Modular graph functions and modular graph forms map decorated graphs to complex-valued functions on the Poincaré upper half plane with definite transformation properties. Specifically, modular graph functions are SL(2,Z)-invariant functions, while modular graph forms may be identified with SL(2,Z)-invariant differential forms. Modular graph functions and forms generalize, and at the same time unify, holomorphic and non-holomorphic Eisenstein series, almost holomorphic modular forms, multiple zeta-functions, and iterated modular integrals. For example, non-holomorphic Eisenstein series may be associated with one-loop graphs and represent a special class of modular graph functions. The expansion of modular graph forms at the cusp includes Laurent polynomials whose coefficients are combinations of Riemann zeta-values and multiple zeta-values, while each modular graph form may be expanded in a basis of iterated modular integrals. Eisenstein series and modular graph functions and forms beyond Eisenstein series occur naturally and pervasively in the study of the low-energy expansion of superstring amplitudes. Here we shall present a purely mathematical approach with only minimal reference to physics.

Elliptic functions are introduced via the method of images following a review of periodic functions, Poisson summation, the unfolding trick, and analytic continuation applied to the Riemann zeta-function. The differential equations and addition formulas obeyed by periodic and elliptic functions are deduced from their Kronecker–Eisenstein series representation. The classic constructions of elliptic functions, in terms of their zeros and poles, are presented in terms of the Weierstrass elliptic function, the Jacobi elliptic functions, and the Jacobi theta-functions. The elliptic function theory developed here is placed in the framework of elliptic curves, Abelian differentials, and Abelian integrals.

In this appendix, we shall define and study complex line bundles over an arbitrary compact Riemann surface, provide their topological classification in terms divisors, and give the Riemann–Roch theorem. We shall prove various dimension formulas, including for the dimension of the moduli space of complex or conformal structures on a Riemann surface. We then discuss sections of line bundles from a more physics-oriented point of view in terms of spaces of vector fields, differential forms, and spinor fields.

In this appendix, we present detailed solutions to each one the 75 exercises provided in the body of the text, namely 5 exercises for each one of the Chapters 2–16. When appropriate, for the more advanced exercises, we also provide references to the literature where the corresponding problems were discussed.

The starting point for string theory is the idea that the elementary constituents of the theory, which in quantum field theory are assumed to be point-like, are in fact one-dimensional objects, namely strings. As time evolves, a string sweeps out a Riemann surface whose topology governs the interactions that result from joining and splitting strings. The Feynman–Polyakov prescription for quantum mechanical string amplitudes amounts to summing over all topologies of the Riemann surface, for each topology integrating over the moduli of the Riemann surface, and for each value of the moduli solving a conformal field theory. Modular invariance plays a key role in the reduction of the integral over moduli to an integral over a single copy of moduli space and, in particular, is responsible for rendering string amplitudes well behaved at short distances. In this chapter, we present a highly condensed introduction to key ingredients of string theory and string amplitudes, relegating the important aspects of toroidal compactification and T-duality to Chapter 13 and a discussion of S-duality in Type IIB string theory to Chapter 14.

A Riemann surface is a connected complex manifold of two real dimensions or equivalently a connected complex manifold of one complex dimension, also referred to as a complex curve. In this appendix, we shall review the topology of Riemann surfaces, their homotopy groups, homology groups, uniformization, construction in terms of Fuchsian groups, as well as their emergence from two-dimensional orientable Riemannian manifolds. All these ingredients provide crucial mathematical background for two-dimensional conformal field theory on higher genus Riemann surfaces and its application to string theory.

In this chapter, we shall draw together a number of different strands of inquiry addressed in Chapters 5, 12, and 13. We shall study the interplay between superstring amplitudes, their low-energy effective interactions, Type IIB supergravity, and the S-duality symmetry of Type IIB superstring theory. We begin with a brief review of Type IIB supergravity which, in particular, provides the massless sector of Type IIB superstring theory. We then discuss how the SL(2,R) symmetry of Type IIB supergravity is reduced to the SL(2,Z) symmetry of Type IIB superstring theory via an anomaly mechanism. We conclude with a discussion of how the low-energy effective interactions induced by string theory on supergravity may be organized in terms of modular functions and modular graph forms under this SL(2,Z) symmetry, and match the predictions provided by perturbative calculations of Chapter 12.

