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Gauge Theoretic Approach to Geometric Langlands Program and Khovanov Homology

Moosavian, Faroogh | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 44333 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Arfaei, Hessamaddin
  7. Abstract:
  8. Langlands Program or Langlands Correspondence is a unified theme for many important results of number theory, which proposed by Robert Langlands in the late 1960's. The aim of this program is to relate intricate problems of arithmetic to analysis and use powerful methods of analysis to investigate these problems. One of the main aspects of this duality is the appearance of two dual groups in two sides of it that can be obtained from one another. In the 1980's, by the work of many mathematicians, it turns out that a geometrical version of this duality can be formulated. In the last ten years, a categorical version of duality is worked out. On the other hand, by the work of physicist like Goddard, Nuyts, Olive and Montonen, it turns out that in some situation, two quantum field theory with two dual gauge group can be related by a non-abelian version of electric-magnetic duality called strong-weak duality or S-duality for short. Sir Micheal Atiyah proposed that this two dualities must be related. In a long paper by A.Kapustin and Ed.Witten in 2007, This proposal was proved correct, at the physical level of rigor, for unramified version of Geometric Langlands Duality. By the works of Witten, Gukov and Kapustin, this proof was extended to the tamely ramified, wildly ramified and quantum verion of the duality. The particular quantum field theory which is used in these physical proofs is a supersymmetric extension of Yang-Mills Theory. The second theme of this thesis is gauge theoretical interpretation of Khovanov Homology. In the 1980's by the work of V.Jones and others, a new view was obtained about many important problems related to links. By Investigating 2D models of statistical physics, Jones obtained an important class of links invariant that leads him to Fields medal in 1990. This invariants are Laurant polynomials with integer coefficients. In a groundbreaking paper in 1989, Ed.Witten gave a beautiful 3D picture of this and many other invariants of links using Chern-Simons gauge theory. The reason for integer coefficients troubled mathematicians for a decate until 2000 that, based on the work of I.Frenkel, his student M.Khovanov related these coefficients to dimensions of some graded vector spaces that can be assigned naturaly to links. Since then, physicist tried to obtain a physical picture of Khovanov's interpretation. This leads to a picture by Gukov, Schwarz and Vafa in terms of some BPS equations in topological string theory and a gauge theoretical approach by Ed.Witten using dualities of string theory. In the first chapter of this thesis, we briefly review the relation of S-duality and Geometric Langlands Program. In the second chapter, we, using Witten's paper, turn to gauge theoretical interpretation of Khovanov Homology and briefly review this relation. In the third chapter, we pose a question about solution of Localization Equations arising from topological twisting of N=4 supersymmetic Yang-Mills Theory. Some facts that is used throughout the thesis, is collected in appendices
  9. Keywords:
  10. Gauge Theory ; Geometric Lang Lands Program ; Khovanov Homology

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