Pentagon Equation Research Articles

On quadrirational pentagon maps

Abstract We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps R : ( x , y ) ↦ ( u ( x , y ) , v ( x , y ) ) , where u , v are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang–Baxter maps from entwining pentagon maps.

Finite-dimensional Hopf superalgebras and graded pentagon equation

Suppose that H is an arbitrary finite-dimensional Hopf superalgebra. Let H(H) be the Heisenberg double of H and let R be the canonical matrix of H(H) that satisfies the graded pentagon equation R12R13R23=R23R12. It is established that H is isomorphic to the Hopf superalgebra P(H(H),R) of left coefficients of R. This result can be regarded as a generalisation of Militaru's result [10] from the non-super situation to the super situation.

On the Structure of Set-Theoretic Polygon Equations

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation, have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work, we explore the corresponding reductions of (dual) polygon equations, which lead to relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.

Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation s:S×S→S×S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s:S\ imes S\\rightarrow S\ imes S$$\\end{document} on a set S and writing s(a,b)=(a·b,θa(b))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s(a, b)=(a\\cdot b,\\, \ heta _a(b))$$\\end{document}, with ·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cdot $$\\end{document} a binary operation on S and θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} a map from S into itself, for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}, one naturally obtains that S,·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( S,\\,\\cdot \\right) $$\\end{document} is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups S,·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( S,\\,\\cdot \\right) $$\\end{document} satisfying special properties on the set of all idempotents E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document}. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} remains invariant in E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document}, for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}. Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} fixes every element in E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document} for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}, from solutions given on each maximal subgroup of S.

Self-duality under gauging a non-invertible symmetry

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ2 × ℤ2 symmetry, and has Rep(H8) and Rep(D8) fusion category symmetries as a result. We find that gauging the entire Rep(H8) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/R. At R = 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{2} $$\\end{document}, which corresponds to two decoupled Ising CFTs (Ising2 in short), the theory is self-dual under gauging the Rep(H8) symmetry. This implies the existence of a topological defect line in the Ising2 CFT obtained from half-space gauging of the Rep(H8) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep(H8) symmetry to form a bigger fusion category which is a ℤ2-extension of Rep(H8). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising2 CFT. Finally, we show that the torus partition functions of the Monster2 CFT and Ising×Monster CFT are also invariant under gauging the Rep(H8) symmetry.

The photography method: Solving pentagon equation

In this paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labeled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general data (say, edge lengths or areas) associated with states (say, triangulations) and a general data transformation rule (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data.

Idempotent set-theoretical solutions of the pentagon equation

AbstractA set-theoretical solution of the pentagon equation on a non-empty set X is a function $$s:X\times X\rightarrow X\times X$$ s : X × X → X × X satisfying the relation $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$$ s 23 s 13 s 12 = s 12 s 23 , with $$s_{12}=s\times \,{{\,\textrm{id}\,}}_X$$ s 12 = s × id X , $$s_{23}={{\,\textrm{id}\,}}_X \times \, s$$ s 23 = id X × s and $$s_{13}=({{\,\textrm{id}\,}}_X\times \, \tau )s_{12}({{\,\textrm{id}\,}}_X\times \,\tau )$$ s 13 = ( id X × τ ) s 12 ( id X × τ ) , where $$\tau :X\times X\rightarrow X\times X$$ τ : X × X → X × X is the flip map given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) , for all $$x,y\in X$$ x , y ∈ X . Writing a solution as $$s(x,y)=(xy,\theta _x(y))$$ s ( x , y ) = ( x y , θ x ( y ) ) , where $$\theta _x: X \rightarrow X$$ θ x : X → X is a map, for every $$x\in X$$ x ∈ X , one has that X is a semigroup. In this paper, we study idempotent solutions, i.e., $$s^2=s$$ s 2 = s , by showing that the idempotents of X have a crucial role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $$\theta _1$$ θ 1 is a monoid homomorphism.

Classifying phases protected by matrix product operator symmetries using matrix product states

We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.

The free field representation for the GL(1∣1) WZW model revisited

The GL(1∣1) WZW model in the free field realization that uses the bc system is revisited. By bosonizing the bc system we describe the Neveu-Schwarz and Ramond sector modules and in terms of the subspaces of a given fermion number l. We show that there are two sectors of mutually local operators, each consists of all Neveu-Schwarz operators and of Ramond operators with either integer or half-integer spins. Conformal blocks and structure constants are found for operators that correspond the highest weight vectors of the spaces . The crossing and braiding matrices are considered and the hexagon and pentagon equations are shown to be satisfied for typical modules. The degenerate case of conformal blocks with atypical (logarithmic) modules as intermediate states is considered. The known conformal block decomposition of correlation functions in the degenerate case is shown to be related to the degeneration splitting in the crossing and braiding relations. The scalar product in atypical modules is discussed. The decomposition of unity in the full correlation functions in the degenerate case in terms of this scalar product is explained.

