Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^{2} \times \mathrm{O}(D)$ multi-matrix models

In the previous papers, it is pointed out that a supersymmetric double-well matrix model corresponds to a two-dimensional type IIA superstring theory on a Ramond-Ramond background at the level of correlation functions. This was confirmed by agreement between their planar correlation functions. The supersymmetry in the matrix model corresponds to the target space supersymmetry and it is shown to be spontaneously broken by nonperturbative effect. Furthermore, in the matrix model we computed one-point functions of single-trace operators to all order of genus expansion in its double scaling limit. We found that this expansion is stringy and not Borel summable and hence there arises an ambiguity in applying the Borel resummation technique. We confirmed that resurgence works here, namely this ambiguity in perturbative series in a zero-instanton sector is exactly canceled by another ambiguity in a one-instanton sector obtained by instanton calculation. In this paper we extend this analysis and study resurgence structure of the two-point functions of the single trace operators. By using results in the random matrix theory, we derive two-point functions at arbitrary genus and see that the perturbative series in the zero-instanton sector again has an ambiguity. We find that the two-point functions inevitably have logarithmic singularity even at higher genus. In this derivation we obtain a new result of the two-point function expressed by the one-point function at the leading order in the soft-edge scaling limit of the random matrix theory. We also compute an ambiguity in the one-instanton sector by using the Airy kernel, and confirm that ambiguities in both sectors cancel each other at the leading order in the double scaling limit. We thus clarify resurgence structure of the two-point functions in the supersymmetric double-well matrix model.

We study the double- and triple-scaling limits of a complex multi-matrix model, with \mathrm{U}(N)^2\times \mathrm{O}(D) symmetry. The double-scaling limit amounts to taking simultaneously the large- N (matrix size) and large- D (number of matrices) limits while keeping the ratio N/\sqrt{D}=M fixed. The triple-scaling limit consists in taking the large- M limit while tuning the coupling constant \lambda to its critical value \lambda_c and keeping fixed the product M(\lambda_c-\lambda)^\alpha , for some value of \alpha that depends on the particular combinatorial restrictions imposed on the model. Our first main result is the complete recursive characterization of the Feynman graphs of arbitrary genus which survive in the double-scaling limit. Next, we classify all the dominant graphs in the triple-scaling limit, which we find to have a plane binary tree structure with decorations. Their critical behavior belongs to the universality class of branched polymers. Lastly, we classify all the dominant graphs in the triple-scaling limit under the restriction to three-edge connected (or two-particle irreducible) graphs. Their critical behavior falls in the universality class of Liouville quantum gravity (or, in other words, the Brownian sphere).

We study the six-particle amplitude in planar mathcal{N} = 4 super Yang-Mills theory in the double scaling (DS) limit, the only nontrivial codimension-one boundary of its positive kinematic region. We construct the relevant function space, which is significantly constrained due to the extended Steinmann relations, up to weight 13 in coproduct form, and up to weight 12 as an explicit polylogarithmic representation. Expanding the latter in the collinear boundary of the DS limit, and using the Pentagon Operator Product Expansion, we compute the non-divergent coefficient of a certain component of the Next-to-Maximally-Helicity-Violating amplitude through weight 12 and eight loops. We also specialize our results to the overlapping origin limit, observing a general pattern for its leading divergences.

We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa (2016 Lett. Math. Phys. 106 1531). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the two-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

Basso-Dixon integrals evaluate rectangular fishnets — Feynman graphs with massless scalar propagators which form a m × n rectangular grid — which arise in certain one-trace four-point correlators in the ‘fishnet’ limit of mathcal{N} = 4 SYM. Recently, Basso et al. explored the thermodynamical limit m → ∞ with fixed aspect ratio n/m of a rectangular fishnet and showed that in general the dependence on the coordinates of the four operators is erased, but it reappears in a scaling limit with two of the operators getting close in a controlled way. In this note I investigate the most general double scaling limit which describes the thermodynamics when one of two pairs of operators become nearly light-like. In this double scaling limit, the rectangular fishnet depends on both coordinate cross ratios. I show that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square. A direct evaluation of the fishnet in the null limit is presented any m and n.

