In this article we review the world-sheet scattering theory of strings on AdS 5 x S5. The asymptotic spectrum of this world-sheet theory contains both fundamental particles and bound states of the latter. We explicitly derive the S-matrix that describes scattering of arbitrary bound states. The key feature that enables this derivation is the so-called Yangian symmetry which is related to the centrally extended su(2|2) superalgebra. Subsequently, we study the universal algebraic properties of the found S-matrix. As in many integrable models, the S-matrix plays a key role in the determination of the energy spectrum. In this context, we employ the Bethe ansatz approach to compute the large volume energy spectrum of string bound states.
In this paper we show that Ultradistributions of Exponential Type (UET) are appropriate for the description in a consistent way world sheet superstring and superstring field theories. A new Lagrangian for the closed world sheet superstring is obtained. We also show that the superstring field is a linear superposition of UET of compact support (CUET), and give the notion of anti-superstring. We evaluate the propagator for the string field, and calculate the convolution of two of them.
We construct a manifestly $N=(4,0)$ world-sheet supersymmetric twistor-like formulation of the $D=6$ Green-Schwarz superstring, using the principle of double (target-space and world-sheet) Grassmann analyticity. The superstring action contains two Lagrange multiplier terms and a Wess-Zumino term. They are written down in the analytic subspace of the world-sheet harmonic $N=(4,0)$ superspace, the target manifold being too an analytic subspace of the harmonic $D=6\;\; N=1$ superspace. The kappa symmetry of the $D=6$ superstring is identified with a Kac-Moody extension of the world-sheet $N=(4,0)…
String theories with (N,N') local world-sheet supersymmetries are related to each other by marginal deformations. This connects N=1 and N=0 theories in which the target-spaces are interpreted as space-times, N=2 theories in which the target spaces can be interpreted as world-volumes, and theories with $N\ge 3$, in which the central charge vanishes -- theories with zero target-space dimensions.
The supercoherent states of the RNS string are constructed using the covariant quantization and analogously the light cone quantization formalisms. Keeping intact the original definition of coherent states of harmonic oscillators, we extend the bosonic annihalation operator into the superspace by inclusion of fermionic contribution to oscillator modes thus construct the supercoherent states with supersymetric harmonic oscillator. We analyse the statistics of these states by explicitly calculating the Mandel parameter and obtained interesting results about the nature of distribution of the stat…
In the framework of the four dimensional heterotic superstring with free world-sheet fermions we study the GUSTs which contain non-abelian horizontal symmetry. To solve the problem of breaking the G_GUT gauge symmetry of the level one current algebra we extend the gauge symmetry to G x G. Using the autoduality of the fermionic charge lattice we study the possibilities of constructing of some GUST gauge groups. We also study the string amplitudes and for one realistic model explicitly calculate the renormalizable and non-renormalizable (N=4,5,6) contributions to the superpotential. We observe t…
Given a family of world-sheet superconformal field theories related by marginal deformation, we can formulate superstring field theory based on any of these world-sheet theories. Background independence is the statement that these different superstring field theories are related to each other by field redefinition. We prove background independence of closed superstring field theory.
We discuss several closely related concepts in the NSR formulation of superstring theory. We demonstrated that recently proposed NSR model for superstrings on $AdS_5 \times S^5$ is described by the world-sheet logarithmic conformal field theory (LCFT). The origin of LCFT on a world-sheet is closely connected to the matter-ghost mixing in the structure of a brane-like vortex operators. We suggest a dynamical origin of M theory as a string theory with an extra dimension given by bosonised superconformal ghosts.
We initiate the computation of the 2-loop quantum AdS_5 x S^5 string corrections on the example of a certain string configuration in S^5 related by an analytic continuation to a folded rotating string in AdS_5 in the ``long string'' limit. The 2-loop term in the energy of the latter should represent the subleading strong-coupling correction to the cusp anomalous dimension and thus provide a further check of recent conjectures about the exact structure of the Bethe ansatz underlying the AdS/CFT duality. We use the conformal gauge and several choices of the κ-symmetry gauge. While we are unable …
We show that the super D-string action is canonically equivalent to the type IIB superstring action with a world-sheet gauge field. Canonical transformation to the type IIB theory with dynamical tension is also constructed to establish the SL(2,Z) covariance beyond the semi-classical approximations.
