International audienceThe computation of a certain class of polarised four-point functions of hea... more International audienceThe computation of a certain class of polarised four-point functions of heavily charged BPS in \(\mathcal {N}=4\) SYM operators boils down to the computation of a special form factor - the octagon. Here I review the representation of the octagon in terms of free fermions and the determinant formulas that follow. The presentation is based mainly on a common work with Valentina Petkova and Didina Serban[1, 2], but I also mention some recent developments obtained by other authors
We review the basics of the dynamics of closed strings moving along the infinite discretized line... more We review the basics of the dynamics of closed strings moving along the infinite discretized line Z Z. The string excitations are described by a field ϕ x (τ) where x ∈ Z Z is the position of the string in the embedding space and τ is a semi-infinite "euclidean time" parameter related to the longitudinal mode of the string. Interactions due to splitting and joining of closed strings are taken into account by a local potential and occur only along the edge τ = 0 of the semi-plane (x, τ).
We derive a set of bilinear functional equations of Hirota type for the partition functions of th... more We derive a set of bilinear functional equations of Hirota type for the partition functions of the sl(2) related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy, which are satisfied by the partition function of the ensemble of planar graphs. SPhT-96/029
The microscopic theories of quantum gravity related to integrable lattice models can be construct... more The microscopic theories of quantum gravity related to integrable lattice models can be constructed as special deformations of pure gravity. Each such deformation is defined by a second order differential operator acting on the coupling constants. As a consequence, the theories with matter fields satisfy a set of constraints inherited from the integrable structure of pure gravity. In particular, a set of bilinear functional equations for each theory with matter fields follows from the Hirota equations defining the KP (KdV) structure of pure gravity.
We present the exact solution of the Baxter's three-color problem on a random planar graph, using... more We present the exact solution of the Baxter's three-color problem on a random planar graph, using its formulation in terms of three coupled random matrices. We find that the number of three-colorings of an infinite random graph is 0.9843 per vertex.
We propose a new method for the computation of quantum three-point functions for operators in su(... more We propose a new method for the computation of quantum three-point functions for operators in su(2) sectors of N = 4 super Yang-Mills theory. The method is based on the existence of a unitary transformation relating inhomogeneous and long-range spin chains. This transformation can be traced back to a combination of boost operators and an inhomogeneous version of Baxter's corner transfer matrix. We reproduce the existing results for the one-loop structure constants in a simplified form and indicate how to use the method at higher loop orders. Then we evaluate the one-loop structure constants in the quasiclassical limit and compare them with the recent strong coupling computation.
In these notes we explain how the CFT description of random matrix models can be used to perform ... more In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an explicit operator construction of the corresponding collective field theory in terms of a bosonic field on a hyperelliptic Riemann surface, with special operators associated with the branch points. The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are then easily obtained as correlation functions of current, fermionic and twist operators. The result for the spectral kernel is valid both in macroscopic and microscopic scales. At the end we briefly consider generalizations in different directions.
We compute bulk 3-and 4-point tachyon correlators in the 2d Liouville gravity with non-rational m... more We compute bulk 3-and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c < 1, following and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground ring generators deformed by Liouville and matter "screening charges". A by-product general formula for the matter 3-point OPE structure constants is derived. We also consider a "diagonal" CFT of 2D quantum gravity, in which the degenerate fields are restricted to the diagonal of the semi-infinite Kac table. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points.
We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense... more We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known (p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently
We apply the recently developped analytical methods for computing the boundary entropy, or the g-... more We apply the recently developped analytical methods for computing the boundary entropy, or the g-function, in integrable theories with non-diagonal scattering. We consider the particular case of the current-perturbed SU (2) k WZNW model with boundary and compute the boundary entropy for a specific boundary condition. The main problem we encounter is that in case of non-diagonal scattering the boundary entropy is infinite. We show that this infinity can be cured by a subtraction. The difference of the boundary entropies in the UV and in the IR limits is finite, and matches the known g-functions for the unperturbed SU (2) k WZNW model for even values of the level.
We construct the boundary ground ring in c ≤ 1 open string theories with non-zero boundary cosmol... more We construct the boundary ground ring in c ≤ 1 open string theories with non-zero boundary cosmological constant (FZZT brane), using the Coulomb gas representation. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions, which involve shifts of the the target space momenta of the boundary fields as well as the boundary parameters on the different segments of the boundary. * contribution to the proceedings of the conference lie theory and its applications in physics-5, june 2003, varna, bulgaria
In these notes we review the method to construct integrable deformations of the compactified c = ... more In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy*.
