conformal field theory and elliptic cohomology abstract in this paper we use conformal field theory to construct a generalized cohomology theory which has some properties of elliptic cohomology theory which was some properties of elliptic cohomology a part of our presentation is a rigorous definition of conformal field theory following segal s axioms and some examples such as lattice theories associated with a unimodular even lattice we also include certain examples and formulate conjectures on modular forms and monstrous moonshine related to the present work

permutation 2 groups i structure and splitness abstract by a 2 group we mean a groupoid equipped with a weakened group structure it is called split when it is equivalent to the semidirect product of a discrete 2 group and a one object 2 group by a permutation 2 group we mean the 2 group s ym g of self equivalences of a groupoid g and natural isomorphisms between them with the product given by composition of self equivalences these generalize the symmetric groups s n n 1 obtained when g is a finite discrete groupoid after introducing the wreath 2 product s n g of the symmetric group s n with an arbitrary 2 group g it is shown that for any finite type groupoid g the permutation 2 group s ym g is equivalent to a product of wreath 2 products of the form s n s ym g for a group g thought of as a one object groupoid this is next used to compute the homotopy invariants of s ym g which classify it up to equivalence using a previously shown splitness criterion for strict 2 groups it is then proved that s ym g can be non split and that the step from the trivial groupoid to an arbitrary one object groupoid is the only source of non splitness various examples of permutation 2 groups are explicitly computed in particular the permutation 2 group of the underlying groupoid of a finite type 2 group it also follows from well known results about the symmetric groups that the permutation 2 group of the groupoid of all finite sets and bijections between them is equivalent to the direct product 2 group z 2 1 z 2 0 where z 2 0 and z 2 1 stand for the group z 2 thought of as a discrete and a one object 2 group respectively

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

higher traces noncommutative motives and the categorified chern character abstract we propose a categorification of the chern character that refines earlier work of toen and vezzosi and of ganter and kapranov if x is an algebraic stack our categorified chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over x which we introduce to s 1 equivariant perfect complexes on the derived free loop stack l x as an application of the theory we show that toen and vezzosi s secondary chern character factors through secondary k theory our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal n categories which is of independent interest

translation by the full twist and deligne lusztig varieties abstract we prove several conjectures about the cohomology of deligne lusztig varieties invariance under conjugation in the braid group behaviour with respect to translation by the full twist parity vanishing of the cohomology for the variety associated with the full twist in the case of split groups of type a and using previous results of the second author this implies broue michel s conjecture on the disjointness of the cohomology for the variety associated to any good regular element that conjecture was inspired by broue s abelian defect group conjecture and the specific form broue conjectured for finite groups of lie type 4 reves 1 et 2

schemas sur les categories abeliennes monoidales symetriques et faisceaux quasi coherents resume soit x un schema sur une categorie abelienne monoidale symetrique c 1 verifiant certaines conditions dans cet article nous developpons la theorie de la categorie derivee d o x qcoh des faisceaux quasi coherents sur x ou x n est pas necessairement noetherien en particulier nous montrons les deux resultats suivants sur la categorie d o x qcoh 1 la categorie d o x qcoh est munie d un produit tensoriel derive et un hom interne et 2 soit x la plus petite sous categorie triangulee contenant tous les objets j m ou j u x est une immersion ouverte avec u affine et m d o u qcoh alors x d o x qcoh

affineness of deligne lusztig varieties for minimal length elements abstract we prove that the deligne lusztig varieties associated to elements of the weyl group which are of minimal length in their twisted class are affine our proof differs from the proof of he and orlik rapoport and it is inspired by the case of regular elements which correspond to the varieties involved in broue s conjectures

extended deligne lusztig varieties for general and special linear groups abstract we give a generalisation of deligne lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non archimedean local field previously a generalisation was given by lusztig by attaching certain varieties to unramified maximal tori inside borel subgroups in this paper we associate a family of so called extended deligne lusztig varieties to all tamely ramified maximal tori of the group moreover we analyse the structure of various generalised deligne lusztig varieties and show that the unramified varieties including a certain natural generalisation do not produce all the irreducible representations in general on the other hand we prove results which together with some computations of lusztig show that for sl 2 f q 2 with odd q the extended deligne lusztig varieties do indeed afford all the irreducible representations

