String Sigma Model Research Articles

Infinite spin limit of semiclassical string states

Motivated by recent works of Hofman and Maldacena and Dorey we consider a special infinite spin limit of semiclassical spinning string states in AdS5 x S5. We discuss examples of known folded and circular 2-spin string solutions and demonstrate explicitly that the 1-loop superstring correction to the classical expression for the energy vanishes in the limit when one of the spins is much larger that the other. We also give a general discussion of this limit at the level of integral equations describing finite gap solutions of the string sigma model and argue that the corresponding asymptotic form of the string and gauge Bethe equations is the same.

Spin chain with a magnetic field and spinning string in a magnetic field background

We analyze the fast-moving string in the magnetic Melvin field background and find that the associated effective Lagrangian of string sigma model describes the spin chain model with external magnetic field. The spin vector in the spin chain has been properly deformed and is living on the deformed two-sphere or deformed two-dimensional hyperboloid, depending on the direction around which the string is spinning. We describe in detail the characters of spin deformation and, in particular, see that this is a general property for a string moving in a class of deformed background.

Green-Schwarz Strings in TsT-transformed backgrounds

We consider classical strings propagating in a background generated by a sequence of TsT transformations. We describe a general procedure to derive the Green-Schwarz action for strings. We show that the U(1) isometry variables of the TsT-transformed background are related to the isometry variables of the initial background in a universal way independent of the details of the background. This allows us to prove that strings in the TsT-transformed background are described by the Green-Schwarz action for strings in the initial background subject to twisted boundary conditions. Our construction implies that a TsT transformation preserves integrability properties of the string sigma model. We discuss in detail type IIB strings propagating in the \g_i-deformed AdS_5 x S^5 space-time, find the twisted boundary conditions for bosons and fermions, and use them to write down an explicit expression for the monodromy matrix. We also discuss string zero modes whose dynamics is governed by a fermionicgeneralization of the integrable Neumann model.

Logarithmic scaling in gauge/string correspondence

We study anomalous dimensions of (super)conformal Wilson operators at weak and strong coupling making use of the integrability symmetry on both sides of the gauge/string correspondence and elucidate the origin of their single-logarithmic behavior for long operators/strings in the limit of large Lorentz spin. On the gauge theory side, we apply the method of the Baxter Q-operator to identify different scaling regimes in the anomalous dimensions in integrable sectors of (supersymmetric) Yang–Mills theory to one-loop order and determine the values of the Lorentz spin at which the logarithmic scaling sets in. We demonstrate that the conventional semiclassical approach based on the analysis of the distribution of Bethe roots breaks down in this domain. We work out an asymptotic expression for the anomalous dimensions which is valid throughout the entire region of variation of the Lorentz spin. On the string theory side, the logarithmic scaling occurs when two most distant points of the folded spinning string approach the boundary of the AdS space. In terms of the spectral curve for the classical string sigma model, the same configuration is described by an elliptic curve with two branching points approaching values determined by the square root of the 't Hooft coupling constant. As a result, the anomalous dimensions cease to obey the BMN scaling and scale logarithmically with the Lorentz spin.

A universality test of the quantum string Bethe ansatz

We show that the quantum corrected string Bethe ansatz passes an important universality test by demonstrating that it correctly incorporates the non-analytical terms in the string sigma model one-loop correction for rational three-spin strings with two out of the three spins identical. Subsequently, we use the quantum corrected string Bethe ansatz to predict the exact form of the non-analytic terms for the generic rational three-spin string.

γi-deformed Lax pair for rotating strings in the fast motion limit

A 3-parameter generalization of the Lunin-Maldacena background has recently been constructed by Frolov. This gamma_i-deformed background is non-supersymmetric. We consider strings in this gamma_i-deformed R \times S^5 background rotating in three orthogonal planes (the 3-spin sector) in a fast motion limit, in which the total angular momentum J is assumed to be large. We show that there exists a consistent transformation which takes the undeformed equations of motion into the gamma_i-deformed equations of motion. This transformation is used to construct a Lax pair for the bosonic part of the gamma_i-deformed theory in the fast motion limit. This implies the integrability of the bosonic part of the gamma_i-deformed string sigma model in the fast motion limit.

Linking Bäcklund and monodromy charges for strings on AdS5× S5

We find an explicit relation between the two known ways of generating an infinite set of local conserved charges for the string sigma model on AdS_5 x S^5: the Backlund and monodromy approaches. We start by constructing the two-parameter family of Backlund transformations for the string with an arbitrary world-sheet metric. We then show that only for a special value of one of the parameters the solutions generated by this transformation are compatible with the Virasoro constraints. By solving the Backlund equations in a non-perturbative fashion, we finally show that the generating functional of the Backlund conservation laws is equal to a certain sum of the quasi-momenta. The positions of the quasi-momenta in the complex spectral plane are uniquely determined by the real parameter of the Backlund transform.

