arxiv: 2604.15420 · v1 · submitted 2026-04-16 · ✦ hep-th · cond-mat.stat-mech· math-ph· math.MP

Recognition: unknown

Local CFTs extremise F

Ludo Fraser-Taliente

Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:25 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechmath-phmath.MP

keywords conformal field theorylong-range CFTsphere free energyextremizationO(N) modelepsilon expansionF-theoremnonlocal theories

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The pith

Local CFTs lie at the extrema of the sphere free energy along lines of long-range theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that local conformal field theories appear as extrema of the universal sphere free energy when embedded in continuous families of nonlocal CFTs. These families are obtained by varying the scaling dimension Delta of the fundamental field via a nonlocal kinetic term with exponent zeta. At the value of Delta where the theory becomes local, the derivative of tilde F with respect to Delta is exactly zero. For unitary CFTs the local point is a maximum. The argument follows because only the nonlocal part of the action contributes to the derivative, and that part disappears in the local limit.

Core claim

Local CFTs lie at the extrema of the (universal part of the) sphere free energy tilde F(Delta) of the long-range CFTs: d tilde F / d Delta at Delta equal to Delta_local equals zero, 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_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.

What carries the argument

The universal sphere free energy tilde F(Delta) equal to sin(pi d / 2) times log of the sphere partition function Z_S^d, whose derivative with respect to the parameter Delta vanishes when the long-range theory reaches its local point.

If this is right

The scaling dimension of the fundamental field is encoded as the zero of d tilde F / d Delta.
The result explains the derivative structure appearing in the large-N expansion of Delta_phi in the O(N) model.
It supplies a non-supersymmetric analogue of the known c, F, a extremisation mechanisms.
The relation is confirmed by direct calculation for the O(N) phi^4 theory and cubic CFTs in both the epsilon expansion and the large-N limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same variational property might locate local fixed points inside other families of nonlocal deformations without separate computation of scaling dimensions.
It could be tested numerically in long-range models studied by bootstrap or Monte Carlo methods.
Higher derivatives of tilde F or other thermodynamic quantities on the sphere might obey related stationarity conditions at the local point.

Load-bearing premise

The derivative of tilde F with respect to Delta receives contributions only from the nonlocal terms in the action, which disappear as Delta approaches the local value.

What would settle it

An explicit computation of d tilde F / d Delta at the local Delta in any unitary model, such as the 3d Ising CFT or the O(N) vector model in epsilon expansion, that yields a nonzero result would falsify the claim.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows local CFTs extremize tilde F on nonlocal lines because the derivative only sees nonlocal action terms that vanish at the local point, with epsilon and large-N checks supporting it.

read the letter

This paper proves that local CFTs sit at extrema of the universal sphere free energy tilde F along families of nonlocal CFTs parametrized by the scaling dimension Delta of the fundamental field. The core argument is that d tilde F / d Delta picks up contributions only from the nonlocal pieces in the action, which disappear exactly when Delta reaches the local value, so the derivative is zero there. For unitary cases it claims a local maximum as well. The checks in the O(N) phi^4 theory and cubic models via epsilon expansion, plus the large-N limit, match the expected vanishing and give a clean explanation for some derivative patterns already seen in large-N expansions of Delta_phi. That part is solid and directly useful for anyone computing conformal data in these models. The maximization step rests on a subleading generalized F-theorem in conformal perturbation theory, but the abstract only states that it is needed without sketching the proof or the order-by-order cancellation. The decoupled sector that appears when recovering the local CFT from the nonlocal construction is not discussed in terms of its own possible Delta dependence, so it is not obvious whether it could feed into the derivative even after the interacting nonlocal terms are gone. This is a minor gap rather than a fatal one, since the basic derivative claim follows from the action definition. The work is aimed at people already thinking about long-range or nonlocal CFTs and F-extremization ideas outside supersymmetry. It is short, focused, and has enough explicit checks to be worth refereeing; the central result is new and the argument is transparent enough that a referee could quickly test the missing pieces.

Referee Report

2 major / 2 minor

Summary. The paper claims that local CFTs recovered at a specific value of the parameter Delta in families of nonlocal long-range CFTs (with kinetic term exponent zeta) extremize the universal sphere free energy tilde F(Delta) = sin(pi d / 2) log Z_{S^d}. Specifically, it proves d tilde F / d Delta vanishes at Delta = Delta_local (the local CFT value), with local maximisation for unitary theories. The argument is that the derivative receives contributions only from nonlocal terms in the action, which are absent at the local point; maximisation requires a subleading generalised F-theorem in conformal perturbation theory. The result is checked via epsilon-expansion and large-N limits for O(N) phi^4 and cubic models, and interpreted as a non-supersymmetric analog of c,F,a-extremisation.

