Deformations of chiral algebras and quantum cohomology of toric varieties

Fyodor Malikov^{1} and Vadim Schechtman

Introduction

Let X be a smooth complex variety. It was shown in [MSV] that the complex cohomology algebra H^{∗}(X) can be obtained as a cohomology of a certain vertex algebra H^{ch}(X) canonically associated with X. By definition, H^{ch}(X) = H^{∗}(X; Ω_{X}^{ch}), where Ω_{X}^{ch} is a sheaf of vertex superalgebras constructed in [MSV] and called the chiral de Rham complex. (If X is compact, then H^{ch}(X) may be called the chiral Hodge cohomology algebra of X.) The algebra H^{ch}(X) is equipped with a canonical odd derivation Q of square zero, and the cohomology of H^{ch}(X) with respect to Q is equal to H^{∗}(X). We shall record this as follows: H^{∗}(X) = H_{Q}(H^{ch}(X)); and whenever we encounter a vector space W with differential d, we let H_{d}(W) denote the corresponding cohomology

In the very interesting paper [B] Borisov defined for a smooth complete toric variety X a certain vertex superalgebra V (X) equipped with an odd derivation D, D^{2} = 0, so that H^{ch}(X) = H_{D}(V (X)). It follows that H^{∗}(X) = H_{Q}(H_{D}(V (X))). In fact, [Q, D] = 0 and it is true that H^{∗}(X) = H_{Q+D}(V (X)).

In the present note we consider a family (V_{q}(X), Q(q)+D(q)), q ∈ H^{2}(X), of vertex superalgebras with differential (also mentioned in [B]) satisfying (V_{0}(X), Q(0)+ D(0)) = (V (X), Q + D) and prove that H_{Q(q)+D(q)}(V_{q}(X)) is equal to the quantum cohomology algebra of X. This is first done in sect. 1.1 - 2.4 in the case when X = ℙ^{N} and then generalized to an arbitrary X in §4.

We also get similar partial results for Fano hypersurfaces in P^{N}. Their exposition in §3 is somewhat sketchy reflecting the patchy status quo.

In sect. 2.5 we present a simpler (making no use of Q) version of this construction in the case of ℙ^{N} (Theorem 2.5A) and apply the deformation technique to compute H^{∗}(ℙ^{N}; Ω_{ℙ^{N}}^{ch}) (Theorem 2.5B). According to Theorem 2.5A, H_{D(q)}(V_{q}(ℙ^{N})), q ∈ ℂ, is a family of vertex algebras over ℂ with fiber that equals the quantum cohomology over q ≠ 0 and blows up to the infinite dimensional vertex algebra H^{ch}(X) over 0 ∈ ℂ. The proof Theorem 2.5A is the most technically involved part of the text; it heavily relies on the previous work, in particular, on the Lsl_{N+1}-symmetry of Ω_{ℙ^{N}}^{ch} introduced in [MS1].

A by-product of this proof, Theorem 2.5B says that the intermediate cohomology functors unchiralize the chiral de Rham complex, that is,

H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) = H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{∗}), 0 < i < N,

where Ω_{ℙ^{N}}^{∗} stands for the usual de Rham complex. Due to the chiral Poincaré duality [MS2], the 0th and Nth cohomology groups of Ω_{ℙ^{N}}^{ch} are naturally dual to each other. All this combined with the result of [BL] relating the super Euler character of Ω_{ℙ^{N}}^{ch} (details are to be found in 2.5.3) and the Ochanine-Witten elliptic genus of ℙ^{N} shows that the supercharacter of H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) satisfies the following functional equation:

1partially supported by an NSF grant

sch^{0}(y, q) + sch^{0}(y^{−1}, q) = Ell_{ℙ^{N}} (y, q) − \frac{y^{(N−1)/2} − y^{−(N−1)/2}}{y^{1/2} − y^{−1/2}} ;

here sch^{0}(y, q) is the supercharacter of H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) and Ell_{ℙ^{N}} (y, q) is the elliptic genus of ℙ^{N}.

Acknowledgments. This work was done while V.S. was visiting the Institut des Hautes Études Scientifiques. The last strokes were laid while F.M. was visiting the Max-Planck-Institut für Mathematik. We are grateful to both the institutions for the support and excellent working conditions.

§1. Borisov’s construction

1.1. Lattice vertex algebras. Let L be a free abelian group on 2N generators A^{i}, B^{i}, 1 ≤ i ≤ N. Give L an integral lattice structure by defining a bilinear symmetric ℤ-valued form

(., .) : L × L → ℤ

so that

(A^{i}, B^{j}) = δ_{ij}, (A^{i}, A^{j}) = (B^{i}, B^{j}) = 0.

Introduce the complexification of L:

𝔥_{L} = L ⊗_{ℤ} ℂ.

The bilinear form (., .) carries over to 𝔥_{L} by bilinearity. Let

𝔥̂_{L} = 𝔥_{L} ⊗ ℂ[t, t^{−1}] ⊕ ℂK

be a Lie algebra with bracket

[x ⊗ t^{i}, y ⊗ t^{j}] = i(x, y)δ_{i+j}K, [x ⊗ t^{i}, K] = 0.

Associated with L there is a group algebra ℂ[L] with basis e^{α}, α ∈ L, and multipliciation

e^{α} · e^{β} = e^{α+β}, e^{0} = 1, α, β ∈ L.

Denote by S_{𝔥_{L}} the symmetric algebra of the space 𝔥_{L} ⊗ t^{−1}ℂ[t^{−1}]. The space S_{𝔥_{L}} ⊗ ℂ[L] carries the well-known vertex algebra structure, see for example [K]. Borisov proposes to enlarge this lattice vertex algebra by fermions as follows.

We tacitly assumed that 𝔥_{L} is a purely even vector space: 𝔥_{L}^{(0)} = 𝔥_{L}, 𝔥_{L}^{(1)} = 0. Let Π𝔥_{L} satisfy the relations Π𝔥_{L}^{(1)} = 𝔥_{L}, Π𝔥_{L}^{(0)} = 0. Thus Π𝔥_{L} is a purely odd vector space with basis Ψ^{i}, Φ^{i} carrying the following odd bilinear form:

(., .) : Π𝔥_{L} × Π𝔥_{L} → ℂ,

(Ψ^{i}, Φ^{j}) = δ_{ij}, (Ψ^{i}, Ψ^{j}) = (Φ^{i}, Φ^{j}) = 0.

Given all this, one defines the Clifford algebra, Cl_{𝔥_{L}}, to be the vector superspace

Cl_{𝔥_{L}} = Π𝔥_{L} ⊗ ℂ[t, t^{−1}] ⊕ ℂK′, Cl_{𝔥_{L}}^{(1)} = Π𝔥_{L} ⊗ ℂ[t, t^{−1}], Cl_{𝔥_{L}}^{(0)} = ℂK′,

with (super)bracket [x ⊗ t^{i}, y ⊗ t^{j}] = (x, y)δ_{i+j}K′.

Let Λ_{𝔥_{L}} be the symmetric algebra of the superspace

⊕_{i=1}^{N}(Φ^{i} ⊗ ℂ[t^{−1}] ⊕ Ψ^{i} ⊗ t^{−1}ℂ[t^{−1}].

(If we had been allowed to forget about the parity, we would have equivalently defined Λ_{𝔥_{L}} to be the exterior algebra of the indicated space.) The space Λ_{𝔥_{L}} carries the well-known vertex algebra structure, see for example [K].

Finally let

V_{L} = Λ_{𝔥_{L}} ⊗ S_{𝔥_{L}} ⊗ ℂ[L].

Being a tensor product of vertex algebras, V_{L} is also a vertex algebra.

1.2. Explicit description of the vertex algebra structure on V_{L}. To simplify the notation, we identify ℂ[L] with the subspace 1⊗1⊗ℂ[L]. As an 𝔥̂_{L} ⊕Cl_{𝔥_{L}}-module, V_{L} is a direct sum of irreducibles and there is one irreducible module, V_{L}(α), for each α ∈ L. V_{L}(α) is freely generated by the supercommutative associative algebra S_{𝔥_{L}} ⊗ Λ_{𝔥_{L}} from the highest weight vector e^{α}. The words “highest weight vector” mean that the following relations hold:

A_{n}^{i}e^{α} = Ψ_{n}^{i}e^{α} = B_{n}^{i}e^{α} = Φ_{n+1}^{i}e^{α} = 0, n ≥ 0,

Ke^{α} = K′e^{α} = e^{α}, xe^{α} = (x, α)e^{α}, x ∈ 𝔥_{L}.

Thus, V_{L}(α), α ∈ L, are different as 𝔥̂_{L} ⊕ Cl_{𝔥_{L}}-modules, but isomorphic as 𝔥̂_{L}_{1}⊕Cl_{𝔥_{L}}-modules, where 𝔥̂_{L}_{1} ⊂ 𝔥̂_{L} is the subalgebra linearly spanned by x⊗t^{i}, i ≠ 0, x ∈ 𝔥_{L}. In fact, the multiplication by e^{β} provides an isomorphism of 𝔥̂_{L}_{1} ⊕ Cl_{𝔥_{L}}- modules:

e^{β} : V_{L}(α) → V_{L}(α + β), x ⊗ e^{α} ↦ x ⊗ e^{α+β}.

Let us now define the state-field correspondence, that is, attach a field x(z) ∈ End(V_{L})((z, z^{−1})) to each state x ∈ V_{L}. As has become customary, we shall write x_{i} for x ⊗ t^{i} (x ∈ 𝔥_{L} or Π𝔥_{L}). We have:

(x_{−n−1}e^{0})(z) = \frac{1}{n!}x(z)^{(n)}, x ∈ 𝔥_{L},

where

x(z) = ∑_{j∈ℤ} x_{j}z^{−j−1}.

In particular, (x_{−1}e^{0})(z) = x(z).

We continue in the same vein:

(Φ_{−n}^{i}e^{0})(z) = \frac{1}{n!}Φ^{i}(z)^{(n)},

where

Φ^{i}(z) = ∑_{j∈ℤ} Φ_{j}^{i}z^{−j};

(Ψ_{−n−1}^{i}e^{0})(z) = \frac{1}{n!}Ψ^{i}(z)^{(n)},

where

Ψ^{i}(z) = ∑_{j∈ℤ} Ψ_{j}^{i}z^{−j−1};

e^{α}(z) = e^{α} · exp (− ∑_{n<0} \frac{α_{n}}{n} z^{−n}) · exp (− ∑_{n>0} \frac{α_{n}}{n} z^{−n}) · z^{α_{0}}(−1)^{pr_{A}(α)}.

