Higher order conformal perturbation theory is studied for theories with and without boundaries. We identify systematically the universal quantities in the beta function equations, and we give explicit formulae for the universal coefficients at next-to-leading order in terms of integrated correlation functions. As an example, we analyse the radius-dependence of the conformal dimension of some boundary operators for the case of a single Neumann brane on a circle, and for an intersecting brane configuration on a torus, reproducing in both cases the expected geometrical answer. *
what follows. We therefore hope to provide a bird's-eye view of the different directions that research into conformal field theories (CFTs) with boundaries or defects is taking. Quantum field theory (QFT) lies at the heart of much of modern theoretical physics: it describes systems in particle physics, condensed matter physics, and even quantum gravity, via holographic duality. Combined with the machinery of the renormalization group (RG), one can in principle systematically study an enormous variety of phenomena at different length scales in QFT-and therefore in nature. CFTs occupy privileged places in the space of QFTs-they lie at the endpoints of RG flows. Therefore, they often characterize the ultraviolet (UV) and infrared (IR) limits of QFTs. Moreover, CFTs describe critical phenomena and the worldsheet theory in string theory. They also provide important testing grounds for integrability, duality, and other general phenomena in QFT. CFTs are powerful because they are highly symmetric. In particular, CFTs are scale-invariant by definition, and so all correlation lengths are infinite. CFTs are also invariant under translations, rotations, boosts, and inversions. These symmetries constrain correlation functions, sometimes completely, providing a powerful non-perturbative approach to many aspects of QFT. However, no real-world system has infinite size-boundary effects are always important. Moreover, all real-world systems involve impurities, differently-ordered regions separated by domain walls, and other types of defects that break translational and rotational symmetries to subgroups 2. In short, the application of CFT to the real world necessarily entails studying boundary CFTs (bCFTs) and defect CFTs (dCFTs). The reduced symmetry of bCFTs and dCFTs compared to CFTs loosens spectral constraints, making correlation functions richer and more intricate. For example, in a bCFT, scalar operators can have non-zero one-point functions, which are generally forbidden in a CFT. However, bCFTs and dCFTs often retain enough symmetry to provide calculable non-perturbative information about many systems. For example, bCFT provides a solution to the single-impurity Kondo problem [1] and a fully non-perturbative definition of D-branes and other spacetime defects in string theory [2]. Remarkably, bCFT and dCFT can also provide insight into RG flows beyond the critical endpoints. Indeed, consider a CFT with a relevant deformation that triggers an RG flow to another CFT. Now, imagine integrating that relevant deformation over half the spacetime. Then, in the IR, the result is an interface between the UV and IR CFTs, called an RG interface or RG domain wall [11]. Such a construction, though clearly sacrificing some of the spacetime symmetry, is potentially very powerful: the problem of classifying RG flows between CFTs maps onto the problem of classifying defects between CFTs-which should be simpler, because much of the machinery of CFT can be brought to bear. Similarly, if the relevant deformation produces a mass gap, then the ...
In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. We show that the entropy at inverse temperature 2π satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for completely general CFT. We then restrict our attention to infraredstable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. For CFT in the range c L + c R < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2π. Under the same conditions we also prove that a CFT can have no more than c L +c R 48−c L −c R · exp{+4π} − 2 marginal deformations.
The behaviour of boundary conditions under relevant bulk perturbations is studied for the Virasoro minimal models. In particular, we consider the bulk deformation by the least relevant bulk field which interpolates between the mth and (m − 1)th unitary minimal model. In the presence of a boundary, this bulk flow induces an RG flow on the boundary, which ensures that the resulting boundary condition is conformal in the (m − 1)th model. By combining perturbative RG techniques with insights from defects and results about nonperturbative boundary flows, we determine the endpoint of the flow, i.e. the boundary condition to which an arbitrary boundary condition of the mth theory flows to.
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