Off-shell strings I: S-matrix and action

Abstract

We explain why Tseytlin’s off-shell formulation of string theory is well-defined. Although quantizing strings on an off-shell background requires an arbitrary choice of Weyl frame, this choice is not physically significant since it can be absorbed into a field redefinition of the target space fields. The off-shell formalism is particularly subtle at tree-level, due to the treatment of the noncompact conformal Killing group SL(2, \mathbb{C} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℂ</mml:mi> </mml:math> ) of the sphere. We prove that Tseytlin’s sphere prescriptions recover the standard tree-level Lorentzian S-matrix, and show how to extract the stringy i\varepsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> prescription from the UV cutoff on the worldsheet. We also demonstrate that the correct tree-level equations of motion are obtained to all orders in perturbation theory in g_s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> and \alpha^{\prime} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>α</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> , and illuminate the close connection between the string action and the c-theorem.

Topics & Concepts