false theta functions and the verlinde formula abstract we discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of w algebras and vertex algebras in general more precisely motivated by clues from conformal field theory first we are able to determine modular like transformation properties of regularized partial and false theta functions then after suitable identification of regularized partial false theta functions with the characters of atypical modules for the singlet vertex algebra w 2 2 p 1 we formulate a verlinde type formula for the fusion rules of irreducible w 2 2 p 1 modules

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

measurement of force and moment mobilities using a finite difference technique measuring only force mobilities it is possible to derive a moment mobility without the need for an externally applied moment the method uses finite differences of both forces and velocities about a central point in this way pure forces and moments can be extracted as can pure velocities and angular velocities there are however errors associated with finite difference techniques in this case the purity of the extracted quantities depends on the area over which they are measured and its significance in terms of wavelength meanwhile experimental errors such as noise dictate that the area is sufficiently large to provide differences which are incorruptible the paper aims to quantify the finite difference error in a general way free of situation specifics

an approximation for the point mobility at the intersection of two perpendicular plates abstract the problem of point excitation at a t intersection of two perpendicular plates is studied in order to establish expressions for the point mobility it is found that the theory for point excitation of the free surface of a semi infinite elastic solid is applicable in the frequency range associated with structure borne sound transmission from this theory the mobility for an infinite system is derived based on this model and on an experimental investigation an estimation procedure for the point mobility in the finite dimension case is developed the agreement with measurements performed in situ is quite acceptable both the theoretical and the experimental investigations reveal that the real part of this mobility is small although it is larger in the experimental results this indicates that other components of excitation are difficult to eliminate and may contribute to the power input in practice because of the small real part of the mobility it is advantageous with respect to the reduction of structure borne sound power transmission to locate the contact points between a source and the receiver at such intersections corrections are deduced for the measured magnitude of the mobility for the case when separate force and motion transducers are used

logarithmic widehat s ell 2 cft models from nichols algebras i we construct chiral algebras that centralize rank 2 nichols algebras with at least one fermionic generator this gives logarithmic w algebra extensions of a fractional level algebra we discuss crucial aspects of the emerging general relation between nichols algebras and logarithmic conformal field theory cft models i the extra input beyond the nichols algebra proper needed to uniquely specify a conformal model ii a relation between the cft counterparts of nichols algebras connected by weyl groupoid maps and iii the common double bosonization u x of such nichols algebras for an extended chiral algebra candidates for its simple modules that are counterparts of the u x simple modules are proposed as a first step toward a functorial relation between u x and w algebra representation categories

on cohomology isomorphisms of groups abstract we show how mislin s theorem on group homomorphisms that induce an isomorphism in cohomology can be proved based on ideas of alperin and the use of lannes s t functor this enables us to extend the class of groups to include groups of finite virtual cohomological dimension and profinite groups

symplectic fermions and a quasi hopf algebra structure on bar u _i sl 2 abstract we consider the finite dimensional restricted quantum group u q s l 2 at q i we show that u i s l 2 does not allow for a universal r matrix even though u v v u holds for all finite dimensional representations u v of u i s l 2 we then give an explicit coassociator and a universal r matrix r such that u i s l 2 becomes a quasi triangular quasi hopf algebra our construction is motivated by the two dimensional chiral conformal field theory of symplectic fermions with central charge c 2 there a braided monoidal category sf has been computed from the factorisation and monodromy properties of conformal blocks and we prove that rep u i s l 2 r is braided monoidally equivalent to sf

holomorphic symplectic fermions let v be the even part of the vertex operator super algebra of r pairs of symplectic fermions up to two conjectures we show that v admits a unique holomorphic extension if r is a multiple of 8 and no holomorphic extension otherwise this is implied by two results obtained in this paper 1 if r is a multiple of 8 one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice d r with shifted stress tensor 2 we classify lagrangian algebras in mathcal s mathcal f mathfrak h a ribbon category associated to symplectic fermions the classification of holomorphic extensions of v follows from 1 and 2 if one assumes that mathcal s mathcal f mathfrak h is ribbon equivalent to mathrm rep v and that simple modules of extensions of v are in one to one relation with simple local modules of the corresponding commutative algebra in mathcal s mathcal f mathfrak h

