Holographic Complexity in Hybrid De Sitter Spacetime

The research delves into holographic complexity in a hybrid de Sitter spacetime, exploring the AdS/CFT correspondence, quantum information in the bulk, and computational complexity. It also examines the volume of the ERB, evolution of complexity in CFT, and probes cosmological horizons using holographic tools, shedding light on volume complexity in dS spacetime.

• Holographic Complexity
• AdS/CFT Correspondence
• Quantum Information
• Computational Complexity
• De Sitter Spacetime

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1. Holographic Holographic complexity in (hybrid) de Sitter in (hybrid) de Sitter spacetime complexity spacetime Based on arXiv:2302.03584 [hep-th], in collaboration with R. Auzzi, G. Nardelli, and G. Pedde Ungureanu arXiv:2303.05025 [hep-th], in collaboration with T. Anegawa, N. Iizuka, and S. K. Sake Nicol Zenoni (group B01) Osaka University April 4th, 2023

2. AdS/CFT [J. M. Maldacena, 1997] Theory of gravity in anti-de Sitter (AdS) Conformal Field Theory (CFT) on the AdS boundary boundary Gravity in (d+1)-dimensional AdS d-dimensional CFT (no gravity) Quantum information may unveil how the bulk emerges from the boundary! Nicol Zenoni (Osaka University) - April 4th, 2023 1

3. The volume of the ERB ?2 2 + 1 ?? 2 ??2 +1 ?? 2 2 ??2+ + ?2? ? 1 for ? > 2 ?2 2 ?? 2 ???? Eternal two-sided BH in AdSd+1spacetime ?? 2 : ??2= ???? ??2 ?2 ? 2 2 ?2 ? ???? ??2+???? 2 + ?2??2 for ? = 2 2 The volume ? of maximal ?-dimensional surfaces gives: 2 ?? ??? =16?????? ? 1 Linear growth at late times! lim ?? ? ?: black hole mass Which quantity in the dual CFT represents the growth of the ERB? Boundary time: ??= ??= ?? Nicol Zenoni (Osaka University) - April 4th, 2023 2

4. Computational complexity Complexity is the minimum number of simple operations required to build a quantum state ?(?) Target state Reference state ? = ?? ?1 Chaotic quantum circuit of ? qubits ? ?? Evolution of the ERB inside an AdSd+1black hole Evolution of complexity of the dual state in CFTd t Complexity = Volume (CV) conjecture [L. Susskind, 2014] ? ? ???? CFT2 ? ??(?) = max ? =? AdS3 Nicol Zenoni (Osaka University) - April 4th, 2023 3

5. A probe of cosmological horizons? It is tempting to employ holographic tools to studying cosmological horizons ? ? The stretched horizon acts as a holographic screen! ? ? ??? CV conjecture in dS ??(?) = max ? =? Nicol Zenoni (Osaka University) - April 4th, 2023 4

6. Volume complexity in dS 2 ?2 ??? 2 2 ?2+ ?2? ? 1 ??2 Pure dSd+1 spacetime: ??2= ??? ??2+??? 2 for ? > 1 2 ??? Volume complexity ??gives [E. J rstad et al., 2022]: 2(? 1) ? 1 ? ? ?? Hyperfast growth at a critical time! 2 ? ? ??? ??? ??? lim ?? ? 2(? 1) ? 1 Complexity diverges at a critical time! 2 ? ? ??? ? (? 1) ? ?? 1 lim ?? ? ?? Boundary time: ??= ??= ?? Volume complexity probes the expansion of dS! Nicol Zenoni (Osaka University) - April 4th, 2023 5

7. A "Frankenstein" geometry An alternative way of investigating dS holography is to exploit the AdS timelike boundary, by embedding dSd+1into AdSd+1[B. Freivogel et al., 2005] ? = ?(?) ?? ? ? = ?(?) ?? AdS black hole dS dS bubble in AdS The full extremal surface is obtained by gluing along the bubble surface an AdS part attached to the boundary and a dS part Refraction-like condition: ??, ??, and ? must be chosen so that the full surface has maximal volume Nicol Zenoni (Osaka University) - April 4th, 2023 6

8. Hybrid holography? If we introduce the stretched horizon, maximal surfaces extend beyond the cosmological horizon 1. The dS stretched horizon time is fixed as ??= ?? 2. The AdS boundary time is fixed as ??= ???? ?? 2? ??? ??? ??= 1 As in pure dS! ?? dS AdS black hole ?? 2? ??? ??? ??= 1 ?? Nicol Zenoni (Osaka University) - April 4th, 2023 7

9. A probe of expansion When the maximal surface is anchored at the stretched horizon, both in pure dS and in the hybrid dS geometry volume complexity probes the dS inflationary expansion TAKE AWAY Is this also true in the dS2case? [E. J rstad et al., 2022, S. Chapman et al., 2021] Hyperfast growth at a critical time! ??? ??? ????? ? ? Complexity remains finite! lim ?? ? ??= ?(1) lim ?? ? = Nicol Zenoni (Osaka University) - April 4th, 2023 8

10. The two-dimensional puzzle dS2is a solution of Jackiw-Teitelboim gravity, and can be built by (half) dimensional reduction starting from dS3[A. Svesko et al., 2022]: 2 ?2 2 ??2 ?(?)+ ?2??2, ? ? = ??? ??2= ?(?) ??2+ dS3spacetime ??? dimensional reduction ??2= ? ? ??2+??2 2?(?)2??2 ?(?)+ ??? Dilaton: replaces the higher-dimensional transverse sphere! dS2spacetime Volume contains information just on the metric, but dilaton is important! Nicol Zenoni (Osaka University) - April 4th, 2023 9

11. Dilaton matters Volume is not invariant under Weyl field redefinition of the metric: ??? ? ??? ? ?? ??? , To illustrate this, we can choose ? = ?? ? It is possible to reproduce the same result as in dS3spacetime! ?? ? Don t forget dilaton! Nicol Zenoni (Osaka University) - April 4th, 2023 10

12. Thank Thank you for for your your attention attention! ! you