Lattice models and conformal field theory

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School of Science | Doctoral thesis (article-based) | Defence date: 2025-02-25
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Date

2025

Department

Matematiikan ja systeemianalyysin laitos
Department of Mathematics and Systems Analysis

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Degree programme

Language

en

Pages

30 + app. 157

Series

Aalto University publication series Doctoral Theses, 25/2025

Abstract

Within the realm of Mathematical Physics, this thesis studies the connection between statistical mechanics and conformal field theory (CFT) in two dimensions. More precisely, we contribute to the mathematical understanding of the scalinglimit convergence of critical discrete models to conformal field theories. In two dimensions, the conformal symmetries of the theory are encoded into its space of local fields in an algebraic manner: it constitutes a representation of the Virasoro algebra. In this thesis, this algebraic feature is exploited to establish the emergence of conformal field theories in the scaling limit of critical models. The key insight that enables our results is that, in critical models with enough integrability, the relevant observables can be proven to carry the algebraic structure of a two-dimensional CFT by means of tools of discrete complex analysis. In Publication I, the discrete model of interest is the (double) dimer model. Within this model, we consider certain fermionic observables whose correlation functions present suitable discrete holomorphicity. The main result can then be stated as follows. The space of the fermionic observables carries a representation of the symplectic-fermions algebra and, furthermore, it constitutes a Virasoro representation with central charge −2.In Publication II, we establish, for the first time, the scaling-limit convergence of a discrete model to a fully-fledged CFT. The space of local observables of (the gradient of) the discrete gaussian free field (DGFF) is proven to be in one-toone correspondence with the space of local fields of the free boson CFT. Then, the (suitably renormalised) correlation functions of DGFF local observables are proven to converge to the CFT correlation functions of the corresponding local fields in the scaling limit. In Publication III, the symplectic fermions CFT in general domains of the complex plane is expounded in detail. The main motivation for considering this logarithmic FT is the scaling limit of the observables introduced in Publication I.

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Supervising professor

Kytölä, Kalle, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland

Keywords

mathematical physics, statistical mechanics, conformal field theory

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Parts

  • [Publication 1]: D. ADAME-CARRILLO. Discrete symplectic fermions on double dimers and their Virasoro representation, Annales Henri Poincaré (2024).
    DOI: 10.1007/s00023-024-01455-w View at publisher
  • [Publication 2]: D. ADAME-CARRILLO, D. BEHZAD, AND K. KYTÖLÄ. Fock space of local fields of the discrete GFF and its scaling limit bosonic CFT, arXiv:2404.15490. Submitted to a journal in July 2024.
  • [Publication 3]: D. ADAME-CARRILLO. Symplectic fermions on general domains, arXiv:2409.12823. Submitted to a journal in November 2024.

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https://urn.fi/URN:ISBN:978-952-64-2388-3

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[diss] Perustieteiden korkeakoulu / SCI