strongly coupled ising chain under a weak random field abstract we present here an analytical method based on the transfer matrix m to study quenched disorder in one dimensional spin systems in the limit of strong couplings and weak disorder the procedure is formulated for the random field ising chain of finite length l and its properties represented as functions of m satisfy a differential equation of the fokker planck type this equation describes evolution with l and a central limit theorem of a novel kind provides the equation and its solutions with universal character we obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit we find that the random field free energy f h and the edwards anderson order parameter m 2 satisfy a simple relation we discuss our results in connection to previous work by luck and nieuwenhuizen

an action of the hecke monoid on rational modules for the borel subgroup of a quantised general linear group abstract we construct an action of the hecke monoid on the category of rational modules for the quantum negative borel subgroup of the quantum general linear group we also show that this action restricts to the category of polynomial modules for this quantum subgroup and induces an action on the category of modules for the quantised borel schur algebra s n r

on autoequivalences of the infty 1 category of infty operads we study the infty 1 category of autoequivalences of infty operads using techniques introduced by toen lurie and barwick and schommer pries we prove that this infty 1 category is a contractible infty groupoid our calculation is based on the model of complete dendroidal segal spaces introduced by cisinski and moerdijk similarly we prove that the infty 1 category of autoequivalences of non symmetric infty operads is the discrete monoidal category associated to mathbf z 2 mathbf z we also include a computation of the infty 1 category of autoequivalences of infty n categories based on rezk s theta n spaces

a note on words in braid monoids abstract the purpose of this note is to give a self contained apart from simple facts about coxeter groups and we hope a bit shorter and more understandable account of some results of c1 c2 on normal forms of braids which are themselves based on the papers d1 t in particular a motivation was to give a proof of proposition 5 1 that we use in b m some proofs and results from section 2 onwards seem to be new i thank several people for improvements from earlier versions of the manuscript m geck for pointing out some errors f digne for pointing out that some results don t need the braid group to be of finite type and j y hee for suggesting and providing further improvements in that direction

derived azumaya algebras and generators for twisted derived categories we introduce a notion of derived azumaya algebras over ring and schemes generalizing the notion of azumaya algebras of grothendieck le groupe de brauer i algebres d azumaya et interpretations diverses dix exposes sur la cohomologie des schemas pp 46 66 north holland amsterdam 1968 we prove that any such algebra b on a scheme x provides a class b in h 1 et x mathbb z times h 2 et x mathbb g m we prove that for x a quasi compact and quasi separated scheme defines a bijective correspondence and in particular that any class in h 2 et x mathbb g m torsion or not can be represented by a derived azumaya algebra on x our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories with coefficients in any local system of reasonable dg categories generalizing the well known existence of compact generators in derived categories of quasi coherent sheaves of bondal and van den bergh mosc math j 3 1 1 36 2003 a huge part of this paper concerns the treatment of twisted derived categories as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator we present explicit examples of derived azumaya algebras that are not represented by classical azumaya algebras as well as applications of our main result to the localization for twisted algebraic k theory and to the stability of saturated dg categories by direct push forwards along smooth and proper maps

poisson bivectors and poisson brackets on affine derived stacks abstract let a be a commutative dg algebra concentrated in degrees m and let spec a be the associated derived stack we give two proofs of the existence of a canonical map from the moduli space of shifted poisson structures in the sense of 16 on spec a to the moduli space of homotopy shifted poisson algebra structures on a the first makes use of a more general description of the poisson operad and of its cofibrant models while the second is more computational and involves an explicit resolution of the poisson operad

werte hypergeometrischer funktionen a partial result on a problem of siegel is given hypergeometric functions with rational parameters have transcendental values in almost all algebraic points up to some natural exceptions these exceptions are the well known algebraic functions and an unexpected second class of examples related to certain shimura curves