We explain, in a slightly modified form, an old construction allowing to reformulate the U(N) gau... more We explain, in a slightly modified form, an old construction allowing to reformulate the U(N) gauge theory defined on a D-dimensional lattice as a theory of lattice strings (a statistical model of random surfaces). The world surface of the lattice string is allowed to have pointlike singularities (branch points) located not only at the sites of the lattice, but also on its links and plaquettes. The strings become noninteracting when N → ∞. In this limit the statistical weight a world surface is given by exp[ − area] times a product of local factors associated with the branch points. In D = 4 dimensions the gauge theory has a nondeconfining first order phase transition dividing the weak and strong coupling phase. From the point of view of the string theory the weak coupling phase is expected to be characterized by spontaneous creation of “windows” on the world sheet of the string.
In these notes we review the method to construct integrable deformations of the compactified c = ... more In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy*.
We give a sequence of equivalent formulations of the ADE and ÂD̂Ê height models defined on a rand... more We give a sequence of equivalent formulations of the ADE and ÂD̂Ê height models defined on a random triangulated surface: random surfaces immersed in Dynkin diagrams, chains of coupled random matrices, Coulomb gases, and multicomponent Bose and Fermi systems representing soliton τ -functions. We also formulate a set of loop-space Feynman rules allowing to calculate easily the partition function on a random surface with arbitrary topology. The formalism allows to describe the critical phenomena on a random surface in a unified fashion and gives a new meaning to the ADE classification.
We study the boundary correlation functions in Liouville theory and in solvable statistical model... more We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary twopoint function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach.
The path integral is calculated that describes the dynamics of the supersymmetric relativistic pa... more The path integral is calculated that describes the dynamics of the supersymmetric relativistic particle with an action proportional to the world line superlength. The kernel of the superparticle propagation is shown to coincide with the Green function of the quantum-field theory with the same supersymmetry.
We construct and study a matrix model that describes two dimensional string theory in the Euclide... more We construct and study a matrix model that describes two dimensional string theory in the Euclidean black hole background. A conjecture of V. Fateev, A. and Al. Zamolodchikov, relating the black hole background to condensation of vortices (winding modes around Euclidean time) plays an important role in the construction. We use the matrix model to study quantum corrections to the thermodynamics of two dimensional black holes.
International audienceThe computation of a certain class of polarised four-point functions of hea... more International audienceThe computation of a certain class of polarised four-point functions of heavily charged BPS in \(\mathcal {N}=4\) SYM operators boils down to the computation of a special form factor - the octagon. Here I review the representation of the octagon in terms of free fermions and the determinant formulas that follow. The presentation is based mainly on a common work with Valentina Petkova and Didina Serban[1, 2], but I also mention some recent developments obtained by other authors
We review the basics of the dynamics of closed strings moving along the infinite discretized line... more We review the basics of the dynamics of closed strings moving along the infinite discretized line Z Z. The string excitations are described by a field ϕ x (τ) where x ∈ Z Z is the position of the string in the embedding space and τ is a semi-infinite "euclidean time" parameter related to the longitudinal mode of the string. Interactions due to splitting and joining of closed strings are taken into account by a local potential and occur only along the edge τ = 0 of the semi-plane (x, τ).
We derive a set of bilinear functional equations of Hirota type for the partition functions of th... more We derive a set of bilinear functional equations of Hirota type for the partition functions of the sl(2) related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy, which are satisfied by the partition function of the ensemble of planar graphs. SPhT-96/029
The microscopic theories of quantum gravity related to integrable lattice models can be construct... more The microscopic theories of quantum gravity related to integrable lattice models can be constructed as special deformations of pure gravity. Each such deformation is defined by a second order differential operator acting on the coupling constants. As a consequence, the theories with matter fields satisfy a set of constraints inherited from the integrable structure of pure gravity. In particular, a set of bilinear functional equations for each theory with matter fields follows from the Hirota equations defining the KP (KdV) structure of pure gravity.
We present the exact solution of the Baxter's three-color problem on a random planar graph, using... more We present the exact solution of the Baxter's three-color problem on a random planar graph, using its formulation in terms of three coupled random matrices. We find that the number of three-colorings of an infinite random graph is 0.9843 per vertex.
We propose a new method for the computation of quantum three-point functions for operators in su(... more We propose a new method for the computation of quantum three-point functions for operators in su(2) sectors of N = 4 super Yang-Mills theory. The method is based on the existence of a unitary transformation relating inhomogeneous and long-range spin chains. This transformation can be traced back to a combination of boost operators and an inhomogeneous version of Baxter's corner transfer matrix. We reproduce the existing results for the one-loop structure constants in a simplified form and indicate how to use the method at higher loop orders. Then we evaluate the one-loop structure constants in the quasiclassical limit and compare them with the recent strong coupling computation.
