Interconnections between type II superstrings, M theory and N=4 supersymmetric Yang--Mills

TL;DR

Green's notes unify M theory, type II strings, and ${\cal N}=4$ SYM through a derivative-expansion framework, showing that protected higher-derivative terms like ${\cal R}^4$ are fixed by nonlinear and modular-SUSY constraints and are captured by modular forms $f^{(w,-w)}$ invariant under $SL(2,\mathbb{Z})$. They demonstrate that these terms arise from one-loop dynamics in eleven dimensions and map to specific strong-coupling instanton contributions in the boundary Yang–Mills theory via AdS/CFT, including precise D-instanton measures and Witten indices that match zero-dimensional matrix-model partitions. The work illuminates how D-instantons and Yang–Mills instantons correspond across the bulk/boundary duality, with Fourier expansions encoding nonperturbative effects and large-$N$ saddle points reproducing the expected moduli spaces. Collectively, the results provide a cohesive, symmetry-driven picture of nonperturbative effects in string/M-theory and their gauge-theory avatars, with exact modular structures constraining the nonperturbative sectors.

Abstract

These lecture notes begin with a review of the first nonleading contributions to the derivative expansion of the M theory effective action compactified on a two-torus. The form of these higher-derivative interactions is shown to follow from ten-dimensional type IIB supersymmetry as well as from one-loop quantum corrections to classical eleven-dimensional supergravity. The detailed information concerning D-instanton effects encoded in these terms is related to the problem of evaluating the Witten index for $N$ D-particles in the type IIA theory. Using the AdS/CFT conjecture, it also leads to very specific predictions of multi-instanton contributions in $\cal N=4$ supersymmetric SU(N) Yang--Mills theory in the limit of strong 't Hooft coupling. [Extended version of lectures given at 22nd Johns Hopkins Workshop (Gothenberg, August 20-22 1998); `Quantum Aspects of Gauge Theories, Supersymmetry and Unification', TMR meeting (Corfu, September 20-26 1998); Andrjewski lectures (Berlin, November 1-6 1998).]