discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

tilting bundles on toric fano fourfolds this paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth 4 dimensional toric fano varieties the tilting bundles lead to a large class of explicit calabi yau 5 algebras obtained as the corresponding rolled up helix algebra a database of the full strong exceptional collections can be found in the package quiverstoricvarieties for the computer algebra system macaulay2

minimal resolutions of algebras abstract a method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself the method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible with some additional finiteness restrictions in the latter case in particular it applies to any finite dimensional algebra over an algebraically closed field the method is illustrated by a number of examples viz truncated algebras monomial algebras and koszul algebras with the aim of unifying existing treatments of these in the literature

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

tropical coamoeba and torus equivariant homological mirror symmetry for the projective space we introduce the notion of a tropical coamoeba which gives a combinatorial description of the fukaya category of the mirror of a toric fano stack we show that the polyhedral decomposition of a real n torus into n 1 permutohedra gives a tropical coamoeba for the mirror of the projective space mathbb p n and we prove a torus equivariant version of homological mirror symmetry for the projective space as a corollary we obtain homological mirror symmetry for toric orbifolds of the projective space

a computational criterion for the kac conjecture abstract we give a criterion for the kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated kac moody algebra our criterion suits very well for computer tests

percolation threshold is not a decreasing function of the average coordination number it is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension this paper provides counterexamples a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order these counterexamples confirm the existence of this counterintuitive phenomenon which was observed in one case in numerical estimates by van der marck

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

spectral decompositions of isometries oncp the spatial action of the stone type spectral family for a certain type of strongly continuous one parameter groups of surjective isometries as well as of the spectral decomposition of their single elements oncp 1 p is examined the umd structure for 1 p and triangular truncations and duality forp 1 are involved in various cases concrete descriptions of the associated riesz nagy operators and arveson s spectral subspaces are derived as byproducts of independent interest

geometric reid s recipe for dimer models crepant resolutions of three dimensional toric gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models by choosing a special stability parameter and hence a distinguished crepant resolution y this derived equivalence generalises the fourier mukai transform relating the g hilbert scheme and the skew group algebra mathbb c x y z g for a finite abelian subgroup of mathrm sl 3 mathbb c we show that this equivalence sends the vertex simples to pure sheaves except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus this generalises results of cautis logvinenko j reine angew math 636 193 236 2009 and cautis craw logvinenko j reine angew math arxiv 1205 3110 2014 to the dimer setting though our approach is different in each case we also describe some of these pure sheaves explicitly and compute the support of the remainder providing a dimer model analogue of results from logvinenko j algebra 324 2064 2087 2010

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

an explicit construction of the mckay correspondence for a hilb c 3 abstract for a finite abelian subgroup a sl 3 c let y a hilb c 3 denote the scheme parametrising a clusters in c 3 ito and nakajima proved that the tautological line bundles indexed by the irreducible representations of a form a basis of the k theory of y we establish the relations between these bundles in the picard group of y and hence following a recipe introduced by reid construct an explicit basis of the integral cohomology of y in one to one correspondence with the irreducible representations of a

calabi yau algebras and weighted quiver polyhedra dimer models have been used in string theory to construct path algebras with relations that are 3 dimensional calabi yau algebras these constructions result in algebras that share some specific properties they are finitely generated modules over their centers and their representation spaces are toric varieties in order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3 dimensional calabi yau algebras can be constructed from dimer models on a torus toric orders are examples of a much broader class of algebras positively graded cancellation algebras for these algebras the cy 3 condition implies the existence of a weighted quiver polyhedron which is an extension of dimer models obtained by replacing the torus with any two dimensional compact orientable orbifold

flops of g hilb and equivalences of derived categories by variation of git quotient for a finite subgroup g in sl 3 c bridgeland king and reid proved that the moduli space of g clusters is a crepant resolution of the quotient c 3 g this paper considers the moduli spaces m theta introduced by kronheimer and further studied by sardo infirri which coincide with g hilb for a particular choice of the git parameter theta for g abelian we prove that every projective crepant resolution of c 3 g is isomorphic to m theta for some parameter theta the key step is the description of git chambers in terms of the k theory of the moduli space via the appropriate fourier mukai transform we also uncover explicit equivalences between the derived categories of moduli m theta for parameters lying in adjacent git chambers

the alternating syzygy behavior of monomial algebras abstract in this paper we present a minimal projective resolution of a monomial algebra over its enveloping algebra the syzygies for this resolution exhibit an alternating behavior which is explained by the construction of a special sequence of paths from the quiver of

