CFT Dual for Timelike Geodesic in Lorentzian dS

Abstract: We construct the Euclidean CFT$<em>{d}$ dual of a generic massive scalar in Lorentzian dS$</em>{d+1}$ via analytic continuation. The resulting $PT$ defect defines a $PT$-invariant state that reproduces the Bunch-Davies Wightman function. However, the entanglement entropy captures only the real part of the central charge. This motivates a single-geodesic dual based on the timelike geodesic-integrated Wightman function, which yields the correlators between a bulk operator and a linear combination of an OPE block and its Casimir partner. We also derive the associated conformal defect and anomaly from an integral identity of the dS/CFT symmetry group.

• The paper constructs a PT-invariant bulk state that reproduces the Bunch-Davies vacuum via an infinite sum over conformal descendants.
• It rigorously maps timelike geodesic-integrated correlators to OPE blocks, revealing sensitivity to the imaginary part of the complex central charge.
• The study extends to defect anomalies and entanglement entropy, offering a unified holographic framework for non-unitary dS/CFT dualities.

Holography for Timelike Geodesics in Lorentzian de Sitter and CFT Dual Structure

Introduction and Motivation

This work presents a systematic investigation of the conformal field theory (CFT) dual of massive scalar fields in $(d+1)$-dimensional Lorentzian de Sitter (dS$_{d+1}$) spacetime, focusing on the formal, symmetry-based mapping of bulk correlation functions, including those integrated along timelike geodesics, to structures in a dual Euclidean CFT$_d$. The authors address the significant challenge posed by de Sitter’s lack of a conventional global timelike boundary and the resulting non-unitary features of the conjectured dual theory.

The main contributions include the explicit construction of $PT$-invariant bulk states that reproduce the Bunch-Davies vacuum Wightman function, characterization of the Hilbert space and adjoint structure consistent with dS isometries, a detailed analysis of entanglement entropy in CFT$_2$ with complex central charge, and a novel treatment of geodesic-integrated correlators and their interpretation in terms of OPE blocks and their Casimir partners.

AdS/dS Analytic Continuation and dS/CFT Symmetry Analysis

The basic holographic paradigm relates AdS$_{d+1}$ to CFT$_d$ by virtue of their shared SO($d$,2) symmetry. Lorentzian de Sitter emerges via analytic continuation of AdS coordinates, yielding an SO($d$+1,1) isometry interpreted as the conformal group acting on the spacelike past and future boundaries $(t\to\pm\infty)$.

For AdS/CFT, the Hermiticity structure of the CFT is standard, leading to unitary representation theory and, in three dimensions, central charge $_{d+1}$0. Under analytic continuation $_{d+1}$1, the central charge becomes complex, resulting in a non-unitary Euclidean CFT dual for dS$_{d+1}$2.

The algebra of conformal generators is preserved, but the adjoint operation is non-standard in dS/CFT: e.g., for sl(2,$_{d+1}$3), $_{d+1}$4 rather than $_{d+1}$5. This structure, which depends on the regime of the conformal dimensions, necessitates a doubled Hilbert space and $_{d+1}$6-symmetry as a substitute for unitarity.

Bulk State Construction and $_{d+1}$7-Invariant Dual

The dual bulk local state is constructed as an infinite sum over descendants of a conformal primary:

$_{d+1}$8

where $_{d+1}$9 is fixed by representation theory and $_d$0 are translation generators. This state satisfies a bulk localization constraint and reproduces bulk two-point functions via analytic continuation from AdS to dS. The crucial observation is that the correct bulk-invariant structure requires a $_d$1-twisted inner product, with $_d$2 acting (in bulk coordinates) as the antipodal map.

The resulting construction naturally encodes the doubled static patch and leads to the generalized Hilbert space used to compute dS$_d$3 correlation functions. The overlap of antipodally mapped local states, properly normalized, yields the Bunch-Davies Wightman function.

Entanglement and Central Charge Structure

A key, and perhaps counter-intuitive, result is that for the $_d$4-invariant states, the reduced density matrix and entanglement entropy (computed in dual CFT$_d$5 for a single interval) are only sensitive to the real part of the complex central charge:

$_d$6

with $_d$7 the interval length and $_d$8 the UV cutoff. The classical bulk contribution ($_d$9 for $PT$0) is not detected; instead, only the one-loop shift (the real $PT$1) contributes.

