Abstract: String backgrounds solve Einstein's equations. The worldsheet theory 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. It follows that the worldsheet theory is a QFT rather than a CFT. This means that, in addition to the Lagrangian, a UV cutoff with dimensions of length must be specified.
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. Instead, at the n-th order of perturbation theory, Tseytlin integrates all n vertex operators over the sphere to obtain an n-point correlator. This introduces log divergences as (n-1) points come together on the sphere. To obtain the correct spherical string amplitude Z_0, Tseytlin's formalism therefore differentiates rather than divides by the log of the UV cutoff.
In this talk, I will explain Tseytlin’s formalism for constructing classical 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 then explain how the classical off-shell action was used by Susskind and Uglum to calculate the classical black hole entropy on a conical manifold in Rindler background. Link to the Event Video |