Does a conformal manifold imply the existence of exactly marginal operators? We answer this question affirmatively under the assumption that there exists a conformal interface with certain properties connecting nearby conformal field theories. We show that the exactly marginal operator that connects the conformal field theories can be reconstructed from the interface displacement operator. Our construction is model independent and based on the general principles of conformal symmetry.

CERN

Shota Komatsu

Japan

Yuya Kusuki

Italy

Marco Meineri

United States

Hirosi Ooguri

Continuous Family of Conformal Field Theories and Exactly Marginal Operators

We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi></mrow></math> conformal field theories, related to each other by exactly marginal boundary deformations, forming a “boundary conformal manifold.” Our definition builds upon previous works on tensor networks, such as matrix product states, where the triple inner product or multiwavefunction overlap plays a key geometric role. On the one hand, our boundary conformal field theory formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the Neveu-Schwarz-Neveu-Schwarz (NS-NS) B-field in string theory. The general definition does not require such an interpretation and is formulated purely field theoretically, in terms of correlation functions of boundary-condition-changing operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.

Yichul Choi

Hyunsoo Ha

Dongyeob Kim

Austria

Shuhei Ohyama

Shinsei Ryu

Higher structures on boundary conformal manifolds: Higher Berry phase and boundary conformal field theory

A representative quantity that characterizes the dynamics of conformal interfaces is the transmission coefficient, which is defined through correlation functions of the stress tensor. Typically, this coefficient is complicated and highly dependent on its details. In this work, we introduce a new perspective based on the notion of a “phantom current”. We have shown that a spin-2 phantom current arising from the folding trick determines the transmission coefficient. As a result, our framework provides a unified explanation of known results in minimal models and the free boson, while also yielding concrete predictions for previously unexplored interfaces.

Yuma Furuta

Toshiki Onagi

Transmission coefficients from phantom currents

We investigate crosscap states in 2D rational conformal field theories (RCFTs), with an emphasis on the role of noninvertible symmetries. In particular, we argue for the existence of crosscap states labeled by each Verlinde line in the RCFT, extending previous constructions involving simple currents. Evidence for the existence of these new states is obtained by deriving a generalized Cardy condition incorporating both crosscaps and topological defects, which we check in some concrete examples. Finally, we briefly discuss how these crosscap states transform under the action of Verlinde lines, as well as the connection to mixed anomalies between parity and internal symmetries.

Wataru Harada

Justin Kaidi

Yuefeng Liu

New Crosscap States

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.

Sridip Pal

Universality of Rényi Entropy in Conformal Field Theory

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

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.

Andreas Karch

Hao-Yu Sun

Mianqi Wang

Universal Bound on Effective Central Charge and Its Saturation

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

While recent progress in the black hole information problem has shown that the entropy of Hawking radiation follows a unitary Page curve, the quantum state of Hawking radiation prior the Page time is still treated as purely thermal, containing no information about the microstructure of the black hole. We demonstrate that there is significant quantum information regarding the quantum state of the black hole in the Hawking radiation prior to the Page time. By computing of the quantum fidelity in a 2D boundary conformal field theory (BCFT) model of black hole evaporation, we demonstrate that an observer outside of an evaporating black hole may distinguish different black holes via measurements of the Hawking radiation at any time during the evaporation process, albeit with an exponentially large number of measurements. Furthermore, our results are universal, applicable to general BCFTs including those with large central charge and rational BCFTs. The techniques we develop for computing the fidelity are more generally applicable to excited states in CFT. As such, we are able to characterize more general aspects of thermalization in 2D conformal field theory.