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Defects and Quantum Seiberg-Witten Geometry

Mathew Bullimore, Hee-Cheol Kim, Peter Koroteev

Abstract

We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R^2 x S^1. We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.

Defects and Quantum Seiberg-Witten Geometry

Abstract

We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R^2 x S^1. We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.

Paper Structure

This paper contains 60 sections, 3 theorems, 252 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Twisted Chiral Rings
  3. The Nekrasov-Shatashvili Correspondence
  4. Quantum/Classical Duality
  5. Electric Frame
  6. Magnetic Frame
  7. Line Operators and Interfaces in $\mathcal{N}=4$ SYM
  8. Quantum Equivariant K-theory
  9. More Genetic Quiver Varieties
  10. Higgsing $T[U(2)]$
  11. Higgsing $T[U(3)]$
  12. 3d Partition Functions
  13. $S^3$ Partition function
  14. 't Hooft Operators
  15. Interfaces
  16. ...and 45 more sections

Key Result

Proposition 2.1

The $T$-equivariant quantum K-ring of the cotangent bundle to the complete N-dimensional complex flag variety is given by where ideal $\mathcal{I}$ is given by relations (eq:tRSRelationsEl ) and $T$ is the maximal torus of $U(N)\times U(1)$ with equivariant parameters $\mu_1,\dots \mu_N$ for $U(N)$ and equivariant parameter $\eta$ for $U(1)$. The correspondence between physical and geometrical pa

Figures (8)

  • Figure 1: 3d $\mathcal{N}=4$ quiver gauge theory of type $A_{L}$ with gauge group $U(N_1)\times\dots\times U(N_L)$ and $M_i$ fundamental hypermultiplets at $i$-th gauge node.
  • Figure 2: A Lagrangian description of the $T[U(N)]$ theory consists of a sequence of gauge groups $U(1)\times\dots \times U(N-1)$ with bifundamental matter and $N$ hypermultiplets at the final node.
  • Figure 3: Higgsing $T[U(3)]$ theory using Type IIB brane construction. Vertical black lines denote NS5 branes along directions 012789, horizontal blue lines denote D3 branes along 0123 directions, and oval red circles stand for D5 branes stretched along 012456 directions. The left side of the figure describes $T[U(3)]$ theory, whereas the right side shows how to obtain $A_2$ quiver with labels $(1,1)(1,1)$ by Higgsing flavor branes 1 and 2. After applying the S-duality to the right figure D5 and NS5 branes switch roles, and D3 branes are self dual. The newly obtained structure of two NS5 branes and three D5 branes describes $U(1)$ theory with three flavors.
  • Figure 4: A Lagrangian description of the $U(1)\times\dots \times U(N-1)$ theory with fundamental matter and $N$ chiral multiplets at the final node.
  • Figure 5: Classification of integrable many-body systems according to their periodicity properties in coordinates $q$ (columns) and momenta $p$ (rows).
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Corollary 2.2
  • Theorem 5.1: Givental, Kim 1993