Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups

TL;DR

The paper advances a unified framework for three-dimensional gauge theories with orthogonal groups by carefully treating global forms, discrete θ-parameters, and background fields for discrete symmetries. It derives explicit level-rank dualities for SO(N) and related groups, including precise counterterm maps and anomaly considerations, and extends these dualities to Chern-Simons–matter theories via gauging of zero-form symmetries. It then applies these results to phase diagrams for tensor and adjoint matter, providing consistency checks through conformal embeddings, operator dimensions, and line counts. The work connects TQFT dualities with CS–Matter dualities across multiple global forms, offering a robust toolkit for analyzing 3D gauge dynamics and related topological phases in condensed matter contexts.

Abstract

We study three-dimensional gauge theories based on orthogonal groups. Depending on the global form of the group these theories admit discrete $θ$-parameters, which control the weights in the sum over topologically distinct gauge bundles. We derive level-rank duality for these topological field theories. Our results may also be viewed as level-rank duality for $SO(N)_{K}$ Chern-Simons theory in the presence of background fields for discrete global symmetries. In particular, we include the required counterterms and analysis of the anomalies. We couple our theories to charged matter and determine how these counterterms are shifted by integrating out massive fermions. By gauging discrete global symmetries we derive new boson-fermion dualities for vector matter, and present the phase diagram of theories with two-index tensor fermions, thus extending previous results for $SO(N)$ to other global forms of the gauge group.

Figures (7)

• Figure 1:
• Figure 2: The phase diagram of $Spin(N)$ gauge theory coupled to an adjoint fermion. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate transitions from the semiclassical phase to a quantum phase. Each transition is weakly coupled in a dual theory, which covers part of the phase diagram. The dual theory has symmetric tensor fermions ($S$ and $\widehat{S}$). Across these transitions the superscript level of the TQFT jumps, i.e. $O(N)^{0}$ and $O(N)^{1}$ are exchanged. The fractional levels represent the effective contributions from the massless fermions. At a special value of the mass in the quantum phase, the ultraviolet theory is supersymmetric. This symmetry is spontaneously broken leading to a massless Goldstino Witten:1999ds. This phase diagram is obtained from that of the $SO(N)$ theory proposed in Gomis:2017ixy, by tracking the global symmetries and counterterms and then gauging.
• Figure 3: A map of possible gauge groups obtained by starting with $SO(N)$ and gauging symmetries ($\mathbb{Z}_{2}$ levels suppressed). In Appendix \ref{['gauging']} we discuss some details of this map.
• Figure 4: The phase diagrams of $SO(N)$ gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears. These figures are identical to those in Gomis:2017ixy except that we now add the map of the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$ global symmetry.
• Figure 5: The phase diagrams of $Spin(N)$ gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. Across these tensor transitions $O(L)^{0}$ and $O(L)^{1}$ are exchanged. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears.