Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Geometry behind N=2 Supersymmetric Effective Actions in Four Dimensions

Albrecht Klemm

TL;DR

This work develops a geometrical framework for exact non-perturbative data in four-dimensional $N=2$ supersymmetric Yang-Mills theories by encoding the low-energy dynamics in holomorphic prepotentials and period integrals on auxiliary curves. It combines perturbative inputs with monodromy/Schwarzian uniformization to determine the exact gauge couplings $\tau(a)=\partial^2{\cal F}/\partial a^2$ and BPS spectra, and extends naturally to higher rank groups via Seiberg-Witten curves and special Kähler geometry. When gravity is included, Calabi–Yau compactifications and mirror symmetry relate gauge dynamics to geometric moduli, enabling exact non-perturbative results through period data and topological string methods. The paper provides rigorous consistency checks, including instanton coefficients, dyon spectra, and Picard-Fuchs equations, reinforcing the deep link between non-perturbative field theory and complex geometry. Overall, it demonstrates how intricate geometric structures govern exact non-perturbative phenomena in quantum field theory and string theory.

Abstract

An introduction to Seiberg-Witten theory and its relation to theories which include gravity.

On the Geometry behind N=2 Supersymmetric Effective Actions in Four Dimensions

TL;DR

This work develops a geometrical framework for exact non-perturbative data in four-dimensional supersymmetric Yang-Mills theories by encoding the low-energy dynamics in holomorphic prepotentials and period integrals on auxiliary curves. It combines perturbative inputs with monodromy/Schwarzian uniformization to determine the exact gauge couplings and BPS spectra, and extends naturally to higher rank groups via Seiberg-Witten curves and special Kähler geometry. When gravity is included, Calabi–Yau compactifications and mirror symmetry relate gauge dynamics to geometric moduli, enabling exact non-perturbative results through period data and topological string methods. The paper provides rigorous consistency checks, including instanton coefficients, dyon spectra, and Picard-Fuchs equations, reinforcing the deep link between non-perturbative field theory and complex geometry. Overall, it demonstrates how intricate geometric structures govern exact non-perturbative phenomena in quantum field theory and string theory.

Abstract

An introduction to Seiberg-Witten theory and its relation to theories which include gravity.

Paper Structure

This paper contains 9 sections, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Electric-magnetic duality and BPS-states
  3. $N=2$ supersymmetric Yang-Mills theory
  4. Definition of the $N=2$ Wilsonian action
  5. Reconstruction of the Wilsonian action
  6. The uniformisation problem
  7. N=2 versus N=4 conventions
  8. The symmetries on the dyon spectrum.
  9. Consistency checks:

Figures (4)

  • Figure 1: Charges of dyons, which fulfill the Dirac-Zwanziger quantization condition, lie on a lattice $\Lambda$ spanned by $e\tau$ and $e$ in the complex plane ($e$ is set to one).
  • Figure 2: Wave function renormalization of the effective coupling.
  • Figure 3: One-loop running of the effective coupling.
  • Figure 4: The strip of width 2 above the two largest arcs is the fundamental region of monodromy group $\Gamma(2)$ as found by the method of isometric cycles described in app. B. Its area is by (\ref{['area']}) is $A=2 \pi$ hence six times the one of $SL(2,\hbox{Z Z})$. Because of the identification (\ref{['symmetryi']}) the fundamental region of the quantum symmetry group of pure $SU(2)$ is given by the hatched region, which corresponds to the group $\Gamma_0(2)$, the subgroup of $SL(2,\hbox{Z Z})$ with $C=0\ {\rm mod} \ 2$. The marked point at $\tau_0=-3/2+i/2$ is the $Z_2$ orbifold point of $\Gamma_0(2)$, hence by (\ref{['area']}) $A=\pi$. In particular the identification by the $T$ generator (\ref{['sdual']}) $\tau\rightarrow \tau + 1$ is realized in the $N=2$ theory, while the $S$ generator is not realized.