, n ) and ( k ′ , n ′ ) contains no latticepoints inside and on one of the edges. These special cases are usually presented interms of the generators P k,n , H k,n , Q k,n defined as follows for all gcd( k, n ) = 1: ∞ X s =0 E ks,ns ( − x ) s = exp " − ∞ X s =1 P ks,ns sx s (2.14) ∞ X s =0 H ks,ns x s = exp " ∞ X s =1 P ks,ns sx s (2.15) ∞ X s =0 Q ks,ns x s = exp " ∞ X s =1 P ks,ns sx s (1 − q − s ) (2.16)For fixed slope nk , the P ’s, H ’s and Q ’s are in relation to E ’s as power sum, completesymmetric, and plethystically modified complete symmetric polynomials are in re-lation to elementary symmetric polynomials. In other words, presenting relationsbetween E ’s is equivalent with presenting relations between the other generators.These relations were first constructed in [3] (see [24] for our conventions):(2.17) [ P k,n , P k ′ ,n ′ ] = 0if nk = n ′ k ′ , and:(2.18) [ P k,n , P k ′ ,n ′ ] = (1 − q s )(1 − q s )1 − q − · Q k + k ′ ,n + n ′ when the triangle (0 , k, n ), ( k + k ′ , n + n ′ ) is oriented clockwise, and containsno lattice points inside or on one of the edges (we write s = gcd( k, n ) gcd( k ′ , n ′ )). Proposition 2.6. Relations (2.17) and (2.18) imply the following relations interms of the E generators, for all s ∈ N : (2.19) [ E ks,ns , E k ′ ,n ′ ] = ∆ s X t =1 q t − q t q − q ( − t − E kt + k ′ ,nt + n ′ E k ( s − t ) ,n ( s − t ) (2.20) [ E k,n , E k ′ s,n ′ s ] = ∆ s X t =1 q t − q t q − q ( − t − E k ′ ( s − t ) ,n ′ ( s − t ) E k + k ′ t,n + n ′ t whenever the triangle spanned by the vectors ( k, n ) and ( k ′ , n ′ ) is oriented clockwise,and has the property that gcd( k, n ) = gcd( k ′ , n ′ ) = gcd( k + k ′ , n + n ′ ) = 1 . Under –ALGEBRAS ASSOCIATED TO SURFACES 7 the same assumptions, but allowing gcd( k + k ′ , n + n ′ ) = s ≥ , we have: (2.21) [ E k,n , E k ′ ,n ′ ] = ∆1 − q − (cid:20) coefficient of x s in E ( xq ) E ( x ) (cid:21) where we set E ( x ) = P ∞ t =0 E ( k + k ′ ) ts , ( n + n ′ ) ts ( − x ) t .Proof. Under our assumptions on the vectors ( k, n ), ( k ′ , n ′ ), the triangle spannedby the vectors ( ks, ns ), ( kt + k ′ , nt + n ′ ) satisfies the assumptions of (2.18):[ P ks,ns , P kt + k ′ ,nt + n ′ ] = (1 − q s )(1 − q s ) P k ( s + t )+ k ′ ,n ( s + t )+ n ′ for all s ∈ N , t ∈ Z . Summing this relation over all s ∈ N and t ∈ Z , we obtain: " − ∞ X s =1 P ks,ns sx s , X t ∈ Z P kt + k ′ ,nt + n ′ y t = − ∞ X s =1 (1 − q s )(1 − q s ) y s sx s X t ∈ Z P k ( s + t )+ k ′ ,n ( s + t )+ n ′ y s + t Let us recall the well-known formula:[ X, Y ] = c · Y and [ X, c ] = [ Y, c ] = 0 ⇒ exp( X ) Y = exp( c ) · Y exp( X )and apply it to X = − P ∞ s =1 P ks,ns sx s and Y = P t ∈ Z P kt + k ′ ,nt + n ′ y t = P t ∈ Z E kt + k ′ ,nt + n ′ y t :(2.22) ∞ X s =0 E ks,ns ( − x ) s X t ∈ Z E kt + k ′ ,nt + n ′ y t = ζ (cid:16) yx (cid:17) − X t ∈ Z E kt + k ′ ,nt + n ′ y t ∞ X s =0 E ks,ns ( − x ) s Indeed, we have P kt + k ′ ,nt + n ′ = E kt + k ′ ,nt + n ′ for all t because our assumption on thevectors ( k, n ) and ( k ′ , n ′ ) implies gcd( kt + k ′ , nt + n ′ ) = 1 , ∀ t . Using the expansion: ζ (cid:16) yx (cid:17) − = 1 − (1 − q )(1 − q ) ∞ X s =1 q s − q s q − q · y s x s and taking the coefficient of x − s y in equality (2.22) yields (2.19). Relation (2.20)is proved analogously. As for (2.21), this follows directly from (2.18) since E k,n = P k,n , E k ′ ,n ′ = P k ′ ,n ′ and (2.16) implies: ∞ X t =0 Q ( k + k ′ ) ts , ( n + n ′ ) ts x t = E ( xq ) E ( x ) (cid:3) E ( xq ) /E ( x ) in (2.21), we see that it implies:(2.23) [ E k,n , E k ′ ,n ′ ] = ∆ (cid:16) E k + k ′ ,n + n ′ ( − s − [ q ] s + . . . (cid:17) where [ q ] s = 1 + q − + ... + q − s +1 and the ellipsis in (2.23) stands for a sum ofproducts of the form E k ,n ...E k t ,n t with t > k i , n i ) lyingon the line segment from (0 , 0) to the lattice point ( k + k ′ , n + n ′ ). Proof. of Theorem 2.4: For the first half of the proof, we closely follow [3], whichwill allow us to obtain the following slightly weaker version of (2.13):(2.24) [ E k,n , E k ′ ,n ′ ] = ∆ k i ∈ N , P k i = kn i ∈ Z , P n i = n X n ′ k ′ ≤ n k ≤ ... ≤ ntkt ≤ nk p k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) · E k ,n ...E k t ,n t ANDREI NEGUT , where:(2.25) p k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) ∈ K loc := K (1+ q + ... + q s − ) s ∈ N Then we will use the methods of [27] to show that the expressions (2.25) do nothave any poles at q = non-trivial root of unity, and so we will conclude that theyactually lie in K , which is precisely what the Theorem claims. We call a product:(2.26) e = E k ,n , ..., E k t ,n t K − orderable (respectively K loc − orderable ) if it can be written as in the right-hand side of (2.13) (respectively (2.24)). Consider the assignment: (cid:16) E k ,n ...E k t ,n t ∈ S (cid:17) Υ −→ (cid:16) the lattice path P (cid:17) where P starts at (0 , 0) and is built out of the segments ( k , n ) , ..., ( k t , n t ) inthis order. This is clearly a one-to-one correspondence between products (2.26)and lattice paths starting at the origin and pointing in the right half plane. Theproducts that appear in the right-hand side of (2.13) and (2.24) all correspond toconvex paths. Given any lattice path P , its convexification P conv is defined as thepath built out of the same segments ( k, n ) as P , but in non-decreasing order of slope nk (for segments of equal slope, their relative order may be chosen arbitrarily). Thearea a ( P ) of the path P is defined as the area of the polygon bounded by P and P conv , and we note that it is always a natural number. It was shown in [3] that if:(2.27) ˜ e = E k,n E k ′ ,n ′ is K -orderable whenever a (Υ(˜ e )) ≤ δ , then any product (2.26) such that a (Υ( e )) ≤ δ is also K -orderable. The same proof works if we replace the ring K with K loc .Therefore, to prove (2.24) by induction on δ ∈ N , it suffices to prove the following: Assume that any e as in (2.26) with a (Υ( e )) < δ is K loc − orderable , then any ˜ e as in (2.27) with a (Υ(˜ e )) = δ is K loc − orderable Let us now prove the claim in boldface letters above. Choose any ˜ e as in (2.27)such that a ( P ) = δ , where P is the path with segments v = ( k, n ) and v ′ = ( k ′ , n ′ ).From now on, the phrase the triangle determined by vectors v and v ′ will refer to thetriangle with a vertex at (0 , 0) and with edges given by drawing the vectors v and v ′ in this order. Choose a lattice point v = ( k , n ) inside the triangle determinedby the vectors v and v ′ , such that the area of the triangle determined by v and v − v is minimal. This implies that the latter triangle respects the hypothesis ofrelation (2.21) (or its equivalent form (2.23)), and so we have:(2.28) E v ∈ ( − s − [ q ] s [ E v − v , E v ]∆ + t> , v + ... + v t = v X v ,...,v t divide v K · E v ...E v t where s = gcd( k, n ), and we write E v instead of E k,n if v = ( k, n ). Taking thecommutator of (2.28) with E v ′ yields the following, in virtue of the Jacobi identity:[ E v , E v ′ ] ∈ ( − s − [ q ] s [[ E v − v , E v ] , E v ′ ]∆ + t> , v + ... + v t = v X v ,...,v t divide v K · [ E v ...E v t , E v ′ ] ⊂⊂ ( − s − [ q ] s (cid:18) [[ E v ′ , E v ] , E v − v ]∆ + [[ E v − v , E v ′ ] , E v ]∆ + –ALGEBRAS ASSOCIATED TO SURFACES 9 (2.29) + t> , v + ... + v t = v X v ,...,v t divide v t X s =1 K · E v ...E v s − [ E v s , E v ′ ] E v s +1 ...E v t We claim that all summands in the right-hand side of (2.29) are K loc − orderable.Indeed, by our choice of the vector v , all commutators that appear in theright-hand side correspond to paths whose area is < δ . Therefore, by the inductionhypothesis in boldface letters, we may express such a commutator as a sum overconvex paths, and it was shown in [3] that all paths obtained in this manner inthe right-hand side of expression (2.29) will still have area < δ . The key geometricstatement here, proved in loc. cit. , is that if one takes two consecutive segmentswhich violate convexity in a path P ′ , and one replaces them by an arbitrary convexpath P between the same endpoints, the resulting path P ′′ has a ( P ′′ ) < a ( P ′ ).We conclude that the right-hand side of (2.29) can be written as a sum over convexpaths P of the elements Υ − ( P ). Because every commutator brings down a factorof ∆ (as follows from relation (2.13) when the area of the triangle determined by( k, n ) and ( k ′ , n ′ ) is < δ , which we may assume as part of our induction hypothesis),we see that the coefficient of any Υ − ( P ) in the right-hand side of (2.29) lies in:( − s − ∆[ q ] s · K loc = ∆ · K loc Now assume, for the purpose of contradiction, that a certain p k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q )that appears in the right-hand side of (2.24) lies in K loc \ K , i.e. has a pole when q isa non-trivial root of unity. Then the pole will remain when we change the right-handside of (2.24) from the basis E k ,n ...E k t ,n t to P k ,n ...P k t ,n t , because the matrixtransforming elementary symmetric polynomials E k i ,n i into power-sum functions P k i ,n i is invertible and has rational coefficients. Note that [ E k,n , E k ′ ,n ′ ] ∈ S is arational function of the form:(2.30) R = r ( z , ..., z k ) Q ≤ i = j ≤ k ( z i q − z j )where r ∈ K [ z ± , ..., z ± k ] Sym . By (2.12), we may express any such element as: R = k i ∈ N , n i ∈ Z X n k ≤ ... ≤ ntkt γ k ,...,k t n ,...,n t · P k ,n ...P k t ,n t where γ k ,...,k t n ,...,n t ∈ F . Then all that remains to prove is that the coefficients γ k ,...,k t n ,...,n t do not have any poles at q is a non-trivial root of unity. There exists a pairing:(2.31) h· , ·i : S ⊗ S → F for which the basis vectors P k ,n ...P k t ,n t are orthogonal (see [27]) and the coeffi-cients we wish to express are given by:(2.32) γ k ,...,k t n ,...,n t = h R, P k ,n ...P k t ,n t ih P k ,n ...P k t ,n t , P k ,n ...P k t ,n t i , The goal is to show that the right-hand side of (2.32) does not have poles when q is a non-trivial root of unity. According to formula (7.15) of [24], we have:(2.33) h P k ,n ...P k t ,n t , P k ,n ...P k t ,n t i = integer t Y i =1 (1 − q s i )(1 − q s i )(1 − q − ) k i (1 − q ) k i (1 − q ) k i (1 − q − s i )where s i = gcd( k i , n i ). According to formula (2.8) of [24], we have for all k, n withgreatest common divisor s , the equality:(2.34) P k,n = Sym Q ki =1 z ⌊ ink ⌋ − ⌊ ( i − nk ⌋ i Q k − i =1 (cid:16) − qz i +1 z i (cid:17) s − X t =0 q t z a ( s − ...z a ( s − t )+1 z a ( s − ...z a ( s − t ) Y i For any shuffle element R as in (2.30) and any ρ ∈ K [ z ± , ..., z ± k ] let: (2.37) P = Sym ρ ( z , ..., z k ) Q k − i =1 (cid:16) − z i +1 q z i (cid:17) Y i For any R ( z , ..., z k ) , R ′ ( z , ..., z k ) ∈ S , we have: (2.39) h R, R ′ i = 1 k ! Z | z | = ... = | z k | R ( z , ..., z k ) R ′ (cid:16) z , ..., z k (cid:17)Q ≤ i = j ≤ k h ζ (cid:16) z i z j (cid:17) z i − pz j z i − qz j i k Y i =1 dz i πiz i (cid:12)(cid:12)(cid:12) p q The integral must be computed by residues under the assumptions | q | , | q | > > | p | ,and only after one evaluates the integral, one must specialize p q .Proof. of Claim 2.9: It suffices to show that formula (2.38) matches (2.39) when: R ′ = Sym z d ...z d k k Y ≤ i Let us re-run the argument that proved Claim 2.9 with R ′ replaced by P of (2.37). Because of the extra factors 1 − z i +1 q z i in the denominatorof P , equality (2.40) does not hold as stated anymore. Instead, we have: h R, P i = Z | z | = ... = | z k | R ( z , ..., z k ) ρ (cid:16) z , ..., z k (cid:17)Q k − i =1 (cid:16) − z i q z i +1 (cid:17) Q i 0, where s = gcd( k, n ), and:(3.3) [ P k,n , P k ′ ,n ′ ] = (1 − q s )(1 − q s )1 − q − · c ∗ Q k + k ′ ,n + n ′ –ALGEBRAS ASSOCIATED TO SURFACES 13 if kn ′ < k ′ n and the triangle with vertices (0 , k, n ), ( k + k ′ , n + n ′ ) contains nolattice points inside or on one of the edges, and s denotes gcd( k, n ) gcd( k ′ n ′ ). Theparticular power c ∗ in formula (3.3) can be found in (2.21) of [24], but it will not berelevant to us. Throughout the present paper, we will set c = q r for a natural num-ber r , in order to cancel the denominator of (3.2). Since the Q k,n are still definedby (2.16), they are also multiples of 1 − q , and this cancels the denominator of (3.3).The generators P k,n and Q k,n are connected with the generators E k,n by formulas(2.14) and (2.16), respectively. The observant reader will note that the P ’s and Q ’sare not really in the algebra A because they are polynomials of the E generatorswith rational coefficients, and Q K . However, it is elementary to convert (3.2)and (3.3) into formulas for the commutators [ E k,n , E k ′ ,n ′ ]: akin to Proposition2.6, these commutators will be expressible as products of E ’s with coefficients in K . We leave the case of (3.3) as an exercise to the interested reader (it will differfrom (2.19)–(2.21) by some powers of c ), but we will now show how to convert (3.2): Proposition 3.2. Assume s ∈ − N , k ∈ N and gcd( k, n ) = 1 . Then (3.2) implies: (3.4) [ E ks,ns , E ks ′ ,ns ′ ] (cid:12)(cid:12)(cid:12) c q r == ( if s ′ < P min( s,s ′ ) i =1 γ i E k ( s + i ) ,n ( s + i ) E k ( s ′ − i ) ,n ( s ′ − i ) if s ′ > for some γ i ∈ K .Proof. Formula (3.4) is easy when s, s ′ < 0, since if P k,n , P k, n , P k, n ... all com-mute, then formula (2.14) implies