Mirror symmetry, Langlands duality, and the Hitchin system

By Tamás Hausel, Michael Thaddeus. 2002

281 citations recorded by the source corpus.

Among the major mathematical approaches to mirror symmetry are those of Batyrev-Borisov and Strominger-Yau-Zaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/76ae17333ff062de9676398b6a14ebb4bb042a6d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. E. Witten
2. Sergei Gukov
3. D. Gaiotto
4. Tudor Dimofte
5. N. Nekrasov
6. A. Kapustin
7. Tamás Hausel
8. C. Vafa

In Mathematical Physics

1. E. Witten
2. Sergei Gukov
3. D. Gaiotto
4. Tudor Dimofte
5. N. Nekrasov
6. A. Kapustin
7. Tamás Hausel
8. Carlos Simpson

In Nuclear and High Energy Physics

1. E. Witten
2. Sergei Gukov
3. D. Gaiotto
4. Tudor Dimofte
5. C. Vafa
6. N. Nekrasov
7. N. Seiberg
8. A. Kapustin

Citations in the corpus

• Langlands duality for Hitchin systems (2006, background): “in a non-trivial torsor structure [ HT03 , DP03].”
• Global topology of the Hitchin system (2011, methodology): “ymous referee for helpful comments. The author was supported by a Royal Society University Research Fellowship. 1.1.Mirror symmetry In order to motivate the topological mirror symmetry conjectures of [HT], we give some background information on three of the main topics involved starting with mirror ”
• Chern-Simons theory and S-duality (2011, background): “ition of the mirror transform and a hyper-Kahler rotation [41]: S-duality = (mirror symmetry) J !K K! J : (4.5) Indeed, Y = M at(G C;C) and Ye = M at(LG C;C) make a famous pair of mirror manifolds [81]. Moreover, since SO(3) ˘=SU(2)=Z 2 in our examples the mirror variety Yeis simply a quotient of t”
• Katz's middle convolution algorithm (2006, background): “action on the moduli space look like? Some things to study in low-dimensional cases would be: compactifications [65] and their the dynamics [37] [38] [39] [12], the Hitchin system [70] [ 66 ], the relationship with Painlev´e equations [71] [14] [15] [47], jumps and wall-crossing phenomena such as in”
• The omega deformation, branes, integrability and Liouville theory (2010, methodology): “What happens to the brane Bε if we use a description obtained by T -duality on scalars coming from holonomy around S(1) rather than S̃(1)? In this case, Bε is replaced by a new (9)In the context of generalized quiver gauge theories, if MH is a moduli space of Higgs bundles on a Riemann surface C wit”
• Holomorphic reduction of N = 2 gauge theories, Wilson-'t Hooft operators, and S-duality (2006, background): “The duality of Hitchin fibrations for G and G is known as classical geometric Langlands duality and has been proved for nonsingular fibers in [30, 31, 32, 33].”
• On the motives of moduli of chains and Higgs bundles (2011, background): “which was motivated by the mirror symmetry conjecture of [20].”
• Topology of character varieties of Abelian groups (2013, background): “We remark that these character varieties are also main players in mirror symmetry and the geometric Langlands programme (see, for example [19, 25, 29]).”
• Wall-crossing in coupled 2d-4d systems (2011, result): “This picture fits in very well with the work of Hausel and Thaddeus [53] who showed that for Hitchin fibrations the phenomenon of having a disconnected fiber is mirror dual to having a discrete B-field.”
• Electric-Magnetic Duality And The Geometric Langlands Program (2006, background): “This mirror symmetry was first proposed in [28].”
• Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles (2004, methodology): “fy the variant part of the cohomology of MΛ under the action of Γ 3—this should be relevant for proving the rank 3 parabolic version, stated in [21], of the mirror symmetry Theorem of Hausel–Thaddeus [22]. Theorem 12.22. The variant part of the rational cohomology of MΛ has Poincar´e polynomial Pvar”
• Langlands duality and G2 spectral curves (2006, unknown): “A”
• Lectures on the Langlands program and conformal field theory (2005, background): “ considered in [79], and in fact they are 52These dual Hitchin fibrations (restricted to the open subsets of stable Higgs pairs in T∗Bun◦ G and T∗Bun◦ LG ) have been shown by T. Hausel and M. Thaddeus [77] to be an example of the Strominger-YauZaslow duality. 72 EDWARD FRENKEL closely related to the ”