We construct a world-sheet action for Green-Schwarz superstring in terms of doubled-yet-gauged spacetime coordinates. For an arbitrarily curved NS-NS background, the action possesses $\mathbf{O}(10,10)$ T-duality, $\mathbf{Spin}(1,9)\times\mathbf{Spin}(9,1)$ Lorentz symmetry, coordinate gauge symmetry, spacetime doubled-yet-gauged diffeomorphisms, world-sheet diffeomorphisms and Weyl symmetry. Further, restricted to flat backgrounds, it enjoys maximal spacetime supersymmetry and kappa-symmetry. After the auxiliary coordinate gauge symmetry potential being integrated out, our action can consist…
We examine whether the free energy of N=4 super Yang-Mills theory (SYM) in four dimensions corresponds to the partition function of the AdS_5 x S^5 superstring when corresponding operators are inserted into both theories. We obtain a formal free energy of N=4 U(N) SYM in four dimensions generated by the Feynman graph expansion to all orders of the 't Hooft coupling expansion with arbitrary N. This free energy is written as the sum over discretized closed two-dimensional surfaces that are identified with the world-sheets of the string. We compare this free energy with a formal partition functio…
We discuss the spectrum of a string propagating on eta-deformed AdS_5 x S^5 by treating its world-sheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a q-deformation of the AdS_5 x S^5 world-sheet S-matrix with real deformation parameter. By considering mirror (double Wick-rotated) versions of these world-sheet theories we give the Thermodynamic Bethe Ansatz description of their exact finite size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the level of the string this appears to say that t…
It was previously shown that most of the superstrings can be obtained from those with higher world-sheet supersymmetry as spontaneously broken phases. In this paper, we show that the small $N=4$ superstring, which was left out of this hierarchy of the universal string, can be obtained from the large $N=4$ strings. We also show that the $N=2$ string is a special vacuum of small $N=4$ string. Thus all the known superstring theories can be derived from a universal string by spontaneous breakdown of supersymmetry.
We consider the open string tachyon action in a world-sheet sigma model approach. We present explicit calculations up to order 8 in derivatives for the bosonic string, and mimic these to order 6 for the superstring, including terms with multiple derivatives acting on the tachyon field. We reproduce lower derivative terms obtained elsewhere, and speculate on the role of the world-sheet contact terms regularizing the action for the superstring tachyon.
We review the recent advances towards finding the spectrum of the AdS_5 x S^5 superstring. We thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem. In certain cases our exposition is original and cannot be found in the existing literature. The present Part I deals with foundations of classical string theory in AdS_5 x S^5, light-cone perturbative quantization and derivation of the exact light-cone world-sheet scattering matrix.
A bosonic string in twenty six dimensions is effectively reduced to four dimensions by eleven Majorana fermions which are vectors in the bosonic represetation SO(d-1,1). By dividing the fermions in two groups, actions can be written down which are world sheet supersymmetric, 2-d local and local 4-d supersymmetric. The novel string is anomally free, free of ghosts and the partition function is modular invariant.