that E k,n , E k, n , E k, n ... also all commute. Onthe other hand, when s < < s ′ , relation (3.2) reads:[ P ks,ns , P ks ′ ,ns ′ ] (cid:12)(cid:12)(cid:12) c q r = δ s + s ′ s (1 − q s )(1 − q s )(1 + q − s + ... + q − s ( kr − ) ⇒⇒ " − X s = −∞ P ks,ns sx − s , ∞ X s ′ =1 P ks ′ ,ns ′ − s ′ y s ′ = ∞ X s =1 (1 − q s )(1 − q s )(1 + q − s + ... + q − s ( kr − ) sx s y s We leave the following claim as an easy exercise: if [ P, P ′ ] = γ with γ central, thenexp( P ) exp( P ′ ) = exp( γ ) exp( P ′ ) exp( P ). With this in mind, we obtain:(3.5) − X s = −∞ E ks,ns ( − x ) − s ∞ X s ′ =1 E ks ′ ,ns ′ ( − y ) s ′ (cid:12)(cid:12)(cid:12) c q r = φ ( xy ) ∞ X s ′ =1 E ks ′ ,ns ′ ( − y ) s ′ − X s = −∞ E ks,ns ( − x ) − s (cid:12)(cid:12)(cid:12) c q r where: φ ( z ) = exp ∞ X s =1 (1 − q s )(1 − q s )(1 + q − s + ... + q − s ( kr − ) sz s ! is equal to 1 + a power series in z − whose coefficients are all in ∆ · K . Taking thecoefficient of ( − x ) s ( − y ) − s ′ in (3.5), we obtain (3.4). (cid:3) , Proposition 3.3. Recall that P ,k ∈ A diag are to E ,k ∈ A diag as power-sumfunctions are to elementary symmetric polynomials. Then for all k ∈ Z , we have: (3.6) [ P ,k , R ( z , ..., z n )] = (1 − q k )(1 − q k )( z k + ... + z kn ) R ( z , ..., z n ) for all R ( z , ..., z n ) ∈ S op ∼ = A → , and: (3.7) [ P ,k , R ( z , ..., z n )] = − (1 − q k )(1 − q k )( z k + ... + z kn ) R ( z , ..., z n ) for all R ( z , ..., z n ) ∈ S ∼ = A ← .Proof. Since the algebra A is a free K -module, it is enough to prove the requiredformulas over F . According to Theorem 2.5 of [27], the shuffle algebra is generatedby { z k ′ } k ′ ∈ Z over F , and so the required formulas follow from the particular case n = 1. In this case, the required relations boil down to:[ P ,k , E ,k ′ ] = (1 − q k )(1 − q k ) E ,k + k ′ [ P ,k , E − ,k ′ ] = − (1 − q k )(1 − q k ) E − ,k + k ′ which are just particular cases of (3.3). (cid:3) K -basis of A :(3.8) A (cid:12)(cid:12)(cid:12) c q r = M ( n ,k ) y ... y ( n t ,k t ) K · E n ,k ...E n t ,k t where the sum goes over all collections ( n , k ) , ..., ( n t , k t ) ∈ Z \ (0 , 0) ordered clock-wise. Here and below, we say that two lattice points ( n, k ) and ( n ′ , k ′ ) are orderedclockwise, denoted by:(3.9) ( n, k ) y ( n ′ , k ′ )if one can reach the latter from the former by turning clockwise around the origin,without crossing the negative y axis. If we wish to exclude the situation when ( n, k )and ( n ′ , k ′ ) lie on the same ray through the origin, then we will use the notation:(3.10) ( n, k ) y ( n ′ , k ′ )instead. Formulas (3.9) and (3.10) are simply inequalities ≤ and < on the slopes,appropriately defined, of the lattice points ( n, k ) and ( n ′ , k ′ ). Just like one candeduce (2.12) from Theorem 2.4, one can deduce (3.8) from the following Theorem: Theorem 3.5. For any lattice points ( n ′ , k ′ ) y ( n, k ) , we have: (3.11) [ E n,k , E n ′ ,k ′ ] (cid:12)(cid:12)(cid:12) c q r = ∆ k i ∈ Z , P k i = k + k ′ n i ∈ Z , P n i = n + n ′ X ( n ′ ,k ′ ) y ( n ,k ) y ...... y ( n t ,k t ) y ( n,k ) p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q ) · E n ,k ...E n t ,k t for certain p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q ) ∈ K , and ∆ = (1 − q )(1 − q ) . Moreover, theequations (3.11) generate the ideal of relations between the elements E n,k ∈ A . –ALGEBRAS ASSOCIATED TO SURFACES 15 Note that we specialize c = q r in (3.11) for two reasons: firstly, when kn ′ = k ′ n ,we make the convention that formula (3.11) be identical to formula (3.4), whichrequires c = q r in order to not have denominators. Secondly, formula (3.3) andthe equivalent formula with P ’s replaced with E ’s, feature certain powers of c in the right-hand side, which only become elements of K upon specialization c = q r . Proof. The generators E n,k (alternatively P n,k ) of the algebra A are permuted by SL ( Z ) acting on the indices. As observed in [3], the relations (3.2) and (3.3) arenot quite invariant under this SL ( Z ) due to the various powers of c that appear inthe right-hand sides of these formulas, but they are invariant under the universalcover of SL ( Z ). As a consequence of this fact, the subalgebras:(3.12) A < ba = K h E n,k i nb 16 ANDREI NEGUT , We will often extend the subalgebra A ↑ by adding the elements E n, on the x -axis: A ↑ ext := K -subalgebra generated by (cid:10) c ± , E n,k (cid:11) k ∈ N ⊔ n ∈ Z ⊂ A As a consequence of (3.8), we have: A ↑ = M −∞ < n k ≤ ... ≤ ntkt < ∞ K · E n ,k ...E n t ,k t (3.18) A ↑ ext (cid:12)(cid:12)(cid:12) c q r = M −∞≤ n k ≤ ... ≤ ntkt ≤∞ K · E n ,k ...E n t ,k t (3.19)(note that we do not need to specialize c = q r in (3.18) because the commutationrelations (3.3) do not involve any powers of c if the indices ( n, k ) and ( n ′ , k ′ ) areboth in the upper half plane, see [24]). The algebra A ↑ is Z × N graded:deg E n,k = ( n, k )and the graded pieces A ↑ n,k have infinite rank over K . However, the subspaces:(3.20) A ↑ , ≤ µn,k = k i ∈ N , P k i = kn i ∈ Z , P n i = n X − µ ≤ n k ≤ ... ≤ ntkt ≤ µ K · E n ,k ...E n t ,k t are finite rank free K -modules. Consider the K -linear map A ↑ , ≤ µ +1 n,k ։ A ↑ , ≤ µn,k whichsends every product E n ,k ...E n t ,k t of (3.18) either to 0 or to itself, and define: b A ↑ n,k = lim ← ,µ A ↑ , ≤ µn,k b A ↑ = k ∈ N M n ∈ Z b A ↑ n,k In more practical terms, we may think of b A ↑ as consisting of infinite K -linearcombinations of basis monomials (3.18) for bounded above k + ... + k t :(3.21) b A ↑ = dM −∞ < n k ≤ ... ≤ ntkt < ∞ K · E n ,k ...E n t ,k t Similarly, we define b A ↑ ext ⊃ b A ↑ by allowing k i = 0, and the analogue of (3.21) is:(3.22) b A ↑ ext (cid:12)(cid:12)(cid:12) c q r = dM −∞≤ n k ≤ ... ≤ ntkt ≤∞ K · E n ,k ...E n t ,k t Proposition 3.7. b A ↑ and b A ↑ ext are closed under multiplication, and thus algebras.Proof. We will prove the statement for b A ↑ , as the case of b A ↑ ext is analogous. Fromnow on, we will consider paths v in the upper half plane that start at the origin, andare built up of steps { ( n , k ) , ..., ( n t , k t ) } ⊂ Z × N . Such a path is called convex if: n k ≤ ... ≤ n t k t which corresponds to ( n , k ) y ... y ( n t , k t ). The size of the path is the lattice point( n, k ) with n = P n i and k = P k i , where the path ends. As in Section 2, there isa one-to-one correspondence between paths and basis vectors of A ↑ , given by: v E v = E n ,k ...E n t ,k t –ALGEBRAS ASSOCIATED TO SURFACES 17 Formulas (3.18) and (3.22) say that elements of the algebras A ↑ and b A ↑ are linearcombinations (finite in the former case, infinite in the latter case) of the elements E v corresponding to convex paths. Let us recall from [3] that formula (3.11)(see also the proof of Theorem 2.4) implies that we can “convexify” any path v ,i.e. write E v as a linear combination P v convex c v v · E v for convex paths v .The main thing we will need to take from their argument is that the coefficients c v v are non-zero only if the path v lies to the left of v . More precisely, we will show: Claim 3.8. For any convex paths v, v ′ of sizes ( n, k ) , ( n ′ , k ′ ) , let v ⊔ v ′ denote theirconcatenation, and let ( v ⊔ v ′ ) conv denote the convexification of v ⊔ v ′ . Then: (3.23) E v · E v ′ = E v ⊔ v ′ ∈ E ( v ⊔ v ′ ) conv + ∆ v convex path of size X ( n + n ′ ,k + k ′ ) to the left of v ⊔ v ′ K · E v For lattice paths that start at (0 , and end ( n + n ′ , k + k ′ ) , the notion of one being tothe left of another is unambiguous, once left is defined as “the negative n direction”. Indeed, the Claim implies Proposition 3.7, because it establishes the followingfact: given an infinite sum of E v ’s (resp. E v ′ ’s) over paths v (resp. v ′ ) of fixed size( n, k ) (resp. ( n ′ , k ′ )), then any given convex path v appears in the right-handside of (3.23) with non-zero coefficient only for finitely many v and v ′ . This meansthat the product of infinite sums of E v ’s and E v ′ ’s is a well-defined infinite sum. Proof. of Claim 3.8: Let v = { ( n , k ) , ..., ( n t , k t ) } , v ′ = { ( n ′ , k ′ ) , ..., ( n ′ t ′ , k ′ t ′ ) } . If: n t k t ≤ n ′ k ′ then v ⊔ v ′ is already a convex path, and the claim holds trivially. If the oppositeinequality holds, then we may apply (3.11) to obtain:(3.24) E v E v ′ = E v \ ( n t ,k t ) E n t ,k t E n ′ ,k ′ E v ′ \ ( n ′ ,k ′ ) ∈∈ E v \ ( n t ,k t ) E n ′ ,k ′ E n t ,k t E v ′ \ ( n ′ ,k ′ ) + ∆ X v convex path K · E v \ ( n t ,k t ) E v E v ′ \ ( n ′ ,k ′ ) where the sum goes over convex paths v of size ( n t + n ′ , k t + k ′ ) which stay to theleft of the path spanned by the two vectors ( n t , k t ) , ( n ′ , k ′ ). Consider the paths:˜ v obtained by concatenating v \ ( n t , k t ) , ( n ′ , k ′ ) , ( n t , k t ) , v ′ \ ( n ′ , k ′ )˜ v obtained by concatenating v \ ( n t , k t ) , v , v ′ \ ( n ′ , k ′ )for any convex path v that appears in the right-hand side of (3.24). All such paths˜ v and ˜ v are strictly to the left of v ⊔ v ′ . If any of these paths is convex, we stop,otherwise we choose two consecutive vectors in the path which spoil convexity:(˜ n, ˜ k ) and (˜ n ′ , ˜ k ′ ) such that ˜ n ˜ k > ˜ n ′ ˜ k ′ and repeat the argument of (3.24). The reason why this recursive procedure willend after finitely many steps is that all our paths have fixed size, and the slopes ofall the vectors in the paths obtained at every step cannot be greater (respectivelysmaller) than the greatest (respectively smallest) of the slopes of the vectors in the , paths v and v ′ . Finally, note that E ˜ v is the only summand in the right-hand sideof (3.24) which does not have a prefactor ∆ in front. Since ˜ v consists of the samevectors as v ⊔ v ′ , but in some other order, at the end of the recursive procedure thissummand will have transformed into E ( v ⊔ v ′ ) conv modulo ∆. (cid:3)(cid:3) Definition 3.9. A representation F of A is called good if it has a grading: (3.25) F = ∞ M n = n F n for some n ∈ Z , such that every E n,k ∈ A acts of F by decreasing the grading by n . Proposition 3.10. The action of A on any good module extends to b A ↑ and b A ↑ ext .Proof. Let i n and pr n denote the inclusion and projection, respectively, to the n -thdirect summand of (3.25). Then we note that:pr n ◦ E n ,k ...E n t ,k t ◦ i n ′ = 0as soon as n < n − n or n t > n ′ − n . Since any element of b A ↑ is a finite linearcombination of monomials E n ,k ...E n t ,k t modulo those monomials which satisfyeither n < n − n or n t > n ′ − n , the Proposition follows. (cid:3) b A ↑ of A ↑ is that itcontains the following elements, studied in [24]:(3.26) { W n,k } k ∈ N n ∈ Z ∈ b A ↑ given by:(3.27) W n,k = k i ∈ N , P k i = kn i ∈ Z , P n i = n X −∞ < n k <...< ntkt < ∞ E n ,k ...E n t ,k t · q α ( v ) where the integer α ( v ) associated to the convex path v = { ( n , k ) , ..., ( n t , k t ) } is: α ( v ) = X ≤ i 0, we define the following power series in z :(3.31) f kk ′ ( z ) = exp " ∞ X n =1 z n n · (1 − q n )(1 − q n )( q max(0 ,k − k ′ ) n − q kn )1 − q n W k ( x ), but one mustcarefully interpret it to obtain an infinite family of relations between the coefficients W n,k . The idea is to equate the coefficients of x − n y − n ′ in the left and right-handsides of (3.29), for all n, n ′ ∈ Z . In the left-hand side, this is achieved by expanding:(3.32) W k ( x ) W k ′ ( y ) f kk ′ (cid:16) yx (cid:17) in non-negative powers of yx (3.33) W k ′ ( y ) W k ( x ) f k ′ k (cid:18) xy (cid:19) in non-negative powers of xy This is the only reasonable choice one can make in order for the expansion of eitherterm to be an infinite sum of the form: X a ≥ coefficient · W n − a,k W n ′ + a,k ′ or X a ≥ coefficient · W n ′ − a,k ′ W n + a,k which are acceptable expressions in the completion b A ↑ of A ↑ . What is not imme-diately clear, and will be discussed in the proof of Theorem 3.13 below, is how tomake sense of the product of W –algebra currents in the right-hand side of (3.29).The only exception is when k ′ = 1, in which case the relation unambiguously reads:(3.34) W k ( x ) W ( y ) ζ (cid:18) xyq k (cid:19) − W ( y ) W k ( x ) ζ (cid:18) yqx (cid:19) == ∆1 − q (cid:20) δ (cid:18) yxq (cid:19) W k +1 ( y ) − δ (cid:18) xyq k (cid:19) W k +1 ( x ) (cid:21) Extracting the coefficients of x − n y − n ′ according to the rules (3.32) and (3.33) yields:[ W n,k , W n ′ , ] + ∆ ∞ X a =1 − q a − q (cid:16) q a ( k − W n − a,k W n ′ + a, − W n ′ − a, W n + a,k (cid:17) =(3.35) = ∆ q − n − q − kn ′ − q · W n + n ′ ,k +1 What is surprising about formula (3.35) is that all the coefficients lie in K insteadof F , even though θ ( s ) could a priori have produced poles of the form 1 − q n in(3.29). In fact, this is a general phenomenon, as evidenced by the result below: , Theorem 3.13. Relations (3.29) are equivalent with the following family of equal-ities, which hold for all k, k ′ > and n, n ′ ∈ Z : (3.36) [ W n,k , W n ′ ,k ′ ] = ∆ ml ≤ m ′ l ′ min( l,l ′ ) ≤ min( k,k ′ ) X k + k ′ = l + l ′ m + m ′ = n + n ′ c m,m ′ ,l,l ′ n,n ′ ,k,k ′ ( q , q ) · W m,l W m ′ ,l ′ for certain c m,m ′ ,l,l ′ n,n ′ ,k,k ′ ( q , q ) ∈ K which we will compute algorithmically. Theright-hand side allows min( l, l ′ ) = 0 , in which case we recall that W m, = δ m . A ∞ = K D W n,k E k ∈ N n ∈ Z . relations (3.36)and let us