The quest for extension of holographic correspondence to non-relativistic sectors naturally includes Schrödinger backgrounds and their field theory duals. In this paper we study the holography by probing the correspondence with pulsating strings. The case we consider is pulsating strings in five-dimensional Schrödinger space times five-torus $T^{1,1}$, which has as field theory dual a dipole CFT. First we find particular pulsating string solutions and then semi-classically quantize the theory. We obtain the wave function of the problem and thoroughly study the corrections to the energy, which by duality are supposed to give anomalous dimensions of certain operators in the dipole CFT.
According to AdS/CFT duality semi-classical strings in the Schrödinger spacetime is conjectured to be a holographic dual to dipole CFT. In this paper we consider pulsating strings in five-dimensional Schrödinger space times five-sphere. We have found classical string solutions pulsating entirely in the Schrödinger part of the background. We quantize the theory semi-classically and obtain the wave function of the problem. We have found the corrections to the energy, which by duality are supposed to give anomalous dimensions of certain operators in the dipole CFT.
We explore nonrelativistic (NR) pulsating string configurations over torsion Newton-Cartan (TNC) geometry having topology $ R \times S^2 $ and check the corresponding analytic integrability criteria following Kovacic's algorithm. In the first part we consider pulsating strings propagating over TNC geometry whose world-sheet theory is described by relativistic CFTs. We compute conserved charges associated with the $ 2D $ sigma model and show that the classical phase space corresponding to these NR pulsating string configurations is Liouvillian integrable. Finally, we consider nonrelativisitc sc…
We find the anomalous dimension and the conserved charges of an R-charged string pulsating on AdS_5. The analysis is performed both on the gauge and string side, where we find agreement at the one-loop level. Furthermore, the solution is shown to be related by analytic continuation to a string which is pulsating on S^5, thus providing an example of the close relationship between the respective isometry groups.
We study the class of pulsating strings on AdS_4 x CP3. Using a generalized ansatz for pulsating string configurations we find new solutions of this class. Further we quasi-classically quantize the theory and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual N=6 Chern-Simons theory.
We derive the energy of pulsating strings as a function of adiabatic invariant oscillation number, which oscillates in $S^2_{\varkappa}$. We find similar solutions for the strings oscillating in deformed $AdS_3$. Furthermore, we generalize the result to the oscillating strings in anti-de Sitter space in the presence of extra angular momentum in $(AdS_3 \times S^1)_\varkappa$.
We study the class of pulsating strings on $AdS_5\times T^{1,1}$. Using a generalized ansatz for pulsating string configurations we find new solutions of this class. Further we quasi-classically quantize the theory and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual ${\cal N}=1$ superconformal field theory.
We study the class of pulsating strings in AdS_5 x T^{1,1}. Using a generalized ansatz for pulsating string configurations we find new solutions of this class. Further we semiclassically quantize the theory and obtain the first correction to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators in the dual N=1 superconformal gauge field theory.
We consider pulsating strings in Lunin-Maldacena backgrounds, specifically in deformed Minkowski spacetime and deformed AdS_5xS^5. We find the relation between the energy and the oscillation number of the pulsating string when the deformation is small. Since the oscillation number is an adiabatic invariant it can be used to explore the regime of highly excited string states. We then quantize the string and look for such a sector. For the deformed Minkowski background we find a precise match with the classical results if the oscillation number is quantized as an even number. For the deformed Ad…
We derive and analyse, analytically and numerically, the equations for perturbations around the pulsating two-spin string soliton in AdS(5) x S(5).We show that the pulsation in S(5) indeed improves the stability properties of the two-spin string soliton in AdS(5), in the sense that the more pulsation we have, the higher spin we can allow and still have stability.
We derive the energy of pulsating string, as function of oscillation number and angular momenta, which oscillates in AdS3 with an extra angular momentum along S1. We find similar solutions for the strings oscillating in S3 in addition to extra angular momentum. Further we generalize the result of the oscillating strings in Anti de-Sitter space in the presence of both spin and angular momentum in AdS5 x S1.
We discuss semiclassical quantization of circular pulsating strings in AdS3 x S3 background with and without the Neveu-Schwarz- Neveu-Schwarz (NS-NS) flux. We find the equations of motion corresponding to the quadratic action in bosonic sector in terms of scalar quantities and invariants of the geometry. The general equations for studying physical perturbations along the string in an arbitrary curved spacetime are written down using covariant formalism. We discuss the stability of these string configurations by studying the solutions of the linearized perturbed equations of motion.
