Superconformal field theory and SUSY N=1 KdV hierarchy II: the Q-operator

The algebraic structures related with integrable structure of superconformal field theory (SCFT) are introduced. The SCFT counterparts of Baxter's Q-operator are constructed. The fusion-like relations for the transfer-matrices in different representations and their truncations are obtained.     

Spectrum of boundary states in N=1 SUSY sine-Gordon theory

We consider N=1 supersymmetric sine-Gordon theory (SSG) with supersymmetric integrable boundary conditions (boundary SSG=BSSG). We find two possible ways to close the boundary bootstrap for this model, corresponding to two different choices for the boundary supercharge. We argue that these two bootstrap solutions should correspond to the two integrable Lagrangian boundary theories considered recently by Nepomechie.     

Finite N=1 SUSY field theories and dimensional regularization

A simple algorithm is proposed to construct finite N=1 SUSY field theories within dimensional regularization. This is achieved by choosing the Yukawa couplings to be y_i=g(αⁱ₀+αⁱ₀ϵ+ϵ+αⁱ₂ϵ²+…), where g is the gauge coupling and the coefficients αⁱ₀, αⁱ₁, etc., are calculated order by order of perturbation theory. It is shown that the theory can be made finite in all orders of perturbation theory.     

Quantum equations of motion and the Liouville equation

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if \(\hat F\) and ? are two such functions and \(\hat F^\prime \) and ?′ contain an arbitrary permutation of variables, then $$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$ where  is a remainder commutator. In the classical limit [ \(\hat F\) , ?] is invariant to such permutation.     

Dynamic field theory and equations of motion in cosmology

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equations in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1$\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ?1$\delta \rho /\rho ?1$. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.     

On the renormalization of topological Yang-Mills field theory in N = 1 superspace

We discuss the renormalization aspects of topological super-Yang-Mills field theory in N = 1 superspace. Our approach makes use of the regularization-independent BRS algebraic technique adapted to the case of an N = 1 supersymmetric model. We give the expression of the most general local counterterm to the classical action to all orders of the perturbative expansion. The counterterm is shown to be a BRS coboundary, implying that the cohomological properties of the supertopological theory are not affected by quantum effects. We also demonstrate the vanishing of the Callan-Symanzik β-function of the model by employing a recently discovered supersymmetric antighost Ward identity.     

The low-lying mass spectrum of the N = 1 SU(2) SUSY Yang-Mills theory with Wilson fermions

We analyze the low energy spectrum of bound states of the N = 1 SU(2) SUSY Yang-Mills Theory (SYM). This work continues the investigation of the non-perturbative properties of SYM by Monte Carlo simulations in the Wilson discretization with dynamical gluinos. The dynamics of the gluinos is included by the Two-Step Multi-Bosonic Algorithm (TSMB) for dynamical fermions. A new set of configurations has been generated on a lattice at and . The analysis also includes sets of configurations previously generated on a smaller ( ) lattice at and 0.1955. Guided by predictions from low energy Lagrangians, we consider spin-1/2, scalar and pseudoscalar particles. The spectrum of SYM is a challenging subject of investigation because of the extremely noisy correlators. In particular, meson-like correlators contain disconnected contributions. The larger time-extension of the lattice allows to observe two-state signals in the effective mass. Finite-volume effects are monitored by comparing results from the two lattice sizes.Received: 14 September 2004, Revised: 29 October 2004, Published online: 21 December 2004     

Conformal bootstrap in Liouville field theory

We consider the recently proposed analytic expression for the three-point function in the Liouville field theory on a sphere. It is verified that in the classical limit this expression reduces to what the classical Liouville theory predicts. Using the suggested three-point function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and check numerically that it satisfies the conformal bootstrap equations. The Liouville reflection amplitude which follows explicitly from the structure constants is also considered and compared with the results of the Bethe ansatz technique.     

Modular bootstrap in Liouville field theory

The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toric 1-point functions in the Liouville field theory with DOZZ structure constants is proved.     

Lagrangian of the N = 1 supersymmetric Yang-Mills theory in the formalism of unified field

A method is proposed to use matrix nonlinear equations in order to construct the Lagrangian of the N = 1 supersymmetric Yang-Mills theory and the algebra of generators of transformations, closed off-shell in the 1.5-order formalism. Contrary to the superfield approach, there is no need to impose a Wess—Zumino type gauge fixing.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 16–21, September 1989.     

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q ₁,q ₂)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.     

Exact results on equations of motion in vacuum string field theory

We prove some algebraic relations on the translationally invariant solutions and the lump solutions in vacuum string field theory. We show that up to the subtlety at the midpoint the definition of the half-string projectors of the known sliver solution can be generalized to other solutions. We also find that we can embed the translationally invariant solution into the matrix equation of motion with the zero-mode.     

Dynamics of localized states in N = 4 SUSY QM

A consistent approach is offered for investigating the temporal dynamics of localized states. It is based on exactly solvable quantum mechanical models with multi-well potentials and the associate propagators. The Hamiltonian states with multi-well potentials form an adequate basis for expanding wave packets (WP) of various types and degrees of localization. Special features of WP tunneling have been studied with due regard for all Hamiltonian states with symmetric and asymmetric potentials.     

New quantum treatment of Liouville field theory

We exhibit a new treatment of the quantum Liouville theory in a box. In this treatment, the central charge of the Virasoro algebra is finite at the critical dimension of the associated string model, and we show how to reconstruct conformally covariant quantum field operators, in terms of a set of equally spaced harmonic oscillators.     

Defects and permutation branes in the Liouville field theory

The defects and permutation branes for the Liouville field theory are considered. By exploiting cluster condition, equations satisfied by permutation branes and defects reflection amplitudes are obtained. It is shown that two types of solutions exist, discrete and continuous families.     

Multi-instanton check of the relation between the prepotential F and the modulus u in N = 2 SUSY Yang-Mills theory

By examining multi-instantons in N = 2 supersymmetric SU(2) gauge theory, we derive, on very general grounds, and to all orders in the instanton number, a relationship between the prepotential

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(Φ), and the coordinate on the quantum moduli space u = TrΦ². This relation was previously obtained by Matone in the context of the explicit Seiberg-Witten low-energy solution of the model. Our findings can be viewed as a multi-instanton check of the proposed exact results in supersymmetric gauge theory.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.