ترغب بنشر مسار تعليمي؟ اضغط هنا

The Multi-Orientable Random Tensor Model, a Review

156   0   0.0 ( 0 )
 نشر من قبل Adrian Tanasa
 تاريخ النشر 2015
  مجال البحث
والبحث باللغة English
 تأليف Adrian Tanasa




اسأل ChatGPT حول البحث

After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the $1/N$ expansion and of the large $N$ limit ($N$ being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit.



قيم البحث

اقرأ أيضاً

110 - Eric Fusy , Adrian Tanasa 2014
Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in N, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.
A short review of the Operator/Feynman diagram/dessin denfants correspondence in the rank 3 tensor model is presented, and the cut & join operation is given in the language of dessin denfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin denfants.
The counting of the dimension of the space of $U(N) times U(N) times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. The derivation relies on the dominance in the sum, of partitions with many parts of length $1$. The dominance of other small parts in restricted partition sums leads to related asymptotic results. The result for the $3$-index tensor observables gives the large $n$ asymptotic expansion for the counting of bipartite ribbon graphs with $n$ edges, and for the dimension of the associated Kronecker permutation centralizer algebra. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The large $n$ dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general $d$-index tensors. The coefficients of $ 1/n$ in these expansions involve Stirling numbers of the second kind along with restricted partition sums.
Recently, [JHEP 20 131 (2020)] obtained (a similar, scaled version of) the ($a,b$)-phase diagram derived from the Kazakov--Zinn-Justin solution of the Hermitian two-matrix model with interactions [mathrm{Tr,}Big{frac{a}{4} (A^4+B^4)+frac{b}{2} ABABBi g},,] starting from Functional Renormalization. We comment on something unexpected: the phase diagram of [JHEP 20 131 (2020)] is based on a $beta_b$-function that does not have the one-loop structure of the Wetterich-Morris Equation. This raises the question of how to reproduce the phase diagram from a set of $beta$-functions that is, in its totality, consistent with Functional Renormalization. A non-minimalist, yet simple truncation that could lead to the phase diagram is provided. Additionally, we identify the ensemble for which the result of op. cit. would be entirely correct.
We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of $n$-particle amplitudes in planar $mathcal{N}=4$ Yang-Mills theory and certain polytopes associated to the Grassmannian G(4, $n$). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of $n le 8$, and find that the polytope $mathcal{C}^dagger(4,9)$ encodes all rational letters, and all square roots of the algebraic letters, of nine-particle amplitudes.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا