We develop a general formalism of duality rotations for bosonic conformal spin-$s$ gauge fields, with $s\geq 2$, in a conformally flat four-dimensional spacetime. In the $s=1$ case this formalism is equivalent to the theory of $\mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik. For each integer spin $s\geq 2$ we demonstrate the existence of families of conformal $\mathsf{U}(1)$ duality-invariant models, including a generalisation of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then extended to the $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric cases. We also sketch a formalism of duality rotations for conformal gauge fields of Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.
We have examined quantum theories of electric magnetic duality invariant vector fields enjoying classical conformal invariance in 4-dimensional flat spacetime. We extend Dirac's argument about "the conditions for a quantum field theory to be relativistic" to "those for a quantum theory to be conformal". We realize that electric magnetic duality invariant vector theories together with classical conformal invariance defined in 4-$d$ flat spacetime are still conformally invariant theories when they are quantized in a way that electric magnetic duality is manifest.
We show how SU(2) $N_f=4$ Seiberg-Witten prepotentials are derived form $\hat{\mathfrak{sl}}_{2,k}$ ($k\to2$) four-point conformal blocks via considering Langlands duality.
We study integrable deformations of sine-Liouville conformal field theory. Every integrable perturbation of this model is related to the series of quantum integrals of motion (hierarchy). We construct the factorized scattering matrices for different integrable perturbed conformal field theories. The perturbation theory, Bethe ansatz technique, renormalization group and methods of perturbed conformal field theory are applied to show that all integrable deformations of sine-Liouville model possess non-trivial duality properties.
We present a conformally invariant generalized form of the free particle action by connecting the wave and particle aspects through gravity. Conformal invariance breaking is introduced by choosing a particular configurat$ of dynamical variables. This leads to the geometrization of the quantum aspects of matter.
An outline of the conformality approach to the gauge hierarchy is given including the use of non-abelian orbifolds to give unified models of the left-right type.
The theory described by the sum of the Einstein-Hilbert action and the action of conformal scalar field possesses the duality symmetry which includes some special conformal transformation of the metric, and also inversion of scalar field and of the gravitational constant. In the present paper the conformal duality is generalized for arbitrary space-time dimension $n \neq 2$ and for the general sigma-model type conformal scalar theory. We also consider to apply the conformal duality for the investigation of quantum gravity in the strong curvature regime. The trace of the first coefficient of th…
Non-supersymmetric d=4 gauge theories which arise from superstring duality on a manifold $AdS_5 \times S_5/Z_p$ are cataloged for a range $2 \leq p \leq 41$. A number have vanishing two-loop gauge β-function, a necessary but not sufficient condition to be a conformal field theory.
Using the conformal compensator superfields of N=2 D=4 supergravity, the Type IIB S-duality transformations are expressed as a linear rotation which mixes the compensator and matter superfields. The classical superspace action for D=4 compactifications of Type IIB supergravity is manifestly invariant under this transformation. Furthermore, the introduction of conformal compensators allows a Fradkin-Tseytlin term to be added to the manifestly SL(2,Z)-covariant sigma model action of Townsend and Cederwall.
We examine conformal rescaling and T-duality in the context of four-dimensional HKT geometries. The closure of the torsion forces the conformal factor to satisfy a modified harmonic equation. Because of this equation the conformal factors form non-commutative groups acting on the HKT geometries. Using conformal rescalings and T-duality transformations we generate from flat space new families of HKT geometries with tri-holomorphic Killing vectors. We also find ultraviolet-finite (4,0) supersymmetric sigma models which are not conformally invariant.
Target space duality transformations are considered for bosonic sigma models and strings away from RG fixed points. A set of consistency conditions are derived, and are seen to be nontrivially satisfied at one-loop order for arbitrary running metric, antisymmetric tensor and dilaton backgrounds. Such conditions are sufficiently stringent to enable an independent determination of the sigma model beta functions at this order.
