Quantum Integrable Model of an Arrangement of Hyperplanes

The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.


Quantum integrable models and Bethe ansatz
A quantum integrable model is a vector space V and an "interesting" collection of commuting linear operators K 1 , K 2 , . . . : V → V . The operators are called Hamiltonians or transfer matrices or conservation laws. The problem is to find common eigenvectors and eigenvalues.
The Bethe ansatz is a method to diagonalize commuting linear operators. One invents a vector-valued function v(t) of some new parameters t = (t 1 , . . . , t k ) and fixes the parameters so that v(t) becomes a common eigenvector of the Hamiltonians. One shows that v(t) ∈ V is an eigenvector if t satisfies some system of equations, Ψ j (t) = 0, j = 1, . . . , k. (1.1) The equations are called the Bethe ansatz equations. The vector v(t) corresponding to a solution of the equations is called a Bethe vector. The method is called the Bethe ansatz method.

Gaudin model
One of the simplest and interesting models is the quantum Gaudin model introduced in [7] and [8]. Choose a simple Lie algebra g, an orthonormal basis {J a } of g with respect to a nondegenerate g-invariant bilinear form, and a collection of distinct complex numbers x = (x 1 , . . . , x N ). Then one defines certain elements of the N -th tensor power of the universal enveloping algebra of g, denoted by K 1 (x), . . . , K N (x) ∈ (U g) ⊗N and called the Gaudin Hamiltonians, Here we use the standard notation: if J ∈ g, then J (i) = 1 ⊗(i−1) ⊗ J ⊗ 1 ⊗(N −i) .
The Gaudin Hamiltonians commute with each other and commute with the diagonal subalgebra (U g) diag ⊂ (U g) ⊗N , Let V Λ denote the finite-dimensional irreducible g-module with highest weight Λ. Decompose a tensor product V Λ = ⊗ N b=1 V Λ b into irreducible g-modules, where W Λ,Λ∞ is the multiplicity space of a representation V Λ∞ . The Gaudin Hamiltonians act on V Λ , preserve decomposition (1.2), and by Schur's lemma induce commuting linear operators on each multiplicity space W = W Λ,Λ∞ , These commuting linear operators on a multiplicity space constitute the quantum Gaudin model. Thus, the Gaudin model depends on g, ( , ), highest weights Λ 1 , . . . , Λ N , Λ ∞ and complex numbers x 1 , . . . , x N .

Gaudin model as a semiclassical limit of KZ equations
On every multiplicity space W = W Λ,Λ∞ of the tensor product V Λ = ⊕ Λ∞ V Λ∞ ⊗ W Λ,Λ∞ one has a system of Knizhnik-Zamolodchikov (KZ) differential equations, where x = (x 1 , . . . , x N ), I(x) ∈ W is the unknown function, K b (x) are the Gaudin Hamiltonians and κ ∈ C × is a parameter of the differential equations. KZ equations are equations for conformal blocks in the Wess-Zumino-Novikov-Witten conformal field theory. For any value of κ, KZ equations define a flat connection on the trivial bundle C N ×W → C N with singularities over the diagonals. The flatness conditions, in particular, imply the conditions [K b (x), K c (x)] = 0, which are the commutativity conditions for the Gaudin Hamiltonians. Thus, we observe two related problems: (1) Given a nonzero number κ, find solutions of the KZ equations.
(2) Given x, find eigenvectors of the Gaudin Hamiltonians.
Problem (2) is a semiclassical limit of problem (1) as κ tends to zero. Namely, assume that the KZ equations have an asymptotic solution of the form I(x) = e P (x)/κ w 0 (x) + κw 1 (x) + κ 2 w 2 (x) + · · · as κ → 0. Here P (x) is a scalar function and w 0 (x), w 1 (x), w 2 (x), . . . are some W -valued functions of x. Substituting this expression to the KZ equations and equating the leading terms we get Hence, for any x and b, the vector w 0 (x) is an eigenvector of the Gaudin Hamiltonian K b (x) with eigenvalue ∂P ∂x b (x). Thus, in order to diagonalize the Gaudin Hamiltonians it is enough to construct asymptotic solutions to the KZ equations.
(ii) In the Gaudin model, the vector space W has a symmetric bilinear form S (the tensor Shapovalov form) and Gaudin Hamiltonians are symmetric operators.
(iii) In the Gaudin model, the Bethe vectors assigned to (properly understood) distinct critical points are orthogonal and the square of the norm of a Bethe vector equals the Hessian of the master function at the corresponding critical point, In particular, the Bethe vector corresponding to a nondegenerate critical point is nonzero.
These statements indicate a connection between the Gaudin Hamiltonians and a mysterious master function (which is not present in the definition of the Gaudin model).
One of the goals of this paper is to uncover this mystery and show that the Bethe ansatz can be interpreted as an elementary construction in the theory of arrangements, where the master function is a basic object.

Gaudin model and arrangements
For a weighted arrangement of affine hyperplanes, we will construct (under certain assumptions) a quantum integrable model, that is, a vector space W with a symmetric bilinear form S (called the contravariant form), a collection of commuting symmetric linear operators on W (called the naive geometric Hamiltonians), a master function Φ(t) and vectors v(t) (called the special vectors or called the values of the canonical element) which are eigenvectors of the linear operators if t is a critical point of the master function.
Then for a given Gaudin model (W, S, K 1 (x), K 2 (x), . . . : W → W ), one can find a suitable (discriminantal) arrangement and identify the objects of the Gaudin model with the corresponding objects of the quantum integrable model of the arrangement. After this identification, the master function Φ(t) and the special vectors v(t) of the arrangement provide the Gaudin model with a Bethe ansatz, that is, with a method to diagonalize the Gaudin Hamiltonians.

Bethe algebra
Let W Λ,Λ∞ be the vector space of a Gaudin model. It turns out that the subalgebra of End(W Λ,Λ∞ ) generated by the Gaudin Hamiltonians can be extended to a larger commutative subalgebra called the Bethe algebra. A general construction of the Bethe algebra for a simple Lie algebra g is given in [4]. That construction is formulated in terms of the center of the universal enveloping algebra of the corresponding affine Lie algebraĝ at the critical level. As a result of that construction, for any x one obtains a commutative subalgebra B(x) ⊂ (U g) ⊗N which commutes with the diagonal subalgebra U g ⊂ (U g) ⊗N . To define the Bethe algebra of V Λ or of W Λ,Λ∞ one considers the image of B(x 0 ) in End(V Λ ) or in End(W Λ,Λ∞ ). The Gaudin Hamiltonians K b (x) are elements of the Bethe algebra of V Λ or of W Λ,Λ∞ .
The construction of the Bethe algebra in [4] is not explicit and it is not easy to study the Bethe algebra of W Λ,Λ∞ for a particular Gaudin model. For example, a standard difficult question is if the Bethe algebra of W Λ,Λ∞ is a maximal commutative subalgebra of End(W Λ,Λ∞ ), cf. [12,5].

Algebra of geometric Hamiltonians
In this paper we address the following problem. Is there a geometric construction of the Bethe algebra? For the quantum integrable model (W, S, K 1 , K 2 , . . . : W → W ) of a given weighted arrangement, can one define a geometric "Bethe algebra" A, which is a maximal commutative subalgebra of End(W ), which contains the naive geometric Hamiltonians K 1 , K 2 , . . . , and which can be identified with the Bethe algebra of [4] for discriminantal arrangements?
In this paper (under certain assumptions) we construct an algebra A called the algebra of geometric Hamiltonians and identify it (in certain cases) with the Bethe algebra of the Gaudin model.

Quantum integrable model of a weighted arrangement (or dynamical theory of arrangements)
To define the quantum integrable model of an arrangement we consider in an affine space C k (with coordinates t = (t 1 , . . . , t k )) an arrangement of n hyperplanes, k < n. Each hyperplane is allowed to move parallelly to itself. The parallel shift of the i-th hyperplane is measured by a number z i and for every z = (z 1 , . . . , z n ) ∈ C n we get in C k an affine arrangement of hyperplanes A(z) = (H j (z)), where g j (t) are given linear functions on C k . We assign nonzero numbers a = (a j ) to the hyperplanes of A(z) (the numbers do not depend on z) and obtain a family of parallelly translated weighted hyperplanes. For generic z ∈ C n , the arrangement A(z) has normal crossings only. The discriminant ∆ ⊂ C n is the subset of all points z such that A(z) is not with normal crossings only.
The master function Φ on C n × C k is the function Φ(z, t) = j a j log(g j (t) + z j ).
Let A(A(z)) = ⊕ k p=0 A p (A(z)) be the Orlik-Solomon algebra and F(A(z)) = ⊕ k p=0 F p (A(z)) the dual vector space. We are interested in the top degree components A k (A(z)) and F k (A(z)).
The weights a define on F k (A(z)) a symmetric bilinear form S (a) (called the contravariant form) and a degree-one element ν(a) = j a j (H j (z)) ∈ A 1 (A(z)). Denote Sing F k (A(z)) ⊂ F k (A(z)) the annihilator of the subspace ν(a) · A k−1 (A(z)) ⊂ A k (A(z)).
For z 1 , z 2 ∈ C n − ∆, all combinatorial objects of the arrangements A(z 1 ) and A(z 2 ) can be canonically identified. In particular, the spaces F k (A(z 1 )), F k (A(z 2 )) as well as the spaces Sing F k (A(z 1 )), Sing F k (A(z 2 )) can be canonically identified. For z ∈ C n − ∆, we denote V = F k (A(z)), Sing V = Sing F k (A(z)).
For any nonzero number κ, the hypergeometric integrals γ(z) e Φ(z,t)/κ ω, ω ∈ A k (A(z)), define a Gauss-Manin (flat) connection on the trivial bundle C n ×Sing V → C n with singularities over the discriminant. The Gauss-Manin differential equations for horizontal sections of the connection on C n × Sing V → C n have the form κ ∂I ∂z j (z) = K j (z)I(z), where I(z) ∈ Sing V is a horizontal section, K j (z) : V → V , j ∈ J, are suitable linear operators preserving Sing V and independent of κ. For every j, the operator K j (z) is a rational function of z regular on C n − ∆. Each operator is symmetric with respect to the contravariant form S (a) . These differential equations are our source of quantum integrable models for weighted arrangements. The quantum integrable models are the semiclassical limit of these differential equations similarly to the transition from KZ equations to the Gaudin model in Section 1.6.
The flatness of the connection for all κ implies the commutativity of the operators, Let V * be the space dual to V . If M : V → V is a linear operator, then M * : V * → V * denotes the dual operator. Let W ⊂ V * be the image of V under the map V → V * associated with the contravariant form and Sing W ⊂ W the image of Sing V ⊂ V . The contravariant form induces on W a nondegenerate symmetric bilinear form, also denoted by S (a) . The operators K j (z) * preserve the subspaces Sing W ⊂ W ⊂ V * . The operators K j (z) * | W : W → W are symmetric with respect to the contravariant form. The operators K j (z) * | Sing W : Sing W → Sing W , j ∈ J, commute.
For z ∈ C n − ∆, we define the quantum integrable model assigned to (A(z), a) to be the collection The unital subalgebra of End(Sing W ) generated by operators K 1 (z) * | Sing W , . . . , K n (z) * | Sing W will be called the algebra of geometric Hamiltonians of (A(z), a).
It is clear that any weighted arrangement with normal crossings only can be realized as a fiber (A(z), a) of such a construction and, thus, a weighted arrangement with normal crossings only is provided with a quantum integrable model.
If z 0 is a point of the discriminant, the construction of the quantum integrable model assigned to the arrangement (A(z 0 ), a) is more delicate. The operator valued functions K j (z) may have first order poles at z 0 . We write K j (z) = K 0 j (z) + K 1 j (z), where K 0 j (z) is the polar part at z 0 and K 1 j (z) the regular part. By suitably regularizing the operators K 1 j (z 0 ), we make them (under certain assumptions) preserve a suitable subspace of Sing V , commute on that subspace and be symmetric with respect to the contravariant form. The algebra of regularized operators K 1 j (z 0 ) on that subspace produces the quantum integrable model assigned to the fiber (A(z 0 ), a). It may happen that some linear combinations (with constant coefficients) K ξ (z) = j ξ j K j (z) are regular at z 0 . In this case the operator K ξ (z 0 ) preserves that subspace and is an element of the algebra of Hamiltonians. In this case the operator K ξ (z 0 ) is called a naive geometric Hamiltonian. (It is naive in the sense that we don't need to go through the regularization procedure to produce that element of the algebra of Hamiltonians. In that sense, for z ∈ C n − ∆ all elements of the algebra of geometric Hamiltonians of the arrangement (A(z), a) are naive.) For applications to the Gaudin model one needs a suitable equivariant version of the described construction, the corresponding family of parallelly translated hyperplanes has a symmetry group and the Gaudin model is identified with the skew-symmetric part of the corresponding quantum integrable model of the arrangement.

Bethe ansatz for the quantum integrable model of an arrangement
The Hamiltonians of the model are (suitably regularized) right hand sides of the Gauss-Manin differential equations (1.3). The solutions to the equations are the integrals γ(z) e Φ(z,t)/κ ω. By taking the semiclassical limit of the integrals as κ tends to zero, we obtain eigenvectors of the Hamiltonians, cf. Section 1.3. The eigenvectors of the Hamiltonians are labeled by the critical points of the phase Φ of the integrals due to the steepest descent method.

Geometric interpretation of the algebra of Hamiltonians
It turns out that the solutions γ(z) e Φ(z,t)/κ ω to equations (1.3) produce more than just eigenvectors of the geometric Hamiltonians. They also produce a geometric interpretation of the whole algebra of geometric Hamiltonians. It turns out that the algebra of geometric Hamiltonians is naturally isomorphic (under certain conditions) to the algebra of functions on the critical set of the master function Φ. The isomorphism is established through the semiclassical limit of the integrals. Moreover, this isomorphism identifies the residue bilinear form on the algebra of functions and the contravariant form of the arrangement.
This geometric interpretation of the algebra of geometric Hamiltonians is motivated by the recent paper [15] where a connection between the algebra of functions on the critical set of the master function and the Bethe algebra of the gl r+1 Gaudin model is established, cf. [14].

Byproducts of constructions
The general motive of this paper is the interplay between the combinatorially defined linear objects of a weighted arrangement and the critical set of the corresponding master function (which is a nonlinear characteristics of the arrangement). As byproducts of our considerations we get relations between linear and nonlinear characteristics of an arrangement. For example, we prove that the sum of Milnor numbers of the critical points of the master function is not greater than the rank of the contravariant form on Sing V .
As another example of such an interaction we show that in any Gaudin model (associated with any simple Lie algebra) the Bethe vector corresponding to an isolated critical points of the master function is nonzero. That result is known for nondegenerate critical points, see [27], and for the Gaudin models associated with the Lie algebra gl r+1 , see [15].

Exposition of the material
In Section 2, basic facts of the theory of arrangements are collected. The main objects are the space of flags, contravariant form, master function, canonical element.
In Section 3, a family of parallelly translated hyperplanes is introduced. In Section 4, remarks on the conservation of the number of critical points of the master function under deformations are presented.
In Section 5, the Gauss-Manin differential equations are considered. The quantum integrable model of a weighted arrangement with normal crossings only is introduced. The "key identity" (5.2) is formulated.
In Section 6, the asymptotic solutions to the Gauss-Manin differential equations are discussed. In Section 7, the quantum integrable model of any fiber (A(z 0 ), a), z 0 ∈ ∆, is defined under assumptions of positivity of weights (a j ) and reality of functions (g j (t)). A general conjecture is formulated.
In Section 8, it is shown that the algebra of geometric Hamiltonians is isomorphic to the algebra of functions on the critical set of the master function. That fact is proved for any (A(z), a), z ∈ C n under Assumption 7.4 of positivity of (a j ) and reality of (g j (t)). In Section 9, more results in this direction are obtained, see Theorems 9.16 and 9.17.
In Section 10, an equivariant version of the algebra of geometric Hamiltonians is introduced and in Section 11 relations with the Gaudin model are described.
Let A = (H j ) j∈J , be an arrangement of n affine hyperplanes in C k . Denote the complement. An edge X α ⊂ C k of the arrangement A is a nonempty intersection of some hyperplanes of A. Denote by J α ⊂ J the subset of indices of all hyperplanes containing X α . Denote l α = codim C k X α . We always assume that A is essential, that is, A has a vertex, an edge which is a point. An edge is called dense if the subarrangement of all hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets so that, after a change of coordinates, hyperplanes in different sets are in different coordinates. In particular, each hyperplane of A is a dense edge.
is an algebra with respect to multiplication The algebra is called the Orlik-Solomon algebra of A.

Weights
An arrangement A is weighted if a map a : J → C, j → a j , is given; a j is called the weight of H j . For an edge X α , define its weight as a α = j∈Jα a j . We always assume that a j = 0 for every j ∈ J. Define Multiplication by ν(a) defines a differential 2.4 Space of flags, see [23] For an edge X α , l α = p, a flag starting at X α is a sequence of edges such that l α j = j for j = 0, . . . , p.
For an edge X α , we define F α as the complex vector space with basis vectors F α 0 ,...,αp=α labeled by the elements of the set of all flags starting at X α . Define F α as the quotient of F α by the subspace generated by all the vectors of the form Such a vector is determined by j ∈ {1, . . . , p − 1} and an incomplete flag Denote by F α 0 ,...,αp the image in F α of the basis vector F α 0 ,...,αp . For p = 0, . . . , k, set

Contravariant map and form, see [23]
Weights a determine a contravariant map where the sum is taken over all p-tuples (H j 1 , . . . , H jp ) such that Identifying A p (A) with F p (A) * , we consider the map as a bilinear form, The bilinear form is called the contravariant form. The contravariant form is symmetric. For where the sum is over all unordered p-element subsets.
The contravariant form was introduced in [23]. It is an analog of the Shapovalov form in representation theory. On relations between the contravariant and Shapovalov forms, see [23] and Section 11.  Proof . The theorem is proved in [28,18]. It is also a straightforward corollary of some results in [23]. Namely, in [23] a flag complex d : F p (A) → F p+1 (A) was considered with the differential defined by formula (2.2.1) in [23]. By [23, Corollary 2.8] the cohomology spaces of that flag complex are trivial in all degrees less than the top degree. In [23], it was also proved that the contravariant map defines a homomorphism of the flag complex to the complex d

Orlik-Solomon algebra as an algebra of differential forms
For j ∈ J, fix a defining equation for the hyperplane H j , f j = 0, where f j is a polynomial of degree one in variables t 1 , . . . , t k . Consider the logarithmic differential form ω j = df j /f j on C k . LetĀ(A) be the C-algebra of differential forms generated by 1 and ω j , j ∈ J. The map A(A) →Ā(A), (H j ) → ω j , is an isomorphism. We identify A(A) andĀ(A).

