Local CFTs extremise $F$

This paper proves that local conformal field theories (CFTs) correspond to the extrema, specifically local maxima for unitary theories, of the sphere free energy along lines of nonlocal CFTs parametrized by the scaling dimension of the fundamental field, thereby providing a new extremisation principle for determining scaling dimensions in non-supersymmetric settings.

The Gist

Imagine you are trying to find the perfect recipe for a cake. You have a basic recipe (a Local CFT) that everyone agrees makes a delicious, standard cake. But, you decide to experiment. You start adding weird, "long-range" ingredients—like sprinkling sugar from across the room instead of right on the batter. This creates a whole family of "Long-Range Cakes" (Nonlocal CFTs).

In this family, you can tweak a dial called $\Delta$ (the scaling dimension). Turning this dial changes how the ingredients interact over distance. Most of the time, these cakes are weird, non-local, and a bit chaotic. But, at one specific setting of the dial, the cake magically turns back into the perfect, standard Local Cake you started with.

The Big Question: If you didn't know the standard recipe existed, how could you find it just by looking at all these weird Long-Range cakes? How do you know which setting of the dial is the "special" one?

The Answer: The author of this paper discovered a mathematical "sweet spot." It turns out that the perfect Local Cake sits exactly at the peak (or the very bottom, depending on how you look at it) of a specific landscape called the Free Energy.

Here is the breakdown using simple analogies:

1. The Landscape of Free Energy ($\tilde{F}$)

Think of the "Free Energy" as a measure of how much "stuff" or "complexity" is in your cake.

• Imagine a hilly landscape where the height of the hill represents the Free Energy.
• As you turn the dial ($\Delta$) to change your Long-Range cake, you are walking along a path on this landscape.
• The author proves that the Local Cake (the normal, standard one) is always found exactly at the top of a hill (a maximum) or the bottom of a valley (a minimum) on this path.

2. The "Slope" Trick (Why it works)

Why is the Local Cake always at the peak?

• The Non-Local Ingredients: The weird, long-range ingredients in the cake are the only things that change as you turn the dial.
• The Local Limit: When you reach the perfect Local Cake, those weird long-range ingredients disappear completely. The cake becomes "local" again (ingredients only touch their immediate neighbors).
• The Flat Spot: Because the weird ingredients vanish at that exact point, the "slope" of the hill becomes perfectly flat. If you are at the top of a hill, the ground is flat under your feet. If you are at the bottom, it's also flat.
• The Discovery: The author realized that if you calculate the slope of this Free Energy landscape, it will be zero exactly when you hit the Local Cake. This gives us a simple test: Find the point where the slope is zero, and you've found the Local CFT!

3. The "Maximum" Rule (For Good Cakes)

The paper goes a step further. If the cake is "healthy" (mathematically called Unitary, meaning it follows the rules of physics like positive energy), the Local Cake isn't just any flat spot—it is the highest peak on the hill.

• Analogy: Imagine a mountain range. The "Local" theories are the highest peaks. Any time you try to make the cake "long-range" (move away from the peak), the Free Energy drops.
• Why? The author suggests that a local theory is the most "efficient" way to organize information. When you force ingredients to interact over long distances, you lose some of that efficiency. The local theory is the most robust, holding the most "degrees of freedom" (complexity) possible.

4. Why is this useful?

Usually, finding the properties of these complex theories is like trying to solve a puzzle with missing pieces. You have to do incredibly hard math (perturbation theory) to guess the answer.

• The New Shortcut: This paper says, "Don't guess! Just look for the peak."
• If you have a family of weird, long-range theories, you can calculate the Free Energy for different settings. The moment you find the setting where the Free Energy stops going up and starts going down (the peak), you have automatically found the scaling dimension of the fundamental field for the standard Local theory.
• It explains why certain complicated math formulas in physics look like they are derivatives of something else—they are! They are just the slope of this Free Energy hill.