We consider classical and quantum integrable sigma models and their relations with the solutions of renormalization group equations. We say that an integrable sigma model possesses the "nice" duality property if the dual quantum field theory has the weak coupling region. As an example, we consider the deformed $CP(n-1)$ sigma model with additional quantum degrees of freedom. We formulate the dual integrable field theory and use perturbed conformal field theory, perturbation theory, $S$-matrix, Bethe Ansatz and renormalization group methods to show that this field theory has the "nice" duality …
In this note, we explore the relation between crossing symmetry and modular invariance in conformal field theory and S-duality in gauge theory. It is shown that partition functions of different S dual theories of N=2 SU(2) gauge theory with four fundamentals can be derived from the crossing symmetry of the Liouville four point function. We also show that the partition function of N=4 SU(2) gauge theory can be derived from the Liouville partition function on torus.
We consider possible conformal field theory (CFT) descriptions of the various inertial ranges that exist in $2d$ duality invariant Magnetohydrodynamics. Such models arise as effective theories of dyonic plasmas in 3 dimensions in which all fields are independent of the third coordinate. We find new constraints on the allowed CFT's compared to those that may describe turbulence in $2d$ plasmas of electric charges only. The predictions from CFT concerning equipartition of energy amongst the electric and magnetic fields are discussed, and quantities exhibiting universal scaling are derived.
We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation, the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables. The constraints are solved in terms of two symmetric potentials that are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these fou…
We review the current status of the construction of unitary representations of U-duality groups of supergravity theories in five, four and three dimensions. We focus mainly on the maximal supergravity theories and on the N=2 Maxwell-Einstein supergravity (MESGT) theories defined by Jordan algebras of degree three in five dimensions and their descendants in four and three dimensions. Entropies of the extremal black hole solutions of these theories in five and four dimensions are given by certain invariants of their U-duality groups. The five dimensional U-duality groups admit extensions to spec…
In this article I first give an abbreviated history of string theory and then describe the recently-conjectured field-string duality. This suggests a class of nonsupersymmetric gauge theories which are conformal (CGT) to leading order of 1/N and some of which may be conformal for finite N. These models are very rigid since the gauge group representations of not only the chiral fermions but also the Higgs scalars are prescribed by the construction. If the standard model becomes conformal at TeV scales the GUT hierarchy is nullified, and model-building on this basis is an interesting direction. …
In recent years two intriguing observations have been made for N=4 super Yang-Mills theory and for superstrings on AdS5xS5: In the planar limit the computation of the spectrum is vastly simplified by the apparent integrability of the models. Furthermore, planar scattering amplitudes of the gauge theory display remarkable features which have been attributed to the appearance of a dual superconformal symmetry. Here we review the connection of these two developments from the point of view of the classical symmetry by means of a super-T-self-duality. In particular, we show explicitly how the charg…
We calculate the holographic conformal anomaly and brane Newton potential when bulk is 5d AdS BH. It is shown that such anomaly is the same as in the case of pure AdS or (asymptotically) dS bulk spaces, i.e. it is (bulk) metric independent one. While Newton potential on the static brane in AdS BH is different from the one in pure AdS space, the gravity trapping still occurs for two branes system. This indicates to metric independence of gravity localization.
It is observed that some structures recently uncovered in the study of Calogero-Sutherland models and anyons are close analogs of well-known structures of boundary conformal field theory. These examples of ``boundary conformal quantum mechanics'', in spite of their apparent simplicity, have a rather reach structure, including some sort of T-duality, and could provide useful frameworks for testing general properties of boundary conformal theories. Of particular interest are the duality properties of anyons and Calogero-Sutherland particles in presence of boundary-violations of conformal invaria…
We summarise the status of an intriguing new duality between planar maximally helicity violating scattering amplitudes and light-like Wilson loops in N=4 super Yang-Mills. In particular, we focus on the role played by (dual) conformal symmetry, which is made predictive by deriving anomalous conformal Ward identities for the Wilson loops. Assuming the duality, the conformal symmetry of the dual Wilson loops becomes an unexpected new symmetry of scattering amplitudes in N=4 SYM.
We use localization techniques to study the non-perturbative properties of an N=2 superconformal gauge theory with gauge group SU(3) and six fundamental flavours. The instanton corrections to the prepotential, the dual periods and the period matrix are calculated in a locus of special vacua possessing a Z_3 symmetry. In a semi-classical expansion, we show that these observables are constrained by S-duality via a modular anomaly equation which takes the form of a recursion relation. The solutions of the recursion relation are quasi-modular functions of Gamma_1(3), which is a subgroup of the S-d…
We propose leading order $α'$-corrections to the Poisson-Lie T-duality transformation rules of the metric, $B$-field, and dilaton. Based on Double Field Theory, whose corrections to this order are known, we argue that they map conformal field theories to conformal field theories. Remarkably, Born geometry plays a central role in the construction.
