on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

the method of mothers for non overlapping non matching ddm in this paper we introduce a variant of the three field formulation where we use only two sets of variables considering to fix the ideas the homogeneous dirichlet problem for delta u g in omega our variables are i the approximations u h s of u in each sub domain omega s each on its own grid and ii an approximation psi h of u on the skeleton the union of the interfaces of the sub domains on an independent grid that could often be uniform the novelty is in the way to derive from psi h the values of each trace of u h s on the boundary of each omega we do it by solving an auxiliary problem on each partial omega s that resembles the mortar method but is more flexible under suitable assumptions quasi optimal error estimates are proved uniformly with respect to the number and size of the subdomains a preliminary version of the method and of its theoretical analysis has been presented in bertoluzza et al 15th international conference on domain decomposition methods 2002

graphs and reflection groups it is shown that graphs that generalize the ade dynkin diagrams and have appeared in various contexts of two dimensional field theory may be regarded in a natural way as encoding the geometry of a root system after recalling what are the conditions satisfied by these graphs we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots some non integrally laced graphs are shown to be associated with subgroups of these reflection groups the empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled and the connections with recent developments in the context of mathcal n 2 supersymmetric theories and topological field theories are discussed a claude

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

a mirror symmetric construction of qht g p q abstract let g be a simple simply connected complex algebraic group we give a lie theoretic construction of a conjectural mirror family associated to a general flag variety g p and show that it recovers the peterson variety presentation for the t equivariant quantum cohomology rings q h t g p q with quantum parameters inverted for sl n b we relate our construction to the mirror family defined by givental and its t equivariant analogue due to joe and kim

fusion rings and geometry the algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper a formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings in general and for current algebra theories in particular it is shown that fusion rings lead to a natural orthogonal polynomial structure the rings are expressed through generators and relations the relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties in general the fusion algebras are mapped to affine varieties which are the locus of the relations the connection with modular transformations is investigated in this picture it is explained how chiral algebras arising in n 2 superconformal field theory can be derived from fusion rings in particular it is argued that theories of the type su n k su n 1 are the n 2 counterparts of grassmann manifolds and that there is a natural identification of the chiral fields with schubert varieties which is a graded algebra isomorphism

error estimates for the three field formulation with bubble stabilization in this paper we prove convergence and error estimates for the so called 3 field formulation using piecewise linear finite elements stabilized with boundary bubbles optimal error bounds are proved in l 2 and in the broken h 1 norm for the internal variable u and in suitable weighted l 2 norms for the other variables a and

fusion of irreducible modules in cal wlm p p prime based on symmetry principles we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the extended logarithmic minimal model in addition to the irreducible modules themselves closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules we conjecture that this fusion algebra is the same as the one obtained by application of the nahm gaberdiel kausch algorithm and find that it reproduces the known such results for and for p 1 this fusion algebra does not contain a unit requiring that the spectrum of modules is invariant under a natural notion of conjugation however introduces additional p 1 p 1 reducible yet indecomposable rank 1 modules among which the identity is found still yielding a well defined fusion algebra in this greater fusion algebra the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights kp 1 1 k 1 2 3 we also identify polynomial fusion rings associated with our fusion algebras

quantum computing postselection and probabilistic polynomial time i study the class of problems efficiently solvable by a quantum computer given the ability to postselect on the outcomes of measurements i prove that this class coincides with a classical complexity class called pp or probabilistic polynomial time using this result i show that several simple changes to the axioms of quantum mechanics would let us solve pp complete problems efficiently the result also implies as an easy corollary a celebrated theorem of beigel reingold and spielman that pp is closed under intersection as well as a generalization of that theorem due to fortnow and reingold this illustrates that quantum computing can yield new and simpler proofs of major results about classical computation

morse theory for lagrange multipliers and adiabatic limits abstract given two morse functions f on a compact manifold m we study the morse homology for the lagrange multiplier function on m r which sends x to f x x take a product metric on m r and rescale its r component by a factor 2 we show that generically for large the morse smale witten chain complex is isomorphic to the one for f and the metric restricted to 1 0 with grading shifted by one on the other hand in the limit 0 we obtain another chain complex which is geometrically quite different but has the same homology as the singular homology of 1 0 the isomorphism between the chain complexes is provided by the homotopy obtained by varying our proofs use both the implicit function theorem on banach manifolds and geometric singular perturbation theory