In the last formula the product of exponentials signifies an unambiguously defined generating series of operators acting on V_{L}; whenever e^{α}(z) is applied to an element of V_{L}(β), z^{α_{0}} is changed for z^{(α,β)}; pr_{A} stands for the projection of L onto the sublattice spanned by A’s, and when applied to an element of V_{L}(β), (−1)^{pr_{A}(α)} is changed for (−1)^{(pr_{A}(α),β)}.

Finally,

x_{−n_{1}}^{(1)} · x_{−n_{2}}^{(2)} ⋯ x_{−n_{k}}^{(k)} · e^{α}(z) =: x_{−n_{1}}^{(1)}(z)x_{−n_{2}}^{(2)}(z) ⋯ x_{−n_{k}}^{(k)}(z)e^{α}(z) : .

Remark. Of these formulas that for e^{α}(z) is undoubtedly the most intricate. The factor (−1)^{pr_{A}(α)} is the Frenkel-Kac cocycle [FK,K] needed in order to conform to the vertex algebra axioms. This factor being dropped, the formula gives the vertex operator known since the early days of string theory. The untiring reader will check that all the calculations of this paper go through with (−1)^{pr_{A}(α)} dropped, the light-minded reader may want to take this for granted.

The vertex algebra structure on V_{L} is equivalently described by the following family of n-th products (n ∈ ℤ):

(n) : V_{L} ⊗ V_{L} → V_{L}, x ⊗ y ↦ x_{(n)}y =^{def} (∫ x(z)z^{n})(y),

where ∫ x(z)z^{n} stands for the linear transformation of V_{L} equal to the coefficient of z^{−n−1} in the series x(z).

1.3. Degeneration of V_{L}. Denote by L_{A} the subgroup of L generated by A^{i}, i = 1, ..., N. Any smooth toric variety X can be defined via a fan, Σ, that is, a collection of “cones” lying in L_{A}. Borisov uses such Σ to define a certain degeneration, V_{L}^{Σ}, of the vertex algebra structure on V_{L}. He further shows that the cohomology of V_{L}^{Σ} with respect to a certain differential D^{Σ} : V_{L}^{Σ} → V_{L}^{Σ} equals H^{∗}(X, Ω_{X}^{ch}), where Ω_{X}^{ch} is the chiral de Rham complex of [MSV]. Let us describe the outcome of this construction in the case when X = ℙ^{N}.

Consider the following set of N + 1 elements of L_{A}: ξ_{1} = A^{1}, ξ_{2} = A^{2}, ..., ξ_{N} = A^{N}, ξ_{N+1} = −A^{1} − A^{2} − ⋯ − A^{N}. Define the cone Δ_{i} ⊂ L_{A} to be the set of all non-negative integral linear combinations of the elements ξ_{1}, ..., ξ_{i−1}, ξ_{i+1}, ..., ξ_{N+1}. The fan Σ in this case is the set consisting of all the intersections Δ_{i_{1}} ∩Δ_{i_{2}} ∩⋯∩Δ_{i_{l}} with 1 ≤ i_{1} < ⋯ < i_{l} ≤ N + 1, 1 ≤ l ≤ N + 1.

We now define V_{L}^{Σ} to be a vertex algebra equal to V_{L} as a vector space with n-th product _{(n),Σ} as follows:

if {∑_{i} n_{i}A^{i}, ∑_{i} n′_{i}A^{i}} ⊂ Δ_{j} for some j, then

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),Σ}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})
= (x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n)}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}});

otherwise

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),Σ}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}}) = 0,

where _{(n)} stands for the n-th product on V_{L}. The fact that these new operations satisfy the Borcherds identities can be proved by including both V_{L} and V_{L}^{Σ} in a 1-parameter family of vertex algebras; this will be done in 2.1.

Let

D = ∫ ∑_{i=1}^{N} Ψ^{i}(z)(e^{A^{i}} − e^{− ∑_{j} A^{j}})(z).  (1.1)

It is obvious that D ∈ End(V_{L}^{Σ}) and D^{2} = 0; therefore, the cohomology H_{D}(V_{L}^{Σ}) arises.

Theorem 1.3. ([B])

H_{D}(V_{L}^{Σ}) = H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}).
§2. Deforming H^{∗}(ℙ^{N})

2.1. The family V_{L,q}.

Here we exhibit a family of vertex algebras, V_{L,q}, q ∈ ℂ, so that V_{L,q} is isomorphic to V_{L} if q ≠ 0 and V_{L,0} is isomorphic to V_{L}^{Σ}; cf. the end of sect.8 [B].

Define the height function

ht : L_{A} → ℤ_{>}

as follows. Since L_{A} = ∪_{i}Δ_{i}, see 1.3, each α ∈ L_{A} is uniquely represented in the form

α = ∑_{i=1}^{N+1} n_{i}ξ_{i}  (2.1)

so that all n_{i} ≥ 0 and #{i : n_{i} > 0} ≤ N. Let

ht(α) = ∑_{i} n_{i},

where n_{1}, ..., n_{N} are as in (2.1).

Define the linear automorphism

t_{q} : V_{L} → V_{L}, q ∈ ℂ − {0}

by the formula

t_{q}(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}}) = q^{ht(∑_{i} n_{i}A^{i})}x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}}.

Define V_{L,q} to be the vertex algebra equal to V_{L} as a vector space with the following n-th product:

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),q}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})
= t_{q}^{−1}(t_{q}(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n)}t_{q}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})).

By definition,

t_{q} : V_{L,q} → V_{L}, q ∈ ℂ − {0},

is a vertex algebra isomorphism. It is also easy to see that if ∑_{i} n_{i}A^{i} and ∑_{i} n′_{i}A^{i} belong to the same cone from Σ, then

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),q}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})
= (x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n)}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}});

otherwise

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),q}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})
∈ qℂ[q](x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n)}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}}).

Two things follow at once: first, the operations

(n),0 = lim_{q→0}_{(n),q}, n ∈ ℤ

are well defined and satisfy the Borcherds identities; second, the vertex algebra, V_{L,0}, obtained in this way is isomorphic to V_{L}^{Σ}. By the way, this remark proves that V_{L}^{Σ} is indeed a vertex algebra.

To get a better feel for this kind of deformation, and for the future use, let us consider the subspace ℂ[L_{A}] ⊂ V_{L,q} with basis e^{α}, α ∈ L_{A}. The (−1)-st product makes this space a commutative algebra. The subspace ℂ[Δ_{j}] defined to be the linear span of e^{α}, α ∈ Δ_{j}, is a polynomial ring on generators e^{ξ_{1}}, ..., e^{ξ_{j−1}}, e^{ξ_{j+1}}, ..., e^{ξ_{N+1}}. For example, if we denote x_{i} = e^{A^{i}}, then ℂ[Δ_{N+1}] = ℂ[x_{1}, ..., x_{N}] and this isomorphism identifies e^{∑_{j} n_{j}A^{j}} with the monomial x_{1}^{n_{1}} ⋯ x_{N}^{n_{N}}.

The entire ℂ[L_{A}] is not a polynomial ring. For example, as follows from the definition of the deformation, there is a relation

(e^{−A^{1}−···−A^{N}})_{(−1)}(e^{A^{1}+···+A^{N}}) = q^{N+1}e^{0},

because ht(0) = 0, ht(A^{1} + ⋯ + A^{N}) = N, ht(−A^{1} − ⋯ − A^{N}) = 1. If we let T = e^{−A^{1}−···−A^{N}}, then the last equality rewrites as follows:

T x_{1}x_{2} ⋯ x_{N} = q^{N+1},

and a moment’s thought shows that in fact

ℂ[L_{A}] = ℂ[x_{1}, ..., x_{N}, T]/(T x_{1}x_{2} ⋯ x_{N} − q^{N+1}).

Being a group algebra, ℂ[L_{A}] carries another algebra structure, a priori different from the one we just described and independent of q. We see that the two structures are isomorphic if q ≠ 0; at q = 0, however, the one we just described degenerates in an algebra with zero divizors.

2.2. The algebra H^{∗}(ℙ^{N}).

Let

Q(z) = A^{i}(z)Φ^{i}(z) − ∑_{j} Φ^{j}(z)′,

G(z) = B^{i}(z)Ψ^{i}(z),

J(z) =: Φ^{i}(z)Ψ^{i}(z) : + ∑_{j} B^{j}(z),

L(z) =: B^{i}(z)A^{i}(z) : + : Φ^{i}(z)′Ψ^{i}(z) :, where the summation with respect to repeated indices is assumed.

One checks that the Fourier components of these 4 fields satsify the commutation relations of the N = 2 algebra. It is also easy to see that the fields G(z), L(z) commute with Borisov’s differential D, see (1.1), and therefore define the fields, to be also denoted G(z), L(z), acting on H_{D}(V_{L}^{Σ}).

The field Q(z) does not commute with D, but its Fourier component Q_{0} = ∫ Q(z) does:

[Q_{0}, D] = 0.

Thus we get an operator, to be also denoted Q_{0}, acting on H_{D}(V_{L}^{Σ}). All this is summarized by saying that H_{D}(V_{L}^{Σ}) is a topological vertex algebra.

A glance at the formulas on p. 17 of [B] shows that the isomorphism H_{D}(V_{L}^{Σ}) = H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) (see Theorem 1.3) identifies these G(z), L(z), Q_{0} with the fields (operator) constructed in [MSV] and denoted in the same way. One of the main results of [MSV] then gives

H^{∗}(ℙ^{N}) = H_{Q_{0}}(H_{D}(V_{L}^{Σ})).  (2.2)

Further, the algebra structure of H^{∗}(ℙ^{N}) is restored from the (−1)-st product on H_{D}(V_{L}^{Σ}).

2.3. Deformation of the algebra structure.

It follows from the proof of Theorem 2.3 below that the cohomology (2.2) can be calculated in the reversed order:

H^{∗}(ℙ^{N}) = H_{D}(H_{Q_{0}}(V_{L}^{Σ})).  (2.3)

Note that D and Q_{0} can also be regarded as well-defined operators acting on the deformed algebra:

D = ∫ ∑_{i=1}^{N} Ψ^{i}(z)(e^{A^{i}} − e^{− ∑_{j} A^{j}})(z), Q_{0} = ∫ A^{i}(z)Φ^{i}(z) : V_{L,q} → V_{L,q}.

It is immediate to see that D^{2} = Q_{0}^{2} = 0 on V_{L,q} as well. Moreover,

[D, Q_{0}] = 0.  (2.4)

Indeed, the formulas of 1.2 imply the following OPE:

∑_{i=1}^{N} Ψ^{i}(z)(e^{A^{i}} − e^{− ∑_{j} A^{j}})(z) · A^{j}(w)Φ^{j}(w) = \frac{∑_{j} e^{A^{j}}(w)′ − e^{− ∑_{j} A^{j}}(w)′}{z − w} .

Therefore,

[D, Q_{0}] = ∫ {∑_{j} e^{A^{j}}(w) − e^{− ∑_{j} A^{j}}(w)}′ = 0.

