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4d/2d Correspondence: Instantons and W-Algebras


Song, Jaewon (2012) 4d/2d Correspondence: Instantons and W-Algebras. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WP20-DX98.


In this thesis, we study the 4d/2d correspondence of Alday-Gaiotto-Tachikawa, which relates the class of 4-dimensional N=2 gauge theories (theories of class S) to a 2-dimensional conformal field theory. The 4d gauge theories are obtained by compactifying 6-dimensional N=(2, 0) theory of type A, D, E on a Riemann surface C. On the 2-dimensional side, we have Toda theory on the surface C with W-algebra symmetry, which is an extension of the Virasoro symmetry. In particular, the instanton partition function of the 4d gauge theory is reproduced by a conformal/chiral block of Virasoro/W-algebra. We develop techniques to compute the partition functions on 4d and 2d sides, for various gauge groups and matter fields.

We generalize the Alday-Gaiotto-Tachikawa 4d/2d correspondence to various cases. First, we study N=2 pure Yang-Mills theory with arbitrary gauge groups, including the exceptional groups. We explicitly construct the corresponding W-algebra currents, and confirm the correspondence holds at 1-instanton level. Second, we study the conformal quiver theory with Sp(1)-SO(4) gauge group. Finally, we study Sicilian gauge theories with trifundamental half-hypermultiplets. We also find that the conformal theories with Sp(1) gauge group and SU(2) gauge group have different instanton partition functions in terms of bare gauge couplings. We show this is an artifact of the renormalization scheme, by explicitly constructing a map between the bare couplings and studying its geometrical interpretations. This demonstrates the scheme independence of renormalization at the non-perturbative level.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Supersymmetry; Instantons; W-algebra; Conformal Field Theory; Gauge Theory;
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ooguri, Hirosi
Group:Caltech Theory
Thesis Committee:
  • Ooguri, Hirosi (chair)
  • Schwarz, John H.
  • Kapustin, Anton N.
  • Spiropulu, Maria
Defense Date:17 May 2012
Non-Caltech Author Email:jws611 (AT)
Record Number:CaltechTHESIS:05302012-170816705
Persistent URL:
Song, Jaewon0000-0002-1238-2435
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7103
Deposited By: Jaewon Song
Deposited On:01 Jun 2012 17:50
Last Modified:06 Oct 2021 21:14

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