Summary

Think of the universe of physics theories as a vast, foggy mountain range.

• Local CFTs are the clear, sunny peaks where the air is thin and the rules are simple.
• Long-Range CFTs are the foggy slopes leading up to those peaks.
• This paper provides a compass: It tells us that if we want to find the sunny peaks (the standard theories we know), we just need to climb until the ground stops sloping and becomes flat. That flat spot is the Local CFT.

This is a powerful tool because it turns a difficult search for specific numbers into a simple optimization problem: Find the peak.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in Conformal Field Theory (CFT): How can one identify the specific point in a family of non-local, long-range CFTs that corresponds to a standard local CFT?

• Context: Many CFTs (e.g., the Ising model, $O(N)$ models) can be generalized to "long-range" (LR) CFTs by modifying the kinetic term of the fundamental field $\phi$ from the local Laplacian $(-\partial^2)$ to a fractional derivative $(-\partial^2)^\zeta$.
• The Parameter: This modification is parametrized by the scaling dimension $\Delta$ of the fundamental field. The action typically takes the form:
  $$S = \int d^d x \left[ \frac{1}{2} \phi (-\partial^2)^{\frac{d}{2} - \Delta} \phi + g_0 \mathcal{O}_{local} \right]$$
  where $\Delta$ is treated as a continuous parameter.
• The Puzzle: As $\Delta$ is tuned to the specific value $\Delta_{local}$ (the scaling dimension of the field in the standard local CFT), the theory is known to flow to the local CFT (plus a decoupled Generalized Free Field). However, from the perspective of the long-range theory, there is no obvious signal that $\Delta_{local}$ is special. The paper seeks a principle that singles out this local point within the continuous family of non-local theories.

2. Methodology

The author employs a combination of non-perturbative arguments, conformal perturbation theory (CPT), and diagrammatic checks.

• Two Constructions: The paper utilizes two known routes to construct long-range CFTs:
  1. GFF + Local: Perturbing a Generalized Free Field (GFF) with local operators.
  2. CFT + $\hat{\phi}\chi$: Coupling a local CFT to an auxiliary non-local GFF $\chi$.
  • The paper assumes an IR duality between these two constructions, meaning they flow to the same IR fixed point.
• The Free Energy Functional: The central object of study is the universal part of the sphere free energy, denoted as $\tilde{F}$ (or a normalized version $\hat{F}$).
  $$\tilde{F} = \sin(\pi d/2) \log Z_{S^d}$$
  This quantity is a candidate for a $c$-function (counting effective degrees of freedom).
• Proof Strategy:
  1. Extremisation: The author proves that the derivative of the free energy with respect to the scaling dimension $\Delta$ vanishes if and only if the theory is local. This relies on the fact that the derivative of the free energy receives contributions only from the explicit non-local kinetic term. In the local limit, this term must vanish (or become a total derivative/local term), forcing the derivative to zero.
  2. Maximisation: For unitary theories, the author uses second-order conformal perturbation theory to show that the local CFT corresponds to a local maximum of $\tilde{F}$. This involves computing the beta function and free energy corrections for deformations where the OPE coefficient $C_{OOO} = 0$ (a characteristic of the long-range deformation).
  3. Normalization: A new normalization $\hat{F}$ is proposed to simplify expressions and remove spurious $d$-dependent prefactors, making the gradient flow structure more transparent.

3. Key Contributions

A. The F-Extremisation Principle

The primary result is the proof that local CFTs lie at the extrema of the sphere free energy $\tilde{F}(\Delta)$ within the family of long-range CFTs:
$$\left. \frac{d\tilde{F}}{d\Delta} \right|_{\Delta = \Delta_{local}} = 0$$

• Mechanism: The derivative $\frac{d\tilde{F}}{d\Delta}$ is proportional to the two-point function normalization of the fundamental field. In the local limit, the non-local kinetic term disappears, implying the two-point function normalization (relative to the free theory) must vanish to maintain locality, thus making the derivative zero.
• Converse: If $\tilde{F}$ is extremized, the theory must be local.