Thus it is natural to take the space H_{D}(H_{Q_{0}}(V_{L,q})) for a deformation of H^{∗}(ℙ^{N}).

Theorem 2.3.

H_{D}(H_{Q_{0}}(V_{L,q})) = ℂ[T]/(T^{N+1} − q^{N+1}).

Proof.

1) Computation of H_{Q_{0}}(V_{L,q}). By definition

Q_{0} = ∑_{n∈ℤ} A_{−n}^{i}Φ_{n}^{i}  (2.5)

Therefore,

[Q_{0}, Ψ_{0}^{j}] = A_{0}^{j}, [Q_{0}, G_{0}] = L_{0}.

These relations imply that

H_{Q_{0}}(V_{L,q}) = H_{Q_{0}}(∩_{j}KerA_{0}^{j} ∩ KerL_{0}).  (2.6)

It follows from 1.2 that the space ∩_{j}KerA_{0}^{j} ∩ KerL_{0} is a linear span of elements of the form:

Φ_{0}^{i_{1}} ⋯ Φ_{0}^{i_{m}}e^{∑_{i} n_{i}A^{i}}.

Formula (2.5) shows that the restriction of Q_{0} to this subspace is 0. Thus

H_{Q_{0}}(∩_{j}KerA_{0}^{j} ∩ KerL_{0}) = ∩_{j}KerA_{0}^{j} ∩ KerL_{0}.

The (−1)-st product makes this subspace a supercommutative algebra. In the same way as in 2.1 we get an isomorphism

∩_{j}KerA_{0}^{j} ∩ KerL_{0} = ℂ[x_{1}, ..., x_{N}, T; Φ_{1}, ..., Φ_{N}]/(T x_{1} ⋯ x_{N} − q^{N+1}), where Φ_{1}, ..., Φ_{N} are understood as grassman variables ([x_{i}, Φ_{j}] = [T, Φ_{j}] = 0, Φ_{i}Φ_{j} + Φ_{j}Φ_{i} = 0) and (T x_{1} ⋯ x_{N} − q^{N+1}) stands for the ideal generated by T x_{1} ⋯ x_{N} − q^{N+1}.

2) Computation of H_{D}(H_{Q_{0}}(V_{L,q})). In view of Step 1), we have to restrict D to

∩_{j}KerA_{0}^{j} ∩ KerL_{0}.

The isomorphism

∩_{j}KerA_{0}^{j} ∩ KerL_{0} = ℂ[x_{1}, ..., x_{N}, T; Φ_{1}, ..., Φ_{N}]/(T x_{1} ⋯ x_{N} − q^{N+1}), identifies D with ∑_{i}(x_{i} − T)∂/∂(Φ_{i}). Therefore, the complex

(ℂ[x_{1}, ..., x_{N}, T; Φ_{1}, ..., Φ_{N}]/(T x_{1} ⋯ x_{N} − q^{N+1}), D)

is simply the Koszul resolution of the algebra {ℂ[x_{1}, ..., x_{N}, T]/(T x_{1} ⋯ x_{N} − q^{N+1})}/(x_{1} − T, x_{2} − T, ..., x_{N} − T) associated with the sequence x_{1} − T, x_{2} − T, ..., x_{N} − T. This sequence is regular and we get at once

H_{D}(H_{Q_{0}}(V_{L,q})) = ℂ[T]/(T^{N+1} − q^{N+1}). □

It is easy to infer from Borisov’s proof of Theorem 1.3 that the element T = e^{−A^{1}−···−A^{N}} ∈ V_{L,q} is a cocycle representing the cohomology class proportional to that of a hyperplane in ℙ^{N}. This means that the deformation of H^{∗}(ℙ^{N}) we obtained coincides with the standard one, except that for some reason q happened to be raised to the N + 1.

2.4. Reduction to a single differential.

Of course it would be nicer to get H^{∗}(ℙ^{N}), or its deformation, as the cohomology of this or that vertex algebra with respect to a single differential rather than to compute a repeated cohomology.

Theorem 2.4.

H_{D+Q_{0}}(V_{L,q})) = ℂ[T]/(T^{N+1} − q^{N+1}).

It is no wonder, in view of Theorem 2.3, that this assertion is a result of computation of a certain spectral sequence. We shall use several spectral sequences arising in the following situation, which is slightly different from the standard one. Let

W = ⊕_{n=−∞}^{+∞}W^{n}

be a graded vector space with two commuting differentials

d_{1} : W^{n} → W^{n+1}, d_{2} : W^{n} → W^{n−1}.  (2.7)

There arise the total differential d = d_{1} + d_{2} and the cohomology H_{d_{1}+d_{2}}(W). Note that this cohomology group is not graded since d_{1} and d_{2} map in opposite directions. We can, however, introduce the filtration

W = ∪_{n}W^{≤n}, W^{≤n} = ⊕_{m=−∞}^{n}W^{m}.

Then

(d_{1} + d_{2})(W^{≤n}) ⊆ W^{≤(n+1)}

and there arises a filtration H_{d_{1}+d_{2}}(W)^{≤n} on the cohomology and the graded object GrH_{d_{1}+d_{2}}(W).

It is straightforward to define a spectral sequence

{E(W)_{r}^{n}, d^{(r)} : E(W)_{r}^{n} → E(W)_{r}^{n−2r+1}}, E(W)_{r+1}^{n} = H_{d^{(r)}}(E(W)_{r}^{n}), (2.8)

the first three terms being as follows:

E(W)_{0}^{n} = W^{n}, E(W)_{1}^{n} = H_{d_{1}}(W^{n}), E(W)_{2}^{n} = H_{d_{2}}(H_{d_{1}}(W^{n})),  (2.9)

where

H_{d_{1}}(W^{n}) = \frac{Ker\{d_{1} : W^{n} → W^{n+1}\}}{Im\{d_{1} : W^{n−1} → W^{n}\}},

H_{d_{2}}(H_{d_{1}}(W^{n})) = \frac{Ker\{d_{2} : H_{d_{1}}(W^{n}) → H_{d_{1}}(W^{n−1})\}}{Im\{d_{2} : H_{d_{1}}(W^{n+1}) → H_{d_{1}}(W^{n})\}}.

In the situation pertaining to Theorem 2.4 we take V_{L,q} for W, Q_{0} for d_{1}, and D for d_{2}. The space V_{L,q} is graded by fermionic charge; this grading is defined by letting the degree of Ψ_{j}^{i} be equal −1, the degree of Φ_{j}^{i} be equal 1, the degree of A_{j}^{i}, B_{j}^{i}, e^{α} be equal 0. By definition,

Q_{0}(V_{L,q}^{n}) ⊆ V_{L,q}^{n+1}, D(V_{L,q}^{n}) ⊆ V_{L,q}^{n−1}

and we get a spectral sequence {E(V_{L,q})_{r}^{n}, d^{(r)}}.

Observe that the grading by fermionic charge and the corresponding filtration are infinite in both directions. Therefore, the standard finiteness conditions that guarantee convergence of spectral sequences fail. Nevertheless the following lemma holds true.

Lemma 2.4. The spectral sequence {E(V_{L,q})_{r}^{n}, d^{(r)}} converges to H_{Q_{0}+D}(V_{L,q}) and collapses:

H_{D}(H_{Q_{0}}(V_{L,q})) = H_{Q+D}(V_{L,q}).

Lemma 2.4 combined with Theorem 2.3 gives Theorem 2.4 at once and it remains to prove Lemma 2.4.

Proof of Lemma 2.4. Introduce yet another grading of the space V_{L,q}^{n} as follows. Let α = (α_{1}, ..., α_{N+1}) be an element of the group ℤ^{N+1}. Let

V_{L,q}^{n}[α] = (∩_{i=1}^{N}Ker(A_{0}^{i} − α_{i}Id)) ∩ Ker(L_{0} − α_{N+1}Id).

Of course

V_{L,q}^{n} = ⊕_{α∈ℤ^{N+1}}V_{L,q}^{n}[α]

and both the differentials preserve this grading. Therefore all calculations can be carried out inside V_{L,q}^{n}[α] with a fixed α. Consider the following two cases.

1) α ≠ 0. In this case, as was observed in the beginnning of the proof of Theorem 2.3 (see e.g. (2.6)), H_{Q_{0}}(V_{L,q}[α]) = 0 and, therefore, E[α]_{1} = 0. It remains to show that H_{Q_{0}+D}(V_{L,q}[α]) = 0. Let x ∈ V_{L,q}[α]^{≤n} be a cocycle. This means that there is a “chain” of elements x_{i} ∈ V_{L,q}[α]^{n−2i}, i = 0, 1, ..., k so that

x = ∑_{i=0}^{k} x_{i},

and the following holds

Q_{0}(x_{0}) = 0, Q_{0}(x_{i+1}) + D(x_{i}) = 0, D(x_{k}) = 0, i = 0, ..., k − 1.  (2.10)

We now repeatedly use the condition H_{Q_{0}}(V_{L,q}[α]) = 0 and (2.10) to construct another chain y_{i} ∈ V_{L,q}[α]^{n−2i−1}, i ≥ 0, satisfying

Q_{0}(y_{0}) = x_{0}, Q_{0}(y_{i+1}) + D(y_{i}) = x_{i+1}.  (2.11)

Indeed, since Q_{0}(x_{0}) = 0, there is y_{0} ∈ V_{L,q}[α]^{n−1} so that Q_{0}(y_{0}) = x_{0}.

Since

Q_{0}(−D(y_{0}) + x_{1}) = DQ_{0}(y_{0}) + Q_{0}(x_{1}) = D(x_{0}) + Q_{0}(x_{1}) = 0,

there is y_{1} ∈ V_{L,q}[α]^{n−3} so that Q_{0}(y_{1}) + D(y_{0}) = x_{1}.

In general, having found y_{i} ∈ V_{L,q}[α]^{n−2i−1}, y_{i−1} ∈ V_{L,q}[α]^{n−2i+1} so that Q_{0}(y_{i})+ D(y_{i−1}) = x_{i}, we calculate as follows:

0 = D(0) = D(Q_{0}(y_{i}) + D(y_{i−1}) − x_{i}) = DQ_{0}(y_{i}) − D(x_{i}).

Due to (2.10), the last expression rewrites as DQ_{0}(y_{i}) + Q_{0}(x_{i+1}) and we get

−Q_{0}D(y_{i}) + Q_{0}(x_{i+1}) = 0.

Therefore, Q_{0}(D(y_{i}) − x_{i+1}) = 0 and there is y_{i+1} so that Q_{0}(y_{i+1}) = −D(y_{i}) + x_{i+1} as desired.

