We compute the leading finite-size effects on the normalized structure constants in semiclassical three-point correlation functions of two finite-size giant magnon string states and three different types of "light" states - primary scalar operators, dilaton operator with nonzero momentum and singlet scalar operators on higher string levels. This is done for the case of TsT-transformed, or gamma-deformed, AdS_5 x S^5 string theory background, dual to N = 1 super Yang-Mills theory in four dimensions, arising as an exactly marginal deformation of N = 4 super Yang-Mills.
We consider pure spinor strings that propagate in the background generated by a sequence of TsT transformations. We use the fact that U(1) isometry variables of TsT-transformed background are related to the isometry variables of the initial background in the universal way that is independent of the details of the background. We will argue that after redefinitions of pure spinors and the fermionic variables we can construct pure spinor action with manifest U(1) isometry. This fact implies that the pure spinor string in TsT-transformed background is described by pure spinor string in the origina…
We consider giant magnons propagating in a γ-deformed AdS_5 x S^5 background obtained from AdS_5 x S^5 by means of a chain of TsT transformations. We point out that in the light-cone gauge and in the infinite J limit the deformed and undeformed string models share the same magnon dispersion relation, the \su(2|2)\oplus \su(2|2)-invariant world-sheet S-matrix and the dressing factor. The γ-dependence in the limit is only due to different level-matching conditions. We consider the reduction of the deformed model to R x S^3 and determine the leading γ-dependence of the dispersion relation for a f…
The so-called homogeneous Yang-Baxter (YB) deformations can be considered a non-abelian generalization of T-duality--shift--T-duality (TsT) transformations. TsT transformations are known to preserve conformal symmetry to all orders in $α'$. Here we argue that (unimodular) YB deformations of a bosonic string also preserve conformal symmetry, at least to two-loop order. We do this by showing that, starting from a background with no NSNS-flux, the deformed background solves the $α'$-corrected supergravity equations to second order in the deformation parameter. At the same time we determine the re…
We investigate dyonic giant magnons propagating on $γ$-deformed $AdS_5\times S^5$ by Neumann-Rosochatius reduction method with a twisted boundary condition. We compute finite-size effect of the dispersion relations of dyonic giant magnons which generalizes the previously known case of the giant magnons with one angular momentum found by Bykov and Frolov.
We present general formulae for the TsT transformation (T-duality, shift, T-duality) of type II string backgrounds and open string boundary conditions. The TsT transformation provides a systematic procedure to find string theory duals of gauge theories with deformed products of fields in the lagrangian, and the duals can be analyzed by using transformed D-brane probes. As examples illustrating some features of the deformed theories, we consider the known backgrounds dual to non-commutative, dipole and beta-deformed N=4 Super Yang-Mills as well as new backgrounds dual to deformations of the rec…
We consider classical strings propagating in a background generated by a sequence of TsT transformations. We describe a general procedure to derive the Green-Schwarz action for strings. We show that the U(1) isometry variables of the TsT-transformed background are related to the isometry variables of the initial background in a universal way independent of the details of the background. This allows us to prove that strings in the TsT-transformed background are described by the Green-Schwarz action for strings in the initial background subject to twisted boundary conditions. Our construction im…
Recently Lunin and Maldacena used an SL(3,R) transformation of the AdS_5 x S^5 background to generate a supergravity solution dual to a so-called beta-deformation of N = 4 super Yang-Mills theory. We use a T-duality-shift-T-duality (TsT) transformation to obtain the beta-deformed background for real beta, and show that solutions of string theory equations of motion in this background are in one-to-one correspondence with those in AdS_5 x S^5 with twisted boundary conditions imposed on the U(1) isometry fields. We then apply the TsT transformation to derive a local and periodic Lax pair for the…
We build string backgrounds for Yang-Baxter deformations of the $AdS_4\times\mathbb{CP}^3$ superstring generated by $r$-matrices satisfying the classical Yang-Baxter equation. We obtain the metric and the NS-NS two-form of the gravity dual corresponding to noncommutative and dipole deformations of ABJM theory, as well as a deformed background with Schrödinger symmetry. The first two backgrounds may also be found by TsT transformations while for the last background we get a new family of non-relativistic ABJM theories with Schrödinger symmetry.