note that Theorem 3.13 states that there is a well-defined homomorphism A ∞ → b A ↑ given by (3.27). After tensoring with F , this embedding is completelydetermined by sending W n, E n, , because (3.35) implies that any W n,k can beobtained as a sum of products of W n, , upon inverting ∆ and rational functions in q . Proposition 3.15. The generators of A ∞ → b A ↑ interact with: (3.38) p − n := P − n, ∈ A ↑ ext and p n := (cid:18) cq (cid:19) n P n, ∈ A ↑ ext by the formulas: (3.39) [ W k ( x ) , p − n ] = − (1 − q n )(1 − q n )(1 − q kn )1 − q n · x n W k ( x )(3.40) [ W k ( x ) , p n ] = (1 − q n )(1 − q n )( q − kn − c n − q n · x n W k ( x )(3.41) [ p − n , p n ] = n (1 − q n )(1 − q n ) 1 − c n − q n for all n ∈ N . Define:(3.42) A ext ∞ = K D W n,k , p n E k ∈ N n ∈ Z . relations (3.36),(3.39),(3.40),(3.41)Throughout this paper, we will always specialize the central charge c to q r for some r ∈ N , so the structure constants of the algebra A ext ∞ will all lie in K (if c = q r ,then (3.41) would contradict this fact). By Proposition 3.15, the homomorphism: A ∞ → b A ↑ extends to A ext ∞ → b A ↑ , ext –ALGEBRAS ASSOCIATED TO SURFACES 21 Proof. of Proposition 3.15: We will prove (3.39), as (3.40) is analogous and (3.41)is a trivial application of (3.2). We will do so by induction on k (the case when k = 1 is simply (3.3)), and start by commuting relation (3.34) with p − n :∆1 − q (cid:20) δ (cid:18) yxq (cid:19) [ W k +1 ( y ) , p − n ] − δ (cid:18) xyq k (cid:19) [ W k +1 ( x ) , p − n ] (cid:21) == [ W k ( x ) W ( y ) , p − n ] ζ (cid:18) xyq k (cid:19) − [ W ( y ) W k ( x ) , p − n ] ζ (cid:18) yxq (cid:19) Leibniz rule == W k ( x )[ W ( y ) , p − n ] ζ (cid:18) xyq k (cid:19) + [ W k ( x ) , p − n ] W ( y ) ζ (cid:18) xyq k (cid:19) − W ( y )[ W k ( x ) , p − n ] ζ (cid:18) yxq (cid:19) − [ W ( y ) , p − n ] W k ( x ) ζ (cid:18) yxq (cid:19) induction hypothesis of (3.39) = − (1 − q n )(1 − q n ) (cid:18) y n + 1 − q kn (1 − q n ) x n (cid:19) (cid:20) W k ( x ) W ( y ) ζ (cid:18) xyq k (cid:19) − W ( y ) W k ( x ) ζ (cid:18) yxq (cid:19)(cid:21) = − ∆(1 − q n )(1 − q n )1 − q (cid:18) y n + 1 − q kn (1 − q n ) x n (cid:19) (cid:20) δ (cid:18) yxq (cid:19) W k +1 ( y ) − δ (cid:18) xyq k (cid:19) W k +1 ( x ) (cid:21) If we multiply both sides of the equation above with 1 − yq k /x , the second δ functionvanishes in both sides of the equation above, and we are left with:∆(1 − q k +1 )1 − q δ (cid:18) yxq (cid:19) [ W k +1 ( y ) , p − n ] == − ∆(1 − q k +1 )1 − q (1 − q n )(1 − q n ) y n (cid:20) − q kn (1 − q n ) q − n (cid:21) δ (cid:18) yxq (cid:19) W k +1 ( y )Extracting the coefficient of x from the equation above yields precisely (3.39) with k replaced by k + 1, thus establishing the induction step. (cid:3) W –algebra of type gl r as a quotient of A ∞ . Definition 3.17. For arbitrary r ∈ N , define: (3.43) A r = A ∞ . relation (3.45)(3.44) A ext r = A ext ∞ . relations (3.45) and (3.46) and c = q r where: (3.45) W k ( x ) = 0 , ∀ k > r (3.46) W r ( x ) = u exp " ∞ X n =1 p − n nx − n exp " ∞ X n =1 p n nx n The parameter u in (3.46) will not be crucial (it will be identified with a linebundle in the next Section) and so we will not mention it explicitly in our notation. , Definition 3.17 explains the appeal of presenting the algebra A r in terms of thegenerators W n,k : to show that A r acts on a certain module, it is enough toshow that the module is a good representation of A , and then check that rela-tions (3.45) holds (if one wants an action of A ext r , then one also has to check (3.46)). Proof. of Theorem 3.13: By (3.27), the element W n,k ∈ b A ↑ is a certain infinite K -linear combination of the elements: E v = E n ,k ...E n t ,k t as v = { ( n , k ) , ..., ( n t , k t ) } runs over convex paths of size ( n, k ). Moreover, theleading order coefficient, i.e. the coefficient of E n,k itself, is a power of q . Bysuccessive applications of Claim 3.8, this implies that there exists ∗ ∈ Z such that:(3.47) W v := W n ,k ...W n t ,k t = q ∗ E v + ∆ convex paths v ′ X to the left of v E v ′ · element of K Therefore, the K -basis { W v } v convex is upper triangular in terms of { E v } v convex ,with respect to the partial ordering on convex paths of the same size, where one pathis to the left of another. Since { E v } v convex form a basis of b A ↑ over K , therefore:(3.48) b A ↑ = dM k ∈ N n ∈ Z k i ∈ N , P k i = kn i ∈ Z , P n i = n M −∞ < n k ≤ ... ≤ ntkt < ∞ K · W n ,k ...W n t ,k t Combining this statement with Claim 3.8, we infer that:(3.49) [ W n,k , W n ′ ,k ′ ] = ∆ k i ∈ N , P k i = k + k ′ n i ∈ Z , P n i = n + n ′ X −∞ < n k ≤ ... ≤ ntkt < ∞ r k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) W n ,k ...W n t ,k t for certain Laurent polynomials r k,k ,...,k t ,k ′ n,n ,...,n t ,n ′ ( q , q ) ∈ K . To prove (3.36), we mustshow that the only terms in the right-hand side of (3.49) with non-zero coefficientare products of exactly two generators W n ,k W n ,k , with min( k , k ) ≤ min( k, k ′ ).To do so, we must properly interpret (3.29). It was shown in [24] that: R kk ′ ( x, y ) = W k ( x ) W k ′ ( y ) f kk ′ (cid:16) yx (cid:17) is a linear combination of the basis elements { E v } v convex path , whose coefficientsare rational functions in x and y . These rational functions have poles at:(3.50) y − xq − i for i ∈ { max(0 , k − k ′ ) + 1 , ..., k } (3.51) y − xq i for i ∈ { max(0 , k ′ − k ) + 1 , ..., k ′ } and satisfy the following symmetry relation:(3.52) R kk ′ ( x, y ) = R k ′ k ( y, x )and the evaluation properties:(3.53) Res y = xq − i R kk ′ ( x, y ) y = R k − i,k ′ + i ( xq − i , x ) θ (min( i, k ′ − k + i )) for i as in (3.50) –ALGEBRAS ASSOCIATED TO SURFACES 23 (3.54) Res y = xq i R kk ′ ( x, y ) y = − R k ′ − i,k + i ( x, xq i ) θ (min( i, k − k ′ + i )) for i as in (3.51)Combining (3.52), (3.53) and (3.54), we interpret formula (3.29) as saying that: h R kk ′ ( x, y ) expanded in | y | ≪ | x | i − h R kk ′ ( x, y ) expanded in | y | ≫ | x | i =(3.55) = − X α/ ∈{ , ∞} δ (cid:16) yxα (cid:17) Res y = xα R kk ′ ( x, y ) y which is simply a reformulation of the residue theorem for rational functions.Note that the interpretation given above was known to [1] (see [28] for a review),although in the somewhat different context of operator product expansionscorresponding to the free field realization of the deformed W –algebra.Formula (3.55) shows how to make sense of the relation (3.29), and how to deduceit from (3.52), (3.53) and (3.54). However, the down-side is that the right-handside of (3.55) is not readily expressed in terms of the generators W n,k which appearonce we expand the left-hand side (unless we are in the simple situation of (3.35)).To fix this issue, let us recall the normal-ordered integrals that appeared in [28]:(3.56) : R kk ′ ( x, y ) : = : W k ( x ) W k ′ ( y ) f kk ′ (cid:16) yx (cid:17) : == I | z |≫| x | , | y | W k ′ ( z ) W k ( x ) f k ′ k (cid:16) xz (cid:17) Dz − yz − I | z |≪| x | , | y | W k ( x ) W k ′ ( z ) f kk ′ (cid:16) zx (cid:17) Dz − yz where Dz = dz πiz . On one hand, by explicitly computing the integrals, we obtain:(3.57) : R kk ′ ( x, y ) : == c ∈ Z X a,b ∈ N ⊔{ } (cid:2) x a + c y b f ak ′ k W − a − b,k ′ W − c,k + x − a − c y − b − f akk ′ W c,k W a + b +1 ,k ′ (cid:3) where we consider the power series expansion of (3.31): f kk ′ ( x ) = ∞ X a =0 f akk ′ x a ∈ x ∆ K [[ x ]]On the other hand, as observed in (3.52), the two integrands in equation (3.56)represent the same rational function. Therefore, the difference between the twointegrals is given by the sum of the residues in the variable z , similar to (3.55):: R kk ′ ( x, y ) : = R kk ′ ( x, y ) + X α/ ∈{ , ∞} Res z = xα R kk ′ ( x, z ) xα − y The residues are prescribed by (3.53) and (3.54), so we obtain:(3.58) R kk ′ ( x, y ) = : R kk ′ ( x, y ) : ++ k ′ X i =max(0 ,k ′ − k )+1 R k ′ − i,k + i ( x, xq i )1 − yxq i θ (min( i, k − k ′ + i )) −− k X i =max(0 ,k − k ′ )+1 R k − i,k ′ + i ( xq − i , x )1 − yq i x θ (min( i, k ′ − k + i )) , One can iterate relation (3.58) to express the right-hand side as:(3.59) R kk ′ ( x, y ) = : R kk ′ ( x, y ) : + min( l,l ′ ) < min( k,k ′ ) X k + k ′ = l + l ′ ,a,b ∈ Z : R ll ′ ( xq a , xq b ) : g abll ′ ( q , q ) Q various c (cid:16) − yxq c (cid:17) for some g abll ′ ( q , q ) ∈ F . We conjecture that g abll ′ ( q , q ) ∈ K for all a, b, l, l ′ , butwe will not prove this result. Instead, we observe that relation (3.59) gives thecorrect interpretation of (3.29). Indeed, expanding (3.59) for | y | ≪ | x | (respectively | y | ≫ | x | gives us the first (respectively second) term in the left-hand side of (3.29),and taking the difference of these two expansions gives us: W k ( x ) W k ′ ( y ) f kk ′ (cid:16) yx (cid:17) − W k ′ ( y ) W k ( x ) f k ′ k (cid:18) xy (cid:19) == min( l,l ′ ) < min( k,k ′ ) X k + k ′ = l + l ′ ,a,b ∈ Z : R ll ′ ( xq a , xq b ) : "X c δ (cid:18) yxq c (cid:19) Q c ′ = c (1 − q c − c ′ ) g abll ′ ( q , q )The normal-ordered products in the right-hand side are quadratic expressions in the W –algebra generators according to (3.57), and so taking the coefficient of x − n y − n ′ for any n, n ′ ∈ Z in the above expression gives us:(3.60) [ W n,k , W n ′ ,k ′ ] + ∞ X a =1 [ W n − a,k W n ′ + a,k ′ f akk ′ − W n ′ − a,k W n + a,k f ak ′ k ] == min( l,l ′ ) < min( k,k ′ ) X k + k ′ = l + l ′ X d ≤ W d,l ′ W n + n ′ − d,l · coeff + X d> W n + n ′ − d,l W d,l ′ · coeff where the coefficients denoted by “coeff” lie in F . For any fixed n + n ′ and k + k ′ ,formula (3.60) allows us to prove by induction on min( k, k ′ ) that:(3.61) [ W n,k , W n ′ ,k ′ ] = ml ≤ m ′ l ′ min( l,l ′ ) ≤ min( k,k ′ ) X k + k ′ = l + l ′ m + m ′ = n + n ′ W m,l W m ′ ,l ′ · coeff(the base case, when min( k, k ′ ) = 0, is trivial because W n, = δ n ). Comparing theformula above with (3.49), together with the fact that { W v } v convex form a basis,implies that the coefficients in (3.61) lie in ∆ · K . This precisely establishes (3.36). (cid:3) The moduli space of sheaves W –algebras in the previous Sections,let us now construct the modules on which they are expected to act. Consider asmooth projective surface S over an algebraically closed field (denoted by C ) andan ample divisor H ⊂ S . The Hilbert polynomial of a coherent sheaf F on S is: P F ( n ) := χ ( S, F ⊗ O ( nH )) = an + bn + c where a, b, c are rational numbers that one can compute from the Grothendieck-Hirzebruch-Riemann-Roch theorem. One can find formulas for these numbers inthe Appendix to [23], but the only thing we will need in the present paper is that –ALGEBRAS ASSOCIATED TO SURFACES 25 they can be expressed in terms of S, H and the rank and Chern classes r, c , c of F . The reduced Hilbert polynomial is defined as: p F ( n ) = P F ( n ) a A rank r > F on S is called stable (respectivelysemistable) if for all proper subsheaves G ⊂ F we have: p G ( n ) < p F ( n ) (respectively p G ( n ) ≤ p F ( n ))for all n ≫ 0. Since the reduced Hilbert polynomials are monic and quadratic,stability (respectively semistability) is determined by checking the respectiveinequalities for the linear term and constant term coefficients. Note that sta-bility depends on the polarization H , but we will fix a choice throughout this paper. Definition 4.2. (see [15] ): Let M ( r,c ,c ) denote the quasiprojective variety whichcorepresents the moduli functor of stable sheaves on S with the invariants r, c , c . We will denote the moduli space by M when the particular invariants will not beimportant to us. We impose the following assumption throughout this paper:(4.1) Assumption A: gcd( r, c · H ) = 1This assumption has two important consequences: firstly, any semistable sheaf isstable. Secondly, there exists a universal sheaf:(4.2) U (cid:15) (cid:15) M × S which is flat over M , and its fiber over any closed point {F} × S is isomorphic to F as a coherent sheaf over S . This leads to the following fact (see [15]): Proposition 4.3. Under Assumption A, M ( r,c ,c ) is a projective variety, whichalso represents the moduli functor of stable sheaves on S with the invariants r, c , c . Remark 4.4. Note that the universal sheaf U is only determined up to tensoringwith a line bundle pulled back from M . This means that one has the freedomto choose such a line bundle on any component M ( r,c ,c ) of the moduli space,and this represents the ambiguity in choosing the universal sheaf on the wholeof M . Throughout the present paper, we will fix such a choice, requiring onlythat the chosen universal sheaves on M ( r,c ,c ) and M ( r,c ,c +1) be compatiblewith each other as described in the Appendix to [23] . This compatibility isprecisely what allows us to define Hecke correspondences in the following Subsection. Remark 4.5. Since many of the constructions in the present paper are local onthe moduli space of stable sheaves, one can replace the category of coherent sheavesby that of twisted coherent sheaves (see [4] for an overview). In this case, thereexists a twisted universal sheaf on M × S , even without imposing Asssumption A. , The interested reader may try to generalize the results in the present paper to thatsetting, but we do not expect any fundamentally new constructions to arise. Because the universal sheaf U is flat over