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

TL;DR: In this article, the authors consider supersymmetric surface defects in compactifications of the 6d conformal matter theories on a punctured Riemann surface and derive field theoretic descriptions of the four dimensional models.
Abstract: 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 $BC_1\,(\sim A_1\sim C_1)$ 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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TL;DR: In this article, the authors studied the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus and found that the resulting operator expression belongs to the class of elliptic quantum curves.

4 citations

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TL;DR: The 4D Lagrangian has the form of a "star shaped quiver" with the rank of the central node depending on the 6D theory and the number and type of punctures as mentioned in this paper .
Abstract: We discuss 4D Lagrangian descriptions, across dimensions IR duals, of compactifications of the 6D (D, D) minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a gauge theory with a simple gauge group. The Lagrangian has the form of a "star shaped quiver" with the rank of the central node depending on the 6D theory and the number and type of punctures. Using this Lagrangian one can construct across dimensions duals for arbitrary compactifications (any, genus, any number and type of USp punctures, and any flux) of the (D, D) minimal conformal matter gauging only symmetries which are manifest in the ultraviolet.
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TL;DR: In this article, the authors studied the vacuum structure and spectrum of N = 2 supersymmetric gauge theory in four dimensions, with gauge group SU(2), and obtained exact formulas for electron and dyon masses and the metric on the moduli space of vacua.

4,007 citations

Journal ArticleDOI
TL;DR: In this paper, the vacuum structure and dyon spectrum of N = 2 supersymmetric gauge theory in four dimensions, with gauge group SU(2), were studied. And the theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and Dyon masses and the metric on the moduli space of vacua.
Abstract: We study the vacuum structure and dyon spectrum of N=2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The ``strongly coupled'' vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation.

2,307 citations

Journal ArticleDOI
TL;DR: In this paper, some nonperturbative constraints on supersymmetry breaking are derived and it is demonstrated that dynamical supersymmetric breaking does not occur in certain interesting classes of theories.

1,980 citations

Journal ArticleDOI
TL;DR: In this article, the generalization of S-duality and Argyres-Seiberg duality for a large class of superconformal quiver gauge theories is studied.
Abstract: We study the generalization of S-duality and Argyres-Seiberg duality for a large class of N = 2 superconformal gauge theories. We identify a family of strongly interacting SCFTs and use them as building blocks for generalized superconformal quiver gauge theories. This setup provides a detailed description of the “very strongly coupled” regions in the moduli space of more familiar gauge theories. As a byproduct, we provide a purely four dimensional construction of N = 2 theories defined by wrapping M5 branes over a Riemann surface.

1,507 citations

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TL;DR: In this paper, the authors present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S 2 × S 3 × 3 × 4 time.
Abstract: We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S 3 × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the $${\mathcal N}=4$$ theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS 5 × S 5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS 5, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the $${\mathcal N}=4$$ theory. We note that entropy for some small supersymmetric AdS 5 black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the $${\mathcal N}=4$$ Yang Mills theory.

1,035 citations