Critical points of the master function
Given weights a : J → C, define the (multivalued) master function Φ : U → C, Usually the function e Φ = j f a j j is called the master function, see [25,26,27], but it is more convenient to work with definition (2.1).
A point t ∈ U is a critical point of Φ if dΦ| t = 0. We can rewrite this equation as ν(a)| t = 0 since ν(a) = dΦ. Denote C(t) U the algebra of rational functions on C k regular on U and the ideal generated by first derivatives of Φ. Let A Φ = C(t) U /I Φ be the algebra of functions on the critical set and [ ]: If all critical points are isolated, then the critical set is finite and the algebra A Φ is finitedimensional. In that case, A Φ is the direct sum of local algebras corresponding to points p of the critical set, The local algebra A p,Φ can be defined as the quotient of the algebra of germs at p of holomorphic functions modulo the ideal I p,Φ generated first derivatives of Φ. Denote by m p ⊂ A p,Φ the maximal ideal generated by germs of functions equal to zero at p.
Proof . If H j 1 , . . . , H j k intersect transversally, then 1/f j 1 , . . . , 1/f j k form a coordinate system on U . This remark proves the lemma.
Define a rational function Hess (a) : C k → C, regular on U , by the formula The function is called the Hessian of Φ. Let p be an isolated critical point of Φ. Denote by [Hess (a) ] p the image of the Hessian in A p,Φ . It is known that the image is nonzero and the one-dimensional subspace C[Hess (a) ] p ⊂ A p,Φ is the annihilating ideal of the maximal ideal m p ⊂ A p,Φ .
Let ρ p : A p,Φ → C, be the Grothendieck residue, . . , k} is the real k-cycle oriented by the condition here ǫ s are sufficiently small positive real numbers, see [9]. It is known that ρ p : [Hess (a) ] p → µ p , where µ p = dim C A p,Φ is the Milnor number of the critical point p. Let ( , ) p be the residue bilinear form, That form is nondegenerate.
2.10 Special vectors in F k (A) and canonical element A differential top degree form η ∈ A k (A) can be written as where f is a rational function on C k , regular on U . Define a rational map v : for any η ∈ A k (A).
The map v is called the specialization map and its value v(t) is called the special vector at t ∈ U , see [27].
will be called the canonical element of the arrangement A. It does not depend on the choice of the basis (F m ) m∈M .
For any t ∈ U , we have Denote by [E] the image of the canonical element in A Φ ⊗ F k (A).  Proof . The theorem follows from Lemma 2.6.  (ii) If t 1 , t 2 ∈ U are different isolated critical points of Φ, then the special singular vectors v(t 1 ), v(t 2 ) are orthogonal,

Arrangements with normal crossings only
An essential arrangement A is with normal crossings only, if exactly k hyperplanes meet at every vertex of A. Assume that A is an essential arrangement with normal crossings only. A subset {j 1 , . . . , j p } ⊂ J will be called independent if the hyperplanes H j 1 , . . . , H jp intersect transversally.
A basis of A p (A) is formed by (H j 1 , . . . , H jp ) where {j 1 < · · · < j p } are independent ordered p-element subsets of J. The dual basis of F p (A) is formed by the corresponding vectors F (H j 1 , . . . , H jp ). These bases of A p (A) and F p (A) will be called standard.
In F p (A) we have for any permutation σ ∈ S p . For an independent subset {j 1 , . . . , j p }, we have for distinct elements of the standard basis.

Real structure on A p (A) and F p (A)
We have defined A p (A) and F p (A) as vector spaces over C. But one can define the corresponding spaces over the field R so that A p (A) = A p (A) R ⊗ R C and F p (A) = F p (A) R ⊗ R C. If all weights a are real, then the differential d (a) : A p (A) → A p+1 (A) preserves the real subspaces and one can define the subspace of singular vectors Sing F k (A) R ⊂ F k (A) R so that Sing F k (A) = Sing F k (A) R ⊗ R C.

A real arrangement with positive weights
Let t 1 , . . . , t k be standard coordinates on C k . Assume that every polynomial f j , j ∈ J, has real coefficients, where z j , b i j are real numbers. Denote U R = U ∩ R k . Let U R = ∪ α D α be the decomposition into the union of connected components. Each connected component is a convex polytope. It is known that the number of bounded connected components equals |χ(U )|, see [29]. Note that the contravariant form on Sing F k (A) R is positive definite.

Resolution of a hyperplane-like divisor
Let Y be a smooth complex compact manifold of dimension k, D a divisor. The divisor D is hyperplane-like if Y can be covered by coordinate charts such that in each chart D is a union of hyperplanes. Such charts will be called linearizing. Let D be a hyperplane-like divisor, U be a linearizing chart. A local edge of D in U is any nonempty irreducible intersection in U of hyperplanes of D in U . An edge of D is the maximal analytic continuation in Y of a local edge. Any edge is an immersed submanifold in Y . An edge is called dense if it is locally dense. For 0 i k − 2, let L i be the collection of all dense edges of D of dimension i. The following theorem is essentially contained in Section 10.8 of [25]. Theorem 2.11 ([22]). Let W 0 = Y . Let π 1 : W 1 → W 0 be the blow up along points in L 0 . In general, for 1 s k − 1, let π s : W s → W s−1 be the blow up along the proper transforms of the (s − 1)-dimensional dense edges in L s−1 under π 1 · · · π s−1 . Let π = π 1 · · · π k−1 . Then W = W n−1 is nonsingular and π −1 (D) has normal crossings.
3 A family of parallelly translated hyperplanes 3.1 An arrangement in C n × C k Recall that J = {1, . . . , n}. Consider C k with coordinates t 1 , . . . , t k , C n with coordinates z 1 , . . . , z n , the projection C n × C k → C n .
Fix n nonzero linear functions on C k , where b i j ∈ C. Define n linear functions on C n × C k , In C n × C k define an arrangement For every fixed z = (z 1 , . . . , z n ) the arrangementÃ induces an arrangement A(z) in the fiber over z of the projection. We identify every fiber with C k . Then A(z) consists of hyperplanes H j (z), j ∈ J, defined in C k by the same equations f j = 0. Denote the complement to the arrangement A(z).
In the rest of the paper we assume that for any z the arrangement A(z) has a vertex. This means that the span of (g j ) j∈J is k-dimensional.
A point z ∈ C n will be called good if A(z) has normal crossings only. Good points form the complement in C n to the union of suitable hyperplanes called the discriminant.

Discriminant
The collection (g j ) j∈J induces a matroid structure on J. A subset C = {i 1 , . . . , i r } ⊂ J is a circuit if (g i ) i∈C are linearly dependent but any proper subset of C gives linearly independent g i 's.
For a circuit C = {i 1 , . . . , i r }, let (λ C i ) i∈C be a nonzero collection of complex numbers such that i∈C λ C i g i = 0. Such a collection is unique up to multiplication by a nonzero number.
For every circuit C fix such a collection and denote f C = i∈C λ C i z i . The equation f C = 0 defines a hyperplane H C in C n . It is convenient to assume that λ C i = 0 for i ∈ J − C and write f C = i∈J λ C i z i . For any z ∈ C n , the hyperplanes (H i (z)) i∈C in C k have nonempty intersection if and only if z ∈ H C . If z ∈ H C , then the intersection has codimension r − 1 in C k .
Denote by C the set of all circuits in J. Denote ∆ = ∪ C∈C H C .
Lemma 3.1. The arrangement A(z) in C k has normal crossings only, if and only if z ∈ C n − ∆.
Remark 3.2. If all linear functions g j , j ∈ J, are real, then for any circuit C ∈ C the numbers (λ C i ) i∈C can be chosen to be real. Therefore, in that case every hyperplane H C is real.

Good fibers
For any z 1 , Assume that weights a = (a j ) j∈J are given and all of them are nonzero. Then each arrange- ). The triple (V, Sing V, S (a) ) does not depend on z ∈ C n − ∆ under the above identification.

Bad fibers
Points of ∆ ⊂ C n will be called bad.
Let z 0 ∈ ∆ and z ∈ C n − ∆. By definition, for any p the space A p (A(z 0 )) is obtained from A p (A(z)) by adding new relations. Hence A k (A(z 0 )) is canonically identified with a quotient space of V * = A k (A(z)) and F p (A(z 0 )) is canonically identified with a subspace of V = F p (A(z)).
Let us consider F k (z 0 ) as a subspace of V . Let S (a) | F k (z 0 ) be the restriction of the contravariant form on V to that subspace. Let S (a) (z 0 ) be the contravariant form on F k (A(z 0 )) of the arrangement A(z 0 ).

