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On 6d N=(2,0) theory compactified on a Riemann surface with finite area

Davide Gaiotto, Gregory W. Moore, Yuji Tachikawa

Abstract

We study 6d N=(2,0) theory of type SU(N) compactified on Riemann surfaces with finite area, including spheres with fewer than three punctures. The Higgs branch, whose metric is inversely proportional to the total area of the Riemann surface, is discussed in detail. We show that the zero-area limit, which gives us a genuine 4d theory, can involve a Wigner-Inonu contraction of global symmetries of the six-dimensional theory. We show how this explains why subgroups of SU(N) can appear as the gauge group in the 4d limit. As a by-product we suggest that half-BPS codimension-two defects in the six-dimensional (2,0) theory have an operator product expansion whose operator product coefficients are four-dimensional field theories.

On 6d N=(2,0) theory compactified on a Riemann surface with finite area

Abstract

We study 6d N=(2,0) theory of type SU(N) compactified on Riemann surfaces with finite area, including spheres with fewer than three punctures. The Higgs branch, whose metric is inversely proportional to the total area of the Riemann surface, is discussed in detail. We show that the zero-area limit, which gives us a genuine 4d theory, can involve a Wigner-Inonu contraction of global symmetries of the six-dimensional theory. We show how this explains why subgroups of SU(N) can appear as the gauge group in the 4d limit. As a by-product we suggest that half-BPS codimension-two defects in the six-dimensional (2,0) theory have an operator product expansion whose operator product coefficients are four-dimensional field theories.

Paper Structure

This paper contains 16 sections, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Two easy pieces
  3. Torus
  4. Sphere with two full punctures
  5. Higgs branch for $C$ with finite area
  6. As the Coulomb branch of 5d super Yang-Mills on $C$
  7. As a hyperkähler quotient
  8. The manifold $I_\rho(\mathcal{A})$
  9. The sphere with three full punctures and the manifold $\eta(T_N)$
  10. Dependence on area from the perspective of the quotient
  11. A connection with the bow construction
  12. Application to the 4d analysis
  13. $T_N$ and the bifundamental
  14. The trinion with one full and two simple punctures
  15. A sphere with four punctures of type $[k,k]$
  16. ...and 1 more sections

Figures (5)

  • Figure 1: 6d theory on a sphere with two full punctures, and its reduction to 5d theory. The punctures become boundary conditions.
  • Figure 2: Left: a skeleton-like metric on $C$. Right: its reduction to 5d. The Higgs branch of the each component is named. The Higgs branch of the total system is given by the hyperkähler quotient via the diagonal $\mathrm{SU}(N)$ actions.
  • Figure 3: The property \ref{['moving']} illustrated. The action of $\mathrm{SU}(N)_i$ is labeled by $i$ in the figure.
  • Figure 4: A sphere with punctures $f,f,s,s$ and two ways of decomposition. Every step is understood to be performed at finite area.
  • Figure 5: A sphere with punctures $f,s,[k,k],[k,k]$, and a sphere with four punctures of type $[k,k]$. The appearance of an additional antisymmetric in the second case can be understood naturally in our approach .