on subspace distances determined by the frobenius norm abstract two measures of separation between two subspaces of a finite dimensional complex vector space are introduced and investigated the measures were inspired by known expressions for the cosines of the angle and minimal angle between subspaces and are based on the frobenius norm by using partitioned representations of a pair of orthogonal projectors several properties of the measures are identified and their links with other notions used to characterize a separation between subspaces are pointed out

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

categorified central extensions etale lie 2 groups and lieʼs third theorem for locally exponential lie algebras abstract lie s third theorem asserting that each finite dimensional lie algebra is the lie algebra of a lie group fails in infinite dimensions the modern account on this phenomenon is the integration problem for central extensions of infinite dimensional lie algebras which in turn is phrased in terms of an integration procedure for lie algebra cocycles this paper remedies the obstructions for integrating cocycles and central extensions from lie algebras to lie groups by generalising the integrating objects those objects obey the maximal coherence that one can expect moreover we show that they are the universal ones for the integration problem the main application of this result is that a mackey complete locally exponential lie algebra e g a banach lie algebra integrates to a lie 2 group in the sense that there is a natural lie functor from certain lie 2 groups to lie algebras sending the integrating lie 2 group to an isomorphic lie algebra

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

on angles and distances between subspaces abstract the angles and distances between two given subspaces of c n 1 are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors several new results are established with the particular attention paid to the notions of inclinedness and minimal angle to demonstrate the usefulness of the approach utilized some results known to be valid in hilbert space are reestablished in c n 1 either in generalized form or with considerably shorter proofs than in the original sources

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

topological properties of linked disclinations in anisotropic liquids a connection between the topological properties of the o 3 and so 3 sigma models based on hopf s invariant and the wess zumino term respectively and related topological concepts are indicated this is applied to the o 3 z2 and the so 3 pi sigma models where z2 and pic so 3 are point symmetry groups describing anisotropic liquids it is shown that the hopf invariant for the nematic liquid assumes integer multiples of 1 4 and similar results hold for the other sigma models applications to a topological field theory of o 3 z2 and so 3 pi sigma models are indicated

on the product of orthogonal projectors abstract in this paper we introduce a canonical form to represent the product of two orthogonal projectors this form is useful for deriving many known results which are scattered in the literature we summarize many of those results and present new ones

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

relative quantum field theory we highlight the general notion of a relative quantum field theory which occurs in several contexts one is in gauge theory based on a compact lie algebra rather than a compact lie group this is relevant to the maximal superconformal theory in six dimensions

braid representations from unitary braided vector spaces we investigate braid group representations associated with unitary braided vector spaces focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order we verify this conjecture for the two infinite families of gaussian and group type braided vector spaces as well as the generalization to quasi braided vector spaces of group type

categorical non abelian cohomology and the schreier theory of groupoids by regarding the classical non abelian cohomology of groups from a 2 dimensional categorical viewpoint we are led to a non abelian cohomology of groupoids which continues to satisfy classification interpretation and representation theorems generalizing the classical ones this categorical approach is based on the fact that if groups are regarded as categories then on the one hand crossed modules are 2 groupoids cocycles are lax 2 functors and the cocycle conditions are precisely the coherence axioms for lax 2 functors and on the other hand group extensions are fibrations of categories furthermore n simplices in the nerve of a 2 category are lax 2 functors

a note on borel summability in this article nevanlinna s theorem about borel summability is improved moreover the borel summability of the asymptotic expansion in the method of stationary phase in infinitely many dimensions is discussed

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

functions of orthogonal projectors involving the moore penrose inverse several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors the results were established in various settings including finite and infinite dimensional vector spaces over either real or complex numbers but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors in the present paper these results are unified and reestablished in a generalized form in a complex euclidean vector space with the generalization obtained by relaxing the nonsingularity assumption and use of the moore penrose inverse instead of the ordinary inverse additionally several new formulae of the type are provided

state sum models and simplicial cohomology we study a class of subdivision invariant lattice models based on the gauge groupz p with particular emphasis on the four dimensional example this model is based upon the assignment of field variables to both the 1 and 2 dimensional simplices of the simplicial complex the property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod p flatness condition by explicit computation of the partition function for the manifoldrp3 s1 we establish that the theory has a quantum hilbert space which differs from the classical one

