Kosmic Field Theories: Towards Holographic Duals for Unitary String Cosmologies

Abstract

Recent work on cosmological amplitudes has established reality conditions (derived from unitarity) for general particle-creation processes in flat FLRW cosmologies, in the Bunch-Davies wavefunction. In light of these results, we propose a new class of large-$N$ holographic gauge theories with $d$ spatial dimensions, which we call Kosmic Field Theories (KFTs), of which only a subset are conformal (KCFTs). By imposing an appropriate -- $d$ dependent -- complex phase for the number of colours $N$ and the t' Hooft coupling $λ$, we show that the complex phases of $n$-point functions in a KFT match the reality conditions required by $d + 1$ bulk unitarity, to all orders in bulk and boundary perturbation theory, including loop diagrams. Since $N^2/λ$ is imaginary in all dimensions, the class of (parity-even) KFTs has no overlap with the class of Wick rotations of unitary QFTs. We also make a preliminary investigation of positivity conditions, which might constrain the overall $\pm$ sign of $N^2$ and $λ$. Although SU($N$) models with adjoint matter can be defined in generic real dimensions $d$, several features of the class of KFTs are nicer when $d$ is odd, allowing $N^2$ to be real. This includes the possibility of non-oriented and/or open string duals, reality of the effective number of degrees of freedom, and closure under RG flow beyond the leading order in a $1/N$ expansion. This could explain why we live in a cosmology with an even number of spacetime dimensions.

Figures (4)

• Figure 1: Penrose diagram of de Sitter spacetime illustrating multiple foliations and key geometric features. The blue dashed lines represent the Poincaré slicing, which is conformally flat, covers only half of the spacetime, and asymptotes to the future boundary (shared with the global slicing of de Sitter); this slicing is most relevant for inflationary cosmology, as it naturally describes an expanding universe with flat spatial sections. In this framework, we are metaobservers at $\mathcal{I}^+$, where late-time cosmological correlators are measured. The red dashed lines correspond to the global slicing, which foliates the entire spacetime into spatial $(D-1)$-spheres of constant global time. The thick red horizontal line at the top denotes the future de Sitter boundary $\mathcal{I}^+$, where the holographic dual in the dS/CFT correspondence is proposed to reside and encode information about cosmological correlators. The green shaded region indicates the static patch accessible to an observer at the South Pole. Each point in the interior of the diagram corresponds to a $(D-2)$-sphere whose radius varies across the diagram: it shrinks to zero at the left and right edges, representing the South Pole and North Pole respectively, which are the poles of the spatial slices. This shrinking radius captures the spherical geometry of spatial sections in global coordinates.
• Figure 2: A connected Feynman diagram with Euler characteristic $\chi=\bar{V}-\bar{E}+\bar{F}$ on the boundary metric $\bar{g}_{ab}$ and Weyl factor of $\bar{\Omega}$ is dual to a string worldsheet of genus $g$ with $\chi=2(1-g)$ with bulk metric $g_{ab}$ and Weyl factor of $\Omega$. This motivates the interpretation of large-$N$ gauge theories in the ’t Hooft limit as dual to closed strings, even in cosmological settings. Schematic correspondence between a 2-loop boundary ribbon graph at $\mathcal{I}^+$ and bulk closed-string worldsheet sphere (genus $g=0$) diagram. The simplest ribbon Yang-Mills diagram (with at least 1 vertex) that contributes at bulk tree-level is shown.
• Figure 3: The genus 1 analogue of Figure \ref{['fig:BulkStringBoundaryGenusZero']}, showing the schematic correspondence between a 4-loop boundary ribbon graph at $\mathcal{I}^+$ and bulk closed-string worldsheet torus (genus $g=1$) diagram. This is the simplest non-planar Feynman diagram with $\bar{F} \ge 3$, allowing purely SU($N$) fields to interact (it vanishes in pure Yang-Mills due to Jacobi identities, but could contribute in other theories). Here we have chosen to interpret 2 of the 6 trivalent vertices as single-trace operator insertions (shown as circles) while the other 4 arise from expanding the boundary Lagrangian.
• Figure 4: A worldsheet is pinched on a circle to form a double-trace vertex that creates/annihilates 2 strings. We normalise such vertices in such a way that the $N$ scaling of both diagrams is the same.