Braid groups and Artin groups

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 techniques which make the computation straightforward for a large class of N = 2 QFTs. The group S(F ) is naturally presented as a generalized braid group.The S-duality groups are often larger than expected. In some models the enhancement of S-duality is quite spectacular. For instance, a QFT with a huge S-duality group is the Lagrangian SCFT with gauge group SO(8) × SO(5) 3 × SO(3) 6 and half-hypermultiplets in the bi-and tri-spinor representations.

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“…For a review of Artin's groups see [69]. The center of the affine Artin braid group is trivial, Z(GÃ n ) = 1 [70].…”

Section: Braiding Relations In Aut(mentioning

confidence: 99%

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Caorsi

Cecotti

2018

Advances in Theoretical and Mathematical Physics

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“…See [105] for precise definitions and detailed discussions. See also [339] for discussions on related braid groups and Artin groups.…”

Section: Properties Of Coxeter Groupsmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

IRMA Lectures in Mathematics and Theoretical Physics

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SummaryIn this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups Out(F n ), Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of Mod g = Mod g,0 on the Teichmüller space T g to construct an intrinsic fundamental domain consisting of finitely many cells, * Partially Supported by NSF grant DMS 0905283 2 solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne-Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups.The layout of this chapter is as follows. In §1, the introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups Γ of semisimple Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures, and their actions on symmetri...

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“…consists of d non-intersecting (other than at the end points) paths on L which start and end in the set {λ [35]). We denote this map which takes a loop γ to the corresponding surface braid by f −1 .…”

Section: Preliminariesmentioning

confidence: 99%

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

Korotkin

Shramchenko

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

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In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases. MSC: 35Q15, 53D45Contents

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Braid groups and Artin groups

Cited by 18 publications

References 130 publications

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

A tale of two groups: arithmetic groups and mapping class groups

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

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