The type IIB supergravity AdS_3 x S^3 x T^4 background with mixed RR and NSNS 3-form fluxes is a near-horizon limit of a non-threshold bound state of D5-D1 and NS5-NS1 branes. The corresponding superstring world-sheet theory is expected to be integrable, opening the possibility of computing its exact spectrum for any values of the coefficient q of the NSNS flux and the string tension. In arXiv:1303.1447 we have found the tree-level S-matrix for the massive BMN excitations in this theory, which turned out to have a simple dependence on q. Here, by analyzing the constraints of symmetry and integ…
Scattering amplitudes of the spin-4/3 fractional superstring are shown to satisfy spurious state decoupling and cyclic symmetry (duality) at tree-level in the string perturbation expansion. This fractional superstring is characterized by the spin-4/3 fractional superconformal algebra---a parafermionic algebra studied by Zamolodchikov and Fateev involving chiral spin-4/3 currents on the world-sheet in addition to the stress-energy tensor. Examples of tree scattering amplitudes are calculated in an explicit c=5 representation of this fractional superconformal algebra realized in terms of free bo…
The spin-4/3 fractional superstring is characterized by a chiral algebra involving a spin-4/3 current on the world-sheet in addition to the energy-momentum tensor. These currents generate physical state conditions on the fractional superstring Fock space. Scattering amplitudes of these physical states are described which satisfy both spurious state decoupling and cyclic symmetry (duality). Examples of such amplitudes are calculated using an explicit $c=5$ realization of the spin-4/3 current algebra. This representation has three flat coordinate boson fields and a global SO(2,1) Lorentz symmetr…
We clarify the relation between the vertex operators in type IIB matrix model and superstring. Green-Schwarz light-cone closed superstring theory is obtained from IIB matrix model on two dimensional noncommutative backgrounds. Superstring vertex operators should be reproduced from those of IIB matrix model through this connection. Indeed, we confirm that supergravity vertex operators in IIB matrix model on the two dimensional backgrounds reduce to those in superstring theory. Noncommutativity plays an important role in our identification. Through this correspondence, we can reproduce superstri…
The spin-4/3 fractional superstring is characterized by a world-sheet chiral algebra involving spin-4/3 currents. The discussion of the tree-level scattering amplitudes of this theory presented in hepth/9310131 is expanded to include amplitudes containing two twisted-sector states. These amplitudes are shown to satisfy spurious state decoupling. The restriction to only two external twisted-sector states is due to the absence of an appropriate dimension-one vertex describing the emission of a single twisted-sector state. This is analogous to the ``old covariant'' formalism of ordinary superstri…
A formalism is provided to calculate tree amplitudes in open superstring theory for any multiplicity at any order in the inverse string tension. We point out that the underlying world-sheet disk integrals share substantial properties with color-ordered tree amplitudes in Yang-Mills field theories. In particular, we closely relate world-sheet integrands of open-string tree amplitudes to the Kawai-Lewellen-Tye representation of supergravity amplitudes. This correspondence helps to reduce the singular parts of world-sheet disk integrals -including their string corrections- to lower-point results.…
We propose possible new string theories based on local world-sheet symmetries corresponding to extensions of the Virasoro algebra by fractional spin currents. They have critical central charges $c=6(K+8)/(K+2)$ and Minkowski space-time dimensions $D=2+16/K$ for $K\geq2$ an integer. We present evidence for their existence by constructing modular invariant partition functions and the massless particle spectra. The dimension $4$ and $6$ strings have space-time supersymmetry.
We present evidence that the supersymmetric matrix model of Marinari and Parisi represents the world-line theory of N unstable D-particles in type II superstring theory in two dimensions. This identification suggests that the matrix model gives a holographic description of superstrings in a two-dimensional black hole geometry.
We show that, in the limit of zero string coupling, $g_s \to 0$, the thermodynamic partition function of matrix string theory is identical to that of the finite temperature, discrete light-cone quantised (DLCQ) type IIA superstring. We discuss how the superstring is recovered in the decompactified $R^+\to\infty$ limit.
Space-time N=2 and N=4 superconformal algebras can be built using world-sheet free fields or, equivalently, free field representations of osp(1|2) and sl(2|1), respectively. A prescription for the calculation of space-time correlators is given. As applications we compute one and two-point correlation functions of Virasoro (superVirasoro) generators. We also present a possible scenario for taking into account of NS fivebranes in the framework of our construction for description of superstring on AdS_3. It is nothing but a simplified version of Green's ``world-sheets for world-sheets''.
We investigate the NBI matrix model with the potential $XΛ+X^{-1}+(2η+1)\log X$ recently proposed to describe IIB superstrings. With the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich matrix model for $η\ne0$ and find the explicit transformation between the two models.
