We review the physical meaning of modular invariance for unitary conformal quantum field theories in d = 2 . For quantum field theory models, while T invariance is necessary for locality, S invariance is not mandatory. S 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. Published by the American Physical Society 2024
We study the Renyi and entanglement entropies for free 2d CFT's at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.
We study the Rényi and entanglement entropies for free 2d CFT’s at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.
A bstract We study the relation between the Jones index and the Rényi entropies of two disjoint intervals on the line and of the ground state for a generic value of the Rényi index in the two conformal field theory models given by the Ising model and a free Majorana fermion, where Haag duality is satisfied. The analytic expressions of the crossing asymmetry for all the submodels displaying a violation of the Haag duality that are closed under the fusion rules are obtained. In the limiting regime where the two intervals become adjacent, the leading term of the expansion of the crossing asymmetry provides the Jones global index, for any finite value of the Rényi index.
Certain duality of relative entropy can fail for a chiral conformal net with nontrivial representations. In this paper we quantify such statement by defining a quantity which measures the failure of such duality, and identify this quantity with relative entropy and global index associated with multi-interval subfactors for a large class of conformal nets. As a consequence of such new formulation we show that the duality holds for a large class of conformal nets if and only if they are holomorphic. The same argument also applies to CFT in two dimensions. In particular we show that the duality holds for a large class of CFT in two dimensions if and only if they are modular invariant. We also obtain various limiting properties of relative entropies which naturally follow from our formula.
We study the two point correlation function of a local operator on an
n-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable
w, dual to the Renyi parameter
n. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform
f(w) has bounded behavior at infinity in the complex
w plane, the discontinuity of the Renyi transform disc
f(w) provides the unique analytic continuation in
n of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large
N holographic CFT
d in dimensions
d>2. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in
2d conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in
2d, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.