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.

After a brief review of scale and conformal symmetry, we develop the basic concepts of two-dimensional conformal field theory in a top-down approach. We define primary and quasi-primary fields and the action of conformal transformations on these fields leads to the Virasoro algebra. Operator product expansions, hermitian conjugation, and the state-operator are discussed in the general setting, while results of the previous chapter are recovered as special cases. Null vectors are discussed to prepare for the later discussion of gauge transformations acting on string states. We give an overview of general results about the properties and classification of two-dimensional CFTs.

We use the case of a single string coordinate to introduce two-dimensional conformal field theory in a bottom-up approach, emphasising the role of operator product expansions and of the state-operator correspondence.

String theory has not even come close to a complete formulation after half a century of intense research. On the other hand, a number of features of the theory suggest that the theory, once completed, may be a final theory. It is argued in this chapter that those two conspicuous characteristics of string physics are related to each other. What links them together is the fact that string theory has no dimensionless-free parameters at a fundamental level. The paper analyzes possible implications of this situation for the long-term prospects of theory building in fundamental physics.

In this chapter the linearized Riemann tensor correlator on a de Sitter background including one-loop corrections from conformal fields is derived. The Riemann tensor correlation function exhibits interesting features: it is gauge-invariant even when including contributions from loops of matter fields, but excluding graviton loops as it is implemented in the 1/N expansion, it is compatible with de Sitter invariance, and provides a complete characterization of the local geometry. The two-point correlator function of the Riemann tensor is computed by taking suitable derivatives of the metric correlator function found in the previous chapter, and the result is written in a manifestly de Sitter-invariant form. Moreover, given the decomposition of the Riemann tensor in terms of Weyl and Ricci tensors, we write the explicit results for the Weyl and Ricci tensors correlators as well as the Weyl–Ricci tensors correlator and study both their subhorizon and superhorizon behavior. These results are extended to general conformal field theories. We also derive the Riemann tensor correlator in Minkowski spacetime in a manifestly Lorentz-invariant form by carefully taking the flat-space limit of our result in de Sitter.

In this chapter we derive the full two-point quantum metric perturbations on a de Sitter background including one-loop corrections from conformal fields. We do the calculation using the CTP effective action with the 1/N expansion, and select an asymptotic initial state by a suitable prescription that defines the vacuum of the interacting theory. The decomposition of the metric perturbations into scalar, vector and tensor perturbations is reviewed, and the effective action is given in terms of that decomposition. We first compute the two-point function of the tensor perturbations, which are dynamical degrees of freedom. The relation with the intrinsic and induced fluctuations of stochastic gravity is discussed. We then compute the two-point metric perturbations for the scalar and vector modes, which are constrained degrees of freedom. The result for the full two-point metric perturbations is invariant under spatial rotations and translations as well as under a simultaneous rescaling of the spatial and conformal time coordinates. Finally, our results are extended to general conformal field theories, even strongly interacting ones, by deriving the effective action for a general conformal field theory.