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Higgsing Transitions from Topological Field Theory & Non-Invertible Symmetry in Chern-Simons Matter Theories
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Abstract
Non-invertible one-form symmetries are naturally realized in (2+1)d topological quantum field theories. In this work, we consider the potential realization of such symmetries in (2+1)d conformal field theories, investigating whether gapless systems can exhibit similar symmetry structures. To that end, we discuss transitions between topological field theories in (2+1)d which are driven by the Higgs mechanism in Chern-Simons matter theories. Such transitions can be modeled mesoscopically by filling spacetime with a lattice-shaped domain wall network separating the two topological phases. Along the domain walls are coset conformal field theories describing gapless chiral modes trapped by a locally vanishing scalar mass. In this presentation, the one-form symmetries of the transition point can be deduced by using anyon condensation to track lines through the domain wall network. Using this framework, we discuss a variety of concrete examples of non-invertible one-form symmetry in fixed-point theories. For instance, Chern-Simons theory coupled to a scalar in the symmetric tensor representation produces a transition from an phase to an phase and has non-invertible one-form symmetry at the fixed point. We also discuss theories with Spin(2N) and gauge groups manifesting other patterns of non-invertible one-form symmetry. In many of our examples, the non-invertible one-form symmetry is not a modular invariant TQFT on its own and thus is an intrinsic part of the fixed-point dynamics.
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We introduce a definition and framework for internal topological symmetries in quantum field theory, including “noninvertible symmetries” and “categorical symmetries”. We outline a calculus of topological defects which takes advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called “gauging” and “condensation defects”), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.
In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N) 𝔰 𝔬 ( N ) , and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.
The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.
Given any symmetry acting on a d-dimensional quantum field theory, there is an associated (d+1)-dimensional topological field theory known as the Symmetry TFT (SymTFT). The SymTFT is useful for decoupling the universal quantities of quantum field theories, such as their generalized global symmetries and ’t Hooft anomalies, from their dynamics. In this work, we explore the SymTFT for theories with Kramers-Wannier-like duality symmetry in both (1+1)d and (3+1)d quantum field theories. After constructing the SymTFT, we use it to reproduce the non-invertible fusion rules of duality defects, and along the way we generalize the concept of duality defects to higher duality defects. We also apply the SymTFT to the problem of distinguishing intrinsically versus non-intrinsically non-invertible duality defects in (1+1)d.
A bstract
In axion-Maxwell theory at the minimal axion-photon coupling, we find non-invertible 0- and 1-form global symmetries arising from the naive shift and center symmetries. Since the Gauss law is anomalous, there is no conserved, gauge-invariant, and quantized electric charge. Rather, using half higher gauging, we find a non-invertible Gauss law associated with a non-invertible 1-form global symmetry, which is related to the Page charge. These symmetries act invertibly on the axion field and Wilson line, but non-invertibly on the monopoles and axion strings, leading to selection rules related to the Witten effect. We also derive various crossing relations between the defects. The non-invertible 0- and 1-form global symmetries mix with other invertible symmetries in a way reminiscent of a higher-group symmetry. Using this non-invertible higher symmetry structure, we derive universal inequalities on the energy scales where different infrared symmetries emerge in any renormalization group flow to the axion-Maxwell theory. Finally, we discuss implications for the Weak Gravity Conjecture and the Completeness Hypothesis in quantum gravity.
A bstract
It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ ) ² . We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category C . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.
We sketch a procedure to capture general non-invertible symmetries of a d-dimensional quantum field theory in the data of a higher-category, which captures the local properties of topological defects associated to the symmetries. We also discuss fusions of topological defects, which involve condensations/gaugings of higher-categorical symmetries localized on the worldvolumes of topological defects. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. This is not possible in the standard description of higher-categories. We explain that the dimension-changing fusions are understood as higher-morphisms of the higher-category describing the symmetry.
