We prove that the supersymmetric deformed $ \mathbb{CP}^{1} $ sigma model (the generalization of the Fateev-Onofri-Zamolodchikov model) admits an equivalent description as a generalized Gross-Neveu model. This formalism is useful for the study of renormalization properties and particularly for calculation of the one- and two-loop $ β$-function. We show that in the UV the superdeformed model flows to the super-Thirring CFT, for which we also develop a superspace approach. It is then demonstrated that the super-Thirring model is equivalent to a sigma model with the cylinder $ \mathbb{R} \times S^{1} $ target space by an explicit computation of the correlation functions on both sides. Apart from that, we observe that the original model has another interesting conformal limit, given by the supercigar model, which as well could be described in the Gross-Neveu approach.
N=2 supersymmetric extension of the l-conformal Galilei algebra is constructed. A relation between its representations in flat spacetime and in Newton-Hooke spacetime is discussed. An infinite-dimensional generalization of the superalgebra is given.
Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $AdS_{3}/CFT_{2}$ correspondence, they are related to classical bulk actions and are used to calculate entanglement entropy and geodesic lengths. In this work, we discuss the identification of classical conformal blocks and the Painlevé VI action showing how isomonodromic deformations naturally appear in this context. We recover the accessory parameter expansion of Heun's equation from the isomonodromic $τ$-function. We also discuss how the $c = 1$ expansion of the $τ$-function…
We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of…
Irreducible second-order Darboux transformations are applied to the periodic Schrodinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials with periodicity defects and bound states embedded into the spectral gaps. The method is applied to the Lame and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.
This paper is concerned with a link between central extensions of N=2 superconformal algebra and a supersymmetric two-component generalization of the Camassa--Holm equation. Deformations of superconformal algebra give rise to two compatible bracket structures. One of the bracket structures is derived from the central extension and admits a momentum operator which agrees with the Sobolev norm of a coadjoint orbit element. The momentum operator induces via Lenard relations a chain of conserved hamiltonians of the resulting supersymmetric Camassa-Holm hierarchy.
The N-extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even N one can identify the 1D N-extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main differences…
We consider bosonic supersymmetric backgrounds of ten-dimensional conformal supergravity. Up to local conformal isometry, we classify the maximally supersymmetric backgrounds, determine their conformal symmetry superalgebras and show how they arise as near-horizon geometries of certain half-BPS backgrounds or as a plane-wave limit thereof. We then show how to define Yang-Mills theory with rigid supersymmetry on any supersymmetric conformal supergravity background and, in particular, on the maximally supersymmetric backgrounds. We conclude by commenting on a striking resemblance between the sup…
We study observables and deformations of generalized Chern-Simons action and show how to apply these results to maximally supersymmetric gauge theories. We describe a construction of large class of deformations based on some results on the cohomology of super Lie algebras proved in the Appendix.
We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits the divergence in the central charge term. The other leads the continuous spectrum and contains no closed time-like curve in the system.
In four spacetime dimensions, all ${\cal N} =1$ supergravity-matter systems can be formulated in the so-called $\mathsf{U}(1)$ superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background $\mathsf{U}(1)$ superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields $\ell_{(α_1 \dots α_m) ({\dot α}_1 \dots {\dot α}_n)}$, with $m$ and $n$ non-negative integers, $m+n>0$, and elaborate on their significance in the following cases: (i) $m=n=1$; (…
We make a preliminary algebraic study of supersymmetric deformations of N=1 Yang-Mills theory in dimension ten with an arbitrary gauge group. This is done in a context of Lie algebra deformation theory. The tangent space to the space of deformation is computed.
We present the minimal realization of the $\ell$-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex Pais-Uhlenbeck oscillator equations. We introduce a minimal deformation of the $\ell$=1/2 conformal Galilei (a.k.a. Schrödinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d=1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric.
