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arXiv:hep-th/0306188 · Journal of Statistical Mechanics Theory and Experiment

Integrable circular brane model and Coulomb charging at large conduction

Sergei L. Lukyanov, Alexander B. Zamolodchikov

Published May 20, 2004• 32 citations
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Abstract

We study a model of 2D QFT with boundary interaction, in which a two-component massless Bose field is constrained to a circle at the boundary. We argue that this model is integrable at two values of the topological angle, θ = 0 and π. For θ = 0 we propose an exact partition function in terms of solutions of an ordinary linear differential equation. The circular brane model is equivalent to the model of quantum Brownian dynamics commonly used in describing the Coulomb charging in quantum dots, in the limit of small dimensionless resistance g 0 of the tunnelling contact. Our proposal translates to a partition function of this model at integer charge.

DOI
10.1088/1742-5468/2004/05/p05003

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We study a model of 2D QFT with boundary interaction, in which a two-component massless Bose field is constrained to a circle at the boundary. We argue that this model is integrable at two values of the topological angle, θ = 0 and π. For θ = 0 we propose an exact partition function in terms of solutions of an ordinary linear differential equation. The circular brane model is equivalent to the model of quantum Brownian dynamics commonly used in describing the Coulomb charging in quantum dots, in the limit of small dimensionless resistance g 0 of the tunnelling contact. Our proposal translates to a partition function of this model at integer charge.
Some results for commuting local integrals of motion for free bosonic theories are presented, which can be applied in construction of integrable boundary states. We consider a particular model of two-dimensional quantum field theory on a disk, whose bulk dynamics is that of a two-component free massless Bose field X = (X, Y), and interaction occurs at the boundary, where the boundary values (XB, YB) are constrained to special curve—the “paperclip brane”. The interaction breaks conformal invariance, but we argue that it preserves integrability. We propose exact expression for the disk partition function (and more general overlap amplitudes ⟨P| B⟩ of the boundary state with all primary states) in terms of solutions of certain ordinary linear differential equations. Usable program code for calculation of commuting integrals of motion is also introduced.
In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced.
In the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with Z_{n} symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant.
Abstract We discuss generalized partition function of 2d CFTs on thermal cylinder decorated by higher qKdV charges. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary non-interacting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary non-interacting bosons allows extending thermal one-point functions to the full non-perturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite tower of chemical potentials in the limit of large central charge.
We construct boundary states in SU (2) k WZNW models using the bosonized Wakimoto free-field representation and study their properties. We introduce a Fock space representation of Ishibashi states which are coherent states of bosons with zero-mode momenta (boundary Coulomb-gas charges) summed over certain lattices according to Fock space resolution of SU (2) k . The Virasoro invariance of the coherent states leads to families of boundary states including the B-type D-branes found by Maldacena, Moore and Seiberg as well as the A-type corresponding to trivial current gluing conditions. We then use the Coulomb-gas technique to compute exact correlation functions of WZNW primary fields on the disk topology with A- and B-type Cardy states on the boundary. We check that the obtained chiral blocks for A-branes are solutions of the Knizhnik–Zamolodchikov equations.
Abstract The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS3.