fuchsian triangle groups and grothendieck dessins variations on a theme of belyi according to a theorem of belyi a smooth projective algebraic curve is defined over a number field if and only if there exists a non constant element of its function field ramified only over 0 1 and the existence of such a belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the poincare upper half plane by a subgroup of finite index in a fuchsian triangle group on the other hand fuchsian triangle groups arise in many contexts such as in the theory of hypergeometric functions and certain triangular billiard problems which would appear at first sight to have no relation to the galois problems that motivated the above discovery of belyi in this note we review several results related to belyi s theorem and we develop certain aspects giving examples for preliminary accounts see the preprint wo1 the conference proceedings article bauitz and the comptes rendus note cowo2

inner products of 2 representations abstract we define and calculate inner products of 2 representations along the way we prove that the categorical trace t r of 24 sec 3 is multiplicative with respect to various notions of categorical tensor product and we identify the center of the category of equivariant objects we discuss applications ranging from schur s result about the number of irreducible projective representations to a formula for the hochschild cohomology of a global quotient orbifold

the global derived period map abstract we develop the global period map in the context of derived geometry generalising griffiths classical period map as well as the infinitesimal derived period map we begin by constructing the derived period domain which classifies hodge filtrations and enhances the classical period domain we analyze the monodromy action then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy we conclude the paper by discussing a few examples and a derived torelli problem in the appendix we describe how to present derived analytic artin stacks as hypergroupoids which may be of independent interest

derived smooth manifolds we define a simplicial category called the category of derived manifolds it contains the category of smooth manifolds as a full discrete subcategory and it is closed under taking arbitrary intersections in a manifold a derived manifold is a space together with a sheaf of local c rings that is obtained by patching together homotopy zero sets of smooth functions on euclidean spaces we show that derived manifolds come equipped with a stable normal bundle and can be imbedded into euclidean space we define a cohomology theory called derived cobordism and use a pontrjagin thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory this allows us to define fundamental classes in cobordism for all derived manifolds in particular the intersection a b of submanifolds a b x exists on the categorical level in our theory and a cup product formula a b a b holds even if the submanifolds are not transverse one can thus consider the theory of derived manifolds as a categorification of intersection theory

presenting higher stacks as simplicial schemes abstract we show that an n geometric stack may be regarded as a special kind of simplicial scheme namely a duskin n hypergroupoid in affine schemes where surjectivity is defined in terms of covering maps yielding artin n stacks deligne mumford n stacks and n schemes as the notion of covering varies this formulation adapts to all hag contexts so in particular works for derived n stacks replacing rings with simplicial rings we exploit this to describe quasi coherent sheaves and complexes on these stacks and to draw comparisons with kontsevich s dg schemes as an application we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks

the spiralling self avoiding walk in a random environment the authors study the spiralling self avoiding lattice walk in a random environment upon scaling the temperature appropriately with system size a phase transition appears in the low temperature phase the walk segments occupy a few low energy positions in the high temperature phase they are effectively free an analogy with the random energy model is pointed out the average size of an n step walk is shown to be asymptotically proportional to n1 2 log n as was known for the homogeneous lattice with a coefficient that increases as the temperature is lowered the spatial distribution of the walk segments is qualitatively different above and below the critical temperature the model also allows for a spin glass interpretation and as such helps to clarify the connection between the concepts of frustration and the chaoticity of the pair correlation both above and below the critical point

stein domains in banach algebraic geometry abstract in this article we give a homological characterization of the topology of stein spaces over any valued base field in particular when working over the field of complex numbers we obtain a characterization of the usual euclidean transcendental topology of complex analytic spaces for non archimedean base fields the topology we characterize coincides with the topology of the berkovich analytic space associated to a non archimedean stein algebra because the characterization we used is borrowed from a definition in derived geometry this work should be read as a derived perspective on analytic geometry