We study supersymmetric and super Poincaré invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given …
Using the superconformal (SC) indices techniques, we construct Seiberg type dualities for $\mathcal{N}=1$ supersymmetric field theories outside the conformal windows. These theories are physically distinguished by the presence of chiral superfields with small or negative $R$-charges.
A general two-dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Applying the generators of the closed subalgebra generated by $(L_{-1},L_{0},G_{-1/(F+1)})$ and $(\bar{L}_{-1},\bar{L}_{0},\bar{G}_{-1/(F+1)})$, the two point functions of the component-fields of supermultiplets are calculated. Then the logarithmic superconformal field theories are investigated and the chiral and full two-point functions are obtained.
We investigate a possibility of scale invariant but non-conformal supersymmetric field theories from a perturbative approach. The explicit existence of monotonically decreasing a-function that generates beta-functions as a gradient flow provides a strong obstruction for such a possibility at two-loop order. We comment on the "discovery" of scale invariant but non-conformal renormalization group trajectories via a "change of scheme" in (4-ε) dimension proposed in literatures.
In this thesis exactly marginal deformations of field theories living on D3-branes at low energies are studied. These theories include N=4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure.
We provide the first holographic evidence for the existence of a non-supersymmetric conformal manifold arising from exactly marginal but supersymmetry-breaking deformations of a superconformal three-dimensional field theory. In particular, we construct a 2-parameter non-supersymmetric deformation of a supersymmetric AdS non-geometric background in Type IIB string theory. We prove that the non-supersymmetric backgrounds are perturbatively stable and also do not suffer from various non-perturbative instabilities. Finally, we argue that diffeomorphism symmetry protects our solutions against highe…
In this work, we study the supersymmetric warped conformal field theory in two dimensions. We show that the Hofman-Strominger theorem on symmetry enhancement could be generalized to the supersymmetric case. More precisely, we find that within a chiral superspace $(x^+,þ)$, a two-dimensional field theory with two translational invariance and a chiral scaling symmetry can have enhanced local symmetry, under the assumption that the dilation spectrum is discrete and non-negative. Similar to the pure bosonic case, there are two kinds of minimal models, one being $N=(1,0)$ supersymmetric conformal f…
Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1-dimension, we discuss a Mobius quantization approach to the two-dimensional conformal field theory (CFT), which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Mobius quantization approach.
In this paper a new supersymmetric extension of conformal mechanics is put forward. The beauty of this extension is that all variables have a clear geometrical meaning and the super-Hamiltonian turns out to be the Lie-derivative of the Hamiltonian flow of standard conformal mechanics. In this paper we also provide a supersymmetric extension of the other conformal generators of the theory and find their "square-roots". The whole superalgebra of these charges is then analyzed in details. We conclude the paper by showing that, using superfields, a constraint can be built which provides the exact …
We explore the $J\bar{T}$ and $T\bar{J}$ deformations of two-dimensional field theories possessing $\mathcal N=(0,1),(1,1)$ and $(0,2)$ supersymmetry. Based on the stress-tensor and flavor current multiplets, we construct various bilinear supersymmetric primary operators that induce the $J\bar{T}/T\bar{J}$ deformation in a manifestly supersymmetric way. Moreover, their supersymmetric descendants are shown to agree with the conventional $J\bar T /T\bar J$ operator on-shell. We also present some examples of $J\bar T /T\bar J$ flows arising from the supersymmetric deformation of free theories. Fi…
We study the minimal unitary representation (minrep) of SO(6,2) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO(6,2), which coincides with the minrep of SO*(8) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of "deformations" of the minrep of SO*(8) labeled by the spin t of an SU(2)_T subgroup of the little group SO(4) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO*(8) that d…
We study deformations of the quantum conformal mechanics of De Alfaro-Fubini-Furlan with rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and non-isospectral deformations of the conformal mechanics model with special values of the coupling constant $g=m(m+1)$, $m\in {\mathbb N}$, in the inverse square potential term, and for each completely isospectral or gapped deformation given b…
Three-dimensional N = 4 supersymmetric quantum field theories admit two topological twists, the Rozansky-Witten twist and its mirror. Either twist can be used to define a supersymmetric compactification on a Riemann surface and a corre- sponding space of supersymmetric ground states. These spaces of ground states can play an interesting role in the Geometric Langlands program. We propose a description of these spaces as conformal blocks for certain non-unitary Vertex Operator Algebras and test our conjecture in some important examples. The two VOAs can be constructed respectively from a UV Lag…
We describe new non-supersymmetric conformal field theories in three and four dimensions, using the CFT/AdS correspondence. In order to believe in their existence at large N_c and strong 't Hooft coupling, we explicitly check the stability of the corresponding non-supersymmetric anti-de Sitter backgrounds. Cases of particular interest are the relevant deformations of the N=4 SCFT in SU(3) and SO(5) invariant directions. It turns out that the former is a stable, and the latter an unstable non-supersymmetric type IIB background.
