Twisted De Rham complex on line and $\widehat{\frak{sl}_2}$ singular vectors

We consider two complexes. The first complex is the twisted De Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\frak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient dual Verma modules over the affine Lie algebra $\widehat{\frak{sl}_2}$. In [SV2] a construction of a monomorphism of the first complex to the second was suggested. It was indicated in [SV2] that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the De Rham complex. In this paper we prove the results formulated in [SV2].


Introduction
We consider two complexes. The first complex is the twisted De Rham complex of scalar meromorphic differential forms on projective line, that are holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of sl 2 -valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient dual Verma modules over the affine Lie algebra sl 2 . In [SV2] a construction of a monomorphism of the first complex to the second was suggested. That construction gives a relation between the singular vectors in the Verma modules and resonance relations in the De Rham complex.
That construction of the homomorphism was invented in the middle of 90s, while the paper [SV2] was prepared for publication 20 years later, when the proofs were forgotten, if they existed. The paper [SV2] provides supporting evidence to the results formulated in [SV2], but not the proofs. The goal of this paper is to give the proofs to the results formulated in [SV2], namely, the proofs that the construction in [SV2] indeed gives a homorphism of complexes and relates the resonances in the De Rham complex and the sl 2 singular vectors.
The construction in [SV2] has two motivations. The first motivation was to generalize the principal construction of [SV1]. In [SV1], the tensor products of contragradient dual Verma modules over a semisimple Lie algebra were identified with the spaces of the top degree logarithmic differential forms over certain configuration spaces. Also the logarithmic parts of the De Rham complexes over the configuration spaces were identified with some standard Lie algebra chain complexes having coefficients in these tensor products, cf. in [KS, KV] a D-module explanation of this correspondence.
The second idea was that the appearance of singular vectors in Verma modules over affine Lie algebras is reflected in the relations between the cohomology classes of logarithmic differential forms. This was proved in an important particular case in [FSV], and in [STV] a one-to-one correspondence was established "on the level of parameters". In [SV2] and in the present paper this correspondence is developed for another non-trivial class of singular vectors, namely for (a part of) Malikov-Feigin-Fuchs singular vectors, cf. [MFF]. The paper has the following structure. In Section 2 we introduce the De Rham complex of a master function and resonance relations. In Section 3 we discuss sl 2 Verma modules, the Kac-Kazhdan reducibility conditions. We formulate Theorem 3.2 which describes certain relations in a contragradient dual Verma module. The proof of Theorem 3.2 is the main new result of this paper. In Theorem 3.3 we describe the connection between the relations, described in Theorem 3.2, and the Malikov-Feigin-Fuchs singular vectors. In Section 4 we construct a map of the De Rham complex of the master function to the chain complex of the Lie algebra of sl 2 -valued algebraic functions. Theorem 4.1 says that the map is a monomorphism of complexes. The proof of Theorem 4.1 is the second new result of this paper. Section 5 is devoted to the proof of Theorem 3.2. The proof is straightforward but rather nontrivial and lengthy.
The authors thank V. Schechtman for useful discussions.
where z 1 , . . . , z n , m 1 , . . . , m n , κ ∈ C are parameters. Fix these parameters and assume that z 1 , . . . , z n are distinct. Set Denote U = C − {z 1 , . . . , z n }. Consider the twisted De Rham complex associated with Φ, Here Ω p (U) is the space of rational differential p-forms on C regular on U. The differential ∂ is given by the formula where d is the standard De Rham differential and the second summand is the left exterior multiplication by the form

