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.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.

arXiv:1812.01119Transactions of the American Mathematical SocietyOct 2, 201915 citations

Covariant Anyons via Mackey Machinery

Radhakrishnan Balu

We obtain relativistically consistent description of anyons, that are relevant in topological quantum computation, via subfactors.We proceed with the construction of a Hilbert space for space-like particles and build composite particles by permutations to form hyperfinite factors from a little group and the whole Poincaré group via the Mackey machinery.This leads to a subfactor via Jones basic construction.With the identification of modular data associated with the subfactor we have a description of anyons in terms of rational conformal field theory that is relativistically consistent.We obtain relativistically consistent description of anyons, that are relevant in topological quantum computation, via subfactors.We proceed with the construction of a Hilbert space for space-like particles and build composite particles by permutations to form hyperfinite factors from a little group and the whole Poincaré group via the Mackey machinery.This leads to a subfactor via Jones basic construction.With the identification of modular data associated with the subfactor we have a description of anyons in terms of rational conformal field theory that is relativistically consistent.

Journal of Stochastic AnalysisJan 25, 20240 citations

Fusion rules and modular transformations in 2D conformal field theory

Erik Verlinde

Nuclear Physics BJan 1, 19881,536 citations

The modular energy range of an interval in chiral conformal field theory

Samuel L. Braunstein, Zhi-Wei Wang

We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$ , and states of mean energy at most $E$ , the supremum of the modular energy is $M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})$ , and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]$ , so the range is linear above and logarithmic below, the two meeting as $RE \to 0$ ; it is strictly wider above, by more than $\frac{2}{5}πRE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $πR/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}$ , recovering the original form of that inequality from the modular form that by itself imposes no constraint.We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \to 0$; it is strictly wider above, by more than $\frac{2}{5}πRE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $πR/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.

arXiv:2609.27830Aug 19, 20260 citations

Quantum group interpretation of some conformal field theories

Luis Álvarez Gaumé, César Gómez, Germán Sierra

Physics Letters BMar 1, 1989284 citations

We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length

R

in a theory of central charge

c

, and states of mean energy at most

E

, the supremum of the modular energy is

M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})

, and neither constant can be improved. The infimum is logarithmic,

M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]

, so the range is linear above and logarithmic below, the two meeting as

RE \to 0

; it is strictly wider above, by more than

\frac{2}{5}πRE

at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient

πR/2

is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant

c/6

is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being

nc/6

for

n

disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound,

ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}

, recovering the original form of that inequality from the modular form that by itself imposes no constraint.