[Paper Review] Formal loops IV: Chiral differential operators

Mikhail Kapranov, Éric Vasserot|ArXiv.org|Dec 13, 2006

Nonlinear Waves and Solitons6 references11 citations

TL;DR

This paper establishes a deep geometric correspondence between sheaves of chiral differential operators (CDO) on a complex manifold $X$ and the determinantal gerbe on the formal loop space ${\cal L}X$, via a symplectic action homomorphism $S: \Omega^{2,cl}_X \to \mathcal{O}^*_{{\cal L}X}$. The key result identifies the obstruction class to global CDO sheaves with the determinant anomaly on ${\cal L}X$, resolving a central problem in vertex algebra geometry through infinite-dimensional $\mathcal{D}$-module theory and factorization structures.

ABSTRACT

We relate the gerbe of sheaves of chiral differential operators (CDO) on a algebraic variety X, studied by Gorbounov, Malikov and Schechtman, to the determinantal gerbe of the formal loop space LX introduced in our earlier paper. The liens of the two gerbes are related by a version of the symplectic action homomorphism. The determinantal gerbe of LX has a factorization structure in the spirit of Beilinson and Drinfeld. In our identification, sheaves of CDO correspond to factorizing objects of this factorization gerbe.

Motivation & Objective

• To relate the sheaf of chiral differential operators (CDO) on a complex manifold $X$ to the geometry of its formal loop space ${\cal L}X$.
• To resolve the global obstruction to constructing CDO sheaves by identifying it with the determinantal anomaly on ${\cal L}X$.
• To establish a canonical correspondence between the gerbe of CDOs on $X$ and the determinantal gerbe on ${\cal L}X$ using the symplectic action homomorphism $S$.
• To develop a regularized functor of global sections for $\mathcal{D}$-modules on locally locally compact ind-schemes, enabling the construction of vertex algebras from $\delta$-functions on Taylor loops.

Proposed method

• Introduces a functor $\Gamma_{\cal E}$ of global sections for $\mathcal{D}$-modules on locally locally compact ind-schemes, regularizing the failure of direct image functors due to determinant corrections.
• Uses the symplectic action homomorphism $S: \Omega^{2,cl}_X \to \mathcal{O}^*_{{\cal L}X}$, defined by $\omega \mapsto \exp(\int d^{-1}\omega)$, to relate closed 2-forms on $X$ to factorizable functions on ${\cal L}X$.
• Constructs the sheaf of CDOs as the pushforward of the $\delta$-function $\delta_{{\cal L}^0 X}$ on the space of Taylor loops, regularized via the determinantal gerbe ${\cal D}et_{{\cal L}X}$.
• Applies factorization structures from Beilinson-Drinfeld theory to define vertex algebra structures on $\Gamma_{\cal E}(\delta_{{\cal L}^0 X})$, ensuring compatibility with the symplectic action.
• Establishes a pullback map $ev^*$ and Contou-Carrère symbol $\partial$ to relate $K_2$-classes on $X$ and ${\cal L}X$, linking Chern characters to the obstruction class.
• Proves that the image of the CDO gerbe class under $S_*$ equals the image of the Chern character $ch_2(\Theta_X)$ under $\partial \circ ev^*$, confirming the identification of gerbes.

Experimental results

Research questions

• RQ1 How can the obstruction to the global existence of sheaves of chiral differential operators on a complex manifold $X$ be geometrically interpreted?
• RQ2 What is the precise relationship between the determinantal gerbe on the formal loop space ${\cal L}X$ and the gerbe of chiral differential operators on $X$?
• RQ3 Can the construction of CDO sheaves be regularized using infinite-dimensional $\mathcal{D}$-module theory and factorization structures?
• RQ4 How does the symplectic action homomorphism $S$ encode the anomaly in the change-of-variables formula for CDOs?
• RQ5 To what extent do factorizable functions on ${\cal L}X$ correspond to consistent vertex algebra structures on the global sections of $\delta$-functions on Taylor loops?

Key findings

• The obstruction class $[\mathcal{CD}O_X] \in H^2(X, \Omega^{2,cl}_X)$ to the global existence of sheaves of chiral differential operators is given by $\frac{1}{2}c_1(X)^2 - c_2(X)$.
• The symplectic action homomorphism $S: \Omega^{2,cl}_X \to \mathcal{O}^*_{{\cal L}X}$ induces an isomorphism between the lien of the CDO gerbe and the lien of the determinantal gerbe on ${\cal L}X$, identifying the two gerbes up to this map.
• The functor $\Gamma_{\cal E}$ of global sections, regularized via the determinantal gerbe ${\cal D}et_{{\cal L}X}$, yields a well-defined vertex algebra structure on $\Gamma_{\cal E}(\delta_{{\cal L}^0 X})$.
• The class of the CDO gerbe $[\mathcal{CD}O_X^{lt}]$ is mapped under $S_*$ to the class $\partial(ev^*(ch_2(\Theta_X)))$ in $H^2(X, \pi_*({\cal O}^*_{{\cal L}X})) \otimes \mathbb{Z}[1/2]$, establishing the main identification.
• The construction confirms that the anomaly in the change-of-variables formula for CDOs is geometrically realized as the determinant of the normal bundle in the infinite-dimensional loop space setting.
• The identification of the CDO gerbe with the determinantal gerbe on ${\cal L}X$ via $S$ provides a canonical, anomaly-free framework for constructing vertex algebras from geometric data on $X$.