Monodromy methods for torus conformal blocks and entanglement entropy at large central charge

Abstract

We compute the entanglement entropy in a two dimensional conformal field theory at finite size and finite temperature in the large central charge limit via the replica trick. We first generalize the known monodromy method for the calculation of conformal blocks on the plane to the torus. Then, we derive a monodromy method for the zero-point conformal blocks of the replica partition function. We explain the differences between the two monodromy methods before applying them to the calculation of the entanglement entropy. We find that the contribution of the vacuum exchange dominates the entanglement entropy for a large class of CFTs, leading to universal results in agreement with holographic predictions from the RT formula. Moreover, we determine in which regime the replica partition function agrees with a correlation function of local twist operators on the torus.

Illuminating entanglement shadows of BTZ black holes by a generalized entanglement measure

“…(hp/c,hq/c;A) . For the ordinary entanglement entropy of a spatial subregion defined by (3.1), the calculation proceeds as follows: the replica partition function Z S N α,replica (A) is dominated by the identity zero-point conformal block on the higher genus replica surface at low temperature and its modular transformation τ → −1/τ at high temperature [18]. By identity block we mean a conformal block whose internal operators all have vanishing conformal weight, h p = h q = 0.…”

“…6 This statement is known as vacuum block dominance and holds for CFTs with large central charge c, a sparse spectrum of light operators and at most exponentially growing OPE coefficients [19], including the S N orbifold studied here [16]. The semiclassical conformal blocks are obtained from the solution of an auxiliary differential equation via a monodromy method [18,19], yielding…”

“…The calculation of the entanglement entropy (3.4) considered here is then a straightforward generalization of the above calculation. We refer to [18,19] for a more detailed explanation of the procedure and only explain the differences to the ordinary entanglement entropy of a spatial subregion in following.…”

“…Note that all descendants of a primary operator Σ are in the same twisted sector as Σ, thus the projection does not change the conformal blocks themselves. The modification of the entangling intervals also necessitates a change in the monodromy conditions used to derive the conformal block in [18]: instead of trivial monodromy around an entangling interval A = [0, z 2 ] we impose trivial monodromy around a path encircling k times the spatial circle combined with the interval [0, z 2 ]. Due to the projection onto the subset S of twisted sectors considered, this choice of monodromy conditions is well defined since all of the fields touched by this path are sewn together into a continuous cycle by the twisted boundary conditions.…”

“…To compute the entanglement entropy, we then need to derive the zero-point conformal block on the replica surface for internal operators 1 and Σ. This may be achieved in a perturbation expansion in α − 1 using the monodromy method derived in [18]. The zeroth order in α − 1 gives the h Σ character as expected.…”