Formally, (2.11) means that

(D + Q_{0})(∑_{i=0}^{∞} y_{i}) = x

and what does not allow us to conclude immediately that x = ∑_{i=0}^{∞} x_{i} is a coboundary is that the sum ∑_{i=0}^{∞} y_{i} looks infinite. To complete case 1) it remains to show that y_{i} = 0 for all sufficiently large i. This is achieved by the following dimensional argument. Note that by construction

y_{i} ∈ ⊕_{|m_{j}|<ki}(S_{𝔥_{L}} ⊗ Λ_{𝔥_{L}}^{n−2i−1} ⊗ e^{∑_{j} m_{j}A^{j}+∑_{j} α_{j}B^{j}}),  (2.12)

where k is a number independent of i and n−2i−1 is the fermionic charge. Indeed, each application of D changes m_{j} by at most 1, Q_{0} preserves m_{j}, and the linear estimate of m_{j} follows; that the fermionic charge of y_{i} is n − 2i − 1 follows from the fact that the fermionic charge of Q (D resp.) is 1 (-1 resp.). On the other hand we have an explicit formula for L_{0} (see the beginning of 2.2), and this formula implies that the smallest eigenvalue of L_{0} restricted to Λ_{𝔥_{L}}^{n−2i−1} is nonnegative and grows faster than a polynomial of degree 2, say q(i), as i → +∞. The same formula gives

L_{0}e^{∑_{j} m_{j}A^{j}+∑_{j} α_{j}B^{j}} = ∑_{j} m_{j}α_{j}e^{∑_{j} m_{j}A^{j}+∑_{j} α_{j}B^{j}}.

Therefore, if y_{i} ≠ 0, then it is a sum of eigenvectors associated to eigenvalues of L_{0} greater or equal q(i) − (α_{1} + ⋯ α_{n})ki. Since this number tends to +∞ as i → +∞, we arrive at contradiction with the assumption L_{0}y_{i} = α_{N+1} if i is sufficiently large. Hence, y_{i} = 0 for all sufficiently large i, each cocycle is a coboundary, and case 1) is accomplished.

2) α = 0. As we saw in the beginning of the proof of Theorem 2.3, the restriction of Q_{0} to V_{L,q}[0] is 0; by definition, the complex (V_{L,q}[0], D + Q_{0}) is equal to (E(V_{L,q})[0]_{1}, d^{(1)}) and the cohomology of the latter were computed in step 2) of the proof of Theorem 2.3. □

2.5. The vertex algebra H_{D}(V_{L,q}) and a computation of H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}).

In this section we prove the following two theorems.

Theorem 2.5A If q ≠ 0, then H_{D}(V_{L,q}) equals the quantum cohomology of ℙ^{N}.

Theorem 2.5B The natural embedding of sheaves ([MSV], see also (2.20) below)

Ω_{ℙ^{N}}^{∗} ↪ Ω_{ℙ^{N}}^{ch}

provides an isomorphism

H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{∗}) ⟶^{∼} H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}), 0 < i < N,

where Ω_{ℙ^{N}}^{∗} is the sheaf of all differential forms.

Recall the previously known results on the cohomology of Ω_{ℙ^{N}}^{ch}. Ω_{ℙ^{N}}^{ch} is a sheaf of ŝl_{N+1}-modules [MS1], see also 2.5.2. In particular, if U_{0} = ℂ^{N} ⊂ ℙ^{N} is a big cell, then Γ(U_{0}, Ω_{ℙ^{N}}^{ch}) is a generalized Wakimoto module over ŝl_{N+1} introduced in [FF]. We proved in [MS1] that

H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) = Γ(U_{0}, Ω_{ℙ^{N}}^{ch})^{int},  (2.13)

where Γ(U_{0}, Ω_{ℙ^{N}}^{ch})^{int} stands for the maximal sl_{N+1}-integrable submodule of Γ(U_{0}, Ω_{ℙ^{N}}^{ch}).

On the other hand, it follows from the chiral Serre duality [MS2] that there is a canonical isomorphism

H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) ⟶^{∼} H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})^{d},  (2.14)

where ^{d} stands for the restricted dual.

Unfortunately, little is known about the structure of Γ(U_{0}, Ω_{ℙ^{N}}^{ch}) and Γ(U_{0}, Ω_{ℙ^{N}}^{ch})^{int} if N > 1; see, however, [MS1] for the case of N = 1. Otherwise, Theorem 2.5 and (2.13-14) give a complete description of H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}). Moreover, these results combined with a result of [BL] lead to a functional equation satisfied by the the supercharacter of H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}), see 2.5.3.

The proofs of Theorems 2.5A and B are contained in 2.5.2. In 2.5.1 we collect some well-known material in order to place these results in the proper context and to formulate (2.18-19), two well-known assertions needed in 2.5.2.

2.5.1 A vertex algebra structure on a vector space V comprises a countable family of multiplications:

(n) : V ⊗ V → V, x ⊗ y ↦ x_{(n)}y, n ∈ ℤ,

a map

T : V → V,

and a vacuum vector

1 ∈ V.

These data satisfy the Borcherds identities which imply, in particular, that T and x_{(0)}, x ∈ V, are derivations of the n-th product for all n. Thus,

[T, y_{(j)}] = (T y)_{(j)}, [x_{(0)}, y_{(j)}] = (x_{(0)}y)_{(j)}.  (2.15)

In the case of the vertex algebra V_{L,q}, the n-th multiplication was defined in the end of 1.2, 1 equals e^{0}, and T will be defined below.

Call V commutative (or holomorphic, see [K] 1.4) if _{(n)} = 0 for all n ≥ 0. If V is commutative, then the (-1)-st multiplication gives it the structure of a commutative superalgebra with derivation T, and the functor arising in this way is an equivalence of the category of commutative vertex algebras and the category of commutative superalgebras with derivation, see again [K] 1.4.

If d_{x} : V → V is a differential (d^{2} = 0), then the cohomology H_{d_{x}}(V) arises. We assert that

d_{x} = x_{(0)} for some x ∈ V ⇒ H_{d_{x}}(V) is a vertex algebra,  (2.16)

since all products on V descend to H_{d_{X}} (V) due to (2.15).

All vertex algebras we are concerned with are conformal. This means that there is a Virasoro field L(z) = ∑_{i} L_{i}z^{−i−2}, L_{i} ∈ End(V), such that L_{i} satisfy the Virasoro commutation relations, T = L_{−1}, L_{0} is diagonalizable, and L(z) is the field attached to the state L_{−2}1 ∈ V. The formula at the beginning of 2.2 shows that V_{L,q} is a conformal vertex algebra, the state L_{−2}1 being equal to ∑_{i}(B_{−1}^{i}A_{−1}^{i} +Φ_{−1}^{i}Ψ_{−1}^{i})e^{0}.

The eigenvalues of L_{0} are called conformal weights. Hence a conformal vertex algebra V is graded by conformal weights, V = ⊕_{n}V_{n}, and in the case of V = V_{L,q} this grading (but not the name) has already been used in the proofs of Theorems 2.3 and 2.4.

Returning to the cohomology vertex algebra H_{d_{x}}(V) in the case when V is conformal and x is an eigenvector of L_{0}, we see that

L_{−2}1 ∈ Kerd_{x} ⇒ H_{d_{x}}(V) is conformal,  (2.17)
L_{−2}1 ∈ Imd_{x} ⇒ H_{d_{x}}(V) is commutative.  (2.18)

Indeed, if L_{−2}1 ∈ Kerd_{X}, then the operators L_{i} ∈ End(V) descend to H_{d_{X}} (V) due to (2.15). If, in addition, L_{−2}1 = d_{x}(y), then all L_{i}’s act on H_{d_{x}}(V) trivially again due to (2.15). Hence L_{0} acts on H_{d_{X}} (V) trivially, each element of H_{d_{X}} (V) is represented by a cocycle of conformal weight 0, and the n-th product on H_{d_{X}} (V) vanishes unless n = −1.

If x ∈ V_{1}, then d_{x}(V_{n}) ⊂ V_{n} for all n, and (2.18) can be sharpened as follows:

L_{−2}1 ∈ Imd_{x} and x ∈ V_{−1} ⇒ H_{d_{X}} (V) = H_{d_{X}} (V_{0}).  (2.19)

In our previous work ([MSV], [MS1], [MS2]) we have dealt with conformal vertex algebras having the following properties: all conformal weights are nonnegative; the conformal weight 0 component is a finitely generated supercommutative ring and the corresponding multiplication coincides with the restriction of the (-1)st multiplication. For example, Ω_{X}^{ch} is a sheaf of such vertex algebras over a smooth manifold X: the conformal weight 0 component of Γ(U, Ω_{X}^{ch}) is the algebra of differential forms over U ⊂ X. In other words, there is a natural embedding

Ω_{X}^{∗} ⟶^{∼} Ω_{X,0}^{ch} ⊂ Ω_{X}^{ch},  (2.20)

and it is this embedding that was invoked in Theorem 2.5B.

H^{∗}(X, Ω_{ℙ^{N}}^{ch}) is also a vertex algebra of this kind because its conformal weight 0 component equals the cohomology algebra H^{∗}(X). It is, therefore, natural to ask if there is a conformal vertex algebra with nonnegative conformal weights so that the (-1)-st multiplication identifies its conformal weight 0 component with the quantum cohomology of X.

The quantum cohomology itself is one such vertex algebra due to the equivalence of categories reviewed above. A more appealing possibility seems to be provided by H_{D}(V_{L,q}): it is a vertex algebra due to (2.16) because (1.1) is equivalent to

D = ∑_{i=1}^{N}(Ψ_{−1}^{i}(e^{A^{i}} − e^{− ∑_{j} A^{j}}))_{(0)},  (2.21)

and it is conformal because, as one easily checks, D(L_{−2}e^{0}) = 0.

Even though Theorem 2.5A says that in this way we do not get anything new either, it allows us to observe a curious phenomenon: H_{D}(V_{L,q}), q ∈ ℂ, is a family of vertex algebras over ℂ with fiber that equals H^{∗}(ℙ^{N}) over any non-zero point and blows up to the non-commutative infinite dimensional vertex algebra H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) over 0 ∈ ℂ.

Rather unexpectedly, Theorem 2.5B turns out to be a by-product of the proof of Theorem 2.5A.

