Emergence of space, general relativity and gauge theory from tensor models

1. Emergence of space, general relativity and gauge theoryfrom tensor models Naoki Sasakura Yukawa Institute for Theoretical Physics

2. Kawamoto-san’s education A class guided by Kawamoto-san Text : the original BPZ paper on CFT ・Not allow superficial understanding ・Everything must be understood certainly ・Full of discussions ・No care about time ・Unusual members Students and staff members from other universities Russian style

3. Kawamoto-san loves discussions • 13:30 Class starts • 15:00 Continue (Official end) • 17:00 Continue (End for most classes) • 19:00 End of the class • 19:00 Go to drink at Izakaya Various discussions on physics and non-physics • 22:00 Go to Kawamoto-san’s home Discussions continue • 6:00 Back home

4. --- Kawamoto-san’s philosophy --- Spacetime is lattice(literally) Not new but has potential to solve problems in the frontiers. • Reduce degrees of freedom Free from infinities Incorporate minimal length May prevent physically unwanted fields (e.g. scalar massless moduli fields in string theory) • Unified theory on lattice Matter contents are related to lattice structures Kawamoto-san’s talk at 13th Nishinomiya Yukawa Memorial Symposium (1998) “Non-String Pursuit towards Unified Model on the Lattice” • Reconnection Dynamical spacetime Possible route to quantum gravity Intrinsically background independent

5. Random surface Numerical Simulation Matrix model 2D quantum gravity Kawamoto, Kazakov, Watabiki, …

6. Tensor models • Generalization of matrix models Random surface Random volume Matrix model Tensor model Master thesis under Kawamoto-san (1990) Sasakura, Mod.Phys.Lett.A6,2613,1991

7. Tensor models were not successful • Continuum limit  Large volume  Large Feynman diagram But no analytical methods known for non-perturbative computations in tensor models. • Topological expansions not known. Difficulty in physical interpretation of the partition function.

8. A different interpretation of tensor models --- My proposal --- Tensor models may be regarded as dynamical theory of fuzzy spaces. The structure constant defining a fuzzy space may be identified with the dynamical variable of tensor models. Sasakura, Mod.Phys.Lett.A21:1017-1028,2006

9. Fuzzy space • Defines algebraically a space. No coordinates. • “Points” replaced with operators • Includes noncommutative spaces • Connect distinct topologies and dimensions

10. Fuzzy space Lattice

11. Symmetry of continuous relabeling of “points” : Total number of “points”

12. The symmetry contains local transformations. A background fuzzy space causes symmetry breaking Non-linearly realized local symmetry → Gauge symmetry (& Gen.Coord.Trans.Sym.) Ferrari, Picasso 1971 Borisov, Ogievetsky 1974 Relabeling symmetry → Origin of local gauge symmetries

13. Contents of the following talk • Gaussian fuzzy space　（Flat D-dimensional fuzzy space) • Construction of an action having Gaussian sol. • Fluctuation mode analysis around the sol. --- Emergence of general relativity • Kaluza-Klein set up --- Emergence of gauge theory --- Emergent scalar field is supermassive (“Planck” order) • Summary and future problems

14. Ordinary continuum space Gaussian fuzzy space β：　parameter of fuzziness Gaussian fuzzy space Sasai,Sasakura, JHEP 0609:046,2006.

15. Gaussian fuzzy space • Simplest fuzzy space • Poincare symmetry  Flat D-dimensional fuzzy space • Can naturally generalize to curved space

16. This metric-tensor correspondence derives DeWitt supermetric from the configuration measure of tensor models. Tensor models DeWitt supermetric in general relativity Used in the comparison of modes Sasakura, Int.J.Mod.Phys.A23:3863-3890,2008.

17. Construction of an action • Demand : has Gaussian fuzzy spaces as classical solutions • Infinitely many such actions • Generally very complicated and unnatural --- Future problems • The action in this talk ---- Convenient but singular • (There exists also non-singular but inconvenient one.) • Least number of terms. • The singular property will not harm the fluctuation analysis. • The low-frequency property independent of the actions.

18. (Symmetric, positive definite)

19. A cartoon for the action This action does not depend explicitly on D All the dimensional Gaussian fuzzy spaces are the classical solutions of this single action. --- An aspect of background independence

20. Analysis of the small fluctuations around Gaussian solutions Eigenvalue and eigenmode analysis

21. List of numerical analysis performed Classical sol. : (Gaussian) fuzzy flat D-dimensional torus • Emergence of general relativity D=2 : Results shown D=1,3,4: Similar good results • Kaluza-Klein mechanism D=2+1 : Results shown D=1+1 : Similar good results

22. Emergence of general relativity D=2 , L=10 • 3 states at P=0 • 1 state at each P≠0 • Zero eigenmodes Sasakura, Prog.Theor.Phys.119:1029-1040,2008.

23. The three modes at P=0 Tensor model General Relativity

24. The mode at P≠0 One mode remains. General relativity Tensor model

25. Kaluza-Klein mechanism In continuum theory M×S１: S１with small radius

26. Fuzzy Kaluza-Klein mechanism in tensor models Classical solution 2+1 dimensional flat torus = =

27. Numerical analysis of fluctuation modes L=3 L=6 • Scalar mass does not scale • Slopes of lines scale Scalar Vector L  Large Supermassive scalar field (“Planck” order) Gravity

28. Summary and future problems Tensor models seem physically interesting. ・Emergence of Tensor models are physically interesting • Space • General relativity • Gauge theory • Gauge symmetry (Gen.Cood.Trans.Sym.) from one single dynamical variable Cabc. ・ Background independent ・ Supermassive scalar field in Kaluza-Klein mechanism. Possible resolution to moduli stabilization. • Natural action ? • Fermion ?

29. Thank you very much for many suggestions ! And Happy Birthday !