Algebraic Reduction of the Ising Model

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.     

Quantum mechanics andp-adic numbers

We study the possibility of representing the proposition lattice associated with a quantum system by a linear vector space with coefficients from ap-adic field. We find inconsistencies if the lattice is assumed, as usual, to be irreducible, complete, orthocomplemented, atomic, and weakly modular.     

A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories

A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.     

Minimal orthomodular lattices

The lattices calledminimal orthomodular (MOL) arise in a special exclusion problem concerning the class of all orthomodular lattices (OML) and the subclass of all modular orthocomplemented lattices. This problem was given in G. Kalmbach's book,Orthomodular Lattices. We prove that an exclusion system necessarily must contain an infinite lattice. We prove that, except one, all the finite, irreducible MOLs have only blocks with eight elements. We characterize finite MOLs by a covering property related to equational classes generated by the modular ortholattices MOn.     

Monstrous Moonshine from orbifolds

We consider the Monster Module of Frenkel, Lepowsky, and Meurman as aZ ₂ orbifold of a bosonic string compactified by the Leech lattice. We show that the main Conway and Norton Monstrous Moonshine properties, stating that the Thompson series for each Monster group conjugacy class has a modular invariance group of genus zero, follow from an orbifold construction based on an orbifold group composed of Monster group elements. it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each Thompson series. It is also shown that the Power Map formula of Conway and Norton follows from the same vacuum structure. Finally, we demonstrate the validity of the vacuum conjectures for sectors twisted by Leech lattice automorphisms in many cases.     

Atom interferometry based on light pulses: Application to the high precision measurement of the ratio <Emphasis Type="Italic">h</Emphasis>/<Emphasis Type="Italic">m</Emphasis> and the determination of the fine structure constant

In this paper we present a short overview of atom interferometry based on light pulses. We discuss different implementations and their applications for high precision measurements. We will focus on the determination of the ratio h/m of the Planck constant to an atomic mass. The measurement of this quantity is performed by combining Bloch oscillations of atoms in a moving optical lattice with a Ramsey-Bordé interferometer.     

Proof of Straley's argument for bootstrap percolation

We prove thatP _c =1 for bootstrap percolation with large void instabilities (in particular, ifm=3 on the square lattice).     

Tails of the density of states of two‐dimensional Dirac fermions

The density of states of Dirac fermions with a random mass on a two‐dimensional lattice is considered. We give the explicit asymptotic form of the single‐electron density of states as a function of both energy and (average) Dirac mass, in the regime where all states are localized. We make use of a weak‐disorder expansion in the parameter g/m², where g is the strength of disorder and m the average Dirac mass for the case in which the evaluation of the (supersymmetric) integrals corresponds to non‐uniform solutions of the saddle point equation. The resulting density of states has tails which deviate from the typical pure Gaussian form by an analytic prefactor.     

Pion mass dependence of the <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">l</Emphasis>3</Subscript> semileptonic scalar form factor within finite volume

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer t _m =(m _K −m _π )², using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with L=1.83 fm and L=2.73 fm and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in Boyle et al. (Phys. Rev. Lett. 100:141601, 2008). It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for lattice data to extrapolate at large lattice size.     

Conformal correlation functions,Frobenius algebras and triangulations

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore–Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality.We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular, for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.     

Vertex Operators – From a Toy Model to Lattice Algebras

Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart g _n $ of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras . Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Received: 16 December 1996 / Accepted: 5 May 1997     

Finite Trotter Approximation to the Averaged Mean Square Distance in the Anderson Model

We prove that a finite Trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice ℤ ^d exhibits at most a diffusive behavior in dimensions d≥3 as long as the Fourier transform of the single-site probability, [^(m)]\hat{\mu }, is in L ²(ℝ).     

The Pion Form Factor on the Lattice at Zero and Finite Temperature

We calculate the electromagnetic form factor of the pion in quenched lattice QCD. The non-perturbatively improved Sheikoleslami-Wohlert lattice action is used together with the consistently -improved current. We calculate the pion form factor for masses down to m_π = 360 MeV, extract the charge radius, and extrapolate toward the physical pion mass. In the second part, we discuss results for the pion form factor and charge radius at 0.93 T_c and compare with zero temperature results.     

Theory of pairing symmetry in the vortex states

We investigate pairing symmetry in an Abrikosov vortex and vortex lattice. It is shown that the Cooper pair wave function at the center of an Abrikosov vortex with vorticity m has a different parity with respect to frequency from that in the bulk if m is an odd number, while it has the same parity if m is an even number. As a result, in a conventional vortex with m = 1, the local density of states at the Fermi energy has a maximum (minimum) at the center of the vortex core in an even (odd)-frequency superconductor. In the vortex lattice of s-wave superconductor, we find that only odd-frequency pairing is present at the core centers, while at the midpoint of the vortex lines, only even-frequency pairing exists. Thus, the odd and even-frequency pairings also form the lattice in the vortex lattice state. We also propose a scanning tunneling microscope experiment using a superconducting tip to explore odd-frequency superconductivity.     

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifolds S ³/Gamma where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ.     

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999     

Quantum Invariant for Torus Link and Modular Forms

We consider an asymptotic expansion of Kashaevs invariant or of the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of a modular form of weight 3/2 which is related to the character.     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and n-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.