Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
CCohomological Hall algebras and perverse coherentsheaves on toric Calabi-Yau 3-folds
Miroslav Rapˇc´ak, Yan Soibelman, Yaping Yang, Gufang ZhaoAugust 17, 2020
Abstract
To a smooth local toric Calabi-Yau 3-fold X we associate the Drinfeld double ofthe (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich andSoibelman. This Drinfeld double is a generalization of the notion of the Cartan doubledYangian defined earlier by Finkelberg and others. We extend this “3 d Calabi-Yauperspective” on the Lie theory furthermore by associating a root system to certainfamilies of X .By general reasons, the COHA acts on the cohomology of the moduli spaces ofcertain perverse coherent systems on X via “raising operators”. We conjecture thatthe Drinfeld double acts on the same cohomology via not only by raising operatorsbut also by“lowering operators”. We also conjecture that this action factors throughthe shifted Yangian of the above-mentioned root system. We add toric divisors to thestory and explain the shifts in the shifted Yangian in terms of the intersection numberswith the divisors. We check the conjectures in several examples. Contents C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 COHA of the resolved conifold . . . . . . . . . . . . . . . . . . . . . . . . . . 221 a r X i v : . [ m a t h . QA ] A ug Shifted Yangians of (cid:100) gl (1) and Costello conjectures 25 (cid:100) gl (1) . . . . . . . . . . . . . . . . . . . . 254.2 Relation to the Costello conjectures . . . . . . . . . . . . . . . . . . . . . . . 26 and shifted Yangians 28 C . . . . . . . . . . . . . . . . . . . . . . . 285.2 Action of the shifted Yangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Checking the relations of the shifted Yangian . . . . . . . . . . . . . . . . . . 32 -folds 50 A Residue pushforward formula in critical cohomology 56
A.1 Convolution operators on fixed points . . . . . . . . . . . . . . . . . . . . . . 56A.2 Jeffery-Kirwan residue formula in critical cohomology . . . . . . . . . . . . . 57
The paper is devoted to Cohomological Hall algebras (COHA) of toric Calabi-Yau 3-foldsand their representations.More precisely, let X be a smooth local toric Calabi-Yau 3-fold which is a resolution ofsingularities f : X → Y , where Y is affine. Assume that the fibers have dimension at most 1and each component of the exceptional fiber is P . Let D b Coh( X ) be the bounded derivedcategory of coherent sheaves on X . The map f : X → Y , by general theory of Bridgelandand Van den Bergh [3, 63], provides D b Coh( X ) with a t -structure whose heart is the modulecategory Mod- A . When A is the Jacobian algebra of a quiver with potential ( Q, W ), thegeneral construction of Kontsevich and Soibelman [35] yields a COHA of the pair (
Q, W ).We denote it by H X and consider to be the COHA associated to this t -structure. By generalreasons H X acts on the cohomology of the moduli spaces of certain sheaves (e.g. torsionfree sheaves, or sheaves supported on a toric divisor) on X via “raising operators”. Thegoal of the paper is to construct various “doubles” of H X as well as their representationson the cohomology of these moduli spaces of sheaves by adding the “lowering operators”.We expect that there exists the notion of “generalized root system” associated to theabove data, which characterizes the Drinfeld double of the (equivariant spherical) COHAalgebraically as the Cartan doubled Yangian of this root system. The root system is asubset in the topological K-theory of the category Mod- A (lattice Γ in the notation of236]) endowed with a bilinear form. There is a notion of simple root (in the case of quiverswith potential they correspond to the vertices of the quiver), the notion of real and imaginaryroots. We will discuss generalized root systems in more detail in the future publications,but we will use the above terminology without quotations in this paper. Geometrically,the Drinfeld double is expected to act on the cohomology of the moduli space of perversecoherent system of X in the sense of [42, 43]. These actions factor through those of the so-called shifted Yangians. We also give a prediction for the shift (see § Let us summarize some properties of the Drinfeld double in question. Assuming that H X hasa shuffle description we construct the Drinfeld double D ( H ) (see [56, 40] for the terminology)of H X , as well as the Drinfeld double D ( SH ) of the equivariant spherical subalgebra SH := SH X of H X in §
2. The Drinfeld double D ( SH ) generalizes the notion of the Cartan doubledYangian in [17]. We prove the following properties of D ( SH ):1. There is a triangular decomposition of D ( SH ) ∼ = SH − ⊗SH ⊗SH + , where SH − , SH + are isomorphic to SH , and SH = C [ ψ ( s ) i | i ∈ I, s ∈ Z ] is the algebra of polynomialsof infinitely many variables.2. The generators of SH + are given by { e ( r ) i | i ∈ I, s ∈ N ]. The relations among thegenerators can be computed by the shuffle description of SH .3. Similarly, the generators of SH − are given by { f ( r ) i | i ∈ I, s ∈ N ]. The relationsamong the generators can be computed by the shuffle description of SH .4. The action of SH on SH + is determined by formula (14). This gives the relationsamong { ψ ( s ) i } and { e ( r ) j } . In a similar way we obtain the relations between { ψ ( s ) i } and { f ( r ) j } .5. The commutation relations between { e ( r ) i } and { f ( r ) j } are given in Proposition 2.3.1,which are determined by the definition of Drinfeld double 2.2 (by taking x = b = 1and a = e ( r ) i , y = f ( r ) j in 2.2). For each “simple root” of the above-mentioned root system there is a corresponding subal-gebra of D ( SH X ). This subalgebra is either isomorphic to the Yangian of sl or the infiniteClifford algebra (see [64]). Moreover, our root system always has an “imaginary root”, andis in a sense of affine-type.It follows from § X , or more precisely, perverse coherent systems in the sense of[42, 43], the action of the Drinfeld double D ( SH X ) factors through the shifted Yangian ofthe generalized root system where it shifts the imaginary root.In this paper we refer to the perverse coherent system in [42, 43] as to the rank-1 perversecoherent system . They are pairs consisting of a perverse coherent sheaf supported on theexceptional fiber of X → Y and a homomorphism to it from the structure sheaf of X . On3he other hand, the action of COHA on the cohomology of the moduli spaces associatedwith sheaves supported on a toric divisor in X or on the cohomology of the moduli spacesof perverse coherent systems supported on a toric divisor is expected to lead to shifts of realroots (see [53]). The shifts match with the prediction from physics that shifts of imaginaryroots occur when considering the D6-brane wrapping X (sheaves supported on the whole X )[29, 49] and shifts of real roots occur when considering configurations of D4-branes wrappingvarious four-cycles inside X (sheaves supported on a toric divisor in X ) [1, 22, 47, 48].The remaining part of the Introduction is devoted to examples in which these expec-tations have been verified. Let Y = Y m,n be an algebraic subvariety of C defined by theequation xy = z m w n , where ( x, y, z, w ) are the standard coordinates on C . It is knownthat there is a toric Calabi-Yau resolution of singularities X = X m,n → Y m,n given bya proper morphism. By [42, §
1] there is a tilting vector bundle P on X m,n such that A = End X ( P ) is isomorphic to the Jacobian algebra of the explicitly written quiver withpotential ( Q, W ) = ( Q m,n , W m,n ). The generating function of (non-equivariant) cohomol-ogy of the moduli space of rank-1 perverse coherent systems on X m,n was studied in [42](see also [43] for X = X , ).We consider special cases with Y = Y , , Y = Y , or Y = Y , in this paper. The generalcase f : X → Y as in the second paragraph of § Y = Y , = C then, by definition, the resolution X is C . In the two other case X is the total space of a rank-2 vector bundle on P . Note that the condition that X is aresolution of singularities of its affinization implies that X is one of the following types:a) the resolved conifold X , = Tot( O P ( − ⊕ O P ( − → Y , ;b) resolution of singularities of the orbifold Y , , X , = Tot( O P ( − ⊕ O P ) → Y , = C / ( Z / × C .The quiver with potential for X = C is B B B X = C W = B [ B , B ] (1)The quivers with potential for the other two cases are: • • a a b b X = Tot( O ( − ⊕ O ( − W = a b a b − a b a b • B • ˜ B B B ˜ B ˜ B X = Tot( O ( − ⊕ O ) W = B ˜ B ˜ B − ˜ B ˜ B B + ˜ B B B − B B ˜ B (2)4 .3 Imaginary roots and shifted Yangians The moduli space of rank-1 perverse coherent systems depends on a choice of stabilitycondition. We refer the readers to [42] for the definition of stability condition, which webriefly recall in § ζ is said to be generic if it belongs to the complement of the union of hyperplanes,i.e. to the interior of a chamber. The moduli space M ζ of ζ -stable perverse coherent systemsdepends only on the chamber that contains ζ but not on the particular ζ . For X = X m,n the hyperplane arrangement coincides with the one given by root hyperplanes of the affineLie algebra of type A m + n − . The chamber structure of the space of stability conditions wasdescribed in [42]. Depending on the chamber the moduli space of stable perverse coherentsystems can be identified with the moduli spaces which appear in Donaldson-Thomas (DT),Pandharipande-Thomas (PT) or non-commutative Donaldson-Thomas (NCDT) theories.We will sometimes refer to them as DT, PT or NCDT moduli spaces depending on thechamber under consideration.If X = X , or X = X , , a stability condition is given by a pair of real numbers ζ = ( ζ , ζ ). Walls of the chambers are labeled by non-negative integers, see [43, Figure 1]: mζ + ( m + 1) ζ = 0 , m ∈ Z ≥ , the DT side, (3) ζ + ζ = 0 , the imaginary root hyperplane, (4)( m + 1) ζ + mζ = 0 , m ∈ Z ≥ , the PT side . (5)The DT moduli space corresponds to the chamber which is positioned immediately below theimaginary root hyperplane and above mζ + ( m + 1) ζ = 0, for m (cid:29)
0. For the PT modulispace we consider the chamber which is immediately above the imaginary root hyperplaneand below ( m + 1) ζ + mζ = 0, for m (cid:29) M ζ in terms of a quiver with potential. For X = C this is straight-forward, see § X = X , is discussed in § Q, W ) is the quiver with potential such that A = End X ( P )is isomorphic to the Jacobian algebra of ( Q, W ) [42, § Q, W ) one constructs a framed quiver ˜ Q with the framed potential ˜ W . The framed quiver is obtained from Q byadding an additional framing vertex as well as an arrow from the framing vertex to one ofthe vertices of Q . We define ˜ W = W . The case of X , is depicted below. • V V • a a b b (cid:3) ι Q for X , = Tot( O ( − ⊕ O ( − ζ defines the corresponding stability condition ˜ ζ for the abeliancategory of representations of ( ˜ Q, ˜ W ) (i.e. the abelian category of representations of thecorresponding Jacobi algebra). It is given in terms of the slope function θ ˜ ζ (see § M ζ is isomorphic to moduli space of θ ˜ ζ -stable representations of ( ˜ Q, ˜ W ) with1-dimensional framing.Notice that X admits an action of the torus T (cid:39) ( C ∗ ) which preserves the canonicalclass. Denote by P be the tilting vector bundle on X . The tilting object P is equivariantwith respect to this T -action, and hence induces a T -action on M ζ .The following result should be true in general, but we formulate it in the case of X , . Proposition 1.3.1
Let ζ be a generic stability condition. Then there is an action of H X on H ∗ c,T ( M ζ , ϕ tr W Q ) ∨ . The H X -action in Proposition 1.3.1 is given by the “raising operators”. We expect the actionof SH X in Proposition 1.3.1 extends to its Drinfeld double D ( SH X ) by adding the actionof “lowering operators”. Furthermore, the action of the Drinfeld double factors through theshifted Yangian where it shifts the imaginary root.Let (cid:126)z = ( z , z , · · · , z | l | )( l ∈ Z ) be complex parameters. Definition of the shifted Yangian Y l ( (cid:126)z ) of (cid:91) gl (1) as an associative algebra depending on the parameters (cid:126)z is recalled in § l -positively ( − l -negatively) shifted in parameters (cid:126)z if l > l < l = 0 and there are no parameters, and we obtain the usual affineYangian Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)). Here the parameters (cid:126) i , ≤ i ≤ C ∗ ) , hence (cid:126) + (cid:126) + (cid:126) = 0. We hope the reader will notconfuse the subscripts in the cases of shifted and usual Yangians.The relation between shifted Yangians and COHAs can be summarized in our case inthe following way. Theorem 1.3.2
1. Let X = C . The action of H X on H ∗ c,T ( (cid:116) n ∈ N Hilb n ( C ) , ϕ tr W Q ) ∨ extends to Y − ( z ) .2. Let X = X , be the resolved conifold. Then for a generic stability condition on the PT-side of the imaginary root hyperplane, the action of Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)) + on H ∗ c,T ( M ζ , ϕ tr W Q ) ∨ extends to the one of Y ( z ) . Remark 1.3.3 a) The algebra Y ( z ) can also be realized as the deformed double currentalgebra for gl (1) . The fact that this algebra acts on the PT-moduli space of the resolvedconifold was originally conjectured by Costello from holography considerations in M-theory[6, §
14 Conjecture].b) We expect the same calculation goes through for an arbitrary X m,n . We expect that theaction of H X m,n extends to the action of the shifted versions of affine Yangian of gl ( m | n ) [53, 54]. The above discussion suggests that the choice of stability condition leads to theshift of imaginary roots in the case when we consider the action on the cohomology of themoduli spaces of torsion free sheaves on X m,n . We are going to illustrate the idea in a simple example. Consider the Calabi Yau 3-fold X , = T ∗ P × C and the effective divisor D to be fiber of T ∗ P × C → P over the See also closely related discussion in [14, 21, 23, 35, 37, 38] and references therein. P . Depending on the choice of the toric divisor thecorresponding geometry can be depicted as in the following toric diagram1 1 (7)In Section 7.1 we obtain a quiver Q with potential W associated with the pair consisting ofthe Calabi Yau 3-fold T ∗ P × C and one of the above toric divisors D . The constructionuses a Z / Z symmetry of the quiver with potential associated with the pair consisting ofthe Calabi Yau 3-fold C and the effective divisor which is the coordinate plane.By the dimension reduction, ( Q, W ) can be described as the following “chainsaw quiver” • V B • V ˜ B B ˜ B (cid:52) I J (8)with relations ˜ B ˜ B − B ˜ B = I J , ˜ B B = B B Analogously to the K-theory case in [46], the (1 , (cid:91) gl (2) (see Definitionin § (cid:77) V ,V H ∗ c, GL( V ) × GL( V ) × T (Rep( Q, V , V ) st , ϕ trW ) ∨ . Here the superscript st refers to the locus of stable representations.In physics language the above example describes a stack of D D P corresponding to the real simple root and the divisor, countedwith multiplicity given by the number of D Contents of the paper In § Z .We refer this subalgebra as the Cartan doubled subalgebra. We define the shifted Yangianas the quotient of the Drinfeld double by certain relations among the generators of theCartan doubled subalgebra, which are determined by a coweight. For a specific choiceof the quiver with potential, we show in § C fromour previous paper. Furthermore, we prove that the Drinfeld double corresponding to theresolved conifold has a quotient algebra which is isomorphic to the affine Yangian of (cid:98) gl (1). In § (cid:98) gl (1). We also discuss the relationship of COHAto quantized Coulomb branch algebras of the Jordan quiver gauge theory. We connect thisdiscussion with a special case of the conjecture of Costello.In § C . In § § Z -orbifold.In §
8, we make a proposal in the general case. Namely, we expect the action of theDrinfeld double of the spherical COHA on more general moduli spaces of perverse coherentsheaves on the general toric Calabi-Yau 3-fold X m,n . The action of the Drinfeld doubleis expected to factor through the shifted Yangian, with the shift determined by certainintersection numbers. We verify this proposal in the examples above. Acknowledgments
We thank Kevin Costello, Ben Davison, Hiraku Nakajima, Nikita Nekrasov, Masahito Ya-mazaki for very helpful discussions.Part of the work was done when the last two named authors were visiting at the IPMU.M.R. was supported by NSF grant 1521446 and 1820912, the Berkeley Center for TheoreticalPhysics and the Simons Foundation. Y.S. was partially supported by the Munson-Simu StarFaculty award of KSU. Part of the work was done when he was supported by the PerimeterInstitute for Theoretical physics as a Distinguished Visiting Research Chair. Y.Y. waspartially supported by the Australian Research Council (ARC) via the award DE190101231.G.Z. was partially supported by ARC via the award DE190101222.The work was started when all authors were visiting the Perimeter Institute for Theoret-ical Physics (PI). They are grateful to PI for excellent research conditions. The PerimeterInstitute for Theoretical Physics is supported by the Government of Canada through the De-partment of Innovation, Science and Economic Development and by the Province of Ontariothrough the Ministry of Research, & Innovation and Science.8