2.5.2 Proof Theorems 2.5A and B.

By definition, the complex (V_{L,q}, D) is the constant vector space V_{L} with differential D polynomially depending on q ∈ ℂ. To make this more precise, observe that V_{L} is graded by the function ht defined in 2.1:

V_{L} = ⊕_{n≥0}V_{L}^{n},  (2.22)

where V_{L}^{n} is a linear span of x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}} with ht(∑_{i} n_{i}A^{i}) = n. The differential D then breaks in a sum

D = d_{+} + q^{N}d_{−},  (2.23a)

so that

d_{+}(V_{L}^{n}) ⊂ V_{L}^{n+1},  (2.23b)
d_{−}(V_{L}^{n}) ⊂ V_{L}^{n−N},  (2.23c)

and

(d_{+})^{2} = (d_{−})^{2} = [d_{+}, d_{−}] = 0.  (2.23d)

Again by definition, the complex (V_{L}, d_{+}) coincides with Borisov’s complex (V_{L}^{Σ}, D). It follows from formulas (2.23a-d) and Theorem 1.3 that there is a spectral sequence of the same type as (2.8), the 1st term and the 1st differential being as follows

E_{1} = H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})  (2.24)
d_{1} = q^{N}d_{−} : H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) → H^{∗}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}),
d_{−}(H^{n}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})) ⊂ H^{n−N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}).  (2.25)

Simply because dimℙ^{N} = N, the 2nd term equals

\frac{H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})}{Im\{d_{−} : H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) → H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})\}}

⊕Ker{d_{−} : H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) → H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})} ⊕ ⊕_{i=1}^{N−1}H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}),

and all higher differentials vanish. An argument similar to (and simpler than) the one used in the proof of Lemma 2.4 shows that this spectral sequence converges to H_{D}(V_{L,q}). Therefore

H_{D}(V_{L,q})

= \frac{H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})}{Im\{d_{−} : H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) → H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})\}}

⊕Ker{d_{−} : H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) → H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})} ⊕ ⊕_{i=1}^{N−1}H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}).  (2.26)

Lemma 2.6. There is y ∈ H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) such that d_{−}(y) ∈ H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) equals the Virasoro element L_{−2}e^{0}.

This lemma allows us to complete the proof of Theorems 2.5A and B instantaneously. Our differentials come from elements of V_{L} of conformal weight 1, see (2.21); hence, due to Lemma 2.6, (2.18) and (2.19) apply: H_{D}(V_{L,q}) equals H_{D}((V_{L,q})_{0}), which is known (Theorem 2.4) to be equal to the quantum cohomology. In particular, as follows from (2.26),

H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) = H^{i}(ℙ^{N}, (Ω_{ℙ^{N}}^{ch})_{0}), 0 < i < N,

the latter space being canoncally isomorphic to H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{∗}) due to (2.20). Thus it remains to prove Lemma 2.6.

Proof of Lemma 2.6 To find an appropriate y ∈ H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) and calculate d_{−}(y) we need to take a plunge in [MSV,B].

Let x^{0} : x^{1} : ⋯: x^{N} be homogeneous coordinates on ℙ^{N} and b^{i} = x^{i}/x^{0}. Consider the N-dimensional torus 𝕋^{N} = Specℂ[(b^{1})^{±1}, ..., (b^{N})^{±1}] ⊂ ℙ^{N}.

We shall need the following facts about the sheaf Ω_{ℙ^{N}}^{ch}.

First,

Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch}) = ℂ[(b_{0}^{i})^{±1}, b_{j−1}^{i}, a_{j−1}^{i}; ϕ_{j}^{i}, ψ_{j−1}^{i}; 1 ≤ i ≤ N, j ≤ 0],  (2.27)

where b_{j}^{i}, a_{j−1}^{i} are even, ϕ_{j}^{i}, ψ_{j−1}^{i} odd.

By letting degx_{j}^{i} = −j, x = b, a, ϕ or ψ, we recover the grading by conformal weight. By letting degb_{j}^{i} = dega_{j}^{i} = 0, degϕ_{j}^{i} = 1, degψ_{j}^{i} = −1 we get another grading, that by fermionic charge. Therefore, Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch}) is bigraded and this bigrading extends to the entire sheaf:

Ω_{ℙ^{N}}^{ch} = ⊕_{m=−∞}^{+∞} ⊕_{n=0}^{+∞} Ω_{ℙ^{N} ,n}^{ch,m}.  (2.28)

Next, we discuss “tensor” properties of Ω_{ℙ^{N}}^{ch}. We identify Γ(𝕋^{N}, Ω_{ℙ^{N}}^{∗}) with Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch})_{0} by identifying b^{i} with b_{0}^{i} and db^{i} with ϕ_{0}^{i}. This identification extends to the isomorphism (2.20).

The structure of higher conformal weight components is more complicated, but here is what we can say about the component of conformal weight 1. Consider the following elements of Γ(𝕋^{N}, Ω_{ℙ^{N} ,1}^{ch,0}):

e_{ij} = b_{0}^{i−1}a_{−1}^{j−1} + ϕ_{0}^{i−1}ψ_{−1}^{j−1}, i, j ≠ 1,  (2.29a)
e_{1j} = a_{−1}^{j−1}, j ≠ 1  (2.29b)

e_{i1} = − ∑_{l=1}^{N} b_{0}^{i−1}b_{0}^{l}a_{−1}^{l} − ∑_{l=1}^{N} b_{0}^{i−1}ϕ_{0}^{l}ψ_{−1}^{l}

− ∑_{l=1}^{N} b_{0}^{l}ϕ_{0}^{i−1}ψ_{−1}^{l}, i ≠ 1.  (2.29c)

It was checked in [MS1] III that these elements come from H^{0}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,0}) ⊂ Γ(𝕋^{N}, Ω_{ℙ^{N} ,1}^{ch,0}) and that the Fourier components of the corresponding fields span a Lie subalgebra of End(Ω_{ℙ^{N}}^{ch}) isomorphic to the loop algebra Lsl_{N+1} = sl_{N+1} ⊗ℂ[t, t^{−1}]. Therefore,

Lsl_{N+1} ↪ End(Ω_{ℙ^{N}}^{ch}),  (2.30a)

so that

sl_{N+1} ↪ H^{0}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,0}), E_{ij} ↦ e_{ij}.  (2.30b)

is a morphism of sl_{N+1}-modules, where E_{ij}, i ≠ j, 1 ≤ i, j ≤ N + 1 are the standard generators of sl_{N+1}, and sl_{N+1} operates on H^{0}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,0}) by means of

the composite map sl_{N+1} ⟶^{∼} sl_{N+1} ⊗ 1 ⊂ ŝl_{N+1}

Elements (2.29a-c) have fermionic charge 0. For the fermionic charge N + 1 component there is an isomorphism:

Ω_{ℙ^{N}}^{1} ⊗ Ω_{ℙ^{N}}^{N} ⟶^{∼} Ω_{ℙ^{N} ,1}^{ch,N+1}.  (2.31)

Over 𝕋^{N} it is defined by the assignment

f_{i}(b^{1}, ..., b^{N})db^{i} ⊗ (db^{1} ∧ db^{2} ∧ ⋯ ∧ db^{N}) ↦ f_{i}(b_{0}^{1}, ..., b_{0}^{N})ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N},
f_{i}(b^{1}, ..., b^{N}) ⊂ ℂ[(b^{1})^{±1}, ..., (b^{N})^{±1}].

Isomorphism (2.31) induces the isomorphism

H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{1} ⊗ Ω_{ℙ^{N}}^{N}) ⟶^{∼} H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,N+1}).  (2.32)

By the Serre duality,

H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{1} ⊗ Ω_{ℙ^{N}}^{N}) ⟶^{∼} H^{0}(ℙ^{N}, T )^{∗},  (2.33)

where T is the tangent sheaf. The Lie algebra sl_{N+1} operates on ℙ^{N}, therefore there arises the map sl_{N+1} → H^{0}(ℙ^{N}, T )^{∗}, which is well known to be an isomorphism. Hence, (2.33) combined with (2.32) rewrites as follows

H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,N+1}) ⟶^{∼} sl_{N+1}.  (2.34)

This map is an isomorphism of sl_{N+1}-modules, and it is not hard to find a Cech cochain representing a highest weight vector of H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,N+1})_{1}, that is, a nonzero vector v satisfying

E_{ij}v = 0, i < j.  (2.35)

If we denote by U_{i} the open subset of ℙ^{N} satisfying x_{i} ≠ 0, then {U_{0}, ..., U_{N}} is an affine cover of ℙ^{N}, so that 𝕋^{N} = U_{0} ∩ U_{1} ∩ ... ∩ U_{N}. The N-th term of the Cech complex equals, therefore, Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch}), and it is an exercise to check that

(b_{0}^{1})^{−1}(b_{0}^{2})^{−1} ⋯ (b_{0}^{N−1})^{−1}(b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N}  (2.36)

represents a highest weight vector of H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,N+1}).

Observe that another copy of sl_{N+1} we have discovered earlier has e_{1N+1} for its highest weight vector, see (2.29b,2.30). The assertion crucial for our proof is that d_{−} sends one highest weight vector to another:

d_{−}((b_{0}^{1})^{−1}b_{0}^{2})^{−1} ⋯ b_{0}^{N−1})^{−1}b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N}) = e_{1N+1} ∈ H^{0}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,0}).

(2.37)

Lemma 2.6 follows from (2.37) easily. To explain this implication we have to digress on elementary representation theory of Lsl_{N+1}.

Consider the decomposition

Lsl_{N+1} = L_{−}sl_{N+1} ⊕ sl_{N+1} ⊕ L_{+}sl_{N+1},

where

L_{±}sl_{N+1} = t^{±1}ℂ[t^{±1}].

Let L_{≥}sl_{N+1} = sl_{N+1} ⊕L_{+}sl_{N+1}. Any sl_{N+1}-module becomes an L_{≥}sl_{N+1}-module if the action of sl_{N+1} is extended to the entire L_{≥}sl_{N+1} by the requirement L_{+}sl_{N+1} ↦

0. Therefore for any sl_{N+1}-module U there arises the Weyl module, denoted 𝕎_{U} and defined as follows:

𝕎_{U} = Ind_{L_{≥}sl_{N+1}}^{Lsl_{N+1}}U.

The Weyl module induced from the trivial representation, 𝕎_{ℂ}, is well-known to be a conformal vertex algebra due to [FZ], see also [K] 4.7. Therefore, it has vacuum vector, 1, and Virasoro element, L_{−2}^{aff}1. Other Weyl modules are modules over 𝕎_{ℂ}. This means, in particular, that Fourier components L_{i}^{aff} act on Weyl modules. The action of L_{0}^{aff} is diagonalizable and defines a grading on each Weyl module also called the grading by conformal weight. The aim of this digression was to formulate the following well-known (and easily derived from the Kac-Kazhdan equations) assertion:

I ⊂ 𝕎_{sl_{N+1}} is a proper Lsl_{N+1}-submodule ⇒ I ∩ (𝕎_{sl_{N+1}})_{2} = {0}, (2.38)

where 𝕎_{sl_{N+1}} stands for the Weyl module induced from the adjoint representation, and (𝕎_{sl_{N+1}})_{2} is its conformal weight 2 component.