In these notes we explain how the CFT description of random matrix models can be used to perform ... more In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an explicit operator construction of the corresponding collective field theory in terms of a bosonic field on a hyperelliptic Riemann surface, with special operators associated with the branch points. The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are then easily obtained as correlation functions of current, fermionic and twist operators. The result for the spectral kernel is valid both in macroscopic and microscopic scales. At the end we briefly consider generalizations in different directions.
We compute bulk 3-and 4-point tachyon correlators in the 2d Liouville gravity with non-rational m... more We compute bulk 3-and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c < 1, following and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground ring generators deformed by Liouville and matter "screening charges". A by-product general formula for the matter 3-point OPE structure constants is derived. We also consider a "diagonal" CFT of 2D quantum gravity, in which the degenerate fields are restricted to the diagonal of the semi-infinite Kac table. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points.
We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense... more We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known (p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently
We apply the recently developped analytical methods for computing the boundary entropy, or the g-... more We apply the recently developped analytical methods for computing the boundary entropy, or the g-function, in integrable theories with non-diagonal scattering. We consider the particular case of the current-perturbed SU (2) k WZNW model with boundary and compute the boundary entropy for a specific boundary condition. The main problem we encounter is that in case of non-diagonal scattering the boundary entropy is infinite. We show that this infinity can be cured by a subtraction. The difference of the boundary entropies in the UV and in the IR limits is finite, and matches the known g-functions for the unperturbed SU (2) k WZNW model for even values of the level.
We construct the boundary ground ring in c ≤ 1 open string theories with non-zero boundary cosmol... more We construct the boundary ground ring in c ≤ 1 open string theories with non-zero boundary cosmological constant (FZZT brane), using the Coulomb gas representation. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions, which involve shifts of the the target space momenta of the boundary fields as well as the boundary parameters on the different segments of the boundary. * contribution to the proceedings of the conference lie theory and its applications in physics-5, june 2003, varna, bulgaria
In these notes we review the method to construct integrable deformations of the compactified c = ... more In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy*.
We explain, in a slightly modified form, an old construction allowing to reformulate the U(N) gau... more We explain, in a slightly modified form, an old construction allowing to reformulate the U(N) gauge theory defined on a D-dimensional lattice as a theory of lattice strings (a statistical model of random surfaces). The world surface of the lattice string is allowed to have pointlike singularities (branch points) located not only at the sites of the lattice, but also on its links and plaquettes. The strings become noninteracting when N → ∞. In this limit the statistical weight a world surface is given by exp[ − area] times a product of local factors associated with the branch points. In D = 4 dimensions the gauge theory has a nondeconfining first order phase transition dividing the weak and strong coupling phase. From the point of view of the string theory the weak coupling phase is expected to be characterized by spontaneous creation of “windows” on the world sheet of the string.
In these notes we review the method to construct integrable deformations of the compactified c = ... more In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy*.
We give a sequence of equivalent formulations of the ADE and ÂD̂Ê height models defined on a rand... more We give a sequence of equivalent formulations of the ADE and ÂD̂Ê height models defined on a random triangulated surface: random surfaces immersed in Dynkin diagrams, chains of coupled random matrices, Coulomb gases, and multicomponent Bose and Fermi systems representing soliton τ -functions. We also formulate a set of loop-space Feynman rules allowing to calculate easily the partition function on a random surface with arbitrary topology. The formalism allows to describe the critical phenomena on a random surface in a unified fashion and gives a new meaning to the ADE classification.
We study the boundary correlation functions in Liouville theory and in solvable statistical model... more We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary twopoint function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach.
The path integral is calculated that describes the dynamics of the supersymmetric relativistic pa... more The path integral is calculated that describes the dynamics of the supersymmetric relativistic particle with an action proportional to the world line superlength. The kernel of the superparticle propagation is shown to coincide with the Green function of the quantum-field theory with the same supersymmetry.
We construct and study a matrix model that describes two dimensional string theory in the Euclide... more We construct and study a matrix model that describes two dimensional string theory in the Euclidean black hole background. A conjecture of V. Fateev, A. and Al. Zamolodchikov, relating the black hole background to condensation of vortices (winding modes around Euclidean time) plays an important role in the construction. We use the matrix model to study quantum corrections to the thermodynamics of two dimensional black holes.
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