Some combinatorial aspects of quantum field theory
Adrian Tanasa
备注：
30 pages, 9 figures. An explicit example of the Bogoliubov subtraction operator's action on the bare Feynman integral of a two-loop graph has been added; the resulting formula has then been compared with the corresponding formula obtained within the Connes-Kreimer algebraic setting. An appendix introducing Grassmann variables and the Grassmann representation of determinants have been added
发表信息：
Seminaire Lotharingien de Combinatoire, B65g (2012)
In this short survey we present the appearance of some combinatorial notions in quantum field theory. We first focus on topological graph polynomials (the Tutte polynomial and its multivariate version) and their relation with the parametric representation of the commutative $Φ^4$ field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás-Riordan polynomial with the parametric representation of some $Φ^4$ field theory on the non-commutative Moyal space. We also review the rôle played by the Connes-Kreimer Hopf algebra as the combinatorial backbone of the renormalization process in field theories. We then show how this generalizes to the scalar $Φ^4$ field theory implemented on the non-commutative Moyal space. Finally, some perspectives for the further generalization of these tools to quantum gravity tensor models are briefly sketched.
We discuss here some computational aspects of the Combinatorial Nullstellensatz argument. Our main result shows that the order of magnitude of the symmetry group associated with permutations of the variables in algebraic constraints, determines the performance of algorithms naturally deduced from Alon's Combinatorial Nullstellensatz arguments. Finally we present a primal-dual polynomial constructions for certifying the existence or the non-existence of solutions to combinatorial problems.
This is a review article on the combinatorial aspects of the mixed Hodge structure of a Milnor fibre of the isolated hypersurface singularity. We give a purely combinatorial method to compute spectral pairs of the singularity under the assumption of simplicial Newton boundary and non-degeneracy of the germ.
The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensio…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We illustrate how one can use basic combinatorial theory and computer programming technique (Python) to analyze the combinatorial game: Mahjong. The results confirm some folklore concerning the game, and expose some unexpected results. Related results and possible future research in connection to artificial intelligence are mentioned. Readers interested in the subject may further develop the techniques to deepen the study of the game, or study other combinatorial games.
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.
The prefix exchange distance of a permutation is the minimum number of exchanges involving the leftmost element that sorts the permutation. We give new combinatorial proofs of known results on the distribution of the prefix exchange distance for a random uniform permutation. We also obtain expressions for the mean and the variance of this distribution, and finally, we show that the normalised prefix exchange distribution converges in distribution to the standard normal distribution.
We obtain two combinatorial results: an equality of Weyl groups and an inequality of roots, in the setting of generalised Bott-Samelson resolutions of minuscule Schubert varieties. These results are used in the companion paper [BK19] to describe minimal rational curves on these resolutions, and their relation to lines on the Schubert varieties.
For any simplicial complex on m vertices a moment-angle complex Z_K embedded in C^m can be defined. There is a canonical action of a torus T^m on Z_K, but this action fails to be free. The Buchstaber number is the maximal integer s(K) for which there exists a subtorus of rank s(K) acting freely on Z_K. The similar definition can be given for real Buchstaber number. We study these invariants using certain sequences of simplicial complexes called universal complexes. Some general properties of Buchstaber numbers follow from combinatorial properties of universal complexes. In particular, we inves…
In this paper we determine a number of meaningful compositions of higher order of a set of functions, which is considered in Malesevic (1998), in implicit and explicit form. Results which are obtained are applied to the vector analysis in order to determine the number of meaningful differential operations of higher order.
The paper presents mathematical models of quasicrystals with particular attention given to cut-and-project sets. We summarize the properties of higher-dimensional quasicrystal models and then focus on the one-dimensional ones. For the description of their properties we use the methods of combinatorics on words.
In this article we propose an algebraic system, which is an abelian group $(A,+)$ with a family of non-associative and non-(left)distributive multiplications $\{\cdot_λ\}_{λ\in H}$. We call this algebraic system dynamical brace. The dynamical brace corresponds to a certain dynamical Yang-Baxter map (which is left nondegenerate and satisfy the unitary condition). Combinatorial aspects of the dynamical brace give us a correspondence between the dynamical brace and a certain family of subsets of $A\rtimes Aut(A)$. From this viewpoint we give an interpretation and examples of the dynamical brace.
We survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background to present a formula for generalized KP solitons, compute several explicit examples, and indicate how such a perspective could be used to extend previous research relating line-soliton solutions of the KP equation with Grassmannians.
The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.
This paper studies a single-suit version of the card game War on a finite deck of cards. There are varying methods of how players put the cards that they win back into their hands, but we primarily consider randomly putting the cards back and deterministically always putting the winning card before the losing card. The concept of a \emph{passthrough} is defined, which refers to a player playing through all cards in their hand from a particular point in the game. We consider games in which the second player wins during their first passthrough. We introduce several combinatorial objects related …
Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be q…
Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves an…
Let $(N,Φ)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as \[\mathcal{D}(a;b)=\{x\in Φ(r)+c\mid r\neq 0,\ b\in Φ(r)+c,\ |(Φ(r)+c)\cap (Φ(a)+b)|=1\}.\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if $\mathcal{B}^{\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,\mathcal{B}^{\mathcal{D}},\in)$ is actually a balanced incomplete block design.
Combinatorial aspects of the non-symmetric strong spectral property for graphs
Faculty of Natural Sciences and Mathematics, University of Banja Luka（班雅卢卡大学自然科学与数学学院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics, and Mechanics（数学、物理与力学研究所）
The inverse problem of general rough sets, considered by the present author in some of her earlier papers, in one of its manifestations is essentially the question of when an agent's view about crisp and non crisp objects over a set of objects has a rough evolution. In this research the nature of the problem is examined from number-theoretic and combinatorial perspectives under very few assumptions about the nature of data and some necessary conditions are proved.
To any nodal curve $C$ is associated the degree class group, a combinatorial invariant which plays an important role in the compactification of the generalised Jacobian of $C$ and in the construction of the Néron model of the Picard variety of families of curves having $C$ as special fibre. In this paper we study this invariant. More precisely, we construct a wide family of graphs having cyclic degree class group and we provide a recursive formula for the cardinality of the degree class group of the members of this family. Moreover, we analyse the behaviour of the degree class group under stan…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumer…
We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case $a\in M_{2\times q}(2\mathbb N)$ to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of the "best algebraic normalization" of $I(a)$, in the case $a\in M_{2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a…
Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor in addition to the usual number theoretical interpretations. In particular we prove that $N_k = - W_k(q, - N_1)$ where W_k(q,t) is a (q,t)-analogue of the number of spanning trees of the wheel graph. Additionally we develop a determinantal formula for N_k where the eigenvalues can be explicitly written in terms of q, N_1, and roots of unity. We also discuss …
Quasi-triangular Hopf algebras were introduced by Drinfel'd in his construction of solutions to the Yang--Baxter Equation. This algebra is built upon $\mathcal{U}_h(\mathfrak{sl}_2)$, the quantized universal enveloping algebra of the Lie algebra $\mathfrak{sl}_2$. In this paper, combinatorial structure in $\mathcal{U}_h(\mathfrak{sl}_2)$ is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $n=1$. We apply this formalism to the quasi-triangu…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by systems of polynomial equations satisfying special properties. If the equations are infeasible, Hilbert's Nullstellensatz gives a certificate of this fact. These certificates have been studied and exhibit combinatorial meaning. In this paper, we generalize some known results and show that the Nullstellensatz certificate can be viewed as enumerating combinatorial …
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we review several applications and discuss implementation and computational aspects.
The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provide…
We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model's recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a give…
We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m,n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we…
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct strata,…
We consider the determination of the number $c_k(α)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $α$, into k-cycles such that the factorisations have minimal length and such that the group generated by the factors acts transitively on the n symbols. The case k=2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the alternating group, and this is another case that, in…
When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to …
We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $γ$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (20…
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of collapsible polyhedra in terms of constructible posets (generalizing face posets of constructible simplicial complexes in the sense of Hochster). The relevant constructible posets are also characterized in terms of Reading's zipping of posets, which is a variation of edge contract…
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of these new identities.
Let $p(x_1,...,x_n) =\sum_{(r_1,...,r_n) \in I_{n,n}} a_{(r_1,...,r_n)} \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients. The support of such polynomial $p(x_1,...,x_n)$ is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 \}$ . The convex hull $CO(supp(p))$ of $supp(p)$ is called the Newton polytope of $p$ . We study the following decision problems, which are far-reaching generalizations of the classical perfect matching problem : {itemize} {\bf Problem 1 .} Consider a homogene…
Eight combinatorial identities are listed and proved by counting paths in the one-dimensional random walk. Four of these identities are assumed to be new.
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes …
Three combinatorial matrices are considered and their LU-decompositions were found. This is typically done by (creative) guessing, and necessary proofs are more or less routine calculations.
The purpose of this paper is to show that some combinatorial sequences, such as second-order Eulerian numbers and Eulerian numbers of type $B$, can be generated by context-free grammars.
Many mathematicians have been studying various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This focus stems from their nascent importance for applications in combinatorics, number theory and in other aspects of applied mathematics. we construct a new type of degenerate poly-Euler polynomials by using the degenerate polylogarithm functions. We also show several combinatorial identities related to this polynomials and numbers.
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving classical harmonic numbers. For the binomial coefficients the definition with gamma functions is used, thus also allowing non-integer arguments in the identities. The generalized harmonic numbers in this case are harmonic numbers with a complex offset, where the classical harmonic numbers are a special case with offset zero.
We establish a new simple explicit description of combinatorial wall-crossing for the rational Cherednik algebra applied to the trivial representation. In this way we recover a theorem of P. Dimakis and G. Yue. We also present two conjectures on combinatorial wall-crossing which were found using computer experiments.
In 1962 O. A. Gross proved that the last digits of the Fubini numbers (or surjective numbers) have a simple periodicity property. We extend this result to a wider class of combinatorial numbers coming from restricted set partitions.
Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin numbers, in particular, about the average height of such objects.
