Exact WKB methods in SU(2) Nf = 1

We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) $$ \mathcal{N} $$ N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

Chemical-potential-assisted particle production in FRW spacetimes

We analyze gravitational particle production assisted by chemical potential. By utilizing the uniformly smoothed Stokes-line method and Borel summation, we gain insight into the fine-grained history of enhanced particle production. Analytic/semi-analytic formulae describing the production amount, time and width are obtained for both spin-1 and spin-1/2 particles in various FRW spacetimes. Our work also serves as a concrete demonstration of the uniformly smoothed Stokes-line method applied to cosmology.

Exact Values of the Gamma Function from Stirling’s Formula

In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica.

Describing phase transitions in field theory by self-similar approximants

Self-similar approximation theory is shown to be a powerful tool for describing phase transitions in quantum field theory. Self-similar approximants present the extrapolation of asymptotic series in powers of small variables to the arbitrary values of the latter, including the variables tending to infinity. The approach is illustrated by considering three problems: (i) The influence of the coupling parameter strength on the critical temperature of the O(N)-symmetric multicomponent field theory. (ii) The calculation of critical exponents for the phase transition in the O(N)-symmetric field theory. (iii) The evaluation of deconfinement temperature in quantum chromodynamics. The results are in good agreement with the available numerical calculations, such as Monte Carlo simulations, Padé-Borel summation, and lattice data.