We develop a method to evaluate the partition function and energy density of a massive scalar on a 2-sphere of radius and at finite temperature as power series in $$ \frac{\beta }{r} $$ . Each term in the power series can be written in terms of polylogarithms. We use this result to obtain the gap equation for the large , critical ( ) model with a quartic interaction on × in the large radius expansion. Solving the gap equation perturbatively we obtain the leading finite size corrections to the expectation value of stress tensor for the ( ) vector model on × . Applying the Euclidean inversion formula on the perturbative expansion of the thermal two point function we obtain the finite size corrections to the expectation value of the higher spin currents of the critical ( ) model. Finally we show that these finite size corrections of higher spin currents tend to that of the free theory at large spin as seen earlier for the model on × .

India

Justin David

Srijan Kumar

The large

We evaluate the thermal one point function of higher spin currents in the critical model of U( ) complex scalars interacting with a quartic potential and the U( ) Gross-Neveu model of Dirac fermions at large and strong coupling using the Euclidean inversion formula. These models are considered in odd space time dimensions and held at finite temperature and finite real chemical potential measured in units of the temperature. We show that these one point functions simplify both at large spin and large . At large spin, the one point functions behave as though the theory is free, the chemical potential appears through a simple pre-factor which is either cosh or sinh depending on whether the spin is even or odd. At large , but at finite spin and chemical potential, the 1-point functions are suppressed exponentially in compared to the free theory. We study a fixed point of the critical Gross-Neveu model in = 3 with 1-point functions exhibiting a branch cut in the chemical potential plane. The critical exponent for the free energy or the pressure at the branch point is 3/2 which coincides with the mean field exponent of the Lee-Yang edge singularity for repulsive core interactions.

One point functions in large

We consider linear superpositions of single particle excitations in a scalar field theory on AdS 3 and evaluate their contribution to the bulk entanglement entropy across the Ryu-Takayanagi surface. We compare the entanglement entropy of these excitations obtained using the Faulkner-Lewkowycz-Maldacena formula to the entanglement entropy of linear superposition of global descendants of a conformal primary in a large c CFT obtained using the replica trick. We show that the closed form expressions for the entanglement entropy in the small interval expansion both in gravity and the CFT precisely agree. The agreement serves as a non-trivial check of the FLM formula for the quantum corrections to holographic entanglement entropy as well as the methods developed in the CFT to evaluate entanglement entropy of descendants. Our checks includes an example in which the state is time dependent and spatially in-homogenous as well another example involving a coherent state with a Bañados geometry as its holographic dual.

Barsha Chowdhury

Semanti Dutta

Jyotirmoy Mukherjee

Precision tests of bulk entanglement entropy

Abstract Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the “magnetic charge” in the bosonization, is represented by a “hole” excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context.

Japan

Okuto Morikawa

Soma Onoda

Hiroshi Suzuki

Yet Another Lattice Formulation of 2D (1) Chiral Gauge Theory via Bosonization David Tong: lectures on gauge theory

We apply the OPE inversion formula on thermal two-point functions of fermions to obtain thermal one-point function of fermion bi-linears appearing in the corresponding OPE. We primarily focus on the OPE channel which contains the stress tensor of the theory. We apply our formalism to the mean field theory of fermions and verify that the inversion formula reproduces the spectrum as well as their corresponding thermal one-point functions. We then examine the large N critical Gross-Neveu model in d = 2k + 1 dimensions with k even and at finite temperature. We show that stress tensor evaluated from the inversion formula agrees with that evaluated from the partition function at the critical point. We demonstrate the expectation values of 3 different classes of higher spin currents are all related to each other by numerical constants, spin and the thermal mass. We evaluate the ratio of the thermal expectation values of higher spin currents at the critical point to the Gaussian fixed point or the Stefan-Boltzmann result, both for the large N critical O(N) model and the Gross-Neveu model in odd dimensions. This ratio is always less than one and it approaches unity on increasing the spin with the dimension d held fixed. The ratio however approaches zero when the dimension d is increased with the spin held fixed.

