Implications of an arithmetical symmetry of the commutant for modular invariants

We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU(3)_k, we classify the modular invariant partition functions when k+3 is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants.Comment: 17 pages, plain TeX, DIAS-STP-92-2

Perturbative Corrections for Staggered Four-Fermion Operators

We present results for one-loop matching coefficients between continuum four-fermion operators, defined in the Naive Dimensional Regularization scheme, and staggered fermion operators of various types. We calculate diagrams involving gluon exchange between quark lines, and ``penguin'' diagrams containing quark loops. For the former we use Landau gauge operators, with and without $O(a)$ improvement, and including the tadpole improvement suggested by Lepage and Mackenzie.For the latter we use gauge-invariant operators. Combined with existing results for two-loop anomalous dimension matrices and one-loop matching coefficients, our results allow a lattice calculation of the amplitudes for $K\bar K$ mixing and $K\to\pi\pi$ decays with all corrections of $O(g^2)$ included. We also discuss the mixing of $\Delta S=1$ operators with lower dimension operators, and show that, with staggered fermions, only a single lower dimension operator need be removed by non-perturbative subtraction.Comment: 44 pages latex (uses psfig), 3 ps figures, all bundled using uufiles (correctly this time!), UW/PT-93-

Galois Modular Invariants of WZW Models

The set of modular invariants that can be obtained from Galois transformations is investigated systematically for WZW models. It is shown that a large subset of Galois modular invariants coincides with simple current invariants. For algebras of type B and D infinite series of previously unknown exceptional automorphism invariants are found.Comment: phyzzx macros, 38 pages. NIKHEF-H/94-3

WZW Commutants, Lattices, and Level 1 Partition Functions

A natural first step in the classification of all `physical' modular invariant partition functions \sum N_{LR}\,\c_L\,\C_R lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of Bauer and Itzykson on the commutant from the $SU(N)$ case they consider to the case where the underlying algebra is any semi-simple Lie algebra (and the levels are arbitrary). We then use this analysis to show that the partition functions associated with even self-dual lattices span the commutant. This proves that the lattice method due to Roberts and Terao, and Warner, will succeed in generating all partition functions. We then make some general remarks concerning certain properties of the coefficient matrices $N_{LR}$, and use those to explicitly find all level 1 partition functions corresponding to the algebras $B_n$, $C_n$, $D_n$, and the 5 exceptionals. Previously, only those associated to $A_n$ seemed to be generally known.Comment: 26 page

Automorphisms of the affine SU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras.Comment: 21 pages, plain TeX, DIAS-STP-92-4

Perturbative Corrections for Staggered Fermion Bilinears

We calculate the perturbative corrections to fermion bilinears that are used in numerical simulations when extracting weak matrix elements using staggered fermions. This extends previous calculations of Golterman and Smit, and Daniel and Sheard. In particular, we calculate the corrections for non-local bilinears defined in Landau gauge with gauge links excluded. We do this for the simplest operators, i.e. those defined on a $2^4$ hypercube, and for tree level improved operators which live on $4^4$ hypercubes. We also consider gauge invariant operators in which the ``tadpole'' contributions are suppressed by projecting the sums of products of gauge links back in to the gauge group. In all cases, we find that the variation in the size of the perturbative corrections is smaller than those with the gauge invariant unimproved operators. This is most strikingly true for the smeared operators. We investigate the efficacy of the mean-field method of Lepage and Mackenzie at summing up tadpole contributions. In a companion paper we apply these results to four-fermion operators.Comment: 29 pages latex, 4 postscript figures included, UW/PT-92-13 and CEBAF-TH-92-2

Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras

The characters $\chi_\mu$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $SL_2(Z)$. The matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular matrices, and describe the modular behaviour of the characters. In this paper we consider all levels of $(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$, and for each of these find all permutations of the highest weights which commute with the corresponding Kac-Peterson matrices. This problem is equivalent to the classification of automorphism invariants of conformal field theories, and its solution, especially considering its simplicity, is a major step toward the classification of all Wess-Zumino-Witten conformal field theories.Comment: 16 pp, plain te