We study the dynamics of Green-Schwarz superstring on the gravitational wave background corresponding to the Matrix string theory and the supersymmetry transformation rules of the superstring. The dynamics is obtained in the light-cone formulation and is shown to agree with that derived from the Matrix string theory. The supersymmetry structure has the corrections due to the effect of the background and is identified with that of the low energy one-loop effective action of Matrix string theory in two superstring background in the weak string coupling limit.
We discuss properties of D-brane configurations in the matrix model of type IIB superstring recently proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya. We calculate central charges in supersymmetry algebra at infinite N and associate them with one- and five-branes present in IIB superstring theory. We consider classical solutions associated with static three- and five-branes and calculate their interactions at one loop in the matrix model. We discuss some aspects of the matrix-model formulation of IIB superstring.
We use the pure spinor formalism to obtain the supersymmetric massless five-point amplitude at tree-level in a streamlined fashion. We also prove the equivalence of an OPE identity in string theory with a subset of the Bern-Carrasco-Johansson five-point kinematic relations, and demonstrate how the remaining BCJ identities follow from the different integration regions over the open string world-sheet, therefore providing a first principles derivation of the (supersymmetric) BCJ identities.
We show that the classical S-matrix calculated from the recently proposed superstring field theories give the correct perturbative S-matrix. In the proof we exploit the fact that the vertices are obtained by a field redefinition in the large Hilbert space. The result extends to include the NS-NS subsector of type II superstring field theory and the recently found equations of motions for the Ramond fields. In addition, our proof implies that the S-matrix obtained from Berkovits' WZW-like string field theory then agrees with the perturbative S-matrix to all orders.
We show that the tree-level S-matrices of the superstring field theories based on the homotopy-algebra structure agree with those obtained in the first-quantized formulation. The proof is given in detail for the heterotic string field theory. The extensions to the type II and open superstring field theories are straightforward.
We compute one-loop correction to the string field theory action of the tachyon for unstable D-branes in the framework of the boundary superstring field theory. We would expect that the one-loop correction comes from the partition function of the two-dimensional world-sheet theory on the annulus. The annulus correction suggests that the genus expansion is, somehow, governed by the effective string coupling defined in terms of the tachyon λ=g_s exp(-T^2/4).
We derive fully covariant expressions for disk scattering amplitudes of any two massless closed strings in which mixed Neumann and Dirichlet world-sheet boundary conditions are included. From the two-point amplitudes, we derive the long range background fields and verify that they correspond to D$p$-brane bound state. Also, from the scattering amplitudes, we calculate the linear coupling of closed string fields to D-brane world-volume and show that they are consistent with Born-Infeld and Chern-Simons actions in the presence of a background field.
I review the properties of a matrix action of relevance for IIB superstrings. This model generalizes the action proposed by Ishibashi, Kawai, Kitazawa, and Tsuchiya by introducing an auxillary field Y, which is the matrix version of the auxillary field g in the Schild action.
We apply the algebraic Bethe ansatz technique to compute the eigenvalues of the transfer matrix constructed from the general bound state S-matrix of the light-cone AdS5 x S5 superstring. This allows us to verify certain conjectures on the quantum characteristic function, and to extend them to the general case.
We address the problem of a non-perturbative formulation of superstring theory by means of the recently proposed matrix models. For the model by Ishibashi, Kawai, Kitazawa and Tsuchiya (IKKT), we perform one-loop calculation of the interaction between operator-like solutions identified with D-brane configurations of type IIB superstring (in particular, for parallel moving and rotated static p-branes). Comparing to the superstring calculations, we show that the matrix model reproduces the superstring results only at large distances or small velocities, corresponding to keeping only the lowest m…
We compute the one-loop world-sheet correction to partition function of $AdS_5 \times S^5$ superstring that should be representing $k$-fundamental circular Wilson loop in planar limit. The 2d metric of the minimal surface ending on $k$-wound circle at the boundary is that of a cone of $AdS_2$ with deficit $2π(1-k)$. We compute determinants of 2d fluctuation operators by first constructing heat kernels of scalar and spinor Laplacians on the cone using the Sommerfeld formula. The final expression for the k-dependent part of the one-loop correction has simple integral representation but is differ…
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 construct Green-Schwarz (GS) light-cone closed superstring theory from type IIB matrix model. A GS light-cone string action is derived from two dimensional N=8 U(n) noncommutative Yang-Mills (NCYM) by identifying noncommutative scale with string scale. Supersymmetry transformation for the light-cone gauge action is also derived from supersymmetry transformation for IIB matrix model. By identifying the physical states and interaction vertices, string theory is perturbatively reproduced.
