We study the single interval entanglement and relative entropies of conformal descendants in 2d CFT. Descendants contain non-trivial entanglement, though the entanglement entropy of the canonical primary in the free boson CFT contains no additional e
ntanglement compared to the vacuum, we show that the entanglement entropy of the state created by its level one descendant is non-trivial and is identical to that of the $U(1)$ current in this theory. We determine the first sub-leading corrections to the short interval expansion of the entanglement entropy of descendants in a general CFT from their four point function on the n-sheeted plane. We show that these corrections are determined by multiplying squares of appropriate dressing factors to the corresponding corrections of the primary. Relative entropy between descendants of the same primary is proportional to the square of the difference of their dressing factors. We apply our results to a class of descendants of generalized free fields and descendants of the vacuum and show that their dressing factors are universal.
We study the crossing equations in $d=3$ for the four point function of two $U(1)$ currents and two scalars including the presence of a parity violating term for the $s$-channel stress tensor exchange. We show the existence of a new tower of double t
race operators in the $t$-channel whose presence is necessary for the crossing equation to be satisfied and determine the corresponding large spin parity violating OPE coefficients. Contrary to the parity even situation, we find that the parity odd $s$-channel light cone stress tensor block do not have logarithmic singularities. This implies that the parity odd term does not contribute to anomalous dimensions in the crossed channel at this order in light cone expansion. We then study the constraints imposed by reflection positivity and crossing symmetry on such a four point function. We reproduce the previously known parity odd collider bounds through this analysis. The contribution of the parity violating term in the collider bound results from a square root branch cut present in the light cone block as opposed to a logarithmic cut in the parity even case, together with the application of the Cauchy-Schwarz inequality.
We derive constraints on three-point functions involving the stress tensor, $T$, and a conserved $U(1)$ current, $j$, in 2+1 dimensional conformal field theories that violate parity, using conformal collider bounds introduced by Hofman and Maldacena.
Conformal invariance allows parity-odd tensor-structures for the $langle T T T rangle$ and $ langle j j T rangle$ correlation functions which are unique to three space-time dimensions. Let the parameters which determine the $langle T T T rangle$ correlation function be $t_4$ and $alpha_T$ , where $alpha_T$ is the parity-violating contribution. Similarly let the parameters which determine $ langle j j T rangle$ correlation function be $a_2$, and $alpha_J$ , where $alpha_J$ is the parity-violating contribution. We show that the parameters $(t_4, alpha_T)$ and $(a_2, alpha_J)$ are bounded to lie inside a disc at the origin of the $t_4$ - $alpha_T$ plane and the $a_2$ - $alpha_J$ plane respectively. We then show that large $N$ Chern-Simons theories coupled to a fundamental fermion/boson lie on the circle which bounds these discs. The `t Hooft coupling determines the location of these theories on the boundary circles.
We derive spectral sum rules in the shear channel for conformal field theories at finite temperature in general $dgeq 3$ dimensions. The sum rules result from the OPE of the stress tensor at high frequency as well as the hydrodynamic behaviour of the
theory at low frequencies. The sum rule states that a weighted integral of the spectral density over frequencies is proportional to the energy density of the theory. We show that the proportionality constant can be written in terms the Hofman-Maldacena variables $t_2, t_4$ which determine the three point function of the stress tensor. For theories which admit a two derivative gravity dual this proportionality constant is given by $frac{d}{2(d+1)}$. We then use causality constraints and obtain bounds on the sum rule which are valid in any conformal field theory. Finally we demonstrate that the high frequency behaviour of the spectral function in the vector and the tensor channel are also determined by the Hofman-Maldacena variables.
We investigate the constraints imposed by global gravitational anomalies on parity odd induced transport coefficients in even dimensions for theories with chiral fermions, gravitinos and self dual tensors. The $eta$-invariant for the large diffeomorp
hism corresponding to the $T$ transformation on a torus constraints the coefficients in the thermal effective action up to mod 2. We show that the result obtained for the parity odd transport for gravitinos using global anomaly matching is consistent with the direct perturbative calculation. In $d=6$ we see that the second Pontryagin class in the anomaly polynomial does not contribute to the $eta$-invariant which provides a topological explanation of this observation in the `replacement rule. We then perform a direct perturbative calculation for the contribution of the self dual tensor in $d=6$ to the parity odd transport coefficient using the Feynman rules proposed by Gaum{e} and Witten. The result for the transport coefficient agrees with that obtained using matching of global anomalies.
We show that planar cal N=4 Yang-Mills theory at zero t Hooft coupling can be efficiently described in terms of 8 bosonic and 8 fermionic oscillators. We show that these oscillators can serve as world-sheet variables, the string bits, of a discretize
d string. There is a one to one correspondence between the on shell gauge invariant words of the free Y-M theory and the states in the oscillators Hilbert space, obeying a local gauge and cyclicity constraints. The planar two-point functions and the three-point functions of all gauge invariant words are obtained by the simple delta-function overlap of the corresponding discrete string world sheet. At first order in the t Hooft coupling, i.e. at one-loop in the Y-M theory, the logarithmic corrections of the planar two-point and the three-point functions can be incorporated by nearest neighbour interactions among the discretized string bits. In the SU(2) sub-sector we show that the one-loop corrections to the structure constants can be uniquely determined by the symmetries of the bit picture. For the SU(2) sub-sector we construct a gauged, linear, discrete world-sheet model for the oscillators, with only nearest neighbour couplings, which reproduces the anomalous dimension Hamiltonian up to two loops. This model also obeys BMN scaling to all loops.