We first perform bosonic T-duality transformation on one of the marginal TsT (T-duality, shift, T-duality)-deformed $AdS_5\times S_5$ spacetime, which corresponds to 4D $\mathcal{N}=4$ non-commutative super Yang-Mills theory (NCSYM). We then construct the solution to killing spinor equations of the resulting background, and perform the fermionic T-duality transformation. The final dual geometry becomes the usual $AdS_5\times S_5$ but with the constant NS-NS B-field depending on the non-commutative parameter. As applications, we study the gluon scattering amplitude and open string (Wilson loop)…
We compute perturbative worldsheet S-matrix of beta-deformed AdS/CFT in the strong and weak `t Hooft coupling limit to compare with exact S-matrix. For the purpose we take near BMN limit of TsT-transformed AdS_5 x S^5 with the twisted boundary condition and compute the S-matrix on worldsheet using light-cone gauge fixed Lagrangian. For the weak coupling side, we compute the S-matrix in the SU(3) sector by applying coordinate Bethe ansatz method to the one-loop dilatation operator obtained from the deformed super Yang-Mills theory. These analysis support the conjectured exact S-matrix in the le…
3d N=2 partition functions on the squashed three-sphere and on the twisted product S2xS1 have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes squashed three-sphere and S2xS1 partition functions but the two cases involve different inner products, the S-pairing and the id-pairing respectively. We define a class of q-deformed CFT correlators where conformal blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing respectively. Applying the boo…
We compute two-point functions of lowest weight operators at the next-to-leading order in the couplings for the beta-deformed N=4 SYM. In particular we focus on the CPO Tr(Phi_1^2) and the operator Tr(Phi_1 Phi_2) not presently listed as BPS. We find that for both operators no anomalous dimension is generated at this order, then confirming the results recently obtained in hep-th/0506128. However, in both cases a finite correction to the two-point function appears.
The gravity dual of $β$-deformed ABJM theory can be obtained by a TsT transformation of $AdS_4\times\mathbb{CP}^3$. We present a supercoset construction of $\mathbb{CP}^3$ to obtain this gravity dual theory as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to get an Abelian $r$-matrix satisfying the classical Yang-Baxter equation. Our results provide another illustration of the relation between Abelian $r$-matrices and TsT transformations.
We compute semiclassical three-point correlation function, or structure constant, of two finite-size (dyonic) giant magnon string states and a light dilaton mode in the Lunin-Maldacena background, which is the γ-deformed, or TsT-transformed AdS_5 x S_γ^5, dual to N = 1 super Yang-Mills theory. We also prove that an important relation between the structure constant and the conformal dimension, checked for the N = 4 super Yang-Mills case, still holds for the γ-deformed string background.
We construct the wave functions in the q-deformed 2d Yang-Mills theory that compute torus correlation functions of affine currents in the VOA associated to a class of 4d $N = 2$ SCFTs. These wave functions are then shown to reduce to the topological correlators of a set of Coulomb branch operators in the $T[SU(N)]$ theory, from which those correlators in the 3d mirror dual of the 4d TN theories can be computed.