the logarithmic triplet theory with boundary the boundary theory for the c 2 triplet model is investigated in detail in particular we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their ope coefficients explicitly we also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization

renormalized hennings invariants and 2 1 tqfts we construct non semisimple 2 1 tqfts yielding mapping class group representations in lyubashenko s spaces in order to do this we first generalize beliakova blanchet and geer s logarithmic hennings invariants based on quantum mathfrak sl 2 to the setting of finite dimensional non degenerate unimodular ribbon hopf algebras the tools used for this construction are a hennings augmented reshetikhin turaev functor and modified traces when the hopf algebra is factorizable we further show that the universal construction of blanchet habegger masbaum and vogel produces a 2 1 tqft on a not completely rigid monoidal subcategory of cobordisms

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

on the triplet vertex algebra w p abstract we study the triplet vertex operator algebra w p of central charge 1 6 p 1 2 p p 2 we show that w p is c 2 cofinite but irrational since it admits indecomposable and logarithmic modules furthermore we prove that w p is of finite representation type and we provide an explicit construction and classification of all irreducible w p modules and describe block decomposition of the category of ordinary w p modules all this is done through an extensive use of zhu s associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms moreover we obtain an upper bound for dim a w p finally for p prime we completely describe the structure of a w p the methods of this paper are easily extendable to other w algebras and superalgebras

the structure of zhu s algebras for certain w algebras abstract we introduce a new approach that allows us to determine the structure of zhu s algebra for certain vertex operator super algebras which admit horizontal z grading by using this method and an earlier description of zhu s algebra for the singlet w algebra we completely describe the structure of zhu s algebra for the triplet vertex algebra w p as a consequence we prove that zhu s algebra a w p and the related poisson algebra p w p have the same dimension we also completely describe zhu s algebras for the n 1 triplet vertex operator superalgebra sw m moreover we obtain similar results for the c 0 triplet vertex algebra w 2 3 important in logarithmic conformal field theory because our approach is internal we had to employ several constant term identities for purposes of getting right upper bounds on dim a v this work is in a way a continuation of the results published in adamovic and milas 2008 4

kazhdan lusztig dual quantum group for logarithimic extensions of virasoro minimal models we derive and study a quantum group gp q that is kazhdan lusztig dual to the w algebra wp q of the logarithmic p q conformal field theory model the algebra wp q is generated by two currents w z and w z of dimension 2p 1 2q 1 and the energy momentum tensor t z the two currents generate a vertex operator ideal r with the property that the quotient wp q r is the vertex operator algebra of the p q virasoro minimal model the number 2pq of irreducible gp q representations is the same as the number of irreducible wp q representations on which r acts nontrivially we find the center of gp q and show that the modular group representation on it is equivalent to the modular group representation on the wp q characters and pseudocharacters the factorization of the gp q ribbon element leads to a factorization of the modular group representation on the center we also find the gp q grothendieck ring which is presumably the logarithmic fusion of the p q model

vertex operator super algebras and lcft we review some of the developments in logarithmic conformal field theory from the vertex algebra point of view several important examples of vertex operator super algebras of the triplet type are discussed including their representation theory particular emphasis is put on c2 cofiniteness of these vertex algebras a description of zhu s algebras and the construction of logarithmic modules

a thin plate model for the moment mobility at the intersection of two perpendicular plates the global response of a t intersection configuration due to a moment excitation has been analyzed the moment is applied at the intersection and the moment vector is parallel with the intersection line a thin plate model has been used to derive the moment mobility for arbitrary combinations of plate elements the imaginary part of the mobility is found to be dependent on both size and rigidity of the indenter whereas the real part is essentially independent numerical results are presented for the practically important case of three identical plate elements constituting the t intersection configuration these results clearly confirm the necessity of taking into account a potential moment excitation for the determination of structure borne sound and vibration transmission at an intersection point