We also discuss how a 0-form sub-symmetry of a higher-categorical symmetry can be gauged and describe the higher-categorical symmetry of the theory obtained after gauging. This provides a procedure for constructing non-invertible higher-categorical symmetries starting from invertible higher-form or higher-group symmetries and gauging a 0-form symmetry. We illustrate this procedure by constructing non-invertible 2-categorical symmetries in 4d gauge theories and non-invertible 3-categorical symmetries in 5d and 6d theories. We check some of the results obtained using our approach against the results obtained using a recently proposed approach based on 't Hooft anomalies.
A bstract
We study U( N ) Chern-Simons theory coupled to massive fundamental fermions in the lightcone Hamiltonian formalism. Focusing on the planar limit, we introduce a consistent regularization scheme, identify the counter terms needed to restore relativistic invariance, and formulate scattering theory in terms of unambiguously defined asymptotic states. We determine the 2 → 2 planar S-matrix element in the singlet channel by solving the Lippmann-Schwinger equation to all orders, establishing a result previously conjectured in the literature.
A bstract
We explore the notion of approximate global symmetries in quantum field theory and quantum gravity. We show that a variety of conjectures about quantum gravity, including the weak gravity conjecture, the distance conjecture, and the magnetic and axion versions of the weak gravity conjecture can be motivated by the assumption that generalized global symmetries should be strongly broken within the context of low-energy effective field theory, i.e. at a characteristic scale less than the Planck scale where quantum gravity effects become important. For example, the assumption that the electric one-form symmetry of Maxwell theory should be strongly broken below the Planck scale implies the weak gravity conjecture. Similarly, the violation of generalized non-invertible symmetries is closely tied to analogs of this conjecture for non-abelian gauge theory. This reasoning enables us to unify these conjectures with the absence of global symmetries in quantum gravity.
A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge c_- c − has to vanish. In this paper, we consider conditions associated with ``higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.
We introduce a class of noninvertible topological defects in (3+1)D gauge theories whose fusion rules are the higher-dimensional analogs of those of the Kramers-Wannier defect in the (1+1)D critical Ising model. As in the lower-dimensional case, the presence of such noninvertible defects implies self-duality under a particular gauging of their discrete (higher-form) symmetries. Examples of theories with such a defect include SO(3) Yang-Mills (YM) at θ=π, N=1 SO(3) super YM, and N=4 SU(2) super YM at τ=i. We also introduce an analogous construction in (2+1)D, and give a number of examples in Chern-Simons-matter theories.
A bstract
It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices : codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.
Makie.jl is a cross-platform plotting ecosystem for the Julia programming language (Bezanson et al., 2012), which enables researchers to create high-performance, GPU-powered, interactive visualizations, as well as publication-quality vector graphics with one unified interface.
The infrastructure based on Observables.jl allows users to express how a visualization
depends on multiple parameters and data sources, which can then be updated live, either
programmatically, or through sliders, buttons and other GUI elements. A sophisticated layout
system makes it easy to assemble complex figures. It is designed to avoid common difficulties
when aligning nested subplots of different sizes, or placing colorbars or legends freely without
spacing issues. Makie.jl leverages the Julia type system to automatically convert many kinds
of input arguments which results in a very flexible API that reduces the need to manually prepare data. Finally, users can extend every step of this pipeline for their custom types through
Julia’s powerful multiple dispatch mechanism, making Makie a highly productive and generic
visualization system.
A bstract
We study a 3d lattice gauge theory with gauge group U(1) N− 1 ⋊ S N , which is obtained by gauging the S N global symmetry of a pure U(1) N− 1 gauge theory, and we call it the semi-Abelian gauge theory. We compute mass gaps and string tensions for both theories using the monopole-gas description. We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire SU( N ) root system. Even though the center symmetry of the semi-Abelian gauge theory is given by ℤ N , we observe that the string tensions do not obey the N -ality rule and carry more detailed information on the representations of the gauge group. We find that this refinement is due to the presence of non-invertible topological lines as a remnant of U(1) N− 1 one-form symmetry in the original Abelian lattice theory. Upon adding charged particles corresponding to W -bosons, such non-invertible symmetries are explicitly broken so that the N -ality rule should emerge in the deep infrared regime.
