Abstract
Topological string theory near the conifold point of a Calabi--Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their Borel plane (also known as ``peacock patterns"), and we conjecture that the corresponding Stokes constants are integer invariants of the Calabi--Yau threefold. We calculate these Stokes constants in some toric examples, confirming our conjecture and providing in some cases explicit generating functions for the new integer invariants, in the form of $q$-series. Our calculations in the toric case rely on the TS/ST correspondence, which promotes the asymptotic series near the conifold point to spectral traces of operators, and makes it easier to identify the Stokes data. The resulting mathematical structure turns out to be very similar to the one of complex Chern--Simons theory. In particular, spectral traces correspond to state integral invariants and factorize in holomorphic/anti-holomorphic blocks.
Highlights
In the case of the topological string, the Stokes constants are precisely the new integer invariants that we define in this paper, we can regard these as analogues of the DGG index
In this paper we have identified an infinite family of asymptotic power series arising naturally in topological string theory, by considering the conifold free energies and “quantizing" the flat coordinates
11We further improved the precision of Borel resummation with the help of conformal maps [92, 93]. Resurgent structure of these series are encoded in “peacock patterns", namely, infinite towers of singularities in the Borel plane with integer Stokes constants. The latter can be regarded as new integer invariants of Calabi–Yau threefolds
Summary
One of the most fruitful applications of supersymmetry to geometry is the possibility to define and compute enumerative invariants based on the counting of BPS states. In [9,10,11,12] it has been advocated that the connection between BPS invariants and analytic problems can be formulated more generally by using the theory of resurgence (see [13] for similar ideas) In this approach, the starting analytic data are asymptotic, perturbative series obtained in an appropriate quantum theory. The asymptotic series obtained in this way are called quantum periods, and, as shown in [10], the corresponding Stokes constants are essentially the BPS invariants studied with the WKB method in [7] Another example is provided by complex Chern–Simons (CS) theory on the complement of a hyperbolic knot. In this case, the asymptotic series are defined by saddle-point expansions of the quantum invariants around classical solutions, and lead to a very rich resurgent structure. The third Appendix summarizes the Hunter–Guerrieri algorithm, which makes it possible to extract trans-series from the large order behavior of a Gevrey-1 series in the oscillatory case, where usual Richardson extrapolation cannot be applied
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