The Drinfeld double of COHA associated to a quiverwith potential
Let R be a commutative integral domain of characteristic zero. Let H be a Hopf algebraover R . By definition, H posses the following R -linear operations1. multiplication µ : H ⊗ H → H ,2. unit η : R → H ,3. comultiplication ∆ : H → H ⊗ H ,4. co-unit (cid:15) : H → R
5. antipode S : H → H which satisfy the following conditions(a) ( H, µ, η ) is an associative R -algebra.(b) ( H, ∆ , (cid:15) ) is an associative R -coalgebra.(c) ∆ , (cid:15) are algebra homomorphisms.(d) µ ( S ⊗ µ (1 ⊗ S )∆ = η(cid:15) .We denote by H the cohomological Hall algebra (COHA for short) associated with aquiver with potential ( Q, W ) defined by Kontsevich and Soibelman [35]. We briefly recallthe definition of H . We write Q = ( I, H ) with I being the set of vertices and H theset of arrows. For a dimension vector v = ( v i ) i ∈ I ∈ N I = Z I ≥ , let Rep( Q, v ) be thealgebraic variety (in fact with non-canonical structure of a vector space) parameterizingrepresentations of Q on the I -graded complex vector space whose degree i ∈ I -piece is C v i .On Rep( Q, v ) there is an action of GL v × T where GL v = (cid:81) i ∈ I GL v i acts by changing basisand the action of the torus T = ( C ∗ ) preserves W (below the T -action is induced fromits Calabi-Yau action on X and hence on End Coh ( X ) ( P )). The trace of W gives rise to theregular function tr W on Rep( Q, v ) which is invariant under the (GL v × T )-action. In thisnotation the equivariant COHA of ( Q, W ), is a N I -graded vector space H = H ( Q,W ) := ⊕ v ∈ N I H v with H v = H ∗ c, GL v × T (Rep( Q, v ) , ϕ tr W ) ∨ , endowed with the multiplication defined in [35, § (cid:116) v ∈ N I Rep(
Q, v ) by Rep( Q ). Let R := C [Lie( T )] be the ring offunctions on Lie( T ). Then H is an R -algebra.We now construct the Drinfeld double of H . To do this, we assume H has a shuffledescription. Explicitly, for v ∈ N I , let g v := (cid:81) i ∈ I gl v i be the Lie algebra, and h v ⊂ g v beits Cartan subalgebra. The Weyl group S v naturally acts on the ring of functions C [ h v ].For each v ∈ N I , C [ h v ] is a polynomial ring with variables { x ( i )1 , x ( i )2 , · · · , x ( i ) v i | i ∈ I } . Forsimplicity, we will denote this set of variables by x [1 ,v ] .9et H be the shuffle algebra defined as follows. As a vector space, we have H = ⊕ v ∈ N I R [ h v ] S v . Consider the embeddings R [ h v ] S v ⊂ R [ h v + v ] S v × S v , x [1 ,v ] (cid:55)→ x [1 ,v ] R [ h v ] S v ⊂ R [ h v + v ] S v × S v , x [1 ,v ] (cid:55)→ x [ v +1 ,v + v ] . For any pair ( p, q ) of positive integers, let Sh( p, q ) be the subset of S p + q consisting of ( p, q )-shuffles (permutations of { , · · · , p + q } that preserve the relative order of { , · · · , p } and { p + 1 , · · · , p + q } ). Set Sh( v , v ) := (cid:81) i Sh( v ( i )1 , v ( i )2 ). The multiplication of H given as (cid:63) : R [ h v ] S v ⊗ R [ h v ] S v → R [ h v + v ] S v v , f (cid:63) g = (cid:88) σ ∈ Sh( v ,v ) σ ( f · g · fac( x [1 ,v ] | x [ v +1 ,v + v ] )) , where fac( x [1 ,v ] | x [ v +1 ,v + v ] ) ∈ ( R [ h v ] S v ⊗ R [ h v ] S v ) loc is an explicit rational functionwith denominator (cid:81) i ∈ I (cid:81) v ( i )1 a =1 (cid:81) v ( i )1 + v ( i )2 b = v ( i )1 +1 ( x ( i ) a − x ( i ) b ). Moreover, it has the property thatfac( x A (cid:116) A | x B ) = fac( x A | x B ) fac( x A | x B ) , fac( x A | x B (cid:116) B ) = fac( x A | x B ) fac( x A | x B ) , for any subsets x A , x A , x A , x B , x B , x B of the collection of variables x [1 ,v ] . In the casethat ( Q, W ) has a cut, the formula of fac( x A | x B ) can be found in [68, Appendix A]. We donot need the explicit expression in this paper. Here the subindex loc stands for localizationwith respect to the divisor defined by fac( x A | x B ).Let H := C [ ψ i,r | i ∈ I, r ∈ N ] be the polynomial ring with infinitely many formalvariables. Let ψ i ( z ) = 1 + (cid:80) r ≥ ψ i,r z − r − ∈ H [[ z − ]] be the generating series of generators ψ i,r ∈ H . Similar to [66], we define the extended shuffle algebra H e := H (cid:110) H using the H –action on H by ψ i ( z ) gψ i ( z ) − := g fac( z | x [1 ,v ] )fac( x [1 ,v ] | z ) , for any g ∈ H v (9)Following the same construction as in [66], we define a localized coproduct on H e ∆ : H e → (cid:88) v + v = v ( H e v ⊗ H e v ) loc . Here loc means the localization away from the union of null-divisors of fac( x [1 ,v ] | x [ v +1 ,v + v ] )over all v + v = v and ψ i ( z ). In particular, the functions ψ i ( z ) , fac( x [1 ,v ] | x [ v +1 ,v + v ] )have inverse in H e loc .On the symmetric algebra H , the map ∆ is given by∆( ψ i ( w )) = ψ i ( w ) ⊗ ψ i ( w ) , i ∈ I. (10)For an element f ( x ( i ) ) ∈ H e i , we define ∆ on H e i by∆( f ( x ( i ) )) = ψ i ( x ( i ) ) ⊗ f ( x ( i ) ) + f ( x ( i ) ) ⊗ ∈ H e e i ⊗ H e e i . We extend ∆ on the entire H by requiring ∆( a (cid:63) b ) = ∆( a ) (cid:63) ∆( b ). For a homogeneouselement f ( x [1 ,v ] ) ∈ H v , ∆ is given as∆( f ( x [1 ,v ] )) = (cid:88) { v + v = v } ψ [1 ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] ) , (11)10here ψ [1 ,v ] ( x [ v +1 ,v ] ) := (cid:81) k ∈ I (cid:81) j = v ( k )1 +1 , ··· ,v ( k ) ψ k ( x ( k ) j ). The subindex [1 , v ] indicatesthat the factor ψ k ( x ( k ) j ) = 1 + (cid:80) r ψ k,r ⊗ ( x ( k ) j ) − r − lies in H e v ,loc ⊗ H e e k ,loc .Define the co-unit (cid:15) : H e → R, ψ i ( z ) (cid:55)→ , f ( x [1 ,v ] ) (cid:55)→ . for v (cid:54) = 0, ψ i ( z ) ∈ H [[ z − ]] , f ( x [1 ,v ] ) ∈ H v .Define the antipode S : H e loc → H e loc as follows. ψ i ( z ) (cid:55)→ ψ − i ( z ) (12) f ( x [1 ,v ] ) (cid:55)→ ( − | v | ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) (13)for ψ i ( z ) ∈ H [[ z − ]] , f ( x [1 ,v ] ) ∈ H v . Here ψ [1 ,v ] ( x [1 ,v ] ) = (cid:81) k ∈ I (cid:81) j = v ( k )1 +1 , ··· ,v ( k ) ψ k ( x ( k ) j ).The subindex [1 , v ] indicates that ψ − ,v ] ( x [1 ,v ] ) lies in H e v . We extend S to H e by requiring S to be an anti-homomorphism. (i.e. S ( a (cid:63) b ) = S ( b ) (cid:63) S ( a )). In particular, when v = e i ,we have S ( E i ( u )) (cid:55)→ − ψ − i ( x i ) E i ( u ) . Clearly, the above assignment (12) defines an anti-homomorphism on H loc . Lemma 2.1.1
1. Choose a = f ( x [1 ,v ] ) ∈ H v , b = g ( x [1 ,v ] ) ∈ H v . We have S ( a (cid:63) b ) = S ( b ) (cid:63) S ( a ) .2. The map S respects the action (14) .Proof. For (1): We have S ( g ( x [1 ,v ] ) (cid:63) S ( f ( x [1 ,v ] )=( − | v | ( − | v | ψ − ,v ] ( x [1 ,v ] ) g ( x [1 ,v ] ) (cid:63) ψ − v +1 ,v ] ( x [ v +1 ,v ] ) f ( x [ v +1 ,v ] )=( − | v | ψ − ,v ] ( x [1 ,v ] ) ψ − v +1 ,v ] (cid:16) g ( x [1 ,v ] ) · fac( x [ v +1 ,v ] | x [1 ,v ] )fac( x [1 ,v ] | x [ v +1 ,v ] ) (cid:17) (cid:63) f ( x [ v +1 ,v ] )=( − | v | ψ − ,v ] ( x [1 ,v ] ) (cid:88) σ ∈ Sh( v ,v ) σ (cid:16) g ( x [1 ,v ] ) · f ( x [ v +1 ,v ] · fac( x [ v +1 ,v ] | x [1 ,v ] ) (cid:17) =( − | v | ψ − ,v ] ( x [1 ,v ] ) (cid:16) f ( x [1 ,v ] ) (cid:63) g ( x [1 ,v ] ) (cid:17) = S (cid:16) f ( x [1 ,v ] ) (cid:63) g ( x [1 ,v ] ) (cid:17) . We now prove (2). We have S ( ψ i ( z ) f ( x [1 ,v ] ) ψ i ( z ) − ) = ψ i ( z ) S ( f ( x [1 ,v ] )) ψ i ( z ) − = ( − | v | ψ i ( z ) ψ − ,v ] ( x [1 ,v ] ) S ( f ( x [1 ,v ] )) ψ i ( z ) − =( − | v | ψ − ,v ] ( x [1 ,v ] ) ψ i ( z ) f ( x [1 ,v ] ) ψ i ( z ) − =( − | v | ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) fac( z | x [1 ,v ] )fac( x [1 ,v ] | z ) )On the other hand, we compute S ( f ( x [1 ,v ] ) fac( z | x [1 ,v ] )fac( x [1 ,v ] | z ) ) = ( − | v | ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) fac( z | x [1 ,v ] )fac( x [1 ,v ] | z )11his completes the proof. (cid:4) . The above Lemma shows the assignments (12) (13) determines a unique anti-homomorphismon H e loc . Lemma 2.1.2
The anti-homomorphism S satisfies the axiom µ ( S ⊗ µ (1 ⊗ S )∆ = η(cid:15) .Proof. We have µ ( S ⊗ ψ i ( w )) = µ ( S ⊗ ψ i ( w ) ⊗ ψ i ( w )) = µ ( ψ − i ( w ) ⊗ ψ i ( w )) = 1 . Similarly, µ (1 ⊗ S )∆( ψ i ( w )) = 1.For a dimension vector v = ( v i ) i ∈ I ∈ N I , we define | v | := (cid:80) i ∈ I v i and (cid:0) vv (cid:1) := (cid:81) i ∈ I (cid:0) v i v i (cid:1) . For f ( x [1 ,v ] ) ∈ H v , we have µ ( S ⊗ f ( x [1 ,v ] ))= µ ( S ⊗ (cid:88) { v + v = v } ψ [1 ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] )= µ (cid:88) { v + v = v } ( − | v | ψ − ,v ] ( x [1 ,v ] ) ψ − ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] )fac( x [1 ,v ] | x [ v +1 ,v ] )= (cid:16) (cid:88) { v + v = v } (cid:18) vv (cid:19) ( − | v | (cid:17) ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) = 0 . In the second equality, the denominator becomes fac( x [1 ,v ] | x [ v +1 ,v ] ) since we switch S ( ψ [1 ,v ] ( x [ v +1 ,v ] ) = ψ − ,v ] ( x [ v +1 ,v ] ) in front of f using ψ − ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] ) = f ( x [1 ,v ] ⊗ x [ v +1 ,v ] ) ψ − ,v ] ( x [ v +1 ,v ] ) fac( x [1 ,v ] | x [ v +1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] ) . Similarly, we compute µ (1 ⊗ S )∆( f ( x [1 ,v ] ))= µ (1 ⊗ S ) (cid:88) { v + v = v } ψ [1 ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] )= (cid:88) { v + v = v } (cid:18) vv (cid:19) ( − | v | ψ [1 ,v ] ( x [ v +1 ,v ] ) ψ − ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] )= (cid:16) (cid:88) { v + v = v } (cid:18) vv (cid:19) ( − | v | (cid:17) f ( x [1 ,v ] ) = 0 . This completes the proof. (cid:4) . In this section, we follow the terminology from the book [30, Chapter 3]. Let A and B betwo Hopf algebras. A skew-Hopf pairing of A and B is an R -bilinear function( · , · ) : A × B → R that satisfies the conditions 12a) (1 , b ) = (cid:15) B ( b ) , ( a,
1) = (cid:15) A ( a ) , (b) ( a, bb (cid:48) ) = (∆ A ( a ) , b ⊗ b (cid:48) ) , (c) ( aa (cid:48) , b ) = ( a ⊗ a (cid:48) , ∆ op B ( b )),(d) ( S A ( a ) , b ) = ( a, S − B ( b )).We consider H e , coop loc that satisfies H e , coop loc = H ⊗ H loc as associative algebras, but isequipped with opposed comultiplication described as follows.1. φ i ( z ) := ( − l i +1 ψ i ( z ), where l i is the number of loops at vertex i ∈ I .2. The H –action on H is by φ i ( z ) gφ − i ( z ) := g fac( z | x [1 ,v ] )fac( x [1 ,v ] | z ) , for any g ∈ H v (14)3. The coproduct on H e , coop loc is given by∆ B : H e , coop loc → H e , coop loc ⊗ H e , coop loc ,φ i ( w ) (cid:55)→ φ i ( w ) ⊗ φ i ( w ) ,g ( x [1 ,v ] ) (cid:55)→ (cid:88) { v + v = v } g ( x [1 ,v ] ⊗ x [ v +1 ,v ] ) φ [ v +1 ,v ] ( x [1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] )4. The antipode S B is given by φ i ( z ) (cid:55)→ φ i ( z ) − g ( x [1 ,v ] ) (cid:55)→ ( − | v | g ( x [1 ,v ] ) φ − ,v ] ( x [1 ,v ] )Similar to the proofs in the case of H e , one easily verify that H e , coop is a Hopf algebra.Let us take A to be the Hopf algebra H e loc constructed in 2.1, and B to be the Hopf algebra H e , coop loc . We now construct a bilinear skew-Hopf pairing between H e loc and H e , coop loc as follows:( · , · ) : H e loc ⊗ H e , coop loc → R • ( f v , g w ) = 0 if v (cid:54) = w , ( f v , φ i ( z )) = 0, ( ψ i ( z ) , g w ) = 0, • ( ψ k ( u ) , ψ l ( w )) = fac( u | w )fac( w | u ) for any k, l ∈ I . • ( f e i , g e i ) := Res x = ∞ f ( x ( i ) ) · g ( − x ( i ) ) dx .where f v ∈ H v , g w ∈ H coop w and ψ i ( z ) ∈ SH [[ z ]]. We extend the pairing to the entire H v × H coop v using the property ( a, bb (cid:48) ) = (∆ A ( a ) , b ⊗ b (cid:48) ) and ( aa (cid:48) , b ) = ( a ⊗ a (cid:48) , ∆ op B ( b )). Anexplicit formula for ( f v , g w ) can be found in [66].We now verify this is a skew-Hopf pairing. (a) is obvious; (b)(c) can be proved the sameway as in [66]. Lemma 2.2.1 ( S A ( a ) , b ) = ( a, S − B ( b )) 13 roof. Let a = ψ k ( z ) , b = ψ l ( w ).( S A ( ψ k ( z )) , ψ l ( w )) = ( ψ − k ( z ) , ψ l ( w )) = fac( w | u )fac( u | w )( ψ k ( z ) , S − B ( ψ l ( w ))) = ( ψ k ( z ) , ψ − l ( w )) = fac( w | u )fac( u | w ) . Thus, ( S A ( ψ k ( z )) , ψ l ( w )) = ( ψ k ( z ) , S − B ( ψ l ( w ))).Let a = f v , b = ψ k ( z ).( S A ( f v ) , ψ k ( z )) = (( − | v | ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) , ψ k ( z ))= (( − | v | ψ − ,v ] ( x [1 ,v ] ) ⊗ f ( x [1 ,v ] ) , ψ k ( z ) ⊗ ψ k ( z )) = 0 . ( f v , S − B ( ψ k ( z ))) = ( f v , ψ − k ( z )) = 0Thus, ( S A ( f v ) , ψ k ( z )) = ( f v , S − B ( ψ k ( z ))).Let a = f v , b = g w .( S A ( f v ) , g w ) = (( − | v | ψ − ,v ] ( x [1 ,v ] ) f ( x [1 ,v ] ) , g w )= ( − | v | ψ − ,v ] ( x [1 ,v ] ) ⊗ f ( x [1 ,v ] ) , (cid:88) { w + w = w } φ [ w +1 ,w ] ( x [1 ,w ] ) g ( x [1 ,w ] ⊗ x [ w +1 ,w ] )fac( x [ w +1 ,w ] | x [1 ,w ] ) = δ v,w ( − | v | (cid:16) ψ − ,v ] ( x [1 ,v ] ) , φ [1 ,w ] ( x [1 ,w ] ) (cid:17)(cid:16) f ( x [1 ,v ] ) , g ( x [1 ,v ] ) (cid:17) ( f v , S − B g w ) = ( f v , ( − | w | φ − ,w ] ( x [1 ,w ] ) g ( x [1 ,w ] ))= (cid:16) (cid:88) { v + v = v } ψ [1 ,v ] ( x [ v +1 ,v ] ) f ( x [1 ,v ] ⊗ x [ v +1 ,v ] )fac( x [ v +1 ,v ] | x [1 ,v ] ) , ( − | w | φ − ,w ] ( x [1 ,w ] ) ⊗ g ( x [1 ,w ] ) (cid:17) = δ v,w ( − | v | (cid:16) ψ [1 ,v ] ( x [1 ,v ] ) , φ − ,w ] ( x [1 ,w ] ) (cid:17)(cid:16) f ( x [1 ,v ] ) , g ( x [1 ,v ] ) (cid:17) Thus, we have ( S A ( f v ) , g w ) = ( f v , S − B g w ), since ( ψ − ,v ] ( x [1 ,v ] ) , φ [1 ,w ] ( x [1 ,w ] )) = ( ψ [1 ,v ] ( x [1 ,v ] ) , φ − ,w ] ( x [1 ,w ] )).This completes the proof. (cid:4) . It is known that if
A, B are Hopf algebras, endowed with a skew-Hopf paring. Then, thereis a unique Hopf structure on A ⊗ B , called the Drinfeld double of ( A, B, ( · , · )), determinedby the following properties [30, Lemma 3.2.2](a) ( a ⊗ a (cid:48) ⊗
1) = aa (cid:48) ⊗ ⊗ b )(1 ⊗ b (cid:48) ) = 1 ⊗ bb (cid:48) ,(c) ( a ⊗ ⊗ b ) = a ⊗ b ,(d) (1 ⊗ b )( a ⊗
1) = (cid:80) ( a , S B ( b )) a ⊗ b ( a , b ),for all a, a (cid:48) ∈ A, b, b (cid:48) ∈ B . We use the notation that ∆ A ( a ) = (cid:80) a ⊗ a ⊗ a and ∆ B ( b ) = (cid:80) b ⊗ b ⊗ b .Let Q (cid:48) = ( I (cid:48) , H (cid:48) ) ⊂ Q be a subquiver. That is, I (cid:48) ⊂ I is a subset and H (cid:48) = { h ∈ H | the incoming and outgoing vertices of h are in I (cid:48) } . It is clear that the shuffle algebra H (cid:48) associated to Q (cid:48) is a subalgebra of the shuffle algebra H associated to I .14 roposition 2.2.2 We have D ( H (cid:48) ) is a subalgebra of D ( H ) .Proof. The Hopf structure of H when restricted to H (cid:48) gives a Hopf structure on H (cid:48) and therestriction of the pairing is still a skew Hopf pairing. The multiplication on D ( H (cid:48) ) by (2.2)is compatible with the multiplication on D ( H ). This completes the proof. (cid:4) . Let SH be the equivariant spherical COHA ([57, 55]). By definition, SH is a subalgebraof H generated by H e i , as i varies in I . We will skip the word “equivariant” if it does notlead to a confusion. Similarly, let SH be the spherical shuffle algebra, which is generated by H e i , as i varies in I . Let D ( SH ) ⊂ D ( H ) be the subalgebra generated by H , H e i H coop e i ,as i varies in I .We assume that there is an algebra homomorphism H → H which is an isomorphism after passing to the localization − ⊗ R [ h v ] Sv R ( h v ) S v , for each v .We have an induced algebra epimorphism SH → SH , which is an isomorphism after passing to the same localization. We define the Drinfelddouble D ( SH ) of the spherical COHA as D ( SH ). Let i ∈ I be a vertex of the quiver, let e i be the dimension vector which is 1 at i ∈ I and 0otherwise. Since any 1-by-1 matrices naturally commute and hence the potential functionvanishes, we have H ∗ c, GL ei × T (Rep( Q, e i ) , ϕ tr W ) ∨ ∼ = R [ x ( i ) ]. Define H i to be the subalgebraof H , generated by R [ x ( i ) ].We compute the relations in the Drinfeld double of COHA. First note that SH e , SH e , coop are two sub-algebras of D ( SH ). Proposition 2.3.1
Let E i ( u ) , F j ( v ) be the generating series E i ( u ) := (cid:88) r ≥ ( x ( i ) ) r u − r − ∈ H e [[ u ]] , F j ( v ) := (cid:88) r ≥ ( − x ( i ) ) r v − r − ∈ H e [[ v ]] , for i, j ∈ I . Then, in D ( SH ) , we have the relation [ E i ( u ) , F j ( v )] = δ ij (cid:16) ψ i ( u ) ⊗ − ⊗ φ i ( u ) (cid:17) − (cid:16) ψ i ( v ) ⊗ − ⊗ φ i ( v ) (cid:17) u − v , if { edge loops at i } is odd , { E i ( u ) , F j ( v ) } = δ ij (cid:16) ψ i ( u ) ⊗ − ⊗ φ i ( u ) (cid:17) − (cid:16) ψ i ( v ) ⊗ − ⊗ φ i ( v ) (cid:17) u − v , if { edge loops at i } is even . where [ a, b ] := ab − ba is the commutator and { a, b } := ab + ba is the super commutator.Proof. By the coproduct formula, we have∆ ( E i ( u )) = ψ i ( x ( i ) ) ⊗ ψ i ( x ( i ) ) ⊗ E i ( u ) + ψ i ( x ( i ) ) ⊗ E i ( u ) ⊗ E i ( u ) ⊗ ⊗ ( F j ( v )) = F j ( v ) ⊗ φ j ( x ( j ) ) ⊗ φ j ( x ( j ) ) + 1 ⊗ F j ( v ) ⊗ φ j ( x ( j ) ) + 1 ⊗ ⊗ F j ( v ) .
