Conformal field theories on Riemann surfaces of genus $g \geq 1$

Abstract

This is an abstract of the PhD thesis CFTs on Riemann Surfaces of genus g ≥ 1 in mathematics, written by Dr. Marianne Leitner under the supervision of Prof. Dmitri Zaitsev at the School of Mathematics, TCD, and submitted in August 2013.The purpose of this thesis is to argue that N -point functions of holomorphic fields in rational conformal field theories can be calculated by methods from algebraic geometry. We establish explicit formulae for the 2-point function of the Virasoro field on hyperelliptic Riemann surfaces of genus g ≥ 1. N -point functions for higher N are obtained inductively, and we show that they have a nice graphical representation. We discuss the Virasoro 3-point function with application to the Virasoro (2, 5) minimal model.The formulae involve a finite number of parameters, notably the 0-point function and the Virasoro 1-point function, which depend on the moduli of the surface and can be calculated by differential equations. We propose an algebraic geometric approach that applies to any hyperelliptic Riemann surface. Our discussion includes a demonstration of our methods to the case g = 1.

An algebraic approach to minimal models in CFTs

“…This section largely uses results obtained for arbitrary genus in [10,11] though Theorem 4 is proved independently using the methods introduced in Subsection 3.3. It is shown (Proposition 8) that the two formulations are equivalent for g = 1.…”

“…We first show that the highest order coefficient α M of the ODE can be normalised to one. For every κ s in the list (10) and for 0 ≤ m ≤ M − 1, we have…”

“…This is the second in a sequence of three papers on a mathematical approach to Conformal Field Theory (CFT) on compact Riemann surfaces, and it covers the second part of the author's PhD thesis in Mathematics [10]. In the first part of the thesis, a working definition of rational CFTs on general Riemann surfaces has been given.…”

Rational CFTs on Riemann surfaces

“…T (z) is the holomorphic field introduced in [10], [11]. 1 For later reference, we note that from the transformation formula of t zz and invariance of T zz (dz) 2 , the following transformation rule follows for T (z): For a coordinate change z → w with w ∈ D(S ), we have…”

“…where the Ľm satisfy the Virasoro algebra (11) (with Ľm in place of L m ). So the Θ k satify the commutation relation…”