We show that a recently proposed Yang-Mills matrix model with an auxiliary field, which is a candidate for a non-perturbative description of type IIB superstrings, captures the Euler characteristic of moduli space of Riemann surfaces. This happens at the saddle point for the Yang-Mills field. It turns out that the large-n limit in this matrix model corresponds to a double scaling limit in the Penner model.
We extend the non-perturbative time-dependent bosonic string action of [3] to a N=1 supersymmetric world sheet action with graviton background, and assume a superpotential, function of the time super coordinate.
It is reviewed how space-time supersymmetry is realized nonlinearly in open superstring theory without making the GSO projection. We show that the world-sheet string dualities, viz. dualities of open-closed strings and of open-open strings, play crucial roles for the existence of 10 dimensional N=2 supersymmetry in a spontaneously broken phase. We also speculate on a possible mechanism of the restoration of supersymmetry from the viewpoint of world-sheet dynamics.
We apply light-cone quantization to a $1+1$ dimensional supersymmetric field theory of large N matrices. We provide some preliminary numerical evidence that when the coupling constant is tuned to a critical value, this model describes a 2+1 dimensional non-critical superstring.
The string equivalent of a massless particle ($m=0$) is the tensionless string ($T=0$). The study of such strings is of interest when trying to understand the high energy limit of ordinary strings. I discuss the classical $T\to 0$ limit of the bosonic string, the spinning string and the superstring. A common feature is the appearence of a space-time (super-)conformal symmetry replacing the world-sheet Weyl invariance. The question of whether this symmetry may survive quantization is addressed. A lightcone analysis of the quantized bosonic tensionless string leads to severe constraints on the p…
The string theory on symmetric product describes the second-quantized string theory. The development for the bosonic open string was discussed in the previous work. In this paper, we consider the open superstring theory on the symmetric product and examine the nature of the second quantization. The fermionic partition functions are obtained from the consistent fermionic extension of the twisted boundary conditions for the non-abelian orbifold, and they can be interpreted in terms of the long string language naturally. In the closed string sector, the boundary/cross-cap states are also construc…
Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS_5 x S^5 superstring world-sheet theory in terms of physical degrees of freedom we construct the Pohlmeyer-reduced version of the corresponding sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the origina…
We compute the massive-sector worldsheet S-matrix for superstring theories in AdS(n) x S(n) x T(10-2n) (with n=2,3,5) in the near BMN expansion up to one-loop order in inverse string tension. We show that, after taking into account the wave function renormalization, the one-loop S-matrix is UV finite. In an appropriate regularization scheme the S-matrix is consistent with the underlying symmetries of the superstring theory, i.e. for the n=3,5 cases it coincides with the one implied by the light-cone gauge symmetries with the dressing phases determined from the crossing equations. For the n=2,3…
A string in four dimensions is constructed by supplementing it with forty four Majorana fermions. The later are represented by eleven vectors in the bosonic representation $SO(D-1,1)$. The central charge is 26. The fermions are grouped in such a way that the resulting action is world sheet supersymmetric. The energy momentum and current generators satisfy the super-Virasoro algebra. GSO projections are necessary for proving modular invariance. Space-time supersymmetry algebra is deduced and is substantiated for specific modes of zero mass. The symmetry group of the model can descend to the low…
In this paper we find the Yangian algebra responsible for the integrability of the AdS2 X S2 X T^6 superstring in the planar limit. We demonstrate the symmetry of the corresponding exact S-matrix in the massive sector, including the presence of the secret symmetry. We give two alternative presentations of the Hopf algebra, along with related discussions on the issue of evaluation representations. We study the classical r-matrix, and re-discover the need for a secret symmetry also in this context. Finally, taking the simplifying zero-coupling limit of the S-matrix as a generating R-matrix for t…
We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large--$N$ limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, possibly is irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.