15n the multiplication formula (2.2), we choose b = F j ( v ), a = E i ( u ). Using the followingpairings( ψ i ( x ( i ) ) ,
1) = 1 , (1 , φ j ( x ( j ) ) = ( − ( e j ,L ) , ( E i ( u ) , − F j ( v ) φ − j ( x ( j ) )) = ( ψ i ( x ( i ) ) ⊗ E i ( u ) + E i ( u ) ⊗ , − F j ( v ) ⊗ φ − j ( x ( j ) )) = − ( − ( e j ,L ) ( E i ( u ) , F j ( v )) , we have(1 ⊗ F j ( v ))( E i ( u ) ⊗
1) = (cid:88) ( a , S B ( b )) a ⊗ b ( a , b )=( ψ i ( x ( i ) ) , ψ i ( x ( i ) ) ⊗ E i ( u ) , F j ( v )) + ( ψ i ( x ( i ) ) , E i ( u ) ⊗ F j ( v )(1 , φ j ( x ( j ) )+ ( E i ( u ) , − F j ( v ) φ − j ( x ( j ) ))1 ⊗ φ j ( x ( j ) )(1 , φ j ( x ( j ) ))=( − ( e j ,L ) E i ( u ) ⊗ F j ( v ) + (cid:16) ψ i ( x ( i ) ) ⊗ − ⊗ φ j ( x ( j ) ) (cid:17) ( E i ( u ) , F j ( v ))=( − ( e j ,L ) E i ( u ) ⊗ F j ( v ) + δ ij Res x ( i ) = ∞ (cid:16) ψ i ( x ( i ) ) ⊗ − ⊗ φ i ( x ( i ) ) (cid:17) u − x ( i ) · v − x ( i ) =( − ( e j ,L ) E i ( u ) ⊗ F j ( v ) + δ ij (cid:16) ψ i ( u ) ⊗ − ⊗ φ i ( u ) (cid:17) − (cid:16) ψ i ( v ) ⊗ − ⊗ φ i ( v ) (cid:17) u − v Here the sign ( − l i +1 is 1 (resp. is − i is has odd number of edge loops (resp.even number of edge loops). Thus, we have the desired relation. (cid:4) . Proposition 2.3.2
The relation between E i ( u ) ⊗ and ⊗ φ j ( v ) is given as (1 ⊗ φ j ( v ))( E i ( u ) ⊗ ⊗ φ − j ( v )) = E i ( u ) fac( v | x ( i ) )fac( x ( i ) | v ) ⊗ . Proof.
By the coproduct formula, we have∆ ( E i ( u )) = ψ i ( x ( i ) ) ⊗ ψ i ( x ( i ) ) ⊗ E i ( u ) + ψ i ( x ( i ) ) ⊗ E i ( u ) ⊗ E i ( u ) ⊗ ⊗ ( φ j ( v )) = φ j ( v ) ⊗ φ j ( v ) ⊗ φ j ( v ) . In the multiplication formula (2.2), we choose b = φ j ( v ) , a = E i ( u ). We have(1 ⊗ φ j ( v ))( E i ( u ) ⊗
1) = (cid:88) ( a , S B ( b )) a ⊗ b ( a , b )=( ψ i ( x ( i ) ) , φ − j ( v )) E i ( u ) ⊗ φ j ( v )(1 , φ j ( v ))= fac( v | x ( i ) )fac( x ( i ) | v ) E i ( u ) ⊗ φ j ( v )This is equivalent to the action(1 ⊗ φ j ( v ))( E i ( u ) ⊗ ⊗ φ − j ( v )) = E i ( u ) fac( v | x ( i ) )fac( x ( i ) | v ) ⊗ . This completes proof. (cid:4) . Proposition 2.3.3
The relation between ψ i ( u ) ⊗ and ⊗ φ j ( v ) is given as (1 ⊗ φ j ( v ))( ψ i ( u ) ⊗
1) = ( ψ i ( u ) ⊗ ⊗ φ j ( v )) . roof. By the coproduct formula, we have∆ ( ψ i ( u )) = ψ i ( u ) ⊗ ψ i ( u ) ⊗ ψ i ( u ) , ∆ ( φ j ( v )) = φ j ( v ) ⊗ φ j ( v ) ⊗ φ j ( v ) . In the multiplication formula (2.2), we choose b = φ j ( v ) , a = ψ i ( u ). We have(1 ⊗ φ j ( v ))( ψ i ( u ) ⊗ ψ i ( u ) , φ − j ( v )) ψ i ( u ) ⊗ φ j ( v )( ψ i ( u ) , φ j ( v )) = ψ i ( u ) ⊗ φ j ( v ) . This completes proof. (cid:4) . By Proposition 2.3.3, in the Drinfeld double D ( SH ), there is a commutative subalgebrawhich is isomorphic to H ⊗ H . For fixed i ∈ I , label the Cartan elements ψ i,r ⊗ ∈H ⊗ r ≥
0) and 1 ⊗ φ i,r ∈ ⊗ H ( r ≥
0) in D ( SH ) by Z as follows ψ ( r ) i := (cid:26) ψ i,r ⊗ − ⊗ φ i,r , if r ≥ ,ψ i, − r − ⊗ ⊗ φ i, − r − , if r < . (15)When g is the Kac-Moody Lie algebra, the Cartan doubled Yangian of g is introduced in[17] (and recalled in Definition 3.2.1). In Proposition 3.2.2, we show that the Cartan doubledYangian of a Kac-Moody algebra surjects to the Drinfeld double of some SH ( X m, ). Notethat the index set of ψ ( r ) i is I × Z . This explains the name “Cartan doubled”.A coweight µ is a Z -linear Z -valued function on Z I . In [17], a shifted Yangian of aKac-Moody algebra [17, Definition 3.5] is defined to be a quotient of the Cartan doubledYangian by a coweight µ . Motivated by the loc.cit. we propose the following definition. Definition 2.4.1
For any coweight µ we define the shifted Yangian Y µ to be the quotientof D ( SH ) by the relations ψ ( p ) i , for all p < −(cid:104) µ, α i (cid:105) . Let (
Q, W ) be a quiver with potential, and let i ∈ I be a vertex of the quiver. Recall H i isthe subalgebra of H generated by the dimension vector e i . Proposition 3.1.1
In the above notation the algebra H i is isomorphic to the positive partof the Clifford algebra if i has no edge loop, Y + (cid:126) ( sl ) if i has 1 edge loop.Proof. The COHA of this subquiver in both cases was computed in [35, § § Y + (cid:126) ( sl ) (see also [67]). (cid:4) . Example 3.1.2
For the Dynkin quiver Q = A endowed with the potential W = 0 , theDrinfeld double D ( H ) is generated by e ( r ) , f ( r ) , ψ ( s ) , r ≥ , s ∈ Z , ubject to the following relations: [ ψ ( s ) , ψ ( t ) ] = 0 { e ( r ) , e ( r ) } = 0 , { f ( r ) , f ( r ) } = 0 , { e ( r ) , ψ ( r ) } = 0 , { f ( r ) , ψ ( r ) } = 0 , { e ( r ) , f ( r ) } = ψ ( r + r ) , where { a, b } := ab + ba , and ψ ( s ) is defined in (15) . Since the A quiver has no arrows, theconjugation action (14) of ψ ( z ) on H is given as ψ ( z ) g ( ψ ( z )) − = fac( z | x )fac( x | z ) g = z − xx − z g = − g. Proposition 2.3.2 implies ψ ( z ) g ( ψ ( z )) − = − g . Therefore, { ψ ( s ) , e ( r ) } = 0 , for s ∈ Z .Similarly, { ψ ( s ) , f ( r ) } = 0 , for s ∈ Z . Proposition 2.3.1 implies that { e ( r ) , f ( r ) } = ψ r + r . Example 3.1.3
Let Q be the Jordan quiver, i.e. it has one vertex and one loop. Then forthe potential W = 0 the Drinfeld double D ( H ) is generated by e ( r ) , f ( r ) , ψ ( s ) , r ≥ , s ∈ Z , subject to the Cartan doubled Yangian relations for sl (see Definition 3.2.1, taking I to bea point). In particular, we have the relation [ e ( u ) , f ( v )] = ψ + ( u ) − ψ + ( v ) u − v , where ψ + ( u ) = ψ ( u ) ⊗ − ⊗ ψ ( u ) which follows from Proposition 2.3.1. Let us start with the following definition.
Definition 3.2.1 [17, Definition 3.1] Let g be a Kac-Moody Lie algebra. The Cartan doubleYangian Y ∞ ( g ) is the C -algebra generated by E ( q ) i , F ( q ) i , H ( p ) i , for i ∈ I, q > , p ∈ Z , subjectto the relations [ H ( p ) i , H qj ] = 0 , (HH)[ E ( p ) i , E ( q ) j ] = δ ij H ( p + q − i , (EF)[ H ( p +1) i , E ( q ) j ] − [ H ( p ) i , E ( q +1) j ] = α i · α j H ( p ) i E ( q ) j + E ( q ) j H ( p ) i ) , (HE)[ H ( p +1) i , F ( q ) j ] − [ H ( p ) i , F ( q +1) j ] = − α i · α j H ( p ) i F ( q ) j + F ( q ) j H ( p ) i ) , (HF)[ E ( p +1) i , E ( q ) j ] − [ E ( p ) i , E ( q +1) j ] = α i · α j E ( p ) i E ( q ) j + E ( q ) j E ( p ) i ) , (EE)[ F ( p +1) i , F ( q ) j ] − [ F ( p ) i , F ( q +1) j ] = − α i · α j F ( p ) i F ( q ) j + F ( q ) j F ( p ) i ) , (FF) i (cid:54) = j, N = 1 − α i · α j , sym [ E ( p ) i , [ E ( p ) i , · · · , [ E ( p N ) i , E qj ] · · · ]] = 0 ,i (cid:54) = j, N = 1 − α i · α j , sym [ F ( p ) i , [ F ( p ) i , · · · , [ F ( p N ) i , F qj ] · · · ]] = 0 , g be a symmetric Kac-Moody Lie algebra with Dynkin diagram Γ. Let ( Q, W )be the tripled quiver with potential as in [24]. Let H be the COHA of ( Q, W ). Proposition 3.2.2
In the above notation we have an epimorphism Y ∞ ( g ) (cid:16) D ( SH ) . Proof.
Consider the following generating series of H ( p ) i ( p ∈ Z ) H >i ( z ) := (cid:88) p ≥ H ( p +1) i z − p − , H i ( z ) (cid:55)→ ψ + i ( z ) = ψ i ( z ) ⊗ − ⊗ ψ i ( z ) ,H
1, let f : X → Y be a resolution of singularities, where X is a smoothlocal toric Calabi-Yau 3-fold and Y is affine. There is a tilting generator P of D b Coh( X )so that the functor R Hom X ( P , − ) : D b Coh( X ) → D b (Mod- A )induces an equivalence of derived categories. Here A := End X ( P ) op is a coherent sheaf ofnon-commutative O Y -algebras, and Mod- A is the abelian category of coherent sheaves of(right) modules over A . Let A ⊂ D b Coh( X ) be a heart of D b Coh( X ) corresponding to theheart Mod- A of the standard t -structure of D b (Mod- A ) under the functor R Hom X ( P , − ).One example of A is Perv − ( X/Y ), the abelian category of perverse coherent sheaves inthe sense of Bridgeland and Van den Bergh [3, 63]. It is known that A can be identifiedwith the Jacobian algebra of a quiver with potential ( Q, W ). We denote the COHA of thepair (
Q, W ) also be H X .Note that each vertex i ∈ I of Q gives rise to a simple object S i in the heart of the induced t -structure on D b Coh( X ). We say S i is bosonic if the simple object S i in D b Coh( X ) has theproperty that Ext ∗ ( S i , S i ) = H ∗ ( P ); we say [ S i ] is fermionic if the simple object S i is aspherical object, that is, Ext ∗ ( S i , S i ) = H ∗ ( S ). We have the following general proposition. Proposition 3.3.1
For a vertex i ∈ I of the quiver Q , assume the corresponding simpleobject S i is a line bundle on j ∗ O P ( m )[ n ] with j : P (cid:44) → X and m, n ∈ Z . Then, S i iseither bosonic or fermionic. Figure 1: The toric diagram of X m, consists of m horizontal lines with orientation (1 , k,
1) for k = 0 , . . . , n . mn Figure 2: The toric diagram of resolved Y m,n consists of m horizontal lines with orienta-tion (1 ,
0) attached from the right together with m horizontal lines with orientation (1 , k, m lines ending from the left followed by n lines ending from theright. Other possible resolutions are associated to different orderings of the left and theright lines. Proof.
It is not difficult to calculate the self-extension of this line bundle. Without loss ofgenerality, assume this line bundle is trivial. Taking the formal neighbourhood of P in X ,we get a rank-2 vector bundle on P the determinant of which is O ( − X is birational to its affinization Y , we get that this rank-2 vector bundle has no ampleline sub-bundles, therefore is either O ⊕ O ( −
2) or O ( − ⊕ O ( − O P has 1-dimensional self-extension. In the second case, O P has no non-trivial self-extensions.The entire Ext ∗ ( S i , S i ) is then determined by the Calabi-Yau condition. (cid:4) . Therefore, when S i is bosonic, it follows from Example 3.1.3 that D ( H i ) is the Cartandoubled Yangian of sl . When S i is fermionic, it follows from Example 3.1.2 that D ( H i ) isthe infinite Clifford algebra.We give three examples of quiver with potentials coming from this setup. Example 3.3.2
Let X = X m, . Then, the defining equation xy = z m of Y m, is a type A -singularity. All the simple objects are bosonic. Indeed, let S m = C / Z m be the type A m − Kleinian surface singularity, and ˜ S m its crepant resolution. Then, X m, ∼ = ˜ S m × C .The toric diagram of X m, is depicted in figure 1. The quiver with potential is the tripledquiver of (cid:92) A m − in the sense of of [24]. Example 3.3.3
Let X = X m,n . A toric diagram of one possible resolution of Y m,n is shownin figure (2) . Other possibilities can be obtained by considering different orderings of the leftand the right (1 , lines ending on a sequence of ( k, lines. All possible choices for Y , are depicted in figure (3) . One can then associate a bosonic simple object to each internalline bounded by two (1 , lines ending from the same side and a fermionic simple object toeach internal line bounded by (1 , lines ending from opposite directions. The quiver withpotential is given explicitly in [42] Y , associated to the two possiblechoices of the root system of gl (2 | Example 3.3.4
Let X = S × C , where S is an elliptically fibered K -surface, with specialfiber C being a collection of P ’s in the (cid:99) A n -type configuration. That is, let ∆ be the collectionirreducible components of C each of which is isomorphic to P . There is one componentof C , removing which results a curve C o an open neighborhood of which is isomorphic to ˜ S m × C as in Example 3.3.2 above. Although there is no contraction of this collection of P ’s with rational singularity, hence the method of constructing a tilting bundle from [63]does not apply here, the result of [25] still suggest that there is a t -structure on D b Coh( X ) that contains O C i (1)[ − with i ∈ I and O C o as simple objects. It is easy to verify that allthese simple objects are bosonic. Let us explain how the notion of shifted Yangians appears later on in our paper. We willconsider the moduli space of stable perverse coherent system on X . As to be explained in 7and 8, the moduli space depends not only on a choice of the stability parameter ζ , but alsoon an algebraic cycle χ of the form N X + r (cid:88) i =1 N i D i , where N i ∈ N , and each D i is a toric divisor. Here χ determines a framing of the quiver withpotential ( Q, W ). We mark the framings associated to [ X ] by (cid:3) and the ones associatedto each D i by (cid:52) . For example, when X = X , , and D is the fiber of the 0 ∈ P , theframed quiver is as follows In physics terminology χ should be thought of as a stack of D D V B • V ˜ B B B ˜ B ˜ B (cid:52) N I J (cid:3) ιN (16)We are going to discuss the framed quiver with potential for general χ in the futurepublications. In this paper we only consider some special cases. In particular, in § ζ and χ determine a coweight µ . We also explain that we expectthat the action of D ( SH ) cohomology of perverse coherent systems factors through Y µ .Furthermore, we expect that the algebra D ( SH ) has a coproduct, which after passing tothe quotient becomes Y µ → Y µ ⊗ Y µ if µ + µ = µ . This coproduct is compatible withhyperbolic restriction to fixed points of a subtorus acting on the framings, similar to [55].Therefore, we can reduce the verification of the action of Y µ for a general χ to the caseswhen the framing dimension is 1. Consider the case when X = C . The quiver with potential is recalled in the Introduction.We denote the quiver by Q and the potential by W . The algebraic properties of the COHAhas been studied [55, § Q , W ) which is called the equivariant sphericalCOHA of ( Q , W ) (or C ). Recall that in [55, Theorem 7.1.1.] the equivariant sphericalCOHA was identified with Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)) + .Furthermore, using the same coproduct as in § SH C is isomorphic to the entire Yangian Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)) [55]. Similar argumentas in Proposition 3.2.2 then implies that the shifted Yangians Y l ( (cid:126)z ) from § D ( SH C ). In this section, the toric Calabi-Yau 3-fold is X = X , , the resolved conifold. The cor-responding quiver with potential was described in the Introduction (see (2)). Recall thatwe denote the quiver by Q , and the potential by W , , and we denote the correspondingCOHA by H X , . By Proposition 3.3.1 there are positive parts of two Clifford subalgebrasin H X , associated to the two vertices of the quiver Q , . The goal of this section is toshow that there is an algebra homomorphism H X , → H C , where H C is the COHA for C from § n ∈ N consider the open subset Rep( Q , , ( n, n )) ⊂ Rep( Q , , ( n, n )) consist-ing of such representations that the map b is an isomorphism. We will show below that22 n ∈ N H ∗ c, GL ( n,n ) × T (Rep( Q , , ( n, n )) , ϕ tr W , ) ∨ has a natural algebra structure, so that therestriction to open subset ⊕ n H ∗ c, GL ( n,n ) × T (Rep( Q , , ( n, n )) , ϕ tr W , ) ∨ → ⊕ n H ∗ c, GL ( n,n ) × T (Rep( Q , , ( n, n )) , ϕ tr W , ) ∨ is an algebra homomorphism.Observe that there is a canonical isomorphism of vector spaces ⊕ n H ∗ c, GL ( n,n ) × T (Rep( Q , , ( n, n )) , ϕ tr W , ) ∨ ∼ = ⊕ n H ∗ c, GL n × T (Rep( Q , n ) , ϕ tr W ) ∨ . Furthermore, with the algebra structures on both sides, the above isomorphism of vec-tor spaces is an isomorphism of algebras with the latter ⊕ n H ∗ c, GL n × T (Rep( Q , n ) , ϕ tr W ) ∨ carrying the multiplication of COHA of C . Proposition 3.6.1
Restriction to the above-defined open subset induces the algebra homo-morphism H X , → H C . Its image contains Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)) + .Proof. Consider the affine space parameterizing representations of Q , on the underlyingvector space V = ( V , V ) with maps a i from V to V and b i backwards for i = 1 ,
2. Wedenote such space by Rep( Q , , V ). We first recall the multiplication of H X , . For this,consider pairs of flags0 → V ⊆ V (cid:48) (cid:16) V (cid:48)(cid:48) → → V ⊆ V (cid:48) (cid:16) V (cid:48)(cid:48) → G/P where G := GL( V (cid:48) ) × GL( V (cid:48) ), and P = P × P is the parabolic subgroupwith P i = { x ∈ GL( V (cid:48) i ) | x ( V i ) ⊂ V i } , i = 1 ,
