Random tensor models in the large N limit: Uncoloring the colored tensor models

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents m = 1 1/m.

[1]  Michael R. Douglas,et al.  STRINGS IN LESS THAN ONE DIMENSION , 1990 .

[2]  J. Ryan,et al.  Colored Tensor Models - a Review , 2011, 1109.4812.

[3]  M. Smerlak,et al.  Scaling behavior of three-dimensional group field theory , 2009, 0906.5477.

[4]  N. Sasakura.  SUPER TENSOR MODELS, SUPER FUZZY SPACES AND SUPER n-ARY TRANSFORMATIONS , 2011, 1106.0379.

[5]  Fabien Vignes-Tourneret,et al.  Renormalisation of Noncommutative ϕ4-Theory by Multi-Scale Analysis , 2006 .

[6]  D. Gross,et al.  Nonperturbative two-dimensional quantum gravity. , 1990, Physical review letters.

[7]  Razvan Gurau,et al.  The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension , 2011, 1102.5759.

[8]  J. Ryan.  TENSOR MODELS AND EMBEDDED RIEMANN SURFACES , 2011, 1104.5471.

[9]  M. Gross.  Tensor models and simplicial quantum gravity in >2-D , 1992 .

[10]  Simone Severini,et al.  Quantum graphity: A model of emergent locality , 2008, 0801.0861.

[11]  P. Francesco.  Rectangular matrix models and combinatorics of colored graphs , 2002, cond-mat/0208037.

[12]  R. Gurau.  Universality for Random Tensors , 2011, 1111.0519.

[13]  N. Sasakura.  Tensor models and 3-ary algebras , 2011, 1104.1463.

[14]  R. Gurau.  The Double Scaling Limit in Arbitrary Dimensions: A Toy Model , 2011, 1110.2460.

[15]  Alexander M. Polyakov,et al.  Fractal Structure of 2D Quantum Gravity , 1988 .

[16]  Gilles Schaeffer,et al.  Bijective Census and Random Generation of Eulerian Planar Maps With Prescribed Degrees , 1997, Electron. J. Comb..

[17]  É. Brézin,et al.  The Ising model coupled to 2D gravity. A nonperturbative analysis , 1990 .

[18]  J. Magnen,et al.  Bosonic colored group field theory , 2009, 0911.1719.

[19]  Zhituo Wang.  Construction of 2-dimensional Grosse-Wulkenhaar Model , 2011 .

[20]  Vincent Rivasseau,et al.  Quantum Gravity and Renormalization: The Tensor Track , 2011, 1112.5104.

[21]  V. Kazakov.  Bilocal Regularization of Models of Random Surfaces , 1985 .

[22]  V. Bonzom.  Multi-critical tensor models and hard dimers on spherical random lattices , 2012, 1201.1931.

[23]  V. Rivasseau,et al.  The Ising model on random lattices in arbitrary dimensions , 2011, Physics Letters B.

[24]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[25]  D. Oriti,et al.  Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds , 2011, 1104.5158.

[26]  Naoki Sasakura,et al.  TENSOR MODEL FOR GRAVITY AND ORIENTABILITY OF MANIFOLD , 1991 .

[27]  V. Rivasseau,et al.  Loop Vertex Expansion for Phi^2k Theory in Zero Dimension , 2010, 1003.1037.

[28]  Fabien Vignes-Tourneret,et al.  Renormalization of Non-Commutative $$\Phi^4_4$$ Field Theory in x Space , 2006 .

[29]  V. Bonzom,et al.  Radiative Corrections in the Boulatov-Ooguri Tensor Model: The 2-Point Function , 2011, 1101.4294.

[30]  N. Sasakura.  Fuzzy spaces from tensor models, cyclicity condition, and n-ary algebras , 2012, 1203.6170.

[31]  Joseph Ben Geloun,et al.  A Renormalizable 4-Dimensional Tensor Field Theory , 2011, 1111.4997.

[32]  S. Salinas.  The Ising Model , 2001 .

[33]  Conformal Random Geometry , 2006, math-ph/0608053.

[34]  A. Baratin,et al.  Diffeomorphisms in group field theories , 2011, 1101.0590.

[35]  G. Hooft.  A Planar Diagram Theory for Strong Interactions , 1974 .

[36]  Valentin Bonzom,et al.  Critical behavior of colored tensor models in the large N limit , 2011, 1105.3122.

[37]  N. Sasakura.  TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS , 2011, 1104.5312.

[38]  V. Rivasseau.  Constructive matrix theory , 2007, 0706.1224.

[39]  M. Smerlak,et al.  Bubble Divergences: Sorting out Topology from Cell Structure , 2011, 1103.3961.

[40]  Vladimir Kazakov,et al.  Ising model on a dynamical planar random lattice: Exact solution , 1986 .

[41]  G. Parisi,et al.  Planar diagrams , 1978 .

[42]  R. Gurau.  A generalization of the Virasoro algebra to arbitrary dimensions , 2011, 1105.6072.

[43]  F. David,et al.  A model of random surfaces with non-trivial critical behaviour , 1985 .

[44]  Bergfinnur Durhuus,et al.  THREE-DIMENSIONAL SIMPLICIAL QUANTUM GRAVITY AND GENERALIZED MATRIX MODELS , 1991 .

[45]  D. Benedetti,et al.  Phase transition in dually weighted colored tensor models , 2011, 1108.5389.

[46]  T. Jónsson,et al.  Summing over all genera for d > 1: a toy model , 1990 .

[47]  R. Gurău,et al.  Lost in translation: topological singularities in group field theory , 2010, 1108.4966.

[48]  R. Gurau.  The 1/N Expansion of Colored Tensor Models , 2010, 1011.2726.

[49]  H. Grosse,et al.  Renormalisation of ϕ4-Theory on Noncommutative ℝ4 in the Matrix Base , 2004, hep-th/0401128.

[50]  J. Distler,et al.  Conformal Field Theory and 2D Quantum Gravity , 1989 .

[51]  D. Oriti.  Foundations of Space and Time: The microscopic dynamics of quantum space as a group field theory , 2011, 1110.5606.

[52]  R. Gurau.  Topological Graph Polynomials in Colored Group Field Theory , 2009, 0911.1945.

[53]  Edouard Brézin,et al.  Exactly Solvable Field Theories of Closed Strings , 1990 .

[54]  D. O. Samary,et al.  3D Tensor Field Theory: Renormalization and One-Loop β-Functions , 2013 .

[55]  Vincent Rivasseau,et al.  The 1/N expansion of colored tensor models in arbitrary dimension , 2011, 1101.4182.

[56]  V. Kazakov.  The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity , 1989 .

[57]  D. O. Samary,et al.  3D Tensor Field Theory: Renormalization and One-loop $\beta$-functions , 2011, 1201.0176.

[58]  Mireille Bousquet-Mélou,et al.  Enumeration of Planar Constellations , 2000, Adv. Appl. Math..

[59]  F. Caravelli,et al.  A simple proof of orientability in the colored Boulatov model , 2010 .

[60]  Vladimir Kazakov,et al.  The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents , 1987 .

[61]  S. Rey,et al.  A Nonperturbative Proposal for Nonabelian Tensor Gauge Theory and Dynamical Quantum Yang-Baxter Maps , 2010, 1002.4636.

[62]  F. David.  CONFORMAL FIELD THEORIES COUPLED TO 2-D GRAVITY IN THE CONFORMAL GAUGE , 1988 .

[63]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[64]  Razvan Gurau,et al.  Colored Group Field Theory , 2009, 0907.2582.