Conformal Field Theories in Fractional Dimensions

TL;DR

The paper investigates conformal field theories in fractional space-time dimensions by applying the conformal bootstrap with $D$ treated as a continuous parameter. It defines a non-perturbative analytic continuation of conformal blocks and crossing symmetry across $2 ext{≤}D ext{≤}4$, positing a line of fixed points interpolating between the 2D Ising model, the 3D Ising model, and the 4D free scalar. Sharp kinks in the bound on $oldsymbol{ riangle_oldsymbol{ ext{ε}}}$ as a function of $oldsymbol{ riangle_oldsymbol{ ext{σ}}}$ align with Ising data and track smoothly with $D$, with $oldsymbol{oldsymbol{ ext{$oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ε}}}}}}}}}$ expansion in good agreement for small $oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ε}}}}}}}$. A central-charge analysis reveals $c_T$ minimized at the kink and below the free-scalar value, consistent with leading $oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ε}}}}}}}$-expansion predictions.

Abstract

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary CFTs connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of phi^4 theory in the epsilon-expansion.

Figures (3)

• Figure 1: Upper bounds on $\gamma_\epsilon$ as a function of $\gamma_\sigma$, plotted for $D=2,2.25,\ldots,4$. For each $D<4$, the bound shows a kink, where a CFT belonging to the Ising model universality class is conjectured to live (black dots, fitted by the blue dashed curve). An example of theories in the bulk of the allowed region are Gaussian models, where $\gamma_\epsilon=2\gamma_\sigma$ (black dotted line).
• Figure 2: Black dots: The anomalous dimensions corresponding to the kinks in Fig. \ref{['fig:summary']}. Red bands: The same dimensions determined by Borel-resumming the $\varepsilon$-expansion series LeGuillou:1987ph. Since $\gamma_\sigma=O(\varepsilon^2)$, we use a square root scale on the $\gamma_\sigma$ axis.
• Figure 3: Black dots: The normalized central charge difference (on a square root scale) corresponding to the kinks in Fig. \ref{['fig:summary']}, interpolated by the blue curve. The dashed red line is the lowest-order $\varepsilon$-expansion prediction.