Conservation of the number of critical points
Let A = (H j ) j∈J be an essential arrangement in C k with weights a. Consider its compactification in the projective space P k containing C k . Assign the weight a ∞ = − j∈J a j to the hyperplane H ∞ = P k − C k and denote by A ∨ the arrangement (H j ) j∈J∪∞ in P k .
The weighted arrangement (A, a) will be called unbalanced if the weight of any dense edge of A ∨ is nonzero.
For example, if all weights (a j ) j∈J are positive, then the weighted arrangement (A, a) is unbalanced. Clearly, the unbalanced weights form a Zarisky open subset in the space of all weights of A. Proof . Let π : W → P k be the resolution (described in Theorem 2.11) of singularities of the divisor D = ∪ j∈J∪∞ H j . Let Φ a be the master function of (A, a). Then locally on W the function π −1 Φ a has the form Here u 1 , . . . , u k are local coordinates, 0 m k, the function φ(u 1 , . . . , u k ) is holomorphic at u = 0, the equation u 1 · · · u m = 0 defines π −1 (D) in this chart. If the image of a divisor u i = 0, 1 i m, under the map π is an s-dimensional piece of an s-dimensional dense edge of A ∨ , then α i equals the weight of that edge. In particular, α i , i = 1, . . . , m, are all nonzero. Let If the critical set of π −1 Φ a on π −1 (U (A)) is infinite, then it contains an algebraic curve. The closure of that curve must intersect π −1 (D). But equations (4.1) show that this is impossible.  5 Hamiltonians of good f ibers

Construction
Consider the master function as a function onŨ ⊂ C n × C k . Let κ be a nonzero complex number. The function e Φ(z,t)/κ defines a rank one local system L κ onŨ whose horizontal sections over open subsets ofŨ are univalued branches of e Φ(z,t)/κ multiplied by complex numbers.
For a fixed z, choose any γ ∈ H k (U (A(z)), L κ | U (A(z)) ). The linear map is an element of Sing F k (A(z)) by Stokes' theorem. It is known that for generic κ any element of Sing F k (A(z)) corresponds to a certain γ and in that case the integration identifies Sing F k (A(z)) and H k (U (A(z)), L κ | U (A(z)) ), see [23].
The vector bundle has a canonical (flat) Gauss-Manin connection. The Gauss-Manin connection induces a flat connection on the trivial bundle C n × Sing V → C n with singularities over the discriminant ∆ ⊂ C n . That connection will be called the Gauss-Manin connection as well.
Theorem 5.1. The Gauss-Manin differential equations for horizontal sections of the connection on C n × Sing V → C n have the form are suitable linear operators preserving Sing V and independent on κ. For every j, the operator K j (z) is a rational function of z regular on C n − ∆. Each operator is symmetric with respect to the contravariant form S (a) .
The flatness of the connection for all κ implies the commutativity of the operators, with the contravariant form and Sing W ⊂ W the image of Sing V . The contravariant form induces on W a nondegenerate symmetric bilinear form, also denoted by S (a) .
For z ∈ C n − ∆, we define the quantum integrable model assigned to (A(z), a) to be the collection The unital subalgebra of End(Sing W ) generated by operators K 1 (z) * | Sing W , . . . , K n (z) * | Sing W will be called the algebra of geometric Hamiltonians of (A(z), a). If the contravariant form S (a) is nondegenerate on V , then this model is isomorphic to the collection It is clear that any weighted essential arrangement with normal crossings only can be realized as a good fiber of such a construction. Thus, every weighted essential arrangement with normal crossings only is provided with a quantum integrable model.

Key identity (5.2)
For any circuit C = {i 1 , . . . , i r } ⊂ J, let us define a linear operator L C : V → V in terms of the standard basis of V , see Section 2.11.
For m = 1, . . . , r, define C m = C − {i m }. Let {j 1 < · · · < j k } ⊂ J be an independent ordered subset and F (H j 1 , . . . , H j k ) the corresponding element of the standard basis. Define . . , j k } ∩ C = C m for some 1 m r, then using the skew-symmetry property (2.5) we can write The map L C is symmetric with respect to the contravariant form.
l, m r. But both sides of this expression are equal to (−1) l+m a i 1 · · · a ir a s 1 · · · a s k−r+1 .
On C n × C k consider the logarithmic differential 1-forms For any circuit C = {i 1 , . . . , i r }, we have Proof . The lemma is a direct corollary of the definition of maps L C .
Identity (5.2) is a key formula of this paper. Identity (5.2) is an analog of the key Theorem 7.2.5 in [23] and it is a generalization of the identity of Lemma 4.2 in [27].

An application of the key identity
Let z → γ(z) ∈ H k (U (A(z)), L κ | U (A(z)) ) be a locally constant section of the Gauss-Manin connection. Then Lemma 5.5. The differential of the function {γ(z)} is given by the formula Proof . The lemma follows from identity (5.2) and the formula of differentiation of an integral.
Lemma 5.6. For every circuit C, the operator L C preserves the subspace Sing V .
Proof . The values of the function {γ(z)} belong to Sing V . Hence, the values of its derivatives belong to Sing V . Now the lemma follows from Lemma 5.5.
Recall that ) be a locally constant section of the Gauss-Manin connection. Then

Another application of the key identity (5.2)
Recall thatŨ is the complement to the union of hyperplanes (H j ) j∈J in C n ×C k , see Section 3.1. Denote by C(z, t)Ũ the algebra of rational functions on C n × C k regular onŨ . For any basis vector (H j 1 , . . . , H j k ) of V * , let us write where f j 1 ,...,j k ∈ C(z, t)Ũ and the z-part is a differential form with zero restriction to any fiber of the projection C n × C k → C n (in coordinates t 1 , . . . , t k , z 1 , . . . , z n , that form has at least one of dz 1 , . . . , dz n as factors in each of its summands). Define the canonical elementẼ ∈ C(z, t)Ũ ⊗ V by the condition for any independent {j 1 , . . . , j k } ⊂ J.
Theorem 5.8. For any j ∈ J, there exist elements h 1 , . . . , h k ∈ C(z, t)Ũ ⊗ V such that Proof . We have where z-part is a V -valued differential k-form with zero restriction to each fiber of the projection C n × C k → C n . Then identity (5.2) and formulas (5.6), (5.7), (5.4) imply the theorem.

Hamiltonians, critical points and the canonical element
Fix z ∈ C n − ∆. Recall that in Section 5.1 we have defined the quantum integrable model assigned to (A(z), a) to be the collection Let p ∈ U (A(z)) be an isolated critical point of the master function Φ(z, ·) : U (A(z)) → C. Let A p,Φ be the local algebra of the critical point and [ ] : Theorem 5.9. We have Proof . Part (i) follows from Theorem 2.8. Part (ii) follows from Theorem 5.8.
Remark 5.10. The elements [a j /f j (z, ·)], j ∈ J, generate A p,Φ due to Lemma 2.5 and the assumption (a j = 0 for all j).

Asymptotic solutions, one variable
Let u be a variable, W a vector space, M (u) ∈ End(W ) an endomorphism depending holomorphically on u at u = 0. Consider a differential equation, depending on a complex parameter κ ∈ C * .
Let P (u) ∈ C, (w m (u) ∈ W ) m∈Z 0 be functions holomorphic at u = 0 and w 0 (0) = 0. The series will be called an asymptotic solution to (6.1) if it satisfies (6.1) to all orders in κ. In particular, the leading order equation is Assume now that has a first order pole at u = 0 and I(u, κ) is a series like in (6.2). The series I(u, κ) will be called an asymptotic solution to equation (6.1) with such M (u) if it satisfies (6.1) to all orders in κ. In particular, the leading order equation is again equation (6.3). Equation (6.3) implies

Critical points of the master function and asymptotic solutions
Let us return to the situation of Section 3. Let t(z) be a nondegenerate critical point of Φ(z, · ) : U (A(z)) → C. Assume that t(z) depends on z holomorphically in a neighborhood of a point z 0 ∈ C n . Fix a univalued branch of Φ in a neighborhood of (z 0 , t(z 0 )) (by choosing arguments of all of the logarithms). Denote Ψ(z) = Φ(z, t(z)).
Let B ⊂ C k be a small ball with center at t(z 0 ). Denote It is well known that H k (B, B − ; Z) = Z, see for example [1]. There exist local coordinates where ǫ is a small positive number. That δ, considered as a k-chain, Recall that any element ω ∈ V * is a linear combination of elements (H j 1 , . . . , H j k ) and such an element (H j 1 , . . . , H j k ) is identified with the differential form In (6.5) we integrate over δ such a differential form multiplied by e Φ(z,t)/κ . The element {δ}(z, κ) as a function of z, κ is holomorphic if z is close to z 0 and κ = 0.
(i) Let κ ∈ R and κ → +0. Then the function {δ}(z, κ) has an asymptotic expansion where (w m (z) ∈ V ) m∈Z 0 are functions of z holomorphic at z 0 and Here v(z, t(z)) is the special vector associated with the critical point (z, t(z)) of the function Φ(z, · ), see Section 2. 10. The sign ± depends on the choice of the orientation of δ.
(iii) The functions (w m (z)) m∈Z 0 take values in Sing V .
Part (i) of the theorem is a direct corollary of the method of steepest descent; see, for example, § 11 in [1]. Part (ii) follows from Lemma 5.4 and formula of differentiation of an integral. Part (iii) follows from Stokes' theorem.
Remark 6.2. The definition of δ depends on the choice of local coordinates u 1 , . . . , u k , but the asymptotic expansion (6.6) does not depend on the choice of δ since the difference of the corresponding integrals is exponentially small.
In that case the operators K j (z 0 )| Sing V : Sing V → Sing V , j ∈ J, are all well-defined, see (5.3), and we have Thus, the special vector v(z 0 , t(z 0 )) is an eigenvector of the geometric Hamiltonians K j (z 0 ).
Remark 6.4. The Gauss-Manin differential equations (5.5) have singularities over the discriminant ∆ ⊂ C n . If z 0 ∈ ∆, then expansion (6.6) still gives an asymptotic solution to equations (5.5) and that asymptotic solution is regular at z 0 .
7 Hamiltonians of bad f ibers

