Papers by Michael Lashkevich
arXiv (Cornell University), Mar 21, 2023
Form factors in the sinh-Gordon model are studied semiclassically for small values of the paramet... more Form factors in the sinh-Gordon model are studied semiclassically for small values of the parameter b ∼ ℏ 1/2 in the background of a radial classical solution, which describes a heavy exponential operator placed at the origin. For this purpose we use a generalization of the radial quantization scheme, well known for a massless boson field. We introduce and study new special functions which generalize the Bessel functions and have a nice interpretation in the Tracy-Widom theory of the Fredholm determinant solutions of the classical sinh-Gordon model. Form factors of the exponential operators in the leading order are completely determined by the classical solutions, while form factors of the descendant operators contain quantum corrections even in this approximation. The construction of descendant operators in two chiralities requires renormalizations similar to those encountered in the conformal perturbation theory.
The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model... more The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model in the breather sector is modified to describe the form factors of descendant operators, which are obtained from the exponential ones, e iαϕ , by means of the action of the Heisenberg algebra associated to the field ϕ(x). As a check of the validity of the construction we count the numbers of operators defined by the form factors at each level in each chiral sector. Another check is related to the so called reflection relations, which identify in the breather sector the descendants of the exponential fields e iαϕ and e i(2α 0 −α)ϕ for generic values of α. We prove the operators defined by the obtained families of form factors to satisfy such reflection relations. A generalization of the construction for form factors to the kink sector is also proposed.
The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model... more The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model in the breather sector is modified to describe the form factors of descendant operators, which are obtained from the exponential ones, $\e^{\i\alpha\varphi}$, by means of the action of the Heisenberg algebra associated to the field~$\varphi(x)$. As a check of the validity of the construction we count the numbers of operators defined by the form factors at each level in each chiral sector. Another check is related to the so called reflection relations, which identify in the breather sector the descendants of the exponential fields $\e^{\i\alpha\varphi}$ and $\e^{\i(2\alpha_0-\alpha)\varphi}$ for generic values of~$\alpha$. We prove the operators defined by the obtained families of form factors to satisfy such reflection relations. A generalization of the construction for form factors to the kink sector is also proposed.
Journal of Physics A: Mathematical and Theoretical, 2020
We study the fused currents of the deformed Virasoro algebra (DVA). By constructing a homotopy op... more We study the fused currents of the deformed Virasoro algebra (DVA). By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfermatrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS (RSOS) models. Though the results of the algebraic approach to lattice models were consistent with the results of other methods, the lack of such proof had been an essential gap in its construction.
Journal of High Energy Physics, 2019
We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approa... more We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approach. The free field procedure for descendant operators is developed by introducing the algebra of screening currents and related algebraic objects. We work out null vector equations in the space of operators. Further we apply the proposed algebraic structures to constructing form factors of the conserved currents T k and Θ k . We propose also form factors of current-current operators of the form T k T −l . Explicit computations of the four-particle form factors allow us to verify the recent conjecture of Smirnov and Zamolodchikov about the structure of the exact scattering matrix of an integrable theory perturbed by a combination of irrelevant operators. Our calculations confirm that such perturbations of the complex sinh-Gordon model and of the ℤ N symmetric Ising models result in extra CDD factors in the S matrix.
Physics Letters B, 2017
We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the... more We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the ZN-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.
Journal of Physics A: Mathematical and Theoretical, 2016
The diagonal matrix elements θ1, θ2|T2nT−2n|θ1, θ2 between two-particle states in the sinh-Gordon... more The diagonal matrix elements θ1, θ2|T2nT−2n|θ1, θ2 between two-particle states in the sinh-Gordon model are computed analytically for all integers n > 0. This confirms the proposal [1] by F. Smirnov and A. Zamolodchikov for these matrix elements and demonstrates effectiveness of the algebraic approach to form factors.
Lectures on the eight-vertex model and bosonization
These are introductory lectures on application of the free field representation (bosonization) te... more These are introductory lectures on application of the free field representation (bosonization) techniques to the solid-on-\S olid (SOS) and eight-vertex models. We start from the very beginnings, including the physical badcground of lattice models and some basic information on quantum integrability. After definitions of the eight-vertex and SOS models, we describe their relation known as the vertex-face correspondence. Then, skipping the Bethe ansatz solution, wc turn to the problem of calculation of correlation functions by mcaiis of the fit.c ficld rcprcscntation. We explain, how the vertex-face correspondence works on the level of vertex operators and bosonization, making it possible to express the correlation functions of the eight-vertex model in terms of the free field representation aimed to describe the SOS model.
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Form factors of descendant operators: [FORMULA] affine Toda theory
The Journal of High Energy Physics, 2010
In the framework of the free field representation we obtain exact form factors of local operators... more In the framework of the free field representation we obtain exact form factors of local operators in the two-dimensional affine Toda theories of the [FORMULA] series. The construction generalizes Lukyanov’s well-known construction to the case of descendant operators. Besides, we propose a free field representation with a countable number of generators for the ‘stripped’ form factors, which generalizes the recent proposal for the sine/sinh-Gordon model. As a check of the construction we compare numbers of the operators defined by these form factors in level subspaces of the chiral sectors with the corresponding numbers in the Lagrangian formalism. We argue that the construction provides a correct counting for operators with both chiralities. At last we study the properties of the operators with respect to the Weyl group. We show that for generic values of parameters there exist Weyl invariant analytic families of the bases in the level subspaces.
