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

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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

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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. 7, 056 (2019) ·
published 28 October 2019

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We use conformal embeddings involving exceptional affine KacMoody algebras
to derive new dualities of threedimensional topological field theories. These
generalize the familiar levelrank duality of ChernSimons theories based on
classical gauge groups to the setting of exceptional gauge groups. For
instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow
SU(9N)_{1}$. 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 bosonboson ChernSimons
matter dualities. We also use them to determine candidate phase diagrams of
timereversal invariant $G_{2}$ gauge theory coupled to either an adjoint
fermion, or two fundamental fermions.
SciPost Phys. 5, 006 (2018) ·
published 20 July 2018

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We study continuum quantum field theories in 2+1 dimensions with
timereversal 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 nontrivial 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 timereversal 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 longdistance 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 twoindex 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 nonabelian symmetry involving timereversal.
SciPost Phys. 4, 021 (2018) ·
published 29 April 2018

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We study threedimensional 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 levelrank duality for these topological
field theories. Our results may also be viewed as levelrank duality for
$SO(N)_{K}$ ChernSimons 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 bosonfermion
dualities for vector matter, and present the phase diagram of theories with
twoindex tensor fermions, thus extending previous results for $SO(N)$ to other
global forms of the gauge group.