Return to the proof of Lemma 2.6. Due to (2.30a), H^{i}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,m}) is an Lsl_{N+1}- module for all i and m. The component H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,N+1}) is an L_{≥}sl_{N+1}-module isomorphic to sl_{N+1}, see (2.34), on which L_{+}sl_{N+1} acts trivially because H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,m}^{ch,N+1}) = 0 for all m < 1. By the universality property of induced modules, 𝕎_{sl_{N+1}} maps onto the Lsl_{N+1}-submodule of H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,N+1}) generated by H^{N}(ℙ^{N}, Ω_{ℙ^{N} ,1}^{ch,N+1}).

Denote this submodule 𝕎̂_{sl_{N+1}}.

Similarly, H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,0}) is an Lsl_{N+1}-module, and the Lsl_{N+1}-submodule generated by 1 is a quotient of 𝕎_{ℂ}. This quotient contains yet another submodule,

the one generated by sl_{N+1} ↪ H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch})_{1}, see (2.30b), to be denoted ˆ 𝕎̂_{sl_{N+1}}. This submodule, again for the same reason, is a quotient of 𝕎_{sl_{N+1}}. By definition,

the above mentioned Virasoro element L_{−2}^{aff}1 belongs to (ˆ𝕎̂_{sl_{N+1}})_{2}

We are practically done. It is easy to derive from [B] that

d_{−} : H^{N}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,N+1}) → H^{0}(ℙ^{N}, Ω_{ℙ^{N}}^{ch,0})

is an Lsl_{N+1}-morphism. Equality (2.37) then means that d_{−}(𝕎̂_{sl_{N+1}}) ⊂ ˆ 𝕎̂_{sl_{N+1}} is non-zero, and is therefore a quotient of 𝕎_{sl_{N+1}} by a proper submodule. Due to (2.38)

(d_{−}(𝕎̂_{sl_{N+1}}))_{2} = (𝕎_{sl_{N+1}})_{2} = (ˆ𝕎̂_{sl_{N+1}})_{2}.

Hence L_{−2}^{aff}1 ∈ d_{−}(𝕎̂_{sl_{N+1}}). To complete the proof of Lemma 2.6 it remains to check that the affine Virasoro element, L_{−2}^{aff}1, coincides with L_{−2}1 and this is easy.

Finally we have to prove (2.37). The difficulty with computation of

d_{−}((b_{0}^{1})^{−1}(b_{0}^{2})^{−1} ⋯ (b_{0}^{N−1})^{−1}(b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N})

lies in that the operator d_{−} is defined in terms of the vertex algebra V_{L}, while

(b_{0}^{1})^{−1}(b_{0}^{2})^{−1} ⋯ (b_{0}^{N−1})^{−1}(b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N}

is an element of Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch}). The vertex algebra embedding

Γ(𝕋^{N}, Ω_{ℙ^{N}}^{ch}) ↪ V_{L},

an important ingredient of Borisov’s proof of Theorem 1.3, is determined by the rules

(b_{0}^{i})^{±} ↦ e^{±B^{i}}, ϕ_{0}^{i} ↦ Φ_{0}^{i}e^{B^{i}}, ψ_{−1}^{i} ↦ Ψ_{−1}^{i}e^{−B^{i}},  (2.39a)
a_{−1}^{i} ↦ A_{−1}^{i}e^{−B^{i}} − Φ_{0}^{i}Ψ_{−1}^{i}e^{−B^{i}},  (2.39b)
x ↦ X ⇒ L_{−1}x ↦ L_{−1}X,  (2.39c)
x ↦ X, y ↦ Y ⇒ x_{(−1)}y ↦ X_{(−1)}Y.  (2.39d)

These rules imply

(b_{0}^{1})^{−1}(b_{0}^{2})^{−1} ⋯ (b_{0}^{N−1})^{−1}(b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N} ↦ e^{−B^{N}} Φ_{−1}^{N}Φ_{0}^{1}Φ_{0}^{2} ⋯ Φ_{0}^{N}.

It follows from Borisov’s proof of Theorem 1.3 that an element of V_{L} representing the class of

(b_{0}^{1})^{−1}(b_{0}^{2})^{−1} ⋯ (b_{0}^{N−1})^{−1}(b_{0}^{N})^{−3}ϕ_{−1}^{i}ϕ_{0}^{1}ϕ_{0}^{2} ⋯ ϕ_{0}^{N}

can be chosen to be equal to

(Ψ_{−1}^{N}e^{A^{N}})_{(0)}(Ψ_{−1}^{N−1}e^{A^{N−1}})_{(0)} ⋯ (Ψ_{−1}^{1}e^{A^{1}})_{(0)}e^{−B^{N}} Φ_{−1}^{N}Φ_{0}^{1}Φ_{0}^{2} ⋯ Φ_{0}^{N}.

The formulas of 1.1-2 imply that (Ψ_{−1}^{i}e^{A^{i}})_{(0)}, 1 ≤ i ≤ N − 1, simply erases Φ_{0}^{i}. Hence

(Ψ_{−1}^{N−1}e^{A^{N}})_{(0)}(Ψ_{−1}^{N−2}e^{A^{N−1}})_{(0)} ⋯ (Ψ_{−1}^{1}e^{A^{1}})_{(0)}e^{−B^{N}} Φ_{−1}^{N}Φ_{0}^{1}Φ_{0}^{2} ⋯ Φ_{0}^{N}
= Φ_{−1}^{N}Φ_{0}^{N}e^{−B^{N} +A^{1}+A^{2}+···A^{N−1}}.

The calculation of the last operation is a little more tedious, but also straightforward; the result is this:

(Ψ_{−1}^{N}e^{A^{N}})_{(0)}(Φ_{−1}^{N}Φ_{0}^{N}e^{−B^{N} +A^{1}+A^{2}+···A^{N−1}})
(Ψ_{−1}^{N}Φ_{−1}^{N}Φ_{0}^{N} − Φ_{−1}^{N}A_{−1}^{N} + \frac{1}{2}Φ_{0}^{N}A_{−2}^{N} + \frac{1}{2}Φ_{0}^{N}(A_{−1}^{N})^{2})e^{−B^{N} +A^{1}+A^{2}+···A^{N}} . (2.40)

To complete our calculation we have to apply d_{−} to this element. Observe that this element comes from the interior of the cone spanned by A^{1}, ..., A^{N} and has height N. It follows from the definition of the spectral sequence and (1.1) or (2.21) that on this element d_{−} equals

−((Ψ_{−1}^{1} + Ψ_{−1}^{2} + ⋯ + Ψ_{−1}^{N})e^{−A^{1}−A^{2}−···−A^{N}})_{(0)}.

Indeed, it is precisely the component of Borisov’s differential (1.1) that decreases the height of the element (2.40). (By the way, it decreases it by N, which explains the assertion (2.23c).) Another calculation similar to those performed shows that

−((Ψ_{−1}^{1} + Ψ_{−1}^{2} + ⋯ + Ψ_{−1}^{N})e^{−A^{1}−A^{2}−···−A^{N}})_{(0)}.

sends the element (2.40) to

A_{−1}^{N}e^{−B^{N}} − Φ_{0}^{N}Ψ_{−1}^{N}e^{−B^{N}} .

According to (2.39b), the latter element corresponds to a_{−1}^{N} and hence to e_{1N+1}, see (2.29b), as desired. □

2.5.3. The sheaf Ω_{ℙ^{N}}^{ch} is bigraded due to (2.28). Therefore to each cohomology group H^{a}(ℙ^{N}, Ω_{ℙ^{N}}^{ch}) we can attach the following shifted supercharacter

sch_{ℙ^{N}}^{a} (y, q) = y^{−N/2} ∑_{j,b}(−1)^{a+b}dimH^{a}(ℙ^{N}, Ω_{ℙ^{N} ,j}^{ch,b})y^{b}q^{j}  (2.41)

The entire sheaf is characterized by its Euler supercharacter

χ_{ℙ^{N}} (y, q) = ∑_{a=0}^{N} sch^{a}(y, q).  (2.42)

Borisov and Libgober proved in [BL] that

χ_{ℙ^{N}} (y, q) = Ell_{ℙ^{N}} (y, q),  (2.43)

where Ell_{ℙ^{N}} (y, q) is the elliptic genus of ℙ^{N}; the definition of and an explicit formula for Ell_{ℙ^{N}} (y, q) can also be found in [BL].

Thanks to Theorem 2.5B, the intermediate supercharacters are known:

sch_{ℙ^{N}}^{a} (y, q) = y^{−N/2+a}, 0 < a < N.  (2.44)

The chiral Poincaré duality [MS2] implies that:

χ^{N}(y, q) = χ^{0}(y^{−1}, q).  (2.45)

Substituting (2.44, 45) in (2.43) we get following identity

χ^{0}(y, q) + χ^{0}(y^{−1}, q) = Ell_{ℙ^{N}} (y, q) − \frac{y^{(N−1)/2} − y^{−(N−1)/2}}{y^{1/2} − y^{−1/2}} .  (2.46)

§3. Deforming cohomology algebras of hypersurfaces in projective spaces

Let L → ℙ^{N} be a degree −n < 0 line bundle, L^{∗} → ℙ^{N} its dual, s ∈ Γ(ℙ^{N}, L^{∗}) a global section so that its zero locus Z(s) ⊂ ℙ^{N} is a smooth hypersurface. The way Borisov calculates the cohomology of the chiral de Rham complex over Z(s) is as follows.

Extend the lattice (L, (., .)) introduced in 1.1 to the lattice (L̂, (., .)) so that

L̂ = L ⊕ ℤA^{u} ⊕ ℤB^{u}, (A^{u}, B^{u}) = 1, (A^{u}, L) = 0, (B^{u}, L) = 0.

There arises the corresponding lattice vertex algebra VL̂. Observe that any subset L′ ⊂ L̂ closed under addition gives rise to the vertex subalgebra V_{L′} ⊂ V_{L} generated by 𝔥L̂ and ClL̂ from the highest weight vectors e^{β}, β ∈ L′; see 1.1-1.2. In our geometric situation let L̂_{n} be the span of B^{i} (i = 1, ..., N), B^{u} with arbitrary integral coefficients and A^{i} (i = 1, ..., N), A^{u}, nA^{u}−A^{1}−⋯−A^{N} with nonnegative integral coefficients.

The vertex algebra VL̂_{n} affords a degeneration, V Σ ; L̂_{n}, and includes in a family, V_{L̂_{n},q}, q ∈ ℂ, in the same way the algebra V_{L} did, see 1.3, 2.1. To construct V Σ ; L̂_{n},

consider the following N + 1 elements of L̂ : ξ_{1} = A^{1}, ξ_{2} = A^{2}, ..., ξ_{N} = A^{N}, ξ_{N+1} = nA^{u} − A^{1} − A^{2} − ⋯ − A^{N}. Define the cone Δ_{i} to be the set of all non-negative integral linear combinations of the elements ξ_{1}, ..., ξ_{i−1}, ξ_{i+1}, ..., ξ_{N+1}, A^{u} and let Σ = {Δ_{1}, ..., Δ_{N+1}}. The vertex algebra V Σ ; L̂_{n} is now defined by repeating word for word the definition of V_{L}^{Σ} in 1.3.