2. Consider Rep( Q , , V (cid:48) ) and the subspace Z consisting of representations such that ( V , V ) is a sub-representation. We have thefollowing diagram of correspondences G × P Z φ (cid:118) (cid:118) (cid:31) (cid:127) η (cid:47) (cid:47) G × P Rep( Q , , V (cid:48) ) ψ (cid:15) (cid:15) G × P (cid:18) Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) (cid:19) Rep( Q , , V (cid:48) )where φ is an affine bundle. The potential W , defines functions tr(( W , ) V (cid:1) ( W , ) V (cid:48)(cid:48) ) onRep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) and tr(( W , ) Z ) on Z by restricting the function tr(( W , ) V (cid:48) ).We have tr(( W , ) V (cid:1) ( W , ) V (cid:48)(cid:48) ) ◦ φ = tr(( W , ) Z ).The map η is a closed embedding and ψ is a projection, both of which are compatiblewith the potential functions.Let Z be the intersection of Z with Rep( Q , , V (cid:48) ) ◦ . Note that P acts on Z , and thatthe natural map η : Z → Rep( Q , , V (cid:48) ) is a pullback Z (cid:47) (cid:47) (cid:15) (cid:15) Rep( Q , , V (cid:48) ) (cid:15) (cid:15) Z (cid:47) (cid:47) Rep( Q , , V (cid:48) )23n particular, restriction to the open subsets intertwines η ∗ and ( η ) ∗ . That is, we have thecommutative diagram H ∗ c,G × T ( G × P Z, ϕ tr W , ) ∨ η ∗ (cid:47) (cid:47) (cid:15) (cid:15) H ∗ c,G × T ( G × P Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ (cid:15) (cid:15) H ∗ c,G × T ( G × P Z , ϕ tr W , ) ∨ η ∗ (cid:47) (cid:47) H ∗ c,G × T ( G × P Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ Similarly, ψ : G × P Rep( Q , , V (cid:48) ) → Rep( Q , , V (cid:48) ) is pullback of ψ , and hence aftertaking cohomology the restrictions to the open subsets intertwines ψ ∗ and ( ψ ) ∗ . H ∗ c,G × T ( G × P Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ ψ ∗ (cid:47) (cid:47) (cid:15) (cid:15) H ∗ c,G × T (Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ (cid:15) (cid:15) H ∗ c,G × T ( G × P Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ ( ψ ) ∗ (cid:47) (cid:47) H ∗ c,G × T (Rep( Q , , V (cid:48) ) , ϕ tr W , ) ∨ Also, clearly the map φ induces the map φ by restriction to open subsets as in the followingdiagram. Again after taking cohomology restrictions to the open subsets intertwines φ ∗ and( φ ) ∗ . H ∗ c,G ( G × P (cid:0) Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) (cid:1) ) ∨ φ ∗ (cid:47) (cid:47) (cid:15) (cid:15) H ∗ c,G ( G × P Z ) ∨ (cid:15) (cid:15) H ∗ c,G ( G × P (Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) )) ∨ ( φ ) ∗ (cid:47) (cid:47) H ∗ c,G ( G × P Z ) ∨ To summarize, we have the following commutative diagram G × P Z φ (cid:119) (cid:119) (cid:31) (cid:127) η (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) G × P Rep( Q , , V (cid:48) ) ψ (cid:39) (cid:39) (cid:127) (cid:95) (cid:15) (cid:15) G × P (cid:18) Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) (cid:19) (cid:127) (cid:95) (cid:15) (cid:15) G × P Z φ (cid:120) (cid:120) (cid:31) (cid:127) η (cid:47) (cid:47) G × P Rep( Q , , V (cid:48) ) ψ (cid:38) (cid:38) Rep( Q , , V (cid:48) ) (cid:127) (cid:95) (cid:15) (cid:15) G × P (cid:18) Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) (cid:19) Rep( Q , , V (cid:48) ) (17)For V (cid:48) = ( V (cid:48) , V (cid:48) ), we now impose a condition that b : V (cid:48) → V (cid:48) is an isomorphism. Thespace of isomorphisms Isom ( V (cid:48) , V (cid:48) ) is a torsor over GL( V (cid:48) ). In particular, GL( V (cid:48) ) actstransitively on Rep( Q , , V (cid:48) ) by the change of basis on V (cid:48) . The quotientRep( Q , , V (cid:48) ) / GL( V (cid:48) ) ∼ = Rep( Q , V (cid:48) )is canonically identified with Rep( Q , V (cid:48) ) with B = a ◦ b , B = a ◦ b , and B = b − ◦ b .The action of GL( V (cid:48) ) on the left hand side and the action of GL( V (cid:48) ) on the right hand sideare compatible. 24imilarly, the parabolic subgroup P acts on Z transitively. Thus, we have the identi-fication( G × P Z ) / GL( V (cid:48) ) = GL ( V (cid:48) ) × P ( Z /P ) ∼ = GL ( V (cid:48) ) × P Z Q ∼ = GL ( V (cid:48) ) × P Z Q , where Z Q is the correspondence used in the multiplication of H Q .Note that the action of P on Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) factors through the pro-jection P (cid:16) GL ( V ) × GL ( V (cid:48)(cid:48) ). Therefore, we have G × P (Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) ) GL ( V (cid:48) ) ∼ = GL ( V (cid:48) ) × P Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) P ∼ = GL ( V (cid:48) ) × P Rep( Q , , V ) × Rep( Q , , V (cid:48)(cid:48) ) GL ( V ) × GL ( V (cid:48)(cid:48) ) ∼ = GL ( V (cid:48) ) × P (cid:16) Rep( Q , V ) × Rep( Q , V (cid:48)(cid:48) ) (cid:17) ∼ = GL ( V (cid:48) ) × P (cid:16) Rep( Q , V ) × Rep( Q , V (cid:48)(cid:48) ) (cid:17) To finish the proof, it remains to notice that quotient of the top correspondence of diagram17 by GL( V (cid:48) ) becomes GL ( V (cid:48) ) × P (cid:18) Rep( Q , V ) × Rep( Q , V (cid:48)(cid:48) ) (cid:19) GL ( V (cid:48) ) × P Z Q φ Q (cid:111) (cid:111) ψ Q ◦ η Q (cid:47) (cid:47) Rep( Q , V (cid:48) )This is the correspondence that defines multiplication of H C . This finishes the proof. (cid:4) . (cid:100) gl (1) and Costello conjectures (cid:100) gl (1) Let (cid:126) , (cid:126) , (cid:126) be the deformation parameters with (cid:126) + (cid:126) + (cid:126) = 0. Set σ := (cid:126) (cid:126) + (cid:126) (cid:126) + (cid:126) (cid:126) , and σ = (cid:126) (cid:126) (cid:126) . Let (cid:126)z = ( z , z , · · · , z | l | ) be the parameters ( l ∈ Z ). Definition 4.1.1
Let Y l ( (cid:126)z ) be a C [ (cid:126) , (cid:126) ] -algebra generated by D ,m ( m ≥ , e n , f n ( n ≥ with the relations [ D ,m , D ,n ] = 0( m, n ≥
1) (18)[ D ,m , e n ] = − e m + n − ( m ≥ , n ≥
0) (19)[ D ,m , f n ] = − f m + n − ( m ≥ , n ≥
0) (20)3[ e m +2 , e n +1 ] − e m +1 , e n +2 ] − [ e m +3 , e n ] + [ e m , e n +3 ]+ σ ([ e m +1 , e n ] − [ e m , e n +1 ]) + σ ( e m e n + e n e m ) = 0 (21)3[ f m +2 , f n +1 ] − f m +1 , f n +2 ] − [ f m +3 , f n ] + [ f m , f n +3 ]+ σ ([ f m +1 , f n ] − [ f m , f n +1 ]) − σ ( f m f n + f n f m ) = 0 (22)Sym S [ e i , [ e i , e i +1 ]] = 0 , (23)Sym S [ f i , [ f i , f i +1 ]] = 0 . (24)[ e m , f n ] = h m + n (25)1 − σ (cid:88) n ≥ h n z − ( n +1) = (cid:40) (cid:81) li =1 ( z − z i ) · ψ ( z ) , l ≥ , (cid:81) li =1 1( z − z i ) · ψ ( z ) , l ≤ . (26)25 here the equality (26) means the coefficients of z − i on both sides are equal for each i ≥ .Here ψ ( z ) := exp (cid:16) − (cid:80) n ≥ D ,n +1 ϕ n ( z ) (cid:17) and the function ϕ n ( t ) is a formal power seriesin t depending on (cid:126) , (cid:126) , (cid:126) . It is given by the following formula. exp( (cid:88) n ≥ ( − n +1 a n ϕ n ( z )) = ( z + a − (cid:126) )( z + a − (cid:126) )( z + a − (cid:126) )( z + a + (cid:126) )( z + a + (cid:126) )( z + a + (cid:126) ) , where a is any element in C . When l = 0, this is the Yangian of (cid:91) gl (1) when z i = 0. We denote it by Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)).When l > l < l -positively ( − l -negatively) shifted Yangian inparameters (cid:126)z . The positively shifted Yangian Y l ( (cid:126)z ) is first defined in [34, Section 6]. Remark 4.1.2
In the above presentation, we have the correspondence compared with thenotation in [34, Section 6]. (cid:126) (cid:55)→ (cid:126) , (cid:126) (cid:55)→ t , (cid:126) (cid:55)→ − (cid:126) − t , D ,n (cid:55)→ D ,n (cid:126) . We also take the central elements c = c = · · · = 0 in the central extension SH c of Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)) , so that the factor (1 − ( (cid:126) + t ) x )(1+ wtx )1 − ( (cid:126) +(1 − w ) t ) x in [34] does not show up in the formulaof ψ ( z ) We explain how Theorem 1.3.2 implies a special case of Costello conjectures [6].Let X l = (cid:94) C / ( Z /l Z ) be a crepant resolution of C / ( Z /l Z ). We have the Hilbert scheme X [ k ] l of k points on X l , which is smooth and quasi-projective. We denote the space of quasi-maps from P to X [ k ] l of degree d by M k,d,l . We follow the notation of quasi-maps from[50, Section 2.3].In [6] the author defined the algebra A l, (cid:126) ,(cid:15) which is a deformation of the envelopingalgebra U (Diff (cid:15) ( C ) ⊗ gl ( l )). Here Diff (cid:15) ( C ) denotes the algebra of differential operators onthe line C with [ ∂ z , z ] = (cid:15) , and (cid:126) is the deformation parameter.It is conjectured in [6, P73] that A l, (cid:126) ,(cid:15) acts on (cid:76) d H ∗ T ( M k,d,l , P ). The space M k,d,l can be realized as the critical locus of a regular function (“potential”) defined similarly tothe one in [50, Section 4.3]. The sheaf of vanishing cycles of this function is the perverseconstructible sheaf P on M k,d,l [6, P72]. Here (cid:126) , (cid:15) are equivariant parameters correspondingto the 2-dimensional torus T = C ∗ (cid:126) × C ∗ (cid:15) action.It is explained in [6] that the algebra A l, (cid:126) ,(cid:15) is a certain limit of Coulomb branch algebra ofthe cyclic quiver gauge theory, which in turn has been algebraically described as cyclotomicrational Cherednik algebra in [34]. Since we need a more detailed description, we brieflyrecall this algebra.First recall the spherical cyclotomic rational Cherednik algebra (see [34, Section 5]). LetΓ N := S N (cid:110) ( Z /l Z ) N be the wreath product of the symmetric group and the cyclic groupof order l . Denote a fixed generator of the i -th factor of ( Z /l Z ) N by α i . The group Γ N actson C (cid:104) ξ , · · · , ξ N , η , · · · , η N (cid:105) by α i ( ξ i ) = (cid:15)ξ i , α i ( ξ j ) = ξ j , α i ( η i ) = (cid:15) − η i , α i ( η j ) = η j ( i (cid:54) = j )with the obvious S N -action. Here (cid:15) denotes a primitive l -th root of unity.26 efinition 4.2.1 The cyclotomic rational Cherednik algebra H cycN,l for gl ( N ) is the quotientof the algebra C [ (cid:126) , (cid:126) , c , · · · , c l − ] (cid:104) ξ , · · · , ξ N , η , · · · , η N (cid:105) (cid:111) Γ N by the relations [ ξ i , ξ j ] = 0 = [ η i , η j ] , ( i, j = 1 , · · · , N ) (27)[ η i , ξ j ] = (cid:40) − (cid:126) + (cid:126) (cid:80) k (cid:54) = i (cid:80) l − m =1 s ik α mi α − mk + (cid:80) l − m =1 c m α mi if i = j − (cid:126) (cid:80) l − m =0 s ij (cid:15) m α i α − mj if i (cid:54) = j (28) Let e Γ N be the idempotent for the group Γ N . The spherical cyclotomic rational Cherednikalgebra is defined as SH cycN,l = e Γ N H cycN,l e Γ N . Theorem 4.2.2 [34, Theorem 1.1, Theorem 6.14, Theorem 1.5]1. Let A (cid:126) be the quantized Coulomb branch algebra of the gauge theory ( G, N ) = (GL( N ) , gl ( N ) ⊕ ( C N ) ⊕ l ) . If l > then A (cid:126) is isomorphic to SH cycN,l with an explicitly given correspon-dence between parameters in both algebras in loc.cit..2. Let A (cid:126) be the quantized Coulomb branch algebra for dim( V ) = N , dim( W ) = l . Then,we have a surjective homomorphism of algebras Ψ : Y l ( (cid:126)z ) → A (cid:126) . It follows from the above theorem that we have an epimorphism Y l ( (cid:126)z ) → SH cycN,l , for all N , such that e (cid:55)→ e Γ N (cid:32) N (cid:88) i =1 ξ li (cid:33) e Γ N .The relation of the shifted Yangian Y l ( (cid:126)z ) and the spherical cyclotomic rational Cherednikalgebra SH cycN,l is Y l ( (cid:126)z ) = lim N →∞ SH cycN,l . (29)Here according to [6, § S n to complex values of n . Then the index N becomes a centralelement in the limit algebra. See also [13] for an analogue of this limit process via Delignecategories. It is expected that the generators and relations in [26] give another presentationof this algebra.According to [6, Theorem 1.6.1] there is an epimorphism A l, (cid:126) ,(cid:15) → SH cycN,l , for all N .Thus, A l, (cid:126) ,(cid:15) is also isomorphic to the limit lim N →∞ SH cycN,l .This gives the isomorphisms Y l ( (cid:126)z ) ∼ = A l, (cid:126) ,(cid:15) ∼ = lim N →∞ SH cycN,l . In this paper, we focus on the case where l = 1, thus X l = C , and X [ k ] l is the Hilbertscheme Hilb k ( C ). The above-mentioned algebra A , (cid:126) ,(cid:15) [34] is isomorphic to Y ( z ).Let X , be the resolved conifold. The PT moduli space of X , space is identified withthe space of O ( − P to Hilb k ( C ) [51, § O X , → F of sheaves on X , , with the condition that the sheaf F is pure 1-dimensional (i.e., has no 0-dimensionalsubsheaves), and Cokernel( s ) is 0-dimensional. The sheaf F is shown to be an extension of27he structure sheaf of a curve in X , by a zero-dimensional sheaf supported on this curve[52, § X , is the total space of a rank-2 vector bundle on P , the projectionto P gives a map from the curve supporting F to P . Pushing-forward via the projection π : X , → P , the sheaf F gives a vector bundle on P the rank of which is equal tothe degree of the map from this curve to P . The action of π ∗ O X , on the vector bundle π ∗ F is equivalent to a pair of commuting O ( − F isequivalent to a section of the vector bundle π ∗ F , which generates the bundles under theaction of π ∗ O X , hence of the two Higgs fields. This is the same as a quasi-map from P to the Hilbert scheme of points on C .By Theorem 1.3.2, we have an action of the shifted Yangian Y ( z ) on the cohomologyof the moduli space of perverse coherent systems on X , with stability parameter on thePT-side of the imaginary root hyperplane. This can be viewed as an action of A l, (cid:126) ,(cid:15) on (cid:76) d H ∗ T ( M k,d,l , P ). and shifted Yangians In this section we prove Theorem 1.3.2(1). Consider the following framed quiver ˜ Q with potential (cid:3) n B B B I The potential is W = ˜ W = B ([ B , B ])A representation of the quiver Q of dimension n is stable framed if the following addi-tional property is satisfied C (cid:104) B , B , B (cid:105) I ( C ) = C n . Here C (cid:104) B , B , B (cid:105) is the algebra of non-commutative polynomials in the variables B , B , B .The set of stable framed representations of dimension n is denoted by M ( n, st . Itconsists of triples of n × n matrices B , B , B ∈ End( C n ), together with a cyclic vector v = I (1) ∈ C n . The group GL n acts by conjugation. Cyclicity of v means that it generates C n under the action of B (cid:48) i s .Note that the critical locus Crit( ˜ W ) of the potential tr ˜ W in M ( n, st consists of triplesof commuting matrices ( B , B , B ) satisfying the property that Im( I ) is a cyclic vectorof C n under the three matrices. Therefore, the quotient stack Crit( ˜ W ) / GL n is a schemeisomorphic to Hilb n ( C ). Here Hilb n ( C ) is the Hilbert scheme of n -points on C definedby Hilb n ( C ) = { J ⊂ C [ x , x , x ] | J is an ideal, and dim( C [ x , x , x ] /J ) = n } . W ) / GL n ∼ = Hilb n ( C ) is given such as follows. To an ideal J ⊂ C [ x , x , x ] let us choose an isomorphism C [ x , x , x ] /J ∼ = C n . Then the linear maps B , B , B ∈ End( C n ) are given by multiplications by x , x , x mod J . Furthermore I ∈ Hom( C , C n ) is I (1) = 1 mod J . Different choices of basis of C [ x , x , x ] /J give isomorphicrepresentations. Conversely, if we have ( B , B , B , I ) ∈ Crit( ˜ W ) / GL n , then, the ideal J isdefined as the kernel of the following map φ : C [ x , x , x ] → C n , f (cid:55)→ f ( B , B , B ) I (1) . Note that φ is surjective by the framed stability condition. Hence dim( C [ x , x , x ] /J ) = n .There is an action of ( C ∗ ) on Hilb n ( C ) induced by rescaling of the coordinates x , x , x . Fixed points (Hilb n ( C )) ( C ∗ ) of this action are in one-to-one correspondencewith 3-dimensional partitions of n .By [55, Theorem 5.1.1], we have an action of SH ( Q ,W ) on (cid:77) n ≥ H ∗ c, GL n × T ( M ( n, st , ϕ ˜ W Q ) ∨ ∼ = (cid:77) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ given by natural correspondences. In this section, we will construct an action of the shifted Yangian Y − ( z ) on (cid:77) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ . In general the scheme Hilb n ( C ) is singular which makes considerations more complicatedthan in the case n = 2. However, we have the following isomorphism of cohomology groups H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ ∼ = H ∗ c, GL n × T ( M ( n, st , ϕ ˜ W Q ) ∨ , where M ( n, st and M ( n, st / GL n are smooth.Let V denote the standard coordinate vector space of dimension n + 1. Fix ξ ⊂ V ,a one dimensional subspace and let V := V /ξ be the quotient. Consider the followingcorrespondence M ( n, n + 1 , stp (cid:117) (cid:117) q (cid:41) (cid:41) M ( n, st M ( n + 1 , st where M ( n, n + 1 , st = { ( B , B , B , I ) ∈ M ( n + 1 , st | B i ( ξ ) ⊂ ξ } . The parabolic subgroup P := { x ∈ GL n +1 = End( V ) | x ( ξ ) ⊂ ξ } acts on M ( n, n + 1 , st .The map p is given by ( B , B , B , I ) (cid:55)→ ( B , B , B , I ) mod ξ , where I mod ξ is thecomposition of I : C → V with the projection V (cid:16) V = V /ξ . The map q is the naturalinclusion.It induces a correspondence M ( n, n + 1 , st /P p (cid:116) (cid:116) q (cid:43) (cid:43) M ( n, st / GL n M ( n + 1 , st / GL n +1 ξ ⊂ V gives a tautological line bundle on the correspondence M ( n, n +1 , st /P , which will be denoted by L . Recall that the Levi subgroup of P is C ∗ × GL n where ξ above is the standard weight 1 representation of the C ∗ -factor. Hence, c ( L ) coincide withthe equivariant variable of the C ∗ -factor. For any rational function G ( x ) in one variable withcoefficients in H ∗ T (pt) we define the class G ( c ( L )) in a localization of H ∗ c,P × T ( M ( n, n +1 , st , ϕ ˜ W Q ) ∨ . Here the localization is taken with respect to H c,T × C ∗ (pt) ∼ = H c,T (pt)[ x ].Define the actions of the operators E ( G ) : H ∗ c, GL n × T ( M ( n, st , ϕ ˜ W Q ) ∨ loc → H ∗ c, GL n +1 × T ( M ( n + 1 , st , ϕ ˜ W Q ) ∨ loc , (30) F ( G ) : H ∗ c, GL n +1 × T ( M ( n + 1 , st , ϕ ˜ W Q ) ∨ loc → H ∗ c, GL n × T ( M ( n, st , ϕ ˜ W Q ) ∨ loc (31)by the following convolutions: E ( G ) := q ∗ ( G ( c ( L )) ⊗ p ∗ ) , F ( G ) := p ∗ ( G ( c ( L )) ⊗ q ∗ ) . Here the subscript loc means localization of H ∗ T (pt)-modules. Note that the fixed point lociwith respect to the T -actions are finite for all three spaces above. Hence the push-forward p ∗ is well-defined after passing to the localization despite of the fact that p is not a propermap.Let λ be a 3-dimensional Young diagram with n boxes, which we denote as a partition λ (cid:96) n . The dual of the compactly supported cohomology group H ∗ c, GL n × T ( M ( n, st , ϕ ˜ W Q ) ∨ loc has a basis given by { λ | λ (cid:96) n } . Let ( λ + (cid:4) ) (cid:96) ( n + 1) be the Young diagram obtained byadding a box (cid:4) to λ . We use the notation (cid:104) λ | ( E ( G )) | λ + (cid:4) (cid:105) for the coefficient of λ + (cid:4) inthe expansion of E ( G )( λ ). Similar convention is used for (cid:104) λ + (cid:4) | ( F ( G )) | λ (cid:105) .We denote by (cid:3) i,j,k the box in the 3-dimensional Young diagram λ which has coordinates( i, j, k ). Here we follow the French convention in writing the coordinates of boxes. Forexample, all boxes has non-negative coordinates and the corner box has coordinates (0 , , C ∗ ) naturally acts on C and H ∗ c, ( C ∗ ) (pt) = Q [ (cid:126) , (cid:126) , (cid:126) ] with (cid:126) i being theequivariant variable of the i th C ∗ -factor of ( C ∗ ) . We also have the 2-dimensional torus T ⊆ ( C ∗ ) preserving the canonical line bundle, and H ∗ c,T (pt) = Q [ (cid:126) , (cid:126) , (cid:126) ] / ( (cid:126) + (cid:126) + (cid:126) = 0).We also have an C ∗ acting on the framing, and let χ be the C ∗ -equivariant variable. For anybox (cid:3) i,j,k , we consider the following element x (cid:3) i,j,k := χ + i (cid:126) + j (cid:126) + k (cid:126) ∈ H ∗ c,T × C ∗ (pt).In the proof, we show this element is equal to the equivariant Chern root of some vectorbundle. Proposition 5.2.1
