Dual field theories of quantum computation

We consider ($3+1$)-dimensional $\mathcal{N}=2$ supersymmetric QED with two flavors of fundamental hypermultiplets. This theory supports $1/2$-Bogomol'nyi-Prasad-Sommerfield (BPS) domain walls and flux tubes (strings), as well as their $1/4$-BPS junctions. The effective ($2+1$)-dimensional theory on the domain wall is known to be a U(1) gauge theory. Previously, the wall-string junctions were shown to play the role of massive charges in this theory. However, the field theory of the junctions on the wall (for semi-infinite strings) appears to be inconsistent due to infrared problems. All these problems can be eliminated by compactifying one spatial dimension orthogonal to the wall and considering a wall-antiwall system ($W\overline{W}$) on a cylinder. We argue that for certain values of parameters this setup provides a controllable analog of bulk-brane duality in field theory. Dynamics of the 4D bulk are mapped onto 3D boundary theory: 3D $\mathcal{N}=2$ supersymmetric quantum electrodynamics (SQED) with two matter superfields and a weak-strong coupling constant relation in 4D and 3D, respectively. The cylinder radius is seen as a ``real mass'' in 3D $\mathcal{N}=2$ SQED. We work out (at weak coupling) the quantum version of the world-volume theory on the walls. Integrating out massive matter (strings in the bulk theory) one generates a Chern-Simons term on the wall world volume and an interaction between the wall and antiwall that scales as a power of distance. Vector and scalar (classically) massless excitations on the walls develop a mass gap at the quantum level; the long-range interactions disappear. The above duality implies that the wall and its antiwall partner (at strong coupling in the bulk theory) are stabilized at the opposite sides of the cylinder.

To gain insight in the quantum nature of the big bang, we study the dual field theory description of asymptotically anti-de Sitter solutions of supergravity that have cosmological singularities. The dual theories do not appear to have a stable ground state. One regularization of the theory causes the cosmological singularities in the bulk to turn into giant black holes with scalar hair. We interpret these hairy black holes in the dual field theory and use them to compute a finite temperature effective potential. In our study of the field theory evolution, we find no evidence for a "bounce" from a big crunch to a big bang. Instead, it appears that the big bang is a rare fluctuation from a generic equilibrium quantum gravity state.

We consider a five-dimensional constant curvature black hole, which is constructed by identifying some points along a Killing vector in a five-dimensional AdS space. The black hole has the topology M4×S1, its exterior is time-dependent and its boundary metric is of the form of a three-dimensional de Sitter space times a circle, which means that the dual conformal field theory resides on a dynamical spacetime. We calculate the quasilocal stress-energy tensor of the gravitational background and then the stress-energy tenor of the dual conformal field theory. It is found that the trace of the tensor does indeed vanish, as expected. Further we find that the constant curvature black hole spacetime is just the “bubble of nothing” resulting from Schwarzschild-AdS black holes when the mass parameter of the latter vanishes.

We construct analytical Lifshitz massive black brane solutions in massive Einstein-Maxwell-dilaton gravity theory. We also study the thermodynamics of these black brane solutions and obtain the thermodynamical stability conditions. On the dual nonrelativistic boundary field theory with Lifshitz symmetry, we analytically compute the DC transport coefficients, including the electric conductivity, thermoelectric conductivity, and thermal conductivity. The novel property of our model is that the massive term supports the Lifshitz black brane solutions with $z\ensuremath{\ne}1$ in such a way that the DC transport coefficients in the dual field theory are finite. We also find that the Wiedemann-Franz law in this dual boundary field theory is violated, which indicates that it may involve strong interactions.

We explore the description of bulk causal structure in a dual field theory. We observe that in the spacetime dual to a spacelike non-commutative field theory, the causal structure in the boundary directions is modified asymptotically. We propose that this modification is described in the dual theory by a modification of the micro-causal light cone. Previous studies of this micro-causal light cone for spacelike non-commutativite field theories agree with the expectations from the bulk spacetime. We describe the spacetime dual to field theories with lightlike non-commutativity, and show that they generically have a drastic modification of the light cone in the bulk: the spacetime is non-distinguishing. This means that the spacetime while being devoid of closed timelike or null curves, has causal curves that are ``almost closed''. We go on to show that the micro-causal light cone in the field theory agrees with this prediction from the bulk.

