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Register a new userNon-semisimple TQFTs and BPS q-series
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We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides -- which until recently were developed independently, using very different methods -- opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) $q$-series invariants labeled by spin$^c$ structures in terms of non-semisimple invariants. In the opposite direction, it offers new insights and perspectives on various elements of non-semisimple TQFTs, bringing the latter into one unifying framework with other invariants of knots and 3-manifolds that recently found realization in quantum field theory and in string theory.
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Many BPS partition functions depend on a choice of additional structure: fluxes, Spin or Spin$^c$ structures, etc. In a context where the BPS generating series depends on a choice of Spin$^c$ structure we show how different limits with respect to the expansion variable $q$ and different ways of summing over Spin$^c$ structures produce different invariants of homology cobordisms out of the BPS $q$-series.
A generalised orbifold of a defect TQFT $mathcal{Z}$ is another TQFT $mathcal{Z}_{mathcal{A}}$ obtained by performing a state sum construction internal to $mathcal{Z}$. As an input it needs a so-called orbifold datum $mathcal{A}$ which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of $3$-dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category $mathcal{W}_{mathcal{A}}$ that we canonically associate to $mathcal{Z}$ and $mathcal{A}$. We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with $mathcal{Z}_{mathcal{A}}$ easier to handle in specific examples.
The physical 3d $mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $hat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $hat{Z}_a(q)$ for some hyperbolic 3-manifolds.
We initiate a systematic study of 3-dimensional `defect topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.
We study the 19-vertex model associated with the quantum group $U_q(hat{sl_2})$ at critical regime $|q|=1$. We give the realizations of the type-I vertex operators in terms of free bosons and free fermions. Using these free field realizations, we give the integral representations for the correlation functions.
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