The space of generalized G_2-theta functions of level one

TL;DR

This paper establishes a connection between the space of G_2-theta functions of level one and certain invariant spaces of SL_2-theta functions, revealing new relationships in the theory of moduli of principal bundles.

Contribution

It proves an isomorphism between G_2-theta functions and invariant spaces of SL_2-theta functions, and links G_2-theta functions to SL_3-theta functions, advancing understanding of these moduli spaces.

Findings

Isomorphism between H^0(M(G_2),L) and invariant space of H^0(M(SL_2),L) ensor H^0(M(SL_2),L^3)

Explicit connections between G_2-theta functions and SL_3-theta functions

New insights into the structure of moduli spaces of principal G-bundles

Abstract

Let G_2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a generic, smooth, connected, projective curve over $\mathbb{C}$ of genus at least 2. For a complex Lie group G, let H^0(M(G),L^k) be the space of generalized G-theta functions over C of level k, where M(G) denotes the moduli stack of principal G-bundles over C and L the ample line bundle that generates the Picard group Pic(M(G)). Using the map obtained from extension of structure groups, we prove that the space H^0(M(G_2),L) of generalized G-2-theta functions over C of level one and the invariant space of H^0(M(SL_2), L) \otimes H^0(M(SL_2), L^3) under the action of 2-torsion elements of the Jacobian JC[2] are isomorphic. We also prove explicit links between H^0(M(G_2),L) and the space of generalized SL_3-theta functions of level one.