Naive geometric Hamiltonians
Let us return to the situation of Section 3. Let z 0 ∈ ∆ and z ∈ C n − ∆. We have Consider the map V → V * corresponding to the contravariant form. In Section 5.1 we denoted the images of V and Sing V by W and Sing W , respectively. We denote the images of F k (A(z 0 )) and Sing F k (A(z 0 )) by W (z 0 ) and Sing W (z 0 ), respectively. We have Recall that ∆ is the union of hyperplanes H C , C ∈ C. Denote Consider the operator-valued functions K j (z) : V → V , j ∈ J, given by formula (5.3). Denote Each of the summands of K 0 j (z) tends to infinity as z tends to z 0 in C n −∆. The operator-valued function K 1 j (z) is regular at z 0 . The operators K j (z) * , L * C preserve the subspaces Sing W ⊂ W ⊂ V * and are symmetric operators on W with respect to the contravariant form on W . The operators K j (z) * restricted to Sing W commute.
The point z 0 ∈ ∆ defines an edge X z 0 of the arrangement (H C ) C∈C , where X z 0 = ∩ C∈C 0 H C . Denote by T z 0 the vector space of constant vectors fields on C n which are tangent to X z 0 , considered as a function of z, is regular at z 0 , moreover, (iii) The dual linear operator considered as a function of z, is regular at z 0 , moreover, Proof . Parts (iii), (iv) follow from parts (i), (ii). Part (i) is clear. Part (ii) follows from a straightforward calculation.
The operators K ξ (z 0 ) * preserve the subspace Sing W (z 0 ). The operators form a commutative family of linear operators. The operators are symmetric with respect to the contravariant form. These operators will be called naive geometric Hamiltonians on Sing W (z 0 ). (i) The space F k (A(z 0 )) lies in the kernel of L C : V → V for any C ∈ C 0 .

Space
(ii) The space W (z 0 ) lies in the kernel of L * C | W : W → W for any C ∈ C 0 . (iii) For any C ∈ C 0 , the image of L * C | W is orthogonal to W (z 0 ) with respect to the contravariant form.
Proof . Part (i) follows from a straightforward easy calculation. Part (ii) follows from part (i). Part (iii) follows from part (ii) and the fact that L * C is symmetric.

Conjecture
commute and are symmetric with respect to the contravariant form.
For z 0 ∈ ∆, we define the quantum integrable model assigned to (A(z 0 ), a) to be the collection The unital subalgebra of End(Sing W (z 0 )) generated by operators will be called the algebra of geometric Hamiltonians of (A(z 0 ), a). Note that the naive geometric Hamiltonians are elements of the algebra of geometric Hamiltonians, since for any ξ = j∈J ξ j ∂ ∂z j ∈ T X z 0 , we have In the next section we prove the conjecture under Assumption 7.4 of certain positivity and reality conditions, see Theorem 7.5. In Section 9 more results in this direction will be obtained, see Theorems 9.16 and 9.17. For applications to the Gaudin model an equivariant version of the conjecture is needed, see Sections 10 and 11. 7.4 Positive (a j ) j∈J , real (g j ) j∈J Assumption 7.4. Assume that all weights a j , j ∈ J, are positive and all functions g j = b 1 j t 1 + · · · + b k j t k , j ∈ J, have real coefficients b i j . The space V has a real structure, V = V R ⊗ R C, see Section 2.12. Under Assumption 7.4 all subspaces in (7.1) are real (can be defined by real equations). The contravariant form S (a) is positive definite on V R and is positive definite on the real parts of all of the subspaces in (7.1).
Denote by pr : Sing V → Sing F k (A(z 0 )) the orthogonal projection.
Theorem 7.5. Assume that Assumption 7.4 is satisfied and z 0 ∈ ∆. Then the operators commute and are symmetric with respect to the contravariant form. 7.5 Proof of Theorem 7.5 for z 0 ∈ ∆ ∩ R n Assume that z 0 ∈ ∆ ∩ R n . Let r : (C, R, 0) → (C n , R n , z 0 ) be a germ of a holomorphic curve such that r(u) ∈ C n − ∆ for u = 0. For u ∈ R >0 , the arrangement A(r(u)) is real. Denote U (r(u)) R = (C k − ∪ j∈J H j (r(u))) ∩ R k . Let U (r(u)) R = ∪ α D α (r(u)) be the decomposition into the union of connected components. We label components so that for any α, the component D α (r(u)) continuously depends on u > 0. Let A be the set of all α such that D α (r(u)) is bounded. Let A 1 be the set of all α such that D α (r(u)) is bounded and vanishes as u → +0 (the limit of D α (r(u)) is not a domain of A(z 0 )). Let A 2 be the set of all α such that D α (r(u)) is bounded and the limit of D α (r(u)) as u → 0 is a domain of A(z 0 ). We have A = A 1 ∪ A 2 and A 1 ∩ A 2 = ∅. All critical points of Φ(r(u), · ) lie in ∪ α∈A D α (r(u)). Each domain D α (r(u)) contains a unique critical point (r(u), t(u) α ) and that critical point is nondegenerate. Denote by v(r(u), t(u) α ) ∈ Sing V the corresponding special vector. That vector is an eigenvector of the geometric Hamiltonians, If α ∈ A 2 , then all eigenvalues 1/f j (r(u), t(u) α ), j ∈ J, are regular functions at u = 0. If α ∈ A 1 , then there is an index j ∈ J such that a j /f j (r(u), t(u) α ) → ∞ as u → 0.
Proof . The lemma follows from Theorem 2.9 and Corollary 2.10.
Assume now that a curve r(u) = (z 1 (u), . . . , z n (u)) is linear in u. Then for any j we have K 0 j (r(u)) = N j /u where N j : Sing V → Sing V is an operator independent of u.
Lemma 7.7. The image of N j is a subspace of (Sing F k (z 0 )) ⊥ .
Proof . The lemma follows from Theorem 2.9 and Corollary 2.10.
7.6 Proof of Theorem 7.5 for any z 0 ∈ ∆ Consider the arrangement (H C ) C∈C in C n and its arbitrary edge X. The arrangement (H C ) C∈C is real, see Remark 3.2. The edge X is the complexification of X ∩ R n . Denotê For any z 1 , z 2 ∈X, the subspaces For z ∈X, the operators prK 1 j (z) : Sing F k (A(X)) → Sing F k (A(X)), j ∈ J, depend on z holomorphically. The operators commute and are symmetric for z ∈X ∩ R n , by reasonings in Section 7.5. Hence they commute and are symmetric for all z ∈X.

Critical points and eigenvectors
Proof . If z 0 ∈ ∆ ∩ R n , then the theorem is just a restatement of formula (7.2). Assume that z 0 is an arbitrary point of ∆. Then there exists an edge X of the arrangement (H C ) C∈C such that z 0 ∈X, see (7.3). For z 0 ∈X, all objects in formula (7.4) depend on z 0 algebraically. Hence, the fact, that formula (7.4) holds for all critical points if z 0 ∈X ∩ R n , implies Theorem 7.8 for any z 0 ∈X.

Hamiltonians, critical points and the canonical element
Let Assumption 7.4 be satisfied. Fix z 0 ∈ ∆. We have defined the quantum integrable model assigned to (A(z 0 ), a) to be the collection Let p ∈ U (A(z 0 )) be an isolated critical point of the function Φ(z 0 , · ) : U (A(z 0 )) → C. Let A p,Φ be the local algebra of the critical point and [ ] : C(t) U (A(z 0 )) → A p,Φ the canonical projection. Denote by Hess (a) the Hessian of Φ(z 0 , · ).
Let E ∈ C(t) U (A(z 0 )) ⊗ F k (A(z 0 )) be the canonical element associated with A(z 0 ), see Section 2.10. Denote by [E] the projection of the canonical element to A p,Φ ⊗ F k (A(z 0 )). By Lemma 2.6, we have  Let F be a germ of a holomorphic function at a point p ∈ C k . Assume that p is an isolated critical point of F with Milnor number µ p . Let A p,F be the local algebra of the critical point and ( , ) p the residue bilinear form on A p,F , see (2.3). Denote by [Hess F ] the projection to A p,F of the germ det 1 l,m k (∂ 2 F/∂t l ∂t m ).
Let h 1 , . . . , h µp be a C-basis of A p,F . Let g 1 , . . . , g n ∈ A p,F be a collection of elements such that the unital subalgebra of A p,F generated by g 1 , . . . , g n equals A p,F .
Let W be a vector space with a symmetric bilinear form S. Let M j : W → W , j ∈ J, be a collection of commuting symmetric linear operators, Assume that an element Denote by Y ⊂ W the vector subspace generated by w 1 , . . . , w µp . By property (8.1), every M j , j ∈ J, preserves Y . Denote by A Y the unital subalgebra of End(Y ) generated by M j | Y , j ∈ J. The subspace Y is an A Y -module. Define a linear map Theorem 8.1.
(i) The map α : A p,F → Y is an isomorphism of vector spaces. The form S restricted to Y is nondegenerate.
(ii) The map g j → M j | Y , j ∈ J, extends uniquely to an algebra isomorphism β : (iii) The maps α, β give an isomorphism of the regular representation of A p,F and the A Ymodule Y , that is M j α(f ) = α(g j f ) for any f ∈ A p,F and j ∈ J.
(iv) Define the value w(p) of w at p as the image of w under the natural projection A p,F ⊗W → A p,F /m p ⊗ W = W . Then w(p) = α(Hess F )/µ p and the value w(p) is nonzero. The vector w(p) is the only (up to proportionality) common eigenvector of the operators M j | Y , j ∈ J, and we have M j w(p) = g j (p)w(p).
This theorem is an analog of Theorem 5.5 and Corollary 5.6 in [15]. The proof is analogous to the proofs in [15].
8.2 Proof of Theorem 8.1 Lemma 8.4. There exists a unique element s ∈ A p,F such that (f, g) S = (sf, g) p for all f, g ∈ A p,F .