The matrix coefficients of the operators E ( G ) , F ( G ) in the basis of fixedpoints are as follows: (cid:104) λ | ( E ( G )) | λ + (cid:4) (cid:105) = Res z = x (cid:4) G ( z ) 1 z − x (cid:3) (cid:89) (cid:3) ∈ λ z − x (cid:3) ( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) (cid:104) λ + (cid:4) | ( F ( G )) | λ (cid:105) = G ( z ) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) ) z − x (cid:3) | z = x (cid:4) Here for a box (cid:3) or (cid:4) in a 3-dimensional Young diagram, x (cid:3) or x (cid:4) stands for the T -weightof the corresponding box, which is an element in H ∗ T ( pt ) .Proof. Both formulas can be derived from (59) and (56). Formula (56) gives (cid:104) λ + (cid:4) | ( f ( G )) | λ (cid:105) = G ( c ( L )) e ( T λ ( M ( n + 1 , st / GL n +1 ) e ( T λ,λ + (cid:4) ( M ( n, n + 1 , st /P )30ere e stands for the T -equivariant Euler class.In order to use (59) we recall that the correspondence G × P M ( n, n + 1 , st has a free G -action. The Levi subgroup of P is C ∗ × GL( n ). We denote the equivariant variablecorresponding to the C ∗ -action by z . Then at the point ( λ, λ + (cid:4) ) ∈ G × P M ( n, n + 1 , st the tautological bundle L becomes the standard 1-dimensional representation of C ∗ , andhence c ( L ) | ( λ,λ + (cid:4) ) = z . Now (59) gives (cid:104) λ | ( e ( G )) | λ + (cid:4) (cid:105) = Res z = x (cid:4) G ( z ) e ( T λ ( M ( n, st / GL n ) e ( T λ,λ + (cid:4) ( M ( n, n + 1 , st /P )Here e denote the T × C ∗ -equivariant Euler class, where C ∗ is the above-mentioned factorof the Levi subgroup of P .Next we calculate all the tangent spaces at the fixed points. We start by calculating thetangent space for M ( n + 1 , st / GL n . Let V be the tautological bundle of rank n + 1. Forsimplicity, for any vector bundle X , we write the dual bundle as X ∨ . Hence, for two vectorbundes X, Y we have Hom(
X, Y ) = X ∨ ⊗ Y .The tangent complex T ( M ( n + 1 , st / GL n +1 ) is given byEnd( V ) → (cid:16) C ⊗ End( V ) + Hom( W, V ) (cid:17) . Its class in the Grothendieck group of M ( n + 1 , st / GL n +1 is then given by T ( M ( n + 1 , st / GL n +1 ) = C ⊗ V ∨ ⊗ V + W ∨ ⊗ V − End( V ) . The tangent complex of the correspondence M ( n, n + 1 , st /P has the following classin the Grothendieck group, where we do not include the terms coming from obstructiontheory. T ( M ( n, n + 1 , st /P ) = C ⊗ P + W ∨ ⊗ V − P. Now formula (59) gives the following (cid:104) λ | ( E ( G )) | λ + (cid:4) (cid:105) = Res z = x (cid:4) G ( c ( L )) e ( T λ ( M ( n, st / GL n ) e ( T λ,λ + (cid:4) ( M ( n, n + 1 , st /P )= Res z = x (cid:4) G ( c ( L )) e ( C ⊗ V ∨ ⊗ V + W ∨ ⊗ V − End( V )) e ( C ⊗ P + W ∨ ⊗ V − P )= Res z = x (cid:4) G ( c ( L )) e ( V ∨ ⊗ ξ ) e ( C ⊗ V ∨ ⊗ ξ ) 1 e ( W ∨ ⊗ ξ )Here as before z is the equivariant Chern root of ξ . We take the basis of V to be { (cid:3) | (cid:3) ∈ λ } .Then the equivariant Chern root of the line spanned by (cid:3) is the T -weight of (cid:3) , which is x (cid:3) in our notation. The three loops B , B , B has weights (cid:126) , (cid:126) , (cid:126) respectively. Let χ bethe Chern roots of W . Thus, we have (cid:104) λ | ( E ( G )) | λ + (cid:4) (cid:105) = Res z = x (cid:4) (cid:0) G ( z ) (cid:89) (cid:3) ∈ λ z − x (cid:3) ( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) 1 z − χ (cid:1) (cid:104) λ + (cid:4) | ( F ( G )) | λ (cid:105) = G ( c ( L )) e ( T λ ( M ( n + 1 , st / GL n +1 ) e ( T λ,λ + (cid:4) ( M ( n, n + 1 , st /P )= G ( c ( L )) e ( C ⊗ V ∨ ⊗ V + W ∨ ⊗ V − End( V )) e ( C ⊗ P + W ∨ ⊗ V − P )= G ( c ( L )) e ( C ⊗ ξ ∨ ⊗ V ) e ( ξ ∨ ⊗ V )= G ( x (cid:4) ) (cid:89) (cid:3) ∈ λ ( x (cid:4) − x (cid:3) + (cid:126) )( x (cid:4) − x (cid:3) + (cid:126) )( x (cid:4) − x (cid:3) + (cid:126) ) x (cid:4) − x (cid:3) This completes the proof. (cid:4)
Now in the definition (30) above we take the function G ( x ) to be x i . For e i ∈ Y − ( z ),we define the action of e i on (cid:76) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ by the operator E ( x i ) above.Similarly, for f i ∈ Y − l ( (cid:126)z ), we define the action of f i by the operator F ( x i ) above. Theparameter z acts by χ .We define the action of ψ ( z ) on (cid:76) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ via the Chern polyno-mial of the following class in the Grothendieck group( q q ) V + ( q q ) V + ( q q ) V − q V + q V + q V. Here q , q , q are the three coordinate lines of C , considered as T -representations. Werefer to [55, § Proposition 5.2.2
Let λ (cid:96) n be a 3d partition. The eigenvalue of ψ ( z ) on [ λ ] equals (cid:89) (cid:3) ∈ λ z − x (cid:3) + (cid:126) z − x (cid:3) − (cid:126) z − x (cid:3) + (cid:126) z − x (cid:3) − (cid:126) z − x (cid:3) + (cid:126) z − x (cid:3) − (cid:126) . Definition 5.2.3
For a 3d Young diagram λ , an addible box is a box in another 3d Youngdiagram λ (cid:48) , such that λ (cid:48) is obtained by adding this box to λ . A removable box in λ is a boxsuch that λ with this box removed is still a 3d Young diagram. Lemma 5.2.4
Assume (cid:126) + (cid:126) + (cid:126) = 0 . Then h ( z ) | λ = z − x (cid:3) ψ ( z ) | λ is multiplication bya rational function, the poles of which are at the addible and removable boxes of λ .Proof. Follows from a careful examination of the cancellations. (cid:4) (21) (22) (23) (24)This follows from the epimorphism Y + − ( z (cid:48) ) → SH ( Q ,W ) , e n (cid:55)→ λ n ,Y −− ( z ) (cid:55)→ SH ( Q ,W ) , ( − n + l f n + l (cid:55)→ λ n and the compatibility of the COHA action and the Y + − ( z ) , Y −− ( z ) actions on the Hilbertscheme. 32 .3.2 Checking the relation (19)This is exactly the same as the affine Yangian case.Let e ( y ) be the generating series (cid:80) ∞ n =0 e n y n +1 , where y is a formal variable. The relation(19) is equivalent to the following relation (see [34]) ψ ( z ) e ( y ) = (cid:18) e ( y ) ψ ( z ) ( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) ) (cid:19) + (32)where () + denotes the part with positive powers in y . Here ( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) ) is regarded as an element in C [ (cid:126) , (cid:126) ][[ z − ]].The action of the series e ( y ) on (cid:76) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ is given by the operator E ( y − x ). By Proposition 5.2.1 the matrix elements of e ( y ) are given by (cid:104) λ | ( e ( y )) | λ + (cid:4) (cid:105) = y − x (cid:4) y Res z = x (cid:4) z − x (cid:3) (cid:89) (cid:3) ∈ λ z − x (cid:3) ( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) . Let A := Res z = x (cid:4) z − x (cid:3) (cid:81) (cid:3) ∈ λ z − x (cid:3) ( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) . Applying both sides of (32)to the partition λ , and looking at the coefficient of λ + (cid:4) , we obtain (cid:104) λ | ( ψ ( z ) e ( y )) | λ + (cid:4) (cid:105) = ψ ( z ) | λ + (cid:4) (cid:104) λ | ( e ( y )) | λ + (cid:4) (cid:105) = ψ ( z ) | λ + (cid:4) y − x (cid:4) y A We have following inductive formula of ψ ( z ) ψ ( z ) | λ + (cid:4) = ψ ( z ) | λ · z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) . Therefore, the left hand side of (32) becomes (cid:104) λ | ( ψ ( z ) e ( y )) | λ + (cid:4) (cid:105) = ψ ( z ) | λ · z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) z − x (cid:4) − (cid:126) z − x (cid:4) + (cid:126) y − x (cid:4) y A Applying the right hand side of (32) to the partition λ , we obtain the right hand side is y − x (cid:4) y ψ ( z ) | λ ( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − − (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) )( z − y − + (cid:126) ) A The equality (32) follows from applying the operator Res y =( x (cid:4) ) − y i on both sides. (25)Let us introduce the operator h i + j := [ e i , f j ] on (cid:76) n ≥ H ∗ c,T (Hilb n ( C ) , ϕ ˜ W Q ) ∨ . By theformula of Proposition 5.2.1, it is clear that [ e i , f j ] = [ e i (cid:48) , f j (cid:48) ] , if i + j = i (cid:48) + j (cid:48) . Define thegenerating series h ( z ) := 1 − σ (cid:88) i ≥ h i z − i − The relation (25) is equivalent to (see [61, Proposition 1.5]) σ ( w − z )[ e ( z ) , f ( w )] = h ( z ) − h ( w ) . h ( z ) = z − x (cid:3) ψ ( z ) . Let λ (cid:96) n . By Proposition 5.2.1, we have (cid:104) λ + (cid:4) | ( f j ) | λ (cid:105)(cid:104) λ | ( e i ) | λ + (cid:4) (cid:105) = Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )On the other hand, using again Proposition 5.2.1, we have (cid:104) λ − (cid:4) | ( e i ) | λ (cid:105)(cid:104) λ | ( f j ) | λ − (cid:4) (cid:105) = Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ − (cid:4) ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )= − Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )This implies that[ e i , f j ] λ = − (cid:88) addible boxes Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) − (cid:88) removable boxes Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )= − (cid:88) addible boxes ∪ removable boxes Res z = x (cid:4) z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )= Res z = ∞ z i + j z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )= Res z = ∞ z i + j z − x (cid:3) ψ ( z ) | λ . where the last equality follows from the residue theorem and Lemma 5.2.4.Then we have h ( z ) = 1 z − x (cid:3) (cid:89) (cid:3) ∈ λ ( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) + (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) )( z − x (cid:3) − (cid:126) ) . (33)We understand the equality (33) as the equality of all coefficients for powers z − ( i +1) , i ≥ Remark 5.3.1
More generally, consider the quiver with potential given by 3 loops quiver,with framing vector space C l . (cid:3) ln et M ( n, l ) st be the set of stable representations of the above quiver, let Coh l ( C , n ) = M ( n, l ) st / GL n . Let T l be the maximal torus of GL l acting by change of basis on the framing.By [55, Theorem 5.1.1], we have an action of SH ( Q ,W ) on (cid:77) n ≥ H ∗ c, GL n × T l × T ( M ( n, l ) st , ϕ ˜ W Q ) ∨ ∼ = (cid:77) n ≥ H ∗ c,T l × T (Coh l ( C , n ) , ϕ ˜ W Q ) ∨ given by natural correspondences. The same calculation as above shows that this actionextends to an action of the shifted Yangian Y − l ( z , z , · · · , z l ) . In this section we prove Theorem 1.3.2(2).
In this section, we are going to prove that on cohomology of the moduli space associatedwith a stability condition chosen on the PT side of the imaginary root hyperplane, there isan action of the equivariant spherical COHA H C . Furthermore, the above action can belifted to the Drinfeld double D ( SH C ). The latter gives rise to the action of the 1-shiftedaffine Yangian of gl (1).Let ( Q, W ) be the quiver with potential from (2). Let ˜ Q be the framed quiver (see (6))obtained by adding to the quiver Q a framing vertex ∞ and an extra edge i ∞ going from ∞ to 0. Here 0 denotes one of the vertices of Q . We extend the potential for ˜ Q by theformula ˜ W = W . For ˜ ζ = ( ζ , ζ , ζ ∞ ) a triple of real numbers and a triple of vector spaces( V , V , V ∞ ) associated to the following 3 vertices of extended quiver ˜ Q , a representation F = ( a , a , b , b , ι ) of ˜ Q with dimension vector (dim( V ) , dim( V ) , dim( V ∞ )), we define theslope of F to be θ ˜ ζ ( F ) := ζ dim( V ) + ζ dim( V ) + ζ ∞ dim( V ∞ )dim( V ) + dim( V ) + dim( V ∞ ) . A representation F of ˜ Q is said to be θ ˜ ζ -(semi-)stable if θ ˜ ζ ( F (cid:48) ) < ( ≤ ) θ ˜ ζ ( F )for any nonzero proper ˜ Q -subrepresentation F (cid:48) of F .For any pair of real numbers ζ = ( ζ , ζ ), define ζ ∞ := − ζ dim( V ) − ζ dim( V ) sothat we have a triple ˜ ζ = ( ζ , ζ , ζ ∞ ). We say the representation F with dimension vector(dim( V ) , dim( V ) ,
1) is ζ -stable, if it is θ ˜ ζ -stable. Note that in the special case ζ = ζ = − Q is θ ˜ ζ -stable if every ˜ Q -subrepresentation of ( V , V , V ∞ ) containing theframing V ∞ is either V ∞ or the entire ( V , V , V ∞ ). For this choice of ζ we also call a ζ -stablerepresentation cyclicly stable .Let M ζ ( v , v ) denote the moduli space of ζ -stable ˜ Q -representations, such that dim( V ) = v , dim( V ) = v , dim( V ∞ ) = 1. The space (cid:96) v ,v M ζ ( v , v ) is isomorphic to the modulispace M ζ of ζ -stable perverse coherent systems in the sense of Nagao and Nakajima [43, § ζ = ( ζ , ζ ) as the stability parameter, and the 2-dimensional real vectorspace { ζ = ( ζ , ζ ) | ζ i ∈ R } the space of stability conditions.Let us recall some terminologies from the Introduction. A stability parameter ζ is called generic if every ζ -semi-stable point is in fact ζ -stable. The locus of non-generic stability35onditions is a union of hyperplanes (which in this case is a union of lines) called walls .The complement of the hyperplane arrangement is the union of open connected componentswhich we call chambers . The moduli space M ζ in the case of generic ζ depends only on thechamber that contains ζ . This is the case we are mostly interested in.The chamber structure of the space of stability conditions for X , considered here isdescribed in [43, Figure 1]. In particular, one wall is given by the imaginary root hyperplane ζ + ζ = 0. We consider the chambers above this imaginary root hyperplane, i.e. thosewhere ζ + ζ >
0. Those walls are given by( m + 1) ζ + mζ = 0 , where m ∈ Z ≥ .Consider a generic stability condition ζ m which is above this wall but close to it. We usea different description of the moduli space M ζ m more relevant for our purpose. For this,we consider the following quiver ˜ Q ζ with potential ˜ W ζ (see [43, Figure 9]). Note that ˜ Q ζ is also obtained from Q by adding the vertex ∞ which by an abuse of terminology we stillrefer to as the framing, but the set of arrows depends on m . We consider representation of˜ Q ζ on a triple of vector spaces ( V , V , V ∞ ) associated to the 3 vertices as before. Again arepresentation of ˜ Q ζ is said to be cyclicly stable if every ˜ Q ζ -subrepresentation of ( V , V , V ∞ )containing the framing V ∞ is either V ∞ or the entire ( V , V , V ∞ ) [43]. The moduli space M m of cyclically stable ˜ Q ζ -representations with V ∞ = C is again isomorphic to M ζ m . • • a a b b V V (cid:3) · · · q , · · · , q m · · · p , · · · , p m +1 The quiver ˜ Q ζ m with potential ˜ W ζ given by a b a b − a b a b + p b q + p ( b q − b q ) + · · · + p m ( b q m − b q m − ) − p m +1 b q m There is a 4-dimensional torus ( C ∗ ) acting by scaling ( a , a , b , b ). However, the 1-dimensional subtorus T = { ( t, t, t − , t − ) ∈ ( C ∗ ) } acts trivially. Then we have the actionof the 3-dimensional subtorus of ( C ∗ ) with the first coordinate being 1 ∈ C ∗ . Note thatthis 3-dimensional torus maps isomorphically to ( C ∗ ) /T under the quotient map. Wewrite an element in this subtorus as (1 , t, q, h ) ∈ ( C ∗ ) . We consider T ⊆ ( C ∗ ) consistingof (1 , t, q, h ) with tqh = 1. The action of T on X preserves the canonical line bundle. Inparticular, the induced action on the moduli spaces preserves the potential.The weights of p i and q i are determined by the conditions wt ( p i ) = wt ( q − i ) , wt ( p i +1 ) = wt ( q − i ) t − . Therefore, we have wt ( b ) = 1 , wt ( b ) = t, wt ( a ) = q, wt ( a ) = t − q − , wt ( p i ) = t − i , wt ( q i ) = t i − . M m contains an open subset M m singled out by the condition that b is surjective. The natural flat pull-back to the open subset induces the isomorphism H ∗ c,T ( M m , ϕ tr ˜ W ζ Q ) ∨ → H ∗ c,T ( M m , ϕ tr ˜ W ζ Q ) ∨ (up to localization). Theorem 6.1.1
1. There is a natural action of H C on H ∗ c,T ( M m , ϕ tr ˜ W ζ Q ) ∨ .2. This action extends to an action of Y ( z ) . We prove here only part 1 of the statement. In order to prove part 2 it suffices to find thecommutation relations of the raising and lowering operators acting on H ∗ c,T ( M m , ϕ tr ˜ W ζ Q ) ∨ ,which we leave until § Proof.
