Chern-Simons-Ghost Theories and de Sitter Space

TL;DR

The paper investigates CS theories coupled to ghost-like fundamental matter in the large-$N$ limit, identifying an exact $N\to -N$ mapping to ordinary CS-matter on $\mathbb{R}^3$ that enforces a bosonization duality in this setting and supports a dS/CFT interpretation with a higher-spin bulk. It then shows that at high temperature on $S^1\times S^2$, the mapping and bosonization fail due to competing large-$N$ saddles and holonomy dynamics, yielding a rich phase structure with topological transitions and partial dualities. The analysis employs perturbative two-loop RG calculations in Landau gauge, nonperturbative light-cone gauge methods, and a detailed saddle-point treatment of thermal holonomies to connect boundary CS-ghost theories to bulk de Sitter gravity wavefunction data. Overall, the work clarifies when ghost CS theories reproduce dual bosonic/fermionic descriptions and highlights the distinct finite-temperature behavior that challenges straightforward holographic intuitions in the dS context. These results advance our understanding of non-unitary holography and the role of high-spin de Sitter duals in quantum gravity.

Abstract

We explore Chern-Simons theories coupled to fundamental ghost-like matter in the large $N$ limit at 't Hooft coupling $λ$. These theories have been conjectured to be holographically dual to parity-violating, asymptotically dS$_4$ universes with a tower of light higher-spin fields. On $\mathbb{R}^3$, to all orders in large-$N$ perturbation theory, we show that Chern-Simons-ghost theories are related to ordinary Chern-Simons-matter theories by mapping $N \rightarrow - N$ and keeping $λ$ fixed. Consequently, the bosonization duality of ordinary Chern-Simons-matter theories extends to a bosonization duality of Chern-Simons-ghost theories on $\mathbb R^3$. On $S^1 \times S^2$, in the small-$S^1$ limit, neither $N \rightarrow -N$ nor bosonization hold, as we show by extensively studying large-$N$ saddles of the theories with both ghost and ordinary matter. The partition functions we compute along the way can be viewed as pieces of the late-time Hartle-Hawking wavefunction for the bulk dS$_4$ gravity theories.

Figures (6)

• Figure 1: (Color online) All currently known duality maps between $U(N)$ Chern-Simons theories on $\mathbb{R}^3$ and parity-violating all-spin Vasiliev theories in de Sitter and anti-de Sitter, including the results in this paper. It should be noted that the bosonization maps hold along the RG flow and not just at the endpoints where the arrows are displayed. Similarly, for the bulk theories one can get from Type A with mixed boundary conditions to Type B with mixed boundary conditions by changing the parity-violating phase (although not in the simple way stated in the legend). An analogous set of dualities exists for the even-spin Vasiliev theories.
• Figure 2: The diagrams that contribute to the renormalization of the six-point coupling at two loops.
• Figure 3: The diagrams that contribute to the wavefunction renormalization at two loops.
• Figure 4: Free energies of the critical boson (left) and regular fermion (right) at all applicable saddle points, plotted with arbitrary normalization as functions of $\lambda_B$ (left) and $\lambda_F$ (right). On the right (regular fermion) plot, the blue/bottom-most line corresponds to the "$+$" saddle which is dominant in this $\lambda_F \geq 0$ regime, and the purple/top-most line corresponds to the "$-$" saddle, which is sub-dominant.
• Figure 5: The free energy of the regular bosonic model, plotted in arbitrary normalization as a function of $\lambda_B$ at $\lambda_6^b = 0$ (i.e. at $\hat{\lambda}_B = \lambda_B$). The purple/bottom line corresponds to the dominant, "$-$" saddle, and the blue/top line corresponds to the sub-dominant, "$+$" saddle.