scholarly journals Geometric Langlands duality and the equations of Nahm and Bogomolny

2010 ◽  
Vol 140 (4) ◽  
pp. 857-895 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Lie Group ◽  
Moduli Space ◽  
Dual Group ◽  
Langlands Duality ◽  
Geometric Langlands

Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of Gv makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

scholarly journals Langlands duality and global Springer theory

2012 ◽  
Vol 148 (3) ◽  
pp. 835-867 ◽  
Author(s):  
Zhiwei Yun
Keyword(s):  
Chern Class ◽  
Class Action ◽  
Dual Group ◽  
The Other ◽  
Support Theorem ◽  
Langlands Duality ◽  
Dual Groups ◽  
Geometric Langlands ◽  
Springer Theory ◽  
Singular Fibers

AbstractWe compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the author. We prove that the stable parts of the parabolic Hitchin complexes for Langlands dual group are naturally isomorphic after passing to the associated graded of the perverse filtration. Moreover, this isomorphism intertwines the global Springer action on one hand and Chern class action on the other. Our result is inspired by the mirror symmetric viewpoint of geometric Langlands duality. Compared to the pioneer work in this subject by T. Hausel and M. Thaddeus, R. Donagi and T. Pantev, and N. Hitchin, our result is valid for more general singular fibers. The proof relies on a variant of Ngô’s support theorem, which is a key point in the proof of the Fundamental Lemma.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

Moduli Space, of supersymmetric gauge theory

2004 ◽  
pp. 244-244
Author(s):  
David Berman ◽  
Hugo Garcia-Compean ◽  
Paulius Miškinis ◽  
Miao Li ◽  
Daniele Oriti ◽  
...  
Keyword(s):  
Supersymmetric Gauge Theory ◽  
Gauge Theory ◽  
Moduli Space ◽  
Supersymmetric Gauge

scholarly journals Spectra of elliptic potentials and supersymmetric gauge theories

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Wei He
Keyword(s):  
Gauge Theory ◽  
Strong Coupling ◽  
Moduli Space ◽  
Gauge Theories ◽  
Duality Transformations ◽  
Spectral Solution ◽  
Supersymmetric Gauge ◽  
Strong Coupling Expansions ◽  
Surface Operator ◽  
Function Potential

Abstract We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to the strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.

scholarly journals THE ENHANÇON, MULTIMONOPOLES AND FUZZY GEOMETRY

2001 ◽  
Vol 16 (05) ◽  
pp. 990-1001 ◽  
Author(s):  
CLIFFORD V. JOHNSON
Keyword(s):  
Gauge Theory ◽  
String Theory ◽  
Moduli Space ◽  
Fuzzy Sphere ◽  
Fuzzy Geometry ◽  
Space Geometry ◽  
The Difference ◽  
A Charge ◽  
Insight Into

The presentation at Strings 2000 was intended to be in two main parts, but there was only time for part one. However both parts appeared on the online proceedings, and are also included in this document. The first part concerns an exploration of the connection between the physics of the "enhançon" geometry arising from wrapping N D6–branes on the K3 manifold in Type IIA string theory and that of a charge N BPS multi–monopole. This also relates to the physics of 2+1 dimensional SU(N) gauge theory with eight supercharges. The main results uncovered by this exploration are: a) better insight into the non–perturbative geometry of the enhançon; b) the structure of the moduli space geometry, and its characterisation in terms of a family of Atiyah–Hitchin–like manifolds; c) the use of Nahm data to describe aspects of the geometry, showing that the enhançon locus itself has a description as a fuzzy sphere. Part two discusses the addition of extra D2–branes into the geometry. Two probe computations show the difference between the geometry as seen by D2–branes and that seen by wrapped D6–branes, and the accompanying gauge theory interpretations are discussed.

scholarly journals Quantum Moduli Space of the Cascading Sp(p + M) x Sp(p) Gauge Theory

2007 ◽  
Vol 117 (3) ◽  
pp. 487-499
Author(s):  
F. Koyama ◽  
F. Yagi
Keyword(s):  
Gauge Theory ◽  
Moduli Space
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