We keep the notation of Section 3.6. To avoid confusion, here in the proof, wedenote by W in place of V ∞ the vector space at the framing vertex of the quiver. Let Surj ( V (cid:48) , V (cid:48) ) = { x ∈ Hom( V (cid:48) , V (cid:48) ) | x is surjective } be the set of surjective morphisms from V (cid:48) to V (cid:48) . Let M ( V (cid:48) , W ) := (cid:16) Surj( V (cid:48) , V (cid:48) ) ⊕ Hom( V (cid:48) , V (cid:48) ) ⊕ Hom( V (cid:48) , V (cid:48) ) ⊕ ⊕ Hom( V (cid:48) , W ) ⊕ m +1 ⊕ Hom(
W, V (cid:48) ) ⊕ m (cid:17) Note that the quotient
Surj ( V (cid:48) , V (cid:48) ) / GL( V (cid:48) ) is a Grassmannian. Consider the followingspace:GL( V (cid:48) ) × P M ( V (cid:48) , W ) = { ( ξ, b , b , a , a ) | ξ ⊂ V (cid:48) , b : V (cid:48) (cid:16) V (cid:48) , b : V (cid:48) → V (cid:48) , a i : V (cid:48) → V (cid:48) , i = 1 , } Denote by M ( V (cid:48) , W ) st, ⊂ M ( V (cid:48) , W ) the stable locus.We define the following spaces (cid:102) M = { ( ξ , ξ , b , b , a , a ) | ξ ⊂ V (cid:48) , ξ ⊂ V (cid:48) , b : V (cid:48) (cid:16) V (cid:48) , such that b ( ξ ) ⊂ ξ , b is an isomorphism , b : V (cid:48) → V (cid:48) , a i : V (cid:48) → V (cid:48) , i = 1 , } ,G × P Z = { ( ξ , ξ , b , b , a , a ) ∈ (cid:102) M | b ( ξ ) ⊂ ξ , a i ( ξ ) ⊂ ξ , i = 1 , } . We have a closed embedding i : G × P Z (cid:44) → (cid:102) M and an affine bundle projection π : (cid:102) M →
GL( V (cid:48) ) × P M ( V (cid:48) , W ) .We have the following correspondence G × P Z φ (cid:119) (cid:119) (cid:31) (cid:127) i (cid:47) (cid:47) (cid:102) M π (cid:47) (cid:47) GL( V (cid:48) ) × P M ( V (cid:48) , W ) ψ (cid:15) (cid:15) G × P (cid:18) Rep( Q , , V ) ×M ( V (cid:48)(cid:48) , W ) (cid:19) M ( V (cid:48) , W ) (34)We have the following isomorphisms G × P (Rep( Q , , V ) × M ( V (cid:48)(cid:48) , W ) st, )GL( V (cid:48) ) ∼ = GL ( V (cid:48) ) × P (Rep( Q , , V ) × M ( V (cid:48)(cid:48) , W ) st, ) P ∼ = GL ( V (cid:48) ) × P (Rep( Q , , V ) × M ( V (cid:48)(cid:48) , W ) st, )GL( V ) × GL( V (cid:48)(cid:48) ) ∼ = GL ( V (cid:48) ) × P (Rep( Q , V ) × ( M ( V (cid:48)(cid:48) , W ) st, / GL( V (cid:48)(cid:48) ))37aking the quotient of the correspondence (34) by GL( V (cid:48) ), we have the correspondence GL ( V (cid:48) ) × P ( Z /P ) φ (cid:116) (cid:116) (cid:31) (cid:127) i (cid:47) (cid:47) (cid:102) M / GL( V (cid:48) ) π (cid:47) (cid:47) M ( V (cid:48) , W ) /P ψ (cid:15) (cid:15) GL ( V (cid:48) ) × P (cid:18) Rep( Q , V ) ×M ( V (cid:48)(cid:48) , W ) / GL( V (cid:48)(cid:48) ) (cid:19) M ( V (cid:48) , W ) / GL( V (cid:48) )(35)The action is defined by ψ ∗ ◦ ( π − ) ∗ ◦ i ∗ ◦ φ ∗ , (36)on the stable locus of (35), where π − exists since π is an affine bundle. (cid:4) In this section, we describe the tangent spaces to the torus fixed points in the open locus M m ⊂ M m , as well as the correspondence used in (35). Again in this section to avoidconfusion we denote by W the vector space at the framing vertex of the quiver in place of V ∞ .Recall that M m ( V ) is the quotient M ( V , W ) st, / (GL( V ) × GL( V )), where M ( V , W ) st, is the cyclically stable locus of M ( V , W ) = (cid:16) Surj( V , V ) ⊕ Hom( V , V ) ⊕ Hom( V , V ) ⊕ ⊕ Hom( V , W ) ⊕ m +1 ⊕ Hom(
W, V ) ⊕ m (cid:17) . The quotient
Surj ( V , V ) / GL( V ) is a Grassmannian, denoted by Grass( v , V ), pa-rameterizing quotients b : V (cid:16) V with the dimension of V being v . Therefore, M ( V , W ) / GL( V ) is an affine bundle over the Grassmannian with fiber F := Hom( V , V ) ⊕ Hom( V , V ) ⊕ ⊕ Hom( V , W ) ⊕ m +1 ⊕ Hom(
W, V ) ⊕ m . The tangent space of M m ( V ) is an extension of the above vector space F by the tangentspace of Grass( v , V ), taking into account of the GL( V )-action. Again we denote by0 → V → V → V → . the tautological sequence of vector bundles on Grass( v , V ). Then, the tangent space ofGrass( v , V ) is Hom( V , V ). The tangent bundle T to M ( V , W ) / GL( V ) has a filtrationwith associated graded given by Gr ( T ) = H om( V , V ) ⊕ ( H om( V , V ) ⊕ H om( V , V ) ⊕ ⊕ H om( V , W ) ⊕ ( m +1) ⊕ H om( W, V ) ⊕ m )The action of GL( V ) induces a Lie algebra map from End( V ) to the sheaf of sections of T endowed with the standard Lie bracket on vector fields.The tangent bundle of the quotient M ( V , W ) / (GL( V ) × GL( V )) is given by the cok-ernel of the above action map.We now compute the tangent space of the correspondence Z st, / ( P × P ) in (35). Inthe above notation let us consider the space (cid:101) Fl := { ( ξ , ξ , b ) | ξ ⊂ V (cid:48) , ξ ⊂ V (cid:48) , b : V (cid:48) (cid:16) V (cid:48) such that b ( ξ ) ⊂ ξ , b is an isomorphism } / GL( V (cid:48) )38hich parameterizes diagrams of vector spaces as follows0 (cid:47) (cid:47) ξ (cid:47) (cid:47) V (cid:48) (cid:47) (cid:47) V (cid:47) (cid:47) (cid:47) (cid:47) ξ b ∼ = (cid:79) (cid:79) (cid:47) (cid:47) V (cid:48) b (cid:79) (cid:79) (cid:79) (cid:79) (cid:47) (cid:47) V b (cid:79) (cid:79) (cid:79) (cid:79) (cid:47) (cid:47) V (cid:48) (cid:63)(cid:31) (cid:79) (cid:79) V (cid:63)(cid:31) (cid:79) (cid:79) where dim( V i ) = v i and dim( V (cid:48) i ) = v i + 1, for i = 1 ,
2. Note that ( G × P Z ) / GL( V (cid:48) ) is anaffine bundle over (cid:101) Fl.The fiber of the affine bundle is isomorphic to P ( V (cid:48) , V (cid:48) ) ⊕ P ( V (cid:48) , V (cid:48) ) ⊕ ⊕ H om( V (cid:48) , W ) ⊕ ( m +1) ⊕ H om( W, V (cid:48) ) ⊕ m where P ( V (cid:48) , V (cid:48) ) := { x ∈ H om( V (cid:48) , V (cid:48) ) | x ( ξ ) ⊂ ξ } ,P ( V (cid:48) , V (cid:48) ) := { x ∈ H om( V (cid:48) , V (cid:48) ) | x ( ξ ) ⊂ ξ } . Let Fl( v + 1 , , V (cid:48) ) be the flag variety of pairs consisting of a quotient b : V (cid:48) (cid:16) V (cid:48) with dim( V (cid:48) ) = v + 1 and a line ξ ⊆ V (cid:48) . Then the base space (cid:101) Fl maps to Fl( v + 1 , , V (cid:48) ),where the map is given by forgetting the subspace ξ . This map realizes (cid:101) Fl as an affinebundle with each fiber isomorphic to Hom( ξ , V (cid:48) ).The tangent bundle of the flag variety Fl( v +1 , , V (cid:48) ) has the same class in the Grothendieckgroup as H om( V , V (cid:48) ) ⊕ H om( ξ, V ) . Hence there is a filtration on the tangent sheaf of ( G × P Z ) / GL( V (cid:48) ) with associated gradedbeing H om( V , V (cid:48) ) ⊕ H om( ξ, V ) ⊕ P ( V (cid:48) , V (cid:48) ) ⊕ P ( V (cid:48) , V (cid:48) ) ⊕ ⊕ H om( V (cid:48) , W ) ⊕ ( m +1) ⊕ H om( W, V (cid:48) ) ⊕ m ⊕ Hom( ξ , V (cid:48) ) . Note that the affine bundle π : (cid:102) M →
GL( V (cid:48) ) × P M ( V (cid:48) , W ) (see (34)) has fiber isomorphicto Hom( ξ , V (cid:48) ).In the action defined by (36), we only focus on the contribution of the tangent sheaf of( G × P Z ) / GL( V (cid:48) ) without the term Hom( ξ , V (cid:48) ).As before, the action of GL( V (cid:48) ) induces a Lie algebra mapEnd( V (cid:48) ) → T ( G × P Z ) / GL( V (cid:48) ) . The tangent bundle of the quotient Z st, / ( P × P ) = ( G × P Z ) / (GL( V (cid:48) ) × GL( V (cid:48) ) is againgiven by the cokernel of the above action map.39 .3 Fixed points In this subsection we will use the terminology and some results of [43]. The following arethe pictures for the empty room configurations (ERC) for the finite type pyramid partitionswith length 3 and length 4 ([43, Figure 12]).The ERC for finite typepyramid partitions with length 3 The ERC for finite typepyramid partitions with length 4 (37)In general, for ERC for the finite type pyramid partitions with length m , there are 1 × m black stones on the first layer, 1 × ( m −
1) white stones on the second layer, 2 × ( m − × ( m −
2) white stones on the fourth, and so on until we reach m × Definition 6.3.1
A finite type pyramid partition of length m is a finite subset Π of theERC of length m in which, for every stone in Π , the stones directly above it are also in Π . Proposition 6.3.2 [43, Proposition 4.14] The set of T -fixed points in M m is isolated andparameterized by finite type pyramid partitions of length m . By our conventions on the torus action, the weights of the black stones on the top layerin the empty room partition are 1 , t, t , · · · , t m − . The weights of the black stones on the layer 3 are qt, ht ; qt , ht ; · · · , qt m − , ht m − . The weights of the black stones on the layer 5 are q t , qht , h t , q t , qht , h t , · · · , q t m − , h t m − . The last one is the 2( m −
1) + 1 = (2 m − q m − t m − , q m − ht m − , q m − h t m − · · · , h m − t m − . By the definition of the torus weight of b , for each pair of black and white stones as inFigure (38), the weight of the white stone is the same as that of the black stone.40et us consider a pair consisting of a black stone and a white stone such that the blackstone is right above of the white stone (as in the the following picture ): (38)We say that a pair of stones as in (38) is removable , if after removing the pair, we still get apyramid partition. Similarly, we say that a pair of stones as in (38) is addible to a pyramidpartition, if they are not in the pyramid partition but after adding the pair to the pyramidpartition, we still get one.Let us analyze what the addible and removable pairs are for a given pyramid partition.For this we use the following terminology in order to describe the relative position of onestone with respect to another one: frontback( t )left( q ) right( h )Furthermore, in this terminology the word above will mean up with respect to the papersurface , and below will mean down with respect the paper surface .A pair of black and white stones is addible, if the following conditions hold for the blackstone in this pair:1. (Black condition) There is a black stone in front of it in the pyramid partition.2. (White condition) If there are white stones above it in the empty room partition (seethe following picture), then these white stones have to be in the pyramid partition.or orFor a pair of black and white stones let us analyze the above two conditions for the blackstone in this pair. In order to avoid confusion we refer to the black stone in the pair as black-one . If the black condition holds for the black-one, then there is another black stone,which we call black-zero , which is positioned in front of the black-one. Moreover, the whitecondition holds automatically for black-zero. Unless one of the white stones above black-zerois the end of a chain of the whites, the white condition also holds for black-one.Similarly, a pair of black and white stones as in (38) is removable, if the pair is at the41nd of a chain like this ... No black stone below itand there is no black stone below the last white stone. Proposition 6.3.3
Assume t + q + h = 0 . Poles of the function h ( z ) =( − |{ black only stones }| ( − m +1 ( z − χ − mt ) (cid:89) b ∈ black stones and white stones ( z − x b + t )( z − x b + q )( z − x b + h )( z − x b − t )( z − x b − q )( z − x b − h ) (cid:89) b ∈ black stones only ( z − x b )( z − x b + q )( z − x b + h )( z − x b − t ) are at the addible and removable pairs.Proof. By definition the function h ( z ) is product of factors coming from each stone. Letus analyze the contribution of each chain of stones. From the above discussion of addibleand removable pairs, in order to prove the proposition it suffices to show that h ( z ) has thefollowing zeros and poles. Here the formula of h is given as a product over all the stonesin λ . We calculate the contribution of one chain of stones in the product, by taking theproduct of the corresponding factors over all the stones in this chain.Zeros:1. For a chain of black stones, we want the product of factors from this chain to havetwo zeros above the end stone. That is, assume the end stone is x , then we want twozeros at x − h and x − q respectively.2. For a chain of white stones we want the product of factors from this chain to havetwo zeros, under the stones in front of the end stone of the chain. That is, let the endstone of this chain be x , then the two zeros are at x + t + q and x + t + h . ( The stonein front of x is x + t . The two places x + t + q and x + t + h , where we want the zerosto be, are the two stones underneath x + t .3. Similarly, a chain of black stones should contribute one zero at the beginning stone.Poles:1. A chain of black stones (with or without white stones) should contribute a pole at theend stone of the chain. 42. A chain of black stones and white stones should contribute a pole at the end stone.Now we verify that h ( z ) does have the aforementioned zeros and poles from the con-tribution of each chain of stones. Suppose we have a chain of black stones (with markedweights) as on the following picture xxtxt ... xt a − The function h ( z ) has the following factor h ( z ) = a − (cid:89) i =0 ( z − x − it )( z − x − it + q )( z − x − it + h )( z − x − it − t )= z − xz − x − at a − (cid:89) i =0 ( z − x − it + q )( z − x − it + h ) . Thus, the pole of h ( z ) is xt a . It is clear that xt a is an addible place.Suppose we have a chain of black and white stones (with marked weights) as on thefollowing picture xxtxt To illustrate the idea of the proof, we compute this case in the explicit example as in thepicture. This can be made in general. In this case, the function h ( z ) has the following factor( z − x )( z − x + q )( z − x + h )( z − x − t ) ( z − x − t + t )( z − x − t + q )( z − x − t + h )( z − x − t − t )( z − x − t − q )( z − x − t − h )( z − x − t + t )( z − x − t + q )( z − x − t + h )( z − x − t − t )( z − x − t − q )( z − x − t − h )= ( z − x ) ( z − x − t + q )( z − x − t + h )( z − x − t )( z − x − t )Thus, the poles of h ( z ) are x + 2 t and x + 3 t . It is clear that xt is an addible place, and xt is a removable place. (cid:4) . 43 .4 Raising and lowering operators Taking into account weights and using the canonical isomorphism Hom( V , V ) (cid:39) V ∨ ⊗ V ,the matrix coefficients of raising operator can be calculated using the Appendix § A.1 in thefollowing way. e (cid:18) tV ∨ ⊗ V ⊕ qV ∨ ⊗ V ⊕ hV ∨ ⊗ V ⊕ V ∨ ⊗ V ⊕ ⊕ mi =1 ( t i − W ∨ ⊗ V ) ⊕ ⊕ m +1 i =1 ( t − i V ∨ ⊗ W ) − End( V ) (cid:19) e (cid:18) tP ( V (cid:48) , V (cid:48) ) ⊕ qP ( V (cid:48) , V (cid:48) ) ⊕ hP ( V (cid:48) , V (cid:48) ) ⊕ V ∨ ⊗ V (cid:48) ⊕ ξ ∨ ⊗ V ⊕ ⊕ mi =1 ( t i − W ∨ ⊗ V (cid:48) ) ⊕ ⊕ m +1 i =1 ( t − i V ∨ ⊗ W ) − End( V (cid:48) ) (cid:19) = 1 e ( tV ∨ ⊗ ξ + qV ∨ ⊗ ξ + hV ∨ ⊗ ξ ) e ( V ∨ ⊗ ξ + ξ ∨ ⊗ V ) e ( ξ ∨ ⊗ V ) 1 e ( ⊕ mi =1 ( t i − W ∨ ⊗ ξ )) . We denote a pyramid partition by λ , and we denote by λ + (cid:4) the one obtained from λ byadding a pair (38) denoted by (cid:4) . Let { x w | w = white stone } be the Chern roots of V , andlet { x b | b = black stone } be the Chern roots of V . Let χ be the Chern root of W . Thenthe above formula gives (cid:104) λ | e i | λ + (cid:4) (cid:105) = Res z = (cid:4) ( − |{ black only stones }| z i (cid:89) b ∈ black stones z − x b z − x b − t (cid:89) w ∈ white stones z − x w − q )( z − x w − h ) m (cid:89) i =1 z − χ − ( i − t Similarly, the lowering operator is given by e (cid:18) tV (cid:48)∨ ⊗ V (cid:48) ⊕ qV (cid:48)∨ ⊗ V (cid:48) ⊕ hV (cid:48)∨ ⊗ V (cid:48) ⊕ V ∨ ⊗ V (cid:48) ⊕ ⊕ mi =1 ( t i − W ∨ ⊗ V (cid:48) ) ⊕ ⊕ m +1 i =1 ( t − i V (cid:48)∨ ⊗ W ) − End( V (cid:48) ) (cid:19) e (cid:18) tP ( V (cid:48) , V (cid:48) ) ⊕ qP ( V (cid:48) , V (cid:48) ) ⊕ hP ( V (cid:48) , V (cid:48) ) ⊕ V ∨ ⊗ V (cid:48) ⊕ ξ ∨ ⊗ V ⊕ ⊕ mi =1 ( t i − W ∨ ⊗ V (cid:48) ) ⊕ ⊕ m +1 i =1 ( t − i V ∨ ⊗ W ) − End( V (cid:48) ) (cid:19) = e ( tξ ∨ ⊗ V + qξ ∨ ⊗ V + hξ ∨ ⊗ V ) e ( ξ ∨ ⊗ V ) e ( ⊕ m +1 i =1 ( t − i ξ ∨ ⊗ W ))The matrix coefficient is given by (cid:104) λ + (cid:4) | f j | λ (cid:105) = z j (cid:89) b ∈ black stones ( z − x b + q )( z − x b + h ) (cid:89) w ∈ white stones z − x w + tz − x w m +1 (cid:89) i =1 ( − z + χ − (1 − i ) t ) | z = (cid:4) h ( z ) = ( − |{ black only stones }| (cid:89) b ∈ black stones ( z − x b )( z − x b + q )( z − x b + h ) z − x b − t (cid:89) w ∈ white stones z − x w + t ( z − x w )( z − x w − q )( z − x w − h ) m (cid:89) i =1 z − χ − ( i − t m +1 (cid:89) i =1 ( − z + χ − (1 − i ) t )= ( − |{ black only stones }| ( − m +1 ( z − χ − mt ) (cid:89) b ∈ black stones and white stones ( z − x b + t )( z − x b + q )( z − x b + h )( z − x b − t )( z − x b − q )( z − x b − h ) · (cid:89) b ∈ black stones only ( z − x b )( z − x b + q )( z − x b + h )( z − x b − t )Similar to the proof in § e i , f j , and h ( z ) satisfy the relations of Y ( z ),where z = χ + mt . This concludes the proof of Theorem 6.1.1(2). Let us now move to the discussion of sheaves supported on toric divisors inside of the toricCalabi-Yau 3-fold X = X m,n . As we mentioned in the Introduction, in physics languagethese divisors correspond to configurations of D4-branes. We also mentioned that they areexpected to give rise to shifts of real roots of the affine Yangian. The shifts are determinedby the intersection numbers of the corresponding divisor with the projective lines P ’s inside X = X m,n . As we will see in the examples below this is indeed the case.For any embedded smooth rational curve P ⊆ X its formal neighbourhood is isomorphicto the one of the embedded smooth rational curve P in either X , or X , . For any toricdivisor D ⊆ X , the formal completion at P is isomorphic to the one for the standard toricdivisors in either X , or X , . Therefore, we study these cases as the basic building blocks. Consider the Calabi Yau 3-fold T ∗ P × C = Tot( O P ( − ⊕ O P ). We take the effectivedivisor D to be fiber of T ∗ P × C → P over the north pole or over the south pole of P .This is illustrated by the toric diagram (7).Let us discuss one of these diagrams (say D is the fiber of T ∗ P × C → P over thenorth pole of P ), since the other is similar. Note that the ( x, z )-coordinate hyperplanedivisor C xz ⊆ C xyz is invariant under the natural Z / C / ( Z / T ∗ P × C → C / ( Z / D , the fiber of T ∗ P × C → P over the north pole of P . Therefore we candescribe the moduli space of stable framed sheaves on T ∗ P × C that supported on D suchas follows.Start with the quiver with potential ( Q fr , W fr ), which is quiver description for themoduli space of framed sheaves on C that supported on the divisor C xz . Note that the45raming ( Q fr , W fr ) has to do with the toric divisor C xz , which is different from the framing( ˜ Q , ˜ W ) considered in § N K B B B (cid:3) N I J The potential is W fr = B (cid:0) [ B , B ] + I J (cid:1) . Explicitly, the group Z / Z acts on C in such a way that its generator σ ∈ Z / Z isrepresented by the matrix σ (cid:55)→ − − The torus ( C ∗ ) ⊂ ( C ∗ ) acts on C by rescaling the coordinates by t , t , t subject to theCalabi-Yau condition t t t = 1. The above formula gives an embedding of the group Z / Z into the subtorus ( C ∗ ) . Under the Z / Z action the weights of the arrows of Q fr are givenby B (cid:32) − , B (cid:32) − , B (cid:32) I (cid:32) +1 , J (cid:32) − . We decompose the vector spaces