We study the beta-deformation of N=4 SYM on S^3 with chemical potentials for the U(1)_R as well as the two global U(1) symmetries. The one-loop effective potential at weak coupling is computed for both the Coulomb and Higgs branches. At near critical chemical potential and small finite temperature, we find a metastable state at the origin of moduli space. On the Higgs branch, this has the interpretation in terms of deconstruction as an extra-dimensional torus which becomes metastable for infinite size and decays to zero size through quantum tunnelling and thermal activation. At strong coupling…
We study the integrable $η$ and $λ$-deformations of supercoset string sigma models, the basic example being the deformation of the $AdS_5 \times S^5$ superstring. We prove that the kappa symmetry variations for these models are of the standard Green-Schwarz form, and we determine the target space supergeometry by computing the superspace torsion. We check that the $λ$-deformation gives rise to a standard (generically type II*) supergravity background; for the $η$-model the requirement that the target space is a supergravity solution translates into a simple condition on the R-matrix which ente…
In this Letter, based on the notion of Gauge/Gravity duality we explore the low frequency behaviour associated with the retarded two point correlators in the ground state of the strongly correlated quantum liquid that is dual to $ η$- deformed background in ($ 2+1 $)D. The massless charge carriers in the dual gauge theory are sourced due to some \textit{probe} $ N_f $ flavour $ Dp $ brane configurations in the bulk. In our analysis we stick to the NS sector and compute the two point correlators by turning on fluctuations associated with the worldvolume gauge fields in the bulk spacetime. Our a…
We study chiral deformations of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories obtained by turning on $τ_J \,{\rm tr} \, Φ^J$ interactions with $Φ$ the ${\cal N}=2$ superfield. Using localization, we compute the deformed gauge theory partition function $Z(\vecτ|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere. In the case of the deformed ${\cal N}=4$ theory, exact formulas for $Z$ and $W$ are derived in terms of an underlying $U(N)$ interacting matrix model replacing the free Gaussian model describing the ${\cal N}=4$ theory. Using the AGT corre…
We compute instanton corrections to correlators in the genus-zero topological subsector of a (0,2) supersymmetric gauged linear sigma model with target space P1xP1, whose left-moving fermions couple to a deformation of the tangent bundle. We then deduce the theory's chiral ring from these correlators, which reduces in the limit of zero deformation to the (2,2) ring. Finally, we compare our results with the computations carried out by Adams et al.[ABS04] and Katz and Sharpe[KS06]. We find immediate agreement with the latter and an interesting puzzle in completely matching the chiral ring of the…
Symmetries, operators and correlators in $J\bar{T}$ deformed CFTs
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；Center for the Fundamental Laws of Nature, Harvard University（哈佛大学自然基本定律中心）
We compute correlators of chiral operators in half-twisted (0,2) supersymmetric gauged linear sigma models. Our results give simple algebraic formulas for a (0,2) generalization of genus zero Gromov-Witten invariants of compact toric varieties. We derive compact expressions for deformed quantum cohomology relations and apply our general method to several examples.
Correlators in $T\bar{T}$ and Root-$T\bar{T}$ Deformed CFTs
Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Gansu Provincial Research Center for Basic Disciplines of Quantum Physics, Lanzhou University（兰州大学）
In this paper we compute the vacuum expectation value of the Wilson loop and its correlators with chiral primary operators in $\mathcal{N}=2, 4$ superconformal $U(N)$ gauge theories at large $N$. After localization these quantities can be computed in terms of a deformed $U(N)$ matrix model. The Wilson loops we deal with are in the fundamental and symmetric representations.
After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function $Z(\vecτ|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere will be computed. The partition function is deformed by turning on $τ_J \,{\rm tr} \, Φ^J$ interactions with $Φ$ the ${\cal N}=2$ superfield. For the ${\cal N}=4$ theory SUSY gauge theory exact formulae for $Z$ and $W$ in terms of an underlying $U(N)$ interacting matrix model can be derived thus rep…
Sharpening the holographic correspondence of Jackiw-Teitelboim (JT) gravity and its dual matrix model description at a finite radial cutoff $λ$ through the $T\overline{T}$ deformation is of interest. To proceed, we simplify the problem by considering the Airy model and deform Airy correlators in the same way as in $T\overline{T}$-deformed JT gravity. We use those correlators to compute the annealed and quenched free energies for both $λ> 0$ and $λ< 0$ from an integral representation of the replica trick. At the leading order in $λ$ and low temperatures, we confirm that the genus-zero quenched …
We compute the 2- and 3-point functions of currents and primary fields of $λ$-deformed integrable $σ$-models characterized also by an integer $k$. Our results apply for any semisimple group $G$, for all values of the deformation parameter $λ$ and up to order $1/k$. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable $λ$-deformed $σ$-models. We also present analogous results in two limiting cases of specia…
We consider non-perturbative effects in the beta-deformed N=4 supersymmetric gauge theory in the context of the AdS/CFT correspondence. We concentrate on the correlators of the Yang-Mills operators which correspond to the lowest Kaluza-Klein modes propagating on the dual supergravity background found by Lunin and Maldacena in hep-th/0502086. In particular, we calculate all multi-instanton contributions to these correlators in the beta-deformed SYM and find a compelling agreement with the results expected in supergravity.
We use a holographic model to compute the quark scalar current correlator. Already used in other contexts, our model consists in a bulk field theory defined on a deformed Anti de Sitter space and allows to derive analogous results to those of the soft wall model. One issue of such computations is to understand relevant holographic physical constraints which give predictive QCD results rather than to make the inverse step as done until now.
We study string theory on the analytically continued $β$ deformed background proposed in hep-th/0509036. This non-static model provides a solvable conformal field theory which describes time-dependent twisted string dynamics. With the mini-superspace approach, we examine disk one-point correlators of D-branes and compute the winding string pair production rate. We find that these results are consistent with the CFT computation.
Recently a prescription to compute the four-dimensional N = 2 superconformal index in the presence of certain BPS surface defects has been given. These surface defects are labelled by symmetric representations of SU(N). In the present paper we give a prescription to compute the superconformal index in the presence of surface defects labelled by arbitrary representations of SU(N). Furthermore, we extend the dictionary between the N = 2 superconformal Schur-index and correlators of q-deformed Yang-Mills to incorporate such surface defects.
Several calculations of 2- and 3-point correlation functions in the deformed theory are presented. The central charge in the Lunin-Maldacena gravity dual is shown to be independent of the deformation parameter. Calculations show that 2- and 3-point functions of chiral primary operators have no radiative corrections to lowest order in the interactions. Correlators of the operator tr(Z_1Z_2), which has not previously been identified as chiral primary, also have vanishing lowest order corrections.
Yang-Baxter (YB) deformations of string sigma model provide deformed target spaces. We propose that homogeneous YB deformations always lead to a certain class of $β$-twisted backgrounds and represent the bosonic part of the supergravity fields in terms of the classical r-matrix associated with the YB deformation. We then show that various $β$-twisted backgrounds can be realized by considering generalized diffeomorphisms in the undeformed background. Our result extends the notable relation between the YB deformations and (non-commuting) TsT transformations. We also discuss more general deformat…
We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to op…
In this article we probe the proposed holographic duality between $T\bar{T}$ deformed two dimensional conformal field theory and the gravity theory of $\rm AdS_3$ with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $T\bar{T}$ CFT in a Euclidean plane and in a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in $\rm AdS_3$ with the corresponding cutoff surface, given a hologra…
In this work, we investigate the quantum chaos in various $T\bar{T}$-deformed SYK models with finite $N$, including the SYK$_4$, the supersymmetric SYK$_4$, and the SYK$_2$ models. We numerically study the evolution of the spectral form factor (SFF), the out-of-time ordered correlator (OTOC), and the Krylov complexity. We find that the characteristic evolution of the SFF, OTOC and K-complexity of both the SYK$_4$ and SSYK$_4$ models remains unchanged under the deformation, which implies that the properties of quantum chaos is preserved. We also identify a many-body localization behavior in the…
We study the Feynman graph structure and compute certain exact four-point correlation functions in chiral CFT$_4$ proposed by Ö.Gürdoğan and one of the authors as a double scaling limit of $γ$-deformed $\mathcal{N}=4$ SYM theory. We give full description of bulk behavior of large Feynman graphs: it shows a generalized "dynamical fishnet" structure, with a dynamical exchange of bosonic and Yukawa couplings. We compute certain four-point correlators in the full chiral CFT$_4$, generalizing recent results for a particular one-coupling version of this theory -- the bi-scalar "fishnet" CFT. We sum …
In the framework of the semiclassical approach, we compute the normalized structure constants in three-point correlation functions, when two of the vertex operators correspond to "heavy" string states, while the third vertex corresponds to a "light" state. This is done for the case when the "heavy" string states are finite-size giant magnons, carrying one or two angular momenta. The "light" states are taken to be singlet scalar operators on higher string levels. We first consider the case of string theory on AdS_5xS^5 dual to N = 4 super Yang-Mills. Then we extend the obtained results to the g…
We study the second quantization of field theory on the q-deformed fuzzy sphere for real q. This is performed using a path-integral over the modes, which generate a quasiassociative algebra. The resulting models have a manifest U_q(su(2)) symmetry with a smooth limit q -> 1, and satisfy positivity and twisted bosonic symmetry properties. A systematic way to calculate n-point correlators in perturbation theory is given. As examples, the 4-point correlator for a free scalar field theory and the planar contribution to the tadpole diagram in ϕ^4 theory are computed. The case of gauge fields is als…