Significance. If the central claim holds, the result offers a simple, minimal encoding of the fundamental scaling dimension Delta_phi in many CFTs through extremisation of tilde F, explains the derivative structure of large-N expansions in the O(N) model, and provides a new perspective on F-theorems without supersymmetry. The explicit checks in controlled perturbative regimes add concrete support, and the connection to the referenced nonlocal construction strengthens the conceptual link between local and long-range CFTs.

major comments (2)

[Abstract and §2] Abstract and §2 (proof sketch): the claim that d tilde F / d Delta receives contributions only from nonlocal terms in the action, which vanish as Delta -> Delta_local, does not yet address the decoupled sector recovered in the construction of arXiv:1703.05325. If this sector's action or sphere free energy retains any Delta dependence (e.g., via its own kinetic term or coupling to zeta), then its contribution to the derivative need not vanish, undermining the proof that d tilde F / d Delta = 0 at the local point.
[Abstract and §4] Abstract and §4 (maximisation): demonstrating that the stationary point is a local maximum requires a proof of the generalised F-theorem to subleading order in conformal perturbation theory. The manuscript must supply the explicit steps of this argument (including any assumptions on unitarity or the form of the perturbation), as the abstract only states that it is needed; without these details the maximisation claim is not fully established.

minor comments (2)

[Introduction] Clarify the precise definition of the universal part tilde F versus the full log Z_{S^d} and its relation to the standard F-theorem, especially in the presence of the decoupled sector.
[§5] In the epsilon-expansion and large-N checks, provide explicit numerical values or a table showing that the derivative vanishes to the computed order, rather than only stating that the result is confirmed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major point below and have revised the text accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (proof sketch): the claim that d tilde F / d Delta receives contributions only from nonlocal terms in the action, which vanish as Delta -> Delta_local, does not yet address the decoupled sector recovered in the construction of arXiv:1703.05325. If this sector's action or sphere free energy retains any Delta dependence (e.g., via its own kinetic term or coupling to zeta), then its contribution to the derivative need not vanish, undermining the proof that d tilde F / d Delta = 0 at the local point.

Authors: In the construction of arXiv:1703.05325, the limit Delta → Delta_local recovers the local CFT in the interacting sector together with a decoupled free scalar whose scaling dimension is fixed by the local theory. This decoupled sector has a strictly local kinetic term whose form is independent of the deformation parameter Delta (or zeta). Because the sector is completely decoupled, its contribution to the sphere partition function (and thus to tilde F) does not depend on Delta. The derivative d tilde F / d Delta therefore receives contributions exclusively from the nonlocal kinetic term of the interacting sector, which is absent at the local point. We will add a short clarifying paragraph in the revised §2 making this decoupling explicit. revision: yes
Referee: [Abstract and §4] Abstract and §4 (maximisation): demonstrating that the stationary point is a local maximum requires a proof of the generalised F-theorem to subleading order in conformal perturbation theory. The manuscript must supply the explicit steps of this argument (including any assumptions on unitarity or the form of the perturbation), as the abstract only states that it is needed; without these details the maximisation claim is not fully established.

Authors: We agree that the maximisation statement requires an explicit subleading argument. The present manuscript notes the necessity of a generalised F-theorem in conformal perturbation theory but does not derive the sign of the second derivative. In the revision we will expand §4 with the following steps: (i) the first-order variation of tilde F vanishes by the stationarity condition already established; (ii) the second-order term is proportional to the integral of the two-point function of the nonlocal perturbing operator; (iii) unitarity implies that this two-point function is positive definite, so the quadratic correction to tilde F is negative. The perturbation is taken to be the long-range deformation, which is relevant or marginal near the local fixed point. These assumptions will be stated clearly. The expanded derivation will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained via external citation and independent F-theorem proof

full rationale

The central claim rests on two elements: (1) the cited result from arXiv:1703.05325 that nonlocal terms vanish at Delta_local (recovering the local CFT plus decoupled sector), and (2) a new proof of the generalized F-theorem in conformal perturbation theory to subleading order for the maximisation statement. Neither reduces to a self-definition, a fitted parameter renamed as prediction, nor a load-bearing self-citation chain; the derivative argument follows directly from the action's structure once the external vanishing is granted. The decoupled-sector concern raised in the skeptic note is a potential correctness issue, not a circularity reduction. The paper's checks in epsilon expansion and large-N are independent verifications, not inputs. Overall the derivation chain is non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of nonlocal CFTs via fractional kinetic terms and the cited fact that Delta equals Delta_local recovers the local CFT. The generalized F-theorem is invoked for maximisation but its proof is not detailed in the abstract. No free parameters or invented entities are introduced.

axioms (1)

domain assumption The derivative of the sphere free energy tilde F with respect to Delta receives contributions only from the nonlocal terms in the action.
This is the key fact used in the simple proof as stated in the abstract.

pith-pipeline@v0.9.0 · 5691 in / 1468 out tokens · 49765 ms · 2026-05-10T10:25:16.192338+00:00 · methodology

discussion (0)

Reference graph

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