Exact WKB analysis of ℂℙ1 holomorphic blocks

Abstract

We study holomorphic blocks in the three dimensional N = 2 gauge theory that describes the CP 1 model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the 1 φ 1 q-hypergeometric function. We show in detail how the q-Borel resummation methods reproduce the results obtained previously by using block-integral methods.

Stokes phenomena in 3d $$ \mathcal{N} $$ = 2 SQED2 and $$ \mathbbm{CP} $$1 models

“…Now we look at the Stokes regions and find that the correct CP 1 Stokes regions [26,34] are obtained in the decoupling limit. The limiting procedure picks out three regions from the top-rear region (upper half of the red-blue quadrants) of figure 3 and is shown in figure 6.…”

“…symmetry. But it is more general, as shown in [34] by following an algebraic approach of constructing the holomorphic blocks. This algebraic approach involves solving certain qdifference equations called line operator identities (LOIs) which are known to annihilate the holomorphic blocks as discussed in [26].…”

“…We conjecture that this can be achieved via transformation of the SQED holomorphic blocks (obtained in the first step around regular singularities) by certain discrete transformations that leave invariant the supersymmetric parameter space L SUSY of the SQED model compactified on S 1 . This second step is reminiscent of the mirror symmetry transformations acting on CP 1 blocks which is related to Stokes phenomenon [26,34].…”

“…Let us recall that the CP 1 model [26,34] is described by a U(1) gauged linear sigma model with two chiral fields having same charge under the gauge group. This GLSM flows to a non-linear sigma model with CP 1 target space in the infrared.…”

“…But in this case, one can use q-Borel resummation and express the solutions in terms of the solutions found near the regular singular point [36]. Using this technique, it is possible (though arduous) to find two more pairs of holomorphic blocks [34]. These holomorphic blocks can also be found by using the Z 3 symmetry transformations, better known as the mirror symmetry transformations [26]:…”