Vertex algebras and elliptic genus

Arbeitstagung, Bonn, June 1999

Vadim Schechtman

This is a report on a joint work with V. Gorbounov, F. Malikov and A. Vaintrob, cf. [MSV], [MS1], [MS2], [GMS].

1. Let X be a smooth complex variety (algebraic or analytic). One can associate with X a canonically defined gerbe (champ of groupoids) 𝔇_{X}^{ch} called the gerbe of chiral differential operators on X. For an open U ⊂ X, the objects of the groupoid 𝔇_{X}(U) are sheaves of vertex algebras over X, called sheaves of chiral differential operators on X. The gerbe 𝔇_{X} is bounded by the lien isomorphic to the sheaf Ω_{X}^{2,cl} of closed two-forms.

The equivalence class

c(𝔇_{X}) ∈ H^{2}(X, Ω_{X}^{2,cl})

is equal to

2ch_{2}(T_{X}) := c_{1}^{2}(T_{X}) − 2c_{2}(T_{X})  (1.1)

where c_{i}(T_{X}) ∈ H^{i}(X, Ω_{X}^{i,cl}) are the Chern classes.

For example, (1.1) vanishes for products of curves, or for flag spaces G/B.

2. Let E be a vector bundle over X. Generalizing the previous construction, one can define the gerbe 𝔇_{ΛE} of chiral differential operators on the exterior algebra ΛE, also bounded by the sheaf Ω_{X}^{2,cl}.

It is plausible that its equivalence class is equal to

c(𝔇_{ΛE}) = 2ch_{2}(T_{X}) − 2ch_{2}(E)  (2.1)

For example, (2.1) vanishes for E = Ω_{X}^{1} or T_{X}.

3. Consider the case E = Ω_{X}^{1}. Let us denote the gerbe 𝔇_{ΛE} of chiral differential operators on the de Rham algebra of differential forms by 𝔇_{Ω}. The groupoid 𝔇_{Ω}(X) is non-empty.

In fact, one can define certain canonical object D_{Ω} of 𝔇_{Ω}(X). It was introduced in [MSV], and called chiral de Rham complex there. Recently A. Beilinson gave a nice coordinate-free construction of D_{Ω} and some of its generalizations in the language of chiral algebras (in the sense of Beilinson - Drinfeld).

D_{Ω} is a sheaf of conformal vertex superalgebras, ℤ-graded by fermionic charge. For a compact X, the cohomology algebra

H^{∗}(X, D_{Ω}) = ⊕_{p=0}^{n} ⊕_{q=−∞}^{∞} ⊕_{i=0}^{∞} H^{p}(X, D_{Ω,i}^{q})  (3.1)

is called the chiral Hodge cohomology of X and has some nice properties. Here D_{Ω,i}^{q} denotes the component of conformal weight i and fermionic charge q, n = dim(X).

The space (3.1) is a conformal vertex superalgebra. It is a N = 2 superalgebra if X is Calabi-Yau (i.e. when the canonical bundle is trivial). We have the ”Poincaré duality”:

H^{p}(X, D_{Ω,i}^{q})^{∗} = H^{n−p}(X, D_{Ω,i}^{n−q})  (3.2)

As was noted in [BL], if X is Calabi-Yau then the generating series

χ_{X}(q, y) := y^{n/2} ∑_{a,b,i} (−1)^{a+b} dim H^{a}(X, D_{Ω,i}^{b})y^{b}q^{i}  (3.3)

coincides with the Ochanine — Witten elliptic genus of X.

References

[BL] L. Borisov, A. Libgober, Elliptic genera and applications to Mirror symmetry, math/9906126.

[GMS] V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators, math.AG/9906117.

[MSV] F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys. (1999), to appear; math.AG/9803041.

[MS1] F. Malikov, V. Schechtman, Chiral de Rham complex. II, D.B. Fuchs’ 60-th Anniversary volume (1999), to appear; math.AG/9901065.

[MS2] F. Malikov, V. Schechtman, Chiral Poincaré duality, math.AG/9905008.

Department of Mathematics, Glasgow University, 15 University Gardens, Glasgow G12 8QW, UK; vs@maths.gla.ac.uk, vadik@mpim-bonn.mpg.de