[Paper Review] Nonvanishing of conformal blocks divisors on $\bar{M}_{0,n}$

Prakash Belkale, Angela Gibney+1 more|arXiv (Cornell University)|Oct 8, 2014

Algebraic Geometry and Number Theory25 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions for the nonvanishing of conformal blocks divisors on the moduli space $̅{M}_{0,n}$, focusing on type A Lie algebras. It introduces additive identities among divisors that depend on the rank of the underlying bundle, which amplify vanishing and nonvanishing results, and proves that nonvanishing holds when theta and critical levels coincide.

ABSTRACT

We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on $\bar{M}_{0,n}$ is nonzero. We give necessary conditions in type A, which are sufficient when theta and critical levels coincide. We show that divisors are subject to additive identities, dependent on ranks of the underlying bundle. These identities amplify vanishing and nonvanishing results and have other applications.

Motivation & Objective

• To determine necessary and sufficient conditions for the nonvanishing of conformal blocks divisors on $̅{M}_{0,n}$, particularly in type A Lie algebras.
• To understand the geometric and representation-theoretic conditions under which the first Chern class of a conformal blocks bundle is nonzero.
• To explore additive identities among conformal blocks divisors that arise from the rank of the bundle, enabling amplification of vanishing and nonvanishing results.
• To clarify when the morphism associated to a conformal blocks divisor is nontrivial, by analyzing when the divisor is nonzero.
• To resolve cases where the rank of the conformal blocks bundle is positive but the divisor vanishes, despite nontrivial morphisms being expected.

Proposed method

• Uses the Verlinde formula and representation theory to compute the rank of conformal blocks bundles $̅{V}_{\mathfrak{g},\vec{\lambda},\ell}$ on $̅{M}_{0,n}$.
• Applies the quantum Horn conjecture and matrix factorization techniques to characterize nonvanishing of the bundle via existence of certain unitary matrices with prescribed conjugacy classes.
• Introduces additive identities for conformal blocks divisors by decomposing weight tuples into sub-weights and analyzing the resulting rank conditions on smaller Lie algebras.
• Employs the coinvariant space $̅{A}_{\mathfrak{g},\vec{\lambda}}$ and its dimension to compare with the rank of the bundle, identifying cases where the divisor vanishes even when the bundle is nontrivial.
• Uses the Pl"ucker embedding and morphism $φ_{\mathbb{D}}$ to analyze when the divisor is trivial (i.e., contracts the entire moduli space to a point).
• Leverages the equivalence between nonvanishing of the conformal blocks bundle and the existence of solutions to a system of linear conditions on homomorphisms between vector bundles on $×mathbb{P}^1$.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on $(\mathfrak{g}, \vec{\lambda}, \ell)$ for the conformal blocks divisor $\mathbb{D}_{\mathfrak{g},\vec{\lambda},\ell}$ on $\u0305{M}_{0,n}$ to be nonzero?
• RQ2 How do additive identities among conformal blocks divisors arise from the rank of the underlying bundle, and what do they reveal about vanishing and nonvanishing?
• RQ3 Under what conditions does the morphism $\phi_{\mathbb{D}}$ associated to a conformal blocks divisor factor through a point, indicating divisor vanishing?
• RQ4 When do nontrivial conformal blocks bundles arise with zero first Chern class, and how can such cases be systematically identified?
• RQ5 In type A, when do the theta level and critical level coincide, and what implications does this have for the nonvanishing of the divisor?

Key findings

• In type A, necessary conditions for nonvanishing of the conformal blocks divisor are given, and they are sufficient when the theta level and critical level coincide.
• Additive identities for conformal blocks divisors are established, which depend on the rank of the bundle and allow for amplification of vanishing and nonvanishing results.
• The paper constructs explicit examples where $0 < \operatorname{rk}\mathbb{V}_{\mathfrak{sl}_4,\vec{\lambda},3} = 1$ but $\dim \mathbb{A}_{\mathfrak{sl}_4,\vec{\lambda}} = 2$, and shows that the divisor is still zero.
• The divisor $\mathbb{D}_{\mathfrak{sl}_4,\{\omega_1,(2\omega_1+\omega_3)^3\},3}$ is shown to be zero via a decomposition into two trivial divisors: $\mathbb{D}_{\mathfrak{sl}_4,\{\omega_1,\ldots,\omega_1\},1} + \mathbb{D}_{\mathfrak{sl}_4,\{0,\omega_1+\omega_3,\omega_1+\omega_3,\omega_1+\omega_3\},2}$, both of which are trivial.
• Nonvanishing of the conformal blocks bundle is equivalent to the existence of a solution to a system of linear conditions on homomorphisms between vector bundles on $\mathbb{P}^1$, which is linked to the quantum Horn problem.
• The nonvanishing of the bundle is characterized by the existence of unitary matrices $A_i \in \operatorname{U}(r+1)$ with $A_1 A_2 \cdots A_n = \gamma \operatorname{Id}$, where each $A_i$ is conjugate to a diagonal matrix with entries $\exp(2\pi i \lambda_i^{(a)}/\ell)$, and $\gamma = \exp(2\pi i \sum |\lambda_i| / \ell(r+1))$.