The pentagon equation and the confluence relations

We show an equivalence of Drinfeld's pentagon equation and Hirose-Sato's confluence relations. As a corollary, we obtain a `pentagon-free' presentation of the Grothendieck-Teichm\"{u}ller group $GRT_1$ and associators.

Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.

Matrix product operator symmetries and intertwiners in string-nets with domain walls

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

Characterizing Topological Order with Matrix Product Operators

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of {mathsf {G}}- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

Set-Theoretic Solutions of the Pentagon Equation

A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times\mathrm{id}$, $s_{23}=\mathrm{id}\times s$ and $s_{13}=(\tau\times\mathrm{id})(\mathrm{id}\times s)(\tau\times\mathrm{id})$ are mappings from $S^3$ to itself and $\tau\colon S^2\to S^2$ is the flip map, i.e., $\tau (x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A \times G$ and a map $\sigma\colon A\to\mathrm{Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $\sigma$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\geq 0$ then, on $S$, there are precisely $\binom{n+2}{2}$ non-isomorphic solutions of the Pentagon Equation.

Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups

The Yang–Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map $$s:S\times S\rightarrow S\times S$$ is said to be a set-theoretical solution of the quantum Yang–Baxter equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, \end{aligned}$$ where $$s_{12}=s\times {{\,\mathrm{id}\,}}_S$$ , $$s_{23}={{\,\mathrm{id}\,}}_S\times s$$ , and $$s_{13}=({{\,\mathrm{id}\,}}_S\times \tau )\,s_{12}\,({{\,\mathrm{id}\,}}_S\times \tau )$$ and $$\tau $$ is the flip map, i.e., the map on $$S\times S$$ given by $$\tau (x,y)=(y,x)$$ . Instead, s is called a set-theoretical solution of the pentagon equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}. \end{aligned}$$ The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang–Baxter equation. Specifically, we present a new construction of solutions of the Yang–Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.

Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory.

We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

Microscopic definitions of anyon data

We present microscopic definitions of both the $F$-symbol and $R$-symbol -- two pieces of algebraic data that characterize anyon excitations in (2+1)-dimensional systems. An important feature of our definitions is that they are operational; that is, they provide concrete procedures for computing these quantities from microscopic models. In fact, our definitions, together with known results, provide a way to extract a complete set of anyon data from a microscopic model, at least in principle. We illustrate our definitions by computing the $F$-symbol and $R$-symbol in several exactly solvable lattice models and edge theories. We also show that our definitions of the $F$-symbol and $R$-symbol satisfy the pentagon and hexagon equations, thereby providing a microscopic derivation of these fundamental constraints.

Set-theoretical solutions of the pentagon equation on groups

Let M be a set. A set-theoretical solution of the pentagon equation on M is a map such that where and and τ is the flip map, i.e., the permutation on M × M given by for all In this paper, we give a complete description of the set-theoretical solutions of the form when either or is a group; moreover, we raise some questions.

Matrix Kadomtsev–Petviashvili Equation: Tropical Limit, Yang–Baxter and Pentagon Maps

In the tropical limit of matrix KP-II solitons, their support at a fixed time is a planar graph with “polarizations” attached to its linear parts. We explore a subclass of soliton solutions whose tropical limit graph has the form of a rooted and generically binary tree and also solutions whose limit graph comprises two relatively inverted such rooted tree graphs. The distribution of polarizations over the lines constituting the graph is fully determined by a parameter-dependent binary operation and a Yang–Baxter map (generally nonlinear), which becomes linear in the vector KP case and is hence given by an R-matrix. The parameter dependence of the binary operation leads to a solution of the pentagon equation, which has a certain relation to the Rogers dilogarithm via a solution of the hexagon equation, the next member in the family of polygon equations. A generalization of the R-matrix obtained in the vector KP case also solves a pentagon equation. A corresponding local version of the latter then leads to a new solution of the hexagon equation.

Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g. in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as $p= \langle F, F\rangle$ where $F$ is the curvature 2-form and $\langle \cdot, \cdot\rangle$ is an invariant scalar product on the corresponding Lie algebra $\mathfrak{g}$. The descent for $p$ gives rise to an element $\omega=\omega_3 + \omega_2 + \omega_1 + \omega_0$ of mixed degree. The 3-form part $\omega_3$ is the Chern-Simons form. The 2-form part $\omega_2$ is known as the Wess-Zumino action in physics. The 1-form component $\omega_1$ is related to the canonical central extension of the loop group $LG$. In this paper, we give a new interpretation of the low degree components $\omega_1$ and $\omega_0$. Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class $p$. In more detail, we define a 1-cocycle $C$ which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation $F$ is mapped to $\omega_1=C(F)$. Furthermore, the component $\omega_0$ is related to the associator corresponding to $F$. It is surprising that while $F$ and $\Phi$ satisfy the highly non-linear twist and pentagon equations, the elements $\omega_1$ and $\omega_0$ solve the linear descent equation.