Similarly, the family V_{L̂_{n},q}, q ≠ 0, is defined by repeating word for word the definition of V_{L,q} in 2.1. This family extends “analytically” to q = 0 if n ≤ N + 1 and we again obtain an isomorphism

V_{L,0} = V Σ ; L̂_{n} if n < N + 1.  (3.1)

(The condition n < N + 1 will be clarified below.)

Borisov’s differential is as follows:

D = ∫ {∑_{i=1}^{N} Ψ^{i}(z)(e^{A^{i}} − e^{nA^{u}−∑_{j} A^{j}})(z) + Ψ^{u}(z)(ne^{nA^{u}−∑_{j} A^{j}} − e^{A^{u}})(z)}. (3.2a)

(For the future use let us note that the right hand side of this equality can be rewritten as a sum over lattice points:

D = ∫ { ∑_{i=1}^{N+1} Ψ^{ξ_{i}}(z)e^{ξ_{i}}(z) − Ψ^{u}(z)e^{A^{u}}(z)},  (3.2b)

where Ψ^{ξ_{i}} = Ψ^{i} (i ≤ N) and Ψ^{ξ_{N+1}} = nΨ^{u} − ∑_{j} Ψ^{j}.)

It is obvious that D ∈ End(V_{L̂_{n},q}) and D^{2} = 0; therefore there arise the cohomology groups H_{D}(V_{L̂_{n},q}) and H_{D}(V Σ ; L̂_{n}) = H_{D}(V_{L̂_{n},0}).

Theorem 3.1. ([B])

H_{D}(V Σ ; L̂_{n}) = H^{∗}(L, Ω_{L}^{ch}).

Borisov proposes to calculate the chiral de Rham complex over the hypersurface Z(s) ⊂ ℙ^{N} by means of a certain Koszul-type resolution of the complex Ω_{L}^{ch}. The combinatorial data that determine s ∈ Γ(ℙ^{N}, L^{∗}) consists of a finite set

Δ^{∗} = {β = B^{u} + ∑_{j=1}^{N} n_{j}B^{j} s.t. (β, ξ_{i}) ≥ 0, i = 1, 2, ..., N + 1},  (3.3)

and a function

g : Δ^{∗} → ℤ_{≥}.  (3.4)

Define

K_{g} = ∑_{β∈Δ^{∗}} ∫ g(β)Φ^{β}(z)e^{β}(z),  (3.5)

where Φ^{β} = Φ^{u} + ∑_{j} n_{j}Φ^{j} provided β = B^{u} + ∑_{j} n_{j}B^{j}. It is easy to see that

K_{g} ∈ End(V_{L̂_{n},q}), K_{g}^{2} = 0, [K_{g}, D] = 0.

Therefore, there arise the cohomology groups H_{D+K_{g}}(V_{L̂_{n},q}) and H_{D+K_{g}}(V Σ ; L̂_{n}) = H_{D+K_{g}}(V_{L̂_{n},0}).

Theorem 3.2. ([B])

H_{D+K_{g}}(V Σ ; L̂_{n}) = H^{∗}(Z(s), Ω_{Z(s)}^{ch}).

All the vertex algebras in sight being topological (see the beginning of 2.2), Theorem 3.2 and the main result of [MSV] give

H^{∗}(Z(s)) = H_{Q_{0}}(H_{D+K_{g}}(V Σ ; L̂_{n})),  (3.6a)

or, equivalently,

H^{∗}(Z(s)) = H_{D+K_{g}}(V Σ ; L̂_{n})_{0},  (3.6b)

where H_{D+K_{g}}(V Σ ; L̂_{n})_{0} stands for the kernel of L_{0}.

This prompts the following

Conjecture 3.3. If n < N, then the algebra H_{D+K_{g}}(V_{L̂_{n},q})_{0} is isomorphic to the quantum cohomology algebra of Z(s).

Unfortunately we do not have a proof of this conjecture; we cannot even prove that H_{D+K_{g}}(V_{L̂_{n},q})_{0} is a deformation of H^{∗}(Z(s)). What we know is collected in the following Lemma 3.4. (i) The element e^{nA^{u}−∑_{j} A^{j}} satisfies

(D + K_{g})(e^{nA^{u}−∑_{j} A^{j}}) = 0,

and, therefore, determines an element of H_{D+K_{g}}(V_{L̂_{n},q})_{0} for all q. If q = 0, then this element, considered as an element of H^{∗}(Z(s)) (see (3.6b)), is proportional to the cohomology class of a hyperplane section.

(ii) Due to (i), e^{nA^{u}−∑_{j} A^{j}} determines an element of H_{D+K_{g}}(V_{L̂_{n},q})_{0} to be denoted x. It satisfies the following relation:

x^{N+1} − q^{N+1−n}n^{n}x^{n} = 0.  (3.7)

(iii) If Z(s) is a hyperplane (i.e. n = 1), then Conjecture 3.3 is correct.

(iv) If Z(s) is the image of the embedding

ℙ^{1} × ℙ^{1} → ℙ^{3}, (u : v), (w : t) ↦ (uw : ut : vw : vt),

then H_{D+K_{g}}(V_{L̂_{2},q})_{0} is isomorphic to ℂ[x, y]/(x^{2} −1, y^{2} −1). Hence Conjecture 3.3 is true in this case.

Proof of Lemma 3.4.

The first part of assertion (i) is a result of the obvious calculation using the formulas of 1.1-1.2. The fact that at q = 0 the element e^{nA^{u}−∑_{j} A^{j}} is proportional to the cohomology class of a hyperplane section follows from Borisov’s proof of Theorem 3.2; this observation is completely analogous to the one made in the end of 2.3.

(ii) By definition (as in the end of 2.1), the subalgebra of V_{L̂_{n},q} generated by e^{A^{i}} (i = 1, ..., N), e^{A^{u}}, and Φ_{0}^{i} (i = 1, ..., N), Φ_{0}^{u} is isomorphic to

ℂ[x_{1}, ..., x_{N}, T, u; Φ_{1}, ..., Φ_{N}, Φ_{u}]/(x_{1}x_{2} ⋯ x_{N}T − q^{N+1−n}u^{n}),

and the restriction of D to this supercommutative algebra coincides with the Koszul differential associated with the regular sequence x_{i} − T, u − nT (i = 1, ..., N). The relation (3.7) follows at once. By the way, the appearance of N + 1 − n as a power of q in (3.8) explains why the condition n < N + 1 was imposed in (3.1).

(iii) Observe that we have the constant family of vector spaces V_{L̂_{n},q}, q ∈ ℂ, with differential D + K_{g} depending on q. At q = 0 the complex (V_{L̂_{n},q}, D + K_{g}) degenerates in Borisov’s complex (V Σ ; L̂_{n}, D +K_{g}). As it always happens in situations of this kind, and has already happened in the beginning of 2.5.2, the differential d = D + K_{g} breaks in a sum d = d_{−}(q) + d_{+} so that [d_{−}(q), d_{+}] = 0, d_{−}(0) = 0, and d_{+} equals Borisov’s differential on V Σ ; L̂_{n}. There arises a spectral sequence, analogous to the one described in (2.24-25), converging to H_{D+K_{g}}(V_{L̂_{n},q}) with 1st term equal to H^{∗}(Z(s), Ω_{Z(s)}^{ch}). The 2-nd term equals the cohomology of the complex (H^{∗}(Z(s), Ω_{Z(s)}^{ch}), d^{(1)}) with d^{(1)} = d_{−}(q)) satisfying:

d_{−}(q)(H^{i}(Z(s), Ω_{Z(s),n}^{ch})) ⊂ H^{i−N+n}(Z(s), Ω_{Z(s),n}^{ch}); n ≥ 0.  (3.8)

All this is true for an arbitrary Z(s). In the case of a hyperplane we have n = 1. It follows immediately that then all higher differentials vanish and it remains to calculate the cohomology (H^{∗}(Z(s), Ω_{Z(s)}^{ch})_{0}, d_{−}(q)). By Theorem 3.2, the space

H^{∗}(Z(s), Ω_{Z(s)}^{ch})_{0} is an algebra generated by the class of e^{nA^{u}−∑_{j} A^{j}}, which d_{−}(q) sends to 0, as follows from either (3.8) or assertion (i). Hence d_{−}(q) is 0 on the indicated space and the algebra H_{D+K_{g}}(V_{L̂_{n},q})_{0} is indeed a deformation of the cohomology of ℙ^{N−1}

The quantum cohomology algebra of the hyperplane is isomorphic to the algebra of functions on an N-point set. Because of (ii), H_{D+K_{g}}(V_{L̂_{n},q})_{0} is a quotient of the algebra of functions on an (N + 1)-point set, hence the algebra of functions on an (N + 1 − k)-point set. As we have just proved, k can only be 1 and this quotient can only be the algebra of functions on an N-point set.

(iv) We make use of the spectral sequence introduced in the proof of (iii). In the case at hand, N = 3, n = 2 and (3.8) rewrites as follows

d_{−}(q)(H^{i}(Z(s), Ω_{Z(s),n}^{ch})) ⊂ H^{i−1}(Z(s), Ω_{Z(s),n}^{ch}); n ≥ 0.  (3.9)

It is again immediate to see that d^{(r)} = 0 if r ≥ 2. Vanishing of d^{(1)} = d_{−}(q) when restricted to H^{i}(Z(s), Ω_{Z(s)}^{ch})_{0} is, however, less obvious than the same fact in (iii).

Let a and b be the generators of the second cohomology of ℙ^{1}×ℙ^{1} coming from the second cohomology of each of the factors. Thus H^{i}(Z(s), Ω_{Z(s)}^{ch})_{0} is linearly spanned

by 1 ∈ H^{0}(Z(s), Ω_{Z(s),0}^{ch,0}), a and b ∈ H^{1}(Z(s), Ω_{Z(s),0}^{ch,1}), ab ∈ H^{2}(Z(s), Ω_{Z(s),0}^{ch,2})

satisfying the relation a^{2} = b^{2} = 0; here Ω_{Z(s),0}^{ch,i} stands for the component of fermionic charge i and conformal weight 0. One checks that

d_{−}(q)(H^{i}(Z(s), Ω_{Z(s),0}^{ch,i})) ⊂ H^{i−1}(Z(s), Ω_{Z(s),0}^{ch,i−2}).  (3.10)

Due to (2.20), Ω_{Z(s),0}^{ch,i−2} = Ω_{Z(s)}^{i−2}; hence

H^{i−1}(Z(s), Ω_{Z(s),0}^{ch,i−2}) = H^{i−1}(Ω_{Z(s)}^{i−2}) = 0  (3.11).

