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.
SciPost Phys. 5, 006 (2018) ·
published 20 July 2018

· pdf
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. 5, 006 (2018) ·
published 20 July 2018

· pdf
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

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

· pdf
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.
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