In this chapter, we shall discuss modular forms for the congruence subgroups introduced in Chapter 6. We shall obtain the dimension formulas for the corresponding rings of modular forms and cusp forms, describe the fields of modular functions on the modular curves introduced in Chapter 6, and construct the associated Eisenstein series. Throughout the chapter, we shall make use of the correspondence between modular forms and differential forms, viewed as sections of holomorphic line bundles on the compact Riemann surface of the modular curve. We shall provide concrete examples of modular forms for the standard congruence subgroups and apply the results to the theorems of Lagrange and Jacobi on counting the number of representations of an integer as a sum of squares.

In this appendix, we collect some basic results in number theory, including the Chinese remainder theorem, its application to solving polynomial equations, the Legendre and Jacobi quadratic residue symbols, quadratic reciprocity, its application to solving quadratic equations modulo N, and a brief introduction to Dirichlet characters and Dirichlet L-functions.

We close with a brief introduction to Galois theory and illustrate the application of these mathematical ideas in physics through examples from conformal field theory.

In this chapter, we construct differential equations in the modular parameter and find solutions to these equations in simple cases. The solutions can generically be assembled into vector-valued modular forms, which have proven fruitful in recent works in mathematics and physics. We will establish that, in general, each component of a vector-valued modular form is a modular form for a congruence subgroup.

Closely related variants of modular forms, including quasi-modular forms, almost-holomorphic modular forms, Maass forms, non-holomorphic Eisenstein series, mock modular forms, and quantum modular forms, are introduced and their properties are analyzed.

In this penultimate chapter, we shall discuss dualities in Yang–Mills theories with extended supersymmetry in four-dimensional Minkowski space-time. We briefly review supersymmetry multiplets of states and fields and the construction of supersymmetric Lagrangian theories with N = 1, 2, and 4 Poincaré supersymmetries. We then discuss the SL(2,Z) Montonen–Olive duality properties of the maximally supersymmetric N = 4 theory and the low-energy effective Lagrangians for N = 2 theories via the Seiberg–Witten solution. We shall close this chapter with a discussion of dualities of N = 2 superconformal gauge theories, which possess interesting spaces of marginal gauge couplings. In some cases, these spaces of couplings can be identified with the moduli spaces for Riemann surfaces of various genera.

In this final appendix, we shall review the modular geometry of the Siegel half-space at higher rank, Riemann theta-functions of higher rank, the embedding of higher-genus Riemann surfaces into the Jacobian variety via the Abel map, and use these ingredients to construct the prime form, the Szego kernel, and other meromorphic functions and differential forms on higher-genus Riemann surfaces.

Some immediate applications of the theory of elliptic functions and modular forms to problems of physical interest are presented, including the construction of the Green functions and functional determinants for the two-dimensional quantum field theories of the bc fields, the scalar field, and the spinor fields on the torus. In particular, it will be shown how the singular terms in the operator product expansion of holomorphic fields for the bc system essentially determine arbitrary correlation functions on the torus in terms of elliptic functions. Along the way, a brief but reasonably systematic introduction will be presented of two-dimensional conformal field theory methods.

The goal of this book is to exhibit the profound and myriad interrelations between the mathematics of modular forms and the physics of string theory. Our presentation is intended to be informal but mathematically precise, logically complete, and reasonably self-contained. The exposition is kept as simple as possible so as to be accessible to adventurous undergraduates, motivated graduate students, and dedicated professionals interested in the interface between theoretical physics and pure mathematics. Assuming little more than a knowledge of complex function theory, we introduce elliptic functions and elliptic curves as a lead-in to modular forms and their various deep generalizations. Following an economical introduction to string theory, its perturbative expansion, toroidal compactification, and supergravity limit are used to illustrate the power of modular invariance in physics. Dualities and their realization via modular forms in Yang–Mills theories with extended supersymmetry are studied both via the Seiberg–Witten solution and via their superconformal phase. Appendices are included to review foundational topics, and 75 exercises with detailed solutions give the reader ample opportunity for practice.