Thermal one-point functions: CFT’s with fermions, large d and large spin

The partition functions of free bosons as well as fermions on AdS 2 are not smooth as a function of their masses. For free bosons, the partition function on AdS 2 is not smooth when the mass saturates the Breitenlohner-Freedman bound. We show that the expectation value of the scalar bilinear on AdS 2 exhibits a kink at the BF bound and the change in slope of the expectation value with respect to the mass is proportional to the inverse radius of AdS 2. For free fermions, when the mass vanishes the partition function exhibits a kink. We show that expectation value of the fermion bilinear is discontinuous and the jump in the expectation value is proportional to the inverse radius of AdS 2. We then show the supersymmetric actions of the chiral multiplet on AdS 2 × S 1 and the hypermultiplet on AdS 2 × S 2 demonstrate these features. The supersymmetric backgrounds are such that as the ratio of the radius of AdS 2 to S 1 or S 2 is dialled, the partition functions as well as expectation of bilinears are not smooth for each Kaluza-Klein mode on S 1 or S 2. Our observation is relevant for evaluating one-loop partition function in the near horizon geometry of extremal black holes.

Italy

Edi Gava

Rajesh Gupta

K. Narain

Discontinuities of free theories on AdS 2

We study the time dependence of Rényi/entanglement entropies of locally excited states created by fields with integer spins s ≤ 2 in 4 dimensions. For spins 0, 1 these states are characterised by localised energy densities of a given width which travel as a spherical wave at the speed of light. For the spin 2 case, in the absence of a local gauge invariant stress tensor, we probe these states with the Kretschmann scalar and show they represent localised curvature densities which travel at the speed of light. We consider the reduced density matrix of the half space with these excitations and develop methods which include a convenient gauge choice to evaluate the time dependence of Rényi/entanglement entropies as these quenches enter the half region. In all cases, the entanglement entropy grows in time and saturates at log 2. In the limit, the width of these excitations tends to zero, the growth is determined by order 2s + 1 polynomials in the ratio of the distance from the co-dimension-2 entangling surface and time. The polynomials corresponding to quenches created by the fields can be organized in terms of their representations under the SO(2) T  × SO(2) L  symmetry preserved by the presence of the co-dimension 2 entangling surface. For fields transforming as scalars under this symmetry, the order 2s + 1 polynomial is completely determined by the spin.

Entanglement entropy of local gravitational quenches

We study thermal one point functions of massive scalars in AdS  d+1 black holes. These are induced by coupling the scalar to either the Weyl tensor squared or the Gauss-Bonnet term. Grinberg and Maldacena argued that the one point functions sourced by the Weyl tensor exponentiate in the limit of large scalar masses and they contain information of the black hole geometry behind the horizon. We observe that the one point functions behave identically in this limit for either of the couplings mentioned earlier. We show that in an appropriate large d limit, the one point function for the charged black hole in AdS  d+1 can be obtained exactly. These black holes in general contain an inner horizon. We show that the one point function exponentiates and contains the information of both the proper time between the outer horizon to the inner horizon as well as the proper length from the inner horizon to the singularity. We also show that Gauss-Bonnet coupling induced one point functions in AdS  d+1 black holes with hyperbolic horizons behave as anticipated by Grinberg-Maldacena. Finally, we study the one point functions in the background of rotating BTZ black holes induced by the cubic coupling of scalars.

Thermal one point functions, large d and interior geometry of black holes

We consider the linearised graviton in 4d Minkowski space and decompose it into tensor spherical harmonics and fix the gauge. The Gauss law of gravity implies that certain radial components of the Riemann tensor of the graviton on the sphere labels the superselection sectors for the graviton. We show that among these 6 normal components of the Riemann tensor, 2 are related locally to the algebra of gauge-invariant operators in the sphere. From the two-point function of these components of the Riemann tensor on S 2 we compute the logarithmic coefficient of the entanglement entropy of these superselection sectors across a spherical entangling surface. For sectors labelled by each of the two components of the Riemann tensor these coefficients are equal and their total contribution is given by   <math> <mo>−</mo> <mfrac> <mn>16</mn> <mn>3</mn> </mfrac> </math>  $$ -\frac{16}{3} $$ . We observe that this coefficient coincides with that extracted from the edge partition function of the massless spin-2 field on the 4-sphere when written in terms of its Harish-Chandra character. As a preliminary step, we also evaluate the logarithmic coefficient of the entanglement entropy from the superselection sectors labelled by the radial component of the electric field of the U(1) theory in even d dimensions. We show that this agrees with the corresponding coefficient of the edge Harish-Chandra character of the massless spin-1 field on S  d .

Entanglement entropy of gravitational edge modes

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 entanglement 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.