M , it inherits certain properties fromthe stable sheaves it parametrizes, such as having projective dimension 1 (indeed,any torsion free sheaf on a smooth projective surface has projective dimension 1,see Example 1.1.16 of [15]): Proposition 4.6. There exists a short exact sequence: (4.3) 0 → W → V → U → with W and V locally free sheaves on M × S (see [23] for a proof ). Z = n ( F , F ′ ) s.t. F ⊃ x F ′ for some x ∈ S o ⊂ M × M ′ where the notation: F ⊃ x F ′ means that F ⊃ F ′ and F / F ′ ∼ = C x Consider the natural projection maps:(4.5) ¯ Z p − ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ p S (cid:15) (cid:15) p + ! ! ❇❇❇❇❇❇❇❇ M S M ′ ( F ⊃ x F ′ ) p − z z ✉✉✉✉✉✉✉✉✉✉ p S (cid:15) (cid:15) p + $ $ ❏❏❏❏❏❏❏❏❏❏ F x F ′ Moreover, we have the tautological line bundle on ¯ Z :(4.6) L (cid:15) (cid:15) ¯ Z L| ( F , F ′ ) = F x / F ′ x = Γ( S, F / F ′ )The space ¯ Z was denoted by ¯ Z in [23], but in the present paper we have changedthe notation because we will shortly encounter a related space, denoted by ¯ Z .Remark 4.4 refers to fixing choices of the universal sheaves U on all componentsof the moduli space M , so that these choices are compatible with ¯ Z . Specifically,this entails the existence of a short exact sequence:(4.7) 0 → ( p + × Id) ∗ ( U ) → ( p − × Id) ∗ ( U ) → Γ ∗ ( L ) → Z × S , where ¯ Z −→ ¯ Z × S denotes the graph of p S . We will often abuse notationand denote the first two terms in (4.7) by U ′ and U , thought of as sheaves on ¯ Z × S . –ALGEBRAS ASSOCIATED TO SURFACES 27 X (defined over a base scheme Y , which will be clearfrom context) is locally defined by a dg algebra: O X = h ... d −→ O (2) X d −→ O (1) X d −→ O (0) X i (the ring of functions on the base scheme Y will map into O (0) X ). The scheme ¯ X whose coordinate ring is locally given by: O ¯ X = O (0) X / Im d will be called the support of the dg scheme X . This means that there exists amap of dg schemes ¯ X → X , and we will therefore say that X is supported on ¯ X . Example 4.9. If V is a locally free sheaf on the scheme Y , and we are given asection σ ∈ Γ( V ) , then the derived zero locus of σ is defined as the dg scheme: Z = Spec Y h ... σ ∨ −−→ ∧ ( V ∨ ) σ ∨ −−→ V ∨ σ ∨ −−→ O Y i Alternatively, we will say that Z is cut out (in a derived sense) by the vanishing of σ . Meanwhile, the scheme-theoretic zero locus of σ refers to the scheme: ¯ Z = Spec Y ( O Y / Im σ ∨ ) It is easy to see that Z is supported on ¯ Z . Z supported on the scheme ¯ Z . Definition 4.11. Let Sym denote the symmetric algebra. Consider the dg scheme: (4.8) Z = P M× S ( U ) = Proj (cid:0) Sym •M× S ( U ) (cid:1) Since U is not a vector bundle, but a coherent sheaf of projective dimension 1 as inProposition 4.6, relation (4.8) is taken to mean that: (4.9) Z $ $ ■■■■■■■■■■ (cid:31) (cid:127) ι − / / P M× S ( V ) ρ − (cid:15) (cid:15) (cid:15) (cid:15) M × S The vertical arrow is a projective bundle, and the horizontal arrow is the dg sub-scheme cut out by the vanishing of the following map of vector bundles on P M× S ( V ) : (4.10) σ ∨ : ρ ∗− ( W ) → ρ ∗− ( V ) ։ O (1) Define L = ι ∗− ( O (1)) , where O (1) is the tautological line bundle on P M× S ( V ) . Similarly, the non-derived zero locus of the section (4.10) is the scheme ¯ Z , whichwill therefore be the support of Z . The line bundles denoted by L in (4.6) andDefinition 4.11 are compatible under the natural map ¯ Z → Z . , p − × p S : Z → M × S as the diagonal arrow in (4.9). Thefollowing Proposition claims that there also exists a map p + × p S : Z → M ′ × S . Inorder to define it, let us write U ′ = V ′ / W ′ for the analogue of (4.3) on M ′ × S . Wewill write ω S both for the canonical line bundle on S , and for its pull-back to M ′ × S . Proposition 4.13. We have the diagram below: (4.11) Z ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ (cid:31) (cid:127) ι + / / P M ′ × S ( W ′∨ ⊗ ω S ) ρ + (cid:15) (cid:15) (cid:15) (cid:15) M ′ × S where the vertical arrow is a projective bundle, and the horizontal arrow ι + is thedg subscheme cut out by the vanishing of the following map of vector bundles: (4.12) σ ′ : O ( − ֒ → ρ ∗ + ( W ′ ⊗ ω − S ) −→ ρ ∗ + ( V ′ ⊗ ω − S ) on P M ′ × S ( W ′∨ ⊗ ω S ) . Moreover, we have L ∼ = ι ∗ + ( O ( − in the notation of (4.11) . We refer the reader to [23] for the proof of Proposition 4.13, which states thatthe derived zero loci of the sections (4.10) and (4.12) are isomorphic dg schemes.Combining Definition 4.11 and Proposition 4.13, we conclude that there exist mapsas in the diagram below, which are compatible with those of (4.5) under the supportmap ¯ Z → Z (we will use the same symbols, namely p − , p S , p + , for these maps):(4.13) Z p − ~ ~ ⑤⑤⑤⑤⑤⑤⑤ p S (cid:15) (cid:15) p + ! ! ❈❈❈❈❈❈❈ M S M ′ M × M ′ × M ′′ :(4.14) ¯ Z = n ( F , F ′ , F ′′ ) s.t. F ⊃ x F ′ ⊃ x F ′′ for some x , x ∈ S o Moreover, we define the dg scheme Z that completes the derived fiber square below:(4.15) S × S Z π + " " ❊❊❊❊❊❊❊❊❊ π − | | ②②②②②②②②② p S × p S O O Z p − ~ ~ ⑤⑤⑤⑤⑤⑤⑤ p + " " ❋❋❋❋❋❋❋❋ Z p − | | ①①①①①①①① p + ! ! ❈❈❈❈❈❈❈❈ M M ′ M ′′ Since Z is supported on ¯ Z , it is easy to see that Z is supported on ¯ Z . Themap p S × p S in (4.15) records the support points ( x , x ) of the quotients in (4.14). –ALGEBRAS ASSOCIATED TO SURFACES 29 Consider the derived restriction of Z to the diagonal ∆ : S ֒ → S × S , i.e. the derivedfiber product of p S × p S and ∆. Definition 4.11 and Proposition 4.13 imply:(4.16) Z | ∆ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ (cid:31) (cid:127) / / P M ′ × S ( V ′ ) × P M ′ × S (cid:0) W ′∨ ⊗ ω S (cid:1) ρ (cid:15) (cid:15) (cid:15) (cid:15) M ′ × S where the embedding is defined as the derived zero locus of the section:(4.17) O σ ⊕ σ ′ −−−→ ρ ∗ ( W ′∨ ) ⊗ O (1) M ρ ∗ ( V ′ ⊗ ω − S ) ⊗ O (1) =: e E The notation O (1), O (1) stands for the tautological line bundles on theprojectivizations that appear in (4.16), and σ , σ ′ are the sections in (4.10), (4.12). Proposition 4.15. The composition of the section σ ∨ ⊕ σ ′ with the map: ρ ∗ ( W ′∨ ) ⊗ O (1) M ρ ∗ ( V ′ ⊗ ω − S ) ⊗ O (1) taut ⊖ taut −−−−−−−−→ O (1) ⊗ O (1) ⊗ ρ ∗ ( ω − S ) vanishes. Above, taut , refers to the tautological surjective homomorphisms: (4.18) ρ ∗ ( V ′ ) taut −−−→ O (1) and ρ ∗ ( W ′∨ ⊗ ω S ) taut −−−→ O (1) on P M ′ × S ( V ′ ) and P M ′ × S ( W ′∨ ⊗ ω S ) , respectively (we abuse notation by alsowriting taut , for the homomorphisms (4.18) tensored by arbitrary line bundles).Proof. Explicitly, the Proposition states that the following compositions coincide: O ( − taut ∨ −−−→ ρ ∗ ( V ′∨ ) j ∨ −→ ρ ∗ ( W ′∨ ) | {z } σ taut −−−→ O (1) ⊗ ρ ∗ ( ω − S ) O ( − taut ∨ −−−→ ρ ∗ ( W ′ ⊗ ω − S ) j −→ ρ ∗ ( V ′ ⊗ ω − S ) | {z } σ ′ taut −−−→ O (1) ⊗ ρ ∗ ( ω − S )up to tensoring with a suitable line bundle. The map j is induced by the mapof vector bundles W ′ → V ′ that features in (4.3), and j ∨ denotes its dual. Thenthe equality of the two compositions above is a consequence of the standard linearalgebra fact that: λ · A · v = v T · A T · λ T where C v −→ C m is a vector, C m A −→ C n is a matrix, and C n λ −→ C is a covector. (cid:3) Definition 4.16. Define the derived scheme: (4.19) Z • ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:31) (cid:127) ι / / P M ′ × S ( V ′ ) × P M ′ × S (cid:0) W ′∨ ⊗ ω S (cid:1) ρ (cid:15) (cid:15) (cid:15) (cid:15) M ′ × S where the embedding is the derived zero locus of the section: (4.20) O σ ⊕ σ ′ −−−→ E , given by the formula (4.17) , but mapping into the vector bundle E given by: (4.21) 0 −→ E i −→ e E taut ⊖ taut −−−−−−−−→ O (1) ⊗ O (1) ⊗ ρ ∗ ( ω − S ) −→ (the maps taut , are defined in Proposition 4.15). Z • ֒ → ¯ Z consisting of flags (4.14) with x = x is thesupport of Z • . Moreover, we have a natural map of dg schemes:(4.22) Z • → Z | ∆ arising from the fact that the former (respectively latter) dg scheme is the derivedzero locus of the section (4.20) (respectively (4.17)). Moreover, the map of dgschemes (4.22) is compatible with the equality of their supports ¯ Z • = ¯ Z | ∆ . Theline bundles O (1), O (1) on the two projective bundles P ( V ′ ) × P ( W ′∨ ⊗ ω S ) in(4.19) restrict to the line bundles L , L − on either of the dg schemes Z | ∆ and Z • .In terms of the support schemes ¯ Z | ∆ = ¯ Z • , these line bundles are given by: L | ( F⊃ x F ′ ⊃ x F ′′ ) = F ′ x / F ′′ x L | ( F⊃ x F ′ ⊃ x F ′′ ) = F x / F ′ x Finally, note that we have forgetful maps: Z • π − −−→ Z (4.23) Z • π + −−→ Z (4.24)which underlie the following forgetful maps between their supports:¯ Z • π − −−→ ¯ Z , ( F ⊃ x F ′ ⊃ x F ′′ ) ( F ⊃ x F ′ )¯ Z • π + −−→ ¯ Z , ( F ⊃ x F ′ ⊃ x F ′′ ) ( F ′ ⊃ x F ′′ )At the level of dg structures, the map (4.23) arises from the commutativity of thefollowing diagram of projective bundles on M ′ × S , coupled with (4.11) and (4.19): E t t projection (cid:15) (cid:15) Z • π − (cid:15) (cid:15) (cid:31) (cid:127) ι / / P M ′ × S ( V ′ ) × P M ′ × S (cid:0) W ′∨ ⊗ ω S (cid:1) projection (cid:15) (cid:15) V ′ ⊗ ω − S ⊗ O (1) t t Z (cid:31) (cid:127) ι + / / P M ′ × S ( W ′∨ ⊗ ω S )In other words, the homomorphism between the dg structure sheaves of Z and Z • is induced by the duals of the maps labeled “projection” in the formula above. Thesame kind of diagram defines the map (4.24). At the level of K –theory, the map π − : Z • → Z is the projectivization of the virtual vector bundle:(4.25) [ V ′ ] − [ W ′ ] + [ L ⊗ ω S ]where the locally free sheaves W ′ , V ′ on Z correspond to the sheaf denoted F ′ in:¯ Z = supp Z = {F ⊃ F ′ } –ALGEBRAS ASSOCIATED TO SURFACES 31 Similarly, the map π + : Z • → Z is the projectivization of the virtual vector bundle:(4.26) h W ′∨ ⊗ ω S i − h V ′∨ ⊗ ω S i + [ L − ⊗ ω S ] where supp Z = {F ′ ⊃ F ′′ } where the locally free sheaves W ′ , V ′ on Z correspond to the sheaf denoted F ′ in:¯ Z = supp Z = {F ′ ⊃ F ′′ } Assumption S : either ( ω S ∼ = O S or c ( ω S ) · H < H -stable sheaf F is either 0 or an isomorphism, weconclude that Hom( F , F ) = C . Meanwhile, Serre duality implies that:Ext ( F , F ) ∼ = Hom( F , F ⊗ ω S ) ∨ The Hom space in the right-hand side is canonically isomorphic to either C or 0,depending on which of the two options of (4.27) holds. Since the tangent spaceto M at the point F is isomorphic to Ext ( F , F ), the descriptions of Hom( F , F )and Ext ( F , F ) above imply that M is smooth (see [15] for details). Proposition 4.19. Under Assumption S, the section (4.10) is regular, hence thedg scheme Z coincides with its support ¯ Z (and the latter scheme is smooth). Proposition 4.19 was proved in [23]. The method of attack was to estimate thedimensions of the tangent spaces to ¯ Z . Since ¯ Z parametrizes nested sheaves, it iswell-known that its tangent spaces are given by the formula:(4.28) Tan ( F⊃F ′ ) ¯ Z = Ker (cid:18) Ext ( F , F ) ⊕ Ext ( F ′ , F ′ ) ( h, − v ) −−−−→ Ext ( F ′ , F ) (cid:19) where the maps h and v are given as in the following diagram, all of whose rowsand columns are exact (we write C x for the length 1 quotient F / F ′ ):Tan x S / / Ext ( F , C x ) / / Ext ( F ′ , C x ) / / / / C Ext ( C x , F ) (cid:31) (cid:127) / / O O Ext ( F , F ) h / / O O O O Ext ( F ′ , F ) / / O O Ext ( C x , F ) O O O O Ext ( C x , F ′ ) / / O O Ext ( F , F ′ ) h ′ / / v ′ O O Ext ( F ′ , F ′ ) v O O / / / / Ext ( C x , F ′ ) O O C (cid:31) (cid:127) / / (cid:31) ? O O Hom( F , C x ) / / O O Hom( F ′ , C x ) / / (cid:31) ? O O Tan x S O O The vector spaces in the corners of the diagram are:Tan x S ∼ = Ext ( C x , C x ) , and C ∼ = Hom( C x , C x ) ∼ = Ext ( C x , C x ) , Moreover, it was shown in [23] that the differential of the map ¯ Z p S → S is surjective:(4.29) p S ∗ : Tan ( F⊃ x F ′ ) ¯ Z ։ Tan x S and that the kernel of this map consists of pairs of Exts that come from Ext ( F , F ′ ):(4.30) ( v ′ , h ′ ) : Ext ( F , F ′ ) ∼ = −→ Ker p S ∗ ⊂ right-hand side of (4.28)We will use (4.29) and (4.30) to prove the following analogue of Proposition 4.19. Proposition 4.20. Under Assumption S, the section (4.20) is regular, hence thedg scheme Z • coincides with its support ¯ Z • (and the latter scheme is smooth).Proof. By analogy with (4.16), the definition of Z as the derived fiber product Z × M ′ Z implies that we have the following commutative triangle of maps:(4.31) Z * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ (cid:31) (cid:127) / / P M ′ × S × S (cid:0) W ′ ∨ ⊗ ω S (cid:1) × P M ′ × S × S ( V ′ ) ˜ ρ (cid:15) (cid:15) (cid:15) (cid:15) M ′ × S × S where W ′ (respectively V ′ ) refers to the pull-back of the vector bundle W ′ (respec-tively V ′ ) from M × S to M × S × S via the first (respectively second) projectionof S × S onto S . The embedding in (4.31) is the derived zero locus of the section:(4.32) O σ ′ ⊕ σ −−−−→ ˜ ρ ∗ ( V ′ ⊗ ω − S ) ⊗ O (1) M ˜ ρ ∗ ( W ′ ∨ ) ⊗ O (1)and it is easy to see that the scheme-theoretic zero locus of the section (4.32) is¯ Z of (4.14). Because of the Grothendieck-Hirzebruch-Riemann-Roch theorem, thedimension of the smooth variety M ′ at any point F ′ of second Chern class c is:(4.33) dim Ext ( F ′ , F ′ ) = 1 + ε − dim χ ( F ′ , F ′ ) = 1 + ε + const + 2 rc where the integer labeled “const” depends only on S, H, r, c , and the number ε isequal to the dimension of Ext ( F ′ , F ′ ), which is 1 or 0 depending on which of thetwo cases in Assumption S holds. The dimension of the derived scheme Z is thedimension of the projective bundle P × P in (4.31) minus the rank of the vectorbundle in the right-hand side of (4.32). Since rank V ′ − rank W ′ = r , we have:(4.34) dim Z = 1 + ε − dim χ ( F ′ , F ′ ) = 1 + ε + const + 2 rc + 2(the two extra dimensions in the right-hand side come from 4 − 2: there are 4 extradimensions in (4.31) due to the copies of the surface S , but we must subtract twodimensions from each of the projectivizations). As for the scheme-theoretic zerolocus, we conclude that the dimensions of all of its local rings satisfies:(4.35) dim ¯ Z ≥ RHS of (4.34) = 1 + ε + const + 2 rc + 2 –ALGEBRAS ASSOCIATED TO SURFACES 33 By a well-known deformation-theory argument, the tangent space to a point of ¯ Z parametrizing a flag ( F ⊃ x F ′ ⊃ x F ′′ ) equals the space of triples of extensions:(4.36) 0 / / F / / S / / F / / / / F ′ / / ?