Clay Cordova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg
SciPost Phys. 8, 002 (2020) ·
published 7 January 2020

· pdf
We extend our earlier work on anomalies in the space of coupling constants to
fourdimensional gauge theories. Pure YangMills theory (without matter) with a
simple and simply connected gauge group has a mixed anomaly between its
oneform 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 timereversal). 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 recentlydiscovered phenomena in these
theories and controls phase transitions and the dynamics on interfaces.
Clay Cordova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg
SciPost Phys. 8, 001 (2020) ·
published 6 January 2020

· 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 fourdimensional 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
fourdimensional gauge theories showing how our view unifies and extends many
recently obtained results.
SciPost Phys. 6, 039 (2019) ·
published 29 March 2019

· pdf
We study 3d and 4d systems with a oneform global symmetry, explore their
consequences, and analyze their gauging. For simplicity, we focus on
$\mathbb{Z}_N$ oneform symmetries. A 3d topological quantum field theory
(TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate
it. The braiding of these lines and their spins are characterized by a single
integer $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes
$\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. Here $\mathcal{T}'$ is a
decoupled TQFT, whose lines are neutral under the global symmetry and
$\mathcal{A}^{N,p}$ is a minimal TQFT with the $\mathbb{Z}_N$ oneform symmetry
of label $p$. The parameter $p$ labels the obstruction to gauging the
$\mathbb{Z}_N$ oneform symmetry; i.e.\ it characterizes the 't Hooft anomaly
of the global symmetry. When $p=0$ mod $2N$, the symmetry can be gauged.
Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with
gauge fields extended to the bulk. This understanding allows us to consider
$SU(N)$ and $PSU(N)$ 4d gauge theories. Their dynamics is gapped and it is
associated with confinement and oblique confinement  probe quarks are
confined. In the $PSU(N)$ theory the lowenergy theory can include a discrete
gauge theory. We will study the behavior of the theory with a spacedependent
$\theta$parameter, which leads to interfaces. Typically, the theory on the
interface is not confining. Furthermore, the liberated probe quarks are anyons
on the interface. The $PSU(N)$ theory is obtained by gauging the $\mathbb{Z}_N$
oneform symmetry of the $SU(N)$ theory. Our understanding of the symmetries in
3d TQFTs allows us to describe the interface in the $PSU(N)$ theory.
Mr Lam: "Question 1 is a good one, and ..."
in Report on Anomalies in the Space of Coupling Constants and Their Dynamical Applications I