A general two dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Then, applying the generators of the closed subalgebra generated by $(L_{-1}, L_{0}, G_{-1/3})$ and $(\bar{L}_{-1}, \bar{L}_{0}, \bar{G}_{-1/3})$, the two point functions of the component fields of supermultiplets are calculated.
The quasiconformal method provides us with a unified approach to the construction of minimal unitary representations (minrep) of noncompact groups, their deformations as well as their supersymmetric extensions. We review the quasiconformal construction of the minrep of SO(d,2), its deformations and their applications to unitary realizations of AdS_{(d+1)}/CFT_d higher spin algebras and their deformations for arbitrary d and supersymmetric extensions for dimensions d less than seven. AdS_{(d+1)}/CFT_d higher spin algebras, their deformations and supersymmetric extensions are given by the envelo…
Logarithmic spin-1/3 superconformal field theories are investigated. the chiral and full two-point functions of two-(or more-) dimensional Jordanian blocks of arbitrary weights, are obtained.
Minimal unitary representation of SO*(8) = SO(6,2) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU(2) subgroup correspond to massless conformal fields in six dimensions as was shown in arXiv:1005.3580. In this paper we study the minimal unitary supermultiplet of OSp(8*|2N) with the even subgroup SO*(8) x USp(2N) and its deformations using quasiconformal methods. We show that the minimal unitary supermultiplet of OSp(8*|2N) admits deformations labeled uniquely by the spin t of an SU(2) subgroup of the little group SO(4) of light…
We construct supersymmetric irregular vertex operators of arbitrary rank, appearing in the colliding limit of primary fields. We find that the structure of the supersymmetric irregular vertices differs significantly from the bosonic case: upon supersymmetrization, the irregular operators are no longer the eigenstates of positive Virasoro and $W_N$ generators but block-diagonalize them. We relate the block-diagonal structure of the irregular vertices to contributions of the Ramond sector to the colliding limit.
We introduce a computational technique for studying non-supersymmetric deformations of domain wall solutions of interest in AdS/CFT. We focus on the Klebanov-Strassler solution, which is dual to a confining gauge theory. From an analysis of asymptotics we find that there are three deformations that leave the ten-dimensional supergravity solution regular and preserve the global bosonic symmetries of the supersymmetric solution. Also, we show that there are no regular near-extremal deformations preserving the global symmetries, as one might expect from the existence of a gap in the gauge theory.
We study the SO(3)-invariant relevant deformations of N=4 SU(N) gauge theory using the methods of Polchinski and Strassler. We present the region of parameter space where the non-supersymmetric vacuum is still described by stable ``dielectric'' five branes within the supergravity approximation.
We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg-de Vries equation. Moreover, we show that these deformation…
We study certain supersymmetry breaking deformations of linear dilaton backgrounds in different dimensions. In some cases, the deformed theory has bulk closed strings tachyons. In other cases there are no bulk tachyons, but there are localized tachyons. The real time condensation of these localized tachyons is described by an exactly solvable worldsheet CFT. We also find some stable, non-supersymmetric backgrounds.