(cid:31) O O S ′ / / ?(cid:31) O O F ′ ?(cid:31) O O / / / / F ′′ / / ?(cid:31) O O S ′′ / / ?(cid:31) O O F ′′ ?(cid:31) O O / / ( F⊃ x F ′ ⊃ x F ′′ ) ¯ Z == Ker h Ext ( F , F ) ⊕ Ext ( F ′ , F ′ ) ⊕ Ext ( F ′′ , F ′′ ) λ −→ Ext ( F ′ , F ) ⊕ Ext ( F ′′ , F ′ ) i where the map λ is the alternating sum of the 4 natural maps induced by theinclusions F ⊃ F ′ ⊃ F ′′ on the Ext spaces in question. By analogy with (4.33), wehave the following formulas for the dimensions of the following Ext spaces:dim Ext ( F , F ) = 1 + ε + const + 2 rc − r dim Ext ( F ′′ , F ′′ ) = 1 + ε + const + 2 rc + 2 r dim Ext ( F ′ , F ) = 1 + 0 + const + 2 rc − r dim Ext ( F ′′ , F ′ ) = 1 + 0 + const + 2 rc + r dim Ext ( F , F ′ ) = 0 + ε + const + 2 rc − r dim Ext ( F ′ , F ′′ ) = 0 + ε + const + 2 rc + r (the first two summands in each RHS represents the dimension of the correspondingHom and Ext spaces, subject to Assumption S). Therefore, the tangent space(4.37) has expected dimension (4.34) if and only if λ has cokernel of dimension 2 − ε . Claim 4.21. The map λ of (4.37) has cokernel of dimension: (4.38) ( − ε if x = x and F / F ′′ ∼ = C x ⊕ C x − ε otherwiseIn the latter case, ¯ Z is smooth of expected dimension at the point ( F ⊃ F ′ ⊃ F ′′ ) ,while in the former case its tangent space has dimension 1 greater than expected.Proof. From the long exact sequences corresponding to the Ext functor, we obtain: λ (cid:0) Ext ( F , F ) , , (cid:1) = (cid:16) Ker Ext ( F ′ , F ) p → Ext ( C x , F ) , (cid:17) λ (cid:0) , , Ext ( F ′′ , F ′′ ) (cid:1) = (cid:16) , Ker Ext ( F ′′ , F ′ ) p → Ext ( F ′′ , C x ) (cid:17) where C x and C x denote F / F ′ and F ′ / F ′′ , respectively. Therefore, the cokernelof the map λ is isomorphic to the cokernel of the map:(4.39) Ext ( F ′ , F ′ ) ( µ ,µ ) −→ Im p ⊕ Im p 24 ANDREI NEGUT , where the maps µ and µ are as in the diagrams below:Hom( F ′ , C x ) (cid:15) (cid:15) / / Ext ( F ′ , F ′ ) µ ( ( r (cid:15) (cid:15) (cid:15) (cid:15) / / Ext ( F ′ , F ) p (cid:15) (cid:15) / / / / Ext ( F ′ , C x ) (cid:15) (cid:15) (cid:15) (cid:15) Ext ( C x , C x ) / / Ext ( C x , F ′ ) q / / Ext ( C x , F ) t (cid:15) (cid:15) (cid:15) (cid:15) s / / / / Ext ( C x , C x ) C ε Ext ( C x , F ′ ) / / (cid:15) (cid:15) Ext ( F ′ , F ′ ) / / r (cid:15) (cid:15) (cid:15) (cid:15) µ ( ( Ext ( F ′′ , F ′ ) p (cid:15) (cid:15) / / Ext ( C x , F ′ ) (cid:15) (cid:15) (cid:15) (cid:15) Ext ( C x , C x ) / / Ext ( F ′ , C x ) q / / Ext ( F ′′ , C x ) t (cid:15) (cid:15) (cid:15) (cid:15) s / / / / Ext ( C x , C x ) C ε In each of the above diagrams, the space in the bottom right is ∼ = C . • If ε = 1, it is easy to see that Ker s i = Ker t i for both i ∈ { , } . Therefore, wemay rewrite (4.39) by saying that Coker λ is isomorphic to the cokernel of:(4.40) Ext ( F ′ , F ′ ) ( µ ,µ ) −→ Im q ⊕ Im q We want to show that the cokernel of (4.40) has dimension 1 (respectively 0) inthe first (respectively second) case of (4.38). Since µ i = q i ◦ r i , we will solve thisproblem by noting that the following square commutes:(4.41) Ext ( F ′ , F ′ ) r / / / / r (cid:15) (cid:15) (cid:15) (cid:15) Ext ( C x , F ′ ) y (cid:15) (cid:15) (cid:15) (cid:15) Ext ( F ′ , C x ) y / / / / Ext ( C x , C x )where the horizontal maps are induced by the sequence 0 → F ′ → F → C x → → F ′′ → F ′ → C x → r , r ) = Ker ( y , − y )Therefore, the row and column of the following diagram are exact:Ext ( C x , C x ) ⊕ Ext ( C x , C x ) ρ * * (cid:15) (cid:15) Ext ( F ′ , F ′ ) ( r ,r ) / / ( µ ,µ ) * * Ext ( C x , F ′ ) ⊕ Ext ( F ′ , C x ) ( q ,q ) (cid:15) (cid:15) (cid:15) (cid:15) ( y , − y ) / / Ext ( C x , C x )Im q ⊕ Im q –ALGEBRAS ASSOCIATED TO SURFACES 35 If x = x , then Ext ( C x , C x ) = 0, and the surjectivity of ( r , r ) impliesthe surjectivity of ( µ , µ ), as required. If x = x , then Ext ( C x , C x ) ∼ = C ,and the cokernel of ( µ , µ ) is isomorphic to the cokernel of ρ . The latter is1 or 0 dimensional depending on whether F / F ′′ is isomorphic to C x ⊕ C x or not. • If ε = 0, then Im q i has codimension 1 inside Im p i , and so we may rewrite (4.39)by saying that Coker λ is isomorphic to the cokernel of:(4.43) Ext ( F ′ , F ′ ) ( µ ,µ ) −→ Im q ⊕ Im q ⊕ C where the right-hand side of (4.43) should more appropriately be called “a vectorspace containing Im q ⊕ Im q as a codimension 2 subspace”. One can repeat theargument following equation (4.40) in the previous bullet to give us the desiredvalues (4.38) for the dimension of the cokernel of the map (4.43). (cid:3) The dg scheme Z • defined as the derived zero locus of the section (4.20) has:(4.44) dim Z • = 1 + ε + const + 2 rc + 1Note that this is 1 less than the dimension of Z from (4.34): two dimensions fewerbecause we require the support points x and x to be the same, but one dimensionmore because the section (4.20) maps to a vector bundle of rank 1 less than (4.32).To conclude the proof of Proposition 4.20, it is therefore enough to show that thetangent spaces to the support scheme:(4.45) ¯ Z • = n ( F , F ′ , F ′′ ) s.t. F ⊃ x F ′ ⊃ x F ′′ for some x ∈ S o have dimension less than or equal to (4.44). But note that the composition of thefollowing linear maps:(4.46) Tan ( F⊃ x F ′ ⊃ x F ′′ ) ¯ Z • ֒ → Tan ( F⊃ x F ′ ⊃ x F ′′ ) ¯ Z p S ∗ × p S ∗ −→ Tan x S ⊕ Tan x S lands in the 2-dimensional diagonal subspace Tan x S ֒ → Tan x S ⊕ Tan x S . Becauseof the last sentence in the statement of Claim 4.21, in order to obtain the desiredestimate on the dimensions of the tangent spaces to ¯ Z • , it suffices to prove that:the map p S ∗ × p S ∗ is surjective if F / F ′′ ∼ = C x ⊕ C x (4.47) the map p S ∗ × p S ∗ has 1-dimensional cokernel if F / F ′′ = C x ⊕ C x (4.48)Let us first tackle the case in (4.47). It suffices to show that for any tangent vector v ∈ Tan x S = Tan x S , there exist tangent vectors to ¯ Z which map to either ( v, , v ). We will only prove the former statement, as the latter is analogous andwe leave it to the interested reader. By (4.29), there exists:( w, w ′ ) ∈ Ker (cid:0) Ext ( F , F ) ⊕ Ext ( F ′ , F ′ ) −→ Ext ( F ′ , F ) (cid:1) which maps to v under p S ∗ . By (4.29), we may change w and w ′ by the image ofone and the same element of Ext ( F , F ′ ) without changing the vector v . The taskis to complete the datum above to a triple:( w, w ′ , w ′′ ) ∈ Tan ( F⊃ x F ′ ⊃ x F ′′ ) ¯ Z ⊂ Ext ( F , F ) ⊕ Ext ( F ′ , F ′ ) ⊕ Ext ( F ′′ , F ′′ ) We must have x = x in order for ( F ⊃ x F ′ ⊃ x F ′′ ) to lie in ¯ Z • , but we will use differentnotations in order to not confuse the length 1 quotients F / F ′ ∼ = C x and F ′ / F ′′ ∼ = C x , However, in order to ensure that ( p S × p S ) ∗ ( w, w ′ , w ′′ ) = ( v, w ′ and w ′′ must come from one and the same element of Ext ( F ′ , F ′′ ). This is equivalent torequiring that in the diagram below with exact row and columns:(4.49) Ext ( F , F ′ ) / / / / (cid:15) (cid:15) Ext ( F , C x ) (cid:15) (cid:15) Ext ( F ′ , F ′′ ) (cid:15) (cid:15) (cid:15) (cid:15) / / Ext ( F ′ , F ′ ) r (cid:15) (cid:15) (cid:15) (cid:15) r / / / / φ ( ( Ext ( F ′ , C x ) y (cid:15) (cid:15) (cid:15) (cid:15) Ext ( C x , F ′′ ) / / Ext ( C x , F ′ ) y / / / / Ext ( C x , C x )we may choose w ′ ∈ Ext ( F ′ , F ′ ) to map to 0 in Ext ( F ′ , C x ). But remember that w ′ may only be changed by adding the image of an arbitrary element in Ext ( F , F ′ ),and so chasing through (4.49) implies that it is enough to show that φ ( w ′ ) = 0. Tothis end, recall that the extensions w and w ′ have the property that they map toone and the same element in Ext ( F ′ , F ). Consider the following diagram:(4.50) Ext ( F , F ) (cid:15) (cid:15) Ext ( F ′ , F ′ ) / / (cid:15) (cid:15) ( ( Ext ( F ′ , F ) (cid:15) (cid:15) Ext ( C x , F ′ ) / / Ext ( C x , F ) ( ( Ext ( C x , C x )The map φ in diagram (4.49) is the composition of the two dotted maps in diagram(4.50), for the following reason: because we assumed that F / F ′′ ∼ = C x ⊕ C x ,we may extend the homolorphism F ′ ։ C x to a homomorphism F ։ C x . Butthen the fact that φ ( w ′ ) = 0 is clear because w and w ′ map to the same el-ement in Ext ( F ′ , F ) and the composition of the two vertical maps in (4.50) is zero.Let us now treat the case in (4.48). We may ask for which v ∈ Tan x S = Tan x S can we choose a tangent vector ( w, w ′ , w ′′ ) to ¯ Z which maps to either ( v, 0) or(0 , v ) under the map (4.46). Chasing through the proof on the previous page showsthat such a tangent vector ( w, w ′ , w ′′ ) can be chosen if and only if φ ( w ′ ) = 0,where φ is the diagonal arrow in (4.49). Since the image of φ is one-dimensional,this places a single linear condition on the vector w ′ , and therefore there exists aone-dimensional family of v ’s such that ( v, 0) and (0 , v ) lie in the image of p S ∗ × p S ∗ .To prove (4.48), we must show that the image of p S ∗ × p S ∗ is three-dimensional, andso we must show that there exists at least one more vector in the image. In fact,we will show that vectors of the form ( v, v ) are always in the image of p S ∗ × p S ∗ , –ALGEBRAS ASSOCIATED TO SURFACES 37 for any v ∈ Tan x S = Tan x S . By (4.29), we may choose vectors: (cid:18) ww ′ (cid:19) ∈ Ker Ext ( F , F ) ⊕ −→ Ext ( F ′ , F )Ext ( F ′ , F ′ ) = Tan ( F⊃ x F ′ ) ¯ Z (4.51) (cid:18) ¯ w ′ w ′′ (cid:19) ∈ Ker Ext ( F ′ , F ′ ) ⊕ −→ Ext ( F ′′ , F ′ )Ext ( F ′′ , F ′′ ) = Tan ( F ′ ⊃ x F ′′ ) ¯ Z (4.52)which map to the same vector v ∈ Tan x S = Tan x S under the map p S ∗ . Accordingto (4.30), we may modify the vectors w ′ and ¯ w ′ by arbitrary elements coming fromExt ( F , F ′ ) and Ext ( F ′ , F ′′ ), respectively, without modifying the images of p S ∗ .We claim that we can perform these modifications so as to ensure w ′ = ¯ w ′ , andin this case we are done, because the tangent vector ( w, w ′ , w ′′ ) will map to ( v, v )under p S ∗ × p S ∗ . The elements w ′ and ¯ w ′ lie in the space Ext ( F ′ , F ′ ) in the middleof the diagram (4.49). Chasing through the aforementioned diagram, we see that wecan make w ′ equal to ¯ w ′ upon modification by elements coming from Ext ( F , F ′ )and Ext ( F ′ , F ′′ ), respectively, if and only if:(4.53) φ ( w ′ ) = φ ( ¯ w ′ )The fact that equality (4.53) holds is a direct consequence of the following claim: Claim 4.22. The element φ ( w ′ ) coincides with the image of v under the map: (4.54) Ext ( C x , C x ) −→ Ext ( C x , C x ) induced by the short exact sequence: −→ F ′ / F ′′ ∼ = C x −→ F / F ′′ −→ F / F ′ ∼ = C x −→ The analogous statement holds for φ ( ¯ w ′ ) , which implies equality (4.53) . We will prove the claim about φ ( w ′ ), and leave the analogous result for φ ( ¯ w ′ ) as anexercise for the interested reader. Assume the pair ( w, w ′ ) of (4.51) corresponds toa commutative diagram:(4.55) 0 / / F ι / / S π / / F / / / / F ′ ι ′ / / ?(cid:31) O O S ′ π ′ / / ?(cid:31) O O F ′ ?