noncommutative geometry and quiver algebras abstract we develop a new framework for noncommutative differential geometry based on double derivations this leads to the notion of moment map and of hamiltonian reduction in noncommutative symplectic geometry for any smooth associative algebra b we define its noncommutative cotangent bundle t b which is a basic example of noncommutative symplectic manifold applying hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras b that includes preprojective algebras associated with quivers our formalism of noncommutative hamiltonian reduction provides the space with a lie algebra structure analogous to the poisson bracket on the zero fiber of the moment map in the special case where is the preprojective algebra associated with a quiver of non dynkin type we give a complete description of the gerstenhaber algebra structure on the hochschild cohomology of in terms of the lie algebra

functional calculus for infinitesimal generators of holomorphic semigroups we give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on banach spaces which involves the bochner riesz kernels of such generators the rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup the strength of the formula is tested on poisson and gauss semigroups inl1 rn andl1 g for a stratified lie groupg we give also a self contained theory of smooth absolutely continuous functions on the half line 0

homologically homogeneous rings in this paper we study the structure of a right noetherian ring r of finite right global dimesion integral over a central subring c and satisfying the following condition if v w are irreducible right r modules with rc v rc w then pr dim v

consistency and derangements in brane tilings brane tilings describe lagrangians vector multiplets chiral multiplets and the superpotential of four dimensional supersymmetric gauge theories these theories written in terms of a bipartite graph on a torus correspond to worldvolume theories on n d3 branes probing a toric calabi yau threefold singularity a pair of permutations compactly encapsulates the data necessary to specify a brane tiling we show that geometric consistency for brane tilings which ensures that the corresponding quantum field theories are well behaved imposes constraints on the pair of permutations restricting certain products constructed from the pair to have no one cycles permutations without one cycles are known as derangements we illustrate this formulation of consistency with known brane tilings counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

a note on dimer models and mckay quivers we give one formulation of a procedure of hanany and vegh j high energy phys 0710 029 35 2007 which takes a lattice polygon as an input and produces a set of isoradial dimer models we study the case of lattice triangles in detail and discuss the relation with coamoebas following feng et al adv theor math phys 12 3 489 545 2008

spectral families of projections in hardy spaces strongly continuous one parameter groups of isometries in the reflexive hardy spaces of the disc d and the half plane are considered in the light of the author s previous joint result with h benzinger and t a gillespie which generalizes stone s theorem for unitary groups to arbitrary banach spaces it is shown that every such group tt of hardy space isometries has a spectral decomposition with respect to a suitable projection valued function on the real line r as in the classical statement of stone s theorem in hilbert space the relevant type of projection valued function is known as a spectral family this circle of ideas is intimately bound up with harmonic analysis particularly in hpr in particular if the group tt acts in hpd and is associated as in forelli s theorem with a group of parabolic mobius transformations of d then it can be analyzed by way of the translation group on hpr the stone type spectral family of the latter is shown to be obtained by restriction of the m riesz projections to hpr by this means a concrete description of the stone type spectral family for a parabolic isometric group on hpd is obtained a pleasant by product of the parabolic case is absorption of the classical paley wiener theorem for hpr 1 p 2 into the framework of the generalized stone s theorem

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

a derived approach to geometric mckay correspondence in dimension three we propose a three dimensional generalization of the geometric mckay correspondence described by gonzales sprinberg and verdier in dimension two we work it out in detail in abelian case more precisely we show that the bridgeland king reid derived category equivalence induces a natural geometric correspondence between irreducible representations of g and subschemes of the exceptional set of g hilb this correspondence appears to be related to reid s recipe

stone s theorem for strongly continuous groups of isometries in hardy spaces for 1 p p 2 it is well known that a strongly continuous one parameter group of isometries in the hardy space hp d does not satisfy the classical stone theorem that is there is no additive spectral measure on the borel sets of the real line whose fourier stieltjes transform is the given group however if we relax the topology of hp d by imbedding it into some larger space then it is shown for the case of 1 p 2 that any such group does satisfy stone s theorem when considered as acting in some appropriate larger space the situation for 2 p is more complicated if the group of isometries is of elliptic type then it again satisfies stone s theorem in a suitable larger space however for parabolic groups the situation is fundamentally different such a group can never be interpreted as a fourier stieltjes transform in any space containing hp d