It is known that 2D field theories admit several sectors of mutually local fields so as two field... more It is known that 2D field theories admit several sectors of mutually local fields so as two fields from different sectors are mutually nonlocal. We show that any one-partical integrable model with ${\bf Z}_2$ symmetry contains three sectors: bosonic, fermionic and `disorder' one. We generalize the form factor axioms to fermionic and `disorder' sectors. For the particular case of the sinh-Gordon model we obtain several form factors in these sectors.
Scaling limits of the SOS and RSOS models in the regime~III are considered. These scaling limits ... more Scaling limits of the SOS and RSOS models in the regime~III are considered. These scaling limits are believed to be described by the sine-Gordon model and the restricted sine-Gordon models (or perturbed minimal conformal models) respectively. We study two different scaling limits and establish the correspondence of the scaling local height operators to exponential or primary fields in quantum field theory. An integral representation for form factors is obtained in this way. In the case of the sine-Gordon model this reproduces Lukyanov's well known representation. The relation between vacuum expectation values of local operators in the sine-Gordon model and perturbed minimal models is also discussed.
The boundary conditions with diagonal boundary $S$ matrix and the boundary form factors for the S... more The boundary conditions with diagonal boundary $S$ matrix and the boundary form factors for the Smirnov--Fateev model on a half line has been considered in the framework of the free field representation. In contrast to the case of the sine-Gordon model, in this case the free field representation is shown to impose severe restrictions on the boundary $S$ matrix, so that a finite number of solutions is only consistent with the free field realization.
Journal of High Energy Physics, 2015
In the framework of the algebraic approach to form factors in two-dimensional integrable models o... more In the framework of the algebraic approach to form factors in two-dimensional integrable models of quantum field theory we consider the reduction of the sine-Gordon model to the Φ13-perturbation of minimal conformal models of the M (2, 2s + 1) series. We find in an algebraic form the condition of compatibility of local operators with the reduction. We propose a construction that make it possible to obtain reduction compatible local operators in terms of screening currents. As an application we obtain exact multiparticle form factors for the compatible with the reduction conserved currents T ±2k , Θ ±(2k−2) , which correspond to the spin ±(2k−1) integrals of motion, for any positive integer k. Furthermore, we obtain all form factors of the operators T 2k T −2l , which generalize the famous TT operator. The construction is analytic in the s parameter and, therefore, makes sense in the sine-Gordon theory.
It is shown that zero ghost conformal blocks of coset theory G/H are determined uniquely by those... more It is shown that zero ghost conformal blocks of coset theory G/H are determined uniquely by those of G and H theories. G/G theories are considered as an example, their structure constants and correlation functions on sphere are calculated.
Journal of High Energy Physics, 2013
We are developing the algebraic construction for form factors of local operators in the sinh-Gord... more We are developing the algebraic construction for form factors of local operators in the sinh-Gordon theory proposed in [1]. We show that the operators corresponding to the null vectors in this construction are given by the degenerate Macdonald polynomials with rectangular partitions and the parameters t = −q on the unit circle. We obtain an integral representation for the null vectors and discuss its simple applications.
Physics Letters A, 1997
The BRST property of Lukyanov's screening operators in the bosonic representation of the deformed... more The BRST property of Lukyanov's screening operators in the bosonic representation of the deformed Virasoro algebra is proven.
Nuclear Physics B, 2013
We continue the study of form factors of descendant operators in the sinh-and sine-Gordon models ... more We continue the study of form factors of descendant operators in the sinh-and sine-Gordon models in the framework of the algebraic construction proposed in [1]. We find the algebraic construction to be related to a particular limit of the tensor product of the deformed Virasoro algebra and a suitably chosen Heisenberg algebra. To analyze the space of local operators in the framework of the form factor formalism we introduce screening operators and construct singular and cosingular vectors in the Fock spaces related to the free field realization of the obtained algebra. We show that the singular vectors are expressed in terms of the degenerate Macdonald polynomials with rectangular partitions. We study the matrix elements that contain a singular vector in one chirality and a cosingular vector in the other chirality and find them to lead to the resonance identities already known in the conformal perturbation theory. Besides, we give a new derivation of the equation of motion in the sinh-Gordon theory, and a new representation for conserved currents.
Nuclear Physics B, 2002
The free field realization of the eight-vertex model is extended to form factors. It is achieved ... more The free field realization of the eight-vertex model is extended to form factors. It is achieved by constructing offdiagonal with respect to the ground state sectors matrix elements of the Λ operator which establishes a relation between corner transfer matrices of the eight-vertex model and of the SOS model. As an example, the two-particle form factor of the σ z operator is evaluated explicitly.
Nuclear Physics B, 1998
A free field representation for the type I vertex operators and the corner transfer matrices of t... more A free field representation for the type I vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to express correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a nonlocal insertion. This new nonlocal insertion admits of a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter-Kelland formula for the average staggered polarization have been reproduced.
Nuclear Physics B, 2004
A two-parametric family of integrable models (the SS model) that contains as particular cases sev... more A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine-cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles.
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Papers by Michael Lashkevich