arXiv subjects

Ludo Fraser-Taliente

Publications and source records attributed to Ludo Fraser-Taliente.

7 recordsLinked to original sources

arXiv2026

Local CFTs extremise $F$

Many CFTs can be extended to lines of nonlocal CFTs parametrised by the scaling dimension $\Delta$ of the fundamental field appearing in the action. $\Delta=\frac{d}{2}-\zeta$ is set by the exponent of the kinetic term $(-\partial^2)^{\zeta}$, which is nonlocal for noninteger $\zeta$. If $\Delta$ is tuned to $\Delta_\mathrm{local}$, the scaling dimension of the fundamental field in the local CFT, arXiv:1703.05325 showed that we recover the conformal data of that CFT (plus a decoupled sector). One natural question is: how is the local point special on this line of nonlocal CFTs? We prove that these local CFTs lie at the extrema of the (universal part of the) sphere free energy $\tilde{F}(\Delta)=\sin(\frac{\pi d}{2}) \log Z_{S^d}$ of the long-range CFTs: $d\tilde{F}/d\Delta|_{\Delta=\Delta_\mathrm{local}}=0$; and for unitary CFTs they locally maximise it. The simple proof uses the fact that the derivative of $\tilde{F}$ with respect to $\Delta$ receives contributions only from the nonlocal terms in the action. The nonlocal terms must be absent in the limit $\Delta \to \Delta_\mathrm{local}$, and hence the derivative is zero. Demonstrating maximisation then requires a proof of the generalised $F$-theorem in conformal perturbation theory to subleading order. We check our result with the O$(N)$ $\phi^4$ and cubic CFTs in the $\epsilon$ expansion and the large-$N$ limit. This result provides a minimal encoding of the scaling dimension of the fundamental field in many CFTs, and also explains the curious derivative structure of the large-$N$ expansion of $\Delta_\phi$ in the O$(N)$ model. Finally, this nonlocal $F$-extremisation can be viewed as a non-supersymmetric version of the known $c,F,a$-extremisation mechanisms.

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arXiv2026

Quantum field theories with many fields

The large-$N$ quantum field theories provide a window into the regime of strongly-coupled physics. Our principal object of study in this thesis is the large-$N$ family of melonic QFTs, which contain the Sachdev-Ye-Kitaev-like models, tensor models, and vector models. We begin with a review of this limit of a large number of degrees of freedom (large-$N$) as an approach to the solution of QFTs. Two toy models are used to clarify this approach: a zero-dimensional field theory and the flow of a generalized free field theory. Both models are solvable, and so we can explicitly demonstrate: using the former, the simplifications at large $N$; using the latter, the tools used to study scale-dependence of physics -- the renormalization group. We develop $\tilde{F}$-extremization, a simple method of solution for an arbitrary large-$N$ melonic QFT in its strongly-coupled limit. The infrared conformal field theories show remarkable simplicity, in that they are entirely solved by the requirement that they have as many degrees of freedom as possible, up to a simple constraint arising from the interaction between the fields. We measure the number of degrees of freedom of the conformal infrared theory via $\tilde{F}$, the universal part of the free energy. We then present the example of the quartic Yukawa model in continuous dimension. This model is considered as a tensor field theory, and solved for its conformal limit; we then illustrate its multiplicity of fixed points and their stability, as well as its operator spectrum, matching the data between the large-$N$ and dimensional expansions. These features reflect general characteristics of melonic conformal field theories: their existence, stability, and spectral characteristics. We conclude with future directions of exploration for the melonic theories.

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arXiv2026

The $T^{\mu\nu}$ of the conformal scalars

We construct the unique primary energy-momentum tensor $T^{\mu\nu}$ for the conformal free scalar with scaling dimension $\Delta=d/2-\zeta$ as a sum of Gegenbauer polynomials. For integer $\zeta$, the sum truncates at order $\zeta$, compactly reproducing all known results; for the nonlocal case of real $\zeta$, it is an infinite sum, with a two-parameter extension that reflects the nonuniqueness of the nonlocal geometric coupling. We find $T^{\mu\nu}$ by imposing off-shell conservation and tracelessness, and then directly solving the primary condition in momentum space. In the integer $\zeta$ case, we reproduce the known two-point function, and confirm the match with the $T^{\mu\nu}$ computed from Juhl's formulae for the GJMS operators (the Weyl-covariant upgrades of $(-\partial^2)^\zeta$), an equality following from the descent of Weyl covariance to conformal invariance.