We study a bottom-up, holographic description of a field theory yielding the spontaneous breaking of an approximate $SO(5)$ global symmetry to its $SO(4)$ subgroup. The weakly coupled, six-dimensional gravity dual has regular geometry. One of the dimensions is compactified on a circle that shrinks smoothly to zero size at a finite value of the holographic direction, hence introducing a physical scale in a way that mimics the effect of confinement in the dual four-dimensional field theory. We study the spectrum of small fluctuations of the bulk fields carrying $SO(5)$ quantum numbers, which can be interpreted as spin-0 and spin-1 bound states in the dual field theory. This work supplements an earlier publication focused only on the $SO(5)$ singlet states. We explore the parameter space of the theory, paying particular attention to composite states that have the right quantum numbers to be identified as pseudo-Nambu-Goldstone bosons (PNGBs). We find that in this model the PNGBs are generally heavy, with masses of the same order as other bound states, indicating the presence of a sizeable amount of explicit symmetry breaking in the field-theory side. Nevertheless, we also find a qualitatively new, unexpected result. When the dimension of the field-theory operator inducing $SO(5)$ breaking is close to half the space-time dimensionality, there exists a region of parameter space in which the PNGBs and the lightest scalar are both parametrically light in comparison to all other bound states of the field theory. Although this region is known to yield metastable classical backgrounds, this finding might be relevant to model building in the composite Higgs context.

In this thesis we investigate on the role that dualities play nowadays in our under-standing of string and Quantum Field Theories.The first part is devoted to holographic dualities, where the case of the string dual of certain one-, two- and four dimensional field theories is explored in detail. We begin in Chapter 1 by discussing the holographic dual of a broad class of four-dimensional field theories. We specialise to the case of quiver field theories where a number of formulas computing charges, number of branes and Linking Numbers are given. We then introduce marginal deformations in the supergravity backgrounds to uncover infinite new families of solutions to type II supergravity and M-theory. In Chapter 2, we discuss spin 2 fluctuations around a class of warped AdS3 backgrounds. We identify explicitly the dual operators of a given protected sector. Also, a formula for the central charge of the dual two-dimensional field theories is derived. In Chapter 3, we introduce two new classes of geometries with an AdS2 factor. We outline the importance to black hole physics and in one case give a prescription for the dual field theories.The second part is devoted to QFT/QFT dualities in Chern-Simons theories, and how dualities between Chern-Simons theories in three dimensions can be motivated in string theory. It is found that a known duality between three-dimensional Chern-Simons theories with unitary gauge symmetry can be motivated by a brane setup in Type 0B string theory with an orientifold. In particular, the phase diagram of unitary QCD3 is inferred from a dual theory.

We construct a series of new hyperbolic black hole solutions in Einstein–Scalar system and we apply holographic approach to investigate the spherical Rényi entropy in various deformations of dual conformal field theories (CFTs). Especially, we introduce various powers of scalars in the scalar potentials for massive and massless scalar. These scalar potentials correspond to deformation of dual CFTs. Then we solve asymptotically hyperbolic AdS black hole solutions numerically. We map the instabilities of these black hole solutions to phase transitions of field theory in terms of CHM mapping between hyperbolic hairy AdS black hole and spherical Rényi entropy in dual field theories. Based on these solutions, we study the temperature dependent condensation of dual operator of massive and massless scalar respectively. These condensations show that there might exist phase transitions in dual deformed CFTs. We also compare free energy between asymptotically hyperbolic AdS black hole solutions and hyperbolic AdS Schwarz (AdS-SW) black hole to test phase transitions. In order to confirm the existence of phase transitions, we turn on linear in-homogeneous perturbation to test stability of these hyperbolic hairy AdS black holes. In this paper, we show how potential parameters affect the stability of hyperbolic black holes in several specific examples. For general values of potential parameters, it needs further study to see how the transition happens. Finally, we comment on these instabilities associated with spherical Rényi entropy in dual deformed CFTs.

