We determined the inner products of two conformal vectors with central charge 1 2 whose τ -involutions generates S3 if none of τ -involutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge 1 2 + 20 21 or 4 5 + 6 7 and has a Griess algebra of dimension three or four, respectively.

We present a family of conformal field theories (or candidates for CFTs) that is build on extremal partition functions. Spectra of these theories can be decomposed into the irreducible representations of the Fischer-Griess Monster sporadic group. Interesting periodicities in the coefficients of extremal partition functions are observed and interpreted as a possible extension of Monster moonshin...

In forensic ballistics, gunshot residue (GSR) particles can be detected using screening or presumptive tests which are mainly focused on the chromophoric reaction. Most serve as an initial indication for a investigator at crime scene before instrumental analysis definitive identification. The methods known to convenient, have fast analysis, and minimal preparation. GSR well-known method of dete...

Inspired by a formal resemblance of certain q-expansions of modular forms and the master field formalism of matrix models in terms of Cuntz operators, we construct a Hermitian onematrix model, which we dub the “modular matrix model.” Together with an N = 1 gauge theory and a special Calabi-Yau geometry, we find a modular matrix model that naturally encodes the Klein elliptic j-invariant, and he...

In this paper, we study the subalgebra generated by two Ising vectors in the Griess algebra of a vertex operator algebra. We show that the structure of it is uniquely determined by some inner products of Ising vectors. We prove that the order of the product of two τ -involutions is less than or equal to 6 and we determine the inner product of two Ising vectors.

In January 1975, Jacques Tits gave a lecture during which he wrote down the order, 246 · 320 · 59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71, or about 8 · 1053, of a sporadic simple group M that Bernd Fischer and, independently, Robert Griess had predicted in 1973. Andrew Ogg, an expert in modular functions who happened to be in the audience, was uniquely qualified to recognize something peculia...