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

polynomial fusion rings of w extended logarithmic minimal models the countably infinite number of virasoro representations of the logarithmic minimal model lm p p can be reorganized into a finite number of w representations with respect to the extended virasoro algebra symmetry w using a lattice implementation of fusion we recently determined the fusion algebra of these representations and found that it closes albeit without an identity for p 1 here we provide a fusion matrix realization of this fusion algebra and identify a fusion ring isomorphic to it we also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings one particular extension is reminiscent of the fundamental fusion algebra of lm p p and offers a natural way of introducing the missing identity for p 1 working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual moore penrose inverses intertwining bet

fusion matrices generalized verlinde formulas and partition functions in cal wlm 1 p the infinite series of logarithmic minimal models is considered in the extended picture where they are denoted by as in the rational models the fusion algebra of is described by a simple graph fusion algebra the corresponding fusion matrices are mutually commuting but in general not diagonalizable nevertheless they can be simultaneously brought to jordan form by a similarity transformation the spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into jordan canonical form consisting of jordan blocks of rank 1 2 or 3 the various similarity transformations and jordan forms are determined from the modular data this gives rise to a generalized verlinde formula for the fusion matrices its relation to the partition functions in the model is discussed in a general framework by application of a particular structure matrix and its moore penrose inverse this verlinde formula reduces to the generalized verlinde formula for the associated grothendieck ring

representations of u qsl 2 1 at even roots of unity we construct all projective modules of the restricted quantum group u qsl 2 1 at an even 2p th root of unity this 64p4 dimensional hopf algebra is a common double bosonization of two rank 2 nichols algebras x1d505 x with fermionic generator s we show that the category of u qsl 2 1 modules is equivalent to the category of yetter drinfeld x1d505 x modules in c hhy kern 1ptd for h 2p 2p where the coaction is defined by a universal r matrix h h as an application of the projective module construction we study the basic algebra of u qsl 2 1 and find the associative algebra structure and the dimension 5p2 p 4 of its center

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

integrable boundary conditions and w extended fusion in the logarithmic minimal models lm 1 p we consider the logarithmic minimal models lm 1 p as rational logarithmic conformal field theories with extended w symmetry to make contact with the extended picture starting from the lattice we identify 4p 2 boundary conditions as specific limits of integrable boundary conditions of the underlying yang baxter integrable lattice models specifically we identify 2p integrable boundary conditions to match the 2p known irreducible w representations these 2p extended representations naturally decompose into infinite sums of the irreducible virasoro representations r s a further 2p 2 reducible yet indecomposable w representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank 2 virasoro representations the fusion rules in the extended picture are deduced from the known fusion rules for the virasoro representations of lm 1 p and are found to be in agreement with previous works the closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach comment 15 pages

a braided monoidal category for free super bosons the chiral conformal field theory of free super bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian lie super algebra h with non degenerate super symmetric pairing the mode algebras of a single free boson and of a single pair of symplectic fermions arise for even odd dimension 1 0 and 0 2 of h respectively in this paper the representations of the untwisted mode algebra of free super bosons are equipped with a tensor product a braiding and an associator in the symplectic fermion case i e if h is purely odd the braided monoidal structure is extended to representations of the z 2z twisted mode algebra the tensor product is obtained by computing spaces of vertex operators the braiding and associator are determined by explicit calculations from three and four point conformal blocks

centralizers of elementary abelian p subgroups and mod p cohomology of profinite groups 1 1 let g be a profinite group and p be a fixed prime in this paper we will be concerned with h c g fp the continuous cohomology of g with coefficients in the trivial module fp we will abbreviate h c g fp by h g fp or simply by h g if p is understood from the context we recall that if g is the inverse limit of finite groups gi then h g colim h gi throughout this paper we will assume that h g is finitely generated as fp algebra by work of lazard la it is known that this holds for many interesting groups for example for profinite p analytic groups like gl n zp the general linear groups over the p adic integers in case h g is finitely generated as fp algebra quillen has shown q1 that there are only finitely many conjugacy classes of elementary abelian p subgroups of g i e groups isomorphic to z p for some natural number n in other words the following category a g is equivalent to a finite category objects of a g are all elementary abelian p subgroups of g and if e1 and e2 are elementary abelian p subgroups of g then the set of morphisms from e1 to e2 in a g consists precisely of those homomorphisms e1 e2 of abelian groups for which there exists an element g g with e geg 1 e e1 the category a g plays an important role both in quillen s results and in the work presented here this category entered into quillen s work as follows the assignment e 7 h e extends to a functor from the opposite category a g op to graded fp algebras and the restriction homomorphisms h g h e for e running through the elementary abelian p subgroups of g induce a canonical map of algebras q h g lima g op h e

