target category of extended field theory

Question

For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (ie its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (physical) considerations constrain the choice of target category? The targets of invertible field theories are (by definition) $\infty$-groupoids and (therefore) can be represented as spectra. What constraints can we impose on target spectra of invertible theories; in particular, on their homotopy groups?

This post imported from StackExchange Physics at 2014-08-03 09:26 (UCT), posted by SE-user user151696

Comments

What is the definition of TFT and the definiton of A-S theory you are using?

This post imported from StackExchange Physics at 2014-08-03 09:26 (UCT), posted by SE-user ACuriousMind

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An A-S TFT is a functor from $\text{Bord}_{<n-1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor from $\text{Bord}_n(\mathcal{F})$ to a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ that restricts to an A-S TFT on $(n-1)$- and $n$-manifolds.

This post imported from StackExchange Physics at 2014-08-03 09:26 (UCT), posted by SE-user user151696

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Comment to the question (v2): Consider adding some references to make the question more accessible to a wider audience.

This post imported from StackExchange Physics at 2014-08-03 09:26 (UCT), posted by SE-user Qmechanic

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10 years old question.