spectral decompositions of one parameter groups of isometries on hardy spaces spectral decompositions of strongly continuous one parameter groups of surjective isometries on hardy spaces of the disk d and the torus t2 are examined a concrete descriptiofi of the pointwise action of these decompositions is presented mainly in the parabolic case leading to a complete description of the action of the partial sum operators of m riesz when carried from lp r to hp d 1 p 2 the pointwise action of the spectral decompositions of these isometric groups on hp t2 1 p oo is also examined and concrete descriptions are derived mainly in the parabolic case

derived reid s recipe for abelian subgroups of sl3 ℂ for any finite subgroup g sl 3 c work of bridgeland king reid constructs an equivalence between the g equivariant derived category of c 3 and the derived category of the crepant resolution y g hilb c 3 of c 3 g when g is abelian we show that this equivalence gives a natural correspondence between irreducible representations of g and exceptional subva rieties of y thereby extending the mckay correspondence from two to three dimensions this categorifies reid s recipe and extends earlier work from cl09 and log10 which dealt only with the case when c 3 g has one isolated singularity

erratum to a derived approach to geometric mckay correspondence in dimension three j reine angew math 636 2009 193 236

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

noncommutative mirror symmetry for punctured surfaces in 2013 abouzaid auroux efimov katzarkov and orlov showed that the wrapped fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities we generalize this result to other punctured riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models in particular given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is mathtt a infty isomorphic to a subcategory of the wrapped fukaya category of a punctured riemann surface the connection between the dimer model and the punctured riemann surface then has a nice interpretation in terms of a duality on dimer models

multigraded linear series and recollement given a scheme y equipped with a collection of globally generated vector bundles e 1 ldots e n we study the universal morphism from y to a fine moduli space mathcal m e of cyclic modules over the endomorphism algebra of e mathcal o y oplus e 1 oplus cdots oplus e n this generalises the classical morphism to the linear series of a basepoint free line bundle on a scheme we describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category when the morphism is surjective this gives a fine moduli space interpretation of the image and as an application we show that for a small finite subgroup g subset mathrm gl 2 mathbb k every sub minimal partial resolution of mathbb a 2 mathbb k g is isomorphic to a fine moduli space mathcal m e c where e c is a summand of the bundle e defining the reconstruction algebra we also consider applications to gorenstein affine threefolds where reid s recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

the kadison kaplansky conjecture for word hyperbolic groups in this paper we prove the kadison kaplansky idempotent conjecture for torsion free word hyperbolic groups the conjecture asserts that the following equivalent statements hold for a torsion free discrete group the reduced group c algebra c r contains no idempotents except 0 and 1 the spectrum of every element of the reduced group c algebra is connected the canonical trace on c r takes integer values on idempotents the last assertion can be viewed as a statement about the pairing between the k theory and the local cyclic cohomology of the group c algebra it is in this setting that we will prove the conjecture our proof is based on a partial analysis of the assembly maps in k theory and local cyclic homology we compare these assembly maps by means of an equivariant bivariant chern connes character before going into details we recall some previous work on the conjecture the first progress was achieved by pimsner and voiculescu pv who proved the kadison kaplansky conjecture for free groups as a consequence of their computation of the k theory of the group c algebra subsequently it was realized that more generally the kadison kaplansky conjecture was a consequence of the baum connes conjecture which gives a geometric description of k c r for any torsion free discrete group in fact the baum connes conjecture states that the k theoretic assembly map

homological mirror symmetry for punctured spheres we prove that the wrapped fukaya category of a punctured sphere s with an arbitrary number of points removed is equivalent to the triangulated category of singularities of a mirror landau ginzburg model proving one side of the homological mirror symmetry conjecture in this case by investigating fractional gradings on these categories we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the landau ginzburg model

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

derived equivalences of gentle algebras via fukaya categories following the approach of haiden katzarkov kontsevich publ math inst hautes etudes sci 126 247 318 2017 to any homologically smooth mathbb z graded gentle algebra a we associate a triple sigma a lambda a eta a where sigma a is an oriented smooth surface with non empty boundary lambda a is a set of stops on partial sigma a and eta a is a line field on sigma a such that the derived category of perfect dg modules of a is equivalent to the partially wrapped fukaya category of sigma a lambda a eta a modifying arguments of johnson and kawazumi we classify the orbit decomposition of the action of the symplectic mapping class group of sigma a on the homotopy classes of line fields as a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent our criterion uses numerical invariants generalizing those given by avella alaminos geiss in avella et al j pure appl algebra 212 1 228 243 2008 as well as some other numerical invariants as an application we find many new cases when the aag invariants determine the derived morita class as another application we establish some derived equivalences between the stacky nodal curves considered in lekili and polishchuk j topology 11 615 444 2018