This paper studies a special class of states for the dual conformal field\ntheories associated with supersymmetric $AdS_5\\times X$ compactifications,\nwhere $X$ is a Sasaki-Einstein manifold with additional $U(1)$ symmetries.\nUnder appropriate circumstances, it is found that elements of the chiral ring\nthat maximize the additional $U(1)$ charge at fixed R-charge are in one to one\ncorrespondence with multitraces of a single composite field. This is also\nequivalent to Schur functions of the composite field. It is argued that in the\nformal zero coupling limit that these dual field theories have, the different\nSchur functions are orthogonal. Together with large $N$ counting arguments, one\npredicts that various extremal three point functions are identical to those of\n${\\cal N}=4 $ SYM, except for a single normalization factor, which can be\nargued to be related to the R-charge of the composite word. The leading and\nsubleading terms in $1/N$ are consistent with a system of free fermions for a\ngeneralized oscillator algebra. One can further test this conjecture by\nconstructing coherent states for the generalized oscillator algebra that can be\ninterpreted as branes exploring a subset of the moduli space of the field\ntheory and use these to compute the effective K\\"ahler potential on this subset\nof the moduli space.\n

We provide a first derivation of the Bekenstein-Hawking entropy of 3D flat cosmological horizons in terms of the counting of states in a dual field theory. These horizons appear in the flat limit of nonextremal rotating Banados-Teitleboim-Zanelli black holes and are remnants of the inner horizons. They also satisfy the first law of thermodynamics. We study flat holography as a limit of AdS(3)/CFT(2) to semiclassically compute the density of states in the dual theory, which is given by a contraction of a 2D conformal field theory, exactly reproducing the bulk entropy in the limit of large charges. We comment on how the dual theory reproduces the bulk first law and how cosmological bulk excitations are matched with boundary quantum numbers.

Using the counterterm subtraction technique we calculatehe stress-energy tensor, action, and other physical quantities for Kerr-AdS black holes in various dimensions. For Kerr-AdS_5 with both rotation parameters non-zero, we demonstrate that stress-energy tensor, in the zero mass parameter limit, is equal to the stress tensor of the weakly coupled four dimensional dual field theory. As a result, the total energy of the generalKerr-AdS_5 black hole at zero mass parameter, exactly matches the Casimir energy of the dual field theory. We show that at high temperature, the general Kerr-AdS_5 and perturbative field theory stress-energy tensors are equal, up to the usual factor of 3/4. We also use the counterterm technique to calculate the stress tensors and actions for Kerr-AdS_6, and Kerr-AdS_7 black holes, with one rotation parameter, and we display the results. We discuss the conformal anomalies of the field theories dual to the Kerr-AdS_5 and Kerr-AdS_7 spacetimes. In these two field theories, we show that the rotation parameters break conformal invariance but not scale invariance, a novel result for a non-trivial field theory. For Kerr-AdS_7 the conformal anomalies calculated on the gravity side and the dual (0,2) tensor multiplet theory are equal up to 4/7 factor. We expect that the Casimir energy of the free field theory is the same as the energy of the Kerr-AdS_7 black hole (with zero mass parameter), up to that factor.