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

the ring structure for equivariant twisted k theory we prove under some mild conditions that the equivariant twisted k theory group of a crossed module admits a ring structure if the twisting 2 cocycle is 2multiplicative we also give an explicit construction of the transgression map t1 h a h 1 n o a for any crossed module n and prove that any element in the image is multiplicative as a consequence we prove that under some mild conditions for a crossed module n and any e s that the equivariant twisted k theory group k e n admits a ring structure as an application we prove that for a compact connected and simply connected lie group g the equivariant twisted k theory group k c g g is endowed with a canonical ring structurek c g g k j d c g g k i j d c g g where d dimg and c h gog s

further results on generalized and hypergeneralized projectors abstract the notions of generalized and hypergeneralized projectors introduced by gros and trenkler j gro s j trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 are revisited on the one hand the present paper provides several new characterizations of these sets and on the other the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered moreover the paper demonstrates the usefulness in studying the properties of generalized and hypergeneralized projectors of the representation of complex matrices given in corollary 6 by hartwig and spindelbock r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

index theory gerbes and hamiltonian quantization we give an atiyah patodi singer index theory construction of the bundle of fermionic fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known schwinger terms faddeev mickelsson cocycle for the gauge group action we relate the aps construction to the bundle gerbe approach discussed recently by carey and murray including an explicit computation of the dixmier douady class an advantage of our method is that it can be applied whenever one has a form of the aps theorem at hand as in the case of fermions in an external gravitational field

extension theory and the calculus of butterflies abstract this paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions the first uses weak monoidal functors the second classifies extensions by means of suitable h 2 actions we develop our theory formally by making explicit a connection between non abelian g torsors and fibrations then we apply our general framework to the classification of extensions in a semi abelian context by means of butterflies 1 between internal crossed modules as a main result we get an internal version of dedecker s theorem on the classification of extensions of a group by a crossed module in the semi abelian context bourn s intrinsic schreier mac lane extension theorem 13 turns out to be an instance of our theorem 6 3 actually even just in the case of groups our approach reveals a result slightly more general than classical schreier mac lane theorem

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

lattice topological field theory on nonorientable surfaces the lattice definition of the two dimensional topological quantum field theory fukuma et al commun math phys 161 157 1994 is generalized to arbitrary not necessarily orientable compact surfaces it is shown that there is a one to one correspondence between real associative algebras and the topological state sum invariants defined on such surfaces the partition and n point functions on all two dimensional surfaces connected sums of the klein bottle or projective plane and g tori are defined and computed for arbitrary algebras in general and for the group ring a r g of discrete groups g in particular

the method of stationary phase for oscillatory integrals on hilbert spaces we develop the method of stationary phase for the normalized oscillatory integral on hilbert space giving borel summable expansions the developments that we obtain hold for more general situations than the ones of previous papers on the same subject

a decomposition of the brauer picard group of the representation category of a finite group abstract we present an approach of calculating the group of braided autoequivalences of the category of representations of the drinfeld double of a finite dimensional hopf algebra h and thus the brauer picard group of h mod we consider two natural subgroups and a subset as candidates for generators in this article h is the group algebra of a finite group g as our main result we prove that any element of the brauer picard group fulfilling an additional cohomological condition decomposes into an ordered product of our candidates for elementary abelian groups g our decomposition reduces to the bruhat decomposition of the brauer picard group which is in this case a lie group over a finite field our results are motivated by and have applications to symmetries and defects in 3d tqft and group extensions of fusion categories

op lax natural transformations twisted quantum field theories and even higher morita categories abstract motivated by the challenge of defining twisted quantum field theories in the context of higher categories we develop a general framework for lax and oplax transformations and their higher analogs between strong n functors we construct a double n category built out of the target n category governing the desired diagrammatics we define op lax transformations as functors into parts thereof and an op lax twisted field theory to be a symmetric monoidal op lax natural transformation between field theories we verify that lax trivially twisted relative field theories are the same as absolute field theories as a second application we extend the higher morita category of e d algebras in a symmetric monoidal n category c to an n d category using the higher morphisms in c