We discuss a regular non-supersymmetric deformation of the Klebanov-Strassler solution related to gaugino mass terms in the dual gauge theory. We also describe the Penrose limit of the new background and the corresponding plane wave string theory.
We define a manifestly supersymmetric version of the $T \overline{T}$ deformation appropriate for a class of $(0+1)$-dimensional theories with $\mathcal{N} = 1$ or $\mathcal{N} = 2$ supersymmetry, including one presentation of the super-Schwarzian theory which is dual to JT supergravity. These deformations are written in terms of Noether currents associated with translations in superspace, so we refer to them collectively as $f(\mathcal{Q})$ deformations. We provide evidence that the $f(\mathcal{Q})$ deformations of $\mathcal{N} = 1$ and $\mathcal{N} = 2$ theories are on-shell equivalent to th…
We discuss non-conformal non-supersymmetric large N gauge theories with vanishing vacuum energy density to all orders in perturbation theory. These gauge theories can be obtained via a field theory limit of Type IIB D3-branes embedded in orbifolded space-times. We also discuss gravity in this setup.
The leading classical asymptotics of Virasoro conformal blocks on the Riemann sphere with n generic and n-3 "heavy" degenerate field insertions can be described in terms of the geometry of Garnier system describing the monodromy preserving deformations of second order Fuchsian differential equations on an n-punctured sphere. This allows us to characterise the leading classical asymptotics of Virasoro conformal blocks completely, and to clarify in which sense conformal field theory represents a quantisation of the isomonodromic deformation problem.
We study N=2 supersymmetric U(1) gauge theory in non(anti)commutative N=2 harmonic superspace with the singlet deformation, which preserves chirality. We construct a Lagrangian which is invariant under both the deformed gauge and supersymmetry transformations. We find the field redefinition such that the N=2 vector multilplet transforms canonically under the deformed symmetries.
We show how to give the expression for periods, Higgs field and its dual of N=2 supersymmetric Yang-Mills theory around the conformal point. This is achieved by evaluating the integral representation in the weak coupling region, and by using analytic continuation to the conformal point. The explicit representation is shown for the SU(2) theory with matter fields and also for pure SU(N) and pure SO(2N) theory around the conformal point where the relation to the beta function of the theory is clarified. We also discuss a relation between the fixed points in the SU(2) theories with matter fields …
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 deform the AdS/CFT Correspondence by the inclusion of a non-supersymmetric scalar mass operator. We discuss the behaviour of the dual 5 dimensional supergravity field then lift the full solution to 10 dimensions. Brane probing the resulting background reveals a potential consistent with the operator we wished to insert.
We set up the bootstrap procedure for supersymmetric Galilean Conformal (SGC) field theories in two dimensions by constructing the SGC blocks in the $\mathcal{N}=1$ and two possible $\mathcal{N} =2$ extensions of the Galilean conformal algebra. In all analyzed cases, we present the bootstrap equations by crossing symmetry of the four point function. In addition, we compute the global SGC blocks analytically by solving the differential equations obtained by acting with the Casimirs of the global subalgebras inside the four point function. These global blocks agree with the general SGC blocks in…
Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by $i$ and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a $\mathcal{PT}$-invariant supersymmetric model with $N$ subsystems by using the conformal generators of su…
We consider instanton effects in a non-supersymmetric gauge theory obtained by marginal deformations of the N=4 SYM. This gauge theory is expected to be dual to type IIB string theory on the AdS_5 times deformed-S^5 background. From an instanton calculation in the deformed gauge theory we extract the prediction for the dilaton-axion field τin dual string theory. In the limit of small deformations where the supergravity regime is valid, our instanton result reproduces the expression for τof the supergravity solution found by Frolov.
