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Generalized Additivity in Unitary Conformal Field Theories

Gideon Vos

TL;DR

The paper studies generalized additivity in twist space for 4D unitary CFTs through the conformal bootstrap, showing that large-spin double-trace operators accumulate at twists $\tau=2\Delta+2N$ for any integer $N$ and extending the known $N=0$ result to all $N$.Using conformal partial waves and crossing symmetry, it derives a leading anomalous dimension $\gamma_s=-\frac{2c_{\tau_m}}{s^{\tau_m}}$ with an explicit coefficient $c_{\tau_m}$ tied to minimal-twist data and OPE coefficients, and then generalizes to arbitrary $N$ via a recursion that preserves the $N=0$ limit.An AdS perspective shows these CNFT results correspond to gravitational interactions in the bulk, with stress-tensor exchange reproducing the leading energy shifts and matching gravity-side predictions, including polynomial $N$-dependence in suitable regimes.Overall, the work demonstrates a robust, non-perturbative mechanism by which twist additivity emerges in unitary CFTs without relying on large-N, offering insights into the holographic structure and potential limitations tied to the twist gap and spacetime dimension.

Abstract

It was demonstrated in recent work that $d=4$ unitary CFT's satisfy a special property: if a scalar operator with conformal dimension $Δ$ exists in the operator spectrum, then the conformal bootstrap demands that large spin primary operators have to exist in the operator spectrum of the CFT with a conformal twist close to $2Δ+2N$ for any integer $N$. In this paper the conformal bootstrap methods that were used to find the anomalous dimension of the $N=0$ operators have been generalized to find the anomalous dimension of all large spin operators of this class. In AdS these operators can be interpreted as the excited states of the product states of objects that were found in other works.

Generalized Additivity in Unitary Conformal Field Theories

TL;DR

The paper studies generalized additivity in twist space for 4D unitary CFTs through the conformal bootstrap, showing that large-spin double-trace operators accumulate at twists $\tau=2\Delta+2N$ for any integer $N$ and extending the known $N=0$ result to all $N$.Using conformal partial waves and crossing symmetry, it derives a leading anomalous dimension $\gamma_s=-\frac{2c_{\tau_m}}{s^{\tau_m}}$ with an explicit coefficient $c_{\tau_m}$ tied to minimal-twist data and OPE coefficients, and then generalizes to arbitrary $N$ via a recursion that preserves the $N=0$ limit.An AdS perspective shows these CNFT results correspond to gravitational interactions in the bulk, with stress-tensor exchange reproducing the leading energy shifts and matching gravity-side predictions, including polynomial $N$-dependence in suitable regimes.Overall, the work demonstrates a robust, non-perturbative mechanism by which twist additivity emerges in unitary CFTs without relying on large-N, offering insights into the holographic structure and potential limitations tied to the twist gap and spacetime dimension.

Abstract

It was demonstrated in recent work that unitary CFT's satisfy a special property: if a scalar operator with conformal dimension exists in the operator spectrum, then the conformal bootstrap demands that large spin primary operators have to exist in the operator spectrum of the CFT with a conformal twist close to for any integer . In this paper the conformal bootstrap methods that were used to find the anomalous dimension of the operators have been generalized to find the anomalous dimension of all large spin operators of this class. In AdS these operators can be interpreted as the excited states of the product states of objects that were found in other works.

Paper Structure

This paper contains 13 sections, 73 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The operator product expansion
  3. Conformal partial wave decomposition
  4. The conformal bootstrap
  5. A short review of KZ.
  6. Evaluating the s-channel CPW decomposition
  7. The anomalous dimension coefficient
  8. Generalizing to arbitrary $N$
  9. The full t-channel
  10. The full s-channel
  11. The anomalous dimension
  12. An AdS point of view
  13. Conclusion

Figures (4)

  • Figure 1: In this picture the coordinate system is displayed in which $z$ and $\bar{z}$ take on a more clear meaning as lightcone coordinates. The point $x_1$ is located at (0,0), $x_3$ at (1,1) and $x_4$ at (L,L). The point $x_2$ is free and is considered at $z\rightarrow0$, $\bar{z}\rightarrow1$.
  • Figure 2: The difference between the t-channel (left) and s-channel (right) is in the way the pairs of operators in the four-point function have been contracted by taking the OPE's
  • Figure 3: The region where we will apply the conformal bootstrap is marked with the thick red line, compare this with the point in figure \ref{['coordinatepicture']}
  • Figure 4: A plot of the reduced anomalous dimension coefficients in table (\ref{['GravityCorrespondence']}). The solid lines are the exact 4th-order polynomials that match the data, see eqs (\ref{['polyfit']}), (\ref{['polyfit2']}) and (\ref{['polyfit3']}).