Describing quantum field theory using the AdS/CFT duality can broaden string theory applications in the context of the holographic principle. This research study supersymmetric solutions of matter-coupled SO(4) and maximal gauged supergravities in seven dimensions of which the dual field theories are sixdimensional superconformal field theories (SCFTs). We find a large class of domain wall (DW) solutions in the maximal theory with various gauge groups. For SO(5) gauge group admitting an AdS7 vacuum, the solutions describe holographic renormalization group (RG) flows from an N = (2, 0) SCFT to non-conformal field theories (SQFTs) in six dimensions. For other gauge groups without AdS7 vacua, these DWs are supersymmetric vacua dual to six-dimensional N = (2, 0) SQFTs. By coupling DWs to three-form fields, we find charged DW solutions in both theories. The solutions with AdS3 ? S 3 slices are interpreted as conformal surface defects within the dual field theories in six dimensions. We also find twisted solutions describing holographic RG flows across dimensions from six-dimensional field theories to SCFTs in lower dimensions. We consider solutions of the mattercoupled theory in the presence of a three-form field and extend twisted solutions of the maximal theory to include singularities in the form of curved DWs. In many cases, these singularities are physically acceptable and can be interpreted as SQFTs in lower dimensions. Many solutions can be uplifted to ten and eleven dimensions resulting in new classes of solutions in string and M-theories.

According to two-time physics, there is more to space and time than can be garnered with the ordinary formulation of physics. Two-time physics has shown that the standard model of particles and forces is successfully reproduced by a two-time field theory in four space and two time dimensions projected as a holographic image on an emergent space-time in $3+1$ dimensions. Among the successes of this approach is the resolution of the strong $CP$ problem of QCD as an outcome of the restrictions imposed by the higher symmetry structures in $4+2$ dimensions. In this paper we launch a program to construct the duals of the standard model as other holographic images of the same $4+2$ dimensional theory on a variety of emergent spacetimes in $3+1$ dimensions. These dual field theories are obtained as a family of gauge choices in the master $4+2$ field theory. In the present paper we deal with some of the simpler gauge choices which lead to interacting Klein-Gordon field theories for the conformal scalar with a predicted $\mathrm{SO}(d,2)$ symmetry in a variety of interesting curved spacetimes in $(d\ensuremath{-}1)+1$ dimensions. More challenging and more interesting gauge choices (including some that relate to mass) which are left to future work are also outlined. Through this approach we discover a new realm of previously unexplored dualities and hidden symmetries that exist both in the macroscopic and microscopic worlds, at the classical and quantum levels. Such phenomena predicted by Two time physics (2T-physics) can in principle be confirmed both by theory and experiment. One time physics (1T-physics) can be used to analyze the predictions but in most instances gives no clue that the predicted phenomena exist in the first place. This point of view suggests a new paradigm for the construction of a fundamental theory that is likely to impact on the quest for unification.

We study charged, static, topological black holes in pure Gauss-Bonnet gravity in asymptotically AdS space. As in general relativity, the theory possesses a unique nondegenerate AdS vacuum. It also admits charged black hole solutions which asymptotically behave as the Reissner-Nordstr\"{o}m AdS black hole. We discuss black hole thermodynamics of these black holes. Then we study phase transitions in a dual quantum field theory in four dimensions, with the St\"{u}ckelberg scalar field as an order parameter. We find in the probe limit that the black hole can develop hair below some critical temperature, which suggests a phase transition. Depending on the scalar coupling constants, the phase transition can be first or second order. Analysis of the free energy reveals that, comparing the two solutions, the hairy state is energetically favorable, thus a phase transition will occur in a dual field theory.

We perform a detailed analysis of holographic entanglement Rényi entropy in some modified theories of gravity with four dimensional conformal field theory duals. First, we construct perturbative black hole solutions in a recently proposed model of Einsteinian cubic gravity in five dimensions, and compute the Rényi entropy as well as the scaling dimension of the twist operators in the dual field theory. Consistency of these results are verified from the AdS/CFT correspondence, via a corresponding computation of the Weyl anomaly on the gravity side. Similar analyses are then carried out for three other examples of modified gravity in five dimensions that include a chemical potential, namely Born-Infeld gravity, charged quasi-topological gravity and a class of Weyl corrected gravity theories with a gauge field, with the last example being treated perturbatively. Some interesting bounds in the dual conformal field theory parameters in quasi-topological gravity are pointed out. We also provide arguments on the validity of our perturbative analysis, whenever applicable.