New Braided Endomorphisms from Conformal Inclusions

Various puzzles about subfactors and integrable lattice models associated with conformal inclusions are resolved in the framework of constructive quantum field theory in two dimensions. In particular, a new class of braided endomorphisms are obtained for a general class of conformal inclusions and their properties are analyzed. The existence of subfactors with principal graphs E ₆ or E ₈ follows from a rather simple argument in our construction. The fusion graphs of many new examples are given. Received: 12 September 1996 / Accepted: 3 July 1997     

Quantum Subgroups of the Haagerup Fusion Categories

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the “quantum subgroups” in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.     

Orbifold subfactors from Hecke algebras

We apply the notion of orbifold models ofSU(N) solvable lattice models to the Hecke algebra subfactors of Wenzl and get a new series of subfactors. In order to distinguish our subfactors from those of Wenzl, we compute the principal graphs for both series of subfactors. An obstruction for flatness of connections arises in this orbifold procedure in the caseN=2 and this eliminates the possibility of the Dynkin diagramsD _2n+1 , but we show that no such obstructions arise in the caseN=3. Our tools are the paragroups of Ocneanu and solutions of Jimbo-Miwa-Okado to the Yang-Baxter equation.     

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     

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.     

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     

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     

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT

Three classes of reciprocal graphs, viz. monocycle (G^C_n), linear chain (G^L_n) and star (G^K_n) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (G^C_n) and linear chain (G^L_n) are obtained by putting a pendant vertex to each vertex of simple monocycle (C_n) and simple linear chain (L_n), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n ? 1) peripheral vertices of the star graph (K_{1, (n?1)}). An n-fold rotational axis of symmetry for G^C_n and (n ? 1)-fold rotational axis of symmetry for G^K_n have been exploited for obtaining their respective condensed graphs. The condensed graph for G^L_n has been generated from that of G^C_n incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.     

Subfactors of Index less than 5, Part 1: The Principal Graph Odometer

In this series of papers we show that there are exactly ten subfactors, other than A _∞ subfactors, of index between 4 and 5. Previously this classification was known up to index \({3+\sqrt{3}}\). In the first paper we give an analogue of Haagerup’s initial classification of subfactors of index less than \({3+\sqrt{3}}\), showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.     

Canonical realizations of the lie algebrasp(2n,R)

The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp _i andq _i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.     

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν _n of the n ^th eigenfunction satisfies n ≥ ν _n . Here, we provide a new interpretation for the Courant nodal deficiency d _n = n − ν _n in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning — it is the number of unstable directions in the vicinity of the critical point corresponding to the n ^th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.     

Overlap graph representation of B6 and B7

The virial coefficients B_n of the pressure of a thermodynamic system can be represented in terms of graphs. The recently defined overlap graphs are studied in detail. Furthermore, the overlap graph representation of the sixth and seventh virial coefficientsB ₆ andB ₇) is determined.     

Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains,cycles and stars

Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such pendant graphs become reciprocal graphs for linear chains and cycles if p = n. The n-vertex star graphs follow the same rule as paths and cycles, they become reciprocal on adding a pendant vertex to each of n vertices. The formulas so developed have been expressed in matrix product and in analytical forms for the three classes of graphs that require only the values of n and p for calculation of the respective CP coefficients. Such formulas have the general applicability for a large variety of molecular graphs with varying n and p and have been shown to be reduced to the corresponding formulas for reciprocal graphs that are the special cases of the graphs discussed here.     

The Yamada polynomial of spacial graphs and knit algebras

The Yamada polynomial for embeddings of graphs is widely generalized by using knit semigroups and polytangles. To construct and investigate them, we use a diagrammatic method combined with the theory of algebrasH _N,M(a,q), which are quotients of knit semigroups and are generalizations of Iwahori-Hecke algebrasH _n(q). Our invariants are versions of Turaev-Reshetikhin's invariants for ribbon graphs, but our construction is more specific and computable.This research was supported in part by NSF grant DMS-9100383     

The Nakayama Automorphism of the Almost Calabi-Yau Algebras Associated to <Emphasis Type="Italic">SU</Emphasis>(3) Modular Invariants

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.     

Modular Invariants from Subfactors:¶Type I Coupling Matrices and Intermediate Subfactors

A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent” coupling matrices Z ^± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z ⁺=Z ⁻, then Z is related to Z ⁺ by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z ⁺≠Z ⁻, and that Z need not be related to a type I invariant by such an automorphism. Received: 8 December 1999 / Accepted: 15 February 2000     

Drinfeld Center and Representation Theory for Monoidal Categories

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category \({\mathcal{C}}\) and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of \({\mathcal{C}}\). This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories \({\mathcal{C}}\) of Hilbert bimodules over a II₁-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II₁-factor obtained by the Longo–Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor \({N \subset M}\) is equivalent to Popa’s property (T) for the corresponding SE-inclusion of II₁-factors. In the last part of the paper we study Müger’s notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.     

Graph invariants of Vassiliev type and application to 4D quantum gravity

We consider graph invariants of Vassiliev type extended by the quantum group link invariants. When they are expanded byx whereq=e ^x , the expansion coefficients are known as the Vassiliev invariants of finite type. In the present paper, we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of representations of the quasi-triangular ribbon Hopf algebra extended byU _q (sl(2),C)), which are subject to a quantum group analog of the spinor identity. In terms of them, we obtain the graph invariants of Vassiliev type expressed to be identified with Chern Simons vacuum expectation values of Wilson loops including intersection points. We also consider the 4d canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type satisfy constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.This is not the author's present address.