[Paper Review] Global Symmetries and N=2 SUSY

Jock McOrist, Ilarion V. Melnikov+1 more|arXiv (Cornell University)|Dec 12, 2013

Black Holes and Theoretical Physics3 citations

TL;DR

This paper proves that N=2 supersymmetric gauge theories arising from gauging a subgroup of Sp(n) in n free hypermultiplets retain global symmetries that are direct sums of unitary, symplectic, and special orthogonal algebras, but do not contain isolated su(N) factors without accompanying u(1) symmetries. This rules out simple UV Lagrangian descriptions for T_N theories with only SU(N)^3 global symmetry, such as Gaiotto’s T_N, as they would require an ungauged U(N) enhancement.

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.

Motivation & Objective

• To classify the global symmetry algebras that arise in N=2 supersymmetric gauge theories obtained by gauging subalgebras of Sp(n) in n free hypermultiplets.
• To determine whether theories with only SU(N)^3 global symmetry—such as Gaiotto’s T_N theories—can arise as IR fixed points of weakly coupled N=2 gauge theories.
• To rule out simple UV Lagrangian descriptions for T_N theories by analyzing the structure of their global symmetries and the constraints imposed by gauging Sp(n).
• To provide a general proof of the classification of remaining global symmetries after gauging, filling a gap in the literature despite known results among experts.
• To explore the implications of these symmetry constraints for the existence of asymptotically free UV completions of strongly coupled N=2 SCFTs like T_N theories.

Proposed method

• Start from a theory of n free hypermultiplets, which has a non-R global symmetry algebra 𝔪𝔞𝔱hfrak(spin)(n) ≅ 𝔪𝔞𝔱hfrak(spin)(2n) for the bosonic components.
• Gauge a subalgebra 𝔤 ⊂ 𝔪𝔞𝔱hfrak(spin)(n), reducing the global symmetry to the commutant of 𝔤 in 𝔪𝔞𝑡hfrak(spin)(n).
• Use representation theory of Lie algebras to decompose the matter multiplets under the remaining global symmetry, focusing on how the fundamental representation of Sp(n) decomposes under the gauge group.
• Apply the theorem that the remaining global symmetry algebra is a direct sum of 𝔪𝔞𝑡hfrak(su), 𝔪𝔞𝑡hfrak(sp), and 𝔪𝔞𝑡hfrak(so) factors, with no isolated 𝔪𝔞𝑡hfrak(su)(N) without a corresponding 𝔪𝔞𝑡hfrak(u)(1).
• Analyze specific examples such as T_3 and T_4 theories, showing that their global symmetry algebras (e.g., SU(3)^3 or SU(4)^3) cannot be realized without additional U(1) factors.
• Use the structure of the Seiberg-Witten curve and known duality relations to contextualize the results within the broader class of N=2 SCFTs.

Experimental results

Research questions

• RQ1 Can N=2 gauge theories with only SU(N)^3 global symmetry, such as Gaiotto’s T_N theories, arise as IR fixed points of weakly coupled UV gauge theories?
• RQ2 What is the structure of the remaining global symmetry algebra after gauging a subgroup of Sp(n) in a theory of n free hypermultiplets?
• RQ3 Why do isolated 𝔪𝔞𝑡hfrak(su)(N) factors not appear in the global symmetry algebra of such N=2 gauge theories without accompanying 𝔪𝔞𝑡hfrak(u)(1) factors?
• RQ4 To what extent can the global symmetry structure constrain the existence of a UV Lagrangian description for strongly coupled N=2 SCFTs?
• RQ5 How do Higgs branch dynamics and strong coupling effects affect the global symmetry structure in the IR?

Key findings

• After gauging a subalgebra of the Sp(n) global symmetry of n free hypermultiplets, the remaining global symmetry algebra is a direct sum of 𝔪𝔞𝑡hfrak(so), 𝔪𝔞𝑡hfrak(sp), and 𝔪𝔞𝑡hfrak(u) factors, with no isolated 𝔪𝔞𝑡hfrak(su)(N) algebras.
• Theories with only SU(N)^3 global symmetry, such as Gaiotto’s T_N theories, cannot be realized as IR fixed points of weakly coupled N=2 gauge theories because such symmetries would require an ungauged U(N) enhancement.
• For the T_3 theory, the naive global symmetry 𝔪𝔞𝑡hfrak(su)(3)^3 is enhanced to 𝔪𝔞𝑡hfrak(e)_6, and the matter transforms in the 78-dimensional representation, which cannot be realized from a simple gauging of Sp(4).
• The T_4 theory, with global symmetry 𝔪𝔞𝑡hfrak(su)(4)^3 ≅ 𝔪𝔞𝑡hfrak(so)(6)^3, cannot be realized from a weakly coupled UV Lagrangian because the matter would transform in the 6-dimensional representation of each so(6), not the 4-dimensional spinor representation required.
• The analysis rules out simple UV Lagrangian descriptions for T_N theories that do not explore the Higgs branch or involve strong coupling, as the symmetry structure is incompatible with isolated 𝔪𝔞𝑡hfrak(su)(N) factors.
• The result provides a strong constraint on the possible global symmetry algebras of N=2 asymptotically free and conformal field theories, suggesting a classification framework based on the decomposition of Sp(n) representations under gauging.