nLab boundary field theory

Idea

In FQFT-language, as a (pre-)quantum field theory in dimension $n$ is given by a functor on an (infinity,n)-category of cobordisms, so a field theory with boundaries/branes is given by such a functor on an $(\infty,n)$-category of cobordisms with marked boundaries. The assignments to the boundary data itself defines a kind of field theory of one dimension lower, which is the corresponding boundary field theory.

If the field theory is thought of as a sigma-model that describes the quantum mechanics of a particle or string or membrane, etc., then the boundary condition is also called a brane for that sigma-model.

Definition

See at

• Cobordism theorem – For cobordisms with singuarities (boundaries/branes and defects/domain walls)

• QFT with defects.

Examples

• boundary conformal field theory

Related concepts

• holographic principle

  AdS-CFT correspondence

singularity	field theory with singularities
boundary condition/brane	boundary field theory
domain wall/bi-brane	QFT with defects

References

• Wikipedia, Boundary conformal field theory

For more see the references at defect field theory.

The case of the chiral boson/edge modes WZW model at the boundary of abelian Chern-Simons theory:

• Irais Rubalcava-Garcia: Constructing the theory at the boundary, its dynamics and degrees of freedom [arXiv:2003.06241]

Comments in the context of multisymplectic geometry are in

• José Zapata, Multisymplectic effective General Boundary Field Theory (2013) [pdf, pdf]

Axiomatization and examples of boundary field theories in local prequantum field theory is in

• Local prequantum field theory

• Domenico Fiorenza, Alessandro Valentino, Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors (arXiv:1409.5723)

Discussion of the BV-phase space and Poisson $\infty$-algebras of BF theory and Einstein gravity on corners:

• Alberto Cattaneo, Poisson Structures from Corners of Field Theories, talk at CQTS (22 Nov 2023) [slides: pdf, video: YT]

and application to the standard model of particle physics:

• Giovanni Canepa, Alberto S. Cattaneo, Filippo Fila-Robattino, Manuel Tecchiolli: Boundary structure of the standard model coupled to gravity, Ann. Henri Poincaré (2024) [arXiv:2307.14955, doi:10.1007/s00023-024-01485-4]