g-function in perturbation theory

We present some explicit computations checking a particular form of gradient formula for a boundary beta function in two-dimensional quantum field theory on a disc. The form of the potential function and metric that we consider were introduced in hep-th/9210065, hep-th/9311177 in the context of background independent open string field theory. We check the gradient formula to the third order in perturbation theory around a fixed point. Special consideration is given to situations when resonant terms are present exhibiting logarithmic divergences and universal nonlinearities in beta functions. The gradient formula is found to work to the given order.Comment: 1+14 pages, Latex; v.2: typos corrected; v.3: minor corrections, to appear in IJM

Simple Elliptic Singularities: a note on their G-function

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus terms in various theories. For the simple singularities the G-function is known explicitly: G=0. The next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of three examples, \widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8, X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly constructed for these three examples. The main property is that the function depends on only one variable, the marginal (dimensionless) deformation variable. Other examples are given based on the foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular group is studied, and applications within the theory of integrable systems are discussed.Comment: 15 page

Concise analytic solutions to the quantum Rabi model with two arbitrary qubits

Using extended coherent states, an analytical exact study has been carried out for the quantum Rabi model (QRM) with two arbitrary qubits in a very concise way. The $G$-functions with $2 \times 2$ determinants are generally derived. For the same coupling constants, the simplest $G$-function, resembling that in the one-qubit QRM, can be obtained. Zeros of the $G$-function yield the whole regular spectrum. The exceptional eigenvalues, which do not belong to the zeros of the $G$ function, are obtained in the closed form. The Dark states in the case of the same coupling can be detected clearly in a continued-fraction technique. The present concise solution is conceptually clear and practically feasible to the general two-qubit QRM and therefore has many applications.Comment: 13 pages, 3 figure

Normalization of Off-shell Boundary State, g-function and Zeta Function Regularization

We consider the model in two dimensions with boundary quadratic deformation (BQD), which has been discussed in tachyon condensation. The partition function of this model (BQD) on a cylinder is determined, using the method of zeta function regularization. We show that, for closed channel partition function, a subtraction procedure must be introduced in order to reproduce the correct results at conformal points. The boundary entropy (g-function) is determined from the partition function and the off-shell boundary state. We propose and consider a supersymmetric generalization of BQD model, which includes a boundary fermion mass term, and check the validity of the subtraction procedure.Comment: 21 pages, LaTeX, comments and 3 new references adde