K, N into eigenspaces with eigenvalues 1 , − K = K ⊕ K N = N ⊕ N Setting N to be zero, we obtain the framed quiver Q from (16) (with the square noderemoved). with potential W given by W = − ( B ˜ B ˜ B − ˜ B ˜ B B + ˜ B B B − B B ˜ B ) + B I J Framed stability condition implies that the image of I is invariant under B , B and ˜ B , ˜ B and generates the whole space.We now use the dimensional reduction of a quiver with potential and a cut (see [35,Section 4.8], [9, Appendix]). We take the cut to be the set of arrows consisting of B , ˜ B .Below we present the dimensionally reduced quiver with relations in our case, referring thereader to [68, Appendix] for a description of the COHA for the dimensionally reduced quiverwith relations for a general cut.Let Q (cid:48) := Q \{ B , ˜ B } be the quiver obtained by removing the two arrows B , ˜ B . A pic-ture of the quiver obtained is (8). Then, Rep( Q, V , V ) = Rep( Q (cid:48) , V , V ) × (Hom( V , V ) ⊕ V , V )). We have the following diagram Z × A n (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15) Rep(
Q, V , V ) π (cid:15) (cid:15) (cid:38) (cid:38) Z (cid:31) (cid:127) (cid:47) (cid:47) Rep( Q (cid:48) , V , V ) (cid:47) (cid:47) ptwhere the affine space A n is Hom( V , V ) ⊕ Hom( V , V ), and Z is the algebraic variety Z = { z ∈ Rep( Q (cid:48) , V , V ) | tr( W )( z, l ) = 0 , for all l ∈ Hom( V , V ) ⊕ Hom( V , V ) } = { z ∈ Rep( Q (cid:48) , V , V ) | ∂W∂B ( z ) = 0 , ∂W∂ ˜ B ( z ) = 0 } = Rep( C [ Q (cid:48) ] /J, V , V ) , where J is the ideal generated by ∂W∂B = − ( ˜ B ˜ B − B ˜ B ) + I J , ∂W∂ ˜ B = − ( − ˜ B B + B B ) . (39)Then [35, Section 4.8] and [9, Appendix] give an isomorphism H ∗ c, GL( V ) × GL( V ) × T (Rep( Q, V , V ) , ϕ trW ) ∼ = H ∗ c, GL( V ) × GL( V ) × T ( Z × A n , Q ) . Note that the variety Z is the representation variety of the quiver Q (cid:48) with relations onthe vector spaces ( V , V ). The quiver Q (cid:48) with relation (39) is known as a chainsaw quiver (defined in [20]).The following definition of Yangian of gl ( n ) can be found in [44] and [19]. Definition 7.1.1
The Yangian of gl ( n ) is generated by { e ( r ) i , f ( r ) i , g ( r ) j | ≤ i < n, ≤ j ≤ n, r ≥ } . Define the following generating series of the generators e i ( z ) := (cid:88) r ≥ e ( r ) i z − r , f i ( z ) := (cid:88) r ≥ f ( r ) i z − r , g j ( z ) := 1 + (cid:88) r ≥ g ( r ) j z − r . hey are subject to the following relations [19, Lemma 2.48]. [ g i ( z ) , g j ( w )] = 0; (40)( z − w )[ g i ( z ) , e j ( w )] = ( δ i,j − δ i,j +1 ) g i ( z )( e j ( z ) − e j ( w )); (41)( z − w )[ g i ( z ) , f j ( w )] = ( δ i,j +1 − δ i,j )( f j ( z ) − f j ( w )) g i ( z ); (42)[ e i ( z ) , f j ( w )] = 0 , if i (cid:54) = j ; (43)( z − w )[ e i ( z ) , f i ( w )] = g i +1 ( w ) g i ( w ) − g i +1 ( z ) g i ( z ) ; (44)( z − w )[ e i ( z ) , e i ( w )] = − ( e i ( z ) − e i ( w )) ; (45)( z − w )[ e i ( z ) , e i +1 ( w )] = − e i ( z ) e i +1 ( w ) + e i ( w ) e i +1 ( w ) − [ e (1) i +1 , e i ( w )] + [ e (1) i +1 , e i ( z )]; (46)[ e i ( z ) , e j ( w )] = 0 , if | i − j | >
1; (47)[ e i ( z ) , [ e i ( z ) , e j ( w )]] + [ e i ( z ) , [ e i ( z ) , e j ( w )]] = 0 , if | i − j | = 1; (48)( z − w )[ f i ( z ) , f i ( w )] = ( f i ( z ) − f i ( w )) ; (49)( z − w )[ f i ( z ) , f i +1 ( w )] = f i +1 ( w ) f i ( z ) − f i +1 ( w ) f i ( w ) + [ f i ( w ) , f (1) i +1 ] − [ f i ( z ) , f (1) i +1 ]; (50)[ f i ( z ) , f j ( w )] = 0 , if | i − j | >
1; (51)[ f i ( z ) , [ f i ( z ) , f j ( w )]] + [ f i ( z ) , [ f i ( z ) , f j ( w )]] = 0 , if | i − j | = 1 . (52)Let r = ( r , · · · , r n ) ∈ N n , we define the r -shifted Yangian of (cid:91) gl ( n ) to be the algebragenerated by { e ( r )¯ i , f ( r )¯ i , g ( r )¯ j | ¯ i, ¯ j ∈ Z /n Z , r ≥ } , subject to the same relations (40)-(52), except the relation (44) is modified to be( z − w )[ e ¯ i ( z ) , f ¯ i ( w )] = g ¯ i +1 ( w ) g ¯ i ( w ) − ( z r i +1 − r i ) g ¯ i +1 ( z ) g ¯ i ( z ) . (53)We have identified the cohomology H ∗ c, GL( V ) × GL( V ) × T (Rep( Q, V , V ) , ϕ trW Q ) with thecohomology of the chainsaw quiver H ∗ c, GL( V ) × GL( V ) × T ( Z × A n , Q ) via the dimensionalreduction. It is shown in [46] that the shifted quantum toroidal algebra of gl ( N ) acts onthe K-theory of the chainsaw quiver variety. Following the same calculation, with K -theoryreplaced by cohomology, we obtain Theorem 7.1.2
The (1 , -shifted Yangian of (cid:91) gl (2) acts on (cid:77) V ,V H ∗ c, GL( V ) × GL( V ) (Rep( Q, V , V ) , ϕ trW Q ) ∨ . .2 Few more examples Consider the CY 3-fold T ∗ P × C = Tot( O P ( − ⊕ O P ), and the effective divisor O ( − P . This is illustrated in the following toric diagram.1The quiver description is given by the following figure: • B • ˜ B B B ˜ B ˜ B (cid:3) I J The potential is given by the formula: B ˜ B ˜ B − ˜ B ˜ B B + ˜ B B B − B B ˜ B + B I J . The framed stability condition says that any subspace containing the image of I andinvariant under the action of B , B , ˜ B , ˜ B coincides with the whole space (i.e. the imageof 1 is a cyclic vector). Then using the dimension reduction with respect to the loops, thecorresponding moduli spaces are the Nakajima quiver varieties with framing 1. It is knownthat there is an action of the affine Yangian of sl (without shifting). Let X be the resolved conifold Tot( O P ( − ⊕ O P ( − P . We denote them by O ( − and O ( − respectively. This is illustrated in the following toric diagram.149he corresponding quiver is • • a a b b (cid:3) i j with the potential a b a b − a b a b + b ij. Applying the dimensional reduction to the above quiver with potential with the cut con-sisting of the edge b , we obtain the quiver with relations from [45], which described stableframed sheaves on the blowup of C . We postpone the study of shifted Yangian action tothe future. -folds In this subsection we explain how to put examples from Sections 5, 6, 7 into a more generalframework.Fix a quiver with potential (
Q, W ). Assume Q is symmetric. (It follows from thisassumption that the quadratic form χ R | Γ l defined in [35] is symmetric.) In [35, Section 6.2,Question 6.2 ], Kontsevich-Soibelman conjecture that there exits a Z -graded Lie algebra g (BPS Lie algebra), such that the (non-equivariant) COHA of ( Q, W ) is isomorphic to theuniversal enveloping algebra of the current algebra g ⊗ C [ T ] of g . This conjecture was provedlater by Davison and Meinhardt [10]. In the special case when ( Q, W ) is the “tripled” quiverof a simply-laced Kac–Moody Lie algebra as in [24], it follows from the recent paper [11] ofDavison that the zeroth piece of the perverse filtration on the COHA of (
Q, W ) is isomorphicto the universal enveloping algebra of g . The latter contains the upper-triangular subalgebraof the Kac–Moody Lie algebra. As a consequence, if the tripled quiver Q has exactly oneloop at each vertex, then the ( i, j )-th entry of the Cartan matrix of the Kac-Moody Liealgebra is equal to the number of arrows of Q between i and j if i (cid:54) = j . The diagonal entriesare all equal to 2.We do not know an explicit description of the root system of the BPS Lie algebra forgeneral symmetric quivers with potential. Nevertheless, assuming furthermore that thepotential W is homogeneous, the three examples 3.3.2 3.3.3 and 3.3.4, suggest that theroot system of the BPS Lie algebra should contain a sub root system, the Cartan matrix of50hich is given as (cid:104) α i , α j (cid:105) = − { h : i → j, h ∈ H } if i (cid:54) = j ;2 if i = j , and i has one loop;0 if i = j , and i has no loops. (54)This sub root system is that of a Lie subalgebra of the BPS Lie algebra, corresponding tothe spherical subalgebra of the COHA. The Drinfeld double of the entire COHA is expectedto be a Cartan doubled Yangian of the root system of the entire BPS Lie algebra.Now go back to geometry. Let X be a toric CY 3-fold as in §
1. We are in the settingof § f : X → Y is a resolution with Y affine and the fibers of f are at most1-dimensional. As mentioned in the introduction of [42, pg.2] that such affine toric CY 3fold Y can be classified in terms of the lattice polygon in R . The classification is given bythe family Y = Y m,n , together with finitely many exceptional cases. When Y = Y m,n , thecorresponding quiver is symmetric [42, Section 1.2].Let Mod f - A ⊆ Mod- A be the full subcategory consisting of coherent sheaves of A -modules whose set-theoretical supports as coherent sheaves on Y are zero-dimensional. Sim-ilarly, we have the abelian category A f , which is equivalent to Mod f - A under the functor R Hom X ( P , − ).Let K ( A ) Z be the Grothendieck group of A f . Let { S i } i ∈ I be a collection of pairwisedistinct simple objects in A f . For each S i , let [ S i ] ∈ K ( A ) Z be its class in the Grothendieckgroup, which we call a simple root . Using this terminology, the set { [ S i ] } i ∈ I of simple rootsis an integer basis of K ( A ) Z . Thus the latter can be thought of as a root lattice . Wecall a simple root [ S i ] is bosonic if the simple object S i in D b Coh( X ) is bosonic, that isExt ∗ ( S i , S i ) = H ∗ ( P ). We call a simple root [ S i ] is fermionic if the simple object S i isfermionic, that is, Ext ∗ ( S i , S i ) = H ∗ ( S ). This terminology is explained in § K ( A ) Z (cid:104) [ S i ] , [ S j ] (cid:105) = − dim(Ext ( S i , S j )) if i (cid:54) = j ;2 if i = j , and i is bosonic;0 if i = j , and i is fermionic.Indeed, in all these cases, the algebra A has a grading coming from the C ∗ -action on X which contracts X to its special fiber, so that the tilting bundle is C ∗ -equivariant. Thequiver with potential ( Q, W ), whose Jacobian algebra is A , can be chosen so that thevertices are labeled by I , and the number of arrows from i to j is dim Ext ( S i , S j ) for i, j ∈ I [2, Theorem 3.1 and proof]. Hence, the matrix (54) can be expressed in this way.In particular, this gives rise to a free abelian group K ( A ) Z endowed with a basis and aninteger pairing. In general, this resembles a part of some ”Cartan data”, possibly that of asub root system of the BPS Lie algebra. Example 8.1.1
In Example 3.3.3, reading off the sequence of bosonic and fermionic simpleobjects by following the diagram (2) from the bottom to the top (or equivalently from thetop to the bottom that produces a reflected Dynkin diagram), one recovers different rootsystems of the gl ( m | n ) algebra [18]. The above-defined root system thus coincides with theroot system of (cid:98) gl ( m | n ) . This configuration has been considered by various authors including[39, 54, 62]. n Example 3.3.4, the generalized root system is expected to be of “double affine” type A n . The construction of § A n [7]. The moduli space of perverse coherent system of X in the sense of § ζ , as well as the algebraic cycle χ . Here the stabilityparameter is understood in the same sense as in §
6. We denote by X the space of stabilityparameters of the perverse coherent system. It can be identified with K ( D b Coh( X )) ∨ R . Wesay that ζ ∈ X is generic if it lies in the interior of a chamber. Each generic ζ determinesan abelian subcategory A ζ , which depends only on the chamber containing ζ . In the casewhen X = X m,n this abelian subcategory comes from a tilting bundle explicitly constructedin [42, § χ is an algebraic cycle of the form N X + (cid:80) ri =1 N i D i where N i ∈ N , and each D i is a toric divisor. Taking its class in the Borel-Moore homologywith integral coefficients, we get [ χ ] ∈ H BM ∗ ( X, Z ). As we only consider algebraic cycles,we have [ χ ] ∈ H BM ( X, Z ) ⊕ H BM ( X, Z ).We can think of [ χ ] as a coweight (i.e. an integer functional on the root lattice) inthe following way. The 0-th homology of X is one dimensional, let [pt] be its basis. Let { [ C ] , · · · , [ C m + n − } be the classes of curves C i (cid:39) P which give a basis of H ( X, Z ). Let ch : K ( A ζ ) → H ∗ ( X )be the homological Chern character map [8, Section 5.9]. Each simple object S i ∈ K ( A ζ )gives rise to a class ch ([ S i ]) ∈ H ∗ ( X ). The shift is then determined by the pairing (cid:104) χ, ch [ S i ] (cid:105) := − [ χ ] ∩ ch [ S i ] , for all i ∈ I, (55)where ∩ : H BM6 −∗ ( X ) ⊗ H ∗ ( X ) → Z is the intersection pairing [8, Proposition 2.6.18]. Sincethe set of simple objects { S i | i ∈ I } gives rise to the set of simple roots spanning the rootlattice, the homology class χ gives rise to a coweight via the pairing (55).Let us illustrate the above discussion in the following three examples.1. Consider the PT moduli space of X , . There are two simple objects { S , S } = {O P ( m ) , O P ( m + 1)[ − } They generate the abelian category (heart of the corresponding t -structure) A + m . Theirclasses in K ( A + m ) gives the two simple roots. Consider the short exact sequence0 → O → O ( m ) → C ⊕ m → P , where C is the skyscraper sheafat 0 ∈ P . Thus, in K ( A + m ), we have [ O P ( m )] = [ O ] + m [ C ] . We calculate thehomological Chern character map using the devissage principle [8, Proposition 5.9.3],we have ch [ S ] = ch [ O P ( m )] = [ C ] + m [pt] ,ch [ S ] = ch [ O P ( m + 1)[ − − [ C ] − ( m + 1)[pt] , where [ C ] is the P class in H ( X ). In this case χ = [ X , ]. The intersection pairingis given by [ X , ] ∩ C = 0 , [ X , ] ∩ [pt] = 1. Note that the imaginary root δ = ch [ S ] + ch [ S ]. This implies that (cid:104) χ, δ (cid:105) = − χ ∩ δ = 1 . Thus, we have the +1 shift of the imaginary root, cf. in § X = X , , and D be the fiber of the vector bundle X , → P over eitherthe north pole or the south pole of P , cf. § { S , S } = {O P ( − m ) , O P ( − m − − } in K ( A m ), which corresponds to the two simple roots. Consider the short exactsequence 0 → O ( − m ) → O → C ⊕ m →
0, where C is the skyscraper sheaf at 0 ∈ P .Thus, in K ( A − m ), we have [ O P ( − m )] = [ O ] − m [ C ] . Applying the homologicalChern character map, we have ch [ S ] = ch [ O P ( − m )] = [ C ] − m [pt] ,ch [ S ] = ch [ O P ( − m − − − [ C ] + ( m + 1)[pt] , where [ C ] is the P class in H ( X ). In this case χ = [ D ]. The intersection pairing isgiven by [ D ] ∩ [ C ] = 1 , [ D ] ∩ [pt] = 0 . This implies that (cid:104) χ, ch [ S ] (cid:105) = − χ ∩ ch [ S ] = − , (cid:104) χ, ch [ S ] (cid:105) = − χ ∩ ch [ S ] = 1 . Thus, the shifts of the two simple roots are −
1, 1 respectively, cf. § C . In this case χ = [ C ]. The imaginary root δ isgiven by the class [pt]. The intersection pairing is given by (cid:104) χ, δ (cid:105) = − [ C ] ∩ δ = − . Thus we have the − § Let D b Coh( X ) be the bounded derived category of coherent sheaves on X , and D Z Coh( X )be the derived category of 2-periodic complexes of coherent sheaves on X (see [4] for thedefinition). Then we have a functor F : D b Coh( X ) → D Z Coh( X )by taking direct sum of odd and even degree complexes. We expect that there is a geomet-ric construction of the Cartan doubled Yangian Y ∞ , constructed by taking cohomology ofcertain moduli stack of objects in D Z Coh( X ) endowed with Hall multiplication, similarlyto [4].Let A ⊂ D b Coh( X ) be the heart of D b Coh( X ) corresponding to a tilting bundle asin §