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arXiv2026

Four-point function of the complex Sachdev-Ye-Kitaev model at finite chemical potential

It is known that, for a range of chemical potentials, the infrared behavior of the complex Sachdev-Ye-Kitaev (cSYK) model is governed by a 1D Nearly Conformal Field Theory (NCFT$_{1}$), thereby realizing a continuous line of NCFTs. A finite chemical potential $\mu$ introduces an asymmetry parameter $\mathscr{E}$ into the cSYK fermion two-point function in the conformal limit. In this work, we compute the cSYK four-point function in the conformal limit for an arbitrary value of $\mathscr{E}$ at leading order in $1/N$. We show that the result is fully consistent with the NCFT$_{1}$ structure of the cSYK model and use it to extract the structure constants for correlation functions of two complex fermions with bilinear operators.

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arXiv2025

The sphere free energy of the vector models to order $1/N$

We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $\phi^4$ and the Gross-Neveu $(\bar{\psi} \psi)^2$ CFTs to order $1/N$. Analytic regularization of these theories requires consistently shifting the UV scaling dimension of the auxiliary field: this can only be done by modifying its kinetic term. This modification combines with the counterterms to give the result that matches the $\epsilon$-expansion, resolving a puzzle raised by Tarnopolsky in arXiv:1609.09113. These $F$s can be written compactly in terms of the anomalous dimensions, for both the short-range and the long-range versions of these CFTs. We also provide various technical results including a computation of the counterterms on the sphere and a neat derivation of the sphere free energy of a free conformal field. Finally, we observe that the long-range CFT becomes the short-range CFT at exactly the point where its $\tilde{F} =-\sin \tfrac{\pi d}{2} F$ is maximized as a function of the vector's scaling dimension.

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arXiv2024

$F$-extremization determines certain large-$N$ CFTs

We show that the conformal data of a range of large-$N$ CFTs, the melonic CFTs, are specified by constrained extremization of the universal part of the sphere free energy $F=-\log Z_{S^d}$, called $\tilde{F}$. This family includes the generalized SYK models, the vector models (O$(N)$, Gross-Neveu, etc.), and the tensor field theories. The known $F$ and $a$-maximization procedures in SCFTs are therefore extended to these non-supersymmetric CFTs in continuous $d$. We establish our result using the two-particle irreducible (2PI) effective action, and, equivalently, by Feynman diagram resummation. $\tilde{F}$ interpolates in continuous dimension between the known $C$-functions, so we interpret this result as an extremization of the number of IR degrees of freedom, in the spirit of the generalized $c,F,a$-theorems. The outcome is a complete classification of the melonic CFTs: they are the conformal mean field theories which extremize the universal part of the sphere free energy, subject to an IR marginality condition on the interaction Lagrangian.

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arXiv2024

Melonic limits of the quartic Yukawa model and general features of melonic CFTs

We study a set of large-$N$ tensor field theories with a rich structure of fixed points, encompassing both the melonic and prismatic CFTs observed previously in the conformal limits of other tensor theories and in the generalised Sachdev-Ye-Kitaev (SYK) model. The tensor fields interact via an $O(N)^3$-invariant generalisation of the quartic Yukawa model, $\phi^2\bar{\psi}\psi+\phi^6$. To understand the structure of IR/UV fixed points, we perform a partial four-loop perturbative analysis in $D=3-\epsilon$. We identify the flows between the melonic and prismatic fixed points in the bosonic and fermionic sectors, finding an apparent line of fixed points in both. We reproduce these fixed points non-perturbatively using the Schwinger-Dyson equations, and in addition identify the supersymmetric fixed points in general dimension. Selecting a particular fermionic fixed point, we study its conformal spectrum non-perturbatively, comparing it to the sextic prismatic model. In particular, we establish the dimensional windows in which this theory remains stable. We comment on the structure of large-$N$ melonic CFTs across various dimensions, noting a number of features which we expect to be common to any such theory.

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