Diagrammatic Calculus Research Articles

Braided Scalar Quantum Field Theory

AbstractWe formulate scalar field theories in a curved braided ‐algebra formalism and analyse their correlation functions using Batalin–Vilkovisky quantization. We perform detailed calculations in cubic braided scalar field theory up to two‐loop order and three‐point multiplicity. The divergent tadpole contributions are eliminated by a suitable choice of central curvature for the ‐structure, and we confirm the absence of UV/IR mixing. The calculations of higher loop and higher multiplicity correlators in homological perturbation theory are facilitated by the introduction of a novel diagrammatic calculus. We derive an algebraic version of the Schwinger–Dyson equations based on the homological perturbation lemma, and use them to prove the braided Wick theorem.

Reconstruction of Voronoi diagrams in inverse potential problems

In this paper we propose and analyze a numerical method for the recovery of a piecewise constant parameter with multiple phases in the inverse potential problem. The potential is assumed to be constant in each phase, and the phases are modeled by a Voronoi diagram generated by a set of sites, which are used as control parameters. We first reformulate the inverse problem as an optimization problem with respect to the position of the sites. Combining techniques of non-smooth shape calculus and sensitivity of Voronoi diagrams, we are able to compute the gradient of the cost function, under standard non-degeneracy conditions of the diagram. We provide two different formulas for the gradient, a volumetric and an interface one, which are compared in numerical experiments. We provide several numerical experiments to investigate the dependence of the reconstruction on the problem parameters, such as noise, number of sites and initialization.

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability of Boolean formulas, and prove it to be sound and complete. We then showcase our calculus in a few case studies. First, we consider SAT-solving. Second, we consider Horn clauses, which leads us to a new decision method for propositional logic programs equivalence under Herbrand model semantics.

Legendrian weaves : N–graph calculus, flag moduli and applications

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology. Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.

Seifert-Torres type formulas for the Alexander polynomial from quantum [formula omitted

We develop a diagrammatic calculus for representations of unrolled quantum sl2 at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given here are a skein relation for n-cabled double crossings and a simple proof that the quantum invariant associated with these representations determines the multivariable Alexander polynomial.

Coherence for adjunctions in a 3-category via string diagrams

We construct a 3-categorical presentation Adj(3,1) and define a coherent adjunction in a strict 3-category C as a map Adj(3,1)→C. We use string diagrams to show that any adjunction in C can be extended to a coherent adjunction in an essentially unique way. The results and their proofs will apply in the context of Gray 3-categories after the string diagram calculus is shown to hold in that context in an upcoming paper.

High-level axioms for graphical linear algebra

• We present useful symmetrical axioms for graphical linear algebra. • We use only the diagrammatic language and its associated reasoning principles. • We develop an approach to matrices, proving its equivalence to the classical one. We focus on a modular, graphical language—graphical linear algebra—and use it as high-level language for calculational reasoning. We propose a minimal framework of axioms that highlight the dualities and symmetries of linear algebra, and showcase the resulting diagrammatic calculus. Our work develops a relational approach to linear algebra, closely connected to classical relational algebra. With the symmetrical high-level axioms we are able to provide a fully diagrammatic proof that a fragment of Graphical Linear Algebra is equivalent to matrix algebra.

Goussarov–Polyak–Viro conjecture for degree three case

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give [Formula: see text] Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of [Formula: see text] formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).

On simple modules of cyclotomic quiver Hecke algebras of type A

We give a classification of the graded simple modules of cyclotomic quiver Hecke algebras of type A using the diagram calculus of the diagrammatic Cherednik algebra. We also obtain a non-trivial lower bound for the dimension of the simple modules which is independent of the field characteristic.

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

We study a presentation of the Khovanov - Lauda - Rouquier's candidate 2-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of 2-cells of this 2-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this 2-category is a categorification of Lusztig's idempotented and integral quantum group Uq(g) associated with a symmetrizable simply-laced Kac-Moody algebra g.

The trigonometric $$E_8$$ R-matrix

An expression for the R-matrix associated to $U_q(\widehat{e_8})$ in its 249-dimensional representation is given using the diagrammatic calculus of $U_q(e_8)$ invariants.

String Diagrams for Regular Logic (Extended Abstract)

Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category FRg(T) on a set T. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts -- the 2-category of relations in FRg(T) -- to that of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus because it naturally gives rise to a graphical string diagram calculus in the spirit of Joyal and Street. We shall show that every natural category has an associated regular calculus, and conversely from every regular calculus one can construct a regular category.

Thicker Soergel calculus for B3

Diagrammatic algebra provides a useful way to study Soergel bimodules. This approach proceeds via the simpler category [Formula: see text]Bim of Bott–Samelson bimodules, for which there is a well developed diagrammatic calculus. As Soergel bimodules are summands of Bott–Samelson bimodules, it is important to understand idempotents in the category [Formula: see text]Bim. For Coxeter groups of type [Formula: see text], we analyze this problem for certain important idempotents, namely, the idempotent projecting to the indecomposable Soergel bimodule corresponding to the longest element of the Coxeter group. We use a strategy analogous to one used by Elias in type [Formula: see text]. We present a full explicit calculation for the first nontrivial case [Formula: see text]. The strategy is applicable for general [Formula: see text] but much more involved. We hope that the [Formula: see text] case serves as a stepping stone for the general case.

DECOMPOSING THE TUBE CATEGORY

AbstractThe tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.

Webs for permutation supermodules of type Q

We give a diagrammatic calculus for the intertwiners between permutation supermodules of the Sergeev superalgebra over the complex numbers. We also give a diagrammatic basis for the space of intertwiners between two permutation supermodules.

2-row Springer Fibres and Khovanov Diagram Algebras for Type D

AbstractWe study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections are iterated ℙ1-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type D diagram calculus labelling the irreducible components in a convenient way that relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types.

Fusion categories via string diagrams

We use the string diagram calculus to give graphical proofs of the basic results of Etingof, Nikshych and Ostrik [On fusion categories, Ann. Math. 162 (2005) 581–642; arXiv:math/0203060, doi:10.4007/annals.2005.162.581] on fusion categories. These results include: the quadruple dual is canonically isomorphic to the identity, positivity of the paired dimensions, and Ocneanu rigidity. We introduce the pairing convention as a convenient graphical framework for working with fusion categories. We use this framework to express the pivotal operators as a product of the apex associator monodromy and the pivotal indicators. We also characterize pivotal structures as solutions of an explicit set of algebraic equations over the complex numbers, refining a formula of Wang.

Categorification of tensor powers of the vector representation of $$U_q({\mathfrak {gl}}(1|1))$$ U q ( gl ( 1 | 1 ) )

We consider the monoidal subcategory of finite-dimensional representations of \(U_q(\mathfrak {gl}(1|1))\) generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both \(U_q(\mathfrak {gl}(1|1))\) and the intertwining operators using subquotient categories of the BGG category Open image in new window.

Heisenberg algebra and a graphical calculus

A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.

Genus 0 characteristic numbers of the tropical projective plane

AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.