*Article* **Topologically Protected Duality on The Boundary of Maxwell-BF Theory**

#### **Alberto Blasi 1 and Nicola Maggiore 1,2,***∗*


Received: 4 June 2019; Accepted: 10 July 2019; Published: 15 July 2019

**Abstract:** The Maxwell-BF theory with a single-sided planar boundary is considered in Euclidean four-dimensional spacetime. The presence of a boundary breaks the Ward identities, which describe the gauge symmetries of the theory, and, using standard methods of quantum field theory, the most general boundary conditions and a nontrivial current algebra on the boundary are derived. The electromagnetic structure, which characterizes the boundary, is used to identify the three-dimensional degrees of freedom, which turn out to be formed by a scalar field and a vector field, related by a duality relation. The induced three-dimensional theory shows a strong–weak coupling duality, which separates different regimes described by different covariant actions. The role of the Maxwell term in the bulk action is discussed, together with the relevance of the topological nature of the bulk action for the boundary physics.

**Keywords:** quantum field theory; topological quantum field theory; duality in gauge field theories; boundary quantum field theory

**PACS:** 03.70.+k Theory of quantized fields; 11.10.-z Field theory