We propose a state from the two-dimensional conformal field theory on the orbifold $(T^4)^N/S(N)$ as a dual description for a pulsating string moving in $AdS_3$. We show that, up to first order in the deforming parameter, the energy in both descriptions has the same dependence on the mode number, but with a non-trivial function of the coupling.
In this paper we consider pulsating strings in warped $AdS_6\times S^4$ background, which is a vacuum solution of massive type {\bf IIA} superstring. The case of rotating strings in this background was considered in hep-th/0402202 and it was found that the results significantly differs from those considered in $AdS_5\times S^5$. Motivated by this results we study pulsating strings in the warped spherical part of the type {\bf IIA} geometry and compare the results with those obtained in hep-th/0209047, hep-th/0310188 and hep-th/0404012. We conclude with comments on our solutions and the obtaine…
We point out the existence of some simple string solitons in AdS_5 x S^5, which at the same time are spinning in AdS_5 and pulsating in S^5, or vice-versa. This introduces an additional arbitrary constant into the scaling relations between energy and spin or R-charge. The arbitrary constant is not an angular momentum, but can be related to the amplitude of the pulsation. We discuss the solutions in detail and consider the scaling relations. Pulsating multi spin or multi R-charge solutions can also be constructed.
One of the most important problems in any conformal field theory is the calculation of three-point functions of primary operators. In this paper we provide explicit examples of correlators with two scalar operators in $\,{\cal N}=4$ super-Yang--Mills theory at large $N$, corresponding to pulsating semiclassical strings in AdS$_5\times S^5$, and an operator with small quantum numbers at strong coupling.
We study a general class of spinning pulsating strings in $(AdS_5 \times S^5)_{\varkappa}$ background. For these family of solitons, we examine the scaling relation between the energy, spin or angular momentum. We verify that in $\varkappa \rightarrow 0 $ limit these relations reduce to the undeformed $AdS_5 \times S^5$ case. We further study an example of a string which is spinning in the $\varkappa$-deformed AdS$_5$ and S$^5$ simultaneously and find out the scaling relation among various conserved charges.
In the present paper, which is a sequel to arXiv:1001:4018, we compute the one-loop correction to the energy of pulsating string solutions in AdS_5 x S^5. We show that, as for rigid spinning string elliptic solutions, the fluctuation operators for pulsating solutions can be also put into the single-gap Lame' form. A novel aspect of pulsating solutions is that the one-loop correction to their energy is expressed in terms of the stability angles of the quadratic fluctuation operators. We explicitly study the "short string" limit of the corresponding one-loop energies, demonstrating a certain uni…
In this paper we consider new solutions for pulsating strings. For this purpose we use tha idea of the generalized ansatz for folded and circular strings in hep-th/0311004. We find the solutions to the resulting Neumann-Rosochatius integrable system and the corrections to the energy. To do that we use the approach developed by Minahan in hep-th/0209047 and find that the corrections are quite different from those obtained in that paper and hep-th/0310188. We conclude with comments on our solutions and obtained corrections to the energy, expanded to the leading order in lambda.
Circular strings pulsating in $AdS_3 \times S^3 \times T^4$ with mixed R-R and NS-NS three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in $AdS_{3}$ with pure NS-NS three-form flux, where the action reduces to the $SL(2,\mathbb{R})$ WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow b…
We study the class of pulsating strings in AdS_5 x Y^{p,q} and AdS_5 x L^{p,q,r}. Using a generalized ansatz for pulsating string configurations, we find new solutions for this class in terms of Heun functions, and derive the particular case of AdS_5 x T^{1,1}, which was analyzed in arXiv:1006.1539 [hep-th]. Unfortunately, Heun functions are still little studied, and we are not able to quantize the theory quasi-classically and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual…
In this article the quantum fluctuation of a rigid and static string is reported to be identical to a free quantum particle. Solutions similar to this static string have already been found in the semi-classical quantizaton of pulsating strings, and our results show that the semi-classical quantization of pulsating strings is, in some cases, a perturbation of static strings. We also interpret the energy of the static string as a lower bound for the pulsating string and speculate about a description of quantum mechanics in terms of semi-classical string theory.
We study multiwrapped circular string pulsating in the radial direction of AdS black hole. We compute the energy of this string as a function of a large quantum number n. One then could associate it with energy and a quantum number of states in the dual finite temperature {\cal N}=4 SYM theory as well as three dimensional pure gauge theory. We observe that the n dependence of the energy has a universal form. We have also considered pulsating string in the background of the near-extremal D4-brane solution. Circular pulsating membrane in M-theory on AdS_7\times S^4 has also been studied.
