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Global Symmetries and N=2 SUSY

Jock McOrist, Ilarion V. Melnikov, Brian Wecht

TL;DR

The work addresses how non-R global symmetries persist in ${\cal N}=2$ gauge theories formed by gauging a subalgebra of $\operatorname{Sp}(n)$ acting on $n$ free hypermultiplets. It develops a representation-theoretic framework to analyze the centralizer $\mathfrak{C}_{\mathfrak{g}}$ of the gauged algebra in $\mathfrak{sp}(n)$, proving that $\mathfrak{C}_{\mathfrak{g}}$ decomposes as a direct sum of $\mathfrak{sp}(k_i)$, $\mathfrak{so}(l_p)$, and $\mathfrak{u}(m_q)$ factors, with a corresponding explicit decomposition of the fundamental representation. This result rules out simple realizations of $\,\mathfrak{su}(N)^3$-type symmetries without accompanying $\mathfrak{u}(1)$ factors in such Lagrangian constructions, and it analyzes how discrete gaugings and Higgs-branch physics could alter or enhance symmetries. The findings provide a stringent constraint on possible UV completions for ${\cal N}=2$ theories and have implications for understanding IR fixed points like the $T_N$ theories, guiding future classification efforts of conformal and asymptotically free gauge theories with ${\cal N}=2$ supersymmetry.

Abstract

We prove that N=2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N=2 gauge theories.

Global Symmetries and N=2 SUSY

TL;DR

The work addresses how non-R global symmetries persist in gauge theories formed by gauging a subalgebra of acting on free hypermultiplets. It develops a representation-theoretic framework to analyze the centralizer of the gauged algebra in , proving that decomposes as a direct sum of , , and factors, with a corresponding explicit decomposition of the fundamental representation. This result rules out simple realizations of -type symmetries without accompanying factors in such Lagrangian constructions, and it analyzes how discrete gaugings and Higgs-branch physics could alter or enhance symmetries. The findings provide a stringent constraint on possible UV completions for theories and have implications for understanding IR fixed points like the theories, guiding future classification efforts of conformal and asymptotically free gauge theories with supersymmetry.

Abstract

We prove that N=2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N=2 gauge theories.

Paper Structure

This paper contains 10 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Symmetries of free fields
  3. Representation theory
  4. A few familiar gaugings
  5. Warm-up: decomposing pseudoreal representations
  6. Global symmetries
  7. Discussion
  8. Discrete gauge symmetries
  9. Higgs branch
  10. General discussion and conclusions

Key Result

Theorem 1

Let $\operatorname{\mathfrak{g}}$ be a semisimple subalgebra of $\operatorname{\mathfrak{sp}}(n)$. Then the commutant subalgebra ${{\mathfrak{C}_{\operatorname{\mathfrak{g}}}}}$ of $\operatorname{\mathfrak{g}}$ in $\operatorname{\mathfrak{sp}}(n)$ is and the fundamental of $\operatorname{\mathfrak{sp}}(n)$ decomposes under $\operatorname{\mathfrak{sp}}(n) \supset \operatorname{\mathfrak{g}} \oplu

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4