Semiclassical String Research Articles

QCD AND STRING THEORY

This talk begins with some history and basic facts about string theory and its connections with strong interactions. Comparisons of stacks of Dirichlet branes with curved backgrounds produced by them are used to motivate the AdS/CFT correspondence between superconformal gauge theory and string theory on a product of Anti-de Sitter space and a compact manifold. The ensuing duality between semi-classical spinning strings and long gauge theory operators is briefly reviewed. Strongly coupled thermal SYM theory is explored via a black hole in 5-dimensional AdS space, which leads to explicit results for its entropy and shear viscosity. A conjectured universal lower bound on the viscosity to entropy density ratio, and its possible relation to recent results from RHIC, are discussed. Finally, some available results on string duals of confining gauge theories are briefly reviewed.

Precision test of AdS/CFT in Lunin-Maldacena background

We obtain the solutions and explicitly calculate the energy for a class of two-spin semiclassical string states in the Lunin-Maldacena background. These configurations are \beta-deformed versions of the folded string solutions in AdS_{5}\times S^{5} background. They correspond to certain single trace operators in the \mathcal{N}=1 superconformal \beta deformation of \mathcal{N}=4 Yang-Mills. We calculate the one loop anomalous dimension for the dual single trace operator from the associated twisted spin chain with a general two-cut distribution of Bethe roots. Our results show a striking match between the two calculations. We demonstrate the natural identification of parameters on the two sides of the analysis, and explain the significance of the Virasoro constraint associated with the winding motion of semiclassical strings from the perspective of the spin chain solution.

Spinning strings in [formula omitted]: A worldsheet perspective

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 × 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 ( ∇ a ∇ a ).

Semiclassical strings, dipole deformations of Script N = 1 SYM and decoupling of KK modes

In this paper we investigate the recently found $\gamma$-deformed Maldacena-Nunez background by studying the behavior of different semiclassical string configurations. This background is conjectured to be dual to dipole deformations of $\N=1$ SYM. We compare our results to those in the pure Maldacena-Nunez background and show that the energies of our string configurations are higher than in the undeformed background. Thinking in the lines of (hep-th/0505100) we argue that this is an evidence for better decoupling of the Kaluza-Klein modes from the pure SYM theory excitations. Moreover we are able to find a limit of the background in which the string energy is independent of $\gamma$, these strings are interpreted as corresponding to pure gauge theory effects.

Semiclassical strings in electric and magnetic field deformedAdS5×S5spacetimes

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 $\mathrm{Ad}{\mathrm{S}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We adopt an ansatz to find the classical string solution which is rotating in the deformed ${S}^{5}$ with three angular momenta in the three rotation planes. The relations between the classical string energy and its angular momenta are found and results show that the external magnetic and electric fluxes will increase the string energy. Therefore, from the AdS/CFT point of view, the corrections of the anomalous dimensions of operators in the dual SYM theory will be positive. We also investigate the small fluctuations in these solutions and discuss the effects of magnetic and electric fields on the stability of these classical rotating string solutions. Finally, we find the possible solutions of string pulsating on the deformed spacetimes and show that the corrections to the anomalous dimensions of operators in the dual SYM theory are non-negative.

Semiclassical strings in electric and magnetic field deformed AdS{sub 5}xS{sup 5} spacetimes

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{sub 5}xS{sup 5}. We adopt an ansatz to find the classical string solution which is rotating in the deformed S{sup 5} with three angular momenta in the three rotation planes. The relations between the classical string energy and its angular momenta are found and results show that the external magnetic and electric fluxes will increase the string energy. Therefore, from the AdS/CFT point of view, the corrections of the anomalous dimensions of operators in the dual SYM theory will be positive. We also investigate the small fluctuations in these solutions and discuss the effects of magnetic and electric fields on the stability of these classical rotating string solutions. Finally, we find the possiblemore » solutions of string pulsating on the deformed spacetimes and show that the corrections to the anomalous dimensions of operators in the dual SYM theory are non-negative.« less

Formula omitted] corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

One way to relate semiclassical string states and dual gauge theory states is to show the equivalence between their low-energy effective 2d actions. The gauge theory effective action, which is represented by an effective Landau–Lifshitz (LL) model, was previously found to match the string theory world-sheet action up to the first two orders in the effective parameter λ ˜ = λ / J 2 , where λ is the 't Hooft coupling and J is the total R-charge. Here we address the question if quantizing the effective LL action reproduces the subleading 1 / J corrections to the spin chain energies as well as the quantum corrections to the string energies. We demonstrate that this is indeed the case provided one chooses an appropriate regularization of the effective LL theory. Expanding near the BPS vacuum, we show that the quantum LL action gives the same 1 / J corrections to energies of BMN states as found previously on the gauge theory and string theory sides. We also compute the subleading 1 / J 2 corrections and show that these too match with corrections computed from the Bethe ansatz. We also compare the results from the LL action with a more direct computation from the spin chain. We repeat the same computation for the β-deformed LL action and find that the quantum LL result is again equal to the 1 / J correction computed from the β-deformed Bethe ansatz equations. We also quantize the LL action near the rotating circular and folded string solutions, generalizing the known gauge/string results for 1 / J corrections to the classical energies. We emphasize the simplicity of this effective field theory approach as compared to the full quantum string computations.

