Mirror Symmetry with Branes by Equivariant Verlinde Formulas

2018 ◽  
pp. 189-218
Author(s):  
Tamás Hausel ◽  
Anton Mellit ◽  
Du Pei
Keyword(s):  
Functional Equation ◽  
Riemann Surface ◽  
Moduli Space ◽  
Mirror Symmetry ◽  
Higgs Bundles ◽  
Exterior Powers

This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.

scholarly journals QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES

2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Symplectic Form ◽  
Dense Subset ◽  
Projective Structure ◽  
Open Dense Subset ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Structure Formula

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

scholarly journals A BRIEF SURVEY OF HIGGS BUNDLES

2019 ◽  
Vol 26 (2) ◽  
pp. 197-214
Author(s):  
RONALD A. ZÚÑIGA ROJAS
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Mirror Symmetry ◽  
Higgs Field ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
Self Duality ◽  
Holomorphic Bundle ◽  
The Subject ◽  
Higgs Fields

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

scholarly journals BIRATIONAL EQUIVALENCE OF HIGGS MODULI

2005 ◽  
Vol 16 (04) ◽  
pp. 365-386
Author(s):  
MRIDUL MEHTA
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Compact Riemann Surface ◽  
Holomorphic Section ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
Birational Equivalence

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

scholarly journals INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS

2021 ◽  
pp. 1-40
Author(s):  
Mirko Mauri
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Character Variety ◽  
Degree Zero ◽  
Surface Groups ◽  
Higgs Bundles ◽  
Character Varieties ◽  
Rank 2

Abstract For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

scholarly journals Endoscopic decompositions and the Hausel–Thaddeus conjecture

2021 ◽  
Vol 9 ◽  
Author(s):  
Davesh Maulik ◽  
Junliang Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Decomposition Theorem ◽  
Higgs Bundles ◽  
Hitchin Fibration ◽  
Vanishing Cycle ◽  
Natural Operators

Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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