1. Denote by H A the COHA associated to A f constructed as in §
1. We expect thechoice of the t-structure defines an algebra embedding H A → Y ∞ , as well as a triangulardecomposition Y ∞ ∼ = H A ⊗ H ⊗ H A [1] , where the subalgebra H is a polynomial algebra C [ (cid:126) , (cid:126) ][ ψ i,k ] i ∈ I,k ∈ Z , with infinite variableslabeled by I × Z .For each i ∈ I , let (cid:104) S i (cid:105) ⊆ A be the Serre subcategory of A generated by S i . Then, theCartan doubled Yangian Y α i ∞ , associated to D Z ( (cid:104) S i (cid:105) ), is a subalgebra of Y ∞ .53et δ be the imaginary root defined in the same way as in [31, Chapter 5]. That is, δ isa root, but δ is not in the Weyl group orbit of the simple roots. We expect that a quotientof Y ∞ is isomorphic to Y (cid:126) , (cid:126) , (cid:126) ( (cid:91) gl (1)), which corresponds to the root δ of Y ∞ .We are interested in representations of Y ∞ on the cohomology of the moduli spacesof perverse coherent systems on X . In the case X = X m,n , everything can be spelledout in the language of quivers with potential using [42]. Let M χζ be the moduli space ofstable perverse coherent systems. There is a potential function W defining the symmetricobstruction theory of M χζ . Let us consider the vector space V := H ∗ c ( M χζ , ϕ W C ) ∨ . Thenwe expect the following. Conjecture 8.2.1
The algebra Y ∞ is isomorphic to D ( SH ) constructed in § Y ∞ on V in agreement with the general philosophy of [58]. Moreover, thisaction factors through an action of the shifted affine Yangian, with the shift determined by [ χ ] and ζ as above. Starting from the heart of the t -structure coming from a tilting bundle as above, one canconstruct the braid group action on D b Coh( X ) induced by mutations with respect to sub-categories (cid:104) S i (cid:105) , i ∈ I . We expect that on one hand this braid group action induces an actionon D Z Coh( X ) and hence on the algebra Y ∞ . On the other hand it induces an action on K ( D b Coh( X )), and therefore an action on X . We expect that this action gives rise to anaction of the Weyl group of the generalized root system described in § K ( D b Coh( X )) determines a hyperplane arrangement in X . The braid group acts byreflections with respect to these hyperplanes.In particular, for two adjacent chambers separated by a root hyperplane, the reflectionwith respect to the root hyperplane sends the heart of the t -structure associated to onechamber to that of the other one. These two chambers determine two Borel subalgebras of Y ∞ which differ by the action of a Weyl group element.As an illustration we explicitly calculate below the effect of the action of the affine Weylgroup on Iwahori subgroups in the well-known case of (cid:99) sl . Then we discuss a similar picturein an example of a particular Calabi-Yau 3-fold in Remark 8.3.2 Example 8.3.1
Let G := SL ( C ) and T be its maximal torus of diagonal matrices. Denoteby B the Borel subgroup consisting of upper triangular matrices, and B − the one consistingof lower triangular matrices.Let G (( t )) := SL (( t )) , U − (( t )) := { (cid:20) ∗ (cid:21) } ⊂ G (( t )) be the corresponding groups overthe field of Laurent series C (( t )) . The Coxeter presentation of the affine Weyl group is (cid:99) W = { s , s | s = s = 1 } . It is isomorphic to Z (cid:110) Z , where Z is generated by s and Z is generated by s s . In particular, any element can be uniquely written as w = ( s s ) m or w = s ( s s ) m for some m ∈ Z . In G (( t )) , we have a subgroup, which is the Tits extensionof (cid:99) W , with generators n s , n s defined as n s := (cid:20) − t − t (cid:21) , n s := (cid:20) − (cid:21) . Thus, ( n s n s ) m = (cid:20) t − m t m (cid:21) For any Weyl group element w = ( s s ) m (resp. w = s ( s s ) m ), we write the correspondingelement in G (( t )) as w = ( n s n s ) m (resp. w = n s ( n s n s ) m ) slightly abusing the notation. e have the following three subgroups of G (( t )) ,the positive Iwahori I + = { g ∈ G [[ t ]] | g (0) ∈ B } , the level 0 Iwahori I = T [[ t ]] U − (( t )) , the negative Iwahori I − = { g ∈ G [[ t − ]] | g ( ∞ ) ∈ B − } . Following [41], for c ∈ C , k ∈ Z , we define X α + kδ ( c ) = (cid:20) ct k (cid:21) , X − α + kδ ( c ) = (cid:20) ct k (cid:21) . The following points are in I + : X − α + δ (1) = (cid:20) t (cid:21) , X α (1) = (cid:20) (cid:21) and they determine I + in the following sense. We take X − α + δ (1) and X α (1) to be theset of simple roots. Then, any positive root will be X α + kδ ( c ) if α = α and k ∈ Z ≥ or if α = − α and k ∈ Z > . We have I + = { a = (cid:20) a a a a (cid:21) | a i ∈ C [[ t ]] , a ∈ t C [[ t ]] , det ( a ) = 1 } = ( (cid:89) k> ,c X − α + kδ ( c )) T ([[ t ]])( (cid:89) k ≥ ,c X α + kδ ( c )) . Similarly we can determine I − and I in terms of their simple roots. For any group element g ∈ G (( t )) , the conjugations gI + g − , gI − g − , gI g − are also Iwahori subgroups. Then,we have the following 3 sets, each consisting a collection of Iwahori subgroups of G (( t )) ,[41] G (( t )) /I + , the thin or positive level affine flag variety, G (( t )) /I , the semi-infinite or level 0 affine flag variety, G (( t )) /I − , the thick or negative level affine flag variety.With the structure of Weyl group above, we have the following orbit decompositions G (( t )) = (cid:97) x ∈ (cid:99) W I + xI + , G (( t )) = (cid:97) y ∈ (cid:99) W I yI + , G (( t )) = (cid:97) z ∈ (cid:99) W I − zI + , where in each decomposition the set of orbits are labelled by (cid:99) W . In particular, two Iwa-hori subgroups from two different components are not conjugate to each other. A directcomputation determines the conjugations of I + in terms of simple roots wX − α + δ (1) w − = (cid:40) X − α +(2 m +1) δ (1) if w = ( n s n s ) m ; X α +(2 m +1) δ ( − if w = n s ( n s n s ) m . wX α (1) w − = (cid:40) X α − mδ (1) if w = ( n s n s ) m ; X − α − (2 m ) δ ( − if w = n s ( n s n s ) m .Furthermore, wX − α + δ (1) w − and wX α (1) w − determine another Iwahori subgroup w ( I + ) which is conjugate to I + . A similar calculation can be done for I − and I . Remark 8.3.2
In the example X = C × T ∗ P the root system from § (cid:99) sl .We now describe a bijection between I + -orbits on the thin affine flag variety G (( t )) /I + with the t-structures A + m parametrized by the chambers on the PT side of the imaginary roothyperplane. This bijection is so that the simple roots of I ∈ G (( t )) /I + coincide with thesimple objects in A + m . To describe this bijection, we use the description of the orbits fromthe decomposition G (( t )) = (cid:96) x ∈ (cid:99) W I + xI + and the Tits extension of Weyl group elements asin Example 8.3.1. For simplicity, we can choose m > when w = ( n s n s ) m , and m < ,when w = n s ( n s n s ) m . (The other choice corresponds to the flop of X .) Therefore, theabove computation shows that the simple objects { ch [ S ] = [ C ] − m [ pt ] , ch [ S ] = − [ C ] + ( m + 1)[ pt ] } in A + m match up with the simple roots { X α − mδ (1) , X − α +( m +1) δ (1) } . This bijection is equivariant with respect to the affine Weyl group actions.Similarly, the I + -orbits in G (( t )) /I − are in natural bijection with the t -structures A − m ,so that the simple roots of I ∈ G (( t )) /I − coincide with the simple objects in A − m .The positive level Iwahori and the negative ones are not related by the action of the Weylgroup. Instead they differ by a homological shift [1] , and hence are isomorphic as abstractalgebras. The level zero Iwahori subgroup and its conjugations (as elements in G (( t )) /I )do not occur here in the context of the COHA of perverse coherent sheaves.In general, however, we do not expect two Iwahori subalgebras to be isomorphic in thecase when they come from two t-structures that do not differ by the action of a braid groupelement. A similar idea of exploring COHA’s associated to two different t-structures can befound in [60] in an attempt to “categorify the wall-crossing”. We also mention the idea ofthe construction of the “derived COHA” which contains all “root COHA’s” corresponding torays in R having the common vertex at the point (0 , , which was proposed by Kontsevichand Soibelman in 2012 (unpublished). A Residue pushforward formula in critical cohomology
A.1 Convolution operators on fixed points
Let X be a complex algebraic H -variety, where H is a complex affine algebraic group.We assume that F := X H consists of isolated fixed points F = { F i } and | F | < ∞ . Let i : F (cid:44) → X be the inclusion. Denote by ϕ f the vanishing cycle complex associated to thefunction f on X . Note that, by assumption, we have H ∗ c,H ( F, ϕ i ◦ f C ) ∨ = H ∗ c,H ( F, C ) ∨ = ⊕ i H ∗ c,H ( F i , C ) ∨ The classes i ∗ (1 i ) form a basis of H ∗ c,H ( X, ϕ f C ) ∨ loc . Let us consider the map on localizations i ∗ : H ∗ c,H ( F, ϕ i ◦ f C ) ∨ loc → H ∗ c,H ( X, ϕ f C ) ∨ loc induced by i ∗ : H ∗ c,H ( F, ϕ i ◦ f C ) ∨ → H ∗ c,H ( X, ϕ f C ) ∨ . N F X be the normal bundle of F inside X , and e ( N F X ) be its Euler characteristic. Wealso have i ∗ i ∗ α = e ( N F X ) , and i − ∗ β = i ∗ βe ( N F X )for α ∈ H ∗ c,H ( F, ϕ i ◦ f ) ∨ , and β ∈ H ∗ c,H ( X, ϕ f ) ∨ (see, e.g., [9, Proposition 2.16(1)]).Consider the correspondence F W ⊂ i W q F (cid:125) (cid:125) p F (cid:33) (cid:33) F X ⊂ i X W q (cid:124) (cid:124) p (cid:33) (cid:33) F Y ⊂ i Y X Y
We have i − X ∗ q ∗ p ∗ i Y ∗ (1 F Y ) = q F ∗ i − W ∗ p ∗ i Y ∗ (1 F Y ) (56)= (cid:88) i i ∗ W e ( N F W W ) p ∗ i Y ∗ (1 F Y )= (cid:88) i ( p F ) ∗ i ∗ Y e ( N F W W ) i Y ∗ (1 F Y )= (cid:88) i ( p F ) ∗ e ( N F Y Y ) e ( N F W W ) A.2 Jeffery-Kirwan residue formula in critical cohomology
Here we discuss the Jeffery-Kirwan residue localization formula in the setting of criticalcohomology. We follow the proof of Guillemin and Kalkman [27], also taking into accountthe description of the vanishing cycle complex of Kapranov [32].
A.2.1 De Rham model of equivariant critical cohomology
First, we recall the de Rham model of critical cohomology following [32], taking into accountthe group action as in [24, § Y be smooth complex algebraic variety endowed withan algebraic C ∗ -action as well as with a C ∗ -invariant regular function f . In order to use the C ∞ -de Rham complexes, we consider sheaves in the analytic topology. Recall that a finesheaf is a sheaf with partition of unity, and that a fine sheaf is acyclic under direct imagefunctor, and hence for any sheaf F the higher derived direct image can be calculated byapplying the direct image functor to a resolution of F by fine sheaves. Recall that Poincar´eLemma implies that a resolution of the constant sheaf C Y by fine sheaves is given by the deRham complex (Ω ∗ Y , d ). Here we use subscript notation for the sheaves, and Ω ∗ ( Y ) for thespace of global sections. The space of compactly supported sections is denoted by Ω ∗ c ( Y ).Let ϕ f ( C Y ) denote the sheaf of vanishing cycles. A complex representing ϕ f ( C Y ) was givenin [32]. Let ˜ d := d + df ∧ be the “twisted” differential on Ω Y . The complex (Ω Y , ˜ d ) is quasi-isomorphic to ϕ f ( C Y ). Let π Y : Y → pt be the structure map. Recall that Y is smoothand hence the Verdier dualizing sheaf D Y is C Y homologically shifted to the dimension of57 . The cohomology of Y valued in the vanishing cycle complex H ∗ c ( Y, ϕ f C ) ∨ := π Y ∗ ϕ f D Y is then calculated as the cohomology of the complex obtained by taking global sections ofthe de Rham complex representing ϕ f C Y [dim Y ].Let π : X → Y be a proper flat map of smooth complex algebraic varieties, and f aregular function on Y . We have a map π ∗ : H ∗ c ( X, ϕ f C ) ∨ → H ∗ c ( Y, ϕ f C ) ∨ induced byapplying π ∗ to the map of complexes on Yπ ∗ (Ω X , ˜ d ) → (Ω ∗ Y , ˜ d ) , where the individual map of sheaves π ∗ Ω ∗ X → Ω ∗ Y is given by integration along fibers of π .The algebraic C ∗ -action induces an action of the maximal torus S ⊆ C ∗ when Y isconsidered as a smooth manifold. In the calculations below we follow [24, § S -action via the Cartan complex. Note that the Borel construction usedin [9] differs from the Cartan model by a completion. The cohomology H ∗ c,S ( Y, ϕ f C ) ∨ canbe calculated via the de Rham model Ω ∗ S ( Y, ϕ f ) := Ω ∗ ( Y ) S ⊗ C [ x ] with the differential˜ d = d + i ( v ) ⊗ x + ∧ df . Here d is the de Rham differential on forms, v is the vector fieldinduced by the S -action, and d S := d + i ( v ) ⊗ x is the differential calculating the usualCartan model of equivariant cohomology. A.2.2 An explicit formula of the Kirwan map
Let X be a smooth complex algebraic variety with a C ∗ -action of dimension n . In orderto consider de Rham complex of the GIT quotient X// ξ C ∗ for a character ξ of C ∗ , weuse Kirwan’s theorem to identify X// ξ C ∗ with the Hamiltonian reduction. More precisely,the GIT quotient X// ξ C ∗ is diffeomorphic to µ − ( ξ ) /S where µ is the moment map ofthe compact Lie group S ⊆ C ∗ . Let X = µ − ([ ξ, ∞ )), which is a smooth manifold withboundary and an S -action that preserves the boundary.Abusing the notation, for any smooth manifold N with boundary, endowed with an S -action as well as with a smooth function f which preserves the boundary and f , we denotethe complex Ω ∗ ( N ) S ⊗ C [ x ] with the differential ˜ d = d + i ( v ) ⊗ x + ∧ df by Ω ∗ S ( N, ϕ f ).In the setup above, let X be a smooth manifold with boundary endowed with a smooth S -action. Assume the S -action on ∂X is locally free, so that ∂X/S ∼ = M is again a smoothalgebraic variety. Let f be an S -invariant regular function on X . Then, the restriction of f to ∂X is an S -invariant smooth function, which is again obtained from pulling back aregular function on M , which by an abuse of notation we still denote by f .We define the Kirwan map κ : H ∗ S ,c ( X, ϕ f ) ∨ → H ∗ c ( M, ϕ f ) ∨ to be the following mapat the level of de Rham complexes. We have the restriction of forms i ∗ : Ω ∗ S ( X, ϕ f ) → Ω ∗ S ( ∂X, ϕ f ), as well as the pullback π ∗ : Ω ∗ S ( ∂X, ϕ f ) → Ω ∗ ( M, ϕ f ). Then, for any closedform α ∈ Ω ∗ ( M, ϕ f ), κ ( α ) is the class of any form γ with the property that π ∗ ( γ ) has thesame class as i ∗ α . That is, i ∗ α − π ∗ ( γ ) is a boundary cycle. The calculation of Guilleminand Kalkman [27] gives an explicit formula for γ , which we recall here.Let θ be a S -invariant one form such that i ( v ) θ = 1, which is well-defined on thecomplement of X S . Consider the formal expression ν = θx + dθ + θ ∧ df . From definition, wehave ˜ d ( θ ) = x + dθ + θ ∧ df , in particular ˜ d ( x + dθ + θ ∧ df ) = 0. Therefore, ˜ d ( ν ) = ˜ d ( θ ) x + dθ + θ ∧ df = 1. Let α ∈ Ω ∗ S ( X, ϕ f ) c such that ˜ d ( α ) = 0. Then, ˜ d ( α ∧ ν ) = α . We consider ν = θx (cid:80) n ≥ ( − dθ − θ ∧ dfx ) n as a Laurent power series in x with coefficients in Ω( X ) S .Now let { X i } i =1 ,...,N be the connected components of the fixed point set, and let U i bepairwise disjoint tubular neighbourhoods of X i in X such that U i ∩ ∂X = ∅ . Let i ∗ k : X k → X
58e the embedding. Assume deg α = dim ∂X −
1. Then, by Stocks theorem, we have0 = (cid:90) X α = (cid:90) X ˜ dν ∧ α = N (cid:88) k =1 (cid:90) ∂U k θαx + dθ + θ ∧ df + (cid:90) ∂X θαx + dθ + θ ∧ df . We confine ourselves to the case when X S = (cid:116) X k is a finite set. We now follow thecalculation of [28, § (cid:82) ∂U k θαx + dθ + θ ∧ df converges to i ∗ k αe ( ν k ) where ν k isthe tangent space of X k and e ( ν k ) is the equivariant Euler class, which can be calculatedas the product of the weights with multiplicities on ν k . Indeed, as in loc. cit. , we have α = f ( x ) + d S β where f ( x ) is the restriction of α at X k and β ∈ Ω S ( U k ), hence ν ∧ α = f ( x ) ν + β + d S ( ν ∧ β ). Integrating on ∂U k , the second term vanishes; the third term isof higher order in terms of the radius of U k ; the first term becomes f ( x ) (cid:82) ∂U k ν , where thecalculation in local coordinate in loc. cit. shows that (cid:82) ∂U k ν = x n e ( ν k ) . To summarize, weobtain (cid:90) ∂X θαx + dθ + θ ∧ df = N (cid:88) k =1 i ∗ k αx n e ( ν k ) (57)under the assumption that deg α = dim ∂X −
1. By degree consideration [69, page 21], wealso have α ∧ ν = ν + βx − with β = Res x = ∞ ( α ∧ ν ). In particular, α = ˜ d ( ν ) + i ( v ) β with i ( v ) β = π ∗ γ for some γ ∈ Ω ∗ ( M, ϕ f ). In other words, κ ( α ) = Res x = ∞ π ∗ θαx + dθ + θ ∧ df . (58) A.2.3 Conclusion
Assume that X is endowed with an action of S × S , where S is a compact Lie groupwhich acts in a Hamiltonian way. Let { E m } be a system of S -spaces the limit of whichis the Borel construction of ES . Following the consideration in [69, § α ∈ H ∗ c,S × S ( M, ϕ f C ) ∨ of a certain degree i , we choose a representative in Ω iS ( E m , ϕ ) with i = dim X m −
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