Formula (2.3) is motivated by the computation
The complex Ω • (U) is the complex of global algebraic sections of the De Rham complex of (O an U , ∂), where ∂ = d + α ∧ · is considered as the integrable connection on the sheaf O an U of holomorphic functions on U.
If κ = 0, then the twisted De Rham complex is not defined. If the resonance relation m i + aκ = 0 is satisfied for some a, then the first term in the right-hand side of (2.5) equals zero. Similarly, if the resonance relation m n+1 + 2 − aκ = 0 is satisfied for some a, then the first term in the right-hand side of (2.6) equals zero.
Let e, f , h be standard generators subject to the relations Let sl 2 be the affine Lie algebra sl 2 = sl 2 [T, These are the standard Chevalley generators defining sl 2 as the Kac-Moody algebra corresponding to the Cartan matrix 2 −2 −2 2 .
3.2. Automorphism π. The Lie algebra sl 2 has an automorphism π, 3.3. Verma modules. We fix k ∈ C and assume that the central element c acts on all our representations by multiplication by k.
For m ∈ C, let V (m, k −m) be the sl 2 Verma module with generating vector v. The Verma module is generated by v subject to the relations Letn − ⊂ sl 2 be the Lie subalgebra generated by f 1 , f 2 and Un − its enveloping algebra. 3.4. Reducibility conditions. See Kac-Kazhdan [KK]. Set is reducible if and only if at least one of the following relations holds: where l, a ∈ Z >0 . If (m, κ) satisfies exactly one of the conditions (a,b), then V (m, k − m) contains a unique proper submodule, and this submodule is generated by a singular vector of degree (la, l(a − 1)) for condition (a) and of degree (l(a − 1), la) for condition (b).
These singular vectors are highly nontrivial and are given by the following theorem.
Theorem 3.1 (Malikov-Feigin-Fuchs, [MFF]). For a, l ∈ Z >0 and κ ∈ C, the monomials An explanation of the meaning of complex powers in these formulas see in [MFF]. For example for m = −2 + κ, we have and for m = −2 + 2κ, we have 3.5. Shapovalov form. The Shapovalov form on an sl 2 Verma module V with generating vector v is the unique symmetric bilinear form S(·, ·) on V such that The space V * is an sl 2 -module with the sl 2 -action defined by the formulas: where ϕ ∈ V * , x ∈ V, i = 1, 2. The sl 2 -module V * is called the contragradient dual Verma module.
The Shapovalov form S considered as a map S : V −→ V * is a morphism of sl 2 -modules.
3.6. Bases in V and V * . Let V be an sl 2 Verma module V . For every γ = (p 1 , p 2 ) ∈ Z 2 0 with p 1 = p 2 , we fix a basis in the homogeneous component V γ ⊂ V . For p 1 > p 2 , we fix the basis For p 1 < p 2 , we fix the basis Notice that for any x ∈ sl 2 the elements x T i and x T j commute. These collections of vectors are bases by the Poincaré-Birkhoff-Witt theorem. For any γ with p 1 = p 2 , we fix a basis in the γ-homogeneous component V * γ ⊂ V * as the basis dual of the basis in V γ distinguished above. If 3.7. Main formula.
Theorem 3.2 ([SV2, Theorem 5.12]). For m, k ∈ C and a ∈ Z >0 , the following identities hold in the contragradient dual Verma module Theorem 3.2 was announced in [SV2]. The proof of Theorem 3.2 is the main result of this paper. The theorem is proved in Section 5.
Remark. Theorem 3.2 says that the action of the element f T a−1 of degree (a, a − 1) on the covector (v) * can be expressed in terms of the actions of the elements h T l and e T l of smaller degree on some other covectors. Similarly the action of the element e T a of degree (a − 1, a) on the covector (v) * can be expressed in terms of the actions of the elements h T l , f T l of smaller degree on some other covectors.
3.8. Relation to Malikov-Feigin-Fuchs vectors. Let For generic values of m and k, the Shapovalov form S is non-degenerate and X a and for some a, l ∈ Z >0 , see Section 3.4. Similarly, for a ∈ Z >0 let (m 0 , k 0 ) be a point of the line m+2−a(k +2) = 0, which does not belong to other resonance lines. Then the function Y a (m, k −m) can be analytically continued to the point (m 0 , k 0 ), and Y a (m 0 , k 0 − m 0 ) is a (nonzero) singular vector of V (m 0 , k 0 − m 0 ), hence it is proportional to the Malikov-Feigin-Fuchs vector F 21 (1, a, k 0 + 2).
Let sl 2 (U) be the Lie algebra of sl 2 -valued rational functions on P 1 regular on U, with the pointwise bracket. Thus, an element of sl 2 (U) has the form e ⊗ u 1 + h ⊗ u 2 + f ⊗ u 3 with u i ∈ Ω 0 (U), and the bracket is defined by the formula [x ⊗ u 1 , y ⊗ u 2 ] = [x, y] ⊗ (u 1 u 2 ). 4.2. sl 2 (U)-modules. We say that an sl 2 -module W has the finiteness property, if for any w ∈ W and x ∈ sl 2 , we have xT j · w = 0 for all j ≫ 0. For example, the contragradient dual Verma module has the finiteness property.
Let W 1 , . . . , W n+1 be sl 2 -modules with the finiteness property. Then the Lie algebra sl 2 (U) acts on W 1 ⊗ . . . ⊗ W n+1 by the formula where for x ⊗ u ∈ sl 2 (U) the symbol [x ⊗ u(t + z j )] denotes the Laurent expansion of x ⊗ u at t = z j and [x ⊗ u(1/t)] denotes the Laurent expansion at t = ∞; the symbol π in the last term denotes the sl 2 -automorphism defined in Section 3.2.
The finiteness property of the tensor factors ensures that the actions of the Laurent series are well-defined.
The sl 2 -action gives us a map

4.3.
Chain complex. For a Lie algebra g and a g-module W we denote by C • (g, W ) the standard chain complex of g with coefficients in W , where
We assign degree 0 to the term ⊗ n+1 j=1 V * j of this complex and assign degree 1 to the differential d, so that the whole complex sits in the non-positive area. 4.4.2. Consider the twisted De Rham complex in (2.2) corresponding to κ = k + 2 with degrees shifted by 1, namely, the complex Ω • (U)[1], where the shift [1] means that we assign degree p − 1 to the term Ω p (U).