wall crossing free fermions and crystal melting we describe wall crossing for local toric calabi yau manifolds without compact four cycles in terms of free fermions vertex operators and crystal melting firstly to each such manifold we associate two states in the free fermion hilbert space the overlap of these states reproduces the bps partition function corresponding to the non commutative donaldson thomas invariants given by the modulus square of the topological string partition function secondly we introduce the wall crossing operators which represent crossing the walls of marginal stability associated to changes of the b field through each two cycle in the manifold bps partition functions in non trivial chambers are given by the expectation values of these operators thirdly we discuss crystal interpretation of such correlators for this whole class of manifolds we describe evolution of these crystals upon a change of the moduli and find crystal interpretation of the flop transition and the dt pt transition the crystals which we find generalize and unify various other calabi yau crystal models which appeared in literature in recent years

mirror symmetry for del pezzo surfaces vanishing cycles and coherent sheaves we study homological mirror symmetry for del pezzo surfaces and their mirror landau ginzburg models in particular we show that the derived category of coherent sheaves on a del pezzo surface x k obtained by blowing up 2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration w k m k with k 3 singular fibers equipped with a suitable symplectic form moreover we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of x k and give an explicit correspondence between the deformation parameters for x k and the cohomology class b i h 2 m k

the orbifold topological vertex abstract we develop a topological vertex formalism for computing the donaldson thomas invariants of calabi yau orbifolds the basic combinatorial object is the orbifold vertex v g a generating function for the number of 3d partitions asymptotic to 2d partitions and colored by representations of a finite abelian group g acting on c 3 in the case where g z n acting on c 3 with transverse a n 1 quotient singularities we give an explicit formula for v g in terms of schur functions we discuss applications of our formalism to the donaldson thomas crepant resolution conjecture and to the orbifold donaldson thomas gromov witten correspondence we also explicitly compute the donaldson thomas partition function for some simple orbifold geometries the local football p a b 1 and the local b z 2 gerbe

the geometric dual of a maximisation for toric sasaki einstein manifolds we show that the reeb vector and hence in particular the volume of a sasaki einstein metric on the base of a toric calabi yau cone of complex dimension n may be computed by minimising a function z on mathbb r n which depends only on the toric data that defines the singularity in this way one can extract certain geometric information for a toric sasaki einstein manifold without finding the metric explicitly for complex dimension n 3 the reeb vector and the volume correspond to the r symmetry and the a central charge of the ads cft dual superconformal field theory respectively we therefore interpret this extremal problem as the geometric dual of a maximisation we illustrate our results with some examples including the yp q singularities and the complex cone over the second del pezzo surface

consistency conditions for brane tilings abstract given a brane tiling on a torus we provide a new way to prove and generalise the recent results of szendr i mozgovoy and reineke regarding the donaldson thomas theory of the moduli space of framed cyclic representations of the associated algebra using only a natural cancellation type consistency condition we show that the algebras are 3 calabi yau and calculate donaldson thomas type invariants of the moduli spaces two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover these ideas allow us to generalise the above results to all consistent brane tilings on k 1 surfaces we also prove a converse no consistent brane tiling on a sphere gives rise to a 3 calabi yau algebra

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

on the noncommutative geometry of square superpotential algebras a superpotential algebra is square if its quiver admits an embedding into a two torus such that the image of its underlying graph is a square grid possibly with diagonal edges in the unit squares examples are provided by dimer models in physics such an embedding reveals much of the algebras representation theory through a device we introduce called an impression let a be a square superpotential algebra z its center and m the maximal ideal at the origin of spec z using an impression we give a classification of all simple a modules up to isomorphism and give algebraic and homological characterizations of the simple a modules of maximal k dimension show that z is a 3 dimensional normal toric domain and zm is gorenstein by determining transcendence bases and z regular sequences and show that am is a noncommutative crepant resolution of zm and thus a local calabi yau algebra a particular class of square superpotential algebras the yp q algebras is considered in detail we show that the azumaya and smooth loci of the centers coincide and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size

criteria for evaluation of universal formulas for percolation thresholds several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature the existing universal formulas have been found to have substantial errors in their predictions for some lattices this paper proposes a set of desirable criteria for universal formulas to satisfy and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature the analysis is limited to lattices in two dimensions

reid s recipe and derived categories we prove two conjectures from cautis and logvinenko 2009 cl09 which describe the geometrical mckay correspondence for a finite abelian subgroup of sl3 c we do it by studying the relation between the derived category mechanics of computing a certain fourier mukai transform and a piece of toric combinatorics known as reid s recipe effectively providing a categorification of the latter