We study supersymmetry breaking deformations of the $\mathcal{N}=1$ 5d fixed point known as $E_1$, the UV completion of $SU(2)$ super-Yang-Mills. The phases of the non-supersymmetric theory can be characterized by Chern-Simons terms involving background $U(1)$ gauge fields, allowing us to identify a phase transition at strong coupling. We propose that this may signify the emergence of a non-trivial, non-supersymmetric CFT in $d=4+1$ dimensions.
We examine the marginal deformations of double-trace type in 3d supersymmetric U(N) model with N complex free bosons and fermions. We compute the anomalous dimensions of higher spin currents to the 1/N order but to all orders in the deformation parameters by mainly applying the conformal perturbation theory. The 3d field theory is supposed to be dual to 4d supersymmetric Vasiliev theory, and the marginal deformations are argued to correspond to modifying boundary conditions for bulk scalars and fermions. Thus the modification should break higher spin gauge symmetry and generate the masses of h…
Very special $T\bar{J}$ deformations of a conformal field theory are irrelevant deformations that break the Lorentz symmetry but preserve the twisted Lorentz symmetry. We construct a holographic description of very special $T\bar{J}$ deformations. We give a holographic recipe to study the double trace as well as single trace deformations. The former is obtained from the change of the boundary condition while the latter is obtained from the change of the supergravity background.
We discuss the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e. to be part of a conformal manifold. In particular, using tools from conformal perturbation theory, we derive a sum rule from which one can extract restrictions on the spectrum of low spin operators and on the behavior of OPE coefficients involving nearly marginal operators. We then consider conformal field theories admitting a gravity dual description, and as such a large-$N$ expansion. We discuss the relation between conformal perturbation theory and loop expansion in the bulk, a…
We describe the classical $o(3,2)$ $r$-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 $AdS$ algebra with dimensionless deformation parameters. We describe the quantization of classical $o(3,2)$ $r$-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum $o(3,2)$ algebra in a convenient Hopf algebra form by considering suitable deformation maps from classical to deformed $o(3,2)$ algebra basis. It appears that if we pass from $κ$-deformed D =3 conformal a…
A supersymmetric Lorentz invariant mechanism for superspace deformations is proposed. It is based on an extension of superspace by one $λ_{a}$ or several Majorana spinors associated with the Penrose twistor picture. Some examples of Lorentz invariant supersymmetric Poisson and Mojal brackets are constructed and the correspondence: $θ_{mn}\leftrightarrow iψ_{m}ψ_{n},\quad C_{ab}\leftrightarrow λ_{a}λ_{b},\quad Ψ^{a}_{m}\leftrightarrow ψ_{m}λ^{a}$ mapping the brackets depending on the constant background into the Lorentz covariant supersymmetric brackets is established. The correspondence reveal…
In this paper, we will analyse a supersymmetric field theory deformed by generalized uncertainty principle and Lifshitz scaling. It will be observed that this deformed supersymmetric field theory contains non-local fractional derivative terms. In order to construct such deformed N=1 supersymmetric theory, a harmonic extension of functions will be used. However, the supersymmetry will be only preserved for a free theory and will be broken by the inclusion of interaction terms.
We investigate some aspects of Moyal-Weyl deformations of superspace and their compatibility with supersymmetry. For the simplest case, when only bosonic coordinates are deformed, we consider a four dimensional supersymmetric field theory which is the deformation of the Wess-Zumino renormalizable theory of a chiral superfield. We then consider the deformation of a free theory of an abelian vector multiplet, which is a non commutative version of the rank one Yang-Mills theory. We finally give the supersymmetric version of the $α'\mapsto 0$ limit of the Born-Infeld action with a B-field turned o…
In this paper we study exactly marginal deformations of field theories living on D3-branes at low energies. These theories include N=4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure. We restrict ourselves only to orbifolds and deformations which leave some supersymmetry unbroken. A number of new families of N=1 superconformal field theories are found. We analyze the deformations perturbatively, and also by using general arguments for the dimension of the space of exactly marginal deformations. We find some cases where the space of perturbative exac…