The deconfined quantum critical point (QCP), separating the Néel and valence bond solid phases in a 2D antiferromagnet, was proposed as an example of ð2 þ 1ÞD criticality fundamentally different from standard Landau-Ginzburg-Wilson-Fisher criticality. In this work, we present multiple equivalent descriptions of deconfined QCPs, and use these to address the possibility of enlarged emergent symmetries in the low-energy limit. The easy-plane deconfined QCP, besides its previously discussed self-duality, is dual to N f ¼ 2 fermionic quantum electrodynamics, which has its own self-duality and hence may have an Oð4Þ × Z T 2 symmetry. We propose several dualities for the deconfined QCP with SU(2) spin symmetry which together make natural the emergence of a previously suggested SO(5) symmetry rotating the Néel and valence bond solid orders. These emergent symmetries are implemented anomalously. The associated infrared theories can also be viewed as surface descriptions of ð3 þ 1ÞD topological paramagnets, giving further insight into the dualities. We describe a number of numerical tests of these dualities. We also discuss the possibility of "pseudocritical" behavior for deconfined critical points, and the meaning of the dualities and emergent symmetries in such a scenario.
We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on Z_N one-form symmetries. A 3d topological quantum field theory (TQFT) 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 T=T′⊗A^(N,p). Here T′ is a decoupled TQFT, whose lines are neutral under the global symmetry and A^(N,p) is a minimal TQFT with the Z_N one-form symmetry of label p. The parameter p labels the obstruction to gauging the Z_N one-form 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 low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent θ-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 Z_N one-form 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.
We study continuum quantum field theories in 2+1 dimensions with
time-reversal symmetry . The standard relation 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
with a non-trivial global symmetry. For example, acting on monopole
operators, could be 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 , 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.
We study three-dimensional gauge theories based on orthogonal groups.
Depending on the global form of the group these theories admit discrete
-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
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.
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev’s quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
We discuss in detail level/rank duality in three-dimensional Chern-Simons theories and various related dualities in three-dimensional Chern-Simons-matter theories. We couple the dual Lagrangians to appropriate background fields (including gauge fields, spin connections and the metric). The non-trivial maps between the currents and the line operators in the dual theories is accounted for by mixing of these fields. In order for the duality to be valid we must add finite counterterms depending on these background fields. This analysis allows us to resolve a number of puzzles with these dualities, to provide derivations of some of them, and to find new consistency conditions and relations between them. In addition, we find new level/rank dualities of topological Chern-Simons theories and new dualities of Chern-Simons-matter theories, including new boson/boson and fermion/fermion dualities.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state
of a half-filled Landau level. This theory should be applicable for a Dirac
fermion in magnetic field at charge neutrality, as well as for the
quantum Hall ground state of nonrelativistic fermions in the
limit of negligible inter-Landau-level mixing. We argue that when particle-hole
symmetry is exact, the composite fermion is a massless Dirac fermion,
characterized by a Berry phase of around the Fermi circle. We write down
a tentative effective field theory of such a fermion and discuss the discrete
symmetries, in particular . The Dirac composite fermions
interact through a gauge, but non-Chern-Simons, interaction. The particle-hole
conjugate pair of Jain-sequence states at filling factors and
, which in the conventional composite fermion picture
corresponds to integer quantum Hall states with different filling factors, n
and n+1, is now mapped to the same half-integer filling factor of
the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are
interpreted as d-wave Bardeen-Cooper-Schrieffer paired states of the Dirac
fermion with orbital angular momentum of opposite signs, while s-wave pairing
would give rise to a novel particle-hole symmetric nonabelian gapped phase.
When particle-hole symmetry is not exact, the Dirac fermion has a breaking mass. The conventional fermionic Chern-Simons theory is
shown to emerge in the nonrelativistic limit of the massive theory.
