Geometry of Effective Field Theory Positivity Cones and Kuranishi Moduli Space of Generalized Complex Structures via Hamilton–Nash–Moser Theory

Abstract

This thesis is divided into two parts, each addressing a separate research problem.

In the first part of the thesis we study so-called positivity bounds -- theoretical constraints on the Wilson coefficients of an effective field theory. These bounds arise from the requirement that a given effective field theory must be the low-energy limit of a quantum field theory that satisfies the fundamental principles of unitarity, locality, and causality. The task of deriving these bounds (for Wilson coefficients corresponding to operators that are quartic in momenta and contribute to two-to-two scattering at tree-level) can be reformulated as a convex geometric question, namely the determination of the extremal representation of a closed convex cone. This so-called positivity cone consists of all positive semi-definite tensors in $$W =\left\{ S \in \mathrm{Sym}^2 (\mathrm{Sym}^2\, V^*)\oplus \mathrm{Sym}^2 \left({\Lambda}^2 V^*\right) : \tau S = S \right\} \subset \mathrm{Sym}^2(V^*\otimes V^*),$$ where $\tau$ denotes transposition in the second and fourth tensor factor and $V$ is a finite-dimensional real vector space. The dimension of $V$ has the physical interpretation of the number of (particle) flavors under consideration.

In this thesis, we solve this question up to three flavors, i.e.~$\dim V\leq 3$, proving a full classification of all extremal elements in these cases. We furthermore study the implications of our findings, deriving the full positivity bounds for amplitudes with and without additional symmetries. In the cases with additional symmetries that we consider, we find that the so-called elastic bounds are sufficient to recover the full positivity bounds. In contrast, if $\dim V\geq 3$ and no additional symmetry conditions are imposed on the amplitude, there will be bounds that cannot be deduced from the elastic positivity bounds. These arise from a new class of extremal elements that appears in the extremal representation of the positivity cone starting from $\dim V=3$.

In the second part of this thesis, we study deformations of complex and generalized complex structures using Hamilton–Nash–Moser theory. Our main result is an extension of Gualtieri's theorem on the existence of a locally complete family of deformations of generalized complex structures on exact Courant algebroids to the broader class of $\mathcal{G}$-flat transitive Courant algebroids. To establish this result, it is necessary to revisit certain analytic aspects of both Gualtieri's proof and Kuranishi's classical proof of the existence of a locally complete family of (classical) complex structures.

We therefore provide transparent proofs of Kuranishi's and Gualtieri's theorems using Hamilton's framework of tame Fréchet spaces and the Hamilton–Nash–Moser implicit function theorem. Building on these results, we then extend Gualtieri's theorem to the setting of $\mathcal{G}$-flat transitive Courant algebroids. A key ingredient is the construction of an infinite-dimensional Lie group structure on the group of autoequivalences of a $\mathcal{G}$-flat transitive Courant algebroid, more specifically, a tame Fréchet Lie group structure.