on the noncommutative donaldson thomas invariants arising from brane tilings abstract given a brane tiling that is a bipartite graph on a torus we can associate with it a quiver potential and a quiver potential algebra under certain consistency conditions on a brane tiling we prove a formula for the donaldson thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra we relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling we prove that the same consistency conditions imply that the quiver potential algebra is a 3 calabi yau algebra we also formulate a rationality conjecture for the generating functions of the donaldson thomas type invariants

critical probability bounds for two dimensional site percolation models the authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two dimensional lattices a technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that pc d or 0 5020 for the dice lattice d combining this method with kesten s duality result simplifies toth s derivation of the lower bound pc s or 0 5034 for the square lattice s the authors also present a technique for deriving upper bounds for bipartite graphs a technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice h pc h or 0 8079 and pc h or pc s the grouping technique is applied to the dice lattice to find the upper bound pc d or 0 7937

spectral decompositions of hyperbolic isometric groups in hardy spaces of the disk abstract we introduce a mechanism that can be used to achieve concrete descriptions of the pointwise action of the spectral decompositions for the hyperbolic type groups of isometries on hp d 1

cellular resolutions of noncommutative toric algebras from superpotentials abstract this paper constructs cellular resolutions for classes of noncommutative algebras analogous to those introduced by bayer and sturmfels 1998 2 in the commutative case to achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension for abelian skew group algebras and algebraically consistent dimer model algebras we introduce a cell complex in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of a we illustrate the general construction of for an example in dimension four arising from a tilting bundle on a smooth toric fano threefold to highlight the importance of the incidence function on

morita equivalences and azumaya loci from higgsing dimer algebras abstract let a a be a cyclic contraction of dimer algebras with a non cancellative and a cancellative a is then prime noetherian and a finitely generated module over its center in contrast a is often not prime nonnoetherian and an infinitely generated module over its center we present certain morita equivalences that relate the representation theory of a with that of a we then characterize the azumaya locus of a in terms of the azumaya locus of a and give an explicit classification of the simple a modules parameterized by the azumaya locus furthermore we show that if the smooth and azumaya loci of a coincide then the smooth and azumaya loci of a coincide this provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers with the property that their smooth and azumaya loci coincide

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

topological conformal field theories and calabi yau categories abstract this is the first of two papers which construct a purely algebraic counterpart to the theory of gromov witten invariants at all genera these gromov witten type invariants depend on a calabi yau a category which plays the role of the target in ordinary gromov witten theory when we use an appropriate a version of the derived category of coherent sheaves on a calabi yau variety this constructs the b model at all genera when the fukaya category of a compact symplectic manifold x is used it is shown under certain assumptions that the usual gromov witten invariants are recovered the assumptions are that open closed gromov witten theory can be constructed for x and that the natural map from the hochschild homology of the fukaya category of x to the ordinary homology of x is an isomorphism

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

the a deformation theory of a point and the derived categories of local calabi yaus abstract let a be an augmented algebra over a semi simple algebra s we show that the ext algebra of s as an a module enriched with its natural a infinity structure can be used to reconstruct the completion of a at the augmentation ideal we use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non compact calabi yau three folds since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants

cohomological hall algebras vertex algebras and instantons we define an action of the double of cohomological hall algebra of kontsevich and soibelman on the cohomology of the moduli space of spiked instantons of nekrasov we identify this action with the one of the affine yangian of mathfrak gl 1 based on that we derive the vertex algebra at the corner mathcal w r 1 r 2 r 3 of gaiotto and rapcak we conjecture that our approach works for a big class of calabi yau categories including those associated with toric calabi yau 3 folds

a cohomological interpretation of the reflexive brauer group throughout this note r will be a normal affine domain i e r is the coordinate ring of a normal afftne variety x over an algebraically closed field of characteristic zero let k denote the field of fractions of r and let c be a central simple algebra over k a subring a of c is said to be reflexive azumaya algebra over r iff i the center of n is equal to r 2 n is f g and reflexive as r module 3 a is an azumaya algebra over the discrete valuation ring r for every height one prime p of r two reflexive azumaya algebras in possibly distinct central simple k algebras are said to be equivalent if there exist eg reflexive r modules m and n such that a 0 end m r x end n as r algebras where the modified tensor product 0 is defined to be or the set of equivalence classes of reflexive azumaya algebras equipped with the modified tensor product as a multiplication rule is a group the so called reflexive brauer group l r of r or of the corresponding normal afftne variety x whereas the natural morphism br r br k

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1