A q-form global symmetry is a global symmetry for which the charged
operators are of space-time dimension q; e.g. Wilson lines, surface defects,
etc., and the charged excitations have q spatial dimensions; e.g. strings,
membranes, etc. Many of the properties of ordinary global symmetries (q=0)
apply here. They lead to Ward identities and hence to selection rules on
amplitudes. Such global symmetries can be coupled to classical background
fields and they can be gauged by summing over these classical fields. These
generalized global symmetries can be spontaneously broken (either completely or
to a subgroup). They can also have 't Hooft anomalies, which prevent us from
gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies
can also lead to anomaly inflow on various defects and exotic Symmetry
Protected Topological phases. Our analysis of these symmetries gives a new
unified perspective of many known phenomena and uncovers new results.
We present explicit computations and conjectures for scattering
matrices in large N {\it U(N)} Chern-Simons theories coupled to fundamental
bosonic or fermionic matter to all orders in the 't Hooft coupling expansion.
The bosonic and fermionic S-matrices map to each other under the recently
conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices
presented in this paper may be regarded as relativistic generalization of
Aharonov-Bohm scattering. They have unusual structural features: they include a
non analytic piece localized on forward scattering, and obey modified crossing
symmetry rules. We conjecture that these unusual features are properties of
S-matrices in all Chern-Simons matter theories. The S-matrix in one of the
exchange channels in our paper has an anyonic character; the parameter map of
the conjectured Bose-Fermi duality may be derived by equating the anyonic phase
in the bosonic and fermionic theories.
We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On
d
the new theory differs from the original one by the spectrum of operators. Sometimes the local operators are the same but there are different line operators, surface operators, etc. The effects of the added topological degrees of freedom are more dramatic when we compactify
d
, and they are crucial in the context of electric-magnetic duality. We explore several examples including DijkgraafWitten theories and their generalizations both in the continuum and on the lattice. When we couple them to ordinary quantum field theories the topological degrees of freedom allow us to express certain characteristic classes of gauge fields as integrals of local densities, thus simplifying the analysis of their physical consequences.
Instead of constructing anyon condensation in various concrete models, we
take a bootstrap approach by considering an abstract situation, in which an
anyon condensation happens in a 2-d topological phase with anyonic excitations
given by a modular tensor category C; and the anyons in the condensed phase are
given by another modular tensor category D. By a bootstrap analysis, we derive
a relations between anyons in D-phase and anyons in C-phase from natural
physical requirements. It turns out that the vacuum (or the tensor unit) A in
D-phase is necessary to be a connected commutative separable algebra in C, and
the category D is equivalent to the category of local A-modules as moduar
tensor categories. This condensation also produce a gapped domain wall with
wall excitations given by the category of A-modules in C. More general
situation is also discussed in this paper. We will also show how to determine
such algebra A from the initial and final data. Multi-condensations and 1-d
condensations will also be briefly discussed. Examples will be given in the
toric code model, Kitaev quantum double models, Levin-Wen types of lattice
models and some chiral topological phases.
Non-Abelian anyons promise to reveal spectacular features of quantum
mechanics that could ultimately provide the foundation for a decoherence-free
quantum computer. A key breakthrough in the pursuit of these exotic particles
originated from Read and Green's observation that the Moore-Read quantum Hall
state and a (relatively simple) two-dimensional p+ip superconductor both
support so-called Ising non-Abelian anyons. Here we establish a similar
correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel
two-dimensional superconductor in which charge-2e Cooper pairs are built from
fractionalized quasiparticles. In particular, both phases harbor Fibonacci
anyons that---unlike Ising anyons---allow for universal topological quantum
computation solely through braiding. Using a variant of Teo and Kane's
construction of non-Abelian phases from weakly coupled chains, we provide a
blueprint for such a superconductor using Abelian quantum Hall states
interlaced with an array of superconducting islands. Fibonacci anyons appear as
neutral deconfined particles that lead to a two-fold ground-state degeneracy on
a torus. In contrast to a p+ip superconductor, vortices do not yield additional
particle types yet depending on non-universal energetics can serve as a trap
for Fibonacci anyons. These results imply that one can, in principle, combine
well-understood and widely available phases of matter to realize non-Abelian
anyons with universal braid statistics. Numerous future directions are
discussed, including speculations on alternative realizations with fewer
experimental requirements.