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

the stack of yang mills fields on lorentzian manifolds we provide an abstract definition and an explicit construction of the stack of non abelian yang mills fields on globally hyperbolic lorentzian manifolds we also formulate a stacky version of the yang mills cauchy problem and show that its well posedness is equivalent to a whole family of parametrized pde problems our work is based on the homotopy theoretical approach to stacks proposed in hollander isr j math 163 93 124 2008 which we shall extend by further constructions that are relevant for our purposes in particular we will clarify the concretification of mapping stacks to classifying stacks such as bgcon

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

chern characters and hirzebruch riemann roch formula for matrix factorizations we study the category of matrix factorizations for an isolated hypersurface singularity we compute the canonical bilinear form on the hochschild homology of this category we find explicit expressions for the chern character and the boundary bulk maps and derive an analogue of the hirzebruch riemann roch formula for the euler characteristic of the hom space between a pair of matrix factorizations we also establish g equivariant versions of these results

analytic geometry over f1 and the fargues fontaine curve abstract this paper develops a theory of analytic geometry over the field with one element the approach used is the analytic counter part of the toen vaquie theory of schemes over f 1 i e the base category relative to which we work out our theory is the category of sets endowed with norms or families of norms base change functors to analytic spaces over banach rings are studied and the basic spaces of analytic geometry e g polydisks are recovered as a base change of analytic spaces over f 1 we conclude by discussing some applications of our theory to the theory of the fargues fontaine curve and to the ring of witt vectors

completions of monoid objects and descent results abstract let c 1 be an abelian closed symmetric monoidal category satisfying certain conditions the purpose of this paper is to prove three descent results for schemes over c 1 that are not necessarily noetherian

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

bredon style homology cohomology and riemann roch for algebraic stacks one of the main obstacles for proving riemann roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the k theory and g theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks in this paper we study in detail a family of cohomology and homology theories which we call bredon style theories that are of this type and in the spirit of the classical bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces we establish riemann roch theorems in this setting it is shown elsewhere that such riemann roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg stacks for example moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory we conclude the present paper with a brief application of this nature

monoidal infinity category of complexes from tannakian viewpoint in this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infty categories of quasi coherent complexes it can be viewed as a derived analogue of tannaka duality as a consequence we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable infty category of quasi coherent complexes with tensor operation

cohomologie des varietes de deligne lusztig abstract we study the cohomology of deligne lusztig varieties with aim the construction of actions of hecke algebras on such cohomologies as predicted by the conjectures of broue malle and michel ultimately aimed at providing an explicit version of the abelian defect conjecture we develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them we are able to compute the cohomology of varieties associated to possibly twisted rank 2 groups and powers of the longest element w 0 some indeterminacies remain for g 2 we use this to construct hecke algebra actions on the cohomology of varieties associated to w 0 or its square for groups of arbitrary rank in the subsequent work f digne j michel endomorphisms of deligne lusztig varieties nagoya j math 183 2006 we construct actions associated to more general regular elements and we study their traces on cohomology

combinatorial model categories have presentations abstract we show that every combinatorial model category is quillen equivalent to a localization of a diagram category where diagram category means diagrams of simplicial sets this says that every combinatorial model category can be built from a category of generators and a set of relations

aqft from n functorial qft there are essentially two different approaches to the axiomatization of quantum field theory qft algebraic qft going back to haag and kastler and functorial qft going back to atiyah and segal more recently based on ideas by baez and dolan the latter is being refined to extended functorial qft by freed hopkins lurie and others the first approach uses local nets of operator algebras which assign to each patch an algebra of observables the latter uses n functors which assign to each patch a propagator of states

higher analytic stacks and gaga theorems abstract we develop the foundations of higher geometric stacks in complex analytic geometry and in non archimedean analytic geometry we study coherent sheaves and prove the analog of grauert s theorem for derived direct images under proper morphisms we define analytification functors and prove the analog of serre s gaga theorems for higher stacks we use the language of infinity category to simplify the theory in particular it enables us to circumvent the functoriality problem of the lisse etale sites for sheaves on stacks our constructions and theorems cover the classical 1 stacks as a special case

