<p>We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math> dimensions, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>th Rényi entropy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>S</mi><mi>A</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup></math> behaves as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi>f</mi><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>n</mi><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mtext>Area</mtext><mo stretchy="false">(</mo><mo stretchy="false">∂</mo><mi>A</mi><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">−</mo><mn>2</mn><mo stretchy="false">)</mo><msup><mrow><mi>ε</mi></mrow><mrow><mi>d</mi><mo stretchy="false">−</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mrow></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo stretchy="false">→</mo><mn>0</mn></math> limit when the boundary of the entanglement domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math> is spherical with the UV cutoff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ε</mi></math>. The theory dependence is encapsulated in the cosmological constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math> in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.</p>

Japan

Yuya Kusuki

United States

Hirosi Ooguri

Sridip Pal

Universality of Rényi Entropy in Conformal Field Theory

We derive a formula for the half-BPS interface entropy between any pair of $$ \mathcal{N} $$ = (4, 4) theories on the same conformal manifold. This generalizes the diastasis formula derived in [1] for $$ \mathcal{N} $$ = (2, 2) theories, which is restricted to the conformal submanifolds generated by either chiral or twisted chiral multiples of $$ \mathcal{N} $$ = (2, 2) supersymmetry. To derive the $$ \mathcal{N} $$ = (4, 4) formula, we use the fact that the conformal manifold of $$ \mathcal{N} $$ = (4, 4) theories is symmetric and quaternionic-Kähler and that its isotropy group contains the SU(2) ⊗ SU(2) external automorphism of the $$ \mathcal{N} $$ = (4, 4) superconformal algebra. As an application of the formula, we prove a supersymmetric non-renormalization theorem, which explains the observation in [2] that the interface entropy for half-BPS Janus solutions in type IIB supergravity on $\textit{AdS}$ × $\textit{S}$ × $\textit{T}$ coincides with the corresponding quantity in their free conformal field limits.

Andreas Karch

Mianqi Wang

Nonrenormalization theorem for $$ \mathcal{N} $$ = (4, 4) interface entropy

A state in a quantum system with a given global symmetry, , can be sensitive to the presence of boundaries, which may either preserve or break this symmetry. In this work, we investigate how conformal invariant boundary conditions influence the -symmetry breaking through the lens of the entanglement asymmetry, a quantifier of the “distance” between a symmetry-broken state and its symmetrized counterpart. By leveraging 2D boundary conformal field theory (BCFT), we investigate the symmetry breaking for both finite and compact Lie groups. Beyond the leading order term, we also compute the subleading corrections in the subsystem size, highlighting their dependence on the symmetry group and the BCFT operator content. We further explore the entanglement asymmetry following a global quantum quench, where a symmetry-broken state evolves under a symmetry-restoring Hamiltonian. In this dynamical setting, we compute the entanglement asymmetry by extending the method of images to a BCFT with non-local objects such as invertible symmetry defects.

Sara Murciano

Entanglement asymmetry and symmetry defects in boundary conformal field theory

For any unitary conformal field theory in two dimensions with the central charge c, we prove that, if there is a nontrivial primary operator whose conformal dimension ∆ vanishes in some limit on the conformal manifold, the Zamolodchikov distance t to the limit is infinite, the approach to this limit is exponential ∆ = exp(−αt + O(1)), and the decay rate obeys the universal bounds c −1/2 ≤ α ≤ 1. In the limit, we also find that an infinite tower of primary operators emerges without a gap above the vacuum and that the conformal field theory becomes locally a tensor product of a sigma-model in the large radius limit and a compact theory. As a corollary, we establish a part of the Distance Conjecture about gravitational theories in three-dimensional anti-de Sitter space. In particular, our bounds on α indicate that the emergence of exponentially light states is inevitable as the moduli field corresponding to t rolls beyond the Planck scale along the steepest path and that this phenomenon can begin already at the curvature scale of the bulk geometry. We also comment on implications of our bounds for gravity in asymptotically flat spacetime by taking the flat space limit and compare with the Sharpened Distance Conjecture.

Yifan Wang

Universal bounds on CFT Distance Conjecture

The effective central charge (denoted by <math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>eff</mi></mrow></msub></mrow></math>) is a measure of entanglement through a conformal interface, while the transmission coefficient (encoded in the coefficient <math><msub><mi>c</mi><mrow><mi>L</mi><mi>R</mi></mrow></msub></math> of the two-point function of the energy-momentum tensor across the interface) is a measure of energy transmission through the interface. It has been pointed out that these two are generally different. In this Letter, we propose the inequalities, <math><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>L</mi><mi>R</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>eff</mi></mrow></msub><mo>≤</mo><mi>min</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></mrow></math>. They have the simple but important implication that the amount of energy transmission can never exceed the amount of information transmission. We verify them using the AdS/CFT correspondence, using the perturbation method, and in examples beyond holography. We also show that these inequalities are sharp by constructing a class of interfaces that saturate them.

Hao-Yu Sun

Universal Bound on Effective Central Charge and Its Saturation

Equilibrium finite temperature observables of a CFT can be described by a local effective action for background fields — a “thermal effective action”. This effective action determines the asymptotic density of states of a CFT as a detailed function of dimension and spin. We discuss subleading perturbative and nonperturbative corrections to the density, comparing with free and holographic examples. We furthermore show how to use the thermal effective action on more complicated geometries at special locations called “hot spots”. The hot spot idea makes a prediction for a CFT partition function on a higher-dimensional version of a genus-2 Riemann surface, in a particular high temperature limit. By decomposing the partition function into a novel higher-dimensional version of genus-2 conformal blocks (which we compute at large scaling dimension), we extract the asymptotic density of heavy-heavy-heavy OPE coefficients in a higher-dimensional CFT. We also compute asymptotics of thermal 1-point functions using the same techniques.

Nathan Benjamin

Jaeha Lee

David Simmons-Duffin

Universal asymptotics for high energy CFT data

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Symmetry-resolved entanglement entropy, spectra &amp; boundary conformal field theory

When an interface connects two CFTs, the entanglement entropy between the two CFTs is determined by a quantity called the effective central charge. The effective central charge does not have a simple form in terms of the central charges of the two CFTs, but intricately depends on the transmissive properties of the interface.

Universality of effective central charge in interface CFTs

We consider a <math><mi>d</mi></math>-dimensional unitary conformal field theory with a compact Lie group global symmetry <math><mi>G</mi></math> and show that, at high temperature <math><mi>T</mi></math> and on a compact Cauchy surface, the probability of a randomly chosen state being in an irreducible unitary representation <math><mi>R</mi></math> of <math><mi>G</mi></math> is proportional to <math><mrow><mo>(</mo><mrow><mi>dim</mi><mtext> </mtext></mrow><mi>R</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>exp</mi><mo>[</mo><mo>−</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>/</mo><mo>(</mo><mi>b</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>]</mo></mrow></math>. We use the spurion analysis to derive this formula and relate the constant <math><mi>b</mi></math> to a domain wall tension. We also verify it for free field theories and holographic conformal field theories and compute <math><mi>b</mi></math> in these cases. This generalizes the result in 2109.03838 that the probability is proportional to <math><mo>(</mo><mrow><mi>dim</mi></mrow><mi>R</mi><msup><mo>)</mo><mn>2</mn></msup></math> when <math><mi>G</mi></math> is a finite group. As a byproduct of this analysis, we clarify thermodynamical properties of black holes with non-Abelian hair in anti–de Sitter space.

Monica Jinwoo Kang

Universal formula for the density of states with continuous symmetry

Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S 1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S 1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory.