We study the relativistic limit of the worldsheet S matrix of $AdS_3 \times S^3 \times S^3 \times S^1$ strings in the presence of mixed Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux, and complete the S-matrix bootstrap including the dressing factors for fundamental particles and bound states. Our results agree with the relativistic limit of the recent proposal arXiv:2512.07721.
We construct the operator product expansions (OPE) of the chiral primary operators in the worldsheet theory for strings on AdS_3 x S^3 x T^4. As an interesting application, we will use the worldsheet OPEs to derive a recursion relation for a particular class of extremal p-point correlators on the sphere. We compare our result with the corresponding recursion relation previously found in the symmetric orbifold theory on the boundary of AdS_3.
We present a study of the worldsheets that describe the classical limit of various string scattering processes. Our main focus is on string scattering in AdS spacetime because of its relation via the AdS/CFT correspondence to gluon scattering in {\cal N}=4 supersymmetric Yang-Mills theory. But we also consider analogous processes in flat Minkowski spacetime which we compare to the AdS case. In addition to scattering of string by string we also find and study worldsheets describing the scattering of a string by external sources.
We consider the AdS_4 x CP^3 IIA superstring sigma-model in the background of the "spinning string" classical solution, which possesses two Noether spins. In the limit when one of the spins is infinite there are massless excitations, which govern the infrared worldsheet properties of the model. We obtain a sigma-model of CP^3 with fermions, which describes the dynamics of these massless modes.
In this work we propose a systematic way to compute the logarithmic divergences of composite operators in the pure spinor description of the $AdS_5\times S^5$ superstring. The computations of these divergences can be summarized in terms of a dilatation operator acting on the local operators. We check our results with some important composite operators of the formalism.
In this note we study worldsheet two- and four-point functions at the one-loop level for the type IIA superstring in AdS(3) x S(3) x M(4) . We first address the regularization ambiguity that appears in the dispersion relation derived from integrability. We demonstrate that only the regulator treating all fields equally respects worldsheet supersymmetry. This is done in an implicit regularization scheme where all divergent terms are collected into master tadpole-type integrals. We then investigate one-loop two-body scattering on the string worldsheet and verify that a recent proposal for the dr…
We obtain the all-loop worldsheet S matrix for fundamental excitations on AdS_3 x S^3 x T^4 by studying the off-shell symmetry algebra of the superspace action in lightcone gauge. The massless modes, unaccounted for in earlier works, are automatically included in our treatment. Their exact dispersion relation is found to be non-relativistic, of giant-magnon form and their scattering is naturally well-defined. This opens the way to a complete investigation of AdS_3/CFT_2 integrability.
The integrability-based solution of string theories related to AdS(n)/CFT(n-1) dualities relies on the worldsheet S matrix. Using generalized unitarity we construct the terms with logarithmic dependence on external momenta at one- and two-loop order in the worldsheet S matrix for strings in a general integrable worldsheet theory. We also discuss aspects of calculations at higher orders. The S-matrix elements are expressed as sums of integrals with coefficients given in terms of tree-level worldsheet four-point scattering amplitudes. One-loop rational functions, not determined by two-dimensiona…
We examine the leading Regge string states relevant for semi-classical spinning string solutions. Using elementary RNS techniques, quadratic terms in an effective lagrangian are constructed which describe massive NSNS strings in a space-time with five-form flux. We then examine the specific case of AdS_5 x S^5, finding the dependence of AdS "energy" (E_0) on spin in AdS (S), spin on the sphere (J), and orbital angular momentum on the sphere (\nabla_a \nabla^a).
The pure spinor formalism for the superstring can be formulated as a twisted N=2 worldsheet theory with fermionic generators $j_{BRST}$ and composite $b$ ghost. After untwisting the formalism to an N=1 worldsheet theory with fermionic stress tensor $j_{BRST}+b$, the worldsheet variables combine into N=1 worldsheet superfields $X^m$ and $Θ^α$ together with a superfield constraint relating $DX^m$ and $DΘ^α$. The constraint implies that the worldsheet superpartner of $θ^α$ is a bosonic twistor variable, and different solutions of the constraint give rise to the pure spinor or extended RNS formali…
The AdS_3/CFT_2 correspondence is checked beyond the supergravity approximation by comparing correlation functions. To this end we calculate 2- and 3-point functions on the sphere of certain chiral primary operators for strings on AdS_3 x S^3 x T^4. These results are then compared with the corresponding amplitudes in the dual 2-dimensional conformal field theory. In the limit of small string coupling, where the sphere diagrams dominate the string perturbation series, beautiful agreement is found.
We confront the recently proposed exact S-matrices for AdS(3)/CFT(2) with direct worldsheet calculations. Utilizing the BMN and Near Flat Space (NFS) expansions for strings on AdS(3) x S(3) x S(3) x S(1) and AdS(3) x S(3) x T(4) we compute both tree-level and one-loop scattering amplitudes. Up to some minor issues we find nice agreement in the tree-level sector. At the one-loop level however we find that certain non-zero tree-level processes, which are not visible in the exact solution, contribute, via the optical theorem, and give an apparent mismatch for certain amplitudes. Furthermore we fi…
We determine the off-shell symmetry algebra and representations of Type IIB superstring theory on $AdS_3\times S^3 \times T^4$ with mixed R-R and NS-NS three-form flux. We use these to derive the non-perturbative worldsheet S matrix of fundamental excitations of the superstring theory. Our analysis includes both massive and massless modes and shows how turning on mixed three-form flux results in an integrable deformation of the S matrix of the pure R-R theory.
We explore the connection between the operator product expansion (OPE) in the boundary and worldsheet conformal field theories in the context of AdS$_{d+1}$/CFT$_d$ correspondence. Considering single trace scalar operators in the boundary theory and using the saddle point analysis of the worldsheet OPE [1], we derive an explicit relation between OPE coefficients in the boundary and worldsheet theories for the contribution of single trace spin $\ell$ operators to the OPE. We also consider external vector operators and obtain the relation between OPE coefficients for the exchange of single trace…
The AdS(4)/CFT(3) duality is a new example of an integrable and exactly solvable AdS/CFT system. There is, however, a puzzling mismatch between the number of degrees of freedom used in the exact solution (4B+4F scattering states) and 8B+8F transverse oscillation modes of critical superstring theory. We offer a resolution of this puzzle by arguing that half of the string modes dissolve in the continuum of two-particle states once alpha' corrections are taken into account. We also check that the conjectured exact S-matrix of AdS(4)/CFT(3) agrees with the tree-level worldsheet calculation.
We derive the non-perturbative worldsheet S matrix for fundamental excitations of Type IIB superstring theory on AdS_3 x S^3 x T^4 with Ramond-Ramond flux. To this end, we study the off-shell symmetry algebra of the theory and its representations. We use these to determine the S matrix up to scalar factors and we derive the crossing equations that these scalar factors satisfy. Our treatment automatically includes fundamental massless excitations, removing a long-standing obstacle in using integrability to study the AdS_3/CFT_2 correspondence. The present paper contains a detailed derivation of…
We continue our study of the worldsheet theory of superstrings on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ in the tensionless limit arXiv:1911.00378. We consider the theory on higher genus surfaces. We give evidence that the worldsheet correlators localise on certain worldsheets that cover the boundary of $\mathrm{AdS}_3$ holomorphically. This simplifies the string moduli space integral dramatically to a finite sum. This property shows that the higher genus corrections of the string worldsheet reproduce the structure of the $1/N$ corrections in the dual symmetric orbifold CFT $…
We continue to investigate planar four point worldsheet correlators of string theories which are conjectured to be duals of free gauge theories. We focus on the extremal correlators <Tr(Z^{J_1}(x)) Tr(Z^{J_2}(y)) Tr(Z^{J_3}(z)) Tr(\bar{Z}^{J}(0))> of $N = 4$ SYM theory, and construct the corresponding worldsheet correlators in the limit when the $J_i >> 1$. The worldsheet correlator gets contributions, in this limit, from a whole family of Feynman graphs. We find that it is supported on a {\it curve} in the moduli space parametrised by the worldsheet crossratio. In a further limit of the space…
We investigate the behaviour of the bosonic string on AdS3 with H-flux at stringy scales, looking in particular for a `tensionless' limit in which there are massless higher spin gauge fields. We do this by revisiting the physical spectrum of the sl(2,R)$_k$ WZW model and considering the limit in which k becomes small. At k=3 we find that there is an infinite stringy tower of massless higher spin fields which are part of a continuum of light states. This can be viewed as a novel tensionless limit, which appears to be distinct from that inferred from the symmetric orbifold description of superst…
Integrability of the string worldsheet theory in AdS(5) x S(5) is related to the existence of a flat connection depending on the spectral parameter. The transfer matrix is the open-ended Wilson line of this flat connection. We study the product of transfer matrices in the near-flat space expansion of the AdS(5) x S(5) string theory in the pure spinor formalism. The natural operations on Wilson lines with insertions are described in terms of r- and s-matrices satisfying a generalized classical Yang-Baxter equation. The formalism is especially transparent for infinite or closed Wilson lines with…
It is possible to formulate light-cone gauge string field theory in noncritical dimensions. Such a theory corresponds to conformal gauge worldsheet theory with nonstandard longitudinal part. We study the longitudinal part of the worldsheet theory on higher genus Riemann surfaces. The results in this paper shall be used to study the dimensional regularization of light-cone gauge string field theory.
We study the worldsheet theory of bosonic string from the point of view of the BV formalism. We explicitly describe the derived Poisson structure which arizes when we expand the Master Action near a Lagrangian submanifold. The BV formalism allows us to clarify the mechanism of holomorphic factorization of string amplitudes.
Pohlmeyer reduction has been instrumental both in the program for computing gluon scattering amplitudes at strong coupling, and more recently in the progress towards semiclassical three-point correlators of heavy operators in AdS/CFT. After a detailed review of the method, we combine it with Darboux and Crum transformations in order to obtain a class of string solutions corresponding to an arbitrary number of kinks and breathers of the elliptic sinh-Gordon equation. We also use our construction in order to identify the previously found dressed giant gluon with the single breather solution.
Mason and Skinner proposed the ambitwistor string theory which directly reproduces the formulas for the amplitudes of massless particles proposed by Cachazo, He and Yuan. In this paper we discuss geometries of the moduli space of worldsheets associated to the bosonic or the RNS ambitwistor string. Further, we investigate the factorization properties of the amplitudes when an internal momentum is near on-shell in the abstract CFT language. Along the way, we propose the existence of the ambitwistor strings with three or four fermionic worldsheet currents.
We analyze scattering of string modes at string junctions of type IIB string theory. In the infrared limit, certain orthogonal linear combinations of the fields on the different strings satisfy either Dirichlet or Neumann boundary conditions. We confirm that the worldsheet theory of a general string network has eight conserved supercharges in agreement with target space BPS considerations. As an application, we obtain the band spectrum of some simple string lattices.
A parametrization of (super) moduli space near the corners corresponding to bosonic or Neveu-Schwarz open string degenerations is introduced for worldsheets of arbitrary topology. With this parametrization, Feynman graph polynomials arise as the $α' \to 0$ limit of objects on moduli space. Furthermore, the integration measures of string theory take on a very simple and elegant form.
Quantum non-local charges are central to the quantum integrability of a sigma-model. In this paper we study the quantum consistency and UV finiteness of non-local charges of string theory in AdS(5)xS(5). We use the pure spinor formalism. We develop the near-flat space expansion of the transfer matrix and calculate the one-loop divergences. We find that the logarithmic divergences cancel at the level of one loop. This gives strong support to the quantum integrability of the full string theory. We develop a calculational setup for the renormalization group analysis of Wilson line type of operato…
We study the worldsheet scattering theory of the $η$ deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding e…
Despite recent evidence indicating the existence of a new kind of self-dual six-dimensional superstring, no satisfactory worldsheet formulation of such a string has been proposed. In this note we point out that a theory built from Z_4 parafermions may have the right properties to describe the light-cone conformal field theory of this string. This indicates a possible worldsheet formulation of this theory.
We derive the protected closed-string spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the three-form fluxes. These follow immediately from the all-loop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves length-changing interactions and that protected states correspond to gapless excitations above the Berenstein-Maldacena-Nastase vacuum. In the case of AdS3 x S3 x T4 the degeneracies of such operators precisely mat…
The tensionless limit of string theory has recently been formulated in terms of worldsheet Rindler physics. In this paper, by considering closed strings moving in background Rindler spacetimes, we provide a concrete exemplification of this phenomenon. We first show that strings probing the near-horizon region of a generic non-extremal blackhole become tensionless thereby linking a spacetime Carroll limit to a worldsheet Carroll limit. Then, considering strings in $d$-dimensional Rindler spacetime we find a Rindler structure induced on the worldsheet. Novelties, including folds, appear on the c…
We calculate some extremal and non-extremal four-point functions on the sphere of certain chiral primary operators for strings on AdS_3 x S^3 x T^4. The computation is done for small values of the spacetime cross-ratio where global SL(2) and SU(2) descendants may be neglected in the intermediate channel. Ignoring also current algebra descendants, we find that in the non-extremal case the integrated worldsheet correlators factorize into spacetime three-point functions, which is non-trivial due to the integration over the moduli space. We then restrict to the extremal case and compare our result…
T-Duality invariant worldsheet string actions, recently written down by Schwarz and Sen, are coupled to the worldsheet gauge fields. It is shown that the integration of the dual coordinates gives the conventional, vector, axial and chiral, gauged string actions for the appropriate choice of the gauged isometries. Alternatively, the gauge field integration is shown to give a T-duality invariant action which matches with the corresponding results known earlier.
We generalize the worldsheet derivation of the topological open/closed string duality given in hep-th/0205297 to cases when there are different types of D branes on the open string side. We use the mirror Landau-Ginzburg description to clarify the correspondence between D branes on the open string side and C phases on the closed string side. We also discuss the duality from the point of view of the B model.
We formulate a set of consistency conditions appropriate to worldsheet form factors in the massive, integrable but non-relativistic, light-cone gauge fixed AdS(5) x S**5 string theory. We then perturbatively verify that these conditions hold, at tree level in the near-plane-wave limit and to one loop in the near-flat (Maldacena-Swanson) limit, for a number of specific cases. We further study the form factors in the weakly coupled dual description, verifying that the relevant conditions naturally hold for the one-loop Heisenberg spin-chain. Finally, we note that the near-plane-wave expressions …
D-brane backgrounds are specified in closed string theories by holes with appropriate mixed Dirichlet and Neumann boundary conditions on the string worldsheet. As presently stated, the prescription defining D-brane backgrounds is such that the Einstein equation is not equivalent to the condition for scale invariance on the string worldsheet. A modified D-brane prescription is found, that leads to the desired equivalence, while preserving all known D-brane lore. A possible interpretation is that the worldsheet cutoff is finite. Possible connections to recent work of Maldacena and Strominger, an…
Alternative gauge choices for worldsheet supersymmetry can elucidate dynamical phenomena obscured in the usual superconformal gauge. In the particular example of the tachyonic $E_8$ heterotic string, we use a judicious gauge choice to show that the process of closed-string tachyon condensation can be understood in terms of a worldsheet super-Higgs effect. The worldsheet gravitino assimilates the goldstino and becomes a dynamical propagating field. Conformal, but not superconformal, invariance is maintained throughout.
We extend Gopakumar's prescription for constructing closed string worldsheets from free field theory diagrams with adjoint matter to open and closed string worldsheets arising from free field theories with fundamental matter. We describe the extension of the gluing mechanism and the electrical circuit analogy to fundamental matter. We discuss the generalization of the existence and uniqueness theorem of Strebel differentials to open Riemann surfaces. Two examples are computed of correlators containing fundamental matter, and the resulting worldsheet OPE's are computed. Generic properties of Go…
We study the one loop correction to the closed bosonic string propagator, including the possibile presence of D-branes, by discretizing the light cone worldsheet on an M times N rectangular lattice, with M proportional to P^+ and N+1 proportional to ix^+. The integrals over the moduli then become sums which we evaluate numerically. The main purpose of this study is to assess the reliability of the worldsheet lattice as a regulator of the divergences in string perturbation theory. There are two natural geometrical counterterms for the lightcone worldsheet, one proportional to the area of the wo…
We revisit the construction of the tensionless limit of closed bosonic string theory in the covariant formulation in the light of Galilean conformal symmetry that rises as the residual gauge symmetry on the tensionless worldsheet. We relate the analysis of the fundamentally tensionless theory to the tensionless limit that is viewed as a contraction of worldsheet coordinates. Analysis of the quantum regime uncovers interesting physics. The degrees of freedom that appear in the tensionless string are fundamentally different from the usual string states. Through a Bogoliubov transformation on the…
We consider the noncommutativity parameter of the space-time as a bosonic worldsheet field. By finding a fermionic super-partner for it, we can find star products between the boson-boson, boson-fermion and fermion-fermion fields of superstring worldsheet and also between superfields of the worldsheet superspace. We find a two dimensional action for the noncommutativity parameter and its fermionic partner. We discuss the symmetries of this action.
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…
We investigate type IIB strings on $\mathrm{AdS}_3\times \mathrm{S}^3\times \mathrm{S}^3\times\mathrm{S}^1$ with mixed Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux. By suitably gauge-fixing the closed string Green-Schwarz (GS) action of this theory, we derive the off-shell symmetry algebra and its representations. We use these to determine the non-perturbative worldsheet S-matrix of fundamental excitations in the theory. The analysis involves both massive and massless modes in complete generality. The S-matrix we find involves a number of phase factors, which in turn satisf…
In the presence of the constant background NS two-form gauge field, we construct the worldsheet partition functions, bulk propagators and boundary propagators for the worldsheets with a handle and a boundary. We analyze the noncommutative $ϕ^3$ field theory amplitudes that correspond to the general two-point insertions on the two-loop nonplanar vacuum bubble. By the direct string theory amplitude computations on the worldsheets with a handle, which reduce to the aforementioned field theory amplitudes in the decoupling limit, we find that the stretched string interpretation remains valid for th…
We determine the two--dimensional Weyl, Lorentz and $κ$--anomalies in the $D=10$ Green--Schwarz heterotic string sigma--model, in an $SO(1,9)$-Lorentz covariant background gauge, and prove their cancellation.
Using worldsheet Hamiltonian methods we derive a charge algebra which generalizes the Courant bracket to include fluxes of general index type. This is achieved by coupling a bi-vector to the Hamiltonian of the Polyakov model. This bracket is useful to describe so-called non-geometric backgrounds and has been discussed in the mathematics literature by Dmitry Roytenberg.
It was recently shown that the CFT dual of string theory on ${\rm AdS}_3 \times {\rm S}^3 \times T^4$, the symmetric orbifold of $T^4$, contains a closed higher spin subsector. Via holography, this makes precise the sense in which tensionless string theory on this background contains a Vasiliev higher spin theory. In this paper we study this phenomenon directly from the worldsheet. Using the WZW description of the background with pure NS-NS flux, we identify the states that make up the leading Regge trajectory and show that they fit into the even spin ${\cal N}=4$ Vasiliev higher spin theory. …
We calculate the one-point function of certain chiral operators, known as open Wilson lines, in the holographic Nambu-Jona-Lasinio (NJL) model at finite temperature as well as finite chemical potential for quark (baryon) number density. These operators are holographically dual to Euclidean string worldsheets bounded by the flavor branes. Their one-point functions can serve as order parameters for chiral symmetry breaking in these models. Analyzing the behavior of the dual string worldsheets with respect to temperature and chemical potential (or quark density) enables us to determine how the or…
Worldline instantons provide a particularly elegant way to derive Schwinger's well-known formula for the pair creation rate due to a constant electric field in quantum electrodynamics. In this note, we show how to extend this method to the corresponding problem of open string pair creation.
For a superstring theory in four spacetime dimensions, we propose a modification of the Born-Infeld action that possesses a well-defined tensionless limit. We interpret this as describing the effective target space dynamics of null strings on a D3-brane. We argue that such a modification can be induced by nonperturbative contributions from instantons in the worldsheet sigma-model describing string propagation on the brane.
We extend the lightcone worldsheet lattice approach to string theory, proposed in 1977 by Giles and me, to allow for coincident D-branes. We find a convenient lattice representation of Dirichlet boundary conditions, which the open string coordinates transverse to the D-branes satisfy. We then represent the sum over all planar open string multi-loop diagrams by introducing an Ising spin system on the worldsheet lattice to keep track of the presence or absence of fluctuating boundaries. Finally we discuss a simple mean field treatment of the resulting coupled Ising/coordinate worldsheet system. …
We use supersymmetric localization techniques to study the low-energy dynamics of BPS vortex-strings in four-dimensional N=2 theories. We focus on theories with SU(Nc)xU(1) gauge group and Nf hypermultiplets, all in the fundamental representation of SU(Nc) but with general U(1) charges. Recently, we proposed a condition that determines whether the low-energy string dynamics is captured by a two-dimensional worldsheet theory that decouples from the bulk. For strings for which this decoupling applies, we propose a prescription for extracting the two-sphere partition function of the string worlds…
Time-like linear dilaton backgrounds admit a classical solution that describes a closed folded string that is created at an instant. We refer to such strings as Instant Folded Strings (IFS). We study an exact worldsheet CFT description of an IFS that involves two vertex operators which describe two open string modes that propagate on a time-like FZZT-brane, which plays the role of a regulator to the IFS. We take advantage of this description to calculate the most basic quantity associated with IFSs - their production rate. Some implications of this calculation to stringy cosmology and black ho…
Worldsheet string theory on $\mathrm{AdS}_3 \times \mathbf{S}^3 \times \mathrm{K3}$
University of the Chinese Academy of Sciences（中国科学院大学）；Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Institut für Theoretische Physik, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；Department of Mathematics, King’s College London（伦敦国王学院数学系）；Center for Quantum Mathematics and Physics (QMAP) Department of Physics & Astronomy, University of California, Davis（加州大学戴维斯分校物理学与天文学系量子数学与物理中心）
At first we introduce an action for the string, which leads to a worldsheet that always is curved. For this action we study the Poincaré symmetry and the associated conserved currents. Then, a generalization of the above action, which contains an arbitrary function of the two-dimensional scalar curvature, will be introduced. An extra scalar field enables us to modify these actions to Weyl invariant models.