We review the novel quasiconformal realizations of exceptional U-duality groups whose "quantization" lead directly to their minimal unitary irreducible representations. The group $E_{8(8)}$ can be realized as a quasiconformal group in the 57 dimensional charge-entropy space of BPS black hole solutions of maximal N=8 supergravity in four dimensions and leaves invariant "lightlike separations" with respect to a quartic norm. Similarly $E_{7(7)}$ acts as a conformal group in the 27 dimensional charge space of BPS black hole solutions in five dimensional N=8 supergravity and leaves invariant "ligh…
We examine the boundary conditions associated with extended supersymmetric Maxwell theory in 5-dimensional anti-De Sitter space. Excitations on the boundary are identical to those of ordinary 4-dimensional conformal invariant super electrodynammics. Extrapolations of these excitations give rise to a 5-dimensional topological gauge theory of the singleton type. The possibility of a connection of this phenomenon to the world volume theory of 3-branes in IIB string theory is discussed.
We present a conformal isometry for static extremal black hole solutions in all four-dimensional Einstein-Maxwell-scalar theories with electromagnetic duality groups `of type $E_7$'. This includes, but is not limited to, all supergravity theories with $\mathcal{N}>2$ supersymmetry and all $\mathcal{N}=2$ supergravity theories with symmetric scalar manifolds. The conformal isometry is valid for arbitrary electromagnetic charge configurations and relies crucially on the notion of Freudenthal duality.
We study the positive energy unitary representations of 2N extended superconformal algebras OSp(8*|2N) in six dimensions. These representations can be formulated in a particle basis or a supercoherent state basis, which are labeled by the superspace coordinates in d=6. We show that the supercoherent states that form the bases of positive energy representations of OSp(8*|2N) can be identified with conformal superfields in six dimensions. The massless conformal superfields correspond precisely to the ultra short doubleton supermultiplets of OSp(8*|2N). The other positive energy unitary represent…
We demonstrate that the fusion algebra of conformal defects of a two-dimensional conformal field theory contains information about the internal symmetries of the theory and allows one to read off generalisations of Kramers-Wannier duality. We illustrate the general mechanism in the examples of the Ising model and the three-states Potts model.
We study the significance of T-duality in the context of the gravitational description of gauge theories. We found that T-duality relates the deferents points of the moduli of a given gauge theory always far from the conformal fixed point. Also the described gauge theories seems to flow naturally to the able conformal points, those that naturally saturate all the possible known examples of near horizon geometries. Supersymmetry properties and T-duality breaking of it are discuss.
We discuss non-conformal gauge theories from type IIB D3-branes embedded in orbifolded space-times. Such theories can be obtained by allowing some non-vanishing logarithmic twisted tadpoles. In certain cases with N=0,1 supersymmetry correlation functions in the planar limit are the same as in the parent N=2 supersymmetric theories. In particular, the effective action in such theories perturbatively is not renormalized beyond one loop in the planar limit. In the N=2 as well as such N=0,1 theories quantum corrections in the D3-brane gauge theories are encoded in the corresponding classical highe…
The subject of relativistic hydrodynamics is explored using the tools of gauge/gravity duality. A brief literature review of AdS/CFT and gauge/gravity duality is presented first. This is followed by a pedagogical introduction to the use of these methods in determining hydrodynamic dispersion relations, $w(q)$, of perturbations in a strongly coupled fluid. Shear and sound mode perturbations are examined in a special class of gravity duals: those where the matter supporting the metric is scalar in nature. Analytical solutions (to order $q^4$ and $q^3$ respectively) for the shear and sound mode d…
We construct a duality between several simple physical systems by showing that they are different aspects of the same quantum theory. Examples include the free relativistic massless particle and the hydrogen atom in any number of dimensions. The key is the gauging of the Sp(2) duality symmetry that treats position and momentum (x,p) as a doublet in phase space. As a consequence of the gauging, the Minkowski space-time vectors (x^μ, p^μ) get enlarged by one additional space-like and one additional time-like dimensions to (x^M,p^M). A manifest global symmetry SO(d,2) rotates (x^M,p^M) like d+2 d…
