[Paper Review] Solutions of Strominger system from unitary representations of cocompact lattices of SL(2,C)

Indranil Biswas, Avijit Mukherjee|arXiv (Cornell University)|Jan 3, 2013

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper constructs explicit solutions to the Strominger system on compact complex 3-folds by leveraging irreducible unitary representations of cocompact lattices in SL(2,ℂ). It demonstrates that the resulting Hermitian vector bundles are stable and the solutions satisfy all equations of motion, including the torsion-compatible metric and curvature constraints, via invariant differential forms on SL(2,ℂ).

ABSTRACT

Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.

Motivation & Objective

• To provide explicit, non-perturbative constructions of solutions to the Strominger system on compact complex 3-folds with torsion.
• To establish a novel method for generating solutions without relying on deformation or perturbative techniques.
• To demonstrate that the resulting holomorphic vector bundles are stable and satisfy the physical and geometric constraints of heterotic superstring theory.
• To explore the interplay between representation theory of cocompact lattices in SL(2,ℂ) and the geometry of solutions to the Strominger system.
• To verify that the constructed solutions satisfy the equation of motion and all components of the Strominger system, including anomaly cancellation and supersymmetry conditions.

Proposed method

• Utilizes irreducible unitary representations of cocompact lattices Γ ⊂ SL(2,ℂ) to define geometric structures on the quotient manifold M = Γ\SL(2,ℂ).
• Constructs a Hermitian metric ω on M via right-invariant 1-forms on SL(2,ℂ), invariant under left SU(2)-action.
• Defines a holomorphic vector bundle E with a flat connection ∇ using the unitary representation, ensuring F_A^{2,0} = F_A^{0,2} = 0.
• Employs right-invariant differential forms and SU(2)-invariance to show that d(‖Ω‖_ω · ω²) = 0, satisfying the supersymmetry condition.
• Applies representation-theoretic invariance to prove that the curvature R(∇^ω) ∧ ω² = 0 by showing the trace scalar λ = 0 via integration and Chern class vanishing.
• Uses the fact that c₁(TM) = 0 and ∫_M dβ ∧ ω² = 0 to conclude λ = 0, thereby verifying the curvature condition.

Experimental results

Research questions

• RQ1 Can explicit solutions to the Strominger system be constructed without perturbative or deformation-based methods?
• RQ2 Do irreducible unitary representations of cocompact lattices in SL(2,ℂ) yield stable holomorphic vector bundles satisfying the Strominger system?
• RQ3 Can the torsion-compatible metric and curvature constraints in the Strominger system be satisfied via invariant forms on SL(2,ℂ)?
• RQ4 Is the equation of motion d(‖Ω‖_ω · ω²) = 0 satisfied by geometric structures derived from unitary representations?
• RQ5 Does the curvature condition R(∇^ω) ∧ ω² = 0 hold for the constructed connection on the tangent bundle?

Key findings

• The constructed solution satisfies all equations of the Strominger system, including the anomaly cancellation condition and the supersymmetry condition d(‖Ω‖_ω · ω²) = 0.
• The holomorphic vector bundle E associated with the unitary representation is stable, as required by the physical and geometric consistency of the model.
• The curvature of the Chern connection ∇^ω satisfies R(∇^ω) ∧ ω² = 0, which is proven by showing the trace scalar λ = 0 via integration and vanishing first Chern class.
• The form ‖Ω‖_ω · ω² is a constant multiple of ω², implying d(‖Ω‖_ω · ω²) = 0, thus satisfying the supersymmetry condition.
• The solution is explicitly constructed from group-theoretic data (unitary representation of a cocompact lattice), avoiding deformation or perturbative methods.
• The curvature components F_A^{2,0} and F_A^{0,2} vanish identically due to the flatness of the connection ∇ on E, satisfying the Yang-Mills condition.