dS/CFT Dictionaries and Analytic Continuation

dS/CFT uses analytic continuation to relate an AdS radial functional to a de Sitter late-time wavefunction. The continuation is a powerful calculational map, but it changes signs, phases, regularity conditions, and often the reality of boundary parameters. It does not by itself construct a unitary Euclidean CFT, a Lorentzian de Sitter inner product, or a complete bulk reconstruction map.

Required background. Late-Time Wavefunctions and Boundary Data fixes the target functional; The GKPW Generating-Functional Dictionary fixes the AdS starting point.

Helpful background. Wick Rotation and Analytic Continuation controls contours; Conjugation and Reflection Positivity supplies the positivity test.

What is analytically continued

Euclidean AdS$_{d+1}$ in Poincaré coordinates has

$$ds^2_{\mathrm{EAdS}}=\frac{L_{\mathrm A}^2}{z^2}(dz^2+d\mathbf x^2).$$

The formal continuation

$$z=-i\eta, \qquad L_{\mathrm A}=\frac{i}{H}$$

gives the expanding de Sitter metric. A solution regular in the EAdS interior continues to a positive-frequency/Bunch–Davies solution only with a specified $i\epsilon$ contour. Likewise,

$$Z_{\mathrm{CFT}}[\phi_0]\simeq e^{-S_{\mathrm{EAdS,on-shell}}[\phi_0]} \quad\longrightarrow\quad \Psi_{\mathrm{BD}}[\varphi]\simeq e^{iS_{\mathrm{dS,on-shell}}[\varphi]}.$$

The source is continued as boundary data, while the AdS vacuum regularity condition becomes state data. Counterterms must be continued too; otherwise local phases and finite pieces are compared in inconsistent schemes. Strominger’s proposal organizes de Sitter isometries as a Euclidean conformal group on $\mathcal I^+$ Strominger 2001, §§ 2–4.

First application: a free scalar kernel

An EAdS scalar has near-boundary weights

$$\Delta_\pm^{\mathrm A}=\frac d2\pm\sqrt{\frac{d^2}{4}+m_{\mathrm A}^2L_{\mathrm A}^2}.$$

After continuing the radius and mass parameters consistently, the de Sitter late-time weights are

$$\Delta_\pm^{\mathrm{dS}}=\frac d2\pm\sqrt{\frac{d^2}{4}-\frac{m_{\mathrm{dS}}^2}{H^2}}.$$

Continuing the regular EAdS bulk-to-boundary solution produces the Bunch–Davies Hankel mode and maps the renormalized nonlocal EAdS kernel $k^{2\nu}$ to the corresponding late-time wavefunction coefficient multiplied by a contour-dependent phase. Local analytic terms become local wavefunction phases and remain counterterm dependent. The map between AdS and dS wavefunctions, rather than a termwise equality of observables, is developed explicitly by Harlow and Stanford Harlow and Stanford 2011, §§ 2–4.

For $m_{\mathrm{dS}}^2/H^2>d^2/4$, write $\nu=i\mu$. Then

$$\Delta_\pm=\frac d2\pm i\mu.$$

These principal-series weights are natural representations of the de Sitter group, but a two-point function with complex scaling cannot be interpreted as that of a Hermitian primary in an ordinary reflection-positive Euclidean CFT without additional structure. The boundary object may be nonunitary, complex, or equipped with a nonstandard pairing.

Reality and inner products

The wavefunction is complex. Its conjugate is obtained by the conjugate contour, and probabilities use $\Psi^*\Psi$. A proposed Euclidean partition function for $\Psi$ therefore need not itself be real or positive. Bulk reality imposes relations among coefficient functions and their conjugates; it does not create Osterwalder–Schrader positivity.

The semiclassical continuation assumes $G_NH^{d-1}\ll1$. An AdS large-radius/string hierarchy does not necessarily continue to a controlled stable de Sitter compactification: $g_s$, $\alpha'H^2$, Kaluza–Klein modes, flux quantization, and vacuum decay need an independent model. Analytic continuation also need not commute with loop integration or late-time/infrared limits.

Adversarial control: test reflection positivity

Choose a principal-series mass and smear the candidate boundary two-point kernel with a test function supported at positive Euclidean time. Apply reflection and complex conjugation. If the resulting quadratic form is not real and nonnegative, the object is not an ordinary unitary Euclidean CFT correlator. Keep it as a wavefunction coefficient with its contour and pairing rather than deleting the failed test. Repeat after adding a local counterterm: only contact and phase data may change.

The evidence ceiling is a precise semiclassical analytic-continuation relation for many free and perturbative coefficient functions, plus conformal constraints at late time. It does not prove reflection positivity, a unique state beyond the contour, finite-time reconstruction, or nonperturbative dS/CFT. Higher-spin examples and normalized in-in observables require their own analyses.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Harlow, D., and Stanford, D. (2011). “Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT.” arXiv.
• Strominger, A. (2001). “The dS/CFT Correspondence.” Journal of High Energy Physics 2001(10), 034. DOI.