The q-AGT–W Relations Via Shuffle Algebras

Neguţ, Andrei

doi:10.1007/s00220-018-3102-3

Cited by 48 publications

(135 citation statements)

References 36 publications

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“…When S = A 2 , Theorem 1.2 was proved in [24] using the fact that the group K M is (generically) an irreducible module for the W -algebra. We do not have this feature for a general surface S, and we have little control on the size of the abelian group K M .…”

Section: Introductionmentioning

confidence: 99%

Shuffle algebras associated to surfaces

Neguţ

2019

Sel. Math. New Ser.

123

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We define an integral form of the deformed W -algebra of type gl r , and construct its action on the K-theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S, under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W -algebras by generators and relations is a new feature. Physically, this action encodes the AGT correspondence for 5d supersymmetric gauge theory on S×circle.

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Section: Introductionmentioning

confidence: 99%

Shuffle algebras associated to surfaces

Neguţ

2019

Sel. Math. New Ser.

123

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“…where ε forgets F • , while δ is the embedding of the locus F ′ • = F ′′ • . Lemma 6.2 of [26] states that we have the following equality of K-theory classes on V i;s;j : Meanwhile, we claim that the residue at 0 vanishes if k > r j . This is because e [s;i) (y) and f [s;j) (y) are regular at 0, while and ψ + s (y) has a pole of order exactly r s .…”

Section: 2mentioning

confidence: 82%

“…In [26], we introduced a new approach to proving (1.1), using explicit correspondences to define the W -algebra action. This allowed us to deform the AGT correspondence from cohomology to algebraic K-theory:…”

Section: Introductionmentioning

confidence: 99%

Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

Neguţ

2019

Lecture Notes in Mathematics

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We construct explicit elements W k ij in (a completion of) the shifted quantum toroidal algebra of type A, and show that these elements act by 0 on the K-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements W k ij will be related to q-deformed W -algebras of type A for arbitrary nilpotent, which would imply a q-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.Proposition 4.5. Under the action of Theorem 4.3, the elements (2.21) act on K r as the operators (4.7) with the same name, hence the abuse of notation.

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“…Here K is the full central extension of the double loop algebra g Q0 [s ±1 , t ±1 ]. Beyond these case, shuffle algebras tend to be rather difficult to study and the algebraic structure of H T * (M Q ) (or H T * (Λ Q )) is still mysterious (see, however [Ne1], [Ne2] for some important applications to the geometry of instanton moduli spaces in the case of the Jordan or affine type A quivers).…”

Section: Let Us Say a Few Words About How The Multiplication Map Hmentioning

confidence: 99%

“…[MS] , [GN]), algebraic geometry and mathematical physics of the instanton spaces on A 2 (e.g. [SV3], [SV4], [Ne2]), combinatorics of Macdonald polynomials (e.g. [BGLX], [DFK]), categorification (e.g.…”

Section: Let Us Say a Few Words About How The Multiplication Map Hmentioning

confidence: 99%

Kac Polynomials and Lie Algebras Associated to Quivers and Curves

Schiffmann¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Ah! si loin des carquois, des torches et des flèches, On se sauvait un peu vers des choses... plus fraîches ! E. Rostand, Cyrano de Bergerac, Acte III, Scène 7 1. Kac polynomials for quivers and curves 1.1. Quivers. Let Q be a locally finite quiver with vertex set I and edge set Ω. For any finite field F q and any dimension vector d ∈ N I , let A Q,d (F q ) be the number of absolutely indecomposable representations of Q over F q , of dimension d. Kac proved the following beautiful result :Here , is the Euler form, see Section 2.1. The fact that the number of (absolutely) indecomposable F q -representations behaves polynomially in q is very remarkable : the absolutely indecomposable representations only form a constructible substack of the stack M Q,d of representations of Q of dimension d and we count them here up to isomorphism, i.e. without the usual orbifold measure. Let us briefly sketch the idea of a proof. Standard Galois cohomology arguments ensure that it is enough to prove that the number of indecomposable F q -representations is given by a polynomial I Q,d in q (this polynomial is not as well behaved as or as interesting as A Q,d ). Next, by the Krull-Schmidt theorem, it is enough to prove that the number of all F q -representations of dimension d is itself given by a polynomial in F q . This amounts to computing the (orbifold) volume of the inertia stack IM Q,d of M Q,d ; performing a unipotent reduction in this context, we are left to computing the volume of the stack N il Q,d parametrizing pairs (M, φ) with M a representation of dimension d and φ ∈ End(M ) being nilpotent. Finally, we use a Jordan stratification of N il Q,d and easily compute the volume of each strata in terms of the volumes of the stacks M Q,d ′ for all d ′ . This actually yields an explicit formula for A Q,d (t) (or I Q,d (t)), see [Hu]. We stress that beyond the case of a few quivers (i.e. those of finite or affine Dynkin type) it is unimaginable to classify and construct all indecomposable representations; nevertheless, the above theorem says that we can count them.The positivity of A Q,d (t) was only recently 1 proved : 1 the special case of an indivisible dimension vector was proved earlier by Crawley-Boevey and Van den Bergh, see [CBVdB]1

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The q-AGT–W Relations Via Shuffle Algebras

Cited by 48 publications

References 36 publications

Shuffle algebras associated to surfaces

Shuffle algebras associated to surfaces

Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

Kac Polynomials and Lie Algebras Associated to Quivers and Curves

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