Introduction to AdS Space

In the present work we start with a basic discussion of CFT that only focuses on two point operator in CFT and then we move to basics of AdS/CFT including different coordinate charts, solution of scalar wave equation in different coordinate systems,normalizability and Breitenlohner-Freedman (BF) bounds

Holographic computational complexity in non-relativistic field theories

The aim of the project is to study the holographic complexity of black holes and black branes. We discuss basic concept of complexity. We see how quantum computational complexity is related to black hole physics. In the process we look at the difference between classical and quantum complexity. We solve for the holographic complexity of pure AdS and pure Lifshitz spacetime. We then study the AdS black hole in detail. We first try to find the nature of the maximal volume slice at a finite time, and then see the small black hole limit of it using the perturbation method. We perform similar late time analysis on the Lifshitz black hole as well. Lastly we try to understand the notion of complexity from first principle calculations. We retrace the procedure of calculating the complexity of a system and try to compare with the holographic quantity. However, although the two approaches are similar in a lot of aspects, they don't match perfectly. i

Holographic complexity of LST and single trace TT¯ , JT¯ and TJ¯ deformations

This work is an extension of our previous work [1] where we exploited holography to compute the complexity characteristics of Little String Theory (LST), a nonlocal, nongravitational field theory which flows to a local 2d CFT in the IR under RG via an integrable irrelevant (TT¯) deformation. Here we look at the more general LST obtained by UV deforming the 2d CFT by incorporating Lorentz violating irrelevant JT¯ and TJ¯ deformations on top of TT¯ deformation, in an effort to capture the novel signatures of Lorentz violation (on top of nonlocality) on quantum complexity. In anticipation of the fact that the dual field theory is Lorentz violating, we compute the volume complexity in two different Lorentz frames and the comparison is drawn between the results. It turns out that for this system the nonlocality and Lorentz violation effects are inextricably intertwined in the UV divergence structure of the quantum complexity. The coefficients of the divergences carry the signature of Lorentz boost violation. We also compute the subregion complexity which displays a (Hagedorn) phase transition with the transition point being the same as that for the phase transition of entanglement entropy [2]. These new results are consistent with our previous work [1]. Null warped AdS3 is treated as special case of interest. © 2022, The Author(s)

Constructing New Asymptotically de-Sitter Cosmological Solutions in (2+1)-Dimensions

In this work we find novel warped Kerr-de Sitter cosmological solution to the Einsteins field equa- tions in ( 2 + 1 ) − dimensions, modified by a Chern-Simons term. To start with we review the exact solutions of Einsteins field equations including the Penrose diagrams for highly symmetric space- times like that of Minkowski, de Sitter, anti-de Sitter and FLRW spacetimes. Next we review BTZ Black hole and their geometry for both rotating and non-rotating black holes. Balasubramanian and et. al. had found topologically non-trivial de Sitter (dS) solutions in (2 + 1)- dimensions, which are dubbed Kerr de Sitter and singular de Sitter quotients with Big bang/big crunch type singularities are also reviewed. Gravity in ( 2 + 1 ) − dimensions is reviewed as a simpler warm up problem to the ( 3 + 1 ) − dimensional case. In ( 2 + 1 ) − dimensions an additional Chern-Simon term is added to the Einstein-Hilbert action, this produces new interesting solutions. The introduction of the Chern-Simons term results in the non-zero mass of the graviton, thus giving us topologically massive gravity solutions. The warped AdS 3 and warped dS 3 spaces are also reviewed together with spacelike stretched warped AdS 3 black holes. Finally we write down the\ud war ped Kerr − dS 3 solutions to the topological massive gravity(TMG) equation of motion

Application of Mellin Space Techniques for Bulk Reconstruction in AdS/CFT

We discuss how CFT correlators can be written in terms of Mellin space. Starting from n > 3 correlators become function of cross ratios, which are conformally invariant and the conformal symmetries are not su�cient to determine the form of correlators completely. CFT has an advantage of OPE, where we can expand product of n operators in terms of primary and its descendants. Solving the coe�cients of the series will solve the n-point function. Using conformal block method we can determine OPE coe�cients. Mellin space techniques are more useful than conformal block approach. CFT correlators have nice form in Mellin space. Poles and residues in Mellin amplitude give the information about dimension of factorizing channel and OPE coe�- cients. We try to �nd poles of Mellin amplitude in Mandelstam variables

