On Razamat’s A 2 and A 3 kernel identities

Ruijsenaars, S. N. M.

doi:10.1088/1751-8121/ab97df

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…These are in a sense minimal theories in 6d. For some of them we know what the integrable models are: A 1 RS for the A 1 (2, 0) SCFT, BC 1 van Diejen for the E-string, the models derived in [22] (and further disucssed in [23]) for the minimal SU (3) and SO(8) SCFTs. However, we lack any understanding for other models in the sequence (the minimal F 4 , E 6 , E 7 , E 7 1…”

Section: Discussionmentioning

confidence: 99%

“…For example it is simply the contribution of the SU(2) N = 2 vector multiplet [2] (without the term coming from the Cartan of the adjoint chiral field). 23…”

Section: First Class Corresponds To the Following Partition Kmentioning

confidence: 99%

“…For example, in the case of the (2, 0) theory of type G ∈ ADE the associated integrable model is given in terms of the Ruijsenaars-Schneider elliptic analytic difference operators (A∆Os) [2,8]. 2 In the case of A 2 and D 4 minimal 6d SCFTs [19][20][21] one can derive novel integrable models [22,23] associated to the A 2 and A 3 root systems respectively. In the case of the 6d SCFT being the rank one E-string [24] one obtains [25] the BC 1 van Diejen model [26].…”

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confidence: 99%

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Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

Nazzal,

Nedelin,

Razamat

2021

Preprint

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We consider supersymmetric surface defects in compactifications of the 6d minimal (D N +3 , D N +3 ) conformal matter theories on a punctured Riemann surface. For the case of N = 1 such defects are introduced into the supersymmetric index computations by an action of the BC 1 (∼ A 1 ∼ C 1 ) van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras A N =1 , C N =1 , and (A 1 ) N =1 . The indices of these 4d theories give rise to three different Kernel functions for the BC 1 van Diejen model. We then consider the generalizations with N > 1. The operators introducing defects into the index computations are certain A N , C N , and (A 1 ) N generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal (D N +3 , D N +3 ) conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the A N case, and derive various properties these operators have to satisfy as a consequence of 4d dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal (D 5 , D 5 ) conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.

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Section: Discussionmentioning

confidence: 99%

“…For example it is simply the contribution of the SU(2) N = 2 vector multiplet [2] (without the term coming from the Cartan of the adjoint chiral field). 23…”

Section: First Class Corresponds To the Following Partition Kmentioning

confidence: 99%

mentioning

confidence: 99%

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Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

Nazzal,

Nedelin,

Razamat

2021

Preprint

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“…24 Let us comment in passing that this picture can be further checked by giving vacuum expectation values to derivatives of the operators in 10 which will introduce surface defects into the theory [118]. Such constructions are related intimately to integrable models, and the corresponding indices should satisfy interesting sets of properties [119] which can be mathematically proven [120] providing more evidence for the conjectures.…”

Section: Closing Puncturesmentioning

confidence: 98%

“…Explicitly for compactifications of the ADE (2, 0) theories the relevant models turn out to be elliptic ADE Ruijsenaars-Schneider systems [118,173]; for the rank N E-string this is the BC N van Diejen model [126,174] (this was shown explicitly for N = 1 using the index methods and conjectured for higher N ); for minimal 6d SCFTs with SU (3) and SO(8) gauge groups these were discussed in [119,120]; for A type conformal matter the operators were discussed in [118,175]; for minimal D type conformal matter the operators were computed in [126]. These operators can be obtained independently by studying Seiberg-Witten curves [176,177] directly in 6d [178].…”

Section: Integrable Modelsmentioning

confidence: 99%

Aspects of 4d supersymmetric dynamics and geometry

Razamat¹,

Sabag²,

Sela³

et al. 2022

Preprint

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In this set of five lectures we present a basic toolbox to discuss the dynamics of four dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional supersymmetric models as compactifications of six dimensional SCFTs. We discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be naturally imbedded in the geometric constructions. The lectures mostly review results which previously appeared in the literature but also contain some unpublished derivations.

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$$C_2$$ generalization of the van Diejen model from the minimal $$(D_5,D_5)$$ conformal matter

Nazzal,

Nedelin

2023

Lett Math Phys

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We study superconformal indices of 4d compactifications of the 6d minimal $$(D_{N+3},D_{N+3})$$ ( D N + 3 , D N + 3 ) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with $$A_N, C_N$$ A N , C N and $$\left( A_1\right) ^N$$ A 1 N global symmetries. We mainly concentrate on $$C_2$$ C 2 case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our $$C_2$$ C 2 operator and previously derived $$A_2$$ A 2 generalization of van Diejen model. Finally, we also consider compactifications with $$A_N$$ A N -type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.

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On Razamat’s A ₂ and A ₃ kernel identities

Cited by 7 publications

References 12 publications

Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

Aspects of 4d supersymmetric dynamics and geometry

$$C_2$$ generalization of the van Diejen model from the minimal $$(D_5,D_5)$$ conformal matter

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