We compute the thermal free energy for all renormalizable Chern Simon
theories coupled to a single fundamental bosonic and fermionic field in the 't
Hooft large N limit. We use our results to conjecture a strong weak coupling
duality invariance for this class of theories. Our conjectured duality reduces
to Giveon Kutasov duality when restricted to {\cal N}=2 supersymmetric theories
and to an earlier conjectured bosonization duality in an appropriate decoupling
limit. Consequently the bosonization duality may be regarded as a deformation
of Giveon Kutasov duality, suggesting that it is true even at large but finite
N.
After recalling the main properties of a conformal embedding of Lie algebrasgp, which is defined by the equality of the Sugawara central charges on both sides, we launch a systematic study of their branching rules. The bulk of the paper is devoted to the proof of a general formula in the casesu(mn)
1su(m)
nsu(n)
m. At the end we give some applications to the construction of modular invariant partition functions.
Using the outer automorphisms of the affine algebra , we show how the branching rules for the conformal subalgebra may be simply calculated. We demonstrate that new modular invariant combinations of characters are obtainable from the branching rules.
It is well‐known that gauging a finite 0‐form symmetry in a quantum field theory leads to a dual symmetry generated by topological Wilson line defects. These are described by the representations of the 0‐form symmetry group which form a 1‐category. We argue that for a d‐dimensional quantum field theory the full set of dual symmetries one obtains is in fact much larger and is described by a (d−1)‐category, which is formed out of lower‐dimensional topological quantum field theories with the same 0‐form symmetry. We study in detail a 2‐categorical piece of this (d−1)‐category described by 2d topological quantum field theories with 0‐form symmetry. We further show that the objects of this 2‐category are the recently discussed 2d condensation defects constructed from higher‐gauging of Wilson lines. Similarly, dual symmetries obtained by gauging any higher‐form or higher‐group symmetry also form a (d−1)‐category formed out of lower‐dimensional topological quantum field theories with that higher‐form or higher‐group symmetry. A particularly interesting case is that of the 2‐category of dual symmetries associated to gauging of finite 2‐group symmetries, as it describes non‐invertible symmetries arising from gauging 0‐form symmetries that act on (d−3)‐form symmetries. Such non‐invertible symmetries were studied recently in the literature via other methods, and our results not only agree with previous results, but our approach also provides a much simpler way of computing various properties of these non‐invertible symmetries. We describe how our results can be applied to compute non‐invertible symmetries of various classes of gauge theories with continuous disconnected gauge groups in various spacetime dimensions. We also discuss the 2‐category formed by 2d condensation defects in any arbitrary quantum field theory.
Duality refers to two equivalent descriptions of the same theory from different points of view. Recently there has been tremendous progress in formulating and understanding possible dualities of quantum many body theories in 2+1-spacetime dimensions. Of particular interest are dualities that describe conformally invariant quantum field theories in (2+1)d. These arise as descriptions of quantum critical points in condensed matter physics. The appreciation of the possible dual descriptions of such theories has greatly enhanced our understanding of some challenging questions about such quantum critical points. Perhaps surprisingly the same dualities also underlie recent progress in our understanding of other problems such as the half-filled Landau level and correlated surface states of topological insulators. Here we provide a pedagogical review of these recent developments from a point of view geared toward condensed matter physics.
We introduce a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory, which is based on a coset factorization, leads to a solvable one dimensional model that is extended to two dimensions using a network construction. In addition to providing a description of the Fibonacci phase without parafermions, our theory predicts a closely related "anti-Fibonacci" phase, whose topological order is characterized by the tricritical Ising model. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that generalizes the Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.