(cid:31) O O / / φ ( w ′ ) is given by the extension:(4.56) 0 −→ F ′ / F ′′ ∼ = C x (1 , −→ F ′ / F ′′ ⊕ F ′ S ′ (0 ,π ′ ) −→ F −→ F / F ′ ∼ = C x −→ , and the notation ⊕ F ′ denotes the push-out with respect to the standard projectionmap F ′ ։ F ′ / F ′′ and the map ι ′ : F ′ → S ′ . Because of the commutative diagram:(4.57) 0 / / F ′ / F ′′ (1 , / / F ′ / F ′′ ⊕ F ′ S ′ (0 ,π ′ ) / / ( ι ′ , projection) (cid:15) (cid:15) F / / projection (cid:15) (cid:15) F / F ′ / / / / F ′ / F ′′ ι ′ / / S ′ / S ′′ π ′ / / F / F ′′ / / F / F ′ / / v ∈ Tan x S =Tan x S implies that S ′ / S ′′ ∼ = S / S ′ , hence the bottom row of (4.57) is equal to: (4.58) 0 −→ F ′ / F ′′ −→ F / F ′′ ι −→ S / S ′ π −→ F / F ′ −→ ( C x , C x ). Since (4.58) is nothing butthe map (4.54) applied to the short exact sequence:0 −→ F / F ′ ∼ = C x −→ S / S ′ −→ F / F ′ ∼ = C x −→ v ∈ Tan x S , we are done. (cid:3) Corollary 4.23. The scheme ¯ Z is l.c.i. of dimension (4.34) .Proof. By Claim 4.21, we may stratify ¯ Z into two locally closed subsets: the locus A of points where the tangent space has expected dimension (4.34), and the locus B of points where the tangent space has dimension 1 bigger than expected. It isclear that dim A is no greater than the right-hand side of (4.34), so it remains toshow that the same is true of dim B . However, recall that Claim 4.21 implies: B ⊂ ¯ Z • ⇒ dim B ≤ dim ¯ Z • = RHS of (4.44) < RHS of (4.34) (cid:3) The double shuffle algebra action Z of (4.9) induce operators on the K –theory groups: K M = ∞ M c = ⌈ r − r c ⌉ K ( M ( r,c ,c ) )explicitly given by:(5.1) K M e d −→ K M ′ × S , e d = ( p + × p S ) ∗ (cid:0) L d · p ∗− (cid:1) (5.2) K M ′ f d −→ K M× S , f d = ( − r det U · ( p − × p S ) ∗ (cid:0) L d − r · p ∗ + (cid:1) A non-split length 2 sheaf A supported at a point x of a smooth surface is an extension0 → C x → A → C x → 0, which is completely determined by how the two generators of m x / m x acton A . If A and B are two such extensions, then we obtain two rank 1 maps A → B and B → A by projecting through the quotients C x . It is easy to show that 0 → C x → A → B → C x → → C x → B → A → C x → ( C x , C x ) –ALGEBRAS ASSOCIATED TO SURFACES 39 with the notations in (4.13). Let us focus on the operators { e d } d ∈ Z . Compositions of k such operators can be described via the correspondence (fiber product is derived):(5.3) Z k := Z × M Z × M ... × M k − Z −→ M × ... × M k which is a dg scheme supported on:(5.4) ¯ Z k = n F , ..., F k sheaves , points x , ..., x k ∈ S s.t. F ⊂ x ... ⊂ x k F k o (the notation above is that M i denotes the moduli space that parametrizes the sta-ble sheaf denoted by F i ). Both Z k and ¯ Z k are endowed with line bundles {L i } ≤ i ≤ k ,which parametrize the length 1 quotients F i / F i − . We have projection maps: Z kp + } } ④④④④④④④④ p Sk (cid:15) (cid:15) p − ! ! ❈❈❈❈❈❈❈❈ M S k M k ¯ Z k ¯ p + } } ⑤⑤⑤⑤⑤⑤⑤⑤ ¯ p Sk (cid:15) (cid:15) ¯ p − ! ! ❇❇❇❇❇❇❇❇ M S k M k where ¯ p + , ¯ p S k , ¯ p − send a flag of sheaves (5.4) to F , ( x , ..., x k ), F k , respectively,and they are compatible with p + , p − , p S k under the support map. Then we have:(5.5) e d ◦ ... ◦ e d k = ( p + × p S k ) ∗ (cid:16) L d ... L d k k · p ∗− (cid:17) : K M −→ K M× S k Certain quadratic relations (the case k = 2) between these compositions wereworked out in [23], but the full set of relations was conjectured in loc. cit. tomatch those in the so-called universal shuffle algebra. However, we do not knowhow to describe the full ideal of relations explicitly. The situation is simplified ifwe compose (5.5) with restriction to the smallest diagonal ∆ : S ֒ → S k :(5.6) e d ◦ ... ◦ e d k (cid:12)(cid:12)(cid:12) ∆ : K M −→ K M× S The idea that (5.6) is the “composition” of the operators e d , ..., e d k : K M → K M× S leads to the following notion. Consider the ring homomorphism:(5.7) K = Z [ q ± , q ± ] symmetric in q ,q φ −→ K S given by sending q and q to the Chern roots of Ω S . Let S ∆ ֒ → S × S be the diagonal. Definition 5.2. An “action” of the double shuffle algebra A from (3.1) on theabelian group K M is an abelian group homomorphism: (5.8) A (cid:12)(cid:12)(cid:12) c q r Φ −→ Hom( K M , K M× S ) satisfying the following properties, for any x, y ∈ Hom( K M , K M× S ) and γ ∈ K :(1) Φ(1) = proj ∗ , where proj : M × S → M is the natural projection map(2) Φ( γx ) coincides with the composition: (5.9) K M Φ( x ) −−−→ K M× S Id M × multipilication by φ ( γ ) −−−−−−−−−−−−−−−−−−→ K M× S (3) Φ( xy ) coincides with the composition: (5.10) K M Φ( y ) −−−→ K M× S Φ( x ) × Id S −−−−−−→ K M× S × S Id M × ∆ ∗ −−−−−−→ K M× S , (4) ∆ ∗ Φ (cid:16) [ x,y ](1 − q )(1 − q ) (cid:17) coincides with the difference of compositions: K M Φ( y ) −−−→ K M× S Φ( x ) × Id S −−−−−−→ K M× S × S K M Φ( x ) −−−→ K M× S Φ( y ) × Id S −−−−−−→ K M× S × S Id M × swap ∗ −−−−−−−−→ K M× S × S where swap : S × S → S × S is the permutation map. Conjecture 1.1 states that there exists a homomorphism (5.8) satisfying properties (1)–(4) above, and we will now show how to construct Φ( E n,k ), ∀ ( n, k ) ∈ Z \ (0 , Remark 5.3. There are general reasons why one expects the shuffle algebra to acton K M : as shown by Schiffmann-Vasserot in [29] , their K –theoretic Hall algebra H naturally acts on groups akin to K M . There is a map Υ : H → S big that arisesfrom equivariant localization on the stack of finite length sheaves on S , and it isnatural to conjecture that the image Υ( H ) also acts on the groups K M (see [19] for a detailed and rigorous treatment in a setup very close to ours, which resultsin a similar shuffle algebra). However, it is not clear how to prove that the map Υ is injective, or less ambitiously, that the kernel of this map acts trivially on K M . (cid:0) z d ∈ A ← (cid:1) = e d (5.11) Φ (cid:0) z d ∈ A → (cid:1) = f d (5.12) ∀ d ∈ Z , where we implicitly use the isomorphism A ← ∼ = S and A → ∼ = S op , and:Φ (cid:0) E , ± k ∈ A diag (cid:1) = operator of multiplication by I z ± k ∧ • U ± z ± q − δ −± ! ± ∀ k ∈ N . For example, when the sign is ± = + in the formula above, the contourintegral just picks up the coefficient of z − k in the expansion of the rational function: ∧ • (cid:18) U z (cid:19) = ∧ • (cid:0) V z (cid:1) ∧ • (cid:0) W z (cid:1) where V , W are the vector bundles of (4.3). Let us focus on the operators (5.11).By property (2) of Definition 5.2, the composition (5.6) should correspond to:Φ Sym z d ...z d k k Y ≤ i Consider the following dg scheme, obtained by chains of derivedfiber products of the dg scheme (4.19) via the maps (4.23) and (4.24) : Z • k = Z • × Z Z • × Z ... × Z Z • −→ M × ... × M k –ALGEBRAS ASSOCIATED TO SURFACES 41 which will be supported on the scheme: (5.13) ¯ Z • k = {F , ..., F k sheaves , x ∈ S such that F ⊂ x ... ⊂ x F k } There are line bundles L , ..., L k on Z • k that correspond to the line bundles on ¯ Z • k that parametrize the quotients F / F , ..., F k / F k − , as well as projection maps: (5.14) Z • kp + } } ⑤⑤⑤⑤⑤⑤⑤⑤ p S (cid:15) (cid:15) p − ! ! ❈❈❈❈❈❈❈❈ M S M k ¯ Z • k ¯ p + ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ ¯ p S (cid:15) (cid:15) ¯ p − ! ! ❇❇❇❇❇❇❇❇ M S M k For any vector of integers d • = ( d , ..., d k ) , define the operators: K M e d • −→ K M× S (5.15) K M f d • −→ K M× S (5.16) given by the following formulas: e d • = ( p + × p S ) ∗ (cid:16) L d ... L d k k · p ∗− (cid:17) (5.17) f d • = ( − rk (det U ) ⊗ k · ( p − × p S ) ∗ (cid:16) L d − r ... L d k − rk · p ∗ + (cid:17) (5.18)For any d = ( d , ..., d k ) ∈ Z k , recall the elements E d • , F d • ∈ A from (3.13) and(3.14), respectively. We make Conjecture 1.1 more precise by stipulating that theseelements should act on K –theory groups via the formulas:Φ ( E d • ) = e d • (5.19) Φ ( F d • ) = f d • (5.20)for all d • = ( d , ..., d k ) ∈ Z k . Note that E d • and F d • generate the algebras A ← and A → , because these algebras are isomorphic to S and S op , respectively.5.6. Consider the following particular cases of the operators (5.17) and (5.18): e − k,n = q gcd( k,n ) − e ( d ,...,d k ) (5.21) e k,n = q gcd( k,n ) − f ( d ,...,d k ) (5.22)for all k > n ∈ Z , where d i = (cid:6) nik (cid:7) − l n ( i − k m + δ ki − δ i , together with:(5.23) e , ± k = operator of multiplication by I z ± k ∧ • U ± z ± q − δ −± ! ± By (5.19), (5.20), the operators e n,k should correspond to the generators E n,k (2.8)of the shuffle algebra. Recall from Theorem 3.5 that the relations in the algebra A are generated by the following commutation relation between the generators E n,k :(5.24) [ E n,k , E n ′ ,k ′ ] = ∆ X v convex path p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q ) · E n ,k ...E n t ,k t , for some Laurent polynomials p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q ) ∈ K , where the sum in the right-hand side of (5.24) goes over all P n i = n + n ′ , P k i = k + k ′ such that the path: v = −−−−−−−−−−−−−→ ( n , k ) , ..., ( n t , k t )is sandwiched between the paths v = −−−−−−−−−−→ ( n + n ′ , k + k ′ ) and v = −−−−−−−−−→ ( n ′ , k ′ ) , ( n, k ), andhas the same convexity as the path v . Conjecture 1.1 implies that the analogouscommutation relations hold between the operators (5.21)–(5.23). Conjecture 5.7. For any lattice points ( n, k ) and ( n ′ , k ′ ) , we have: (5.25) [ e n,k , e n ′ ,k ′ ] = ∆ ∗ X v convex path φ ( p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q )) · e n ,k ...e n t ,k t (cid:12)(cid:12)(cid:12) ∆ where φ : K → K S is the map (5.7) . Let us explain the notation in (5.25). The LHS is an operator K M → K M× S × S ,with e n,k acting in the first factor of S × S and e n ′ ,k ′ acting in the second factor.Meanwhile, each summand in the RHS is the following composed operator: K M e nt,kt −−−−→ K M× S e nt − ,kt − × Id S −−−−−−−−−−→ ... e n ,k × Id k − S −−−−−−−−−→ K M× S × ... × S Id M × ∆ ∗ −−−−−−→ K M× S Id M × multiplication by φ ( p n,n ,...,nt,n ′ k,k ,...,kt,k ′ ( q ,q )) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ K M× S and ∆ ∗ in (5.25) denotes the diagonal embedding K M× S → K M× S × S . Proposition 5.8. Conjectures 1.1 and 5.7 are equivalent.Proof. It is clear that Conjecture 1.1 implies Conjecture 5.7, since formula (3.11),together with properties (2), (3), (4) of Definition 5.2, implies formula (5.25).Conversely, let us assume that (5.25) holds and prove Conjecture 1.1. By (3.8), anyelement of A| c q r can be written as:(5.26) x = n ,...,n t X k ,...,k t p n ,...,n t k ,...,k t ( q , q ) · E n ,k ...E n t ,k t for some p n ,...,n t k ,...,k t ( q , q ) ∈ K , where the vectors ( n , k ) , ..., ( n t , k t ) that appearin the right-hand side are ordered clockwise by slope. Properties (2) and (3) ofDefinition 5.2 completely determine the action of x on K M , since they force:(5.27) Φ( x ) = n ,...,n t X k ,...,k t φ ( p n ,...,n t k ,...,k t ( q , q )) · e n ,k ...e n t ,k t (cid:12)(cid:12)(cid:12) ∆ Property (1) of Definition 5.2 holds trivially. Since φ is a ring homomorphism, Φdefined by (5.27) satisfies property (2) of Definition 5.2. As for property (3) , note The notation below means that v is the broken line between (0 , 0) and ( n + ... + n t , k + ... + k t ),formed out of the line segments ( n , k ),...,( n t , k t ) in this order –ALGEBRAS ASSOCIATED TO SURFACES 43 that restricting (5.25) to the diagonal implies:(5.28) e n,k e n ′ ,k ′ | ∆ − e n ′ ,k ′ e n,k | ∆ == X v convex path φ ((1 − q )(1 − q ) p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q )) · e n ,k ...e n t ,k t (cid:12)(cid:12)(cid:12) ∆ and relation (5.28) allows us to iteratively transform any product e n ,k ...e n t ,k t | ∆ into a linear combination of such products where the vectors ( n , k ) , ..., ( n t , k t )are ordered by slope. Moreover, the specific coefficients that appear in the linearcombination match the ones that transform the product E n ,k ...E n t ,k t ∈ A intoa linear combination of such products where the vectors ( n , k ) , ..., ( n t , k t ) areordered by slope. This proves that Φ( x )Φ( y ) | ∆ = Φ( xy ) for any x, y of the form(5.26), by showing that both Φ( x )Φ( y ) | ∆ and Φ( xy ) have the same expansion asin the right-hand side of (5.27). Thus property (3) of Definition 5.2 is established,and the only thing that we used was the associativity property:(5.29) (cid:16)(cid:16) X ◦ Y (cid:12)(cid:12)(cid:12) ∆ (cid:17) ◦ Z (cid:17) (cid:12)(cid:12)(cid:12) ∆ = (cid:16) X ◦ (cid:16) Y ◦ Z (cid:12)(cid:12)(cid:12) ∆ (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ∆ for any K M X,Y,Z −−−−→ K M× S . The fact that assignment (5.27) satisfies property (4) of Definition 5.2 is proved analogously with the computation above, but instead ofthe associativity property (5.29) one uses the following version of the Leibniz rule:(5.30) h X, Y ◦ Z (cid:12)(cid:12)(cid:12) ∆ i red = [ X, Y ] red ◦ Z (cid:12)(cid:12)(cid:12) ∆ + Y ◦ [ X, Z ] red (cid:12)(cid:12)(cid:12) ∆ where if [ X, Y ] = ∆ ∗ ( A ), we write A = [ X, Y ] red . (cid:3) Remark 5.9. In Conjecture 5.7, it suffices to assume that p n,n ,...,n t ,n ′ k,k ,...,k t ,k ′ ( q , q ) of (5.25) are certain Laurent polynomials that do not depend on the surface S ,and one can deduce that they must coincide with the Laurent polynomials thatappear in (5.24) . The argument for this claim is to consider S = A and replaceusual K –theory by equivariant K –theory with respect to the standard torus action C ∗ × C ∗ y A ( q and q correspond to the equivariant parameters) and replace M by the moduli space of framed sheaves. If this is the case, we showed in [26] that Conjecture 1.1 holds and the generators E n,k correspond to the operators e n,k ,thus implying that the coefficients in (5.24) match those in (5.25) . Assumption B: The Kunneth decomposition K S × S ∼ = K S ⊠ K S holds, and theclass of the diagonal splits up as: (5.31) [ O ∆ ] = X c l c ⊠ l c ∈ K S ⊠ K S Following [6] , this implies K M× S ∼ = K M ⊠ K S , and so we may decompose the classof the universal sheaf as: (5.32) [ U ] = X c [ T c ] ⊠ l c ∈ K M ⊠ K S , We assume that {T c } are vector bundles on M whose exterior powers generate K M as a ring. Note that under assumption (5.31), the collections { l c } and { l c } yield dual Z -basesof K S , with respect to the bilinear form: K S ⊗ K S → Z , ([ V ] , [ V ]) = χ ( S, V ⊗ V )The last statement of Assumption B says that any element of K M is of the form:(5.33) Ψ( ..., T c , ... )where Ψ is a Laurent polynomial, symmetric in the Chern roots of each vectorbundle T c separately (if we weakened the assumption by dropping the conditionthat K M is generated by the T c , then we would be only constructing an action of A on the subalgebra of K M generated by the T c ). Under Assumption B, [23] showsthat the operators (5.1) and (5.2) act by: e d Ψ( ..., T c , ... ) = Z ∞− z d Ψ( ..., T c + zl c , ... ) ∧ • (cid:16) zq U (cid:17) (5.34) f d Ψ( ..., T c , ... ) = Z ∞− z d Ψ( ..., T c − zl c , ... ) ∧ • (cid:16) − z U (cid:17) (5.35)where:(5.36) Z ∞− F ( z ) = Res z = ∞ F ( z ) z − Res z =0 F ( z ) z Iterating (5.34) and (5.35) yields:(5.37) e d ...e d k Ψ( ..., T c , ... ) == Z z ≺ ... ≺ z k ∞− z d ...z d k k Ψ( ..., T c + P ki =1 z i l ( i ) c , ... ) Q ≤ i The operators of Definition 5.5 act on tautological classes by: (5.43) e ( d ,...,d k ) Ψ( ..., T c , ... ) == Z z ≺ ... ≺ z k ∞− z d ...z d k k Ψ( ..., T c + ( z + ... + z k ) l c , ... ) (cid:16) − z qz (cid:17) ... (cid:16) − z k qz k − (cid:17) Q i 0. By (5.22), we have e n,k = f d • and e n ′ ,k ′ = f d ′• for certain vectorsof integers d • , d ′• (we ignore the powers of q that appear in (5.22), as they will playno role in what follows). Then (5.44) implies the following formula, for any Ψ:Ψ( ..., T c , ... ) e n,k Z z n ≺ ... ≺ z ∞− z d ...z d n n Ψ( ..., T c − ( z + ... + z n ) l c , ... ) (cid:16) − z qz (cid:17) ... (cid:16) − z n qz n − (cid:17) Q i 0. We will show that for any d • = ( d , ..., d n ) and d ′• = ( d ′− n ′ , ..., d ′ ),we have the following a priori weaker version of Conjecture 5.7:(5.60) [ e d • , f d ′• ] = ∆ ∗ X ˜ d • ,k, ˜ d ′• φ ( p ˜ d • ,k, ˜ d ′• ( q , q )) · e ˜ d • h ± k f ˜ d ′• (cid:12)(cid:12)(cid:12) ∆ where the coefficients p ˜ d • ,k, ˜ d ′• ( q , q ) ∈ K that appear in the right-hand side do notdepend on the surface S . In the formula above: h ± k : K M → K M× S are the coefficients of h ± ( z ) = P ∞ k =0 h ± k z ± k , which acts on K M as multiplication by: ∧ • (cid:18) z ( q − U (cid:19) Its relevance to our purposes is the following simple formula for the n = − n ′ = 1case of (5.60), which follows immediately from (5.43) and (5.44), and was provedin [23] even in the absence of Assumption B: "X d ∈ Z e d z d , X d ′ ∈ Z f d ′ w d ′ = δ (cid:16) zw (cid:17) ∆ ∗ (cid:18) h + ( z ) − h − ( w )1 − q (cid:19) Let us show that (5.60) implies (5.25). By (3.15) and (3.16), the right-hand sideof (5.60) can be written as a product of generators e n,k in increasing order of theslope n/k . Since the coefficients of the resulting decomposition do not depend onthe surface S , and since for equivariant S = A the coefficients match those thatoccur in the algebra A (as shown in [24], [26]), relation (5.25) follows.Therefore, it remains to prove (5.60), which will occupy the remainder of the presentSection. We will use a residue computation, which will require us to assume that:(5.61) 1 − q a is invertible in the ring K S for all a ∈ N . Since this never happens in the algebraic K –theory of a smoothprojective surface (as q is unipotent), we must argue for why this assumption isallowed. By Assumption B, in order to prove (5.60), it is enough to show that:(5.62) [ e d • , f d ′• ]Ψ( ..., T c , ... ) == ∆ ∗ X ˜ d • ,k, ˜ d ′• φ ( p ˜ d • ,k, ˜ d ′• ( q , q )) · e ˜ d • h ± k f ˜ d ′• (cid:12)(cid:12)(cid:12) ∆ Ψ( ..., T c , ... ) –ALGEBRAS ASSOCIATED TO SURFACES 51 for all symmetric Laurent polynomials Ψ as in (5.33). Because of (5.43) and (5.44),both sides of relation (5.62) will be Laurent polynomials in the tautological classes,with coefficients which are Laurent polynomials in q , q ∈ K S . To prove that acertain identity between such polynomials holds, it is enough to do it for a choice of M and S for which there do not exist any relations between the K -theory classes q , q and the various tautological classes T c . To this end: • choose S = A • replace K –theory by the C ∗ × C ∗ equivariant K –theory of S (whichis the ring K of (2.1)), localized with respect to the elements 1 − q a of (5.61) • consider the moduli space M of rank R sheaves on P , for a sufficientlylarge natural number R , framed along the divisor ∞ ⊂ P :(5.63) F| ∞ ∼ = O ⊕ R ∞ (see [24], [26] for a version of our treatment in the setting of framed sheaves).Moreover, let any torus T ⊂ GL R act on M by left multiplication of the isomor-phism (5.63). Therefore, in C ∗ × C ∗ × T equivariant K –theory, the universal sheaf U on M × S splits up as a direct sum ⊕ c U c · t c , where { t c } goes over the elementarycharacters of the torus T . If one chooses R ∈ N large enough, then there will beno relations between the tautological bundles T c = proj ∗ ( U c ), hence there are noalgebraic relations between { Ψ( ..., T c , ... ) } Ψ Laurent polynomial that hold for all R ∈ N .Therefore, it remains to prove formula (5.62) subject to assumption (5.61). Westart by invoking (5.43) and (5.44):(5.64) e d • f d ′• Ψ( ..., T c , ... ) = Z z ≺ ... ≺ z n ≺ w − n ′ ≺ ... ≺ w ∞− R ( z , ..., z n , w , ..., w − n ′ )where:(5.65) R ( z , ..., z n , w , ..., w − n ′ ) = z d ...z d n n w d ′ ...w d ′− n ′ − n ′ ≤ i ≤ n Y ≤ j ≤− n ′ ζ (cid:18) z i w j (cid:19) Ψ( ..., T c + ( z + ... + z n ) l (1) c − ( w + ... + w − n ′ ) l (2) c , ... ) Q ni =1 ∧ • (cid:0) z i q U (1) (cid:1) Q − n ′ i =1 ∧ • (cid:0) − w i U (2) (cid:1)Q n − i =1 (cid:16) − z i +1 q (1) z i (cid:17) Q − n ′ − j =1 (cid:16) − w j +1 q (2) w j (cid:17) Q ≤ i
7→ U and: ζ ( x ) ζ ( x ) = ∧ • (Ω S · x )(1 − x )(1 − xq ) ∈ K S ( x )Explicitly, we have:(5.68) R | ∆ ( z , ..., z n , w , ..., w − n ′ ) = z d ...z d n n w d ′ ...w d ′− n ′ − n ′ ≤ i ≤ n Y ≤ j ≤− n ′ ζ (cid:18) z i w j (cid:19) Ψ( ..., T c + ( z + ... + z n − w − ... − w − n ′ ) l c , ... ) Q ni =1 ∧ • (cid:0) z i q U (cid:1) Q − n ′ i =1 ∧ • (cid:0) − w i U (cid:1)Q n − i =1 (cid:16) − z i +1 qz i (cid:17) Q − n ′ − j =1 (cid:16) − w j +1 qw j (cid:17) Q ≤ ib ζ ( s/w j ) ζ ( w j /s ) (cid:0) − z a +1 qs (cid:1) (cid:16) − sqz a − (cid:17) (cid:0) − w b +1 qs (cid:1) (cid:16) − sqw b − (cid:17)Q i = a z d i i Q j = b w d ′ j j Q i = aj = b ζ (cid:16) z i w j (cid:17) Ψ( ..., T c + ( P i = a z i − P j = b w j ) l c , ... ) Q i = a ∧ • ( z i q/ U ) Q j = b ∧ • ( w j / U ) Q i,i +1 = a (cid:16) − z i +1 qz i (cid:17) Q j,j +1 = b (cid:16) − w j +1 qw j (cid:17) Q i,i ′ = aia ζ ( s/ ( z i q )) ζ ( z i q/s ) Q j
The denominator (1 − q )(1 − q ) in (5.67) simply means that one should removethe initial factor of (1 − q )(1 − q ) from formulas (5.69) and (5.70). Meanwhile,the denominator 1 − q is acceptable because of assumption (5.61). Now recall that(5.69) and (5.70) must be integrated over the contours (5.67). • In the case of (5.70), we can move the variable s to the left of the chain ofvariables ... ≺ w b +1 ≺ ... ≺ z a − ≺ s ≺ z a +1 ≺ ... ≺ w b − ≺ ... and the onlypoles we would encounter are z a − = s (with sign +) and w b +1 = sq (with sign − ), because Ψ has no poles in s other than 0 and ∞ . • In the case of (5.69), we can move the variable s to the right of the chain ofvariables ... ≺ w b +1 ≺ ... ≺ z a − ≺ s ≺ z a +1 ≺ ... ≺ w b − ≺ ... and the onlypoles we would encounter are z a +1 = sq (with sign − ) and w b − = sq (with sign+), together with 0 and ∞ . • However, note that the residues at 0 and ∞ in s can be computed by expanding(5.69) in powers of s , which yields some Laurent polynomial in the variables { z i } i = a , { w j } j = b and some series coefficient of the ratio: ∧ • (cid:0) sq U (cid:1) ∧ • (cid:0) s U (cid:1) expanded in either positive or negative powers of s . In virtue of the inductionhypothesis, the corresponding residue will be a nice summand , which for theremainder of this proof will refer to a summand as in the right hand side of (5.62).Putting all the residues together, we conclude that the RHS of (5.67) equals: n X a =1 − n ′ − X b =1 Z ... ≺ w b +2 ≺ s ≺ z ≺ ... ≺ z a − ≺ z a +1 ≺ ... ≺ z n ≺ w b − ≺ ... ∞− ∆ ∗ Res z a = sq w b = s,w b +1 = sq ( R | ∆ )(1 − q )(1 − q ) −− n X a =2 − n ′ X b =1 Z ... ≺ w b +1 ≺ z ≺ ... ≺ z a − ≺ s ≺ z a +1 ≺ ... ≺ z n ≺ w b − ≺ ... ∞− ∆ ∗ " Res z a − = s,z a = sq w b = s ( R | ∆ )(1 − q )(1 − q ) ++ n − X a =1 − n ′ X b =1 Z ... ≺ w b +1 ≺ z ≺ ... ≺ z a − ≺ s ≺ z a +2 ≺ ... ≺ z n ≺ w b − ≺ ... ∞− ∆ ∗ " Res z a = s,z a +1 = sq w b = s ( R | ∆ )(1 − q )(1 − q ) −− n X a =1 − n ′ X b =2 Z ... ≺ w b +1 ≺ z ≺ ... ≺ z a − ≺ z a +1 ≺ ... ≺ z n ≺ s ≺ w b − ≺ ... ∞− ∆ ∗ (cid:20) Res z a = sw b = s,w b − = sq ( R | ∆ )(1 − q )(1 − q ) (cid:21) plus nice summands. Note that the second and third rows cancel out, becausethey give the same residues with opposite signs (upon relabeling a a − b b − s sq ), but the contours of integration have s to the left/rightof chain of variables in the case of the first/fourth rows. Therefore, we concludethat the formula above is simply the base case (namely k = 1) of the following result: , Claim 5.14. For any k ≥ , the RHS of (5.67) equals the sum over all A, B of: Z ... ≺ w b +1 ≺ ( z i for i/ ∈ A ) ≺ s ≺ w b − k − ≺ ... ∞− ∆ ∗ Res z ai + ε = sq − ε − P j ≤ i cj − aj w b = s,...,w b − k = sq k ( R | ∆ )(1 − q )(1 − q ) − (5.71) − Z ... ≺ w b +1 ≺ s ≺ ( z i for i/ ∈ A ) ≺ w b − k − ≺ ... ∞− ∆ ∗ Res z ai + ε = sq − ε − P j ≤ i cj − aj w b = s,...,w b − k = sq k ( R | ∆ )(1 − q )(1 − q ) plus nice summands, as A ranges over the following k -element subsets of { , ..., n } : A = { a , ..., c − | {z } consecutive , ..., a t , ..., c t − | {z } consecutive } for a < c ≤ a < c ≤ ... ≤ a t < c t and B = { b − k, ..., b } ranges over ( k + 1) -element subsets of { , ..., − n ′ } consistingof consecutive elements. The contours of the variables z i in the two integralsare ordered in increasing order of i . The residues that appear in (5.71) havedenominators − q a , for various a ∈ N , but these are acceptable due to (5.61) . To make the subsequent explanation more vivid, we will think of A as a disjointunion of strings , each string consisting of consecutive numbers a i , ..., c i − 1. In(5.71), the variables z j for a i ≤ j < c i are specialized to sq ∗ where ∗ ∈ { , ..., k − } increases from one string to the next, but decreases with step 1 within a givenstring. First of all, let us note that Claim completes the proof of the Conjecturesubject to Assumption B, because as soon as k > min( n, − n ′ ), the sum of integralsabove is vacuous, and all we are left with are nice summands.We now prove Claim 5.14. For fixed sets A and B , the integrand of the two integralsof (5.71) is one and the same rational function in s, { z i } i/ ∈ A , { w j } j / ∈ B , namely:Res z ai + ε = sq − ε − P j ≤ i cj − aj w b = s,...,w b − k = sq k ( R | ∆ ) = Q ≤ i ≤ ta i ≤ e Consider the following diagram: (6.1) Z • × Z ′ Z • ε (cid:15) (cid:15) Z (cid:31) (cid:127) δ / / Z × M× S Z with maps given in terms of closed points by: (6.2) ( F ⊂ x F ⊂ x e F ) × ( F ⊂ x F ⊂ x e F ) ε (cid:15) (cid:15) {F = F } (cid:31) (cid:127) δ / / ( F ⊂ x F ) × ( F ⊂ x F ) Note that Z = Z ′ are denoted differently in (6.1) to emphasize the fact that theformer has F as the bigger sheaf, while the latter has F as the smaller sheaf: Z = {F ′ ⊂ x F} and Z ′ = {F ⊂ x e F} Consider the line bundles L , L , L , e L on the spaces in (6.1) whose fibers over theclosed points (6.2) are given by the one-dimensional quotients: L = F x / F x , L = F x / F x , e L = e F x / F x and L = δ ∗ ( L ) = δ ∗ ( L ) Consider also the universal sheaf U that parametrizes sheaves denoted F in thediagram above. Then we have the following equality in the K –theory of Z × M× S Z : (6.3) δ ∗ [ L k ] − ε ∗ [ e L k − r ]( − r q k − r (det U ) = Z ∞− z k ∧ • (cid:0) U z (cid:1)(cid:16) − z L (cid:17) (cid:16) − z L (cid:17) for all k ∈ Z . Recall that the notation R ∞− was introduced in (5.36) . X (which for us will be a dg schemesupported over a Noetherian scheme) with structure sheaf O , and a locally freesheaf E endowed with a co-section s : E → O , one may define the Koszul complex: ∧ • ( E , s ) = h ... d s −→ ∧ E d s −→ E d s −→ O i where we define:(6.4) ∧ k E d s −→ ∧ k − E e ∧ ... ∧ e k k X i =1 ( − i − s ( e i ) · e ∧ ... ∧ b e i ∧ ... ∧ e k The Koszul complex is a dg algebra, which can be best seen by packaging it as:(6.5) ∧ • ( E , s ) = ∞ M a =0 ∧ a E [ − a ] , d s ! where the differential d s is the direct sum of the maps (6.4), and the grading ismarked by the number inside the square brackets (so an element v ∈ ∧ a E has , degree | v | = − a ). The multiplication in the dg algebra (6.5) is given by wedgeproduct, and it is commutative (in the dg sense) since: v ∧ w = ( − | v || w | w ∧ v More generally, if ι : E ′ → E is a map of locally free sheaves on the space X suchthat s ◦ ι = 0, then we consider the commutative dg algebra:(6.6) ∧ • (cid:16) E ′ ι −→ E , s (cid:17) = ∞ M a,b =0 S a E ′ ⊗ ∧ b E [ − a − b ] , d ι + d s where the differential is given by summing up the maps:(6.7) S a E ′ ⊗ ∧ b E d ι −→ S a − E ′ ⊗ ∧ b +1 E e ′ ...e ′ a ⊗ e ∧ ... ∧ e b a X i =1 e ′ ... b e ′ i ...e ′ a ⊗ ι ( e ′ i ) ∧ e ∧ ... ∧ e b with the maps d s : S a E ′ ⊗ ∧ b E → S a E ′ ⊗ ∧ b − E given by the same formula as (6.4). Proposition 6.4. Consider a map of locally free sheaves ι : E ′ → E and a cosection s : E → O on a space X , and let s ′ = s ◦ ι : E ′ → O . The natural map of dg algebras: O Z ′ := ∧ • ( E ′ , s ′ ) → ∧ • ( E , s ) =: O Z induced by ι gives rise to a map of dg schemes Z → Z ′ . Moreover, we have: (6.8) O Z q.i.s. ∼ = ∧ • Z ′ (cid:16) E ′ | Z ′ ι −→ E| Z ′ , s (cid:17) as dg modules over O Z ′ .Proof. Note that in order for (6.6) to be complex, one needs the composition of s and ι to vanish. In the case at