Proof . Consider the linear function
Hence for any f, g ∈ A p,F we have (f, g) S = (1, f g) S = (s, f g) p = (sf, g) p .  On the other hand we have Hence, µp l,m=1

Remark on maximal commutative subalgebras
Let A be a commutative algebra with unity element 1. Let B be the subalgebra of End(A) generated by all multiplication operators L f :

Interpretation of the algebra of Hamiltonians of good fibers
Under notations of Section 3 fix a point z ∈ C n − ∆. Recall that in formula (5.1) we have defined the quantum integrable model assigned to (A(z), a) to be the collection Sing W ; S (a) | Sing W ; K 1 (z) * | Sing W , . . . , K n (z) * | Sing W : Sing W → Sing W .
Let p ∈ U (A(z)) be an isolated critical point of the master function Φ(z, · ) : U (A(z)) → C. Let A p,Φ be the local algebra of the critical point and ( , ) p the residue bilinear form on A p,Φ . Let [E] ∈ A p,Φ ⊗ Sing W be the element corresponding to the canonical element, see Section 5.5.
Define a linear map Denote Y p the image of α p .
(ii) The operators K j (z) * preserve Y p . Moreover, for any j ∈ J, g ∈ A p,Φ , we have [E](p).
Theorem 8.11. The linear map α p identifies the contravariant form on Y p and the residue form for any f, g ∈ A p,Φ .
Proof . If the Milnor number of p is one, then the theorem follows from Lemma 8.7. If the Milnor number is greater than one, the theorem follows by continuity from the case of the Milnor number equal to one, since all objects involved depend continuously on the weights a and parameters z. Note that the theorem says that the element s of Lemma 8.7 in our situation equals (−1) k .
Remark 8.12. In formula (8.3), each of α p (f ), α p (g) is given by the Grothendieck residue, so each of α p (f ), α p (g) is a k-dimensional integral. The quantity (f, g) p is also a k-dimensional integral. Thus formula (8.3) is an equality relating a bilinear expression in k-dimensional integrals to an individual k-dimensional integral.
Denote by A Yp the unital subalgebra of End(Y p ) generated by K j (z) * | Yp , j ∈ J.
(ii) The maps α p , β p give an isomorphism of the regular representation of A p,Φ and the A Ypmodule Y p , that is β p (h)α p (g) = α p (hg) for any h, g ∈ A p,Φ .
(iii) The algebra A Yp is a maximal commutative subalgebra of End(Y p ).
(iv) All elements of the algebra A Yp are symmetric operators with respect to the contravariant form S (a) .
Theorem 8.14. Let p 1 , . . . , p d be a list of all distinct isolated critical points of Φ(z, · ). Let Y ps = α ps (A ps,Φ ) ⊂ Sing W , s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct. The subspaces are orthogonal.
Proof . It follows from Theorem 8.10 that for any s = 1, . . . , d and j ∈ J the operator K j (z) * − a j /f j (z, p s ) restricted to Y ps is nilpotent. We also know that the numbers a j /f j (z, p s ) separate the points p 1 , . . . , p d . These observations imply Theorem 8.14. Denote Y = ⊕ d s=1 Y ps . Denote by A Y the unital subalgebra of End(Y ) generated by K j (z) * | Y , j ∈ J. Consider the isomorphisms Corollary 8. 16. We have (iii) The isomorphisms α, β identify the regular representation of the algebra ⊕ d s=1 A ps,Φ and the A Y -module Y . The isomorphism α identifies the contravariant form on Y and the residue form ( , ) = ⊕ d s=1 ( , ) ps on ⊕ d s=1 A ps,Φ multiplied by (−1) k .
(iv) In particular, if the dimension of Sing W equals the sum of Milnor numbers d s=1 µ s , then the module Sing W over the unital subalgebra of End(Sing W ) generated by geometric Hamiltonians K j (z) * | Sing W : Sing W → Sing W , j ∈ J, is isomorphic to the regular representation of the algebra ⊕ d s=1 A ps,Φ . (v) If for z ∈ C n − ∆ the arrangement (A(z), a) is unbalanced, then the module Sing W over the unital subalgebra of End(Sing W ) generated by geometric Hamiltonians K j (z) * | Sing W , j ∈ J, is isomorphic to the regular representation of the algebra ⊕ d s=1 A ps,Φ .
Corollary 8.17. If for z ∈ C n − ∆ the arrangement (A(z), a) is unbalanced, then the contravariant form is nondegenerate on Sing F k (A(z)).
Proof . Indeed in this case the sum of Milnor numbers of critical points of the master function equals |χ(U (A(z)))| and equals dim Sing F k (A(z)).

Interpretation of the algebra of Hamiltonians of bad fibers if Assumption 7.4 is satisfied
Let Assumption 7.4 be satisfied. Fix z 0 ∈ ∆. We have defined the quantum integrable model assigned to (A(z 0 ), a) to be the collection Sing F k A z 0 ; S (a) | Sing F k (A(z 0 )) ; , see Theorem 7.5. Let p ∈ U (A(z 0 )) be an isolated critical point of the master function Φ(z 0 , · ) : U (A(z 0 )) → C. Let A p,Φ be the local algebra of the critical point and ( , ) p the residue bilinear form on A p,Φ . Let [E] ∈ A p,Φ ⊗ Sing F k (A(z 0 )) be the canonical element corresponding to the arrangements A(z 0 ), see Section 7.8.
Define a linear map Denote Y p the image of α p . (ii) For any j ∈ J, the operatorK j (z 0 ) preserves Y p . Moreover, for any j ∈ J and g ∈ A p,Φ , we have α p (ga j /[f j (z 0 , ·)]) =K j (z 0 )α p (g).

. Then the value [E](p) is nonzero. The vector [E](p)
is the only (up to proportionality) common eigenvector of the operatorsK j (z 0 )| Yp : Y p → Y p , j ∈ J, and we havẽ Denote by A Yp the unital subalgebra of End(Y p ) generated byK j (z 0 )| Yp , j ∈ J.
Corollary 8. 19. Let Assumption 7.4 be satisfied. Then (ii) The maps α p , β p give an isomorphism of the regular representation of A p,Φ and the A Ypmodule Y p , that is β p (h)α p (g) = α p (hg) for any h, g ∈ A p,Φ .
(iii) The algebra A Yp is a maximal commutative subalgebra of End(Y p ).
Recall that under Assumption 7.4 all critical points of Φ(z 0 , · ) are isolated, the sum of their Milnor numbers equals dim Sing F k (A(z 0 )) and the form S (a) | Sing F k (A(z 0 )) is nondegenerate.
Denote by A A(z 0 ),a the unital subalgebra of End(Sing F k (A(z 0 ))) generated byK j (z 0 ), j ∈ J. The algebra A A(z 0 ),a is called the algebra of geometric Hamiltonian of the arrangement (A(z 0 ), a), see Section 7.3.
Consider the isomorphisms Corollary 8.21. Let Assumption 7.4 be satisfied. Then ,a is a maximal commutative subalgebra of End(Sing F k (A(z 0 ))).
(iii) The isomorphisms α, β identify the regular representation of the algebra ⊕ d s=1 A ps,Φ and the A A(z 0 ),a -module Sing F k (A(z 0 )). The isomorphism α identifies the contravariant form on Sing F k (A(z 0 )) and the residue form ( , ) = ⊕ d s=1 ( , ) ps on ⊕ d s=1 A ps,Φ multiplied by (−1) k .
9 More on Hamiltonians of bad f ibers 9.1 An abstract setting Let k < n be positive integers and J = {1, . . . , n} as before.
Let B ⊂ C k be a ball with center at a point p. Let F u be a holomorphic function on B depending holomorphically on a complex parameter u at u = 0. Assume that F 0 = F u=0 has a single critical point at p with Milnor number µ. Let C(t) B be the algebra of holomorphic functions on B, I u ⊂ C(t) B the ideal generated by ∂F u /∂t i , i = 1, . . . , k, and A u = C(t) B /I u . Assume that dim C A u does not depend on u for u in a neighborhood of 0. Let [ ] u : C(t) B → A u be the canonical projection, ( , ) u the residue bilinear form on A u and Hess F u = det 1 l,m k (∂ 2 F u /∂t l ∂t m ).
Let h 1 , . . . , h µ ∈ C(t) B be a collection of elements such that for any u the elements [h 1 ] u , . . . , [h µ ] u form a C-basis of A u .
Let g 1,u , . . . , g n,u ∈ C(t) B be elements depending on u holomorphically at u = 0 and such that for any u (close to 0) the unital subalgebra of A u generated by [g 1,u ] u , . . . , [g n,u ] u equals A u .
Let W be a vector space with a symmetric bilinear form S. For u = 0, let M j,u : W → W , j ∈ J, be a collection of commuting symmetric linear operators, for all i, j ∈ J and x, y ∈ W.
We assume that every M j,u depends on u meromorphically (for u close to 0) and has at most simple pole at u = 0, Let w 1,u , . . . , w µ,u ∈ W be a collection of vectors depending on u holomorphically at u = 0. Consider the element Assume that for every nonzero u (close to 0) we have and for every u (close to 0) we have µp l,m=1 For any u, denote by Y u ⊂ W the vector subspace generated by w 1,u , . . . , w µ,u . By property (9.2), for any nonzero u (close to 0) every M j,u , j ∈ J, preserves Y u . For any nonzero u (close to 0) denote by A Yu the unital subalgebra of End(Y u ) generated by M j,u | Yu , j ∈ J. The subspace Y u is an A Yu -module.
For any u (u = 0 included) define a linear map Theorem 9.1. For any u (in particular, for u = 0) the map α u : A u → Y u is an isomorphism of vector spaces. The form S restricted to Y u is nondegenerate.
Proof . Define a bilinear form ( , ) S,u on A u , Proof . For u = 0, the statement follows from Lemma 8.3. For u = 0, the statement follows by continuity.
The next two lemmas are similar to the corresponding analogs in Section 8.2.  Corollary 9.6. The space Y 0 is of dimension µ and w 1,0 , . . . , w l,0 is its basis.
(i) The map [g j,0 ] 0 →L j,0 , j ∈ J, extends uniquely to an algebra isomorphism β 0 : (ii) the algebra A Yu tends to the algebra A Y 0 as u → 0. More precisely, for any j ∈ J and l = 1, . . . , µ, we have M j,u w l,u →L j,0 w l,0 as u → 0.
LetỸ ⊂ W be a vector subspace such that  (ii) Let (ξ j ) j∈J ⊂ C be numbers such that j∈J ξ j M Proof . For j ∈ J, let w l,u = w l,0 + w (1) l u + · · · be the Taylor expansion of w l,u . By part (ii) of Theorem 9.7 we have The operator M Proof . The proof follows from remarks in Section 8.3.
Corollary 9.11. The vector [w] 0 (p) is the only (up to proportionality) common eigenvector of the operators prỸ M Assume that the parameter u is changed holomorphically, u = c 1 v + c 2 v 2 + · · · where c i ∈ C, c 1 = 0, and v is a new parameter. Let be the new Laurent expansion.
Lemma 9.12. For any j ∈ J, we have prỸ M i | Y 0 and the algebra A Y 0 ⊂ End(Y 0 ) does not change under the reparametrization of u.
Proof . One proof of the lemma follows from Theorem 9.7. Another proof follows from the fact thatM