on fusion algebra of chiral models we discuss some algebraic setting of chiral models in terms of the statistical dimensions of their fields in particular the conformal dimensions and the central charge of the chiral models are calculated from their braid matrices furthermore at level k 2 we present the characteristic polynomials of their fusion matrices in a factored form

on total functions existence theorems and computational complexity abstract nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between p and np we show that this class which we call tfnp contains a host of important problems whose membership in p is currently not known these include besides factoring local optimization brouwer s fixed points a computational version of sperner s lemma bimatrix equilibria in games and linear complementarity for p matrices

error estimate for a stabilised domain decomposition method with nonmatching grids summary in this paper we analyse a stabilisation technique for the so called three field formulation for nonoverlapping domain decomposition methods the stabilisation is based on boundary bubble functions in each subdomain which are then eliminated by static condensation the discretisation grids in the subdomains can be chosen independently as well as the grid for the final interface problem we present the analysis of the method and we construct a set of bubble functions which guarantees the optimal rate of convergence

on the complexity of nash equilibria and other fixed points we reexamine what it means to compute nash equilibria and more generally what it means to compute a fixed point of a given brouwer function and we investigate the complexity of the associated problems specifically we study the complexity of the following problem given a finite game gamma with 3 or more players and given epsilon 0 compute an approximation within epsilon of some actual nash equilibrium we show that approximation of an actual nash equilibrium even to within any nontrivial constant additive factor epsilon 1 2 in just one desired coordinate is at least as hard as the long standing square root sum problem as well as a more general arithmetic circuit decision problem that characterizes p time in a unit cost model of computation with arbitrary precision rational arithmetic thus placing the approximation problem in p or even np would resolve major open problems in the complexity of numerical computation we show similar results for market equilibria it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium where the economy is given by explicit algebraic formulas for the excess demand functions we define a class fixp which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits straight line programs over basis max min with rational constants we show that the exact or approximate computation of nash equilibria for 3 or more players is complete for fixp the price equilibrium problem for exchange economies with algebraic demand functions is another fixp complete problem we show that the piecewise linear fragment of fixp equals ppad many other problems in game theory economics and probability theory can be cast as fixed point problems for such algebraic functions we discuss several important such problems computing the value of shapley s stochastic games and the simpler games of condon extinction probabilities of branching processes probabilities of stochastic context free grammars and termination probabilities of recursive markov chains we show that for some of them the approximation or even exact computation problem can be placed in ppad while for others they are at least as hard as the square root sum and arithmetic circuit decision problems

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

a lagrange multiplier method for the finite element solution of elliptic interface problems using non matching meshes in this paper we propose a lagrange multiplier method for the finite element solution of multi domain elliptic partial differential equations using non matching meshes the interface lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes e g we will consider global polynomials as multiplier the ideas are illustrated using poisson s equation as a model and the proposed method is shown to be stable and optimally convergent numerical experiments demonstrating the theoretical results are also presented

fusion potentials i the authors reconsider the conjecture by gepner 1991 that the fusion ring of a rational conformal field theory is isomorphic to a ring of polynomials in n variables quotiented by an ideal of constraints that derive from a potential they show that in a variety of cases this is indeed true with one variable polynomials

threshold computation and cryptographic security threshold machines are turing machines whose acceptance is determined by what portion of the machine s computation paths are accepting paths probabilistic machines are turing machines whose acceptance is determined by the probability weight of the machine s accepting computation paths in 1975 simon proved that for unbounded error polynomial time machines these two notions yield the same class pp perhaps because simon s result seemed to collapse the threshold and probabilistic modes of computation the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored in this paper we compare the bounded error probabilistic class bpp with the analogous threshold class bpppath and more generally we study the structural properties of bpppath we prove that rm bpp path contains both np bpp and p rm np log and that rm bpp path is contained in p rm sigma 2 p log rm bpp np and pp we conclude that unless the polynomial hierarchy collapses bounded error threshold computation is strictly more powerful than bounded error probabilistic computation we also consider the natural notion of secure access to a database an adversary who watches the queries should gain no information about the input other than perhaps its length we show for both bpp and bpppath that if there is any database for which this formalization of security differs from the security given by oblivious database access then p neq pspace it follows that if any set lacking small circuits can be securely accepted then p neq pspace

polynomial fusion rings of logarithmic minimal models we identify quotient polynomial rings isomorphic to the recently found fundamental fusion algebras of logarithmic minimal models