hand, this does not hold over X by assumption, butit does hold over the dg scheme Z ′ on account of the latter being the derived zerolocus of the cosection s ′ = s ◦ ι . Explicitly, for all a ≥ 0, the map: S a E ′ γ −→ S a − E ′ e ′ ...e ′ a a X i =1 s ′ ( e ′ i ) · e ′ ... b e ′ i ...e ′ a is null-homotopic over the dg scheme Z ′ . To see this, we observe that “restrict-ing” to the dg scheme Z ′ is the same thing as tensoring with the Koszul complex ∧ • ( E ′ , s ′ ), and therefore the induced map: S a E ′ ⊗ ∧ • ( E ′ , s ′ ) γ ⊗ Id ∧• ( E′ ,s ′ ) −−−−−−−−−→ S a − E ′ ⊗ ∧ • ( E ′ , s ′ )has the property that γ ⊗ Id ∧ • ( E ′ ,s ′ ) = [ h, d s ′ ], where the homotopy is: S a E ′ ⊗ ∧ c E ′ h −→ S a − E ′ ⊗ ∧ c +1 E ′ e ′ ...e ′ a ⊗ e ∧ ... ∧ e c a X i =1 e ′ ... b e ′ i ...e ′ a ⊗ e ′ i ∧ e ∧ ... ∧ e c –ALGEBRAS ASSOCIATED TO SURFACES 59 Therefore, the right-hand side of (6.8) can be presented as a complex over X as:(6.9) ∞ M a,b,c =0 S a E ′ [ − a ] ⊗ ∧ b E [ − b ] ⊗ ∧ c E ′ [ − c ] , d ι + d s + d s ′ − ( − c h (the fact that this is a chain complex is a consequence of the equality ( d ι + d s ) = γ ⊗ Id ∧ • ( E ′ ,s ′ ) ⊗ Id ∧ • ( E ,s ) = [ h, d s ′ ] ⊗ Id ∧ • ( E ,s ) , which is straightforward). However,the complex above can be filtered according to n = a + c , with associated graded: ∞ M n =0 n M a =0 S a E ′ [ − a ] ⊗ ∧ n − a E ′ , h [ − n ] ⊗ ∧ • ( E , s )It is well-known that the underlined complex is acyclic unless n = 0, so therefore(6.9) is quasi-isomorphic to ∧ • ( E , s ) = O Z . Hence (6.8) holds in the derived cat-egory of coherent sheaves on X . However, it also holds in the derived categoryof coherent sheaves on Z ′ because all complexes involved (including (6.9)) are dgalgebras which receive a natural map from O Z ′ = ∧ • ( E ′ , s ′ ). (cid:3) × (with nofurther subscript) for derived fiber product over the scheme M × S , where M parametrizes the sheaves denoted by F in the statement of Lemma 6.2. Simi-larly, we will write P ( E ) for the projectivization of the locally free sheaf E on M× S . Proof. of Lemma 6.2: We will compute δ ∗ ( L k ) and ε ∗ ( e L k ) as complexes in: D − (Coh ( Z × Z ))and we will show that the complexes are identical, except for finitely many terms atthe right of the complex. These terms will be expressed in terms of U , L , L andtheir powers, and the resulting expression will be shown to match the right-handside of (6.3), thus concluding the proof of the Lemma. We recall that U = V / W where W ֒ → V are vector bundles on M × S , which leads to the presentation (4.9)of the map of dg schemes Z → M × S , ( F ′ ⊂ x F ) ( F , x ):(6.10) Z ֒ → P ( V )where the embedding is the derived zero locus of the co-section: W ⊗ O ( − → V ⊗ O ( − → O (we write O (1) for the tautological line bundle on the projectivization in (6.10), andabuse notation by writing V , W for the aforementioned vector bundles on M× S , aswell as their pull-backs to the projectvization). With this in mind, let us considerthe following diagram of derived fiber products:(6.11) Z (cid:31) (cid:127) δ / / p $ $ ■■■■■■■■■■ Z × Z q (cid:15) (cid:15) P ( V ) × Z 10 ANDREI NEGUT , By definition, the embedding q is cut out by the composition W ⊗ O ( − → V ⊗ O ( − → O (where we use the notation O (1) and O (1) to denote thetautological line bundles on the first and second factors in the fiber products (6.11),respectively). Meanwhile, because p is obtained from the diagonal embedding: P ( V ) ֒ → P ( V ) × P ( V )by derived base change, then Beilinson’s resolution implies that the embedding p is cut out by the composition σ : N ⊗ O ( − → V ⊗ O ( − → O , where: N = Ker V ։ O (1) N = Ker V ։ O (1)We may then invoke Proposition 6.4 to conclude that the embedding δ is cut outby the induced map of complexes [ W ⊗ O ( − → N ⊗ O ( − → O , i.e.: O δ q.i.s. ∼ = ∧ • ( W ⊗ L − → N ⊗ L − , σ )on Z × Z (this is because the line bundle L on Z is the restriction of O (1) on P ( V ), see Definition 4.11). Plugging in (6.6) for the Koszul complex, we obtain: δ ∗ ( O ) ∼ = ∞ M a,b =0 S a W ⊗ ∧ b N ⊗ L − a − b [ − a − b ] , d ι + d σ where d σ and d ι are the differentials (6.4) and (6.7) associated to the maps σ : N ⊗ L − → O and ι : W → N = Ker ( V ։ O (1)). Since L = δ ∗ ( L ) = δ ∗ ( L ):(6.12) δ ∗ ( L k ) = M a,b ≥ S a W ⊗ ∧ b N ⊗ L k − − a − b L [ − a − b ] , d ι + d σ Formula (6.12) gives the first summand in the left-hand side of (6.3), and now wemust obtain a similar expression for the second summand. We have the diagram:(6.13) Z • × Z ′ Z • (cid:31) (cid:127) η / / ε ) ) (cid:22) v ν ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ P Z × Z ( W ∨ ⊗ ω S ) π / / / / (cid:127) _ ρ (cid:15) (cid:15) Z × Z P ( V ) × P ( V ) × P ( W ∨ ⊗ ω S )By Definition 4.16, the closed embedding Z • ֒ → P ( V ) × P ( W ∨ ⊗ ω S ) is cut out by:Coker O ( − ⊗ O ( − ⊗ ω S → W ⊗ O ( − ⊕V ∨ ⊗ O ( − ⊗ ω S → O where the arrows are the tautological maps on P ( V ) and P ( W ∨ ⊗ ω S ). Since: Z ′ ֒ → P ( W ∨ ⊗ ω S ) –ALGEBRAS ASSOCIATED TO SURFACES 61 is cut out by the natural map V ∨ ⊗ O ( − ⊗ ω S → O on P ( W ∨ ⊗ ω S ) in virtue of(4.12), we conclude that the embedding ν in diagram (6.13) is cut out by:(6.14) Coker O ( − ⊗ O ( − ⊗ ω S ⊕O ( − ⊗ O ( − ⊗ ω S → W ⊗ O ( − ⊕W ⊗ O ( − ⊕V ∨ ⊗ O ( − ⊗ ω S σ ′ −→ O on P ( V ) × P ( V ) × P ( W ∨ ⊗ ω S ), where O (1) , O (1) , O (1) denote the tautological linebundles on the three projectivizations, and all arrows in (6.14) are the tautologicalmaps. Similarly, according to Proposition 4.13, the closed embedding ρ in diagram(6.13) is cut out by the co-section: W ⊗ O ( − ⊕W ⊗ O ( − −→ O Combining this fact with (6.14) allows us to apply Proposition 6.4 to the setting of(6.13). We thus conclude that the closed embedding η has the property that: O η q.i.s. ∼ = ∧ • O ( − ⊗ O ( − ⊗ ω S ⊕O ( − ⊗ O ( − ⊗ ω S → V ∨ ⊗ O ( − ⊗ ω S , σ ′ where the Koszul complex is computed on P Z × Z ( W ∨ ⊗ ω S ). Since the cokernelof O ( − → V ∨ is the vector bundle N ∨ , we may simplify the complex above to: O η q.i.s. ∼ = ∧ • ( O ( − ⊗ O ( − ⊗ ω S → N ∨ ⊗ O ( − ⊗ ω S , σ ′ ) == ∞ M a,b =0 ∧ b N ∨ ⊗ O ( − a ) ⊗ O ( − a − b ) ⊗ ω a + bS [ − a − b ] , d σ ∨ + d σ ′ where we recall the section σ : N → O (1). Recall that O i (1) | Z = L i , hence:(6.15) ε ∗ ( e L k ) = η ∗ ◦ π ∗ ( O ( − k )) = π ∗ ( O ( − k ) ⊗ O η ) == π ∗ ∞ M a,b =0 ∧ b N ∨ ⊗ O ( − a − b − k ) ⊗ L − a ⊗ ω a + bS [ − a − b ] , d σ ∨ + d σ ′ The push-forward π ∗ does not affect N , L or ω S , which are pulled back from thebase Z × Z , but it has the following well-known effect on powers of O ( − π ∗ ( O ( l )) = S l W ∨ ⊗ ω lS if l ≥ S − l − w W ⊗ det W ⊗ ω lS [ w − 1] if l ≤ − w w = rank W . For simplicity, let us assume we are dealing with k > ε ∗ ( e L k ) = ∞ M a,b =0 ∧ b N ∨ ⊗ S a + b + k − w W ⊗ L − a ⊗ ω − kS ⊗ det W [ − a − b + w − , (we will not write down the differential explicitly anymore, but note that it matchesthe one in the complex (6.12), because applying π ∗ to the tautological map σ ′ : N ∨ ⊗ ω S → O (1) yields precisely the map ι ). We may use the natural isomorphism ∧ b N ∨ ∼ = (det N ) ∨ ⊗ ∧ v − − b N , where v = rank V = w + r , to obtain the formula: ε ∗ ( e L k ) = ∞ M a,b =0 ∧ v − − b N ⊗ S a + b + k − w W ⊗ L − a ω − kS det W det N [ − a − b + w − Let us relabel indices a ′ = a + b + k − w and b ′ = v − − b , and thus: ε ∗ ( e L k ) = a ′ + b ′ ≥ k + r − M ≤ b ′ ≤ v − S a ′ W ⊗ ∧ b ′ N ⊗ L k + r − − a ′ − b ′ L ω − kS det W det V [ − a ′ − b ′ + 2 k + r − Once again relabeling k k − r , a ′ a , b ′ b gives us: ε ∗ ( e L k − r ) = a + b ≥ k − M ≤ b ≤ v − S a W ⊗ ∧ b N ⊗ L k − − a − b L ω − k + rS det U [ − a − b + 2 k − r − ⇒⇒ ε ∗ ( e L k − r ) ⊗ (det U ) ⊗ ω k − rS [ − k + r + 2] == a + b ≥ k − M ≤ b ≤ v − S a W ⊗ ∧ b N ⊗ L k − − a − b L [ − a − b ] Comparing the right-hand side of the relation above with (6.12) implies that thereis a map in D − (Coh( Z × Z )): δ ∗ ( L k ) −→ ε ∗ ( e L k − r ) ⊗ (det U ) ⊗ ω k − rS [ − k + r + 2]whose cone is the finite complex: ≤ a + b ≤ k − M ≤ b ≤ v − S a W ⊗ ∧ b N ⊗ L k − − a − b L [ − a − b ] , d ι + d σ The class of this complex in K –theory is precisely the right-hand side of (6.3), asa consequence of [ N ] = [ V ] − [ L ], [ U ] = [ V ] − [ W ] and the additivity of ∧ • . (cid:3) A y K M i.e. an assignment A → Hom( K M , K M× S )which sends: E d • ∈ A ← ⊂ A to the operator e d • of (5.17) F d • ∈ A → ⊂ A to the operator f d • of (5.18)for all d • = ( d , ..., d k ). Moreover, we recall that K M is a good module of A ,since the grading on K M by second Chern class is bounded below in virtue of theBogomolov inequality. This implies that f ( d ,...,d k ) annihilates any given element of K M if k is large enough, hence the conjectural action (6.17) induces an action of: b A ↑ y K M and hence an action A ∞ y K M –ALGEBRAS ASSOCIATED TO SURFACES 63 (by the definition of A ∞ in (3.37), there exists a homomorphism A ∞ → b A ↑ ). Theextended algebra A ext ∞ of (3.42) also acts on K M by the same reasons.6.7. It would be rather difficult to prove that relations (3.45) and (3.46) hold in K based on the definition (3.27) of the generating currents of the deformed W –algebra,so we appeal to a different formula that was worked out in (3.20) of [24]:(6.18) W d,k = d − d = d X k + k + k = k T ← d ,k E ,k T → d ,k · q ( k − d where k , k go over the natural numbers, and k , d , d go over the non-negativeintegers. The elements E ,k in (6.18) are simply the standard generators of A diag ⊂A (see Proposition 3.3), while: T ← n,k ∈ A ← ∼ = S and T → n,k ∈ A → ∼ = S op correspond to the following elements of the shuffle algebra:(6.19) T n,k ( z , ..., z n ) = Sym ( − k − z kn Q n − i =1 (cid:16) − z i +1 qz i (cid:17) Y ≤ i 14 ANDREI NEGUT , where f M is the moduli space that parametrizes the sheaves denoted by e F in (6.23).Note that we may rewrite (6.22) as a residue, namely:(6.24) W d,k = ( − k X d − d = d q ( k − r ) d ( π × π S ) ∗ Res z = ∞ z k dz ∧ • (cid:16) e U z (cid:17) ( − rd (det e U ) d Q r z (cid:16) − z L (cid:17) (cid:16) − z L (cid:17) · π ∗ Theorem 1.2 reduces to the following, which we prove regardless of Conjecture 1.1. Theorem 6.9. The RHS of (6.24) vanishes if k > r , while for k = r it equals: (6.25) X d − d = d ( π × π S ) ∗ " ( − rd (det e U ) d Q r · det e U · π ∗ We observe that the operator (6.25) precisely matches the action on K M of: u d ,d ≥ X d − d = d h − d h d ∈ A where u = det e U , and h ± k are to the elements p ± k of (3.38) as complete symmetricpolynomials are to power-sum functions. Indeed, as shown in [24], the elements: h − k = H − k, ∈ A ← and h k = q ( k − r H k, ∈ A → correspond to setting d = ... = d k = 0 in (5.17) and (5.18), respectively (the factthat this statement also holds in the module K M requires assuming Conjecture 1.1). Proof. of Theorem 6.9: We claim that both cases k > r and k = r boil down to:(6.26)0 = X d − d = d q ( k − r ) d ( π × π S ) ∗ ( − rd (det e U ) d Q r Z ∞− z k ∧ • (cid:16) e U z (cid:17)(cid:16) − z L (cid:17) (cid:16) − z L (cid:17) · π ∗ for all k ≥ r . This is a consequence of the fact that the integrand in the right-handside of (6.26) is z k − r times a rational function which is regular at 0. The value ofthis rational function at z = 0 is equal to the right-hand side of (6.25).Therefore, the task has become to prove (6.26). Let us consider the space W d ,d which parametrizes flags:(6.27) e FF ′ ⊂ x ... ⊂ x F ⊂ x F ⊃ x F ⊃ x ... ⊃ x F ′ ?(cid:31) x O O where the total number of inclusions in the horizontal row is d and d , respectively.As a dg scheme, W d ,d is defined as the derived fiber product: –ALGEBRAS ASSOCIATED TO SURFACES 65 W d ,d (cid:15) (cid:15) a ( d ,d / / V d ,d (cid:15) (cid:15) Z × Z ′ Z ε / / Z × M× S Z where the vertical maps remember {F , F ⊂ x F ⊂ x e F} and {F , F ⊂ x F} ,respectively, and the horizontal maps forget e F . We also have a derived fiber square: W d − ,d − (cid:15) (cid:15) (cid:31) (cid:127) b ( d ,d / / V d ,d (cid:15) (cid:15) Z (cid:31) (cid:127) δ / / Z × M× S Z where the vertical projection maps remember {F ⊂ x e F} and {F ⊂ x e F ⊃ x F ′ } ,respectively. Because both squares are Cartesian, we may invoke Lemma 6.2: b ( d ,d ) ∗ [ L k ] − a ( d ,d ) ∗ [ e L k − r ]( − r q k − r det U = Z ∞− z k ∧ • (cid:0) U z (cid:1)(cid:16) − z L (cid:17) (cid:16) − z L (cid:17) where e L parametrizes the one-dimensional vector space e F x / F x . As we sum theformula above over all non-negative integers d , d such that d − d = d , we obtain:(6.28) X d − d = d ( π × π S ) ∗ " ( − rd q ( k − r ) d (det e U ) d Q r b ( d ,d ) ∗ [ L k ] · π ∗ −− X d − d = d ( π × π S ) ∗ " ( − r ( d +1) q ( k − r )( d +1) (det e U ) d +1 Q r a ( d ,d ) ∗ [ e L k ] · π ∗ == X d − d = d ( π × π S ) ∗ q ( k − r ) d ( − rd (det e U ) d Q r Z ∞− z k ∧ • (cid:16) e U z (cid:17)(cid:16) − z L (cid:17) (cid:16) − z L (cid:17) · π ∗ The equality above uses the identity Q d − e L = Q d , where we write Q d for theline bundle parametrizing det F x / F ′ ,x in the diagram (6.27) with d symbols ⊃ x .Moreover, in (6.28) we used the equality of line bundles det U = det e U , which holdsbecause the determinants of all possible universal bundles on the correspondences Z and Z • are canonically isomorphic. 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