Spinors, Lagrangians and rank 2 Higgs bundles

Hitchin, Nigel

doi:10.1112/plms.12034

Cited by 10 publications

(15 citation statements)

References 18 publications

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“…Following the analysis of [14] or proceeding analogously to our previous examples, one may conclude that at above a generic point in the bases of the Hitchin system, the complex Lagrangian is defined by the equations…”

Section: Baa Analysis Of Tri-fundamentalmentioning

confidence: 72%

“…The BAA twist of such boundary conditions will give us a class of complex Lagrangian submanifolds L(C, G, M ) of the space of Higgs bundles labelled by M . The mathematical treatment of such Lagrangians has been initiated recently by [14]. These Lagrangian submanifolds admit a natural quantization to D-modules given by conformal blocks for a free symplectic boson vertex algebra.…”

Section: General Strategymentioning

confidence: 99%

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S-duality of boundary conditions and the Geometric Langlands program

Gaiotto¹

2018

Proceedings of Symposia in Pure Mathematics

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Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions. arXiv:1609.09030v2 [hep-th] 8 Nov 2016 42 B.3.2 The ADHM system 43 B.3.3 Kähler SQM 44 -ii -B.3.4 Hyper-holomorphic SQM 46 B.3.5 Hypermultiplets on a Riemann surface 47 B.3.6 From N = 4 SQM to branes 48 B.3.7 From branes so SQM 48 C The category of BBB branes 49 D Gauge group reductions 50 D.1 Dirichlet boundary conditions 50 D.2 Nahm pole boundary conditions and interfaces 51manner, by tensor product with a certain reference bundle. 'T Hooft operators act by Hecke correspondences, functors which play a key role in Geometric Langlands. Dirichlet boundary conditions D produce "eigenbranes" for the action of Wilson lines, simply because they fix the value at the boundary of the connection and scalar fields which enter the Wilson line path-ordered exponential. S-duality exchanges Wilson lines and 't Hooft lines. As a consequence, the S-dual boundary condition ∨ D will produce "eigenbranes" for the action of 't Hooft lines, i.e. Hecke eigensheaves.The analysis of [1] did not employ directly the S-dual boundary condition ∨ D, as the action of S-duality on boundary conditions for non-Abelian gauge theory was not understood at the time. This provided an important motivation for the work of [4, 5], which described general half-BPS boundary conditions four-dimensional N = 4 gauge theory and their behaviour under the action of S-duality.One conclusion of that work was that ∨ D can be defined with the help of a very special three-dimensional N = 4 SCFT T [G], which unfortunately lacks a simple UV Lagrangian description. For some classical groups a UV Lagrangian description is available, but breaks symmetries which are crucial for GL applications. On the other hand, the analysis also provided us with a large list of S-dual pairs of boundary conditions and interfaces which do admit an effective UV Lagrangian description.The purpose of this paper is to make explicit the map from half-BPS boundary conditions which admit an effective Lagrangian description to objects in the categories relevant for the Geometric Langlands duality (see [6] for some early work on the subject). The expected payoff is a long list of Geometric Langlands dual pairs of mathematical objects which can be defined uniformly on any Riemann surface.The obvious limitation of the computational techniques employed in this paper is the requirement of an effective UV description of the boundary conditions. This prevents us from, say, constructing generic Hecke eigensheaves directly from ∨ D. In a separate paper, [7], we will discuss alternative computational ap...

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Section: Baa Analysis Of Tri-fundamentalmentioning

confidence: 72%

Section: General Strategymentioning

confidence: 99%

S-duality of boundary conditions and the Geometric Langlands program

Gaiotto¹

2018

Proceedings of Symposia in Pure Mathematics

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“…Finally, it should be mentioned that in the last couple of years researchers have found other novel ways in which branes can be constructed within the moduli space of Higgs bundles, and which are yet to be generalized to other settings. Examples of these are Nahm branes [81], branes appearing through spinors [115], through moment maps [90], and through Borel subgroups [84]. However, since Lagrangian branes can appear in any of three types, it is of interest to understand families of Lagrangian branes of each types which are related in some geometric fashion, but it is not yet known of other triples of families of branes appearing within hyperkähler spaces other than the ones obtained thought the methods of [23].…”

Section: Construction Of Branesmentioning

confidence: 99%

Singular Geometry and Higgs Bundles in String Theory

Anderson¹,

Esole²,

Fredrickson³

et al. 2018

SIGMA

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Abstract. This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

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“…representations of the fundamental group of the punctured sphere, associated to the T F T , obtained interpreting [33] the Nicolai map restricted to the lattice divisor Eq. (4.17) as hyper-Kahler reduction [57,58].…”

Section: Introductionmentioning

confidence: 99%

Yang–Mills mass gap at large-N, noncommutative YM theory, topological quantum field theory and hyperfiniteness

Bochicchio

2015

Int. J. Mod. Phys. D

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We review a number of old and new concepts in quantum gauge theories, some of which are well-established but not widely appreciated, some are most recent, that may have analogs in gauge formulations of quantum gravity, loop quantum gravity, and their topological versions, and may be of general interest. Such concepts involve noncommutative gauge theories and their relation to the large-N limit, loop equations and the change to the anti-selfdual (ASD) variables also known as Nicolai map, topological field theory (TFT) and its relation to localization and Morse-Smale-Floer homology, with an emphasis both on the mathematical aspects and the physical meaning. These concepts, assembled in a new way, enter a line of attack to the problem of the mass gap in large-N SU (N ) Yang-Mills (YM), that is reviewed as well.Algebraic considerations furnish a measure of the mathematical complexity of a complete solution of large-N SU(N ) YM: In the large-N limit of pure SU (N ) YM the ambient algebra of Wilson loops is known to be a type II 1 nonhyperfinite factor.Nevertheless, for the mass gap problem at the leading 1/N order, only the subalgebra of local gauge-invariant single-trace operators matters. The connected two-point correlators in this subalgebra must be an infinite sum of propagators of free massive fields, since the interaction is subleading in 1 N , a vast simplification. It is an open problem, determined by the growth of the degeneracy of the spectrum, whether the aforementioned local subalgebra is in fact hyperfinite.Moreover, the sum of free propagators that occurs in the two-point correlators in the aforementioned local subalgebra must be asymptotic for large momentum to the result implied by the asymptotic freedom and the renormalization group: This fundamental constraint fixes asymptotically the residues of the poles of the propagators in terms of the mass spectrum and of the anomalous dimensions of the local operators.For the mass gap problem, in the search of a hyperfinite subalgebra containing the scalar sector of large-N YM, a major role is played by the existence of a TFT underlying the large-N limit of YM, with twisted boundary conditions on a torus or, which is the same by Morita duality, on a noncommutative torus.The TFT is trivial at the leading large-N order and localized on a set of critical points by means of a quantum version of Morse-Smale-Floer homology, that involves loop equations in the ASD variables. 1530017-1 M. BochicchioA hyperfinite sector arises by fluctuations around the trivial TFT, in which the joint spectrum of scalar and pseudoscalar glueballs is linear in the square of the masses m 2 k = kΛ 2 YM with degeneracy k = 1, 2, . . . , and the two-point correlator satisfies the aforementioned fundamental constraint arising by the asymptotic freedom and the renormalization group.

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Spinors, Lagrangians and rank 2 Higgs bundles

Cited by 10 publications

References 18 publications

S-duality of boundary conditions and the Geometric Langlands program

S-duality of boundary conditions and the Geometric Langlands program

Singular Geometry and Higgs Bundles in String Theory

Yang–Mills mass gap at large-N, noncommutative YM theory, topological quantum field theory and hyperfiniteness

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