A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how the general solution can be obtained. Then we demonstrate how the wave-like solutions of the general Toda systems can be simply derived. In the dilaton gravity theory this solutions describe nonlinear waves coupled to gravity. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to apply the idea of the separation of variables to non-integrable theories.
General properties of a class of two-dimensional dilaton gravity (DG) theories with multi-exponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponential model. From the constraint it follows that in DG theories the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. We also show how the wave-like solutions of t…
We consider a $D$-dimensional cosmological model with a dilaton field and two $(D-d-1)$-form field strengths which have nonvanishing fluxes in extra dimensions. Exact solutions for the model with a certain set of couplings are obtained by separation of three variables. Some of the solutions describe accelerating expansion of the $d$-dimensional space. Quantum cosmological aspects of the model are also briefly mentioned.
With the help of the Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field, we find an integrable subsystem of SU(2) Yang-Mills theory coupled to the dilaton. Here integrability means the existence of infinitely many symmetries and infinitely many conserved currents. Further, we construct infinitely many static solutions of this integrable subsystem. These solutions can be identified with certain limiting solutions of the full system, which have been found previously in the context of numerical investigations of the Yang-Mills dilaton theory. In addition, we derive a Bogomolny…
A class of explicitly integrable models of 1+1 dimensional dilaton gravity coupled to scalar fields is described in some detail. The equations of motion of these models reduce to systems of the Liouville equations endowed with energy and momentum constraints. The general solution of the equations and constraints in terms of chiral moduli fields is explicitly constructed and some extensions of the basic integrable model are briefly discussed. These models may be related to high dimensional supergravity theories but here they are mostly considered independently of such interpretations. A brief r…
In this work we study renormalization-group flows by deforming a class of conformal sigma-models. At leading order in $α'$, renormalization-group equations represent a Ricci flow. In the three-sphere background, the latter is described by the Halphen system, which is exactly solvable in terms of modular forms. The round sphere is found to be the unique perturbative infra-red fixed point at one loop order.
The general properties of a class of two-dimensional dilaton gravity (DG) theories with multi-exponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general model. It follows that in DG the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda-Liouville DG is to solve the ener…
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld--Sokolov realization.
A class of integrable models of 1+1 dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to 0+1 dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real space-time dimension are derived. A generalization of the `no-hair' theorem is pointed out and the Hamiltonian formulation that enables quantizing the s-integrable syste…
The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented in terms of matrix elements of fundamental representations of semisimple A_n algebras for a given group element.
In the present paper we obtain some integrable generalisations of the continuous Toda system generated by a flat connection form taking values in higher grading subspaces of the algebra of the area--preserving diffeomorphism of the torus $T^2$, and construct their general solutions. The grading condition which we use here, imposed on the connection, can be realised in terms of some holomorphic distributions on the corresponding homogeneous spaces.
Integrable models of 1+1 dimensional gravity coupled to scalar and vector fields are briefly reviewed. A new class of integrable models with nonminimal coupling to scalar fields is constructed and discussed.
A pedagogical presentation of integrable models with special reference to the Toda lattice hierarchy has been attempted. The example of the KdV equation has been studied in detail, beginning with the infinite conserved quantities and going on to the Lax formalism for the same. We then go on to symplectic manifolds for which we construct the Lax operator. This formalism is applied to Toda Lattice systems. The Zakharov Shabat formalism aimed at encompassing all integrable models is also covered after which the zero curvature condition and its fallout are discussed. We then take up Toda Field The…
Boundary conditions compatible with classical integrability are studied both directly, using an approach based on the explicit construction of conserved quantities, and indirectly by first developing a generalisation of the Lax pair idea. The latter approach is closer to the spirit of earlier work by Sklyanin and yields a complete set of conjectures for permissible boundary conditions for any affine Toda field theory.
Global canonical transformations to free chiral fields are constructed for DG models minimally coupled to scalar fields. The boundary terms for such canonical transformations are shown to vanish in asymptotically static coordinates if there is no scalar field.
