We study Ward identities and selection rules for local correlators in disordered theories where a 0-form global symmetry of a QFT is explicitly broken by a random coupling h h but it re-emerges after quenched average. We consider h h space-dependent or constant. In both cases we construct the symmetry operator implementing the group action, topological after average. In the first case, relevant in statistical systems with random impurities, such symmetries can be coupled to external backgrounds and can be gauged, like ordinary symmetries in QFTs. We also determine exotic selection rules arising when symmetries emerge after average in the IR, explaining the origin of LogCFTs from symmetry considerations. In the second case, relevant in AdS/CFT to describe the dual boundary theory of certain bulk gravitational theories, the charge operator is not purely codimension-1, it can be defined only on homologically trivial cycles and on connected spaces. Selection rules for average correlators exist, yet such symmetries cannot be coupled to background gauge fields in ordinary ways and cannot be gauged. When the space is disconnected, in each connected component charge violation occurs, as expected from Euclidean wormholes in the bulk theory. Our findings show the obstruction to interpret symmetries emergent after average as gauged in the bulk.
The Sachdev--Ye--Kitaev (SYK) model has a wormholelike solution after averaging over the fermionic couplings in the nearly
AdS2 space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as ``half-wormholes.'' In this paper, we will study the fate of these wormholes in a model without quenched disorder, namely, a tensor model with $O(N{)}^{q\ensuremath{-}1}$ gauge symmetry whose correlation function and thermodynamics in the large
N limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson operator, in terms of global charges or nontrivial cobordism classes associated with disconnected wormholes. Therefore for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our ``half-wormholes.'' We will also comment on the late time behavior of the spectral form factor, particularly its leading and subleading order contributions coming from higher genus wormholes in the gravitational sector. Finally we will show how the other nontrivial saddles from half-wormhole dominate and give rise to unusual thermodynamics in the bulk sector due to nonperturbative effects.
We investigate the interplay of generalized global symmetries in 2+1 dimensions in a lattice model that couples a \mathbb{Z}_N ℤ N clock model to a \mathbb{Z}_N ℤ N gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed ’t Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend this lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules.
It is widely believed that there are no exact global symmetries in quantum gravity. We review recent works showing that global symmetries can nevertheless emerge after ensemble averages of theories, as encountered in holography. As a precision case study, we discuss the ensemble average of two-dimensional Narain-type conformal field theories associated with an even quadratic form of general signature, which is computed by the Siegel-Weil formula and interpreted as a sum over geometries in a three-dimensional Abelian Chern-Simons theory. Global symmetries of the bulk anyons can emerge after the average, as T-dualities relating different theories in the ensemble are "folded" into symmetries of a single theory (duality origami). We discuss the relation to the Swampland program, including the emergence of global symmetries at infinite distance. Finally, we recast strong-to-weak spontaneous symmetry breaking (SWSSB) of mixed states as an emergence of symmetries in ensemble averages; in holography, wormholes connecting the two sides of a thermofield-double construction provide its bulk realization. This motivates a new Swampland conjecture.
We analyze
2+1d and
3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group
G by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat
G gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat
G gauge fields to obtain Dijkgraaf-Witten topological
G gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in
2+1d and condensing bosonic gauge charges in
3+1d. In the dual boundary approach, we study
1+1d and
2+1d quantum field theories that have
G 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct
G characters that generate topological data for the bulk topological gauge theory. Finally, we study a
2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where
Z2T/R is time-reversal/reflection symmetry, and the
U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in
3+1d that cancels this anomaly. This signals the existence of SPTs in
3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
We study one-dimensional disordered systems with average non-invertible symmetries, where quenched disorder may locally break part of the symmetry while preserving it upon disorder averaging. A canonical example is the random transverse-field Ising model, which at criticality exhibits an average Kramers-Wannier duality. We consider the general setting in which the full symmetry is described by a
G-graded fusion category
B, whose identity component
A remains exact, while the components with nontrivial
G-grading are realized either exactly or only on average. We develop a topological holographic framework that encodes the symmetry data of the 1D system in a 2D topological order
Z[A] (the Drinfeld center of
A), enriched by an exact or, respectively, average
G symmetry. Within this framework, we obtain a complete classification of anomalies and average symmetry-protected topological (SPT) phases: when the components with nontrivial
G-grading are realized only on average, the symmetry is anomaly-free if and only if
Z[A] admits a magnetic Lagrangian algebra that is invariant under the permutation action of
G on anyons. When an anomaly is present, we show that the ground state of a single disorder realization is long-range entangled with probability one in the thermodynamic limit, and is expected to exhibit power-law Griffiths singularities in the low-energy spectrum. Finally, we present an explicit, exactly solvable lattice model based on a symmetry-enriched string-net construction. It yields trivial ground state ensemble in the anomaly-free case, and exhibits exotic low-energy behavior in the presence of an average anomaly.
We consider Abelian topological quantum field theories (TQFTs) in 3D and show that gaugings of invertible global symmetries naturally give rise to additive codes. These codes emerge as nonanomalous subgroups of the 1-form symmetry group, parametrizing the fusion rules of condensable TQFT anyons. The boundary theories dual to TQFTs with a maximal symmetry subgroup gauged, i.e., with the corresponding anyons condensed, are "code" conformal field theories (CFTs). This observation bridges together, in the holographic sense, results on 1-form symmetries of 3D TQFTs with developments connecting codes to 2D CFTs. Building on this relationship, we proceed to consider the ensemble of maximal gaugings (topological boundary conditions) in a general, not necessarily Abelian 3D TQFT, and propose that the resulting ensemble of boundary CFTs has a holographic description as a gravitational theory: the bulk TQFT summed over topologies.