superspace conformal field theory conformal sigma models and wess zumino witten wzw models on coset superspaces provide important examples of logarithmic conformal field theories they possess many applications to problems in string and condensed matter theory we review recent results and developments including the general construction of wzw models on type i supergroups the classification of conformal sigma models and their embedding into string theory

group cohomology and control of p fusion we show that if an inclusion of finite groups h g of index prime to p induces a homeomorphism of mod p cohomology varieties or equivalently an f isomorphism in mod p cohomology then h controls p fusion in g if p is odd this generalizes classical results of quillen who proved this when h is a sylow p subgroup and furthermore implies a hitherto difficult result of mislin about cohomology isomorphisms for p 2 we give analogous results at the cost of replacing mod p cohomology with higher chromatic cohomology theories

grothendieck ring and verlinde formula for the w extended logarithmic minimal model wlm 1 p we consider the grothendieck ring of the fusion algebra of the w extended logarithmic minimal model wlm 1 p informally this is the fusion ring of w irreducible characters so it is blind to the jordan block structures associated with reducible yet indecomposable representations as in the rational models the grothendieck ring is described by a simple graph fusion algebra the 2p dimensional matrices of the regular representation are mutually commuting but not diagonalizable they are brought simultaneously to jordan form by the modular data coming from the full 3p 1 dimensional s matrix which includes transformations of the p 1 pseudo characters the spectral decomposition yields a verlinde like formula that is manifestly independent of the modular parameter tau but is in fact equivalent to the verlinde like formula recently proposed by gaberdiel and runkel involving a tau dependent s matrix comment 13 pages v2 example comments and references added

calculation of point mobilities at generic multi plate beam junctions a new method is presented for the calculation of point mobilities at generic multi plate beam junctions such a junction is considered to consist of an arbitrary number of semi infinite plates coupled together through a beam which may also be omitted plates and beams are assumed to obey mindlin and timoshenko theories respectively the calculation is based on a numerical wavenumber integration technique a specially adapted numerical integration technique is used to reduce the amount of numerical calculations to be done the method is applied to a number of example structures to test its results and to illustrate its application

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

w extended kac representations and integrable boundary conditions in the logarithmic minimal models wlm 1 p we construct new yang baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit we interpret these extended kac representations as finitely generated extended feigin fuchs modules over the triplet w algebra the extended fusion rules of these representations are inferred from the recently conjectured virasoro fusion rules of the kac representations in the underlying logarithmic minimal model we also introduce the modules contragredient to the extended kac modules and work out the correspondingly extended fusion algebra our results are in accordance with the kazhdan lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q e i p finally polynomial fusion rings isomorphic with the various fusion algebras are determined and the corresponding grothendieck ring of characters is identified

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

characters and a verlinde type formula for symmetric hopf algebras abstract we study certain aspects of finite dimensional non semisimple symmetric hopf algebras h and their duals h we focus on the set i h of characters of projective h modules which is an ideal of the algebra of cocommutative elements of h this ideal corresponds via a symmetrizing form to the projective center higman ideal of h which turns out to be h where is an integral of h and is the left adjoint action of h on itself we describe h via primitive and central primitive idempotents of h we also show that it is stable under the quantum fourier transform our best results are obtained when h is a factorizable ribbon hopf algebra over an algebraically closed field of characteristic 0 in this case h is also the image of i h under a translated drinfel d map we use this fact to prove the existence of a steinberg like character the above ingredients are used to prove a verlinde type formula for h