Anomalies in the space of coupling constants and their dynamical applications I

Abstract

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 Poincaré symmetry) to background gauge fields (and a metric for the Poincaré 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 four-dimensional 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 four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.2 In condensed matter physics, symmetry protected topological orders (SPTs) are also characterized at low energies by such actions. Depending on the precise definitions and context, "SPT" may be synonymous with "invertible field theory", or may instead refer to the deformation class of an invertible field theory, i.e. the equivalence class of invertible theories obtained by continuously varying parameters.3 In certain cases, there is no Y such that BY " X and A on X is extended into Y . Then, one can construct an anomaly free partition function by assuming that X is a component of the boundary of Y and Y has additional boundary components.

Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory

“…In the gauge theory formulation, this is achieved by including particular discrete topological terms in the bulk action. The resulting partition functions are still calculable, and can be expressed in terms of restricted classes of projective representations of the group G. We find that the dual matrix integral again factors into distinct sectors, each now corresponding to a genuinely projective representation of G. These results dovetail nicely with recent discussions in the QFT literature [46][47][48] and are relevant for the complex SYK model, which exhibits a mixed anomaly between the global U p1q symmetry and charge conjugation when the number of Dirac fermions is odd [30,49].…”

“…We denote the connected component of the identity in Op2q by Op2q`, and the other connected component by Op2q´. An Op2q symmetry naturally arises in systems with a U p1q symmetry that are invariant under charge conjugation, and the possible mixed anomaly between U p1q and Z 2 has interesting consequences both in quantum mechanics and in gauge theory [46,48]. 19 By an Op2q BF gauge theory, we mean a sum over all flat Op2q bundles.…”

Generalized ’t Hooft anomalies on non-spin manifolds

“…Here i = 1, ..., 2N f and all λ i are undotted SL(2, C) spinors 12. We stress that while comparing the results in these references to the ones given here, one should keep in mind that N f in this paper denotes the number of Dirac, not Weyl flavors.…”

“…However, more exotic scenarios have recently been discussed in [23,28,[30][31][32]. 12 In [30], we conjectured that the theory with N c = 2 and a single N f = 1 Dirac fermion will form a massless composite, schematically given by (λ) 3 , a doublet under the enhanced SU (2N f ) = SU (2) flavor symmetry, accompanied by the breaking Z 8 → Z 4 , due to an SU (2) invariant four-fermion condensate. This IR scenario has to be supplemented by a TQFT that matches a mixed anomaly between the 0-form discrete chiral and 1-form center symmetries on non-spin backgrounds [23], further studied in [16,19,31].…”

“…The study of these generalized anomalies is a currently active area of research with contributions coming from the high-energy, condensed matter, and mathematical communities. We do not claim to be in command of all points of view and only give a list of references written from a (largely) high-energy physics perspective and pertaining to theories somewhat similar to the ones discussed in this paper [7][8][9][10][11][12][13][14][15][16][17][18][19].…”