The duality symmetry of free electromagnetic field is analyzed within an algebraic approach. To this end, the conformal $c(1,3)$ algebra generators are expressed as operators quadratic in some abstract operators $κ^α$ and $π_β$ which satisfy Heisenberg algebra relations. It is then shown that the duality generator can also be expressed in this manner. Standard issues regarding duality are considered in such a framework. It is shown that duality generator also generates chiral transformations, and the conflict between duality and manifest Lorentz symmetry is analyzed from the viewpoint of symme…
A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P,Q]=-i\hbar between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of th…
AdS/CFT duality is a conjectured dual correspondence between the large $N$ limit of Conformal Field Theory (CFT) in $d$-dimensions and the supergravity (SUGRA) in $d+1$-dimensional Anti de Sitter (AdS) space. By using this conjecture, we can study various properties of large $N$ CFT by simple calculations in SUGRA. Recently much attention has been paid to the Renormalization Group (RG) flow viewed from the SUGRA side. Such RG flow in CFT is known to be characterized by the c-function which connects CFTs with different central charges. Therefore, we are interested in deriving this c-function fr…
The AGT relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that for the four-point conformal block the modular transform up to the non-perturbative contributions can be written in form of the ordinary Fourier transform when $β\equiv-ε_1/ε_2=1$. Here we extend this conjecture to general values of $ε_1,ε_2$. Namely, we argue that for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters $ε_1,ε_2$. The conjecture is based on explicit …
This Ph.D. thesis was submitted to the Catholic University of Leuven. We discuss two main topics: various formulations of duality-symmetric theories and D=5 supergravity (both construction and supersymmetric solutions). The thesis is structured as follows: Ch. 1 Introduction; Ch. 2 String theory toolkit; Ch. 3 Free duality-symmetric theories; Ch. 4 Interacting duality-symmetric theories; Ch. 5 Conformal supergravity in D=5: Weyl multiplets; Ch. 6 Conformal supergravity in D=5: matter and gauge-fixing; Ch. 7 Supersymmetric solutions in D=5 gauged supergravity.
We use gauge/gravity duality to study simultaneously the mass spectrum and the thermodynamics of a generic quasi-conformal gauge theory, specified by its beta function. The beta function of a quasi-conformal theory almost vanishes, and the coupling is almost constant between two widely separated energy scales. Depending on whether the gravity dual has a black hole or not, the mass spectrum is either a spectrum of quasinormal oscillations or a normal T=0 mass spectrum. The mass spectrum is quantitatively correlated with the thermal properties of the system. As the theory approaches conformality…
A self-duality group $\cal G$ in quantum field theory can have anomalies. In that case, the space of ordinary coupling constants $\cal M$ can be extended to include the space $\cal F$ of coefficients of counterterms in background fields. The extended space $\cal N$ forms a bundle over $\cal M$ with fiber $\cal F$, and the topology of the bundle is determined by the anomaly. For example, the ${\cal G}=SL(2,\mathbb{Z})$ duality of the 4d Maxwell theory has an anomaly, and the space ${\cal F}=S^1$ for the gravitational theta-angle is nontrivially fibered over ${\cal M}=\mathbb{H}/SL(2,\mathbb{Z})…
If the standard model is embedded in a conformal theory, what is the simplest possibility? We analyse all abelian orbifolds for discrete symmetry $Z_p$ with $p \leq 7$, and find that the simplest such theory is indeed $SU(3)^7$. Such a theory predicts the correct electroweak unification (sin$^2 θ\simeq 0.231$). A color coupling $α_C(M) \simeq 0.07$ suggests a conformal scale $M$ near to 10 TeV.
Compactification of Type IIB superstring on an $AdS_5 \times S^5/Γ$ background leads to SU(N) gauge field theories with prescribed matter representations. In the 't Hooft limit of large N such theories are conformally finite. For finite N and broken supersymmetry ($\cal N$ = 0) I derive the constraints to be two-loop conformal and examine the consequences for a wide choice of $Γ$ and its embedding $Γ\subset {\cal C}^3 (\supset S^5)$.