$SL(2,\mathbb{Z})$ invariant action for probe $(m,n)$ string in $AdS_3\times S^3\times T^4$ with mixed three-form fluxes can be described by an integrable deformation of an one-dimensional Neumann-Rosochatius (NR) system. We present the deformed features of the integrable model and study general class of rotating and pulsating solutions by solving the integrable equations of motion. For the rotating string, the explicit solutions can be expressed in terms of elliptic functions. We make use of the integrals of motion to find out the scaling relation among conserved charges for the particular ca…
We construct a family of closed string solutions with kinks in a subspace of AdS_5 x S^5 and study their properties. In certain limits these solutions become folded pulsating strings, although in general they are made of multiple pulsating rectangles. One unusual feature of these solutions is that their monodromy matrices are trivial, leading to vanishing quasi-momenta. Exact Backlund transformations of these solutions are found, again giving vanishing higher conserved charges. We also consider the fluctuation modes around these solutions as well as the semiclassical splitting of these strings…
We study semiclassical string solutions on the 1/2 BPS geometry of type IIB string theory characterized by concentric rings on the boundary plane. We consider both folded rotating strings carrying nonzero R-charge and circular pulsating strings. We find that unlike rotating strings, as far as circular pulsating strings are concerned, the dynamics remains qualitatively unchanged when the concentric rings replace AdS_5\times S^5. Using the gravity dual we have also studied the Wilson loop of the corresponding gauge theory. The result is qualitatively the same as that in AdS_5\times S^5 in the gl…
We study rigidly rotating and pulsating $(m,n)$ strings in $AdS_3 \times S^3$ with mixed three form flux. The $AdS_3 \times S^3$ background with mixed three form flux is obtained in the near horizon limit of $SL(2,Z)$-transformed solution, corresponding to the bound state of NS5-branes and fundamental strings. We study the probe $(m,n)-$string in this background by solving the manifest $SL(2,Z)-$covariant form of the action. We find out the dyonic giant magnon and single spike solutions corresponding to the equations of motion of a probe string in this background and find out various relations…
Neumann-Rosochatius system is a well known one dimensional integrable system. We study the rotating and pulsating string in $AdS_4 \times \mathbb{CP}^3$ with a $B_{\rm{NS}}$ holonomy turned on over $\mathbb{CP}^1 \subset \mathbb{CP}^3$, or the so called Aharony-Bergman-Jafferis (ABJ) background. We observe that the string equations of motion in both cases are integrable and the Lagrangians reduce to a form similar to that of deformed Neuman-Rosochatius system. We find out the scaling relations among various conserved charges and comment on the finite size effect for the dyonic giant magnons on…
We further study the correspondence between open semiclassical strings and long defect operators which is discussed in our previous work [hep-th/0410139]. We give an interpretation of the spontaneous symmetry breaking of SO(6)-> SO(3)_H x SO(3)_V from the viewpoint of the Riemann surface by following the argument of Minahan. Then we use the concrete form of the resolvent for a single cut solution and compute the anomalous dimension of operators dual to an open pulsating string at three-loop level. In the string side we obtain the energy of the open pulsating string solution by semiclassical an…
We consider circular strings rotating with equal spins S_1=S_2=S in two orthogonal planes in AdS_5 and suggest that they may be dual to "long" gauge theory operators built out of self-dual components of gauge field strength. As was found in hep-th/0404187, the one-loop anomalous dimensions of the such gauge-theory operators are described by an anti-ferromagnetic XXX_1 spin chain and scale linearly with length L>>1. We find that in the case of rigid rotating string both the classical energy E_0 and the 1-loop string correction E_1 depend linearly on the spin S (within the stability region of th…
Puff Field Theory (PFT) is an example of a non-local field theory which arises from a novel embedding of D-branes in Melvin universe. We study several rotating and pulsating string solutions of the F-string equations of motion in the supergravity dual of the PFT. Further, we find a PP-wave geometry from this nonlocal spacetime by applying a Penrose limit and comment on its similarity with the maximally supersymmetric PP-wave background.
We present a detailed study of the pulsating string solutions in AdS3 x S3 supported by both RR and NS-NS fluxes. This background has recently been proved to be integrable. We find the dispersion relation between the energy, oscillation number and other conserved charges when the NS-NS flux turned on is small. We further discuss the fate of the string solutions in pure RR and NS-NS cases.
