Critical Ising Model with Boundary Magnetic Field: RG Interface and Effective Hamiltonians

Critical 2D Ising model with a boundary magnetic field is arguably the simplest QFT that interpolates between two non-trivial fixed points. We use the diagonalising Bogolyubov transformation for this model to investigate two quantities. Firstly we explicitly construct an RG interface operator that is a boundary condition changing operator linking the free boundary condition with the one with a boundary magnetic field. We investigate its properties and in particular show that in the limit of large magnetic field this operator becomes the dimension 1/16 primary field linking the free and fixed boundary conditions. Secondly we use Schrieffer-Wolff method to construct effective Hamiltonians both near the UV and IR fixed points.Comment: 38 pages; v.2 minor changes, to appear in JHE

On Asymptotic Hamiltonian for SU(N) Matrix Theory

We compute the leading contribution to the effective Hamiltonian of SU(N) matrix theory in the limit of large separation. We work with a gauge fixed Hamiltonian and use generalized Born-Oppenheimer approximation, extending the recent work of Halpern and Schwartz for SU(2). The answer turns out to be a free Hamiltonian for the coordinates along the flat directions of the potential. Applications to finding ground state candidates and calculation of the correction (surface) term to Witten index are discussed.Comment: 13 pages, Latex; v2: a reference added; v3: References to the papers by M.B. Green and M. Gutperle are added. The complete calculation of the Witten index for SU(N) matrix theory follows from combination of the results of our paper with the results of M.B. Green and M. Gutperle and the results obtained by G. Moore, N. Nekrasov, and S. Shatashvil

Renormalization group defects for boundary flows

Recently Gaiotto [1] considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet ones for a given renormalization group flow. In this paper we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the IR conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by \psi_{13} perturbation we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of (p,2)--> (p-1,1)+(p+1,1) flows.Comment: 1+23 pages, 2 Latex figures; v.3: minor corrections throughout the text, references adde

1/4-BPS states on noncommutative tori

We give an explicit expression for classical 1/4-BPS fields in supersymmetric Yang-Mills theory on noncommutative tori. We use it to study quantum 1/4-BPS states. In particular we calculate the degeneracy of 1/4-BPS energy levels.Comment: 15 pages, Latex; v.2 typos correcte

Compactification of M(atrix) theory on noncommutative toroidal orbifolds

It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra B_{\theta} that can be defined as a crossed product of noncommutative torus and the group Z_{2}. Our paper is devoted to the study of projective modules over B_{\theta} (Z_{2}-equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for B_{\theta} algebras working out the two-dimensional case in detail.Comment: 19 pages, Latex; v2: comments clarifying the duality group structure added, section 5 extended, minor improvements all over the tex

Gradient formula for the beta-function of 2d quantum field theory

We give a non-perturbative proof of a gradient formula for beta functions of two-dimensional quantum field theories. The gradient formula has the form \partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric, b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.Comment: LaTex file, 31 pages, no figures; v.2 referencing corrected in the introductio

General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions

We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group flow on the space of boundary conditions is generated by the boundary beta functions \beta^{a}(\lambda) for the boundary coupling constants \lambda^{a}. We prove a gradient formula \partial\ln z/\partial\lambda^{a} =-g_{ab}^{S}\beta^{b} where z(\lambda) is the boundary partition function at given temperature T=1/\beta, and g_{ab}^{S}(\lambda) is a certain positive-definite metric on the space of supersymmetric boundary conditions. The proof depends on canonical ultraviolet behavior at the boundary. Any system whose short distance behavior is governed by a fixed point satisfies this requirement. The gradient formula implies that the boundary energy, -\partial\ln z/\partial\beta = -T\beta^{a}\partial_{a}\ln z, is nonnegative. Equivalently, the quantity \ln z(\lambda) decreases under the renormalization group flow.Comment: 21 pages, Late

Noncommutative supergeometry, duality and deformations

We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\} =0$). We develop the theory of connections on modules over $Q$-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case $SO(d,d,{\bf Z})$-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that $Q$-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar $Q$-algebras can be constructed also in the case when the deformation parameter is not formal.Comment: Extended version of hep-th/991221