[Paper Review] Noncompact CFT and the g-theorem

Shyamoli Chaudhuri|arXiv (Cornell University)|Aug 26, 2004

Quantum Electrodynamics and Casimir EffectPhysics and Astronomy3 references3 citations

TL;DR

This paper clarifies the behavior of vacuum degeneracy g in noncompact two-dimensional conformal field theories (CFTs) under worldsheet renormalization group (RG) flow, confirming that g decreases while the central charge is preserved. Using a pedagogical review of Affleck and Ludwig's g-theorem, it demonstrates that the infrared fixed point of an unstable type II string theory without branes is a noncompact, supersymmetric, zero-temperature vacuum, with g correctly computed via normalizable bosonic states.

ABSTRACT

We clarify recent conjectures that the vacuum degeneracy g of a noncompact worldsheet sigma model with a continuous spectrum of scaling dimensions is lowered under renormalization group flow while preserving the central charge. Starting with a pedagogical review of Affleck and Ludwig's g-theorem, we clarify the precise distinctions necessary in applying this result from unitary and compact two-dimensional conformal field theory (CFT) to the description of worldsheet renormalization group (RG) flows in string theory. As an illustration, we use the worldsheet RG to describe the asymptotic approach to the noncompact, zero temperature, limit of the unstable finite temperature type II string theory in the absence of branes, or of a Yang-Mills sector. At intermediate points along the flow from unstable ultraviolet to stable infrared fixed point, we confirm that the vacuum degeneracy is correctly computed by restricting to the normalizable bosonic states in the worldsheet sigma model. We find that the vacuum degeneracy is indeed lowered under worldsheet RG flow, and that the infrared stable endpoint is the noncompact, and supersymmetric, zero temperature vacuum, in broad agreement with the recent conjectures.

Motivation & Objective

• To clarify the application of the g-theorem—originally formulated for unitary, compact 2D CFTs—to noncompact worldsheet sigma models in string theory.
• To investigate whether vacuum degeneracy g decreases under worldsheet RG flow while preserving the central charge in noncompact, continuous-spectrum CFTs.
• To analyze the asymptotic behavior of unstable type II string theory at finite temperature in the absence of branes or Yang-Mills sectors.
• To confirm that normalizable bosonic states correctly compute g along the RG flow to the infrared fixed point.

Proposed method

• Conducts a pedagogical review of Affleck and Ludwig's original g-theorem for unitary, compact 2D CFTs to establish foundational principles.
• Adapts the g-theorem to noncompact CFTs by focusing on the role of normalizable states in computing vacuum degeneracy g.
• Analyzes the worldsheet RG flow from an unstable ultraviolet fixed point to a stable infrared fixed point in a noncompact sigma model.
• Examines the limit of zero temperature and absence of branes or Yang-Mills sectors to identify the infrared endpoint.
• Uses the spectrum of normalizable bosonic states to compute g at intermediate and infrared points along the RG trajectory.
• Compares the UV and IR values of g to confirm monotonic decrease under RG flow, consistent with the g-theorem.

Experimental results

Research questions

• RQ1 Does the vacuum degeneracy g decrease under worldsheet RG flow in noncompact CFTs, despite a continuous spectrum of scaling dimensions?
• RQ2 Can the g-theorem be consistently extended from compact, unitary CFTs to noncompact, continuous-spectrum CFTs relevant to string theory?
• RQ3 What is the infrared fixed point of an unstable type II string theory in the absence of branes or Yang-Mills sectors?
• RQ4 Is the vacuum degeneracy g correctly captured by restricting to normalizable bosonic states in noncompact worldsheet models?
• RQ5 Does the central charge remain invariant during the RG flow from unstable UV to stable IR fixed point in this noncompact setting?

Key findings

• The vacuum degeneracy g is indeed lowered under worldsheet RG flow in the noncompact CFT, confirming the conjecture.
• The central charge is preserved throughout the RG flow, consistent with the g-theorem's expectations.
• The infrared fixed point is identified as a noncompact, supersymmetric, zero-temperature vacuum state.
• Normalizable bosonic states provide the correct computation of g at all points along the RG trajectory.
• The RG flow from the unstable UV theory to the stable IR theory is consistent with the g-theorem when properly adapted to noncompact settings.
• The results support recent conjectures on vacuum degeneracy in noncompact string theory backgrounds without branes or Yang-Mills sectors.