Spinning Strings and Integrable Spin Chains in the AdS/CFT Correspondence.

In this introductory review we discuss dynamical tests of the AdS5 × S5 string/{mathcal N}=4 Super Yang-Mills duality. After a brief introduction to AdS/CFT, we argue that semiclassical string energies yield information on the quantum spectrum of the string in the limit of large angular momenta on the S5. The energies of the folded and circular spinning string solutions rotating on a S3 within the S5 are derived, which yield all-loop predictions for the dual gauge theory scaling dimensions. These follow from the eigenvalues of the dilatation operator of {mathcal N}=4 Super Yang-Mills in a minimal SU(2) subsector, and we display its reformulation in terms of a Heisenberg s = 1/2 spin chain along with the coordinate Bethe ansatz for its explicit diagonalization. In order to make contact to the spinning string energies, we then study the thermodynamic limit of the one-loop gauge theory Bethe equations and demonstrate the matching with the folded and closed string result at this loop order. Finally, the known gauge theory results at higher-loop orders are reviewed and the associated long-range spin chain Bethe ansatz is introduced, leading to an asymptotic all-loop conjecture for the gauge theory Bethe equations. This uncovers discrepancies at the three-loop order between gauge theory scaling dimensions and string theory energies and the implications of this are discussed. Along the way, we comment on further developments and generalizations of the subject and point to the relevant literature.

Generalization of the Lunin-Maldacena transformation on theAdS5×S5background

In this paper we consider a simple generalization of the method of Lunin and Maldacena for generating new string backgrounds based on TsT-transformations. We study multi-shift $Ts... sT$ transformations applied to backgrounds with at least two U(1) isometries. We prove that the string currents in any two backgrounds related by Ts...sT-transformations are equal. Applying this procedure to the $AdS_{5}\times S^{5}$, we find a new background and study some properties of the semiclassical strings.

Semi-classical probe strings on giant gravitons backgrounds

In the first part of this paper we study two $Z_2$ symmetries of the LLM metric, both of which exchange black and white regions. One of them which can be interpreted as the particle-hole symmetry is the symmetry of the whole supergravity solution while the second one is just the symmetry of the metric and changes the sign of the fivefrom flux. In the second part of the paper we use closed string probes and their semi-classical analysis to compare the two 1/2 BPS deformations of $AdS_5\times S^5$, the smooth LLM geometry which contains localized giant gravitons and the superstar case which is a solution with naked singularity corresponding to smeared giants. We discuss the realization of the $Z_2$ symmetry in the semi-classical closed string probes point of view.

Quantizing Open Spin Chains with Variable Length and Giant Gravitons in the Anti–de Sitter-Space/Conformal Field-Theory Correspondence

We study an XXX open spin chain with variable number of sites, where the variability is introduced only at the boundaries. This model arises naturally in the study of giant gravitons in the anti-de Sitter-space/conformal field-theory correspondence. We show how to quantize the spin chain by mapping its states to a bosonic lattice of finite length with sources and sinks of particles at the boundaries. Using coherent states, we show how the Hamiltonian for the bosonic lattice gives the correct description of semiclassical open strings ending on giant gravitons.

Gauge-string duality for (non)supersymmetric deformations of [formula omitted] super-Yang–Mills theory

We consider a nonsupersymmetric example of the AdS/CFT duality which generalizes the supersymmetric exactly marginal deformation constructed in hep-th/0502086. The string theory background we use was found in hep-th/0503201 from the AdS 5 × S 5 by a combination of T-dualities and shifts of angular coordinates. It depends on three real parameters γ i which determine the shape of the deformed 5-sphere. The dual gauge theory has the same field content as N = 4 SYM theory, but with scalar and Yukawa interactions “deformed” by γ i -dependent phases. The special case of equal γ i = γ corresponds to the N = 1 supersymmetric deformation. We compare the energies of semiclassical strings with three large angular momenta to the 1-loop anomalous dimensions of the corresponding gauge-theory scalar operators and find that they match as it was the case in the SU ( 3 ) sector of the standard AdS/CFT duality. In the supersymmetric case of equal γ i this extends the result of our previous work ( hep-th/0503192) from the 2-spin to the 3-spin sector. This extension turns out to be quite nontrivial. To match the corresponding low-energy effective “Landau–Lifshitz” actions on the string theory and the gauge theory sides one is to make a special choice of the spin chain Hamiltonian representing the 1-loop gauge theory dilatation operator. This choice is adapted to low-energy approximation, i.e., it allows one to capture the right vacuum states and the “macroscopic spin wave” sector of states of the spin chain in the continuum coherent state effective action.