Hamiltonians of bad fibers
Let us return to the situation of Sections 3 and 7 and recall the previous constructions. Let z 0 ∈ ∆. Let V → V * be the map associated with the contravariant form. Let W , Sing W , W (z 0 ), Sing W (z 0 ) be the images of V , Sing V , F k (A(z 0 )), Sing F k (A(z 0 )), respectively. The contravariant form on V induces a nondegenerate symmetric bilinear form on W also denoted by S (a) .
For z ∈ C n − ∆, we have linear operators K j (z) : The dual operators K j (z) * : V * → V * preserve the subspaces Sing W ⊂ W ⊂ V * , commute on the subspace Sing W and are symmetric on W with respect to the contravariant form. The operators L * C : V * → V * , C ∈ C, preserve the subspaces Sing W ⊂ W ⊂ V * . The space W (z 0 ) lies in the kernel of L * C | W : W → W for any C ∈ C 0 . Let T z 0 = {ξ = j∈J ξ j ∂ ∂z j | ξ j ∈ C, ξ(f C ) = 0 for all C ∈ C 0 }. Let p ∈ U (A(z 0 )) be an isolated critical point of the master function Φ(z 0 , · ) : U (A(z 0 )) → C. Let A p,Φ be the local algebra of the critical point and ( , ) p the residue bilinear form on A p,Φ . Let [ ] : C(t) U (A(z 0 )) → A p,Φ be the canonical projection and [E] ∈ A p,Φ ⊗ Sing W (z 0 ) the element corresponding to the canonical element.
Define a linear map Denote Y p the image of α p .
(i) We have ker α p = 0. The isomorphism α p identifies the contravariant form on Y p and the residue form ( , ) p on A p,Φ multiplied by (−1) k . In particular, the contravariant form on Y p is nondegenerate.
(ii) LetỸ ⊂ Sing W be a vector subspace such that (a) Y p ⊂Ỹ ; (b) the contravariant form restricted onỸ is nondegenerate; (c) for any j ∈ J, the subspaceỸ lies in the kernel of L * C , C ∈ C 0 .
Let prỸ : Sing W →Ỹ be the orthogonal projection. Then for any j ∈ J the operator prỸ K 1 j (z 0 ) * | Yp maps Y p to Y p and does not depend on the choice ofỸ . The operators prỸ K 1 j (z 0 ) * | Yp : Y p → Y p , j ∈ J, commute and are symmetric with respect to the contravariant form on Y p .
(iv) The naive geometric Hamiltonians K ξ (z 0 ) * , ξ ∈ T z 0 , preserve the subspace Y p and the operators K ξ (z 0 ) * | Yp are elements of the subalgebra A Yp . ( is the only (up to proportionality) common eigenvector of the operators prỸ K 1 j (z 0 ) * | Yp , j ∈ J, and we have [E](p).
(vi) For any j ∈ J, g ∈ A p,Φ , we have extends uniquely to an algebra isomorphism β p : (vii) The isomorphisms α p , β p identify the regular representation of the algebra A p,Φ and the A Yp -module Y p .
All statements of the theorem (but the second statement of part (i)) follow from the corresponding statements of Section 9.1. The second statement of part (i) has the same proof as Theorem 8.11. Theorem 9.14. Let p 1 , . . . , p d be a list of all distinct isolated critical points of Φ(z 0 , · ). Let Y ps = α ps (A ps,Φ ) ⊂ Sing W (z 0 ), s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct and orthogonal with respect to the contravariant form.
Corollary 9.15. The sum of Milnor numbers of the critical points p 1 , . . . , p d is not greater than the rank of the contravariant form S (a) | Sing F k (A(z 0 )) .
(b) the contravariant form restricted onỸ is nondegenerate; (c) for any j ∈ J, the subspaceỸ lies in the kernel of L * C , C ∈ C 0 .
For example, we can chooseỸ = Y . Let prỸ : Sing W →Ỹ be the orthogonal projection.
(i) For any j ∈ J the operator prỸ K 1 j (z 0 ) * maps Y to Y and does not depend on the choice ofỸ . The operators prỸ K 1 j (z 0 ) * | Y : Y → Y , j ∈ J, commute, preserve each of the subspaces Y ps and are symmetric with respect to the contravariant form on Y .
Denote by A Y be the unital subalgebra of End(Y ) generated by prỸ K 1 j (z 0 ) * | Y , j ∈ J. (ii) The naive geometric Hamiltonians K ξ (z 0 ) * , ξ ∈ T z 0 , preserve the subspace Y and the operators K ξ (z 0 ) * | Y are elements of the subalgebra A Y .
(iii) Consider the isomorphisms is a maximal commutative subalgebra of End(Y ); (c) the isomorphisms α, β identify the regular representation of the algebra ⊕ d s=1 A ps,Φ and the A Y -module Y ; the isomorphism α identifies the contravariant form on Y and the residue form ( , ) = ⊕ d s=1 ( , ) ps on ⊕ d s=1 A ps,Φ multiplied by (−1) k ; (d) in particular, if the rank of the contravariant form S (a) | Sing F k (z 0 ) equals the sum of Milnor numbers of the points p 1 , . . . , p d , then Y = Sing W (z 0 ) and the module Sing W (z 0 ) over the unital subalgebra of End(Sing W (z 0 )) generated by geometric Hamiltonians prỸ K 1 j (z 0 ) * | Sing W (z 0 ) , j ∈ J, is isomorphic to the regular representation of the algebra ⊕ d s=1 A ps,Φ ; The theorem follows from the corresponding statements of Section 9.1.
Theorem 9.17. Assume that the arrangement (A(z 0 ), a) is unbalanced.
(i) Then the contravariant form is nondegenerate on Sing W (z 0 ).
(ii) Let pr Sing W (z 0 ) : Sing W → Sing W (z 0 ) be the orthogonal projection, A Sing W (z 0 ) be the unital subalgebra of End(Sing W (z 0 )) generated by the operators pr Sing W (z 0 ) K 1 j (z 0 ) * | Sing W (z 0 ) , j ∈ J. Then A Sing W (z 0 ) is commutative and its elements are symmetric with respect to the contravariant form on Sing W (z 0 ).
(iii) Let p 1 , . . ., p d be a list of all distinct isolated critical points of Φ(z 0 , · ). Then the A Sing W (z 0 )module Sing W (z 0 ) is isomorphic to the regular representation of the algebra ⊕ d s=1 A ps,Φ . Proof . If (A(z 0 ), a) is unbalanced, then the sum of Milnor numbers of the master function equals |χ(U (A(z 0 )))| and equals dim Sing F k (A(z 0 )). This implies part (i) of the theorem. Parts (ii) and (iii) follow from Theorem 9.16.