universal homotopy theories abstract begin with a small category c the goal of this short note is to point out that there is such a thing as a universal model category built from c we describe applications of this to the study of homotopy colimits the dwyer kan theory of framings sheaf theory and the homotopy theory of schemes

cohomology and deformation theory of monoidal 2 categories i abstract in this paper we define a cohomology theory for an arbitrary k linear semistrict semigroupal 2 category c called for short a gray semigroup and show that its first order unitary deformations up to the suitable notion of equivalence are in bijection with the elements of the second cohomology group fundamental to the construction is a double complex similar to the gerstenhaber schack double complex for bialgebras the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1 morphisms we also identify the cohomologies describing separately the deformations of the tensor product the associator and the pentagonator to obtain the above results a cohomology theory for an arbitrary k linear unitary pseudofunctor is introduced describing its purely pseudofunctorial deformations and generalizing yetter s cohomology for semigroupal functors in m kapranov e getzler eds higher category theory ams contemporary mathematics vol 230 amer math soc providence ri 1998 pp 117 134 the corresponding higher order obstructions will be considered in detail in a future paper

derived galois deformation rings abstract we define a derived version of mazur s galois deformation ring it is a pro simplicial ring r classifying deformations of a fixed galois representation to simplicial coefficient rings its zeroth homotopy group 0 r recovers mazur s deformation ring we give evidence that these rings r occur in the wild for suitable galois representations the langlands program predicts that 0 r should act on the homology of an arithmetic group we explain how the taylor wiles method can be used to upgrade such an action to a graded action of r on the homology

2 groups 2 characters and burnside rings abstract we study 2 representations i e actions of 2 groups on 2 vector spaces our main focus is character theory for 2 representations to this end we employ the technique of extended burnside rings our main theorem is that the ganter kapranov 2 character is a particular mark homomorphism of the burnside ring as an application we give a new proof of osorno s formula for the ganter kapranov 2 character of a finite group

action du groupe des tresses sur une categorie we describe how to define by generators and relations an action of the braid group on a category more generally how to define a functor from a generalized positive braids monoid corresponding to a finite coxeter group to a monoidal category applications to constructions of bondal exceptional systems and of broue michel correspondences on flag manifolds are given

weighted projective lines of tubular type and equivariantization abstract we prove that the categories of coherent sheaves over weighted projective lines of tubular type are explicitly related to each other via the equivariantization with respect to certain cyclic group actions

the 2 group of symmetries of a split chain complex abstract we explicitly compute the 2 group of self equivalences and homotopy classes of chain homotopies between them for any split chain complex a in an arbitrary k linear abelian category k any commutative ring with unit in particular it is shown that it is a split 2 group whose equivalence class depends only on the homology of a and that it is equivalent to the trivial 2 group when a is exact this provides a description of the general linear 2 group of a baez and crans 2 vector space over an arbitrary field f and of its generalization to chain complexes of vector spaces of arbitrary length

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

representation theory of 2 groups on kapranov and voevodsky s 2 vector spaces abstract in this paper the representation theory of 2 groups in 2 categories is considered focusing the attention on the 2 category rep 2 mat k g of representations of a 2 group g in a version of kapranov and voevodsky s 2 category of 2 vector spaces over a field k the set of equivalence classes of such representations is computed in terms of the invariants 0 g 1 g and h 3 0 g 1 g classifying g and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action in particular it is shown that the monoidal category of finite dimensional linear representations more generally the category of z projective representations for any given cohomology class z h 2 0 g k of the first homotopy group 0 g as well as its category of representations on finite sets both live in rep 2 mat k g the first as the monoidal category of endomorphisms of the trivial representation more generally as the category of intertwiners between suitable 1 dimensional representations and the second as a non full subcategory of the homotopy category of rep 2 mat k g

