Contributor info: Dr Clay Córdova

Clay Córdova, Po-Shen Hsin, Carolyn Zhang

SciPost Phys. 17, 131 (2024) · published 12 November 2024 | Toggle abstract · pdf

Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$ obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$ a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.

Clay Córdova, Thomas T. Dumitrescu

SciPost Phys. 16, 139 (2024) · published 29 May 2024 | Toggle abstract · pdf

We study four-dimensional adjoint QCD with gauge group $SU(2)$ and two Weyl fermion flavors, which has an $SU(2)_R$ chiral symmetry. The infrared behavior of this theory is not firmly established. We explore candidate infrared phases by embedding adjoint QCD into ${\mathcal N}=2$ supersymmetric Yang-Mills theory deformed by a supersymmetry-breaking scalar mass $M$ that preserves all global symmetries and 't Hooft anomalies. This includes 't Hooft anomalies that are only visible when the theory is placed on manifolds that do not admit a spin structure. The consistency of this procedure is guaranteed by a nonabelian spin-charge relation involving the $SU(2)_R$ symmetry that is familiar from topologically twisted ${\mathcal N}=2$ theories. Since every vacuum on the Coulomb branch of the ${\mathcal N}=2$ theory necessarily matches all 't Hooft anomalies, we can generate candidate phases for adjoint QCD by deforming the theories in these vacua while preserving all symmetries and 't Hooft anomalies. One such deformation is the supersymmetry-breaking scalar mass $M$ itself, which can be reliably analyzed when $M$ is small. In this regime it gives rise to an exotic Coulomb phase without chiral symmetry breaking. By contrast, the theory near the monopole and dyon points can be deformed to realize a candidate phase with monopole-induced confinement and chiral symmetry breaking. The low-energy theory consists of two copies of a $\mathbb{CP}^1$ sigma model, which we analyze in detail. Certain topological couplings that are likely to be present in this $\mathbb{CP}^1$ model turn the confining solitonic string of the model into a topological insulator. We also examine the behavior of various candidate phases under fermion mass deformations. We speculate on the possible large-$M$ behavior of the deformed ${\mathcal N}=2$ theory and conjecture that the $\mathbb{CP}^1$ phase eventually becomes dominant.

Clay Córdova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg

SciPost Phys. 8, 002 (2020) · published 7 January 2020 | Toggle abstract · pdf

We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $\theta$-parameter. This anomaly is at the root of many recently discovered properties of these theories, including their phase transitions and interfaces. These new anomalies can be used to extend this understanding to systems without discrete symmetries (such as time-reversal). We also study $SU(N)$ and $Sp(N)$ gauge theories with matter in the fundamental representation. Here we find a mixed anomaly between the flavor symmetry group and the $\theta$-periodicity. Again, this anomaly unifies distinct recently-discovered phenomena in these theories and controls phase transitions and the dynamics on interfaces.

Clay Córdova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg

SciPost Phys. 8, 001 (2020) · published 6 January 2020 | Toggle abstract · pdf

It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincar\'{e} symmetry) to background gauge fields (and a metric for the Poincar\'{e} symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.

Clay Córdova, Po-Shen Hsin, Kantaro Ohmori

SciPost Phys. 7, 056 (2019) · published 28 October 2019 | Toggle abstract · pdf

We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow SU(9-N)_{-1}$. Others such as $SO(3)_{8}\leftrightarrow PSU(3)_{-6},$ are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant $G_{2}$ gauge theory coupled to either an adjoint fermion, or two fundamental fermions.

Clay Córdova, Po-Shen Hsin, Nathan Seiberg

SciPost Phys. 5, 006 (2018) · published 20 July 2018 | Toggle abstract · pdf

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry $\cal T$. The standard relation ${\cal T}^2=(-1)^F$ is satisfied on all the "perturbative operators" i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators ${\cal T}^2=(-1)^F {\cal M}$ with $\cal M$ a non-trivial global symmetry. For example, acting on monopole operators, $\cal M$ could be $\pm1$ depending on the magnetic charge. We study in detail $U(1)$ gauge theories with fermions of various charges. Such a modification of the time-reversal algebra happens when the number of odd charge fermions is $2 ~{\rm mod}~4$, e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the long-distance behavior of QED with a single fermion of charge $2$ is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to $SO(N)$ gauge theories with fermions in the vector or in two-index tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising non-abelian symmetry involving time-reversal.

Clay Córdova, Po-Shen Hsin, Nathan Seiberg

SciPost Phys. 4, 021 (2018) · published 29 April 2018 | Toggle abstract · pdf

We study three-dimensional gauge theories based on orthogonal groups. Depending on the global form of the group these theories admit discrete $\theta$-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.