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Contents

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

References

• Wikipedia, Boundary conformal field theory

For more see the references at defect field theory.

Comments in the context of multisymplectic geometry are in

• José Zapata, Multisymplectic effective General Boundary Field Theory (2013) [[pdf](http://relativity.phys.lsu.edu/ilqgs/zapata090313.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]