E-string Quantum Curve

Chen, Jin; Haghighat, Babak; Kim, Hee-Cheol; Sperling, Marcus; Wang, Xin

doi:10.48550/arxiv.2103.16996

Cited by 5 publications

(13 citation statements)

References 86 publications

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“…In the field theory, the defect can be introduced by giving a position-dependent vacuum expectation value (VEV) to the moment map operator breaking Sp(2N −8) flavor symmetry to Sp(2N − 10) and by taking an IR limit, which is analogous to the defect construction in [35]. Similar types of defect Higgsing in 6d theories have been considered in [38,39]. The field theory description of the underlying standard Higgs process is detailed in Appendix B.…”

Section: Codimension 2 Defectmentioning

confidence: 99%

“…In this subsection, we derive the quantum curve from saddle point approach in a path integral formalism developed in [38,39]. We start from the k-string elliptic genus,…”

Section: Path Integral Representationmentioning

confidence: 99%

“…The relevant 4d theories arise from compactifying 6d N = (1, 0) SCFTs on a generic Riemann surface and the surface defects amount to codimension 4 defects in the 6d theories. In particular, the quantum elliptic Hamiltonians and their eigenvalues were shown to be related to the elliptic Seiberg-Witten curves, describing the moduli space of the 6d SCFTs compactified on a torus, in [37][38][39] by quantizing the elliptic SW-curves of some simple 6d N = (1, 0) SCFTs in terms of partition functions with codimension 2 and 4 defects. This motivates a natural route to formulate quantizations of elliptic Seiberg-Witten curves for 6d SCFTs using supersymmetric defects and their partition functions.…”

Section: Introductionmentioning

confidence: 99%

“…The Seiberg-Witten curves for these SCFTs were obtained in [40] by taking the thermodynamic limit of the Ω-background partition functions, with parameters 1,2 . To obtain the quantum SW-curves, we use the properties of the defect operators along the same lines as [39]. We introduce half-BPS codimension 2 and 4 defects coupled to the bulk 6d SO(N c ) gauge symmetry and compute their partition function in the Ω-background using localization.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Quantum Curves of 6d SO(N) theories

Chen,

Haghighat,

Kim

et al. 2021

Preprint

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We discuss supersymmetric defects in 6d N = (1, 0) SCFTs with SO(N c ) gauge group and N c − 8 fundamental flavors. The codimension 2 and 4 defects are engineered by coupling the 6d gauge fields to charged free fields in four and two dimensions, respectively. We find that the partition function in the presence of the codimension 2 defect on R 4 × T 2 in the Nekrasov-Shatashvili limit satisfies an elliptic difference equation which quantizes the Seiberg-Witten curve of the 6d theory. The expectation value of the codimension 4 defect appearing in the difference equation is an even (under reflection) degree N c section over the elliptic curve when N c is even, and an odd section when N c is odd. We also find that RG-flows of the defects and the associated difference equations in the 6d SO(2N + 1) gauge theories triggered by Higgs VEVs of KK-momentum states provide quantum Seiberg-Witten curves for Z 2 twisted compactifications of the 6d SO(2N ) gauge theories.

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Section: Codimension 2 Defectmentioning

confidence: 99%

“…In this subsection, we derive the quantum curve from saddle point approach in a path integral formalism developed in [38,39]. We start from the k-string elliptic genus,…”

Section: Path Integral Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Elliptic Quantum Curves of 6d SO(N) theories

Chen,

Haghighat,

Kim

et al. 2021

Preprint

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“…This was not shown explicitly yet, but the results of [27][28][29] should be helpful to establish this relation. Recently the Q = 1 relation was also derived [30] directly in six dimensions by studying the relevant Seiberg-Witten curve [31,32]. 4 In [38] general compactifications of the 6d SCFT residing on two M5 branes probing a Z k singularity using such methods was also discussed.…”

mentioning

confidence: 99%

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

2022

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We consider supersymmetric surface defects in compactifications of the 6d6d minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a punctured Riemann surface. For the case of N=1N=1 such defects are introduced into the supersymmetric index computations by an action of the BC_1\,(\sim A_1\sim C_1)BC1(∼A1∼C1) 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}AN=1, C_{N=1}CN=1, and (A_1)^{N=1}(A1)N=1. The indices of these 4d4d theories give rise to three different Kernel functions for the BC_1BC1 van Diejen model. We then consider the generalizations with N>1N>1. The operators introducing defects into the index computations are certain A_{N}AN, C_NCN, and (A_1)^{N}(A1)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})(DN+3,DN+3) conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the A_NAN case, and derive various properties these operators have to satisfy as a consequence of 4d4d 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)(D5,D5) conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.

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E-string Quantum Curve

Cited by 5 publications

References 86 publications

Elliptic Quantum Curves of 6d SO(N) theories

Elliptic Quantum Curves of 6d SO(N) theories

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

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

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