on a subspace metric based on matrix rank abstract the metric between subspaces m n c n 1 defined by m n rk p m p n where rk denotes rank of a matrix argument and p m and p n are the orthogonal projectors onto the subspaces m and n respectively is investigated such a metric takes integer values only and is not induced by any vector norm by exploiting partitioned representations of the projectors several features of the metric m n are identified it turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces such as distance also called gap frobenius distance direct distance angle or minimal angle

some twisted self dual solutions for the yang mills equations on a hypertorus thesu n yang mills equations are considered in a four dimensional euclidean box with periodic boundary conditions hypertorus gauge invariant twists can be introduced in these boundary conditions to be labeled with integersn n defined modulon the pontryagin number in this space is often fractional whenever this number is zero there are solutions to the equationsg 0 hereg is the covariant curl when this number is not zero we find a set of solutions to the equations g mu nu tilde g mu nu provided that the periodsa of the box satisfy certain relations

column space equalities for orthogonal projectors in their paper y tian g p h styan rank equalities for idempotent and involutory matrices linear algebra appl 335 2001 101 117 tian and styan established several rank equalities involving a pair of idempotent matrices p and q subsequently these results are reinvestigated from the point of view of the following question provided that idempotent p q are hermitian which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces simultaneously some related results are established which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors

angles between infinite dimensional subspaces with applications to the rayleigh ritz and alternating projectors methods we define angles for infinite dimensional subspaces of hilbert spaces inspired by the work of e j hannan 1961 1962 the angles of dixmier and friedrichs and the gaps are characterized we establish connections between the angles corresponding to orthogonal complements the sensitivity of angles with respect to subspaces is estimated we show that the squared cosines of the angles from one subspace to another can be interpreted as ritz values in the rayleigh ritz method the hausdorff distance between the ritz values corresponding to different trial subspaces is shown to be bounded by a constant times the gap between the subspaces we prove a similar eigenvalue perturbation bound that involves the gap squared an ultimate acceleration of the classical alternating projectors method is proposed its convergence rate is estimated in terms of the angles we illustrate the acceleration for a domain decomposition method with a small overlap for the 1d diffusion equation

structures and diagrammatics of four dimensional topological lattice field theories abstract crane and frenkel proposed a state sum invariant for triangulated 4 manifolds they sketched the definition of a hopf category that was to be used in their construction crane and yetter studied hopf categories and gave some examples using group cocycles that are associated to the drinfeld double of a finite group in this paper we define a state sum invariant of triangulated 4 manifolds using crane yetter cocycles as boltzmann weights our invariant is analogous to the 3 dimensional invariants defined by dijkgraaf and witten and the invariants that are defined via hopf algebras we present diagrammatic methods for the study of such invariants that illustrate connections between hopf categories and moves to triangulations

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

t hooft anomalies of discrete gauge theories and non abelian group cohomology we study discrete symmetries of dijkgraaf witten theories and their gauging in the framework of extended functorial quantum field theory non abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit t hooft anomalies in terms of the lyndon hochschild serre spectral sequence we give an explicit realization of a discrete gauge theory with t hooft anomaly as a state on the boundary of a higher dimensional dijkgraaf witten theory this allows us to calculate the 2 cocycle twisting the projective representation of physical symmetries via transgression we present a general discussion of the bulk boundary correspondence at the level of partition functions and state spaces which we make explicit for discrete gauge theories

