We compute, for any Rényi index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>n</mi></mrow></math>, the exact difference between the mutual Rényi information of a pair of free massless scalars and that of a Maxwell field in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>4</mn></math> dimensions. Using the standard dimensional reduction method in polar coordinates, the problem is mapped to that of a single scalar field in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn></math> with Dirichlet boundary conditions, which in turn can be conveniently related to the algebra of a chiral current on the full line. This latter identification, which maps operator algebras on an interval to two-interval operator algebras, yields exact results that clarify the structure of the long-distance, perturbative operator product expansion of the mutual information. We find that this series has a finite radius of convergence only for integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>, while it becomes only asymptotical for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo>=</mo><mn>1</mn></math> and general noninteger values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>.

Argentina

Nicolás Abate

Horacio Casini

Marina Huerta

Leandro Martinek

Exact mutual information difference: Scalar vs Maxwell fields

We explore the holographic prescription for computing the refined Rényi entropies $$ {\overset{\sim }{S}}_n $$ in the $\textit{n}$ → 0 limit within the AdS /CFT framework. This limit can be interpreted as a high-temperature regime with respect to the energy defined by the modular Hamiltonian of the state reduced to a subregion. To leading order in $\textit{n}$, we find that the system attains local equilibrium and admits a CFT description in terms of a Euclidean, irrotational perfect fluid. This fluid exhibits vortex-like boundary conditions at the entangling surface. Guided by this physical picture, we construct an ansatz for the dual geometry in terms of the boundary fluid variables. We show that our anzats solves the Einstein’s equations coupled to a cosmic brane at leading order in $\textit{n}$, in agreement with Dong’s proposal for the holographic dual to the refined Rényi entropy. The resulting conical singularity, signaling the brane’s location, can be understood from this perspective as the bulk extension of the boundary fluid vortices.

Netherlands

Cesar Agón

Pedro Martinez

Holographic Rényi $\textit{n}$ → 0 entropy and Euclidean fluids

The operator product expansion (OPE) of twist operators in the replica trick framework enables a long-distance expansion of the mutual information (MI) in conformal field theories (CFTs). In this expansion, the terms are labeled by primary operators, as contributions from descendant operators can be resummed into conformal blocks. However, for the MI, the expansion involves primaries from the multi-replica theory, which includes far more operators than those in the original theory. In this work, we develop a method to resum this series, yielding an expansion in terms of the primaries of the original theory, specifically restricted to the two-copy sector. This is achieved by expressing the twist operators in a non-local manner across different replicas and using a modular flow representation to obtain the $\textit{n}$ → 1 limit of the Rényi index. We explicitly compute the resulting “enhanced conformal blocks,” which, surprisingly, provide excellent approximations to the MI of generalized free fields across the full range of cross ratios. Remarkably, this approximation appears to be exact in the limit of large spacetime dimensions.

Umut Gürsoy

Guim Planas

Mutual information from modular flow in CFTs

Minimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn></math> conformal field theories (CFTs) are usually classified through modular invariant partition functions. There is a finer classification of “noncomplete” models when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math> duality is not imposed. We approach this classification by starting with the local chiral algebra and adding primaries sequentially. At each step, we only impose locality (T duality) and closure of the operator algebra. For each chiral algebra, this produces a treelike graph. Each tree node corresponds to a local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn></math> CFT, with an intrinsic Jones index measuring the size of Haag duality violation. This index can be computed with the partition function and is related to the total quantum dimension of the category of superselection sectors of the node and to the relative size between the node and a modular invariant completion. In this way, we find in a very explicit manner a classification of local minimal (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mo>&lt;</mo><mn>1</mn></math>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn></math> CFTs. When appropriate, this matches Kawahigashi-Longo’s previous results. We use this finer classification to constrain renormalization group (RG) flows. For a relevant perturbation, the flow can be restricted to the subalgebra associated with it, typically corresponding to a nonmodular invariant node in the tree. The structure of the graph above such node needs to be preserved by the RG flow. In particular, the superselection sector category for the node must be preserved. This gives selection rules that recover in a unified fashion several known facts while unraveling new ones.

