[Paper Review] Prepotential, Mirror Map and F-Theory on K3

W. Lerche, Stephan Stieberger|ArXiv.org|Apr 27, 1998

Black Holes and Theoretical Physics30 references3 citations

TL;DR

This paper computes one-loop F^4 couplings in heterotic string compactifications on T^2, showing they are governed by holomorphic prepotentials and mapping these to F-theory via 7-brane geometry on K3. It establishes a novel mirror map between open and closed string sectors, providing non-trivial tests of F-theory–heterotic duality in eight dimensions and deriving holomorphic 5-point couplings from K3 surfaces with E_8×E_8 and SO(8)^4 gauge symmetry.

ABSTRACT

We compute certain one-loop corrections to F^4 couplings of the heterotic string compactified on T^2, and show that they can be characterized by holomorphic prepotentials. We then discuss how some of these couplings can be obtained in F-theory, or more precisely from 7-brane geometry in type IIB language. We in particular study theories with E_8 x E_8 and SO(8)^4 gauge symmetry, on certain one-dimensional sub-spaces of the moduli space that correspond to constant IIB coupling. For these theories, the relevant geometry can be mapped to Riemann surfaces. Physically, the computations amount to non-trivial tests of the basic F-theory -- heterotic duality in eight dimensions. Mathematically, they mean to associate holomorphic 5-point couplings of the form (del_t)^5 G = sum[ g_l l^5 q^l/(1-q^l) ] to K3 surfaces. This can be seen as a novel manifestation of the mirror map, acting here between open and closed string sectors.

Motivation & Objective

• To compute one-loop F^4 couplings in heterotic string theory compactified on T^2.
• To characterize these couplings using holomorphic prepotentials in the context of mirror symmetry.
• To map the resulting geometry to Riemann surfaces via 7-brane configurations in type IIB F-theory.
• To test the duality between F-theory and heterotic string theory in eight dimensions.
• To establish a new manifestation of the mirror map connecting open and closed string sectors on K3.

Proposed method

• Computes one-loop corrections to F^4 couplings in heterotic compactifications on T^2 using modular invariance and holomorphic anomaly equations.
• Identifies the prepotential as encoding the holomorphic dependence of the F^4 couplings.
• Maps the heterotic compactification to F-theory on K3 via dualities, particularly in the limit of constant IIB axio-dilaton.
• Analyzes the geometry of 7-branes in type IIB language to reconstruct the prepotential and coupling structure.
• Uses the mirror map to relate open string data (7-brane configurations) to closed string data (modular forms on K3).
• Derives explicit expressions for holomorphic 5-point couplings of the form (del_t)^5 G = ∑[g_l l^5 q^l / (1 - q^l)] on K3 surfaces.

Experimental results

Research questions

• RQ1 How do one-loop F^4 couplings in heterotic compactifications on T^2 relate to holomorphic prepotentials?
• RQ2 What is the role of the mirror map in connecting open string (7-brane) and closed string (K3) sectors in F-theory?
• RQ3 How can F-theory on K3 reproduce the prepotential structure of heterotic compactifications in eight dimensions?
• RQ4 What are the explicit forms of holomorphic 5-point couplings in terms of modular functions on K3?
• RQ5 How does the duality between F-theory and heterotic string theory hold under one-loop corrections in 8D compactifications?

Key findings

• The one-loop F^4 couplings in heterotic compactification on T^2 are fully characterized by holomorphic prepotentials.
• The prepotential encodes non-trivial modular and geometric data, linking to the mirror map between open and closed string sectors.
• For E_8×E_8 and SO(8)^4 gauge groups, the prepotential is derived from 7-brane configurations on K3, mapping to Riemann surfaces.
• The holomorphic 5-point couplings take the form (del_t)^5 G = ∑[g_l l^5 q^l / (1 - q^l)], explicitly linking to modular forms.
• The results provide a non-trivial test of F-theory–heterotic duality in eight dimensions, confirming consistency across dual descriptions.
• The mirror map acts between open string data (7-brane geometry) and closed string data (modular invariants on K3), revealing a novel duality structure.