B. F-Maximisation for Unitary Theories

For unitary CFTs, the paper demonstrates that the local point is not just an extremum but a local maximum:
$$\left. \frac{d^2\tilde{F}}{d\Delta^2} \right|_{local} < 0$$
This is proven using conformal perturbation theory around the local CFT. The result relies on the positivity of the two-point function of the deformation operator and the reality of the action.

C. The New Normalization $\hat{F}$

The author introduces a re-normalized free energy $\hat{F}$:
$$\hat{F} = \frac{\Gamma(d)}{\Gamma(-d/2)\Gamma(d/2)^3} \tilde{F}$$
This normalization:

• Removes overall factors of $\pi$ and large denominators in $\epsilon$-expansions.
• Makes the leading term in CPT exactly the integral of the Zamolodchikov metric times the beta function (gradient flow).
• Improves the convergence of Padé approximants when extrapolating from $d=4$ to $d=3$ and $d=2$.

D. Resolution of Large-N Structure

The paper explains a "suspicious fact" observed in large-$N$ expansions of the $O(N)$ model. The complicated expressions for the anomalous dimension $\gamma_\phi$ in the $1/N$ expansion are shown to be simply the condition that the derivative of the long-range free energy vanishes. This provides a minimal encoding of the scaling dimension.

4. Results and Checks

The theoretical results are verified through several perturbative checks:

1. $O(N)$ $\phi^4$ Theory:

   • Calculated the free energy $\tilde{F}$ in the $\eta$-expansion (where $\Delta = d/2 - \eta/4$) up to order $\eta^5$.
   • Showed that extremizing $\tilde{F}$ with respect to $\eta$ reproduces the standard Wilson-Fisher scaling dimension $\Delta_{SR}$ to high orders in $\epsilon$ ($d=4-\epsilon$).
   • Verified that the two-point function normalization vanishes at the fixed point (in specific normalizations), consistent with the extremisation condition.

2. Cubic Model:

   • Applied the method to the cubic model with $O(N)$ symmetry.
   • Successfully recovered the scaling dimensions for the Yang-Lee edge singularity ($N=0$) and the $OSp(1|2)$ model ($N=-2$) by extremizing $\hat{F}$.

3. Higher-Derivative Theories:

   • Extended the results to higher-derivative Wilson-Fisher theories ($\phi(-\partial^2)^k \phi$).
   • Found that the sign of the second derivative (maxima vs. minima) alternates based on $k$, consistent with the unitarity properties of the parent theories.

4. Numerical Extrapolation:

   • Demonstrated that using $\hat{F}$ instead of $\tilde{F}$ significantly improves the accuracy of Padé approximants when extrapolating the free energy from $d=4$ to $d=3$ and $d=2$ for the Ising model.

5. Significance

• Unification of Local and Non-Local: The work provides a unified framework where local CFTs are not distinct entities but specific "critical points" (extrema) within a broader landscape of non-local, long-range theories.
• Non-Supersymmetric Extremisation: It establishes a non-supersymmetric analogue to the well-known $c$-maximisation (2D), $F$-maximisation (3D), and $a$-maximisation (4D) theorems found in Supersymmetric CFTs (SCFTs). Unlike SUSY cases, this extremisation is over the scaling dimension of the field rather than R-charges.
• Computational Tool: The principle offers a new method for computing conformal data. Instead of solving complex RG equations directly, one can compute the free energy of the long-range theory as a function of $\Delta$ and find its extremum to determine the local scaling dimension.
• Conceptual Insight: It suggests that the "locality" of a theory is a thermodynamic-like stability condition (maximizing degrees of freedom/free energy) within the space of possible scaling dimensions.

In summary, the paper establishes that locality is an emergent property defined by the extremisation of the sphere free energy in the space of long-range deformations, offering both a deep theoretical insight and a practical computational tool for studying CFTs.