Reconnection of colliding cosmic strings

For vortex strings in the Abelian Higgs model and D-strings in superstring theory, both of which can be regarded as cosmic strings, we give analytical study of reconnection (recombination, inter-commutation) when they collide, by using effective field theories on the strings. First, for the vortex strings, via a string sigma model, we verify analytically that the reconnection is classically inevitable for small collision velocity and small relative angle. Evolution of the shape of the reconnected strings provides an upper bound on the collision velocity in order for the reconnection to occur. These analytical results are in agreement with previous numerical results. On the other hand, reconnection of the D-strings is not classical but probabilistic. We show that a quantum calculation of the reconnection probability using a D-string action reproduces the nonperturbative nature of the worldsheet results by Jackson, Jones and Polchinski. The difference on the reconnection -- classically inevitable for the vortex strings while quantum mechanical for the D-strings -- is suggested to originate from the difference between the effective field theories on the strings.

The factorized S-matrix of CFT/AdS

We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.

Rational three-spin string duals and non-anomalous finite size effects

We determine by a one line computation the one-loop conformal dimension and the associated non-anomalous finite size correction for all operators dual to spinning strings of rational type having three angular momenta (J_1,J_2,J_3) on S^5. Finite size corrections are conjectured to encode information about string sigma model loop corrections to the spectrum of type IIB superstrings on AdS_5xS^5. We compare our result to the zero-mode contribution to the leading quantum string correction derived for the stable three-spin string with two out of the three spin labels identical and observe agreement. As a side result we clarify the relation between the Bethe root description of three-spin strings of the type (J,J',J') with respectively J>J' and J<J'.

Semiclassical relativistic strings inS5and long coherent operators inN=4 SYM theory

We consider the low energy effective action corresponding to the 1-loop, planar, dilatation operator in the scalar sector of N=4 SU(N) SYM theory. For a general class of non-holomorphic ``long'' operators, of bare dimension L>>1, it is a sigma model action with 8-dimensional target space and agrees with a limit of the phase-space string sigma model action describing generic fast-moving strings in the S^5 part of AdS_5 x S^5. The limit of the string action is taken in a way that allows for a systematic expansion to higher orders in the effective coupling $\lambda/L^2$. This extends previous work on rigid rotating strings in S^5 (dual to operators in the SU(3) sector of the dilatation operator) to the case when string oscillations or pulsations in S^5 are allowed. We establish a map between the profile of the leading order string solution and the structure of the corresponding coherent, ``locally BPS'', SYM scalar operator. As an application, we explicitly determine the form of the non-holomorphic operators dual to the pulsating strings. Using action--angle variables, we also directly compute the energy of pulsating solutions, simplifying previous treatments.

Large spin limit of AdS5× S5 string theory and low energy expansion of ferromagnetic spin chains

By considering AdS 5× S 5 string states with large S 5 angular momenta one can provide non-trivial quantitative checks of the AdS/CFT duality. A string rotating in S 5 with two angular momenta J 1, J 2 is dual to an operator in N=4 SYM theory whose conformal dimension can be computed by diagonalizing a (generalization of) spin 1/2 Heisenberg chain Hamiltonian. It was recently argued and verified to lowest order in a large J= J 1+ J 2 expansion, that the Heisenberg chain can be described using a non-relativistic low energy effective 2d action for a unit vector field n i which exactly matches the corresponding large J limit of the classical AdS 5× S 5 string action. In this paper we show that this agreement extends to the next order and develop a systematic procedure for computing higher orders in such large angular momentum expansion. This involves several non-trivial steps. On the string side, we need to choose a special gauge with a non-diagonal world-sheet metric which insures that the angular momentum is uniformly distributed along the string, as indeed is the case on the spin chain side. We need also to implement an order by order redefinition of the field n i to get an action linear in the time derivative. On the spin chain side, it turns out to be crucial to include the effects of integrating out short wave-length modes. In this way we gain a better understanding of how (a subsector of) the string sigma model emerges from the dual gauge theory, allowing us to demonstrate the duality beyond comparing particular examples of states with large J.

Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations

We consider AdS_5 x S^5 string states with several large angular momenta along AdS_5 and S^5 directions which are dual to single-trace Super-Yang-Mills (SYM) operators built out of chiral combinations of scalars and covariant derivatives. In particular, we focus on the SU(3) sector (with three spins in S^5) and the SL(2) sector (with one spin in AdS_5 and one in S^5), generalizing recent work hep-th/0311203 and hep-th/0403120 on the SU(2) sector with two spins in S^5. We show that, in the large spin limit and at the leading order in the effective coupling expansion, the string sigma model equations of motion reduce to matrix Landau-Lifshitz equations. We then demonstrate that the coherent-state expectation value of the one-loop SYM dilatation operator restricted to the corresponding sector of single trace operators is also effectively described by the same equations. This implies a universal leading order equivalence between string energies and SYM anomalous dimensions, as well as a matching of integrable structures. We also discuss the more general 5-spin sector and comment on SO(6) states dual to non-chiral scalar operators.

Higher Conserved Charges and Integrability for Spinning Strings inAdS5 S5

We demonstrate the existence of an infinite number of local commuting charges for classical solutions of the string sigma model on AdS_5 x S^5 associated with a certain circular three-spin solution spinning with large angular momenta in three orthogonal directions on the five-sphere. Using the AdS/CFT correspondence we find agreement to one-loop with the tower of conserved higher charges in planar N=4 super Yang-Mills theory associated with the dual composite single-trace operator in the highest weight representation (J_1,J_2,J_2) of SO(6). The agreement can be explained by the presence of integrability on both sides of the duality.

Matching Higher Conserved Charges for Strings and Spins

We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in = 4 gauge theory and energies of spinning strings on AdS5 × S5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Bäcklund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.

Plane waves with weak singularities

We study a class of time dependent solutions of the vacuum Einstein equations which are plane waves with weak null singularities. This singularity is weak in the sense that though the tidal forces diverge at the singularity, the rate of divergence is such that the distortion suffered by a freely falling observer remains finite. Among such weak singular plane waves there is a sub-class which do not exhibit large back reaction in the presence of test scalar probes. String propagation in these backgrounds is smooth and there is a natural way to continue the metric beyond the singularity. This continued metric admits string propagation without the string becoming infinitely excited. We construct a one parameter family of smooth metrics which are at a finite distance in the space of metrics from the extended metric and a well defined operator in the string sigma model which resolves the singularity.

Penrose limit ofAdS3×S3×S3×S1and its associated σ-model

We study the Penrose limit of a supersymmetric IIB background, with non-trivial NS 3-form field strength, obtaining a solution with the smallest number of supercharges (i.e. 16) allowed; we write down explicitly the superalgebra of the theory, build the supersymmetric associated IIB string sigma model and make conjectures on the dual gauge theory.

GAUGE FIELDS AND SPACE-TIME

In this article I attempt to collect some ideas, opinions and formulae which may be useful in solving the problem of gauge/string/space-time correspondence. This includes the validity of D-brane representation, counting of gauge-invariant words, relations between the null states and the Yang-Mills equations and the discussion of the strong coupling limit of the string sigma model. The article is based on the talk given at the "Odyssey 2001" conference.

Linear Sigma Models for Open Strings

We formulate and study a class of massive N=2 supersymmetric gauge field theories coupled to boundary degrees of freedom on the strip. For some values of the parameters, the infrared limits of these theories can be interpreted as open string sigma models describing D-branes in large-radius Calabi-Yau compactifications. For other values of the parameters, these theories flow to CFTs describing branes in more exotic, non-geometric phases of the Calabi-Yau moduli space such as the Landau-Ginzburg orbifold phase. Some simple properties of the branes (like large radius monodromies and spectra of worldvolume excitations) can be computed in our model. We also provide simple worldsheet models of the transitions which occur at loci of marginal stability, and of Higgs-Coulomb transitions.

On off-shell structure of open string sigma model

We analyze several problems related to off-shell structure of open string sigma model by using a combination of derivative expansion and expansion in powers of the fields. According to the sigma model approach to bosonic open string theory, the tachyon effective action $S(T)$ coincides with the renormalized partition function $Z(T)$ of sigma model on a disk, up to a term vanishing on shell. On the other hand, $Z(T)$ is a generating functional of perturbative open string scattering amplitudes. If $S(T) = Z(T)$, then there should be no contribution of exchange diagrams to string amplitudes computed using $S(T)$. We compute the cubic term in the effective action, and show that it vanishes if some but not all external legs are on shell, and, therefore, any exchange diagram involving the cubic term vanishes too. Then, we discuss a problem of turning on nonrenormalizable boundary interactions, corresponding to massive string modes. We compute the quadratic term for a symmetric tensor field, and show that despite nonrenormalizability of the model one can consistently remove all divergent terms, and obtain a quadratic action reproducing the on-shell condition for the field. We also briefly discuss fermionic (NS) sigma model, compute the tachyon quadratic term, and show that it reproduces the correct tachyon mass. We note that turning on a massive symmetric tensor field leads to the appearance of a term linear in it, which can be removed by adding a higher-derivative term to the boundary of the disc.