1. Random volumes from matrices Sotaro Sugishita (Kyoto Univ.) Masafumi Fukuma & Naoya Umeda (Kyoto Univ.) arXiv:1503.08812 (accepted in JHEP) 1504.03532 @Osaka Univ. 7 July, 2015 based on works with

2. Introduction Matrix models have played important roles in 2D quantum gravity string theory The discoveries of various dualities in string theory leads us to think that not only string but also more extended objects are fundamental. For example, it is believed that membrane, whose worldvolume is 3D, is the fundamental object in M-theory. However, membrane theory is less understood than string theory. To understand membrane theory, we propose analogs of matrix models. 1/33

3. Outline  relation between matrix models and string theory  generalization to random volumes  generalities of the triangle-hinge models matrix models generate random surfaces tensor models our new models (the triangle-hinge models)  prescription to put matters on our models  restriction to 3D manifolds The models generate objects which are not 3D manifolds Algebraic construction (the models are characterized by algebras) Matrix ring 2/33

4.  free energy propagator interaction vertex Action  Feynman rules = sum of connected vacuum diagrams = 3/33

5. We can interpret each vacuum diagram as a triangular decomposition of closed 2D surface. #(props) = #(edges), #(int vertices) = #(triangles), #(loops) = #(vertices)  free energy 4/33

6. Matrix model generates random surfaces! In the large N limit, the contribution of sphere dominates. …… 5/33

7. 2D pure gravity = area random surfaces The partition function is given by 6/33

8. worldsheet theory of string = 2D quantum gravity with matters Some matrix models give non-perturbative definitions of some non-critical string theories.  Continuum limit  String theory We can take a continuum limit. Planar contribution critical value We can also consider all topology by taking the double scaling limit. [Brezin-Kazakov, Douglas-Shenker, Gross-Migdal (1990)] 7/33

9. generalization to random volumes It is believed that membranes are also fundamental objects in String or M-theory. Membrane theory If we can obtain generalization of matrix models to 3D random volume theory, it is expected that we can analyze membrane theory. worldvolume theory of membrane = 3D quantum gravity with matters But, understanding of membrane theory is not sufficient, as compared to string theory. 8/33

10. Tensor models Tensor models are natural generalizations of matrix models. We don’t know techniques to solve the models analytically. The models generate singular objects (e.g. pseudo manifolds).  Remarks Tensor models generate random tetrahedral decompositions. [Ambjorn-Durhuus-Jonsson (1991), Sasakura (1991), Gross (1992)] building block is tetrahedron. 9/33

11. We propose a new class of models which generate random volumes. interpret tetrahedral decmp as collection of triangles and multiple hinges The triangle-hinge models Main idea: [Fukuma-Sugishita-Umeda (arXiv:1503.08812)] 2-hinge triangle Matrix modelTensor modelOur new model int termtriangletetrahedrontriangle & hinge cf. [Chung-Fukuma-Shapere (1993)] 3-hinge 10/33

12. dynamical variables are symmetric matrices, not rank-3 tensors. triangle k-hinge Action: 11/33

13. = = cyclic sym reversal sym Similarly, hinges should have cyclic sym. 12/33

14. propagator (Wick contraction)  Propagator glues an edge of a triangle to an edge of a hinge. or This term represents twisted gluing. 13/33

15. Free energy : symmetry factor, : #(triangles), : #(k-hinges), Examples of diagrams This is not tetrahedral decomposition. Diagrams are not generally tetrahedral decomps. 14/33

16.  pseudo manifolds Ex. This consists of 4 tetrahedra but is not manifold. It is known that the Euler number of 3d manifold must be zero. Note that tensor models also generate pseudo manifolds. Another explanation for non-manifoldness 15/33

17. In the limit, we can neglect these diagrams. 16/33

18. Algebraic construction 1 Consider semisimple associative algebra A. Define metric as A is semisimple has inverse Then, we define, 17/33

19. Algebraic construction 2  There is graphical representation for the properties of algebra. associativity: = metric: = = junction 18/33

20.  index lines of triangles and hinges  Free energy We claim: function of 19/33

21. Each connected component of the index network can be regarded as a closed 2D surface enclosing a vertex. (polygonal decomposition of ) Using the properties of algebra A, is invariant under 2D topology-preserving local moves.  index function and index network 1.Factorization of index function: 2.2D topological invariant index network = = [Fukuma-Hosono-Kawai (1992)], 20/33

22. We will see, where is genus of. Note that 2D surface is not always 2-sphere. For example,  From now on, we will investigate the case A is matrix ring. 21/33

23. Matrix ring It is known that any semisimple associative algebra is isomorphic to a direct sum of matrix rings. Here, we consider the most simple case. a basis: multiplication: 22/33

24. In the case of matrix ring, each index function is given by  index lines of triangles and hinges trianglek-hinge polygon junction segment :genus of = 23/33

25. Free energy We can count the number of vertices by considering the direct sum of matrix ring. In this case, the free energy is given by 24/33

26. Restriction to tetrahedral decomposition 1 So far, we cannot neglect the contributions from objects which are not tetrahedral decompositions. Strategy: All index networks of the objects which represent tetrahedral decompositions are always triangular decompositions. 25/33

27. Restriction to tetrahedral decomposition 2 Change the form of tensor. 26/33

28. Restriction to tetrahedral decomposition 3 Furthermore, we can take a limit where only 3-gons (triangles) can remain. Each weight can be rewritten as all index networks represent triangular decompositions. 27/33

29. Restriction to manifolds with tetrahedral decompositions We can single out tetrahedral decompositions of 3D manifolds!  3D pure gravity with CC Our model corresponds to pure gravity. 28/33

30. Putting maters on the triangle-hinge models 1 [Fukuma-Sugishita-Umeda (arXiv:1504.03532)] We can put matter degrees of freedom on any simplices (i.e. tetrahedra, triangles, edges and vertices). General prescriptions Take algebra as Assume a factorized form The “gravity” part is used to restrict diagrams to 3D manifolds. The “matter” part takes specific forms to represent various matters. Then, index functions factorize as 29/33

31. Set and consider the following “matter” interactions: Ex. Coloring tetrahedra Putting maters on the triangle-hinge models 2 labels of colors Each triangle has projection matrices. One triangle has two colors. 30/33

32. The four index triangles in one tetrahedron give the factor Only when four triangles have the same color, the factors can take non-vanishing values. We can say that each tetrahedron has a color. Putting maters on the triangle-hinge models 3 31/33

33. Summary 1/2 We proposed a new class of models (the triangle-hinge models) which generate 3D random volumes. The fundamental building blocks are triangle and multiple hinges. The dynamical variables are symmetric matrices. Thus, unlike tensor models, there is a possibility that we can solve the triangle-hinge models analytically by using the techniques of matrix models (ex. saddle pt analysis) (work in progress). Our models are characterized by semisimple associative algebras. 32/33

34. Although our models generally generate objects which are not 3D manifolds, we can single out 3D manifolds with tetrahedral dcmps by taking specific limits of coupling consts. Summary 2/2 We investigated the properties of our models in the case where the defining algebra is matrix ring. We can put some matters (local spin systems) on any simplices. 33/33

36. Since the leading contribution of free energy is a sum over all 3D topology, it is divergent. The free energies of matrix models also seem to be divergent sum. Future Works But, we can divide the sum according to topology and each sum has a finite radius of convergence. We need to divide the sum of our models according to topology or something (work in progress). Can we single out the contributions from 3-sphere? So far, we don’t know how to take a continuum limit.