Off-Shell Strings I: S-matrix and Action (2211.08607v3)

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,$\mathbb{C}$) 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 $i\varepsilon$ 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 $g_s$ and $\alpha^{\prime}$, and illuminate the close connection between the string action and the c-theorem.

• The paper introduces a consistent off-shell string formulation using Tseytlin’s framework to resolve gauge fixing and renormalization challenges.
• It derives the tree-level S-matrix by addressing the noncompact SL(2,C) symmetry through derivative prescriptions with respect to the UV cutoff.
• The study links renormalization group flows with string propagation, setting the stage for exploring effective actions and quantum gravity implications.

Insightful Overview of "Off-Shell Strings I: S-matrix and Action"

The paper "Off-Shell Strings I: S-matrix and Action" by Amr Ahmadain and Aron C. Wall addresses the longstanding puzzle of constructing a consistent off-shell formulation of string theory. By leveraging Tseytlin's framework for off-shell strings, the authors aim to resolve conceptual issues related to the definition of off-shell string theory, propose a method for calculating the string S-matrix, and derive equations of motion directly from the string worldsheet perspective.

Off-Shell String Theory: A Well-Defined Framework

Traditionally, string theory is defined on-shell, where worldsheet theories are conformally invariant, and physical amplitudes emerge from the moduli space integrals with certain symmetry groups modded out. However, the off-shell formulation, which accommodates non-conformally invariant worldsheet theories, faces questions about gauge fixing and physical interpretation. Tseytlin's off-shell framework, which the authors explore extensively, posits that the addition of non-conformally invariant elements allows for a reformulation where divergences and ambiguities are manageable. Specifically, gauge-fixing ambiguities associated with the Weyl factor on the worldsheet can be wholly absorbed into field redefinitions in the target space, a feature not uniquely problematic to string theory but shared with quantum field theories under renormalization.

Tseytlin's Prescriptions and the SL(2,$\mathbb{C}$) Challenge

A critical aspect addressed in this paper is the treatment of the noncompact SL(2,$\mathbb{C}$) symmetry on the worldsheet, particularly at tree level. While compactifying the integration over insertion points by fixing three vertex operators is a conventional approach, this leaves the sphere partition function zero due to the infinite volume of the SL(2,$\mathbb{C}$) group. Tseytlin circumvented this using derivative prescriptions with respect to $\log \epsilon$, where $\epsilon$ is the UV cutoff, leading to novel representations of the string amplitudes. The authors provide rationale validating Tseytlin's approach, emphasizing that correctly applied, it yields the appropriate tree-level S-matrix, proving consistency with the standard on-shell results.

Implications for String Theory and Beyond

One significant contribution of this paper is the thorough examination of the implications of off-shell string theory in string perturbation theory, especially concerning renormalization group (RG) flows. By integrating conformal perturbation theory with renormalization views, the authors argue that the UV cutoff on the worldsheet corresponds to an IR cutoff in target space. This correspondence not only links RG flows with string propagation but also addresses how locality issues in target space appear and can be managed.

Moreover, this formulation allows for an exploration of target space effective actions directly derivable within the string background, providing crucial insights into potential extensions to quantum gravity. The treatment of c-theorems within this framework connects deep fundamental concepts in theoretical physics, leveraging the flow and stationary points of associated C-functions as proofs of underlying physical processes and principles.

Conclusions and Future Directions

The paper lays a solid foundation for bridging the divide between on-shell and off-shell string formulations, expanding the conceptual toolkit available for theoretical exploration in string theory. Future work could explore facilitating calculations involving massive string modes and extending the results in diverse backgrounds, including non-geometrical compactifications. Additionally, implications for black hole thermodynamics and connections to recent developments in holography merit further paper, promising to shed light on quantum aspects of gravity. The associated second part of the work promises to extend these insights toward calculating black hole entropy, offering a substantive contribution to string cosmology and quantum gravity applications.