correlation function of random field ising chains is it lorentzian or not the two point connected correlation function or wavevector dependent susceptibility of the ferromagnetic ising chain in a random field is calculated exactly at any temperature for a two parameter family of diluted symmetric exponential distributions of the magnetic fields thermodynamic properties of this model have been derived in a previous work by the authors besides the correlation function itself the solution provides exact results for the usual susceptibility the correlation length and the edwards anderson parameter the low temperature regime is examined in full detail the authors obtain in closed form the limit value at zero temperature of various quantities and the first correction to this limit which behaves linearly with temperature the correlation length is discontinuous at zero temperature they also derive the scaling form of the correlation function for small p where the dilution p is the probability for a spin to be subjected to a non zero random field even in this limit the correlation function is more complicated than the simple lorentzian predicted e g by mean field theory

on homotopy varieties given an algebraic theory t a homotopy t algebra is a simplicial set where all equations from t hold up to homotopy all homotopy t algebras form a homotopy variety we will give a characterization of homotopy varieties analogous to the characterization of varieties

k theory and the bridge from motives to noncommutative motives abstract in this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic k theory of schemes and dg categories the work is divided into three main parts in the first part we use the techniques of higher algebra developed in 63 to provide a universal characterization for the symmetric monoidal 1 category underlying the motivic stable a 1 homotopy theory of morel voevodsky 107 67 more precisely given a symmetric monoidal model category v together with an object x v we characterize the underlying symmetric monoidal 1 category of the symmetric monoidal model category sp v x introduced by hovey in 43 by means of a universal property amongst symmetric monoidal 1 categories this characterization trivializes the problem of finding motivic monoidal realizations in the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of kontsevich 56 55 54 the key ingredient is a notion of nisnevich topology in the noncommutative setting compatible with the classical notion this compatibility together with the universal property proved in the first part ensures the existence of a canonical monoidal map from the stable motivic theory of morel voevodsky towards these new noncommutative motives that allow us to compare the two theories in the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic k theory of dg categories more precisely we prove that the non connective k theory of dg categories introduced by schlichting 82 is the non commutative nisnevich sheafification of connective algebraic k theory then we prove that its further non commutative a 1 localization is a tensor unit in our noncommutative motives as a corollary we obtain a precise proof for an original conjecture of kontsevich claiming that k theory gives the correct mapping spaces in noncommutative motives our major application is the discovery of a canonical factorization of our motivic bridge through the 1 category of modules over the commutative algebra object representing homotopy invariant algebraic k theory of schemes the results in 77 imply that this bridge is fully faithful over a field k with resolutions of singularities so that at the motivic level no information below k theory is lost by passing to the noncommutative world

representation and character theory in 2 categories we develop a 2 categorical generalization of the theory of group representations and characters we categorify the concept of the trace of a linear transformation associating to any endofunctor of any small category a set called its categorical trace in a linear situation the categorical trace is a vector space and we associate to any two commuting self equivalences a number called their joint trace for a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors we call this function the 2 character of such functions of commuting pairs and more generally n tuples appear in the work of hopkins kuhn and ravenel michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic on equivariant morava e theories we define the concept of induced categorical representation and show that the corresponding 2 character is given by the same formula as was obtained in michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic for the transfer map in the second equivariant morava e theory

zeros on the temperature axis of spin correlations in a random bond ising chain the authors study the correlation between two spins diametrically opposite in a random ising loop of 2r spins with random nearest neighbour couplings as a function of the inverse temperature beta this correlation undergoes random sign changes with a density rho r beta they determine this density for an arbitrary distribution p k of the coupling constants for i r to infinity and ii the scaling limit beta to infinity r to infinity at r beta fixed for r to infinity the total number of zeros on the beta axis grows asymptotically as 1 2 1 21 zeta 3 1 4 pi 21 2 log r as an application the density of zeros is calculated for the spin spin correlation in a double infinite ising chain with random bonds having with a probability p an infinite transverse coupling between each pair of corresponding sites a connection is made with predictions from spin glass theory

equivariant localization and completion in cyclic homology and derived loop spaces abstract we prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups x g in the setting of derived loop spaces as well as hochschild homology and its cyclic variants we show that the derived loop spaces of the stack x g and its classical z fixed point stack 0 x z g z become equivalent after completion along a semisimple parameter z g g implying the analogous statement for hochschild and cyclic homology of the dg category of perfect complexes perf x g we then prove an analogue of the atiyah segal completion theorem in the setting of periodic cyclic homology where the completion of the periodic cyclic homology of perf x g at the identity e g g is identified with a 2 periodic version of the derived de rham cohomology of x g together these results identify the completed periodic cyclic homology of a stack x g over a parameter z g g with the 2 periodic derived de rham cohomology of its z fixed points