Italy

Valentin Benedetti

Javier M. Magán

Selection rules for RG flows of minimal models

We review the physical meaning of modular invariance for unitary conformal quantum field theories in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math>. For quantum field theory models, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math> invariance is necessary for locality, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math> invariance is not mandatory. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi></mrow></math> invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational conformal field theories (CFTs), the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual information. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index.

Japan

Yasuyuki Kawahigashi

Roberto Longo

Modular invariance as completeness

Twist operators implement symmetries in bounded regions of the space. Standard twists are a special class of twists constructed using modular tools. The twists corresponding to translations have interesting special properties. They can move continuously an operator from a region to a disjoint one without ever passing through the gap separating the two. In addition, they have generators satisfying the spectrum condition. We compute explicitly these twists for the two-dimensional chiral fermion field. The twist generator gives place to a new type of energy inequality where the smeared energy density is bounded below by an operator.

Standard translation twists and an operator-bounded energy inequality

Entanglement temperatures (ET) are a generalization of Unruh temperatures valid for states reduced to any region of space. They encode in a thermal fashion the high-energy behavior of the state around a point. These temperatures are determined by an eikonal equation in Euclidean space. We show that the real-time continuation of these equations implies ballistic propagation. For theories with a free UV fixed point, the ET determines the state at a large modular temperature. In particular, we show that the <math><mi>n</mi><mo>→</mo><mn>0</mn></math> limit of Rényi entropies <math><msub><mi>S</mi><mi>n</mi></msub></math> can be computed from the ET. This establishes a formula for these Rényi entropies for any region in terms of solutions of the eikonal equations. In the <math><mi>n</mi><mo>→</mo><mn>0</mn></math> limit, the relevant high-temperature state propagation is determined by a free relativistic Boltzmann equation, with an infinite tower of conserved currents. For the special case of states and regions with a conformal Killing symmetry, these equations coincide with the ones of a perfect fluid.

Cesar A. Agón

Pedro J. Martinez

Rényi entropies in the <math><mi>n</mi><mo>→</mo><mn>0</mn></math> limit and entanglement temperatures

The entanglement entropy of an arbitrary spacetime region <math><mi>A</mi></math> in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></math>. For general theories, the value of <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></math> is minimized when <math><mi>A</mi></math> is a round disk, <math><msub><mi>F</mi><mn>0</mn></msub></math>, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>/</mo><msub><mi>F</mi><mn>0</mn></msub></math> is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies <math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≤</mo><mn>3</mn><mo>/</mo><mo>(</mo><mn>4</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mn>2</mn><mo>−</mo><mn>6</mn><mi>ζ</mi><mo>[</mo><mn>3</mn><mo>]</mo><mo>)</mo><mo>≃</mo><mn>0.14887</mn></mrow></math> for general CFTs. We verify the validity of this bound for free scalars and fermions, general <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> and Gross-Neveu models, holographic theories, <math><mi>N</mi><mo>=</mo><mn>2</mn></math> Wess-Zumino models and general ABJM theories.

Spain

Pablo Bueno

Ecuador

Oscar Lasso Andino

Israel

Javier Moreno

Conformal Bounds in Three Dimensions from Entanglement Entropy

We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to R  d−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

Ignacio Landea

Gonzalo Torroba

Irreversibility, QNEC, and defects

A theory with a non-compact form-symmetry is described by two closed form fields of degrees k and d – k. Effective theory examples are non-linear electrodynamics, a photon field coupled to a neutron field, and a low energy Goldstone boson. We show these models cannot be completed in the UV without breaking the non-compact form-symmetry down to a compact one. This amounts to the existence of electric or magnetic charges. A theory with an unbroken non-compact k-form symmetry is massless and free.