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Register a new userNonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings
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This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.
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We study a family of models for an $N_1 times N_2$ matrix worth of Ising spins $S_{aB}$. In the large $N_i$ limit we show that the spins soften, so that the partition function is described by a bosonic matrix integral with a single `spherical constraint. In this way we generalize the results of [1] to a wide class of Ising Hamiltonians with $O(N_1,mathbb{Z})times O(N_2,mathbb{Z})$ symmetry. The models can undergo topological large $N$ phase transitions in which the thermal expectation value of the distribution of singular values of the matrix $S_{aB}$ becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.
It is proposed that a family of Jackiw-Teitelboim supergravites, recently discussed in connection with matrix models by Stanford and Witten, can be given a complete definition, to all orders in the topological expansion and beyond, in terms of a specific combination of minimal string theories. This construction defines non-perturbative physics for the supergravity that is well-defined and stable. The minimal models come from double-scaled complex matrix models and correspond to the cases $(2Gamma{+}1,2)$ in the Altland-Zirnbauer $(boldsymbol{alpha},boldsymbol{beta})$ classification of random matrix ensembles, where $Gamma$ is a parameter. A central role is played by a non-linear `string equation that naturally incorporates $Gamma$, usually taken to be an integer, counting e.g., D-branes in the minimal models. Here, half-integer $Gamma$ also has an interpretation. In fact, $Gamma{=}{pm}frac12$ yields the cases $(0,2)$ and $(2,2)$ that were shown by Stanford and Witten to have very special properties. These features are manifest in this definition because the relevant solutions of the string equation have special properties for $Gamma{=}{pm}frac12$. Additional special features for other half-integer $Gamma$ suggest new surprises in the supergravity models.
We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/sqrt{D}$ in $mathrm{U}(N)timesmathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $Ntimes N$ matrices transform in the vector representation of $mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $eta(h)$ on the maximum power $D^{1+eta(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $eta(h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that $eta(h)leq 2h$; the sharper bound $eta(h)=h$ is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that $eta(h)=h$ for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.
This paper deals with various interrelations between strings and surfaces in three dimensional ambient space, two dimensional integrable models and two dimensional and four dimensional decomposed SU(2) Yang-Mills theories. Initially, a spinor version of the Frenet equation is introduced in order to describe the differential geometry of static three dimensional string-like structures. Then its relation to the structure of the su(2) Lie algebra valued Maurer-Cartan one-form is presented; while by introducing time evolution of the string a Lax pair is obtained, as an integrability condition. In addition, it is show how the Lax pair of the integrable nonlinear Schroedinger equation becomes embedded into the Lax pair of the time extended spinor Frenet equation and it is described how a spinor based projection operator formalism can be used to construct the conserved quantities, in the case of the nonlinear Schroedinger equation. Then the Lax pair structure of the time extended spinor Frenet equation is related to properties of flat connections in a two dimensional decomposed SU(2) Yang-Mills theory. In addition, the connection between the decomposed Yang-Mills and the Gauss-Godazzi equation that describes surfaces in three dimensional ambient space is presented. In that context the relation between isothermic surfaces and integrable models is discussed. Finally, the utility of the Cartan approach to differential geometry is considered. In particular, the similarities between the Cartan formalism and the structure of both two dimensional and four dimensional decomposed SU(2) Yang-Mills theories are discussed, while the description of two dimensional integrable models as embedded structures in the four dimensional decomposed SU(2) Yang-Mills theory are presented.
A new class of higher-spin gauge theories associated with various Coxeter groups is proposed. The emphasize is on the $B_p$--models. The cases of $B_1$ and its infinite graded-symmetric product $sym,(times B_1)^infty$ correspond to the usual higher-spin theory and its multi-particle extension, respectively. The multi-particle $B_2$--higher-spin theory is conjectured to be associated with String Theory. $B_p$--higher-spin models with $p>2$ are anticipated to be dual to the rank-$p$ boundary tensor sigma-models. $B_p$ higher-spin models with $pgeq 2$ possess two coupling constants responsible for higher-spin interactions in $AdS$ background and stringy/tensor effects, respectively. The brane-like idempotent extension of the Coxeter higher-spin theory is proposed allowing to unify in the same model the fields supported by space-times of different dimensions. Consistency of the holographic interpretation of the boundary matrix-like model in the $B_2$-higher-spin model is shown to demand $Ngeq 4$ SUSY, suggesting duality with the $N=4$ SYM upon spontaneous breaking of higher-spin symmetries. The proposed models are shown to admit unitary truncations.
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