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I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. Throughout much of their exposition, they rely a lot on the ratio of $P^+$ and $P^-$ equalling 1, such as when discussing the equivalence of modular tensor categories and 3D TQFTs, but it isn't entirely clear to me what these terms mean geometrically.

The definition, as stated, is $$ P^\pm = \sum_{i \in \mathcal{I}} \theta_i^{\pm 1} d_i^2 . $$ Here, $\theta_i$ are the twists and $d_i$ are the quantum dimensions, and the sum is over isotopy classes of the tangles.

While it is hard to share geometric intuition over the internet, any attempt would be greatly appreciated!

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  • $\begingroup$ $P^+$ is related to the central charge $c$ via $c\mod 8= 4\arg(P_+)/\pi$, but I am lacking any intuition, what the geometrical meaning of the central charge is in the RT TQFT. There are some related comments here: mathoverflow.net/questions/68474/… $\endgroup$
    – Marcel Bischoff
    Apr 18, 2014 at 9:14
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    $\begingroup$ The sum is over simple objects of the modular tensor category, not over isotopy classes of tangles. $\endgroup$
    – Kevin Walker
    Apr 19, 2014 at 0:10

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$P^\pm$ is the numerical factor that results from doing $\pm 1$ surgery on a small unknotted framed loop in a 3-manifold (actually the square root of the global dimension times this number, but I'll ignore this overall factor). The surgery operation has no effect on the (homeomorphism class of the) 3-manifold, but it does change the "extended" 3-manifold information (null-bordism, homology framing, or $p_1$-structure). The geometric picture to have in mind is an unknot with framing $\pm 1$, or a bordism from the 3-manifold to itself consisting of a single 4-dimensional 2-handle and signature $\pm 1$.

A more general formula is $$\sum_i J_i(K) d_i ,$$ which is the formula for doing surgery on a framed knot $K$. Here $J_i(K)$ is the generalized Jones polynomial of $K$ with label $i$ and the sum is over simple objects of the modular tensor category. (Depending on one's conventions, there is also an overall constant, which I am ignoring.) When $K$ is an unknot with framing $m$, then $$J_i(K) = \theta_i^m d_i .$$

There are many places to read about this. It's probably explained later in Bakalov and Kirillov (I don't have a copy at hand to check), and it's certainly explained in the book by Kauffman and Lins. It's also explained in many of the journal articles which preceded those two books.

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  • $\begingroup$ Interesting, so what is the geometric significance of having $P^+/P^-=1$? $\endgroup$
    – Benjamin Horowitz
    Apr 20, 2014 at 18:05
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    $\begingroup$ If $P^+ = P^-$ and both are positive real numbers, then the theory has trivial (exponentiated) central charge, and so does not depend on a choice of framing, $p_1$-structure, etc. Turaev-Viro theories have this property. $\endgroup$
    – Kevin Walker
    Apr 20, 2014 at 23:53

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