[Paper Review] Notes on correlation functions in (0,2) theories

Eric Sharpe|ArXiv.org|Feb 5, 2005

Advanced Algebra and Geometry13 references4 citations

TL;DR

This paper develops a mathematical framework for computing correlation functions in (0,2) heterotic string theories by generalizing A-model rational curve counting to perturbative heterotic compactifications. Using gauged linear sigma models, it constructs natural compactifications of moduli spaces of rational curves and extends relevant sheaves over these compactifications, ensuring consistency with (0,2) mirror symmetry and anomaly cancellation, thereby enabling direct verification of physical predictions for $ar{ extbf{27}}^3$ couplings.

ABSTRACT

In this note we shall review recent work on generalizing rational curve counting to perturbative heterotic theories.

Motivation & Objective

• To generalize A-model rational curve counting to perturbative (0,2) heterotic string theories.
• To provide a mathematical formulation of correlation functions involving charged states, specifically analogues of the $ar{\mathbf{27}}^3$ coupling.
• To address the lack of understanding in (0,2) mirror symmetry by constructing tools to verify physical predictions from gauged linear sigma models.
• To ensure consistency with anomaly cancellation and duality symmetries in (0,2) theories.

Proposed method

• Formal translation of correlation function computations into algebraic geometry via sheaf cohomology and moduli space compactification.
• Use of gauged linear sigma models to naturally compactify moduli spaces of rational curves in (0,2) theories.
• Construction of sheaf extensions $\mathcal{F}$ and $\mathcal{F}_1$ over compactified moduli spaces to compute $R^{0,1}\pi_*\alpha^*\mathcal{E}$.
• Verification that the first Chern classes of $\mathcal{F}$ and $\mathcal{F}_1$ satisfy $c_1(\mathcal{F}) - c_1(\mathcal{F}_1) = c_1(T\mathcal{M}) - c_1(\text{Obs})$, ensuring consistency with duality.
• Application of linear sigma model constraints to ensure $\Lambda^{\text{top}}\mathcal{E}^\vee \cong K_X$ and anomaly cancellation.
• Presentation-dependent construction of sheaf extensions, showing consistency across different physical presentations of the same bundle.

Experimental results

Research questions

• RQ1 How can correlation functions in (0,2) heterotic theories be systematically formulated in algebraic geometry terms?
• RQ2 What mechanisms in gauged linear sigma models allow for the compactification of moduli spaces of rational curves and extension of sheaves over them?
• RQ3 How do the first Chern classes of the extended sheaves $\mathcal{F}$ and $\mathcal{F}_1$ relate to the geometry of the moduli space and its obstruction bundle?
• RQ4 Can the mathematical framework verify physical predictions for $\bar{\mathbf{27}}^3$ couplings in (0,2) mirror symmetry?
• RQ5 What role does the condition $\Lambda^{\text{top}}\mathcal{E}^\vee \cong K_X$ play in ensuring consistency of the correlation function computation?

Key findings

• The gauged linear sigma model provides a natural mechanism to compactify the moduli space of rational curves in (0,2) heterotic theories.
• Sheaves $\mathcal{F}$ and $\mathcal{F}_1$ are constructed as extensions of $R^{0,1}\pi_*\alpha^*\mathcal{E}$ over the compactification divisor, with well-defined first Chern classes.
• The first Chern class difference satisfies $c_1(\mathcal{F}) - c_1(\mathcal{F}_1) = c_1(T\mathcal{M}) - c_1(\text{Obs})$, confirming geometric consistency.
• The construction is presentation-dependent, but consistent across different physical presentations of the same bundle $\mathcal{E}$.
• The framework enables direct mathematical verification of physical predictions for $\bar{\mathbf{27}}^3$ couplings from [ABS].
• The condition $\Lambda^{\text{top}}\mathcal{E}^\vee \cong K_X$ ensures the correct duality symmetry of states in the (0,2) theory, matching the A-model property.