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

totally positive toeplitz matrices and quantum cohomology of partial flag varieties we show that the set of totally positive unipotent lower triangular toeplitz matrices in gln form a real semi algebraic cell of dimension n 1 furthermore we prove a natural cell decomposition for its closure the proof uses properties of the quantum cohomology rings of the partial flag varieties of gln c relying in particular on the positivity of the structure constants which are enumerative gromov witten invariants we also give a characterization of total positivity for toeplitz matrices in terms of the quantum schubert classes this work builds on some results of dale peterson s which we explain with proofs in the type a case

generating the fukaya categories of hamiltonian g manifolds abstract let g be a compact lie group and let k be a field of characteristic p geq 0 such that h g has no p torsion if p 0 we show that a free lagrangian orbit of a hamiltonian g action on a compact monotone symplectic manifold x split generates an idempotent summand of the monotone fukaya category mathcal f x k if and only if it represents a nonzero object of that summand slightly more general results are also provided our result is based on an explicit understanding of the wrapped fukaya category mathcal w t g k through koszul twisted complexes involving the zero section and a cotangent fibre and on a functor d b mathcal w t g k to d b mathcal f x times x k canonically associated to the hamiltonian g action on x we explore several examples which can be studied in a uniform manner including toric fano varieties and certain grassmannians

symmetric boundary conditions in boundary critical phenomena conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of lie algebras a g of ade type for each model we consider the action of its discrete symmetry group on the boundary conditions we find that the invariant ones correspond to the nodes in the product graph a otimes g that are fixed by some automorphism we proceed to determine the charges of the fields in the various hilbert spaces but in a general minimal model many consistent solutions occur in the unitary models a a we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group in contrast a solution with this property does not exist in the unitary models of the series a d and a e 6 a possible interpretation of this fact is that a certain large number of invariant boundary conditions have unphysical negative classical boundary boltzmann weights we give a tentative characterization of the problematic boundary conditions comment 13 pages revtex reorganized and expanded version includes a new section on unitary minimal models conjectures reformulated pointing to the generic existence of negative boundary boltzmann weights in unitary models

adjoint affine fusion and tadpoles we study affine fusion with the adjoint representation for simple lie algebras elementary and universal formulas determine the decomposition of a tensor product of an integrable highest weight representation with the adjoint representation using the refined affine depth rule we prove that equally striking results apply to adjoint affine fusion for diagonal fusion a coefficient equals the number of nonzero dynkin labels of the relevant affine highest weight minus 1 a nice lattice polytope interpretation follows and allows the straightforward calculation of the genus 1 1 point adjoint verlinde dimension the adjoint affine fusion tadpole explicit formulas piecewise polynomial in the level are written for the adjoint tadpoles of all classical lie algebras we show that off diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non dominant representations

rank 2 fusion rings are complete intersections abstract we give a non constructive proof that fusion rings attached to a simple complex lie algebra of rank 2 are complete intersections

fusion algebras of logarithmic minimal models we present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models lm p p considering virasoro representations with no enlarged or extended symmetry algebra the generators of fusion are countably infinite in number but the ensuing fusion rules are quasi rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations the fusion rules are commutative associative and exhibit an s 2 structure but require so called kac representations which are typically reducible yet indecomposable representations of rank 1 in particular the identity of the fundamental fusion algebra p 1 is a reducible yet indecomposable kac representation of rank 1 we make detailed comparisons of our fusion rules with the results of gaberdiel and kausch for p 1 and with eberle and flohr for p p 2 5 corresponding to the logarithmic yang lee model in the latter case we confirm the appearance of indecomposable representations of rank 3 we also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3 the conjectured fusion rules are supported within our lattice approach by extensive numerical studies of the associated integrable lattice models details of our lattice findings and numerical results will be presented elsewhere the agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models lm p p with the augmented cp p minimal models defined algebraically

analysis of a mesh dependent stabilization for the three fields domain decomposition method we consider a stabilized version of the three fields domain decomposition method in the finite element framework by a variant of the stabilization proposed in 2 under fairly general conditions on the meshes and on the finite element discretization considered we prove that for both the two dimensional and the three dimensional case the stabilized method retains the optimality sub optimality properties allowed by the interpolation in the underlying discretization spaces

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

computational tameness of classical non causal models we show that the computational power of the non causal circuit model i e the circuit model where the assumption of a global causal order is replaced by the assumption of logical consistency is completely characterized by the complexity class up coup an example of a problem in that class is factorization our result implies that classical deterministic closed timelike curves ctcs cannot efficiently solve problems that lie outside of that class thus in stark contrast to other ctc models these ctcs cannot efficiently solve np complete problems unless np up coup conp which lets their existence in nature appear less implausible this result gives a new characterization of up coup in terms of fixed points