Abstract. The worldsheet theory of string backgrounds is a CFT with zero central charge. This is the definition of on-shell string theory. In off-shell string theory, on the other hand, conformal invariance on the worldsheet is explicitly broken, and the worldsheet theory is therefore a QFT rather than a CFT, with a UV cutoff on the worldsheet. While Tseytlin's off-shell formalism can be used at arbitrary genus g, the treatment of the sphere diagram (genus-0) is particularly subtle. In particular, on the sphere, Tseytlin does not deal with the SL(2,C) group by fixing 3 points, as this prescription does not properly extend to the off-shell case. In this talk, I will explain Tseytlin’s formalism for constructing classical (sphere) off-shell effective actions and provide a general proof that it gives the correct equations of motion, to all orders in perturbation theory and α′. I will also show how Tseytlin's prescriptions are equivalent to quotienting out by the gauge orbits of a regulated moduli space with n operator insertions. In the second part of the talk, I will explain the underlying conceptual structure of the Susskind and Uglum black hole entropy argument. There I will show explicitly how the classical (tree-level) effective action and entropy S = A/4G_N may be calculated from the sphere diagrams. We also discuss the behavior of the Susskind and Uglum entropy under RG flow. Although the conical manifold smooths out under RG flow, moving towards an on-shell configuration, the entropy doesn't change.
I will also compare these off-shell results with the much more popular orbifold method for calculating entropy from the on-shell C/Z_N background. By considering processes involving twisted string states (basically the orbifold conserves twist and hence does not allow twisted strings to pinch off at the tip, while the off-shell NLSM of a slightly smoothed out cone obviously does allow this process), I will conclude that the orbifold backgrounds are fundamentally different from conical backgrounds at the same value of inverse temperature \beta---unless one allows tachyons to condense on the orbifold -- signalling that the analytically continued orbifold is a fundamentally non-geometrical construction. Thus, it is implausible that the orbifold background can be interpreted as the thermal partition function of any unitary statistical mechanical system at inverse temperature \beta.
Time permitting, I will end with some important insights into the ER=EPR hypothesis that can be obtained from the fact that the tachyon condensate (in orbifold backgrounds) at a codimension-2 surface is apparently equivalent to ordinary flat space. I will discuss prospects for deriving the holographic entanglement entropy (the RT formula) using the orbifold method.