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The purpose of this paper is to give a decorated version of the Eynard–Orantin topological recursion using a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic reformulation of a topological recursion to define how to decorate a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a decorated version of the generalized Catalan numbers. We show that the function that counts these decorated graphs, which is a decoration of the counting function of the generalized Catalan numbers by a Frobenius algebra, satisfies a topological recursion with respect to the edge-contraction axioms. The path we follow to pass from the A-model side to the remodeled B-model side is to use a discrete Laplace transform as a mirror symmetry map. We show that a decorated version by a 2D TQFT of the Eynard–Orantin differentials satisfies a decorated version of the Eynard–Orantin recursion formula. We illustrate these results using a toy model for the theory arising from the orbifold cohomology of the classifying space of a finite group. In this example, the graphs are orbifold cell graphs (graphs drawn on an orbifold punctured Riemann surface) defined out of the moduli space M¯¯¯¯¯¯g,n(BG) of stable morphisms from twisted curves to the classifying space of a finite group G. In particular we show that the cotangent class intersection numbers on the moduli space M¯¯¯¯¯¯g,n(BG) satisfy a decorated Eynard–Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective the known result which states that the ψ-class intersection numbers on M¯¯¯¯¯¯g,n(BG) satisfy the Virasoro constraint condition.

Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution., 52 pages, 12 figures, 4 tables

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C3/(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve., Comment: 39 pages, 2 figures

The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck's determinant for finite rank endomorphisms and is equivalent to the classic analytic definitions. The theory can be interpreted as a multiplicative analogue to Tate's formalism of abstract residues in terms of traces of finite potent linear operators on infinite-dimensional vector spaces, and allows us to relate Tate's theory to the Segal-Wilson pairing in the context of loop groups., Version 3. Minor changes

Este trabajo hace un estudio detallado de la construcción de Krichever paradiversos espacios de moduli, que hemos elegido motivados por el Programa deAbelianización de Hitchin. En el año 1988, Hitchin descubre una aplicaciónque va del espacio cotangente al moduli de fibrados (sobre una superficie deRiemann fija) a un espacio de secciones globales, y demuestra que es un sistemaintegrable. Formula entonces la siguiente pregunta a la comunidad científica:¿Pueden darse, de modo concreto, las ecuaciones diferenciales de este sistemaintegrable?Nuestro primer objetivo ha sido profundizar en dicha cuestión utilizandocomo herramientas la Grassmanniana infinita y el morfismo de Krichever. Elsegundo objetivo ha consistido en buscar otro sistema integrable con propiedadesanálogas al de Hitchin, y por último, encontrar esquemas en grupos que losuniformicen. Este último objetivo es un paso importante antes de poder pensarespacios de moduli como variedades solución, variedades integrales, de jerarquías de ecuaciones diferenciales.Así pues, damos explícitamente las ecuaciones que caracterizan el espacio de moduli de pares de Higgs, al que añadimos ciertos datos formales. Para ello, demostramos que el espacio resultante es un esquema,caracterizamos la imagendel morfismo de Krichever (que esta vez valora no en una Grassmanniana, sino entoda una fibración de Grassmannianas infinitas), y traducimos dicha condiciónen una identidad bilineal en términos funciones de Baker-Akhiezer.Generalizamos también la construcción de Krichever para los siguientes espaciosde moduli: fibrados vectoriales y curvas lisas, revestimientos finitos ypunteados entre curvas lisas, y revestimientos finitos con haz de línea sobre la curva que reviste. Apoyados en este estudio, demostramos la existencia de un sistema integrable con propiedades análogas al de Hitchin y damos un relación del mismo con el Programa de Abelianización de Hitchin. Por último, probamos que ciertos esquemas en grupos - entre los que cabe destacar el grupo de automorfismossemilineales - hacen las veces de generadores locales para dichosespacios de moduli., This work makes a detailed study of Krichever's construction for several moduli spaces, which we have chosen motivated by Hitchin's Abelianization Program. In 1988, Hitchin discovered a map from the cotangent space to the moduli space of vector bundles (over a fixed Riemann surface) to an affine space of global sections, and he has shown that it is an integrable system. He addressed then the following question to the scientific comunity: can we compute, in some concrete way, the differential equations of this integrable system?Our first aim has been to study in depth Hitchin's question using as tools the infinite Grassmannian and the Krichever map. The second goal consists of looking for an integrable system with analogue properties to that of Hitchin, and finally, to find out group schemes that uniformizes such a moduli spaces. This last goal is an important step before thinking moduli spaces as solution varieties of hierarchies of differential equations. We have explicitly computed equations characterizing the moduli space of Higgs pairs, to which we add formal trivialization data. To achieve this result, we have shown that this space is a scheme, we have characterized the image of the appropriated Krichever map (which takes values not in a single infinite Grassmannian, but in a fibration of infinite Grassmannians), and we have translated this condition into a bilinear identity in terms of Baker-Akhiezer functions.We also generalize the Krichever contruction for the following moduli spaces: vector bundles and curves, pointed and finite coverings between smooth curves, and finite coverings as before with a line bundle upstairs in addition. This study allows us to find out an integrable system which behaves in an similar way as Hitchin system does, and to formulate a relationship with Hitchin's Abelianization Program. Finally, we have shown that certain group schemes - among which it is worth to point out the group of semilinear automorphisms - play the role of local generators for such moduli spaces.