Abstract:The S-duality group S(F ) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F ) modulo the subgroup acting trivially on the physical quantities. S(F ) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g, Z) (g being the dimension of the Coulomb branch). This identification reduces the determination of the S-duality group of a given N = 2 theory to a problem in homological algebra. In this paper we describe the te… Show more
“…For SU (2) N = 2 * , Aut(C )/(phy.triv.) is a Z 2 extension of the modular group P SL(2, Z) [32,33]. As in the previous examples, a finite subgroup of Aut(C )/(phy.triv.)…”
Section: The Remaining 5 Discrete Gaugingsmentioning
confidence: 96%
“…To a given N = 2 QFT there are attached several triangle categories describing its BPS sector [32,33]. These categories are fairly well understood when the QFT has the so-called BPS-quiver property [23,24,30].…”
Section: Bps Categories: a Cartoonmentioning
confidence: 99%
“…Pragmatically, this amounts to requiring the 4d/2d correspondence to be satisfied by the covering cluster categoryC of our UV 2-CY category C . To keep the statement as simple as possible, we shall not state the criterion in its most general form (the interested reader may look at [32,33,53]) but only its simplified version which applies to rank-1 SCFT. In our tables we shall insert back the rank-1 asymptotically-free categories.…”
Section: Uv Completeness: Our Final Proposalmentioning
confidence: 99%
“…As an auto-equivalence of nil C(4-cycle) its satisfiesτ 4 = Id. 53 For details see §.6.2.1 of [32]. 54 S 1 is the simple module of CD 4 with support on node 1 of the Dynkin quiver (8.1).…”
Section: Discrete Gaugings Of Argyres-douglas Modelsmentioning
confidence: 99%
“…This is not a problem according the discussion in §.7.1, since in this case h = 1. For h ≥ 1 the only condition is that the discrete gauge group G should be an unobstructed subgroup of the homological S-duality group [32,33] equal to the group of auto-equivalences Aut(C ) of the UV category C of the parent theory (modulo its subgroup acting trivially on physical observables).…”
Section: The Remaining 5 Discrete Gaugingsmentioning
Argyres and co-workers started a program to classify all 4d N = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 N = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces.The classification of 4d N = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces.In the follow-up paper we apply the RT methods to higher rank 4d N = 2 SCFT.
“…For SU (2) N = 2 * , Aut(C )/(phy.triv.) is a Z 2 extension of the modular group P SL(2, Z) [32,33]. As in the previous examples, a finite subgroup of Aut(C )/(phy.triv.)…”
Section: The Remaining 5 Discrete Gaugingsmentioning
confidence: 96%
“…To a given N = 2 QFT there are attached several triangle categories describing its BPS sector [32,33]. These categories are fairly well understood when the QFT has the so-called BPS-quiver property [23,24,30].…”
Section: Bps Categories: a Cartoonmentioning
confidence: 99%
“…Pragmatically, this amounts to requiring the 4d/2d correspondence to be satisfied by the covering cluster categoryC of our UV 2-CY category C . To keep the statement as simple as possible, we shall not state the criterion in its most general form (the interested reader may look at [32,33,53]) but only its simplified version which applies to rank-1 SCFT. In our tables we shall insert back the rank-1 asymptotically-free categories.…”
Section: Uv Completeness: Our Final Proposalmentioning
confidence: 99%
“…As an auto-equivalence of nil C(4-cycle) its satisfiesτ 4 = Id. 53 For details see §.6.2.1 of [32]. 54 S 1 is the simple module of CD 4 with support on node 1 of the Dynkin quiver (8.1).…”
Section: Discrete Gaugings Of Argyres-douglas Modelsmentioning
confidence: 99%
“…This is not a problem according the discussion in §.7.1, since in this case h = 1. For h ≥ 1 the only condition is that the discrete gauge group G should be an unobstructed subgroup of the homological S-duality group [32,33] equal to the group of auto-equivalences Aut(C ) of the UV category C of the parent theory (modulo its subgroup acting trivially on physical observables).…”
Section: The Remaining 5 Discrete Gaugingsmentioning
Argyres and co-workers started a program to classify all 4d N = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 N = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces.The classification of 4d N = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces.In the follow-up paper we apply the RT methods to higher rank 4d N = 2 SCFT.
Abstract:We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality.
It is possible to understand whether a given BPS spectrum is generated by a relevant deformation of a 4D $$ \mathcal{N} $$
N
= 2 SCFT or of an asymptotically free theory from the periodicity properties of the corresponding quantum monodromy. With the aim of giving a better understanding of the above conjecture, in this paper we revisit the description of framed BPS states of four-dimensional relativistic quantum field theories with eight conserved supercharges in terms of supersymmetric quantum mechanics. We unveil aspects of the deep interrelationship in between the Seiberg-dualities of the latter, the discrete symmetries of the theory in the bulk, and quantum discrete integrable systems.
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