Recently, Witten showed that there is a natural action of the group SL(2,Z) on the space of 3 dimensional conformal field theories with U(1) global symmetry and a chosen coupling of the symmetry current to a background gauge field on a 3-fold N. He further argued that, for a class of conformal field theories, in the nearly Gaussian limit, this SL(2,Z) action may be viewed as a holographic image of the well-known SL(2,Z) Abelian duality of a pure U(1) gauge theory on AdS-like 4-folds M bounded by N, as dictated by the AdS/CFT correspondence. However, he showed that explicitly only for the gener…
Recent work on the action of T duality on Dirichlet-branes is generalized to the case in which the open string satisfies boundary conditions that are neither Neumann nor Dirichlet. This is achieved by implementing T duality as a canonical transformation of the $σ$-model path integral. A class of boundary interactions that violate conformal symmetry is found to be T-dual of a correspondingly non-conformal class of boundary conditions. The analogy with some problems in boundary-non-critical quantum mechanics of interest for condensed matter is pointed out.
V. Drinfeld proposed conjectures on geometric Langlands correspondence and its quantum deformation. We refine these conjectures and propose their relationship with algebraic conformal field theory.
We study the crossing symmetry of the conformal blocks of the conformal field theory based on the affine Lie superalgebra osp(1|2). Within the framework of a free field realization of the osp(1|2) current algebra, the fusion and braiding matrices of the model are determined. These results are related in a simple way to those corresponding to the su(2) algebra by means of a suitable identification of parameters. In order to obtain the link invariants corresponding to the osp(1|2) conformal field theory, we analyze the corresponding topological Chern-Simons theory. In a first approach we quantiz…
We reconsider the naturalness from the viewpoint of effective field theories, motivated by the alternative scenario that the standard model holds until a high-energy scale such as the Planck scale. We propose a calculation scheme of radiative corrections utilizing a hidden duality, in the expectation that the unnaturalness for scalar masses might be an artifact in the effective theory and it could be improved if features of an ultimate theory are taken in.
We study the action of non-Abelian T-duality in the context of N=1 geometries with well understood field theory duals. In the conformal case this gives rise to a new solution that contains an AdS_5 X S^2 piece. In the case of non-conformal geometries we obtain a new background in massive IIA supergravity that presents similar behaviour to the cascade of Seiberg dualities. Some physical observables are discussed.
This thesis is based on the publications arXiv:1106.4253, arXiv:1206.2920, arXiv:1210.2590. We construct warped "AdS_4 x S^2 x S^2 x Σ" type IIB supergravity solutions dual to the infrared fixed points of all 3d N=4 linear and circular quivers. We provide checks of the correspondence by evaluating the supergravity action in a large N limit and by matching it with the computation of the free energy obtained from matrix models. To complete previous work, we establish the holographic dictionary for the more general supergravity solutions corresponding to domain walls (non-compact Σ) and dual to 4…
We identify a means to explicitly construct primary operators of free conformal field theories (CFTs) in spacetime dimensions $d=2,~3$, and $4$. Working in momentum space with spinors, we find that the $N$-distinguishable-particle Hilbert space $\mathcal{H}_N$ exhibits a $U(N)$ action in $d=4$ ($O(N)$ in $d=2,3$) which dually describes the decomposition of $\mathcal{H}_N$ into irreducible representations of the conformal group. This $U(N)$ is a natural $N$-particle generalization of the single-particle $U(1)$ little group. The spectrum of primary operators is identified with the harmonics of $…
Some properties of the higher grading integrable generalizations of the conformal affine Toda systems are studied. The fields associated to the non-zero grade generators are Dirac spinors. The effective action is written in terms of the Wess-Zumino-Novikov-Witten (WZNW) action associated to an affine Lie algebra, and an off-critical theory is obtained as the result of the spontaneous breakdown of the conformal symmetry. Moreover, the off-critical theory presents a remarkable equivalence between the Noether and topological currents of the model. Related to the off-critical model we define a rea…
The Laboratory for Quantum Gravity & Strings, Department of Mathematics and Applied Mathematics, University of Cape Town（开普敦大学数学与应用数学系量子引力与弦实验室）；Instituto de Física Téorica, UNESP-Universidade Estadual Paulista（圣保罗州立大学理论物理研究所）
