Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

Kamnitzer, Joel; Pham, Khoa; Weekes, Alex

doi:10.1016/j.aim.2022.108281

Cited by 6 publications

(10 citation statements)

References 25 publications

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“…. In [KPW22], we partially proved this statement in the case that µ (1) = −α i . However, we don't currently understand the relationship between comultiplication and Theorem 2.14.…”

Section: Quiver Gauge Theoriesmentioning

confidence: 65%

“…Geometrically, dim M C (L/C × ξ , N ξ 0 ) = dim M C (G, N ) − 2 and so one might expect to obtain M C (L/C × ξ , N ξ 0 ) by Hamiltonian reduction of M C (G, N ) by the action of an additive group. In the quiver situation, it is possible to achieve this (see [KPW22] and Remark 3.2), but not in a way compatible with the above mentioned integrable systems. Thus, in this paper, we pursue a different approach.…”

mentioning

confidence: 99%

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Lie algebra actions on module categories for truncated shifted Yangians

Kamnitzer¹,

Webster²,

Weekes³

et al. 2022

Preprint

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We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools.After this general definition, we focus on quiver gauge theories attached to a quiver Γ. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra gΓ on category O for these Coulomb branch algebras. When Γ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra. * ((G/P ) 2 ) has a convolution structure of its own, and Poincaré duality shows that it is a matrix algebra:When P = B, then this is the nilHecke algebra of W .Lemma 2.3. The inclusion G(O)/I P × N (O) ֒→ R P G,N induces an algebra map(G/P ) be the primitive idempotent in this matrix algebra that projects to the W -invariants. We can formulate the usual abelianization in equivariant cohomology as the statement that for any G-space X, we have e ′ H P * (X) ∼ = H G * (X) where H P ⋊C × *

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“…. In [KPW22], we partially proved this statement in the case that µ (1) = −α i . However, we don't currently understand the relationship between comultiplication and Theorem 2.14.…”

Section: Quiver Gauge Theoriesmentioning

confidence: 65%

mentioning

confidence: 99%

Lie algebra actions on module categories for truncated shifted Yangians

Kamnitzer¹,

Webster²,

Weekes³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This expression, in general, describes the Poisson bracket on the phase space of the 't Hooft line, where we put arbitrary charge µ at 0 and η at ∞. It is shown in [30] that in the case of an ADE group, this Poisson bracket matches the natural Poisson bracket on the Coulomb branch of a three-dimensional N = 4 quiver gauge theory. We will discuss the quiver gauge theory picture in more detail later.…”

Section: Poisson Bracket On the Phase Spacementioning

confidence: 99%

“…At the classical level, this holds because of the results of [30]. In general, a sufficiently strong uniqueness theorem for the shifted Yangian as a quantization of its classical limit (compatible with additional structures such as coproducts) will prove that our algebra Y µ (g) is isomorphic to the shifted Yangian, but such a result seems not to be currently available.…”

Section: Extra Relations From Other Groupsmentioning

confidence: 99%

Q-operators are 't Hooft lines

Costello,

Gaiotto,

Yagi

2021

Preprint

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“…This scheme is a closed subvariety of G((z −1 )). It follows from [KPW,Theorem A.8] that for antidominant µ the embedding W µ ⊂ G((z −1 )) is Poisson so the Poisson algebra O(W µ ) is the quotient of the Poisson algebra O(G((z −1 ))) by the Poisson ideal.…”

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confidence: 99%

Bethe subalgebras in antidominantly shifted Yangians

Крылов¹,

Rybnikov²

2022

Preprint

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The loop group G((z −1 )) of a simple complex Lie group G has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras B(C) ⊂ O(G((z −1 )) depending on the parameter C ∈ G called classical universal Bethe subalgebras. To every antidominant cocharacter µ of the maximal torus T ⊂ G one can associate the closed Poisson subspace Wµ of G((z −1 )) (the Poisson algebra O(Wµ) is the classical limit of so-called shifted Yangian Yµ(g) defined in [BFN, Appendix B]). We consider the images of B(C) in O(Wµ) that we denote by Bµ(C) that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular C centralizing µ, we compute the Poincaré series of these subalgebras. For g = gl n , we define the natural quantization Y(gl n ) rtt of O(gl n ((z −1 )))) and universal Bethe subalgebras B(C) ⊂ Y(gl n ) rtt . Following the ideas of Frassek, Pestun and Tsymbaliuk and using their RTT realization of Yµ(gl n ), we obtain the natural surjections Y(gl n ) rtt ։ Yµ(gl n ) which quantize the embedding Wµ ⊂ gl n ((z −1 ))). Taking the images of B(C) in Yµ(gl n ) we obtain Bethe subalgebras Bµ(C) ⊂ Yµ(gl n ).

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Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

Cited by 6 publications

References 25 publications

Lie algebra actions on module categories for truncated shifted Yangians

Lie algebra actions on module categories for truncated shifted Yangians

Q-operators are 't Hooft lines

Bethe subalgebras in antidominantly shifted Yangians

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