when is the self intersection of a subvariety a fibration abstract we provide a necessary and sufficient condition for the derived self intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme as a consequence we deduce a generalized hkr isomorphism we also investigate the relationship of our result to path spaces in homotopy theory buchweitz flenner formality in algebraic geometry and draw parallels with similar results in lie theory and symplectic geometry

two dimensional algebra in lattice gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher category theory in doing so we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher cat

butterflies in a semi abelian context it is known that monoidal functors between internal groupoids in the category grp of groups constitute the bicategory of fractions of the 2 category grpd grp of internal groupoids internal functors and internal natural transformations in grp with respect to weak equivalences that is internal functors which are internally fully faithful and essentially surjective on objects monoidal functors can be equivalently described by a kind of weak morphisms introduced by b noohi under the name of butterflies in order to internalize monoidal functors in a wide context we introduce the notion of internal butterflies between internal crossed modules in a semi abelian category c and we show that they are morphisms of a bicategory b c our main result states that when in c the notions of huq commutator and smith commutator coincide then the bicategory b c of internal butterflies is the bicategory of fractions of grpd c with respect to weak equivalences

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families

averaging t structures and extension closure of aisles we ask when a finite set of t structures in a triangulated category can be averaged into one t structure or equivalently when the extension closure of a finite set of aisles is again an aisle there is a straightforward positive answer for a possibly infinite set of compactly generated t structures in a big triangulated category for piecewise tame hereditary categories we give a criterion for when averaging is possible and an algorithm that computes truncation triangles in this case a finite group action on a triangulated category gives a natural way of producing a finite set of t structures out of a given one if averaging is possible there is an induced t structure on the equivariant triangulated category

on the regular representation of an essentially finite 2 group the regular representation of an essentially finite 2 group mathbb g in the 2 category mathbf 2vect k of kapranov and voevodsky 2 vector spaces is defined and cohomology invariants classifying it computed it is next shown that all hom categories in mathbf rep mathbf 2vect k mathbb g are 2 vector spaces under quite standard assumptions on the field k and a formula giving the corresponding intertwining numbers is obtained which proves they are symmetric finally it is shown that the forgetful 2 functor boldmath omega mathbf rep mathbf 2vect k mathbb g to mathbf 2vect k is representable with the regular representation as representing object as a consequence we obtain a k linear equivalence between the 2 vector space mathbf vect k mathcal g of functors from the underlying groupoid of mathbb g to mathbf vect k on the one hand and the k linear category mathcal e nd boldmath omega of pseudonatural endomorphisms of boldmath omega on the other hand we conclude that mathcal e nd boldmath omega is a 2 vector space and we partially describe a basis of it

homotopical algebraic geometry i topos theory this is the rst of a series of papers devoted to lay the foundations of algebraic geometry in homotopical and higher categorical contexts in this rst part we investigate a notion of higher topos for this we use s categories i e simplicially enriched categories as models for certain kind of 1 categories and we develop the notions of s topologies s sites and stacks over them we prove in particular that for an s category t endowed with an s topology there exists a model category of stacks over t generalizing the model category structure on simplicial presheaves over a grothendieck site of a joyal and r jardine we also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves by proving that there exists a one to one correspodence between s topologies on an s category t and certain left exact bouseld localizations of the model category of pre stacks on t based on the above results we study the notion of model topos introduced by c rezk and we relate it to our model categories of stacks over s sites in the second part of the paper we present a parallel theory where s categories s topologies and s sites are replaced by model categories model topologies and model sites we prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over s sites is provided by the simplicial localization construction of dwyer and kan as an example of application we propose a denition of etale k theory of ring spectra extending the etale k theory of commutative rings