Timelike T-duality of string theory appears as a symmetry of time evolution in string field theory, exchanging evolution through times t and 1/t, and exchanging boundary states with backgrounds. This is demonstrated by constructing the string field Schrodinger functional, the generator of time evolution, based on Feynman diagram arguments and in analogy with quantum field theory. There, the functional can be described using only properties of first quantised particles on a timelike orbifold. Using new sewing rules applicable to both open and closed strings we generalise this approach to bosoni…
In this paper, we study the semi-classical strings in AdS_4*CP^3 spacetime. We construct various kinds of string solutions, including the point-like, circular, folded and pulsating strings. For the circular and folded strings, we figure out their field theory dual operators. In particular, we discuss the anomalous dimensions of the corresponding operators from decoupled SU(2)*SU(2) spin chain. We find that in the large angular momentum limit, the field theory and string theory results are in very good agreement up to an interpolating function of coupling constant.
We study rigidly rotating and pulsating strings in the background of a 1+1 dimensional intersection of two orthogonal stacks of fivebranes in type IIB string theory by using the Neumann-Rosochatius (NR) model. Starting with the Polyakov action of the probe fundamental string we show that a generalised ansatz reduce the system into the one dimensional NR model in the presence of flux. The integrable construction of the model is exploited to analyze the rotating and oscillating string solution. We render the large $J$ BMN-type expansion for the energy of rotating string states while the same for…
We construct semi-classical string solutions of the Schrödinger $Sch_5 \times S^5$ spacetime, which is conjectured to be the gravity dual of a non-local dipole-deformed CFT. They are the counterparts of the giant magnon and spiky string solutions of the undeformed $AdS_5 \times S^5$ to which they flow when the deformation parameter is turned off. They live in an $S^3$ subspace of the five-sphere along the directions of which the $B$-field has non-zero components having also extent in the $Sch_5$ part of the metric. Finally, we speculate on the form of the dual field theory operators.
We study string theory on the pp-wave geometry obtained by taking the Penrose limit around a certain null geodesic of the non-supersymmetric Schrodinger background. We solve for the spectrum of bosonic excitations and find compelling agreement with the dispersion relation of the giant magnons in the Schrodinger background obtained previously in arXiv:1712.03091. Inspired by the pp-wave spectrum we conjecture an exact in the t'Hooft coupling dispersion relation for the magnons in the original Schrodinger background. We show that the pp-wave background admits exactly 16 Killing spinors. We use t…
We find a new class of spiky solutions for closed strings in flat, $AdS_3\subset AdS_5$ and $R\times S^2(\subset S^5)$ backgrounds. In the flat case the new solutions turn out to be T-dual configurations of spiky strings found by Kruczenski hep-th/0410226. In the case of solutions living in $AdS$, we make a semi classical analysis by taking the large angular momentum limit. The anomalous dimension for these dual spikes is similar to that for rotating and pulsating circular strings in AdS with angular momentum playing the role of the level number. This replaces the well known logarithmic depend…
The quest for extension of holographic correspondence to the case of finite temperature naturally includes Kerr-AdS black holes and their field theory duals. We probe the five-dimensional Kerr-AdS space time by pulsating strings. First we find particular pulsating string solutions and then semi-classically quantize the theory. For the string with large values of energy, we use the Bohr-Sommerfeld analysis to find the energy of the string as a function of a large quantum number. We obtain the wave function of the problem and thoroughly study the corrections to the energy, which according to the…
We study different features of 3D non-relativistic CFT using gravity description. As the corresponding gravity solution can be embedded into the type IIB string theory, we study semi-classical closed/open strings in this background. In particular we consider folded rotating and circular pulsating closed strings where we find the anomalous dimension of the dual operators as a function of their quantum numbers. We also consider moving open strings in this background which can be used to compute the drag force. In particular we find that for slowly moving particles, the energy is lost exponential…
We investigate the question of possible integrability of classical string motion in curved p-brane backgrounds. For example, the D3-brane metric interpolates between the flat and the AdS_5 x S^5 regions in which string propagation is integrable. We find that while the point-like string (geodesic) equations are integrable, the equations describing an extended string in the complete D3-brane geometry are not. The same conclusion is reached for similar brane intersection backgrounds interpolating between flat space and AdS_k x S^k. We consider, in particular, the case of the NS 5-brane-fundamenta…
We treat RR flux backgrounds of type II string theory in the framework of closed superstring field theory based on the NSR formalism, focusing on two examples: (1) the pp-wave background supported by 5-form flux, and (2) $AdS_3\times S^3\times M_4$ supported by mixed 3-form fluxes. In both cases, we analyze the classical string field solution perturbatively, and compute the correction to the dispersion relation of string states to quadratic order in the RR flux. In the first example, our result is in a delicate way consistent with that obtained from lightcone quantization of the Green-Schwarz …