Bose condensation is central to our understanding of quantum phases of matter. Here we review Bose condensation in topologically ordered phases (also called topological symmetry breaking), where the condensing bosons have non-trivial mutual statistics with other quasiparticles in the system. We give a non-technical overview of the relationship between the phases before and after condensation, drawing parallels with more familiar symmetry-breaking transitions. We then review two important applications of this phenomenon. First, we describe the equivalence between such condensation transitions and pairs of phases with gappable boundaries, as well as examples where multiple types of gapped boundary between the same two phases exist. Second, we discuss how such transitions can lead to global symmetries which exchange or permute anyon types. Finally we discuss the nature of the critical point, which can be mapped to a conventional phase transition in some -- but not all -- cases.
The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past 30 years, has generated many ground-breaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the 30 years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of non-Abelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.
Particle-vortex duality is a powerful theoretical tool that has been used to study bosonic systems. Here, we propose an analogous duality for Dirac fermions in 2+1 dimensions. The physics of a single Dirac cone is proposed to be described by a dual theory, QED3, with again a single Dirac fermion but coupled to a gauge field. This duality is established by considering two alternate descriptions of the three-dimensional topological insulator (TI) surface. The first description is the usual Dirac fermion surface state. The dual description is accessed via an electric-magnetic duality of the bulk TI coupled to a gauge field, which maps it to a gauged chiral topological insulator. This alternate description ultimately leads to a new surface theory, QED3, which provides a simple description of otherwise intractable interacting electronic states. For example, an explicit derivation of the T-Pfaffian state, a proposed surface topological order of the TI, is obtained by simply pair condensing the dual fermions. The roles of time-reversal and particle-hole symmetries are exchanged by the duality, which connects some of our results to a recent conjecture by Son on particle-hole symmetric quantum Hall states.
Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in 2+1 dimensions that generalize the known particle/vortex duality. Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the 2+1-dimensional analog of QED with a single Dirac fermion (a theory known as ) is identified with the O(2) Wilson-Fisher fixed point. The gauged version of that fixed point with a Chern-Simons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to 3+1-dimensional S-duality.
There is significant evidence for a duality between (non-supersymmetric) U(N ) Chern-Simons theories at level k coupled to fermions, and U(k) Chern-Simons theories at level N coupled to scalars. Most of the evidence comes from the large N ’t Hooft limit, where many details of the duality (such as whether the gauge group is U(N ) or SU(N ), the precise level of the U(1) factor, and order one shifts in the level) are not important. The main evidence for the validity of the duality at finite N comes from adding masses and flowing to pure Chern-Simons theories related by level-rank duality, and from flowing to the non-supersymmetric duality from supersymmetric dualities, whose finite N validity is well-established. In this note we clarify the implications of these flows for the precise form of the duality; in particular we argue that in its simplest form the duality maps SU(N ) theories to U(k) theories, though there is also another version relating U(N ) to U(k). This precise form strongly affects the mapping under the duality of baryon and monopole operators, and we show, following arguments by Radičević, that their mapping is consistent with our claims. We also discuss the implications of our results for the additional duality between these Chern-Simons matter theories and (the UV completion of) high-spin gravity theories on AdS
4. The latter theories should contain heavy particles carrying electric and/or magnetic charges under their U(1) gauge symmetry.
A state without gapless excitations may be characterized as an
'incompressible quantum fluid', and variational wave functions of
Jastrow form which describe such states have been proposed by Laughlin
(1983) at occupations nu = 1/m, m being an odd integer. The Laughlin
wave functions are not translationally invariant, but describe a cicular
droplet of fluid, which must be confined in an external potential. In
the present investigation, a description is provided of a variant of
Laughlin's scheme with fully translationally invariant wave functions.
The scheme is extended to describe a hierarchy of fluid states with
occupation factors given by a continued fractions expression. A
two-dimensional electron gas of N particles is placed on a spherical
surface of radius R, in a radial (monopole) magnetic field. This
approach allows the construction of homogeneous states with finite N.
