Torres Sandoval, Matias Ignacio (2024) Semiclassical Methods in Conformal Field Theory. [Tesi di dottorato]

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This thesis aims to study large charged sectors of strongly coupled Conformal Field Theories with global symmetries. In Chapter I, we focus on the basic concepts of conformal field theories and the large charge expansion. We illustrate this method using a $U(1)$-invariant CFT in $d=4-\epsilon$. In Chapter II, we investigate the analytic properties of the fixed charge expansion for several conformal field theories in different space-time dimensions. The models investigated are $O(N)$ and $QED_3$. In $d = 3 - \epsilon$ the contribution to the $O(N)$ fixed charge $Q$ conformal dimensions obtained in the double scaling limit of large charge and vanishing $\epsilon$ is non-Borel summable, doubly factorial divergent, and with order $\sqrt{Q}$ optimal truncation order. For $d = 4 - \epsilon$, the situation is different since the same large $Q$ and small $\epsilon$ regime the next order corrections to the scaling dimensions lead to a convergent series. The resummed series displays a new branch cut singularity which is relevant for the stability of the $O(N)$ large charge sector for negative $\epsilon$. In Chapter III we analyze the impact of the $\theta$-angle and axion dynamics for two-color QCD at nonzero baryon charge and as a function of the number of matter fields on the vacuum properties, the pattern of chiral symmetry breaking as well as the spectrum of the theory. We show that the vacuum acquires a rich structure when the underlying CP-violating topological operator is added to the theory. We discover novel phases and analyze the order of their transitions characterizing the dynamics of the odd and even number of flavours. We further determine the critical chemical potential as function of the $\theta$-angle separating the normal from the superfluid phase of the theory. Then, we determine the impact of the $\theta$-angle and axion physics in the conformal phase of the large-charge baryon sector of two-color QCD. We employ an effective approach featuring Goldstone and dilaton degrees of freedom augmented by the topological terms in the theory. We investigate how different dilaton potentials, including the ones for which a systematic counting scheme can be established, affect the results. Via state-operator correspondence, we compute the corrections to the would-be conformal dimensions of the lowest large-charge operators as a function of the $\theta$-term and dilaton potential.

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