We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schroedinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N=2 gauge systems in four di…
It is observed that the (intersecting) branes of M/string theory, which are known to give AdS geometry (directly or upto a conformal transformation) in the near horizon limit, do also lead to Schr\" odinger/Lifshitz dual space-times (without or with hyperscaling violation) upon using appropriate solution generating transformation and dimensional reduction. We show that the dynamical exponents of the Schr\" odinger and the Lifshitz space-times obtained in this way always add upto 2. We illustrate this by several examples, including M2-, M5-branes of M-theory and D$(p+1)$-branes ($p\neq 4$, sinc…
We consider defect composite operators in a defect superconformal field theory obtained by inserting an AdS_4 x S^2-brane in the AdS_5 x S^5 background. The one-loop dilatation operator for the scalar sector is represented by an integrable open spin chain. We give a description to construct coherent states for the open spin chain. Then, by evaluating the expectation value of the Hamiltonian with the coherent states in a long operator limit, a Landau-Lifshitz type of sigma model action is obtained. This action is also derived from the string action and hence we find a complete agreement in both…
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 explore further the Hamiltonian formulation of Yang-Mills theory in 2+1 dimensions in terms of gauge-invariant matrix variables. Coupling to scalar matter fields is discussed in terms of gauge-invariant fields. We analyze how the screening of adjoint (and other screenable) representations can arise in this formalism. A Schrodinger equation is then derived for the gluelump states which are the daughter states when an adjoint string breaks. A variational solution of this Schrodinger equation leads to an analytic estimate of the string-breaking energy which is within 8.8% of the latest lattice…
In this paper we will consider the minisuperspace approach to S-branes dynamics in the Schrödinger picture description. Time-evolution of vacuum wave functional for quantum field theory on S-brane is studied. Open string pair production is calculated. The analysis of density matrix for mixed states is also performed.
We first apply the transformation of mixing azimuthal and internal coordinate or mixing time and internal coordinate to the 11D M-theory with a stack N M2-branes to find the spacetime of a stack of N D2-branes with magnetic or electric flux in 10 D IIA string theory, after the Kaluza-Klein reduction. We then perform the T duality to the spacetime to find the background of a stack of N D3-branes with magnetic or electric flux. In the near-horizon limit the background becomes the magnetic or electric field deformed $AdS_5 \times S^5$. We adopt an ansatz to find the classical string solution whic…
We study a particular class of type II string vacua which become Schrödinger like spacetime in the IR region but are conformally AdS in asymptotic UV region. These solutions are found to possess some unique properties such as the presence of a spacetime `screen'. This Schrödinger (spacetime) screen is however very different from a black-hole horizon. It requires the presence of finite chemical potential and a negative charge density in the Schrödinger CFT. We find that these vacua give rise to reduced entanglement entropy as compared to Lifshitz-AdS counterpart, perhaps due to the screening ef…
We consider a semiclassical multiwrapped circular string pulsating on S_5, whose center of mass has angular momentum J on an S_3 subspace. Using the AdS/CFT correspondence we argue that the one-loop anomalous dimension of the dual operator is a simple rational function of J/L, where J is the R-charge and L is the bare dimension of the operator. We then reproduce this result directly from a super Yang-Mills computation, where we make use of the integrability of the one-loop system to set up an integral equation that we solve. We then verify the results of Frolov and Tseytlin for circular rotati…
We show that the vacuum state functional for both open and closed string field theories can be constructed from the vacuum expectation values it must generate. The method also applies to quantum field theory and as an application we give a diagrammatic description of the equivalance between Schrodinger and covariant repreresentations of field theory.
We consider black probes of Anti-de Sitter and Schrödinger spacetimes embedded in string theory and M-theory and construct perturbatively new black hole geometries. We begin by reviewing black string configurations in Anti-de Sitter dual to finite temperature Wilson loops in the deconfined phase of the gauge theory and generalise the construction to the confined phase. We then consider black strings in thermal Schrödinger, obtained via a null Melvin twist of the extremal D3-brane, and construct three distinct types of black string configurations with spacelike as well as lightlike separated bo…
We present a construction of string--localized covariant free quantum fields for a large class of irreducible (ray) representations of the Poincare group. Among these are the representations of mass zero and infinite spin, which are known to be incompatible with point-like localized fields. (Based on joint work with B.Schroer and J.Yngvason [13].)