Thus, all higher differentials vanish and H_{D+K_{g}}(V_{L̂_{n},q})_{0} is at least a deformation of H^{∗}(ℙ^{1} × ℙ^{1}).

Moreover, the algebras generated by a and b respectively deform. We arrive at the dichotomy: either both a, b generically satisfy a quadratic relation with a double root, or both satsify that with 2 distinct roots. (They cannot behave differently for the automorfizm of ℙ^{3} exchanging x_{1} ↔ x_{2} permutes the two copies of ℙ^{1} and preserves the entire differential vertex algebra structure.) In the first case we get (a + b)^{3} = 0 and this contradicts (3.7), which with N = 3, n = 2 reads as follows:

(a + b)^{4} − 4q^{2}(a + b)^{2} = 0.

The second case is assertion (iv). □

Remarks. (i) By Corollary 9.3 of [G], in the case n < N the cohomology class p of a hyperplane section satsifies in the quantum cohomology of Z(s) the relation

p^{N} = qn^{n}p^{n−1}.

The amusing similarity between this equality and (3.7) suggests that the subalgebra of H_{D+K_{g}}(V_{L̂_{n},q})_{0} generated by x might be equal to ℂ[x]/(x^{N}−q^{N+1−n}n^{n}x^{n−1}).

(ii) Borisov’s suggestion to treat the mirror symmetry as a flip interchanging A’s and B’s seems to be working in our “quantized” situation as well. Compare (3.5) with (3.2b) to note that D and K_{g} are sums over two sets of lattice points defined by self-dual condition (3.3). Hence the A−B flip changes D to a similar differential to be associated with the mirror partner of Z(s) lying in another toric manifold, see the next section. Of course the vertex algebra VL̂_{n} bears a certain asymmetry, since

not all elements of the type e^{∑_{j} n_{j}A^{j}+n_{u}A^{u}} are allowed, but Borisov’s “transition to the whole lattice” (see Theorem 8.3 in [B]) and the spectral sequence introduced in the proof of Lemma 3.4 (iii) seem to straighten things out.

§4. Quantum cohomology of toric varieties

Let us briefly explain how the constructions and results of section 2 carry over to an arbitrary smooth compact toric variety of dimension N. Each such variety is determined by a complete regular fan in L_{A}. This and other relevant concepts can be defined as follows (see [D, Bat] for details).

4.1 Let I ⊂ L_{A}. The cone generated by I is said to be the set of all non-negative integral combinations of elements of I and is denoted Δ_{I}.

A cone generated by part of a basis of L_{A} is called regular.

A complete regular fan Σ is defined to be a collection of regular cones {σ_{1}, ..., σ_{s}} so that the following conditions hold:

(i) If σ′ is a face of σ ∈ Σ, then σ′ ∈ Σ;

(ii) If σ, σ′ ∈ Σ, then σ ∩ σ′ is a face of σ;

(iii) (the completeness condition) L_{A} = σ_{1} ∪ ... ∪ σ_{s}.

We skip the construction of the smooth compact toric manifold X_{Σ} attached to a regular complete fan Σ referring the reader to [D], but formulate Batyrev’s result on H^{2}(X_{Σ}, ℝ), see [Bat].

A function ϕ : L_{A} → ℝ is called piecewise linear if its restriction to any cone in Σ is a morphism of abelian groups. Denote by P L(Σ) the space of all piecewise linear functions.

Let G(Σ) = {ξ_{1}, ..., ξ_{n}} be the set of the generators of all 1-dimensional cones in Σ. Since each piecewise linear function is determined by its values on ξ_{i} (i = 1, ..., n), P L(Σ) is an n-dimensional real vector space. It contains the N-dimensional subspace of globally linear functions; the latter is naturally isomorphic to L_{B} ⊗_{ℤ} ℝ.

Theorem 4.1 ([Bat])

H^{2}(X_{Σ}, ℝ) = P L(Σ)/L_{B} ⊗_{ℤ} ℝ.

4.2 Let us return to the vertex algebra V_{L}. Having fixed an arbitrary ℝ-valued function ϕ on L_{A}, we proceed in much the same way as in 2.1.

Define the linear automorphism

t_{ϕ} : V_{L} → V_{L}

by the formula

t_{ϕ}(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}}) = e^{−ϕ(∑_{i} n_{i}A^{i})}x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}}.  (4.1)

Define V_{L,ϕ} to be the vertex algebra equal to V_{L} as a vector space with the following n-th product:

(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n),ϕ}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})
= t_{ϕ}^{−1}(t_{ϕ}(x ⊗ e^{∑_{i} m_{i}B^{i}+∑_{i} n_{i}A^{i}})_{(n)}t_{ϕ}(y ⊗ e^{∑_{i} m′_{i}B^{i}+∑_{i} n′_{i}A^{i}})).  (4.2)

By definition,

t_{ϕ} : V_{L,ϕ} → V_{L}

is a vertex algebra isomorphism. This provides us with a constant family of vertex algebras parametrized by ϕ and we would like to study the behavior of this family as ϕ tends to ∞. For this we have to impose certain restrictions on ϕ.

Following [Bat], call a piecewise linear function ϕ convex if

ϕ(x) + ϕ(y) ≥ ϕ(x + y) all x, y ∈ L_{A}.  (4.3)

The cone of all convex piecewise linear functions descends to the cone in H^{2}(X_{Σ}, ℝ) = P L(Σ)/L_{B} ⊗_{ℤ} ℝ, see Theorem 4.1. Denote this cone by K(Σ) and its interior by K^{0}(Σ). K^{0}(Σ) consists of classes of all strictly convex piecewise linear functions, that is, of all those functions ϕ for which equality in (4.1) is achieved if and only if x and y belong to the same cone in Σ.

We see immediately that

(i) if ϕ is convex piecewise linear, then the operations

(n),∞ϕ = lim_{τ→+∞}_{(n),τϕ}, n ∈ ℤ

are well defined and satisfy the Borcherds identities; denote the vertex algebra arising in this way by V_{L,∞ϕ};

(ii) if ϕ is strictly convex piecewise linear, then V_{L,∞ϕ} is isomorphic to Borisov’s algebra V_{L}^{Σ}.

These assertions mean that the family V_{L,ϕ} produces a deformation of V_{L}^{Σ} with base equal to the cone of strictly convex piecewise linear functions. It is also immediate to see that if ϕ−ϕ′ is a linear function, then the two deformations V_{L,τϕ} and V_{L,τϕ′}, τ ≥ 0, are equivalent. Therefore we have obtained the family of vertex algebras V_{L,ϕ}, ϕ ∈ K^{0}(Σ), which is a deformation of V_{L}^{Σ} with base K^{0}(Σ).

Denote by QH_{ϕ}^{∗}(X_{Σ}, ℝ) the quantum cohomology of X_{Σ} as defined in section 5 of [Bat]. Borisov’s differential is as follows

D = ∫ ∑_{i=1}^{N} Ψ^{i}(z)(∑_{j=1}^{n}(B^{i}, ξ_{j})e^{ξ_{j}}(z)),  (4.4)

where {ξ_{1}, ..., ξ_{n}} is the set of generators of all 1-dimensional cones in Σ.

Theorem 4.2

H_{Q_{0}+D}(V_{L,ϕ}) = QH_{ϕ}^{∗}(X_{Σ}, ℝ).

Proof. First of all,

(Q_{0})^{2} = 0, D^{2} = 0, [Q_{0}, D] = 0.

(The first two of these assertions are obvious, the last one is obtained in the same way as (2.4).) Hence there arises a spectral sequence completely analogous to the one used in 2.4. It converges and collapses:

H_{Q_{0}+D}(V_{L,ϕ}) = H_{D}(H_{Q_{0}}(V_{L,ϕ}));

this is done in exactly the same way as in the proof of Theorem 2.4.

In part 1) of the proof of Theorem 2.3 the space H_{Q_{0}}(V_{L,ϕ}) was shown to be equal to the group algebra ℝ[L_{A}] extended by grassman variables Φ_{0}^{i} (i = 1, ..., N). Thus H_{Q_{0}}(V_{L,ϕ}) is a Koszul complex, and the restriction of D to this space equals the Koszul differential associated with the sequence

∑_{j=1}^{n}(B^{i}, ξ_{j})e^{ξ_{j}}, i = 1, ..., N.  (4.5)

Therefore, H_{D}(H_{Q_{0}}(V_{L,ϕ})) is the corresponding “Koszul cohomology”.

On the other hand, Batyrev defines QH_{ϕ}^{∗}(X_{Σ}, ℝ) to be the polynomial ring ℝ[z_{1}, ..., z_{n}] modulo the sum of two ideals denoted P (Σ) and Q_{ϕ}(Σ). It follows from the proof of either Theorem 9.5 or Theorem 8.4 in [Bat] that

ℝ[L_{A}] = ℝ[z_{1}, ..., z_{n}]/Q_{ϕ}(Σ).

Under this identification, the image of the ideal P (Σ) in ℝ[L_{A}] is generated by the elements (4.5) as follows from the comparison of (4.5) above and Definition 3.7 in [Bat]. The ring ℝ[L_{A}] is Cohen-Macaulay; hence the sequence (4.5) is regular. □

References

[Bat] V. Batyrev, Quantum cohomology rings of toric manifolds, Journées de Géométrie Algébrique d’Orsay (Orsay 1992), Astérisque, 218 (1993), 9-34; alggeom/9310004.

[B] L. Borisov, Vertex algebras and Mirror symmetry, math.AG/9809094.

[BL] L. Borisov, A.Libgober, Elliptic genera and applications to mirror symmetry, Inv. Math. 140 (2000), 453-485; math/9906126

[D] V.I. Danilov, The geometry of toric varieties, Uspekhi Mat.Nauk, 33, No. 2 (1978), 85-134 (Russian); Russian Math. Surveys, 33, No. 2 (1978), 97-154.

[FK] I.B. Frenkel, V.G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent.Math, 62 (1980), 23-66

[FZ] I.B. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebra, Duke Math. J., vol. 66, No. 1, (1992), 123-168

[G] A. Givental, Equivariant Gromow-Witten invariants, Internat. Math. Res. Notices, 13 (1996), 613-663.

[K] V.G. Kac Vertex algebras for beginners, 2nd edition, University Lecture Series, 10, AMS, Providence, Rhode Island, 1998

[MSV] F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys., 204 (1999), 439-473.

[MS1] F. Malikov, V. Schechtman Chiral de Rham complex. II Amer. Math. Soc. Transl. (2), vol. 194 (1999), 149-188

[MS2] F. Malikov, V. Schechtman Chiral Poincaré duality Math. Res. Lett. vol. 6 (1999), 533-546

F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA; fmalikov@mathj.usc.edu

V.S.: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France; vadik@ihes.fr