Abstract

We consider the order parameter $u=\left<{\rm Tr}\phi^2\right>$ as function of the running coupling constant $\tau \in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $\tau$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $\Gamma$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $\Gamma\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.

Highlights

  • A manifestation of S duality or strong-weak coupling duality is the equivalent dynamics of a quantum field theory at distinct values of its coupling constant [1,2,3,4,5]

  • We address this question for asymptotically free N 1⁄4 2 Yang-Mills theories with gauge group SUð2Þ and Nf ≤ 3 fundamental hypermultiplets

  • On the other hand if we take the appropriate scaling limit near the conformal field theory point [34], we find that the disconnected region is a fundamental domain for the order parameter of the AD theory

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Summary

INTRODUCTION

A manifestation of S duality or strong-weak coupling duality is the equivalent dynamics of a quantum field theory at distinct values of its coupling constant [1,2,3,4,5]. We address this question for asymptotically free N 1⁄4 2 Yang-Mills theories with gauge group SUð2Þ and Nf ≤ 3 fundamental hypermultiplets To this end, we consider the order parameter for the Coulomb branch, which is a function of the running coupling τ invariant under S duality [3,6,7,8,9,10]. (iii) Merging of nonlocal singularities (AD theories): The dynamics is quite different if we tune the masses to special values where singularities corresponding to nonlocal dyons collide in the u plane Such singularities give rise to superconformal AD field theories [33,34,35,36,37,38]. If no other branch points remain in F Nf , the order parameters become modular functions for a congruence subgroup As a by-product of our analyses we propose an expression for the beta functions of the massive Nf ≤ 3 theories, generalizing the results of [42,43]

The SW solutions
Partitioning the upper half-plane
Ramification locus
Partitioning the u plane
MATONE’S RELATION FOR MASSIVE THEORIES
Periods and Weierstraß form
Matone’s relation
Branch points
The massless theory
Type II AD mass
Generic real mass
Generic complex mass
Equal masses
Fundamental domain
Partitioning of the u plane
Two distinct masses
Type III AD mass choose m1
One nonzero mass
Generic masses
Type IV AD mass
Type III AD mass
DISCUSSION
Modular forms
Modular curves

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