CMB Non-Gaussianities from Inflation Models

nflation [1, 2] offers solutions to several open questions in standard cosmology. It proposes a period of extremely rapid (exponential) expansion of the universe prior to the gradual Big Bang expansion. During Inflation, the energy density of the universe was dominated by a cosmological constant-type of vacuum energy which later decayed to produce the matter and radiation that fills the universe today. The simplest model of inflation involves a single scalar field, say, φ , dubbed as the inflaton [3]. Inflationary models are defined by a particular inflaton action (kinetic and potential terms) and its coupling to gravity. The dynamics (self coupling) of the inflaton field depend on the inflationary potential, V ( φ ). The different possibilities for V ( φ ) can lead to different models of inflation. Some well studied models [4] are small field inflation , large field inflation , axion monodromy inflation etc. There also are models with multiple inflaton fields [4, 5], e.g. N inflation . Thus, inflation is not a unique theory and still remains as a paradigm since the possibilities for getting inflationary expansion are (frustratingly) varied. In addition, we would like to distinguish inflation from other alternatives. To break the degeneracies among models, we need higher order correlation functions [6] of the CMB spectrum, namely non-Gaussianities . The ultimate goal of the project was to see how to filter out some inflation models based on the CMB non-Gaussianities. This is a very apt time to work on this as there are huge amounts of data available from experiments like Planck , BICEP 2 and BICEP

Conformal Bootstrap Method

The aim of this project is to study Conformal Field Theories in d ≥ 3 dimensions and derive the Conformal Bootstrap Equation which may be used to solve theories with conformal invariance. The identification conformal derives from the property that the transformation does not affect the angle between two arbitrary curves cross- ing each other at some point, conformal transformations preserve angles. Mathe- matically, a conformal transformation comprises of two steps - doing a conformal isometry, followed by a Weyl rescaling of the metric. The bootstrap approach is an algebraic recursive procedure, which heavily relies on determining the consequences of the symmetries and imposing consistency conditions. We started by considering infinitesimal form of conformal diffeomorphisms for d ≥ 3 spacetime dimensions. We identified translations, dilatations, rotations, boosts and special conformal transfor- mations as members of the conformal group and that the algebra is isomorphic to a pseudo-orthogonal SO ( d + 1 , 1). A number of fundamental systems have been shown to obey conformal invariance and we explicitly worked out the case for the massless scalar field and in a heuristic fashion, for the case of the source-free Maxwell field. Conformally invariant systems, in particular field theories are described by fields which transform homogeneously under conformal transformations i.e. which consti- tute irreducible representations (irreps) of conformal group. Seeking the irreducible representations of the group, we used the little group method and derived the com- plete set of transformation rules for Φ( x ). Next, we saw that the two and three point functions were determined upto some multiplicative constants by imposing confor- mal invariance alone, but this success stopped here and it was seen that for four(or more) points, the functions have an arbitrary dependence on what are known as anharmonic ratios or cross ratios (which are invariants unchanged by all conformal transformations). Next, we establish state operator correspondence by using radial quantization and show that any operator at a point away from the origin is a linear combination of the primary and its descendants which are together known as a conformal family. Then, we will show that in a CFT, radial quantization leads us to a useful tool known as the Operator Product Expansion (OPE). And that by using this tool, we can recursively reduce any n-point function to 2 point functions. And we show that they are completely determined by conformal invariance upto some constants known as OPE coefficients. Following that, we identify a consistency condition for filtering junk CFT data, this is known as crossing symmetry or OPE associativity . Using this condition we derived the very powerful conformal bootstrap equation . For unitary theories, we can have even more constraints on the spectrum, for which we derived the unitarity bounds before finishing with an algorithm which tells us how we can use the bootstrap equation to rule out inconsistent families of spectra

Holographic Computational Complexity as a measure for the Multiverse

The landscape of solutions in String Theory leads to the possibility of di�erent universes with its own physical laws and fundamental constants (like cosmological constant(cc), rank of gauge groups and so on). Also, in cosmology, the process of in ation becomes eternal once quantum uctuations are taken into account and gives rise to many universes. Thus, it is plausible that the various distinct string vacua landscape are populating the di�erent pocket universes of the in ationary multiverse. It is hard to de�ne probabilities in the multiverse. In a new approach for multiverse analysis [9] , the multiverse is taken as a simulation by a quantum supercomputer, and the time taken for simulation is the computational complexity. Universes which have less complexity are produced more easily and are common. A recent conjecture by Susskind [10], relates the computational complexity to the Einstein-Hilbert action. The aim of this project is to compute the complexity for a multiverse with two vacua , a pure dS spacetime separated by a domain wall from a Schwarshild AdS spacetime. Di�erent domain wall trajectories are possible. We compute the Einstein-Hilbert action term of this system for the case when the de Sitter radius is less than the black hole event horizon radius