Gauge Theory and Wild Ramification

Witten, Edward

doi:10.1142/s0219530508001195

Cited by 85 publications

(178 citation statements)

References 38 publications

(235 reference statements)

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“…We now apply this to our situation, which may be viewed as the simplest example of the geometric Langlands correspondence with wild ramification (i.e., connections admitting an irregular singularity). We note that wild ramification has been studied by E. Witten in the context of S -duality of supersymmetric Yang-Mills theory [Wit08].…”

Section: Example Of the Geometric Langlands Correspondence With Wild mentioning

confidence: 93%

A rigid irregular connection on the projective line

2009

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In this paper we construct a connection r on the trivial G-bundle on ‫ސ‬ 1 for any simple complex algebraic group G, which is regular outside of the points 0 and 1, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point 1, with slope 1= h, the reciprocal of the Coxeter number of G. The connection r, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of`-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. Thesè -adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation V of G, and describe the differential Galois group of r as a subgroup of G.

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Section: Example Of the Geometric Langlands Correspondence With Wild mentioning

confidence: 93%

A rigid irregular connection on the projective line

2009

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“…Indeed, there are unresolved technical issues in extending the nonabelian Hodge correspondence to the present setting, as the treatment in [BB04] makes certain semisimplicity assumptions not satisfied by Toda systems. However, it is expected that this assumption can be lifted, see for example [Wit08]. More to the point, these analytic issues are orthogonal to our immediate goals.…”

Section: Toda Spectral Network At Strongmentioning

confidence: 95%

Toda Systems, Cluster Characters, and Spectral Networks

Williams

2016

Commun. Math. Phys.

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Abstract. We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on SLn-character varieties for n ą 2 where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the SL2 case.

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“…In [12] it was shown how the main statements of the Geometric Langlands Program in the unramified case can be deduced by considering a topologically twisted version of the Montonen-Olive duality. This was later extended to the ramified case [7,18,8].…”

Section: Introductionmentioning

confidence: 92%

Gauge Theory, Mirror Symmetry, and the Geometric Langlands Program

Kapustin

2008

Homological Mirror Symmetry

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Abstract. Montonen-Olive duality implies that the categories of A-branes on the moduli spaces of Higgs bundles on a Riemann surface C for groups G and L G are equivalent. We reformulate this as a statement about categories of B-branes on the quantized moduli spaces of flat connections for groups G C and L G C . We show that it implies the statement of the Quantum Geometric Langlands duality with a purely imaginary "quantum parameter" which is proportional to the inverse of the Planck constant of the gauge theory. The ramified version of the story is also considered.

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Gauge Theory and Wild Ramification

Cited by 85 publications

References 38 publications

A rigid irregular connection on the projective line

A rigid irregular connection on the projective line

Toda Systems, Cluster Characters, and Spectral Networks

Gauge Theory, Mirror Symmetry, and the Geometric Langlands Program

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