Putting M theory on computer Jun Nishimura KEK & Graduate University for Advanced Studies (SOKENDAI) based on collaboration with Konstantinos Anagnostopoulos (National Technical University, Athens) Masanori Hanada (RIKEN) Shingo Takeuchi (SOKENDAI) Talk at LATTICE2007, Regensburg, July 31, 2007 Ref: Hanada-J.N.-Takeuchi, arXiv: [hep-lat] Anagnostopoulos-Hanada- J.N.-Takeuchi, arXiv: [hep-th]

0. Introduction - non-pert. formulation of superstring/M theories e.g.) Matrix Theory (Banks-Fischler-Shenker-Susskind ’97) IIB matrix model (Ishibashi-Kawai-Kitazawa-Tsuchiya ’97)  quantum description of black holes etc.  dynamical origin of space-time dimensionality, gauge group, matters, etc. - gauge/gravity duality e.g.) AdS/CFT Maldacena(’97) SUPERSYMMETRY on the lattice… playing more and more important roles in string/M theory large N gauge theories

SUSY matrix quantum mechanics 1 dim. U(N) gauge theory with 16 supercharges 10d N=1 SYM dimensional reduction BFSS conjecture: non-perturbative formulation of M theory dual geometry: black D0 brane solution in type IIA SUGRA gauge/gravity correspondence (non-conformal ver.) Banks-Fischler-Shenker-Susskind ’97 Itzhaki-Maldacena-Sonnenschein-Yankielowicz ’98 finite T Klebanov-Tseytlin ’96 confirm the conjectured duality from first principles Monte Carlo studies can

Plan 0. Introduction 1. SUSY matrix QM with 16 supercharges 2. non-lattice sim. for SUSY QM 3. Monte Carlo results 4. summary and discussions

1. SUSY matrix QM with 16 supercharges low Tstrongly coupled high Tweakly coupled dual gravity description (except for zero modes) high T exp. 1d gauge theory with adjoint matters p.b.c. anti p.b.c. 10d gauge field Maj-Weyl fermion pure SYM Dim.Red. Kawahara-J.N.-Takeuchi in prep.

2. Non-lattice simulation for SUSY QM static diagonal gauge : fixes sym. under large gauge tr. fixes the gauge inv. completely (specific to 1d) Fourier mode expansion :

Advantages of the non-lattice simulation: theoretically clean restoration of SUSY (much faster than cont. lim.) cont. lim. approached faster than naïve expectation from # of d.o.f.. Fourier acceleration requires no extra cost. the gauge-fixed action in the continuum except for e.g.) in the 1d Wess-Zumino model degenerate mass for boson and fermion observed for higher modes : naturally suppressed by the kinetic term compensates superficial increase in computational efforts by factor of removes critical slowing down completely c.f.) lattice approach (Catterall’s talk) Catterall-Wiseman, arXiv: [hep-lat] Catterall-Karamov ’02

3. Monte Carlo results high T exp. (incl. next-leading) internal energy free energy obtained from dual BH geometry

characteristic behavior of the deconfined phase high T exp. (incl. next-leading) Polyakov line consistent with a speculation based on gauge/gravity duality (Barbon et al., Aharony et al.)

c.f.) bosonic model (fermions omitted) high T exp. (incl. next-leading) confined deconfined high T exp. (leading only) Kawahara-Takeuchi-J.N. arXiv:

5. Summary and discussions the first Monte Carlo results for high precision confirmation of gauge/gravity duality from first principles matrix QM with maximal SUSY (16 supercharges) plays important roles in superstring/M theory non-lattice simulation method + Rational HMC method Pfaffianintegrating out fermions real positive to high accuracy in the region of T investigated no phase transition (unlike the bosonic model) no sign problem ! at small T

Implications of our results 1)Microscopic d.o.f. which accounts for the B.H. entropy has been identified. c.f) much stronger results than Strominger-Vafa (96) 2) SUSY gauge theory provides a natural framework for quantum description of B.H. at T not very small (where classical description of B.H. is no more valid.) For those who wish to gain some background… Zwiebach, The first course in string theory Becker-Becker-Schwartz, Superstring theory and M theory