Abstract
In this dissertation, we discuss progress on four topics in algebraic geometry, number theory, and physics. Chapter 1 describes a new approach to explicitly understanding period integrals of Calabi-Yau families using invariant theory. We review the canonical normalization of period integrals for two types of Calabi-Yau families and the tautological systems that govern them. Specializing to the families of Calabi-Yau double covers of projective spaces and Calabi-Yau hypersurfaces in the projective plane, we analyze the geometry of the parameter space and the transformation properties of periods with respect to a Lie group action encoded in the tautological systems. This allows us to prove novel explicit formulas for the holomorphic periods in terms of invariant functions on the parameter spaces of the families. This technique can be further applied to construct similar period formulas in other contexts.
Chapter 2 is an extension of the story in invariant theory which began with the Chevalley restriction theorem and continued on the level of orbit-theoretic statements about restriction properties of representations of reductive groups. Inspired by the role of invariant theory in the study of period integrals in Chapter 1, we extend these ideas to the context of Galois theory. The main result is the construction of a scheme whose closed points parametrize candidates for subvarieties with restriction properties on the level of invariant rings or function fields. Furthermore, we give algebraic and geometric criteria for such a subvariety to satisfy the restriction property on invariant rings, thus generalizing previous restriction theorems inside of our construction. Together, Chapters 1 and 2 should be viewed as an intertwined new perspective on period integrals of CY families: our invariant-theoretic results on the level of function fields are exactly the insight that allowed us construct period formulas on the whole parameter space by lifting periods from certain subvarieties in Chapter 1.
In Chapter 3, we adapt to the setting of Hecke cusp eigenforms a method for approximation of L-functions using finitely many Euler factors. This technique, based on a regularization and symmetrization procedure, emerged less than ten years ago and has been utilized in a variety of other settings. Along the way, we elucidate the novel role of an equidistributional condition involving ratios of logarithms of primes. We also derive via Mellin transforms a new definition for the approximations as a convolution-type integral which stands in contrast to the definition by regularization and symmetrization of the finite Euler product. These new insights clarify aspects of the construction appearing in other works, and suggest the possibility of a unified framework for the approximation procedure.
Finally, in Chapter 4, we discuss the representation theory of a class of deformed conformal field theories which generalize the free boson theory. We derive the action of the global conformal algebra on the space of fields in the deformed theory, then on the Hilbert space of states via the state-operator correspondence. Next, we translate these representations to twisted $\mathcal{D}$-modules via Beilinson-Bernstein localization and show that these actions can be understood through the action of the global conformal algebra on the total space of a line bundle on the Riemann sphere. In this language, the operator insertions appearing in the state-operator correspondence are realized as a skyscraper sheaf.