We evaluate the large-N behavior of the superconformal indices of toric quiver gauge theories, and use it to find the entropy functions of the dual electrically charged rotating AdS 5 black holes. To this end, we employ the recently proposed Bethe Ansatz method, and find a certain set of solutions to the Bethe Ansatz Equations of toric theories. This, in turn, allows us to compute the large-N behavior of the index for these theories, including the infinite families Y pq , X pq and L pqr of quiver gauge theories. Our results are in perfect agreement with the predictions made recently using the Cardy-like limit of the superconformal index. We also explore the index structure in the space of chemical potentials and describe the pattern of Stokes lines arising in the conifold theory case.

We derive explicit expressions for the two-point function of a massless scalar field in the interior region of a Reissner-Nordstrom black hole, in both the Unruh and Hartle-Hawking quantum states. The two-point function is expressed in terms of the standard lmω modes of the scalar field (those associated with a spherical harmonic Y lm and a temporal mode e −iωt ), which can be conveniently obtained by solving an ordinary differential equation, the radial equation. These explicit expressions are the internal analogs of the well known results in the external region (originally derived by Christensen and Fulling), in which the two-point function outside the black hole is written in terms of the external lmω modes of the field. They allow the computation of < Φ 2 >ren and the renormalized stress-energy tensor inside the black hole, after the radial equation has been solved (usually numerically). In the second part of the paper, we provide an explicit expression for the trace of the renormalized stress-energy tensor of a minimally-coupled massless scalar field (which is non-conformal), relating it to the d'Alembertian of < Φ 2 >ren. This expression proves itself useful in various calculations of the renormalized stress-energy tensor. I. INTRODUCTIONIn the framework of semiclassical general relativity, the gravitational field is treated classically as a curved spacetime while other fields are treated as quantum fields residing in this background spacetime. The relation between the spacetime geometry and the stress-energy of the quantum fields is described by the semiclassical Einstein equation( 1.1) where G µν is the Einstein tensor of spacetime, and ren is the renormalized stress-energy tensor (RSET), which is the renormalized expectation value of the stress-energy tensor operatorT , associated with the quantum fields. In Eq. (1.1) and throughout this paper we adopt standard geometric units c = G = 1, and signature (− + ++).The main challenge in analyzing the semiclassical Einstein equation is the computation of the RSET. Even when the background geometry is fixed and the corresponding metric is given, performing a procedure of renormalization is not an easy task. The task becomes much more difficult upon trying to solve the full self-consistent problem represented by Eq. (1.1). One reason (besides the obvious numerical challenge) is that Eq. (1.1) admits runaway solutions [1].An example of a quantum field that is often chosen for its simplicity is that of a scalar field, which satisfies the Klein-Gordon equationwhereΦ is the scalar field operator, m denotes the field's mass, and ξ is the coupling constant. As a first stage towards calculating the RSET, it is customary to begin by calculating <Φ 2 > ren , as this quantity is endowed with many of the essential features of the RSET, but is simpler to compute. In other words, <Φ 2 > ren serves as a simple toy model for the RSET. The standard method to calculate quantities which are quadratic in the field and its derivatives is point-splitting (or covariant point ...

Recently a very interesting three-dimensional N = 2 supersymmetric theory with SU(3) global symmetry was discussed by several authors. We denote this model by T x. This was conjectured to have two dual descriptions, one with explicit supersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using T x as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of N = 2 mirror dualities involving such constructions with the dual being either a simple N = 2 Wess-Zumino model or a discrete gauging thereof.

We numerically compute the renormalized expectation value Φ 2 ren of a minimally-coupled massless quantum scalar field in the interior of a four-dimensional Reissner-Nordstrom black hole, in both the Hartle-Hawking and Unruh states. To this end we use a recently developed mode-sum renormalization scheme based on covariant point splitting. In both quantum states, Φ 2 ren is found to approach a finite value at the inner horizon (IH). The final approach to the IH asymptotic value is marked by an inverse-power tail r −n * , where r * is the Regge-Wheeler "tortoise coordinate", and with n = 2 for the Hartle-Hawking state and n = 3 for the Unruh state. We also report here the results of an analytical computation of these inverse-power tails of Φ 2 ren near the IH. Our numerical results show very good agreement with this analytical derivation (for both the power index and the tail amplitude), in both quantum states. Finally, from this asymptotic behavior of Φ 2 ren we analytically compute the leading-order asymptotic behavior of the trace T µ µ ren of the renormalized stress-energy tensor at the IH. In both quantum states this quantity is found to diverge like b(r − r−) −1 r −n−2 * (with n specified above, and with a known parameter b). To the best of our knowledge, this is the first fully-quantitative derivation of the asymptotic behavior of these renormalized quantities at the inner horizon of a four-dimensional Reissner-Nordstrom black hole.

The full computation of the renormalized expectation values Φ 2 ren and T µν ren in 4D black hole interiors has been a long standing challenge, which has impeded the investigation of quantum effects on the internal structure of black holes for decades. Employing a recently developed mode sum renormalization scheme to numerically implement the point-splitting method, we report here the first computation of Φ 2 ren in Unruh state in the region inside the event horizon of a 4D Schwarzschild black hole. We further present its Hartle-Hawking counterpart, which we calculated using the same method, and obtain a fairly good agreement with previous results attained using an entirely different method by Candelas and Jensen in 1986. Our results further agree upon approaching the event horizon when compared with previous results calculated outside the black hole. Finally, the results we obtained for Hartle-Hawking state at the event horizon agree with previous analytical results published by Candelas in 1980. This work sets the stage for further explorations of Φ 2 ren and T µν ren in 4D black hole interiors.