In this paper we reformulate the dilaton-gravity theory of Callan \etal\ as a new effective conformal field theory which turns out to be a generalization of the so-called $SL_2$-conformal affine Toda (CAT) theory studied some times ago by Babelon and Bonora. We quantize this model, thus keeping in account the dilaton-gravity quantum effects. We then implement a Renormalization Group analysis to study the black hole thermodynamics and the final state of the Hawking evaporation.
A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical close to TC model [7] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the $(2\times 2)$ Lax operators and the associated quantum $R$-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the …
The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\t…
The reduction of higher dimensional supergravities to low dimensional dilaton gravity theories is outlined. Then a recently proposed new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is described in more detail. These models are reducible to systems of independent Liouville equations whose solutions are not independent because they must satisfy the energy and momentum constraints. The constraints are solved, thus giving the explicit analytic solution of the theory in terms of arbitrary chiral fields. In particular, these integrab…
We study the two-dimensional supersymmetric Toda theory based on the Lie superalgebra $B(1,1) \equiv Osp(3|2)$ and construct its quantum W-currents. We also investigate the fermionic affinization of this model: we show that despite the non-unitary form of the Lagrangian the $B^{(1)}(1,1)$ theory has a real particle mass spectrum which is not renormalized at one-loop. We construct the first higher--spin conserved current, prove its conservation to all-loop order, compute one-loop corrections to the corresponding charge and check consistency between charge and mass renormalization.
We calculate normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed conformal field theories. We apply these results to evaluate explicitly the expectation values of order parameters in the field theories associated with statistical systems, like XY, Z_n-Ising and Ashkin-Teller models. The same results are used for the calculation of the asymptotics of cylindrically symmetric solutions of the classical Toda equations which ap…
We present a new quantization scheme for $2D$ gravity coupled to an $SU(2)$ principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a ``clock field''. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the …
Static axisymmetric Einstein-Maxwell-Dilaton and stationary axisymmetric Einstein-Maxwell-Dilaton-Axion (EMDA) theories in four space-time dimensions are shown to be integrable by means of the inverse scattering transform method. The proof is based on the coset-space representation of the 4-dim theory in a space-time admitting a Killing vector field. Hidden symmetry group of the four-dimensional EMDA theory, unifying T and S string dualities, is shown to be Sp(2, R) acting transitively on the coset Sp(2, R)/U(2). In the case of two-parameter Abelian space-time isometry group, the hidden symmet…
We consider D-dimensional Einstein gravity coupled to (n-1) U(1) vector fields and (n-2) dilatonic scalars. We find that for some appropriate exponential dilaton couplings of the field strengths, the equations of motion for the static charged ansatz can be reduced to a set of one-dimensional SL(n,R) Toda equations. This allows us to obtain a general class of explicit black holes with mass and (n-1) independent charges. The near-horizon geometry in the extremal limit is AdS_2 x S^{D-2}. The n=2 case gives the Reissner-Nordstrom solution, and the n=3 example includes the Kaluza-Klein dyon. We st…
The present paper studies a completely integrable conformally invariant model in 1+1 dimensions that corresponds to string propagation on the two-dimensional black hole background (semi-ininite cigar). Besides the two space-time string fields there is a third (internal) field with a very specific Liouville-type interaction leading to the complete integrability. This system is known as non-abelian Toda theory. I give the general explicit classical solution. It realizes a rather involved transformation expressing the interacting string fields in terms of (three) functions $φ_j(u)$ and $\barφ_j(v…
The most prominent class of integrable quantum field theories in 1+1 dimensions is affine Toda theory. Distinguished by a rich underlying Lie algebraic structure these models have in recent years attracted much attention not only as test laboratories for non-perturbative methods in quantum field theory but also in the context of off-critical models. After a short introduction the mathematical preliminaries such as root systems, Coxeter geometry, dual algebras, q-deformed Coxeter elements and q-deformed Cartan matrices are introduced. Using this mathematical framework the bootstrap analysis of …
Following a prescription of \cite{4} for a solitonic specialization of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely integrable systems arising in the framework of the group-algebraic approach, including the \lq\lq non-abelian" version of the affine Toda theory.