Berenstein, Maldacena, and Nastase have recently discovered a particular limit of AdS/CFT correspondence where string theory in a plane wave background is dual to a sector of $mathcal N =4$ SYM in a double scaling limit. It is based on the observation that a plane wave background can be obtained by taking Penrose limit of Anti de Sitter background. The corresponding gauge theory limit is identified via AdS/CFT dictionary. This proposal is especially exciting because string worldsheet theory in a plane wave background is exactly solvable, thereby opening a possibility that one can go beyond supergravity approximation. In the absence of string interactions, the duality made a remarkable prediction for anomalous dimension of gauge theory operators from exact free string spectrum, which was soon verified. In this thesis, we attempt to extend the duality to the interacting theory level. We propose that the correct holographic recipe is to identify the full string field theory Hamiltonian with the dilatation operator of gauge theory. In practice, we must find an identification map between string theory and gauge theory Hilbert spaces and evaluate matrix elements of the two operators accordingly. The requirement that the inner product should be preserved determines a unique identification map assuming that it is hermitian. We show that transition amplitudes of string field theory agree with matrix elements of dilatation operator under this preferred identification for states with two different impurities. We later extend it to states with arbitrary impurities. In doing so, we find a diagrammatic correspondence between string field theory and gauge theory Feynman diagrams thereby providing direct handles on the duality. Our proposal is $universal$ in the sense that it is applicable to any interaction type such as the open-closed interaction, and to all orders in $g_2$ and $l^prime$. Hopefully, this thesis will be a key step towards proving the novel duality and a beginning of an exciting journey to the stringy regime of string/gauge duality.

Using the standard 1/N expansion, we study O(N) vector models in D dimensions with an arbitrary potential. We limit ourselves to renormalizable theories. We show that there exists a value of the coupling constant corresponding to a critical point and that a double scaling limit can be performed as in D=0 and in the case of matrix models in D=0, 1. For D=1 the theory is renormalizable with an arbitrary potential and we find in general a hierarchy of critical theories labeled by an integer k. The universal partition function obtained in the double scaling limit is constructed. Finally, we show that the critical behaviour of those models is the same as a branched polymer model recently constructed by Ambjørn, Durhuus and Jónsson.

The asymptotics of orthogonal polynomials with complex weight on complex contours was studied in [1] and [5]. It was shown that in the asymptotic limit, the zeros of these polynomials accumulate on tree-like graphs in and generically, the density of zeros vanishes like a square-root at each vertex of these tree graphs. In this article, we study the double scaling limit in which the density vanishes to an order higher than in an interior vertex. This is a new critical phenomenon that has no parallel in the case of orthogonal polynomials on the real line. Yet, surprisingly, the behavior of the orthogonal polynomial in the double scaling limit can be modeled by a solution of the Painleve 1 equation that was used previously [15] in the study of a different double scale limit.

Double scaling limit in O(N) vector models

Colored tensor models generalize matrix models in arbitrary dimensions, yielding a statistical theory of random higher-dimensional topological spaces. They admit a $1/N$ expansion dominated by graphs of spherical topology. The simplest tensor model one can consider maps onto a rectangular matrix model with skewed scalings. We analyze this simplest toy model and show that it exhibits a family of multicritical points and a novel double scaling limit. We show in $D=3$ dimensions that only graphs representing spheres contribute in the double scaling limit and argue that similar results hold for any dimension.

Using the saddle point method, we give an explicit form of the planar free energy and Wilson loops of unitary matrix models in the one-cut regime. The multi-critical unitary matrix models are shown to undergo third-order phase transitions at two points by studying the planar free energy. One of these ungapped/gapped phase transitions is multi-critical, while the other is not multi-critical. The spectral curve of the k-th multi-critical matrix model exhibits an A4k−1 singularity at the multi-critical point. Perturbation around the multi-critical point and its double scaling limit are studied. In order to take the double scaling limit, the perturbed coupling constants should be fine-tuned such that all the zero points of the spectral curve approach to the A4k−1 singular point. The fine-tuning is examined in the one-cut regime, and the scaling behavior of the perturbed couplings is determined. It is shown that the double scaling limit of the spectral curve is isomorphic to the Seiberg-Witten curve of the Argyres-Douglas theory of type (A1,A4k−1).

Matrix model calculations beyond the spherical limit

In this paper, we study a double scaling limit of two multi-matrix models: the U(N)2×O(D) -invariant model with all quartic interactions and the bipartite U(N)×O(D) -invariant model with tetrahedral interaction (D being here the number of matrices and N being the size of each matrix). Those models admit a double, large N and large D expansion. While N tracks the genus of the Feynman graphs, D tracks another quantity called the grade. In both models, we rewrite the sum over Feynman graphs at fixed genus and grade as a finite sum over combinatorial objects called schemes. This is a result of combinatorial nature which remains true in the quantum mechanical setting and in quantum field theory. Then we proceed to the double scaling limit at large D, i.e. for vanishing grade. In particular, we find that the most singular schemes, in both models, are the same as those found in Benedetti et al for the U(N)2×O(D) -invariant model restricted to its tetrahedral interaction. This is a different universality class than in the 1-matrix model whose double scaling is not summable.

In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the “stacked” triangulations. For D < 6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits.

Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $D\geq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$ while tuning to criticality, which turns out to be summable in dimension less than six. This double scaling limit is here extended to arbitrary models. This is done by means of the Schwinger--Dyson equations, which generalize the loop equations of random matrix models, coupled to a double scale analysis of the cumulants.