Designing new apartment buildings for strings and conformal field theories. First steps

TL;DR

This paper introduces a novel approach to constructing tachyon-free scattering amplitudes and partition functions using the mathematical framework of apartments and buildings, with implications for string theory and conformal field theories.

Contribution

It develops a new method combining reflection group theory and algebraic geometry to produce tachyon-free amplitudes and partition functions, connecting to supersymmetric quantum mechanics.

Findings

Reproduces tachyon-free spectra of open and closed bosonic strings

Derives a new partition function related to N=2 supersymmetric quantum mechanics

Provides a framework applicable to conformal field theories without altering existing results

Abstract

The concepts of apartments and buildings were suggested by Tits for description of the Weyl-Coxeter reflection groups. We use these and many additional facts from the theory of reflection and pseudo-reflection groups along with results from the algebraic and symplectic geometry of toric varieties in order to obtain the tachyon-free Veneziano-like multiparticle scattering amplitudes and the partition function generating these amplitudes.Although the obtained amplitudes reproduce the tachyon-free spectra of both open and closed boisonic string, the generating (partition) function is not that of the traditional bosonic string. It is argued that it is directly related to the N=2 sypersymmetric quantum mechanical model proposed by Witten in 1982 in connection with his development of the Morse theory.Such partition function can be independently obtained with help of the results by Solomon (published in 1963) on invariants of finite (pseudo) reflection groups. Although the formalism developed in this work is also applicable to conformal field theories (CFT), it leaves all CFT results unchanged