The space of generalised G2-theta functions of level one

TL;DR

This paper investigates the structure of G2-theta functions at level one, revealing that the monodromy representation associated with the WZW-connection has infinite image, which advances understanding of the geometric and representation-theoretic properties of G2.

Contribution

It introduces natural linear maps between G2 and SL_2/SL_3 Verlinde spaces and shows the monodromy representation for G2 at level one is infinite.

Findings

The image of the monodromy representation is infinite.

Linear maps between G2 and SL groups' Verlinde spaces are constructed.

Insights into the structure of G2-theta functions at level one.

Abstract

Let C be a smooth projective complex curve of genus at least 2. For a simply-connected complex Lie group G the vector space of global sections H^0(M(G), L^l) of the l-th power of the ample generator L of the Picard group of the moduli stack of principal G-bundles over C is commonly called the space of generalized G-theta functions or Verlinde space of level l. In the case G = G_2, the exceptional Lie group of automorphisms of the complex Cayley algebra, we study natural linear maps between the Verlinde space H^0(M(G_2), L) of level one and some Verlinde spaces for SL_2 and SL_3. We deduce that the image of the monodromy representation of the WZW-connection for G = G_2 and l=1 is infinite.