Spectral action in Betti Geometric Langlands

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

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On the Notion of Conductor in the Local Geometric Langlands Correspondence

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-016-1 ◽

2017 ◽

Vol 69 (1) ◽

pp. 107-129

Author(s):

Masoud Kamgarpour

Keyword(s):

Irreducible Representation ◽

Galois Representation ◽

Loop Group ◽

Langlands Correspondence ◽

Categorical Representation ◽

Swan Conductor ◽

Geometric Langlands ◽

Meromorphic Connection ◽

Local Langlands Correspondence ◽

Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

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A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/749631 ◽

2009 ◽

Vol 2009 ◽

pp. 1-14

Author(s):

Do Ngoc Diep

Keyword(s):

Symmetry Group ◽

Lie Algebra ◽

Vector Bundles ◽

Sigma Model ◽

Fock Space ◽

Loop Algebra ◽

Target Space ◽

Automorphic Representations ◽

Langlands Correspondence ◽

Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

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Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture

Affine Flag Manifolds and Principal Bundles ◽

10.1007/978-3-0346-0288-4_2 ◽

2010 ◽

pp. 51-90

Author(s):

Tomás L. Gómez

Keyword(s):

Integrable System ◽

Geometric Langlands

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Arbeitsgemeinschaft: The Geometric Langlands Conjecture

Oberwolfach Reports ◽

10.4171/owr/2016/20 ◽

2016 ◽

Vol 13 (2) ◽

pp. 1027-1098

Author(s):

Laurent Fargues ◽

Dennis Gaitsgory ◽

Peter Scholze ◽

Kari Vilonen

Keyword(s):

Geometric Langlands

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A GRID MODEL FOR SHALLOW WATER WAVES

Coastal Engineering Proceedings ◽

10.9753/icce.v20.20 ◽

1986 ◽

Vol 1 (20) ◽

pp. 20 ◽

Cited By ~ 3

Author(s):

Leo H. Holthuijsen ◽

Nico Booij

Keyword(s):

Wave Propagation ◽

Water Waves ◽

Wave Breaking ◽

Traditional Approach ◽

High Wind ◽

Local Wind ◽

Shallow Water Waves ◽

Spectral Action ◽

Mean Wave Period ◽

Constant Period

Waves in coastal regions can be affected by the bottom, by currents and by the local wind. The traditional approach in numerical modelling of these waves is to compute the wave propagation with so-called wave rays for mono-chromatic waves (one constant period and one deep water direction) and to supplement this with computations of bottom dissipation. This approach has two important disadvantages. Firstly, spectral computations, e.g. to determine a varying mean wave period or varying shortcrestedness, would be rather inefficient in this approach. Secondly, interpretation of the results of the refraction computations is usually cumbersome because of crossing wave rays. The model presented here has been designed to correct these shortcomings: the computations are carried out efficiently for a large number of wave components and the effects of currents, bottom friction, local wind and wave breaking are added. This requires the exploitation of the concept of the spectral action balance equation and numerical wave propagation on a grid rather than along wave rays. The model has been in operation for problems varying from locally generated waves over tidal flats to swell penetration into Norwegian fjords. A comparison with extensive measurements is described for young swell under high wind penetrating the Rhine estuary.

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The Spectral Action and Cosmic Topology

Communications in Mathematical Physics ◽

10.1007/s00220-011-1211-3 ◽

2011 ◽

Vol 304 (1) ◽

pp. 125-174 ◽

Cited By ~ 20

Author(s):

Matilde Marcolli ◽

Elena Pierpaoli ◽

Kevin Teh

Keyword(s):

Spectral Action

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Spectral action gravity and cosmological models

Comptes Rendus Physique ◽

10.1016/j.crhy.2017.03.001 ◽

2017 ◽

Vol 18 (3-4) ◽

pp. 226-234 ◽

Cited By ~ 2

Author(s):

Matilde Marcolli

Keyword(s):

Cosmological Models ◽

Spectral Action

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STRINGS, NONCOMMUTATIVE GEOMETRY AND THE SIZE OF THE TARGET SPACE

International Journal of Modern Physics A ◽

10.1142/s0217751x99002116 ◽

1999 ◽

Vol 14 (28) ◽

pp. 4501-4517 ◽

Cited By ~ 1

Author(s):

FEDELE LIZZI

Keyword(s):

String Theory ◽

Dirac Operator ◽

Noncommutative Geometry ◽

Low Energy ◽

Target Space ◽

World Sheet ◽

Crucial Point ◽

Spectral Action ◽

Energy Eigenvalues ◽

The World

We describe how the presence of the antisymmetric tensor (torsion) on the world sheet action of string theory renders the size of the target space a gauge noninvariant quantity. This generalizes the R ↔ 1/R symmetry in which momenta and windings are exchanged, to the whole O(d,d,ℤ). The crucial point is that, with a transformation, it is possible always to have all of the lowest eigenvalues of the Hamiltonian to be momentum modes. We interpret this in the framework of noncommutative geometry, in which algebras take the place of point spaces, and of the spectral action principle for which the eigenvalues of the Dirac operator are the fundamental objects, out of which the theory is constructed. A quantum observer, in the presence of many low energy eigenvalues of the Dirac operator (and hence of the Hamiltonian) will always interpreted the target space of the string theory as effectively uncompactified.

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