2 fixed error pointed out by Noah

You can find some of the details in a draft version of my TQFT notes on my web page. Here's a summary.

Given an n-category with strong duality (by which I mean, more or less, pivotal if n=2 and a higher dimensional version of pivotal for n > 2; this is stronger than what Lurie means by an n-category "with duals"), there's a standard procedure to construct an n+1-dimensional TQFT. This procedure works for free for the 0- through n-dimensional parts of the (extended) n+1-dimensional TQFT. In these dimensions, we need no additional assumptions on the n-cat, and there is no need to choose a decomposition of the manifolds, so there are no combinatorial topology moves to check. The construction is manifestly invariant.

To construct the top, n+1-dimensional part of the TQFT, we need to make some finiteness assumptions on the n-category. (This corresponds to the top-dimensional part of what Lurie et al mean by "fully dualizable".) If the n-category satisfies these assumption, then we get, for each handle decomposition of the n+1-manifold, a state sum type expression for the path integral of the n+1-manifold. It is not hard to show that this state sum is invariant under handle slides and handle cancelation, so we get a well-defined invariant of n+1-manifolds that interacts with the rest of the TQFT (via gluing formulas) in the correct way.

(Small technicality: the path integral construction depends on a choice of element in the Hilbert space of the n-sphere, corresponding to the path integral of the n+1-ball. Multiplying this choice by \lambda changes the path integral by \lambda^\chi, where \chi is the Euler characteristic of the n+1-manifold.)

A modular tensor category is a 3-category with strong duality and the right sort of finiteness properties, so we can apply the above construction to get a 3+1-dimensional TQFT.

In dimension 3 the vector space we construct is an old-fashioned skein module: finite linear combinations of ribbon graphs in M^3 modulo local relations. (Actually, the dual of this vector space.) If M is closed this in 1-dimensional. More generally, if M has boundary then it has the same dimension as the Witten-Reshetikhin-Turaev vector space associated to the boundary of M.

In dimension 4, the type of state sum we get depends on the type of handle decomposition. For a general handle decomposition we get the Crane-Yetter state sum. For 2-handles attached to a 4-ball along a framed link L we get the Reshetikhin-Turaev surgery formula for L. For a 4-dimensional neighborhood of a 2-complex we get the Turaev "shadow" state sum. For a closed 4-manifold we find that the path integral is equal a^\chi b^\sigma, where \chi is the Euler characteristic and \sigma is the signature of the 4-manifold. By choosing \lambda above appropriately we can make a=1. b is related to the total quantum dimension central charge of the MTC (or maybe to the inverse value of that)the RT surgery formula on framing +-1 unknot). For a 4-manifold with boundary we find that the state sum computes the Witten-Reshetikhin-Turaev invariant of the boundary of the 4-manifold.

In dimension 2, the 1-category associated to a closed surface is a full matrix category; i.e. it is Morita trivial. For a surface with boundary k circles the category is Morita equivalent to k copies of the MTC thought of as a 1-category.

In all of the above cases, we find that the TQFT invariant of Z(X), where dim(X) = 2 or 3 or 4, depends strongly on the boundary of X but only weakly (i.e. only up to bordism) in the interior of X. So we can define a new 2+1 dimensional TQFT Z' via the formula

Z'(Y) := Z(boundary^{-1}(Y)).

This TQFT has an anomaly, since we need to enhance Y with enough extra structure to pick out an inverse-boundary, up to bordism.

1

You can find some of the details in a draft version of my TQFT notes on my web page. Here's a summary.

Given an n-category with strong duality (by which I mean, more or less, pivotal if n=2 and a higher dimensional version of pivotal for n > 2; this is stronger than what Lurie means by an n-category "with duals"), there's a standard procedure to construct an n+1-dimensional TQFT. This procedure works for free for the 0- through n-dimensional parts of the (extended) n+1-dimensional TQFT. In these dimensions, we need no additional assumptions on the n-cat, and there is no need to choose a decomposition of the manifolds, so there are no combinatorial topology moves to check. The construction is manifestly invariant.

To construct the top, n+1-dimensional part of the TQFT, we need to make some finiteness assumptions on the n-category. (This corresponds to the top-dimensional part of what Lurie et al mean by "fully dualizable".) If the n-category satisfies these assumption, then we get, for each handle decomposition of the n+1-manifold, a state sum type expression for the path integral of the n+1-manifold. It is not hard to show that this state sum is invariant under handle slides and handle cancelation, so we get a well-defined invariant of n+1-manifolds that interacts with the rest of the TQFT (via gluing formulas) in the correct way.

(Small technicality: the path integral construction depends on a choice of element in the Hilbert space of the n-sphere, corresponding to the path integral of the n+1-ball. Multiplying this choice by \lambda changes the path integral by \lambda^\chi, where \chi is the Euler characteristic of the n+1-manifold.)

A modular tensor category is a 3-category with strong duality and the right sort of finiteness properties, so we can apply the above construction to get a 3+1-dimensional TQFT.

In dimension 3 the vector space we construct is an old-fashioned skein module: finite linear combinations of ribbon graphs in M^3 modulo local relations. (Actually, the dual of this vector space.) If M is closed this in 1-dimensional. More generally, if M has boundary then it has the same dimension as the Witten-Reshetikhin-Turaev vector space associated to the boundary of M.

In dimension 4, the type of state sum we get depends on the type of handle decomposition. For a general handle decomposition we get the Crane-Yetter state sum. For 2-handles attached to a 4-ball along a framed link L we get the Reshetikhin-Turaev surgery formula for L. For a 4-dimensional neighborhood of a 2-complex we get the Turaev "shadow" state sum. For a closed 4-manifold we find that the path integral is equal a^\chi b^\sigma, where \chi is the Euler characteristic and \sigma is the signature of the 4-manifold. By choosing \lambda above appropriately we can make a=1. b is the total quantum dimension of the MTC (or maybe the inverse of that). For a 4-manifold with boundary we find that the state sum computes the Witten-Reshetikhin-Turaev invariant of the boundary of the 4-manifold.

In dimension 2, the 1-category associated to a closed surface is a full matrix category; i.e. it is Morita trivial. For a surface with boundary k circles the category is Morita equivalent to k copies of the MTC thought of as a 1-category.

In all of the above cases, we find that the TQFT invariant of Z(X), where dim(X) = 2 or 3 or 4, depends strongly on the boundary of X but only weakly (i.e. only up to bordism) in the interior of X. So we can define a new 2+1 dimensional TQFT Z' via the formula

Z'(Y) := Z(boundary^{-1}(Y)).

This TQFT has an anomaly, since we need to enhance Y with enough extra structure to pick out an inverse-boundary, up to bordism.