We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against t…
In these notes we study the duality between sigma-models and Toda QFT's. We claim that $\mathfrak{gl}(n|n)$ affine Toda field theory behaves in the strong coupling limit as $η-$deformed $\mathbb{CP}(n-1)$ sigma-model plus a free field.
We show here that matrix Darboux-Toda transformation can be written as a product of a number of mappings. Each of these mappings is a symmetry of the matrix nonlinear Shrodinger system of integro-differential equations. We thus introduce a completely new type of discrete transformations for this system. The discrete symmetry of the vector nonlinear Shrodinger system is a particular realization of these mappings.
We consider the Pohlmeyer reduction for spacelike minimal area worldsheets in AdS$_5$. The Lax pair for the reduced theory is found, and written entirely in terms of the $A_3=D_3$ root system, generalizing the $B_2$ affine Toda system which appears for the AdS$_4$ string. For the $B_2$ affine Toda system, we show that the area of the worlsheet is obtainable from the moduli space Kähler potential of a related Hitchin system. We also explore the Saveliev-Leznov construction for solutions of the $B_2$ affine Toda system, and recover the rotationally symmetric solution associated to Painleve trans…
We study the general form of the equations for isotropic single-scalar, multi-scalar and dyonic $p$-branes in superstring theory and M-theory, and show that they can be cast into the form of Liouville, Toda (or Toda-like) equations. The general solutions describe non-extremal isotropic $p$-branes, reducing to the previously-known extremal solutions in limiting cases. In the non-extremal case, the dilatonic scalar fields are finite at the outer event horizon.
A new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is introduced. These models are reducible to systems of independent Liouville equations whose solutions must satisfy the energy and momentum constraints. The constraints are solved thus giving the explicit analytic solution of the theory in terms of arbitrary chiral fields. In particular, these integrable theories describe spherically symmetric black holes and branes of higher dimensional supergravity theories as well as superstring motivated cosmological models.
Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang-Baxter equations, and second by solving a linear intertwining relation between a finite dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite dimensional representation expressed in terms of sets of generalized `quantum' annihilation and creation operators. The principal examples analysed are based on the $a_2^{(2)}$ and $a_n^{(1)}$ affine Toda models bu…
The bootstrap equations for the ADE series of purely elastic scattering theories have turned out to be intimately connected with the geometry of root systems and the Coxeter element. An informal review of some of this material is given, mentioning also a couple of other contexts -- the Pasquier models, and the simply-laced affine Toda field theories -- where similar structures are encountered. The relevance of twisted Coxeter elements is indicated, and a construction of these elements inspired by the twisted foldings of the affine Toda models is described.
Integrable open elliptic Toda chain with boundaries
Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论与数学物理研究所）
In this letter we discuss the relation of four-dimensional, $N=2$ supersymmetric string backgrounds to integrable models. In particular we show that non-Kählerian gravitational backgrounds with one $U(1)$ isometry plus non-trivial antisymmetric tensor and dilaton fields arise as the solutions of the Liouville equation or, for the case of vanishing central charge deficit, as the solutions of the continual Toda equation. When performing an Abelian duality transformation, a particular class of solutions of the continual Toda equation leads to the well-known gravitational Eguchi-Hanson instanton b…
For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$…
We reconsider dyonic p-brane solutions derivable from Liouville and Toda integrable systems and investigate their geometric structure. It is shown that the non-BPS non-black dyonic branes are not regular on the horizon.
A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable.
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N=3 the $τ$-function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.
We review the origin of anomaly-induced dynamics in theories of $d=2$ gravity from a BRST viewpoint and show how quantum canonical transformations may be used to solve the resulting Liouville or Toda models for the anomalous modes.
Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current conservation. In two dimensions for n < 6 known integrable models (deformed Toda and non-linear sigma, pure affine Toda...) with dual counterparts are obtained in this way from the multicomplex space MC itself and from the natural embedding $\MC_n \subset \MMC_m, n < m$. For $ n \ge 6$ a generic constraint on the space of parametersis obtained from current conserv…
We reorganize the derivative expansion of general (power-counting non-renormalizable) 2d dilaton gravity such that the mass function is integrable. As an example, we consider a three-parameter family of models and provide conditions on the parameters such that the ground state is either Minkowski, Rindler, or (A)dS.
We study $N$=2 supersymmetric integrable theories with spontaneously-broken \Zn\ symmetry. They have exact soliton masses given by the affine $SU(n)$ Toda masses and fractional fermion numbers given by multiples of $1/n$. The basic such $N$=2 integrable theory is the $A_n$-type $N$=2 minimal model perturbed by the most relevant operator. The soliton content and exact S-matrices are obtained using the Landau-Ginzburg description. We study the thermodynamics of these theories and calculate the ground-state energies exactly, verifying that they have the correct conformal limits. We conjecture tha…
The requirement that the laws of physics must be invariant under point-dependent transformations of the units of length, time, and mass is used as a selection principle while studying different generic effective theories of gravity. Thereof theories with non-minimal coupling of the dilaton both to the curvature and to the Lagrangian of the matter fields seem to represent the most viable low-energy [and low-curvature] description of gravity. Consequently, the cosmological singularity problem is treated within the context of string cosmology with non-minimal coupling of the dilaton to a barotrop…
In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.
Integrable boundary Toda theories are considered. We use boundary one-point functions and boundary scattering theory to construct the explicit solutions corresponding to classical vacuum configurations. The boundary ground state energies are conjectured.
The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong-Wong type. Under certain conditions these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes…
A class of time dependent solutions to $(3+1)$ Einstein--Maxwell-dilaton theory with attractive electric force is found from Einstein--Weyl structures in (2+1) dimensions corresponding to dispersionless Kadomtsev--Petviashvili and $SU(\infty)$ Toda equations. These solutions are obtained from time--like Kaluza--Klein reductions of $(3+2)$ solitons.
In this note, we consider the equations of motion for charged branes in AdS space and show that they can be cast into one-dimensional coupled Toda type. Then we solve the equations of motion to construct static charged AdS brane solutions which are invariant under translation along $p$ directions and have toroidal symmetry. The solutions are described by mass, charges and a dilaton coupling constant.
The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.
We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy namely the q-KdV and q-Boussinesq integrable systems. We present also the q-generalization of the conformal transformations of the currents and discuss the primarity condition of the fields by using the Volterra gauge group transformations for …
A general construction of affine Non Abelian (NA) - Toda models in terms of axial and vector gauged two loop WZNW model is discussed. They represent {\it integrable perturbations} of the conformal $σ$-models (with tachyons included) describing (charged) black hole type string backgrounds. We study the {\it off-critical} T-duality between certain families of axial and vector type of integrable models for the case of affine NA- Toda theories with one global U(1) symmetry. In particular we find the Lie algebraic condition defining a subclass of {\it T-selfdual} torsionless NA Toda models and thei…
Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）；The International Joint Institute of Tianjin University, Fuzhou, Tianjin University（天津大学福州国际联合学院）
We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equa…
Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessa…
For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenk…
The $n\to\infty$ continuum limit of super-Toda models associated with the affine $sl(2n|2n)^{(1)}$ (super)algebra series produces $(2+1)$-dimensional integrable equations in the ${\bf S}^{1}\times {\bf R}^2$ spacetimes. The equations of motion of the (super)Toda hierarchies depend not only on the chosen (super)algebras but also on the specific presentation of their Cartan matrices. Four distinct series of integrable hierarchies in relation with symmetric-versus-antisymmetric, null-versus-nonnull presentations of the corresponding Cartan matrices are investigated. In the continuum limit we deri…