group cohomology and gauge equivalence of some twisted quantum doubles the purpose of the present paper is to investigate relationships that exist between certain quasi hopf algebras namely twisted quantum doubles of a finite abelian group and their module categories this is closely related to aspects of group cohomology concerning the relationship of the bar complex to some other complexes defined and studied by eilenberg and maclane mac52 em50a and em50b although our results are purely homological and algebraic in nature the motivation for studying these questions derives mainly from connections with conformal field theory cft and vertex operator algebras dw90 dpr92 ms89 dl93 we will thus review background motivation although it is hardly necessary for an understanding of our main results and then give a more detailed overview of the paper it is not necessary for the reader to appreciate what a vertex operator algebra voa is cf bor86 flm88 fhl93 but only to understand this voas have a representation theory hence a linear module category and it is important to understand the nature of this category it is expected that the module category v mod for a voa v has the structure of a braided monoidal category js93 roughly this means that there is a notion of tensor product of v modules and that the tensor product of three modules is associative by an isomorphism that depends on the particular modules moreover tensor product of modules is commutative again via an isomorphism that depends on the particular modules particularly nice voas called rational have the additional property that v mod is a semisimple category so that v has only a finite number of isomorphism classes of simple modules and every object in v mod is a direct sum of simple objects the simplest voas of all from this perspective are called holomorphic they are rational and have but one simple module namely the adjoint module v itself thus if v is a holomorphic voa then v mod is equivalent as a semi simple braided monoidal category to the category of vector spaces

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

on a pair of vector spaces the paper provides an extensive investigation of a pair of vectors spaces with a particular attention paid to perpendicular subspaces several new properties of the pair are identified some of which enabled to derive known facts about subspaces by means of simpler proofs the approach utilized exploits partitioned representations of orthogonal projectors and is based on the fact that there is one to one correspondence between an orthogonal projector and the subspace onto which it projects

on the localization of topological invariants the approach of formal differential geometry to the topological invariants which can be localized is developed the universal space and universal characteristic forms are constructed they give rise to primary and secondary characteristic forms

on the angular metrics between linear subspaces it has recently been shown that the symmetric gauge functions on the canonical principal angles give a family of unitarily invariant metrics between linear subspaces of the same dimension in this short paper we extend such metrics to subspaces of possibly different dimensions this extension is necessary in addressing some perturbation analysis problems involving subspaces with different dimensions examples of such perturbation analysis problems are also studied in this paper using the extended metrics

feynman integrals and fredholm determinants in this article simple properties of feynman integrals such as the translation invariance fubini theorem and cameron martin formula are used in order to derive a product formula for fredholm determinants this formula enables us to compute explicitly some of such determinants as well as the index of the related fredholm operators some examples are done which have applications in the semiclassical approximation 0

topological properties of static and dynamic defect configurations in ordered liquids abstract the topology of linked disclinations is studied in uniaxial nematic liquids and in anisotropic liquids with an order parameter space so 3 pi 3 in these models pi 3 are the finite point symmetry groups of 3 space with applications to helium 3 pi 3 i biaxial nematic liquid pi 3 d2 and anisotropic super cooled liquids pi 3 groups of platonic solids the topological properties are studied via hopf s invariant of the o 3 model and its relation to the wess zumino term of the so 3 pi 3 model in an orthonormal drei bein representation of so 3 dynamic processes are topology changing during intersection of disclinations and are studied via magnetic n pole singularities and the instanton number in an electromagnetic formalism the connection with tunneling amplitudes in a so 3 yang mills theory is indicated applications of the theory to topological fluid dynamics is worked out for the uniaxial nematic liquid and indicated for the so 3 spin liquid

on the projectors ff and f f abstract a particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors namely ff and f f determined by a complex square matrix f and its moore penrose inverse f various functions of the projectors are considered from the point of view of their nonsingularity idempotency nilpotency or their relation to the known classes of matrices such as ep bi ep gp dr or sr this part of the paper was inspired by benitez and rakocevic j benitez v rakocevic matrices a such that aa a a are nonsingular appl math comput 217 2010 3493 3503 further characteristics of ff and f f with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues are derived as well besides establishing selected exemplary results dealing with ff and f f the paper develops a general approach whose applicability extends far beyond the characteristics provided therein

from 3 algebras to δ groups and symmetric cohomology we introduce groups and show how they fit in the context of lattice field theory to a topological space m we associate a group m we define the symmetric cohomology hsn g a of a group g with coefficients in a g module a the group m is determined by the action of 1 m on 2 m and an element of hs3 1 m 2 m

multimatrix models induced by group extensions multimatrix models for which the index set has a group structure and the interaction obeys a zero curvature condition can be deformed related to central extensions of this group the deformed multimatrix models lead to statistical systems defined on random graphs with a topological action it is shown how these topological theories on graphs can be used to weight graphs according to topological conditions