Abstract
A global symmetry of a quantum field theory is said to have an ’t Hooft anomaly if it cannot be promoted to a local symmetry of a gauged theory. In this paper, we show that the anomaly is also an obstruction to defining symmetric boundary conditions. This applies to Lorentz symmetries with gravitational anomalies as well. For theories with perturbative anomalies, we demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions and current Ward identities in the presence of a boundary. We then recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, also forbid the existence of boundary conditions. This conjecture holds for known gravitational anomalies in D ≤ 3 which allows us to conclude the obstruction result for D ≤ 4.
Highlights
That the combined bulk-boundary system can be coupled consistently to a background gauge field [2]
We recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, forbid the existence of boundary conditions
By analyzing the Wess-Zumino consistency conditions and the anomaly-descent procedure, we show that the existence of a symmetric boundary requires the corresponding Schwinger term in the descent equations to trivialize, which in turn demands the anomaly polynomial for the relevant symmetries to take a factorized form depending on central U(1) factors of the symmetry group
Summary
At the top of the descent equations is Q(20n)+1, a Chern-Simons-type term which represents the action of a D + 1-dimensional bulk theory on the boundary of which T is gauge invariant. It is associated with a degree 2n + 2 anomaly polynomial I2n+2[T ] = dQ(20n)+1, a polynomial in the background gauge field strength F (B) (which includes the Riemann curvature 2-form R). Up to d-exact c-number ambiguities due to redefinitions of the Gauss-law operators by terms involving the background gauge field, the solution is determined by the anomaly and given by the following term in the descent equations [18]. Where Γμνρ is the Christoffel connection and Q2n+1 differs from Q2n+1 by an exact 2n + 1form
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