We calculate on-shell scattering amplitudes involving fermions at the tree level in open superstring field theory. We confirm that four-point and five-point amplitudes in the world-sheet path integral with the standard prescription using picture-changing operators are reproduced. For the four-point amplitudes, we find that the quartic interaction required by gauge invariance adjusts the different assignment of picture-changing operators in the $s$-channel and in the $t$-channel of Feynman diagrams with two cubic vertices. For the five-point amplitudes, the correct amplitudes are reproduced in …
In a recent paper Townsend suggested to associate to the D=11 solitonic membrane of N=1 supergravity a certain thickness, and then to identify this membrane with the fundamental supermembrane. By integrating out the 8 transverse dimensions of the "thick" solitonic membrane, we show that the resulting world-volume action indeed contains all the usual supermembrane terms, as well as background curvature terms and extrinsic curvature terms, which are believed to render the membrane spectrum discrete. We also outline the analog derivation for the "thick" D=10 solitonic string solution of N=IIA sup…
In the previous paper, the authors pointed out correspondence between a supersymmetric double-well matrix model and two-dimensional type IIA superstring theory on a Ramond-Ramond background from the viewpoint of symmetry and spectrum. This was confirmed by agreement between planar correlation functions in the matrix model and tree-level amplitudes in the superstring theory. In order to investigate the correspondence further, in this paper we compute correlation functions to all order of genus expansion in the double scaling limit of the matrix model. One-point functions of operators protected …
With a motivation to find a 2-d Lorentz-invariant solution of the AdS_5 x S^5 superstring we continue the study of the Pohlmeyer-reduced form of this theory. The reduced theory is constructed from currents of the superstring sigma model and is classically equivalent to it. Its action is that of G/H=Sp(2,2)xSp(4)/SU(2)^4 gauged WZW model deformed by an integrable potential and coupled to fermions. This theory is UV finite and is conjectured to be related to the superstring theory also at the quantum level. Expanded near the trivial vacuum it has the same elementary excitations (8+8 massive boso…
Multi-cut two-matrix models are studied in the Z_k symmetry breaking k-cut (\hat p,\hat q) critical points which should correspond to (\hat p,\hat q) minimal k-fractional superstring theory. FZZT-brane or macroscopic loop amplitudes are obtained in all of these critical points and found to have two kinds of solutions in general. Each of these solutions is expressed by hyperbolic cosine or sine functions with proper phase shifts. The algebraic geometries and ZZ-brane disk amplitudes (instanton actions) of these solutions are also studied. In particular, our results suggest that minimal \infty-f…
We construct a kappa-symmetric Green-Schwarz action for type IIA string theory on AdS_2. As a candidate holographic dual, we consider superconformal matrix quantum mechanics, given by the Marinari-Parisi model with vanishing or logarithmic superpotential. We derive that the super-eigenvalues form a consistent subsector, and that their dynamics reduces to that of the supersymmetric Calogero-Moser model. The classical string action and the matrix model both have an infinite set of conserved charges, that include an N=2 target space super-Virasoro symmetry. As a microscopic test of the duality, w…
We show that the recently found S-matrices describing the scattering of two-particle bound states of the light-cone string sigma model on AdS5 x S5 are compatible with Yangian symmetry. In case the invariance with respect to the centrally extended su(2|2) algebra is not sufficient to fully specify the scattering matrix, the requirement of Yangian symmetry provides an alternative to the Yang-Baxter equation and leads to a complete, up to an overall phase, determination of the S-matrix. We then compare the semi-classical limit of the bound state S-matrices with the universal classical r-matrix b…
