We develop a general framework for the description of anomalies using extended functorial field theories extending previous work by Freed and Monnier. In this framework, anomalies are described by invertible field theories in one dimension higher and anomalous field theories live on their boundaries. We provide precise mathematical definitions for all concepts involved using the language of symmetric monoidal bicategories. In particular, field theories with anomalies will be described by symmetric monoidal transformations. The use of higher categorical concepts is necessary to capture the Hamiltonian picture of anomalies. The relation to the path integral and the Hamiltonian description of anomalies will be explained in detail. Furthermore, we discuss anomaly inflow in detail. We apply the general framework to the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We use the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose to explicitly construct an extended invertible field theory encoding the anomaly. This allows us to compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space. As a second application, we study discrete symmetries of Dijkgraaf-Witten theories and their gauging. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory.
In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz-Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have …
We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra in \textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.
Translation-invariant noncommutative gauge theories are discussed in the setting of matrix modeled gauge theories. Using the matrix model formulation the explicit form of consistent anomalies and consistent Schwinger terms for translation-invariant noncommutative gauge theories are derived.
We give a modern geometric viewpoint on anomalies in quantum field theory and illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This is background for the resolution of worldsheet anomalies in orientifold superstring theory.
We study discrete symmetries of Dijkgraaf-Witten theories and their gauging in the framework of (extended) functorial quantum field theory. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory. This allows us to calculate the 2-cocycle twisting the projective representation of physical …
We compute the global gauge and gravitational anomalies of the A-type (2,0) superconformal quantum field theories in six dimensions, and conjecture a formula valid for the D- and E-type theories. We show that the anomaly contains terms that do not contribute to the local anomaly but that are crucial for the consistency of the global anomaly. A side result is an intuitive picture for the appearance of Hopf-Wess-Zumino terms on the Coulomb branch of the (2,0) theories.
We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the state space, or that the latter is really an abelian gerbe rather than an ordinary Hilbert space. We include in the discussion the case of non-invertible anom…
For Chern-Simons-matter theories in three dimensions, gauge invariance may require the Chern-Simons level k to be half-integral, in which case parity is violated. As noted by Pasquetti for abelian theories with N=2 supersymmetry, the partition function on the ellipsoid also admits a suitable holomorphic factorization precisely when the value of k is properly quantized. Using known formulas for the partition function, we investigate analytic aspects of this factorization for non-abelian gauge groups and general matter representations. We verify that factorization occurs in accord with the parit…
We propose a closed formula of the universal part of supersymmetric Rényi entropy $S_q$ for $(2,0)$ superconformal theories in six-dimensions. We show that $S_q$ across a spherical entangling surface is a cubic polynomial of $γ:=1/q$, with all coefficients expressed in terms of the newly discovered Weyl anomalies $a$ and $c$. This is equivalent to a similar statement of the supersymmetric free energy on conic (or squashed) six-sphere. We first obtain the closed formula by promoting the free tensor multiplet result and then provide an independent derivation by assuming that $S_q$ can be written…
We give a functorial definition of $G$-gerbes over a simplicial complex when the local symmetry group $G$ is non-Abelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the space of edge-paths. By computing the curvature of the latter on the faces of an infinitesimal 4-simplex, we recover the cocycle identities satisfied by the curvature of this gerbe. The link with $BF$-theories suggests that gerbes provide a framework adapted to the geometric formulation of stro…
We prove the Adler-Bardeen theorem in a large class of general gauge theories, including nonrenormalizable ones. We assume that the gauge symmetries are general covariance, local Lorentz symmetry and Abelian and non-Abelian Yang-Mills symmetries, and that the local functionals of vanishing ghost number satisfy a variant of the Kluberg-Stern--Zuber conjecture. We show that if the gauge anomalies are trivial at one loop, for every truncation of the theory there exists a subtraction scheme where they manifestly vanish to all orders, within the truncation. Outside the truncation the cancellation o…
We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul-Tate and the full BRST differential in the BV-BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an $L_\infty$ algebra. The de…
In a Hamiltonian approach to anomalies parity and time reversal symmetries can be restored by introducing suitable impure (or mixed) states. However, the expectation values of observables such as the Hamiltonian diverges in such impure states. Here we show that such divergent expectation values can be treated within a renormalization group framework, leading to a set of $β$-functions in the moduli space of the operators representing the observables. This leads to well defined expectation values of the Hamiltonian in a phase where the impure state restores the $P$ and $T$ symmetry. We also show…
We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $θ$-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 time-reversal). We also study…