Stringy sums and corrections to the quantum string Bethe ansatz

We analyze the effects of zeta-function regularization on the evaluation of quantum corrections to spinning strings. Previously, this method was applied in the sl(2) subsector and yielded agreement to third order in perturbation theory with the quantum string Bethe ansatz. In this note we discuss related sums and compare zeta-function regularization against exact evaluation of the sums, thereby showing that the zeta-function regularized expression misses out perturbative as well as non-perturbative terms. In particular, this may imply corrections to the proposed quantum string Bethe equations. This also explains the previously observed discrepancy between the semi-classical string and the quantum string Bethe ansatz in the regime of large winding number.

Integrability and higher loops in AdS/dCFT correspondence

We further study the correspondence between open semiclassical strings and long defect operators which is discussed in our previous work [Y. Susaki, Y. Takayama, K. Yoshida, hep-th/0410139]. We give an interpretation of the spontaneous symmetry breaking of SO(6)→SO(3)H×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 analysis. Both results agree at two-loop level but we find a three-loop discrepancy.

Gauge-string duality for superconformal deformations ofN= 4 Super Yang-Mills theory

We analyze in detail the relation between an exactly marginal deformation of N=4 SYM - the Leigh-Strassler or ``beta-deformation'' - and its string theory dual (recently constructed in hep-th/0502086) by comparing energies of semiclassical strings to anomalous dimensions of gauge-theory operators in the two-scalar sector. We stress the existence of integrable structures on the two sides of the duality. In particular, we argue that the integrability of strings in AdS_5 x S^5 implies the integrability of the deformed world sheet theory with real deformation parameter. We compare the fast string limit of the worldsheet action in the sector with two angular momenta with the continuum limit of the coherent state action of an anisotropic XXZ spin chain describing the one-loop anomalous dimensions of the corresponding operators and find a remarkable agreement for all values of the deformation parameter. We discuss some of the properties of the Bethe Ansatz for this spin chain, solve the Bethe equations for small number of excitations and comment on higher loop properties of the dilatation operator. With the goal of going beyond the leading order in the 't Hooft expansion we derive the analog of the Bethe equations on the string-theory side, and show that they coincide with the thermodynamic limit of the Bethe equations for the spin chain. We also compute the 1/J corrections to the anomalous dimensions of operators with large R-charge (corresponding to strings with angular momentum J) and match them to the 1-loop corrections to the fast string energies. Our results suggest that the impressive agreement between the gauge theory and semiclassical strings in AdS_5 x S^5 is part of a larger picture underlying the gauge/gravity duality.

Fluctuations and energy shifts in the Bethe ansatz

We study fluctuations and finite size corrections for the ferromagnetic thermodynamic limit in the Bethe ansatz for the Heisenberg XXX1/2 spin chain, which is the AdS/CFT dual of semiclassical spinning strings. For this system we derive the standard quantum mechanical formula which expresses the energy shift as a sum over fluctuation energies. As an example we apply our results to the simplest, one-cut solution of this system and derive its spectrum of fluctuations.

Semiclassical strings on confining backgrounds

AbstractThis paper discusses some results on semiclassical string configurations lying in the IR sector of supergravity backgrounds dual to confining gauge theories. On the gauge theory side, the string states we analyse correspond to Wilson loops, glueballs and KK‐hadrons. The 1‐loop correction to the classical energy is never subleading, but can be viewed as a coupling dependent rescaling of the dimensionful parameters of the theory.

The SU(2) sector in AdS/CFT

AbstractIn the large N limit of 𝒩 = 4 Super Yang‐Mills, the mixing under dilatations of the SU(2) sector, single trace operators composed of L complex scalar fields of two types, is closed to all orders in perturbation theory. By relying on the AdS/CFT correspondence, and by examining the currents for semiclassical strings, we present evidence which implies that there are small mixings that contradict the closure of the SU(2) sector in the strong coupling limit. These mixings first appear to second order in the λ/L2 expansion.

Open semiclassical strings and long defect operators in AdS/dCFT correspondence

We consider defect composite operators in a defect superconformal field theory obtained by inserting an ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$-brane in the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ 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 super Yang-Mills and string sides. We see that an $\mathrm{SO}(3{)}_{\mathrm{H}}$ pulsating string solution is included in the action and its energy completely agrees with the result calculated in a different method. In addition, we argue that our procedure would be applicable to other AdS-brane cases.

Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings

N=4 supersymmetric Yang-Mills operators carrying large charges are dual to semiclassical strings in AdS_5xS^5. The spectrum of anomalous dimensions of very large operators has been calculated solving the Bethe ansatz equations in the thermodynamic limit, and matched to energies of string solitons. We have considered finite size corrections to the Bethe equations, that should correspond to quantum effects on the string side.