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     

From Path Representations to Global Morphisms for a Class of Minimal Models

We construct global observable algebras and global DHR morphisms for the Virasoro minimal models with central charge c(2,q), q odd. To this end, we pass from the irreducible highest weight modules to path representations, which involve fusion graphs of the c(2,q) models. The paths have an interpretation in terms of quasi-particles which capture some structure of non-conformal perturbations of the c(2,q) models. The path algebras associated to the path spaces serve as algebras of bounded observables. Global morphisms which implement the superselection sectors are constructed using quantum symmetries: We argue that there is a canonical semi-simple quantum symmetry algebra for each quasi-rational CFT, in particular for the c(2,q) models. These symmetry algebras act naturally on the path spaces, which allows to define a global field algebra and covariant multiplets therein.     

Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers M^N of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns "random" sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S^N_j} of H⁰(M, L^N), we show that for almost every sequence {S^N_j}, the associated sequence of zero currents &1/NZ_S^N_j; tends to the curvature form y of L. Thus, the zeros of a sequence of sections s_N ] H⁰(M, L^N) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S^N_j} of H⁰(M, L^N) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.     

Spectral Measures for <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group G ₂, including the McKay graphs for the irreducible representations of G ₂ and its maximal torus, and fusion modules associated to all known G ₂ modular invariants.     

Reducibility of Quantum Representations of Mapping Class Groups

In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels.     

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two “chiral” induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the “physical spectrum” of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU _{( n )} and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU _{( n )} as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU(3)₅⊂SU(6)₁ is computed for the first time. Received: 24 December 1998 / Accepted: 22 February 1999     

The W-boson mass and precision tests of the Standard Model

We have examined the electroweak radiative corrections in the LEP precision data in view of the new measurements of M_W and m_t as well as the recent progress in the higher order radiative corrections. From the minimal L²-fit to the experimental Z-decay parameters (with the aid of a modified ZFITTER program), we predict that M_W=80.29(4)(2)rGeV where the first error is due to the uncertainty in the fitted m_t for a fixed m_H and the second error comes from the m_H in the range 60􊖸rGeV, which is to be compared with the current world average M_W=80.23(18)rGeV. The current world average value of M_W and the 1994 LEP data definitely favor nonvanishing electroweak radiative corrections and are consistent with a heavy m_t as measured by the recent CDF report but with a heavy Higgs scalar of about 400rGeV within the context of the minimal standard model. The sensitivity of and the errors in the best fit solutions due to the uncertainties in the gluonic coupling !_s(M_Z) and !(M_Z) are also studied carefully. In addition we discuss how the future precision measurements of M_W can provide a decisive test for the standard model with radiative corrections and give a profound implication for the measurement of t-quark and Higgs masses.     

Trace Construction of a Basis for the Solution Space¶of sl N qKZ Equation

The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type A_Nу⁽¹⁾ is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix elements of the composition of intertwining operators are explicitly determined as rational functions up to an overall scalar function. The integral formula for the trace is presented.     

Representations of Vertex Operator Algebra V L + for Rank One Lattice L

We classify the irreducible modules for the fixed point vertex operator subalgebra V_L⁺ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the у isometry of L.     

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Z _p ,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZ _p factors.     

N Dependence of Upper Bounds of Critical Temperatures of 2D O(N) Spin Models

We investigate critical temperature of the classical O(N) spin model in two dimensions. We show that if N is large and there is a phase transition in the system, the critical inverse temperature g_c obeys the bound g_c(N)> const. N log N.     

Collisional broadening and shift of Doppler-free two-photon thallium lines perturbed by rare-gas atoms

The extended Omont-Ueda-Kaulakys treatment of collisional effects on quasi-Rydberg states, in which the perturbation of the lower state is taken into account is applied to thallium-rare gas systems. The pressure broadening and shift coefficients of two-photon transitions in thallium involving the 6P_1/2 -n P_1/2,3/2 (n = 9-14) states are calculated and compared with experimental data obtained by Hermann et al. [Eur. Phys. J. D 1, 129 (1998)].     

On the structure of unitary conformal field theory II: Representation theoretic approach

Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid- and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.     

Inhomogeneous Lattice Paths, Generalized Kostka Polynomials and A n −1 Supernomials

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials, while the other, which we name the A_nу supernomial, is a q-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the A_nу supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.     

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.     

Influence of the substrate voltage on the random telegraph signal parameters in submicron n-channel metal-oxide-semiconductor field-effect transistors under a constant inversion charge density

In this paper, the impact of the substrate bias U_BS on the parameters of a repulsive random telegraph signal in an n-channel metal-oxide-semiconductor field-effect transistor is studied. Particular attention is paid to the variation of the capture time constant F_c with the channel current I in linear operation. It is shown that the strong reduction of F_c with I can be explained by the Coulomb blockade effect. The corresponding Coulomb energy (E of the charged-near-interface oxide trap is shown to be a strong function of the substrate bias. From the analysis of the experimental results considering surface quantization effects follows that the variation of (E with U_BS is caused by the change in both the inversion layer surface charge density N_s and in the surface electric field F_s that influences the distance between the centroid of the inversion layer and the interface. In fact, it will be demonstrated that (E can be expressed in function of a single parameter (N_sF_s²). Finally, the impact of the substrate bias on the other parameters, i.e., the amplitude (I, the emission time constant F_e and the distance d of the trap from the interface, will also be addressed.     

Subfactor Realisation of Modular Invariants

 We study the problem of realising modular invariants by braided subfactors and the related problem of classifying nimreps. We develop the fusion rule structure of these modular invariants. This structure is a useful tool in the analysis of modular data from quantum double subfactors, particularly those of the double of cyclic groups, the symmetric group on 3 letters and the double of the subfactors with principal graph the extended Dynkin diagram D ₅ ⁽¹⁾. In particular for the double of S ₃, 14 of the 48 modular modular invariants are nimless, and only 28 of the remaining 34 nimble invariants can be realised by subfactors. Received: 14 February 2003 / Accepted: 3 April 2003 Published online: 19 May 2003 Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Deformed Algebras from Elliptic sl(N) Algebras

We extend to the sl(N)sl(N) case the results that we previously obtained on the construction of W_q,p{\cal W}_{q,p} algebras from the elliptic algebra A_q,p([^(sl)](2)_c){\cal A}_{q,p}(\widehat{sl}(2)_{c}). The elliptic algebra \elp\elp at the critical level c= m N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(Nу)/2 integers, defining q-deformations of the W_N{\cal W}_{N} algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^\sfrac12)^NM = q^-c-N(-p^\sfrac{1}{2})^{NM} = q^{-c-N} for M ? \ZZM\in\ZZ. It becomes Abelian when in addition p= q^Nh, where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N{\cal W}_{N} algebras depending on the parity of h, characterizing the exchange structures at p p q^Nh as new W_q,p(sl(N)){\cal W}_{q,p}(sl(N)) algebras.     

THE IRREDUCIBLE REPRESENTATIONS AND THEIR OCCUPATION NUMBER IN DIRECT MULTIPLICATION OF IRREDUCIBLE REPRESENTATIONS OF SU3 GROUP

In this paper, the C-G series of SU₃ group is studied using Racah-Speier theorem. A general formula which can be used to calculate irreducible representations and their occupation number in C-G series of SU₃ group is derived. The general formula is very straight forward and effective for actual calculation One can apply the formula to analyse the distribution rule of occupation number in C-G series of SU₃ group and to define the null space of Winger operator of SU₃ group, the latter is important to the calculation of Wigner coefficients when there is a multiplicity.