[论文解读] Universal Bounds on CFT Distance Conjecture

Hirosi Ooguri, Yifan Wang|arXiv (Cornell University)|May 1, 2024

Advanced Algebra and Logic被引用 5

一句话总结

证明在二维单位CFT中，如果一个主算符的维数沿着共形流形消失，则到该极限的 Zamolodchikov 距离为无穷大，且消失呈指数衰减，具有关于衰减速率的普遍界限。

ABSTRACT

For any unitary conformal field theory in two dimensions with the central charge $c$, we prove that, if there is a nontrivial primary operator whose conformal dimension $Δ$ vanishes in some limit on the conformal manifold, the Zamolodchikov distance $t$ to the limit is infinite, the approach to this limit is exponential $Δ= \exp(- αt +O(1) )$, and the decay rate obeys the universal bounds $c^{-1/2} \leq α\leq 1$. In the limit, we also find that an infinite tower of primary operators emerges without a gap above the vacuum and that the conformal field theory becomes locally a tensor product of a sigma-model in the large radius limit and a compact theory. As a corollary, we establish a part of the Distance Conjecture about gravitational theories in three-dimensional anti-de Sitter space. In particular, our bounds on $α$ indicate that the emergence of exponentially light states is inevitable as the moduli field corresponding to $t$ rolls beyond the Planck scale along the steepest path and that this phenomenon can begin already at the curvature scale of the bulk geometry. We also comment on implications of our bounds for gravity in asymptotically flat spacetime by taking the flat space limit and compare with the Sharpened Distance Conjecture.

研究动机与目标

• Motivate and formalize the CFT Distance Conjecture for two-dimensional unitary CFTs.
• Show that infinite-distance limits correspond to vanishing operator dimensions along geodesics on the conformal manifold.
• Establish universal bounds on the exponential decay rate of dimensions toward the limit.
• reveal emergent sigma-model structures and discuss AdS3 gravity implications.
• Provide concrete examples to illustrate the theorems and bounds.

提出的方法

• Define and use the Zamolodchikov metric on conformal manifolds and exactly marginal deformations.
• Prove that if a primary’s conformal dimension Δ→0 along a geodesic, the distance t is infinite (Theorem 1).
• Show that Δ decays exponentially as Δ=exp(−α t+O(1)) with universal bounds c^(-1/2) ≤ α ≤ 1 (Theorem 2).
• Demonstrate that in the limiting CFT there is an emergent sigma-model factor and no gap above the vacuum (Theorem 3).
• Derive a lower bound on α, α ≥ N^(-1/2) with N ≤ c (and stronger for superconformal cases), yielding c^(-1/2) ≤ α ≤ 1 (Theorem 4).
• Provide explicit examples from Narain moduli space and Quintic Calabi-Yau to illustrate the limits and bounds.

实验结果

研究问题

• RQ1 Do infinite-distance limits on 2D CFT conformal manifolds correspond to vanishing operator dimension gaps?
• RQ2 Can one bound the exponential decay rate α of Δ in terms of the central charge c and other data?
• RQ3 What is the structure of the limiting CFT when Δ→0—does it decompose into a sigma-model sector?
• RQ4 How do these 2D results inform the AdS3/CFT2 version of the Distance Conjecture?
• RQ5 Are the bounds on α saturated in explicit examples, and under what conditions?

主要发现

• If a geodesic on the conformal manifold leads to Δ→0, the Zamolodchikov distance to the limit is infinite (Theorem 1).
• The conformal dimension vanishes exponentially with rate Δ=exp(−α t+O(1)) and 1 ≥ α ≥ c^(-1/2) (Theorem 2 and Theorem 4).
• In the limit, the CFT contains a sigma-model subalgebra with no gap above the vacuum (Theorem 3).
• For optimal choices, α saturates the bounds: α=1 in some limits and α=c^(-1/2) in others (as shown in examples).
• Examples include the Narain c=2 toroidal CFT and the c=6 quintic Calabi-Yau model; these demonstrate infinite-distance behavior and the corresponding decay rates.