Averaging the geodesic lengths for $PT$2-related geodesics in the bulk similarly gives the same result, confirming that the entanglement entropy, whether calculated geometrically or via CFT, is blind to the classical piece in dS holography.

Timelike Geodesic Integrals and Bulk-OPE Block Dictionary

The central technical advance is the analysis of the correlator

$PT$3

where $PT$4 is the Bunch-Davies Wightman function and $PT$5 is a timelike geodesic anchored between antipodal boundary points. The explicit integration uses Barnes-type representations and contour integration techniques.

Figure 1: The Barnes clockwise contour $PT$6 encloses the series of poles $PT$7, $PT$8, relevant to the geodesic integral in dS.

Depending on the hyperbolic separation and sign structure, either the clockwise ($PT$9) or anti-clockwise ($_2$0) Barnes contours are used:

Figure 2: The Barnes anti-clockwise contour $_2$1 for integrating over $_2$2, capturing contributions from the conformal dimensions of the OPE block and its partner.

A striking outcome is that the geodesic-integrated object obeys precisely the same hypergeometric equation as the OPE block in CFT$_2$3. Explicitly, the result can be recast as a linear combination of an OPE block for $_2$4 and its "partner" corresponding to $_2$5 (the two roots of the wave equation). The explicit coefficients are determined by analytic continuation and normalization.

This demonstrates that, in contrast to the entanglement entropy, the geodesic-integrated Wightman function is sensitive to the imaginary part of the central charge—the “classical” gravitational effect. Thus, this observable completes the bulk/boundary dictionary for dS/CFT, assigning the real part of $_2$6 to information-theoretic quantities and the imaginary part to bulk geometric data.

Conformal Defects, Tilt Operators, and Anomalies

The paper extends these results to the theory of conformal defects. The presence of the $_2$7-defect and its associated $_2$8 symmetry enables the application of recent developments in the analysis of defect anomalies. The authors derive integrated Ward identities, classify local anomaly functionals $_2$9, and relate them to multilinear combinations of "tilt" operators localized on the defect.

For $_{d+1}$0 defects (lines), the anomaly is captured by Wess-Zumino-type terms and is classified by $_{d+1}$1. The bulk interpretation is as a worldline sigma model coupled to background NS-NS flux, with the defect fluctuations mapping to Goldstone modes on $_{d+1}$2.

Theoretical and Practical Implications

This study clarifies the operation of holography in non-unitary, cosmological settings. Specifically:

• Hilbert space structure: The $_{d+1}$3-twisted adjoint and doubled Hilbert space underpinning dS/CFT leads to new classes of dual states and observables absent in standard (unitary) holographic dualities.
• Bulk reconstruction: Timelike geodesic integrals, and their CFT duals involving OPE blocks and partners, provide a way to probe classically gravitational information not accessible to canonical information-theoretic measures.
• Defect anomalies: The anomaly classification and the connection to bulk worldline effective actions suggest robust mathematical structures that transcend particular spacetime signatures, with possible connections to brane theory and flow of topological charges.
• Future directions: Extending these constructions to fields with spin, interacting QFTs in dS, or to explicit models of dynamical gravity, and understanding the stability of such non-unitary Hilbert spaces, are all pressing directions.

Conclusion

The paper establishes a precise, symmetry-informed framework for dualizing bulk timelike geodesic observables in Lorentzian de Sitter space to composite, $_{d+1}$4-twisted structures in a non-unitary CFT$_{d+1}$5. The explicit treatment of entanglement, timelike geodesic integrals, and defect anomalies provides a unified approach to holographic duality beyond AdS/CFT, with mathematical and physical implications for quantum gravity, cosmology, and non-Hermitian QFT.

Figure 1: The Barnes clockwise contour $_{d+1}$6 used in the analytic continuation of the integral representation.

Figure 2: The Barnes anti-clockwise contour $_{d+1}$7, capturing the poles at $_{d+1}$8.