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On the reduction of 4d N=1 theories on S^2

Abhijit Gadde, Shlomo S. Razamat, Brian Willett

TL;DR

The paper analyzes how 4d N=1 theories compactified on S^2 yield 2d N=(0,2) theories, identifying a necessary condition—non-negative integer U(1)_R charges—for the infrared and small-S^2 limits to commute. It develops a concrete reduction rule mapping 4d chiral multiplets to 2d N=(0,2) multiplets, and leverages the S^2 × T^2 index and the elliptic genus to test and constrain the resulting theories. Through detailed examinations of Seiberg dualities and their 2d reductions, it demonstrates that 4d dualities imply corresponding 2d dualities, including (0,2), (2,2), and (0,4) cases, depending on R-symmetry choices. The work further extends to N=2 models, revealing a spectrum of 2d theories with enhanced supersymmetry and providing multiple consistency checks via elliptic genera. Overall, the approach offers a unified, partition-function–driven framework to derive and verify 2d dualities from higher-dimensional dualities.

Abstract

We discuss reductions of general N=1 four dimensional gauge theories on S^2. The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an N=(0,2) gauge theory. As an application of our general observations, we discuss reductions of N=1 and N=2 dualities and argue that they imply certain two dimensional dualities.

On the reduction of 4d N=1 theories on S^2

TL;DR

The paper analyzes how 4d N=1 theories compactified on S^2 yield 2d N=(0,2) theories, identifying a necessary condition—non-negative integer U(1)_R charges—for the infrared and small-S^2 limits to commute. It develops a concrete reduction rule mapping 4d chiral multiplets to 2d N=(0,2) multiplets, and leverages the S^2 × T^2 index and the elliptic genus to test and constrain the resulting theories. Through detailed examinations of Seiberg dualities and their 2d reductions, it demonstrates that 4d dualities imply corresponding 2d dualities, including (0,2), (2,2), and (0,4) cases, depending on R-symmetry choices. The work further extends to N=2 models, revealing a spectrum of 2d theories with enhanced supersymmetry and providing multiple consistency checks via elliptic genera. Overall, the approach offers a unified, partition-function–driven framework to derive and verify 2d dualities from higher-dimensional dualities.

Abstract

We discuss reductions of general N=1 four dimensional gauge theories on S^2. The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an N=(0,2) gauge theory. As an application of our general observations, we discuss reductions of N=1 and N=2 dualities and argue that they imply certain two dimensional dualities.

Paper Structure

This paper contains 12 sections, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reduction of four dimensional theories to two dimensions
  3. Elliptic genus
  4. $\mathbb S^2\times \mathbb T^2$ partition function
  5. Truncation of sum over fluxes
  6. Reduction of Seiberg dualities
  7. $SU(N_c)$ dualities
  8. Global Symmetries
  9. $U(N)$ version
  10. $Sp(2N_c)$ Seiberg duality
  11. Reductions of ${\cal N}=2$ models
  12. Example: $SU(N)$ $N_f=2N$ $S$-duality

Figures (1)

  • Figure 1: Quivers for the dual theories. Here the central node is an $SU(N)$ gauge group, while the outer nodes are $U(N)$ flavor groups, with $N_c=\frac{1}{2}(N_1+N_2-N_3)$ and $\hat{N}_c=\frac{1}{2}(N_3+N_2-N_1)$ . Here we must impose $N_2 \geq N_1 + N_3$, and as a result we obtain a duality rather than a triality.