Construction. Define a linear map
for a > 0. Define a linear map for a 0.
The homomorphism is injective.
Theorem 4.1 was announced in [SV2]. Here is a proof of the theorem.
4.6. Image of logarithmic subcomplex. Under the monomorphism η of Theorem 4.1 the image of the logarithmic subcomplex (Ω • log (U), ∂) is the chain complex C • (n − , ⊗ n+1 j=1 V * j ) of the nilpotent subalgebra n − ⊂ sl 2 generated by f . More precisely, we have Far-reaching generalizations of this identification of the logarithmic subcomplex with the chain complex of the nilpotent Lie algebra n − see in [SV1].
5. Proof of Theorem 3.2 5.1. Formula (3.6) folows from formula (3.5). The Lie algebra sl 2 has an automorphism ρ, corresponding to the involution of the Dynkin diagram: We have ρ 2 = id. In other words, ρ acts by the formulas For m ∈ C, let σ m : sl 2 → End(V (m, k − m)) be the Verma module structure. Let σ m • ρ : sl 2 → End(V (m, k − m)) be the twisted module structure.
Clearly the sl 2 -modules (σ m • ρ, V (m, k − m)) and (σ m−k , V (m − k, m)) are isomorphic. If v m ∈ V (m, k − m) and v k−m ∈ V (k − m, m) are generating vectors, then an isomorphism for any i 1 , . . . , i l ∈ {1, 2}. The isomorphism χ restricts to isomorphisms of the graded components, In Section 3.6 we fixed bases of the homogeneous components V (p 1 ,p 2 ) with p 1 = p 2 of any Verma module V . By Lemma 5.1, under the isomorphism χ the chosen basis of V (k − m, m) (p 1 ,p 2 ) is mapped to the chosen basis of V (m, k − m) (p 2 ,p 1 ) up to multiplication of the basis vectors by ±1. This ±1 appears due to the formula ρ( h T i ) = − h T i . In particular, we have In Section 3.6 we fixed bases in the homogeneous components V * (p 1 ,p 2 ) with p 1 = p 2 of any contragradient dual Verma module V * . Under the isomorphism χ * , the chosen basis of V (k − m, m) * (p 1 ,p 2 ) is mapped to the chosen basis of V (m, k − m) * (p 2 ,p 1 ) up to multiplication of the basis vectors by ±1. In particular, we have Assume that the relation in formula (3.5) holds in every contragradient dual Verma module V * . Then in V (k − m, m) * it takes the form The isomorphism χ * sends this relation to the relation in which is exactly the relation in formula (3.6). Thus formula (3.5) implies formula (3.6).

Auxiliary lemma. Let
and Proof. The proof is by induction. We prove the first equality, the others are proved similarly.
We 5.3. The structure of the proof of formula (3.5). We reformulate formula (3.5) as and will prove it in this form. Each term in (5.4) is an element of the homogeneous component V * (a,a−1) . In Section 3.6 we distinguished a basis of the dual component V (a,a−1) . We will calculate the value of the right-hand side in (5.4) on an arbitrary basis vector and will obtain the value of the left-hand side on that vector.
The basis in V (a,a−1) consists of the vectors We partition the basis in four groups. Group O consists of the single basis vector f T a−1 v. Group I consists of all basis vectors with r = 1, but different from f T a−1 v. Group II consists of all basis vectors with r = 2. Group III consists of all basis vectors with r 3.
Notice that the value of the left-hand side of (5.4) on the basis vector f T a−1 v equals m + (a − 1)(k + 2). Hence we need to show that the value of the right-hand side on the basis vector f T a−1 v equals m + (a − 1)(k + 2). Similarly the value of the left-hand side on any basis vector of Groups I-III equals zero. Hence we need to prove that the value of the right-hand side on any basis vector of Groups I-III equals zero. These four statements are the content of Propositions 5.3, 5.4, 5.7, and 5.9 below. These propositions prove Theorem 3.2.
Lemma 5.5. In the notation above, if s = 1, then f T a−1 (v) * , w = 2nk, (5.7) if ℓ a − 1 − n, We prove (5.11) by induction on s. For s = 2 we have The second summand gives zero when pairing with so that either hT ℓ is pulled to the right not affecting the number of f 's or it gives f T ℓ−i 2 , for which we apply the same argument as above after pulling it to the right to argue that the pairing of the vector in (5.32) with