AdS 2 /CFT 1, Whittaker vector & Wheeler-DeWitt equation Tadashi Okazaki (NTU) arXiv:

Conclusion Result 2 #(conformal dimention) ↔ #(prime number) Result 1 GKP-Witten relation in AdS 2 /CFT 1

Conformal Symmetry in Quantum Mechanics Conf(R) = SL(2,R)

w/ classical Lagrangian discrete spectrum Works so far on CQM My work w/o classical Lagrangian continuous spectrum DFF model Calogero model ’76 deAlfaro, Fubini, Furlan ’69 Calogero

Irreducible Unitary rep of SL(2,R) ’46 Gelfand & Namark ’47 Bargmann ’52 Harish- Chandra Ex) DFF model discrete series complementary series principal series mock discrete series most analysis so far this talk

Whittaker vector principal series rep of SL(n+1,R) Chevalley basis irrep w/ weight Whittaker vector energy eigenstate in CQM

Whittaker function plays a key role in physics & math ! (sorry, I cannot explain today…) CQM Whittaker fcn Black Hole (AdS 2 gravity) Integrable model Toda Hamiltonian 6 vertex model pfn Combinatorics Macdonald polynomial Kashiwara crystal Number Theory (Eisenstein series) Geometry (Gromov-Witten of P 1 )

AdS d+1 /CFT d pfn of bulk AdS d+1 generating fcn of corr fcn in bdy CFT d GKP-Witten relation

AdS 2 /CFT 1 is special ! I. all the extremal BPS BH contain AdS 2 II. excited states on bdy I. not Fock space but Hilbert space (no radial quantization & no CFT method) II. spatially disconnected bdy but entangled ? CQM side AdS 2 side ’07 Kunduri etal. ’08 Figueras etal. ’10 Balasubramanian etal. ’07 Azeyanagi ’01 Zamolodchikov etal. ’11 Chamon etal’08 Sen

pfn of bulk AdS 2 generating fcn of corr fcn in bdy CQM Result 1 GKP-Witten relation in AdS 2 /CFT 1 generating fcn of expectation value of dilatation operator wavefcn of LFT in minisuperspace approx. == excited states !

CQM 2d gravity wavefc n pfn w/ BC ’06 Kashani-Poor

Wheeler-DeWitt equation !

Dictionary AdS 2 radius cf) AdS 3 radius ’86 Brown Henneaux energy of bdy ground state flat space _ _ ?

Q. What is I. conformal dimension in CQM (physics) II. prime numer in Number Theory (math) Encounter of building blocks in physics & math !

Result 2 #(conformal dimention) ↔ #(prime number) asymptotic behavior of counting fcn of Riemann zeros

Hilbert 8 th Problem & Millennium Problem RH (Riemann Hypothesis) All the non-trivial zeros of zeta are complex numbers with real part ½. zeta knows prime number !

1859 Riemann zeros of zeta # (prime number) # (zeros of zeta) _ _ = Riemann zeros

Hilbert-Pólya conjecture Imaginary part of non-trivial Riemann zeros correspond to eigenvalues of some self-adjoint operator. ’99 Berry, Keating ’99 Connes asymptotic behavior of counting fcn of Riemann zeros

This is not Hamiltonian in CQM ! In DFF model This is rather Dilatation operator !

Q. What is asymptotic behavior of counting fcn of Riemann zeros !

Conclusion Result 2 #(conformal dimention) ↔ #(prime number) Result 1 GKP-Witten relation in AdS 2 /CFT 1

Conformal Quantum Mechanics is unopened treasure box in physics & math!