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Publication# Semiclassical methods in conformal field theories scrutinized by the epsilon-expansion

Abstract

Conformal Field Theories (CFTs) are crucial for our understanding of Quantum Field Theory (QFT). Because of their powerful symmetry properties, they play the role of signposts in the space of QFTs. Any method that gives us information about their structure, and lets us compute their observables, is therefore of great interest. In this thesis we explore the large quantum number sector of CFTs, by describing a semiclassical expansion approach. The idea is to describe the theory in terms of fluctuations around a classical background, which corresponds to a superfluid state of finite charge density. We detail the implementation of the method in the case of U (1)-invariant lagrangian CFTs defined in the epsilon-expansion. After introducing the method for generic correlators, we illustrate it by performing the computation of several observables.First, we compute the scaling dimension of the lowest operator having a given large charge n under the U (1) symmetry. We demonstrate how the semiclassical result in this case bridges the gap between the naive diagrammatic computation (which fails at too large n) and the general large-charge expansion of CFTs (which is only valid for n large enough).Second, we apply the method to the computation of 3- and 4-point functions involving the same operator. This lets us derive some of the OPE (Operator Product Expansion) coefficients.Finally, we consider the rest of the spectrum of charge-n operators, and propose a way to classify them by studying their free-theory equivalent. In the free theory, we construct the complete set of primary operators with number of derivatives bounded by the charge.We also find a mapping between the excited states of the superfluid and the vacuum states of standard quantization, which is valid when the spin of said states is bounded by the square root of the charge.

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Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

Superfluidity

Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two isotopes of helium (helium-3 and helium-4) when they are liquefied by cooling to cryogenic temperatures. It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity.

Scale invariance

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation). Dilatations can form part of a larger conformal symmetry. In mathematics, scale invariance usually refers to an invariance of individual functions or curves.

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