Quasiparticles at the fractional quantized Hall states obey quantization rules appropriate to particles of fractional statistics. A natural set of approximations for the ground-state energies and energy gaps at all levels of the hierarchy of fractional quantized Hall states are obtained by assuming that the dominant interaction between quasiparticles is just the Coulomb interaction between the quasiparticle charges. Stable states at various rational filling factors are iteratively constructed by adding quasiparticles or holes to lower-order states and estimating the corresponding energies.
The coset construction is the most important tool to construct rational
conformal field theories with known chiral data. For some cosets at small
level, so-called maverick cosets, the familiar analysis using selection and
identification rules breaks down. Intriguingly, this phenomenon is linked to
the existence of exceptional modular invariants. Recent progress in CFT, based
on studying algebras in tensor categories, allows for a universal construction
of the chiral data of coset theories which in particular also applies to
maverick cosets.
A previous construction of unitary representations of the Virasoro algebra is extended and interpreted physically in terms of a coset space quark model. The quaternionic projective spaces HPn−1 yield the complete range of possible values for the central charge when it is less than unity, namely 1 − 6/(n + 1)(n + 2). The supersymmetric extension is also found.
A path integral formulation of G/H conformal theories is presented, based on gauged Wess-Zumino-Witten actions. The conformal charge of the model is c=cG−cH, exactly as in the GKO construction. Ghost modes appear in the formalism as a result of gauge fixing. As a consequence, the holomorphic stress-tensor T(z) obtained in our construction has ghost contributions. We find T(z)=TG−TH only for matrix elements of physical states which are also ghost free.
We show that the coset construction for affine algebras ĝ ⊃ ĥ can be realized by coupling a group G WZW model to a gauge field taking values in the Lie algebra h. The partition function of the coset models is computed exactly in terms of the branching functions of ĝ⊃ĥ. Correlation functions may be expressed in terms of those of the G-valued WZW model and of the Hscc/H-valued one, also exactly soluble. The special cases include unitary, superconformal, paramefermionic and other discrete series.
For every tensor category there is a braided tensor category , the ‘center’ of . It is well known to be related to Drinfel'd's notion of the quantum double of a finite dimensional Hopf algebra H by an equivalence of braided tensor categories. In the Hopf algebra situation, whenever D(H)-mod is semisimple (which is the case iff D(H) is semisimple iff H is semisimple and cosemisimple iff S2=id and ) it is modular in the sense of Turaev, i.e. its S-matrix is invertible. (This was proven by Etingof and Gelaki in characteristic zero. We give a fairly general proof in the appendix.) The present paper is concerned with a generalization of this and other results to the quantum double (center) of more general tensor categories.We consider -linear tensor categories with simple unit and finitely many isomorphism classes of simple objects. We assume that is either a ∗-category (i.e. and there is a positive ∗-operation on the morphisms) or semisimple and spherical over an algebraically closed field . In the latter case we assume , where the summation runs over the isomorphism classes of simple objects. We prove that (i) is a semisimple spherical (or ∗-) category and (ii) is weakly monoidally Morita equivalent (in the sense of Müger (J. Pure Appl. Algebrea 180 (2003) 81–157)) to . This implies . (iii) We analyze the simple objects of in terms of certain finite dimensional algebras, of which Ocneanu's tube algebra is the smallest. We prove the conjecture of Gelfand and Kazhdan according to which the number of simple objects of coincides with the dimension of the state space of the torus in the triangulation TQFT built from . (iv) We prove that is modular and we compute . (v) Finally, if is already modular then , where is the tensor category with the braiding .
All known rational conformal field theories may be obtained from (2+1) dimensional Chern-Simons gauge theories by appropriate choice of gauge group. We conjecture that all rational field theories are classified by groups via (2+1)-dimensional Chern-Simons gauge theories.
We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore–Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A–A-bimodules.The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties.We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore–Seiberg data.