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TQFTs by definition satisfy cutting and gluing axioms. Roughly speaking, you should be able to obtain the partition function of the TQFT on a general (closed) manifold by cutting the manifold into small, elementary pieces which we understand, and then the partition function can be calculated from assembling the pieces together. This holds very generally in ...

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The mathematical framework that I am familiar with for abelian p-form gauge theory (the one promoted by Freed, Moore and others) is that of Cheeger-Simons differential forms. In this framework, the space of topologically trivial p-form gauge fields over a manifold $X$ (the analogue of 1-form gauge fields on the trivial $U(1)$ bundle) are identified with the ...

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A very late answer, but for symmetry-protected topological (SPT) phases, I believe it is true (certainly, no counterexamples are known) that the boundary is "non-trivial" if and only if the bulk is a non-trivial SPT phase. Here "non-trivial" boundary has a very specific meaning. A boundary is "non-trivial" if there is NO symmetry-respecting terms that we can ...

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