Remark on critical points of real arrangements
Assume that (g j ) j∈J are real, see Remark 3.2. Assume that z 0 ∈ R n ⊂ C n . Assume that the contravariant form is positive definite on Sing W (z 0 ). Let pr Sing W (z 0 ) : Sing W → Sing W (z 0 ) be the orthogonal projection. Assume that the operators commute and are symmetric with respect to the contravariant form. Proof . On one hand, under assumptions of the theorem all the linear operators preserve Sing W (z 0 ) R and can be diagonalized simultaneously. That means that any element of the algebra of geometric Hamiltonians A Sing W (z 0 ) (generated by operators pr On the other hand, if the Milnor number of p is greater than one, then the local algebra A p,Φ has nilpotent elements and, by Theorem 9.13, the algebra A Sing W (z 0 ) has nondiagonalizable elements.
The second part of the theorem is clear.
Theorem 9.18 is in the spirit of the main theorem of [12] and Conjecture 5.1 in [11]. The main theorem of [12] says that certain Schubert cycles intersect transversally and all intersection point are real. These two statements correspond to the two statements of Theorem 9.18.

A family of prediscriminantal arrangements
In this section we consider a special family of parallelly translated hyperplanes, see Section 3. The members of that family will be called prediscriminantal arrangements. Data 10.1. Let h * be a complex vector space of dimension r with a collection of vectors α 1 , . . . , α r , Λ 1 , . . . , Λ N ∈ h * and a symmetric bilinear form ( , ). We assume that (α i , α i ) = 0 for every i = 1, . . . , r.
Let k = (k 1 , . . . , k r ) be a collection of nonnegative integers. We denote k = i k i and assume that k > 0. We assume that for every b = 1, . . . , N there exists i such that (α i , Λ b ) = 0 and k i > 0.
Consider the expressions such that i = 1, . . . , r and 1 l < l ′ k i ; (10.1) Let J denote the set of all low indices of the letters f in these expressions. So J is the union of three nonintersecting subsets J 1 , J 2 , J 3 where J 1 consists of triples {(i), l, l ′ } from the first line of (10.1), J 2 consists of four-tuples {(i, i ′ ), l, l ′ } from the second line, J 3 consists of triples {(i, b), l} from the third line. Let n be the number of elements in J.
Consider C k with coordinates kr .
Consider C n with coordinates z = (z j ) j∈J and C n × C k with coordinates z, t. For any j ∈ J the expression f j can be considered as a linear function on C n × C k . We have The functions g j , j ∈ J, can be considered as linear functions on C k .
For j ∈ J, the equation f j (z, t) = 0 defines a hyperplaneH j ⊂ C n × C k and we get an We assign (nonzero) weights a j to hyperplanes ofC by putting The weighted arrangementC is an example of a family of parallelly translated hyperplanes considered in Sections 3-9.

Discriminantal arrangements
Let X ⊂ C n be the subset defined by the following equations: z (i),l,l ′ = 0, i = 1, . . . , r and 1 l < l ′ k i ; The subset X is an N -dimensional affine space. We will use the following coordinates x 1 , . . . , x N on X defined by the equations Let us consider the arrangement A(z) = (H j (z)) j∈J in the fiber C k of the projection C n × C k → C n over a point z ∈ U (X) with coordinates x 1 (z), . . . , x N (z). Its hyperplanes are defined by the equations: . . , r and 1 l < l ′ k i ; The weights of these hyperplanes are defined by formula (10.2). This weighted arrangement is called discriminantal, see [23,25].
Clearly every point of X ⊂ C n is a fixed point of the S k -action.
The actions of S k on C n and C k induce an action on C n × C k . The S k -action on C n × C k preserves the arrangementÃ and sends fibers of the projection C n × C k → C n to fibers.
The S k -action onÃ corresponds to an S k -action on the set J = J 1 ∪ J 2 ∪ J 3 . That action preserves the summands. For σ = (σ 1 , . . . , σ r ) ∈ S k we have Consider the discriminant ∆ = ∪ C∈C H C ⊂ C n , see Section 3.2. Here C is the set of all circuits of the matroid of the collection (g j ) j∈J . Clearly the S k -action on C n preserves the discriminant and permutes the hyperplanes The action of S k on the hyperplanes of the discriminant corresponds to the following action on C. If C = {j 1 , . . . , j l } ⊂ J is a circuit and σ ∈ S k , then σ(C) is the circuit {σ(j 1 ), . . . , σ(j l )}.

The S k -action on geometric Hamiltonians
We use notations of Section 3.3 and for z ∈ C n −∆ denote V = F k (A(z)), Sing V = Sing F k (A(z)). The triple (V, Sing V, S (a) ) does not depend on z ∈ C n − ∆ as explained in Section 3.3.
Fix an order on the set J. Recall that the standard basis of V * = A k (A(z)), associated with an order on J, is formed by elements (H j 1 , . . . , H j k ) where {j 1 < · · · < j k } runs through the set of all independent ordered k-element subsets of J. The (dual) standard basis of V is formed by the corresponding vectors F (H j 1 (z), . . . , H j k (z)). We have F (H j 1 (z), . . . , H j k (z)) = (−1) |µ| F (H j µ(1) (z), . . . , H j µ(k) (z)) for any µ ∈ S k , see Section 2.11.
The S k -action on J induces an action on V and V * . For σ ∈ S k , we have and (H j 1 (z), . . . , H j k (z)) → (H σ(j 1 ) (z), . . . , H σ(j k ) (z)). The S k -action on V preserves the subspace Sing V and preserves the contravariant form S (a) on V . Let z 0 ∈ U (X). Then z 0 is S k -invariant and the group S k acts on the fiber C k over z 0 and on the weighted arrangement (A(z 0 ), a) in that fiber. The subspaces F k (A(z 0 )), Sing F k (A(z 0 )) of V are S k -invariant.
Let V → V * be the map associated with the contravariant form. Let W , Sing W , W (z 0 ), Sing W (z 0 ) be the images of V , Sing V , F k (A(z 0 )), Sing F k (A(z 0 )), respectively. All these subspaces are S k -invariant.
An S k -action on a vector space defines an S k -action on linear operators on that space. For σ ∈ S k and a linear operator L we define σ(L) = σLσ −1 .
In Section 5.2 we have defined operators L C : V → V , C ∈ C. Clearly for any σ ∈ S k and any C ∈ C we have σ(L C ) = L σ(C) .
In Section 5.2 we have considered differential 1-forms on C n × C k which were denoted by ω j , j ∈ J, and ω C , C ∈ C. The S k -action on C n × C k preserves this set of differential 1-forms. Namely for any j ∈ J, C ∈ C, σ ∈ S k , we have σ : ω j → ω σ(j) , ω C → ω σ(C) .
Lemma 10.2. The following objects are S k -invariant: see formula (5.4). The functions K j (z) are End(V )-valued meromorphic functions on C n .
Since S k acts on C n and End(V ) it also acts on End(V )-valued functions on C n , σ : F (q) → σF (σ −1 (q))σ −1 for q ∈ C n . Lemma 10.2 allows us to describe the S k -action on functions K j (z). Corollary 10.3. An element σ = (σ 1 , . . . , σ r ) ∈ S k acts on functions K j , j ∈ J, by the formulas: Let z 0 ∈ U (X). Recall that C 0 = {C ∈ C | z 0 ∈ H C }, Corollary 10.4. An element σ ∈ S k acts on operators K 1 j (z 0 ) : V → V , j ∈ J, by the formula where the minus sign is chosen only if j = {(i), l, l ′ } and σ i (l) > σ(l ′ ).

Functions
, ξ(f C ) = 0 for all C ∈ C 0 }, see Section 7.1. Define the following constant vector fields on C n , By Lemma 7.1, the functions K ∂y b (z) are regular at z 0 and their dual operators preserve the subspace Sing W (z 0 ) ⊂ V * . Corollary 10.6. For z 0 ∈ U (X), the operators The operators K ∂x b (z 0 ) * | Sing W (z 0 ) are naive geometric Hamiltonians on Sing W (z 0 ) in the sense of Section 7.1. They commute and they are symmetric operators with respect to the contravariant form.

Naive geometric Hamiltonians on Sing
The space W has the canonical direct sum decomposition into isotypical components corresponding to irreducible representations of S k . One of the isotypical components is the component corresponding to the alternating representation. If L : W → W is an S k -invariant linear operator, then L preserves the canonical decomposition and, in particular, it preserves the subspace W − .
Let z 0 ∈ U (X). Then the subspace Sing W (z 0 ) ⊂ W is an S k -submodule. We define will be called naive geometric Hamiltonians on Sing W − (z 0 ).
Lemma 10.7. An element σ ∈ S k acts on functions f j , j ∈ J, by the formula where the minus sign is chosen only if j = {(i), l, l ′ } and σ i (l) > σ(l ′ ).
The S k -action on C(t) U (A(z 0 )) induces an isomorphism σ : Let us compare the residue bilinear forms on A p,Φ and A σ(p),Φ and projections of the canonical element to A p,Φ and A σ(p),Φ .
By Theorem 9.14, the subspaces Y σ(p) are all orthogonal and the contravariant form on Let Ant = σ∈S k (−1) σ σ : Sing W (z 0 ) → Sing W − (z 0 ) be the anti-symmetrization operator. Denote Define a linear monomorphism The linear map α − p identifies the contravariant form on Y − O(p) and the residue bilinear form on A p,Φ multiplied by (−1) k k 1 ! · · · k r !. In particular, the contravariant form on Y − O(p) is nondegenerate. We also have (c) for any j ∈ J, the subspaceỸ lies in the kernel of L * C , C ∈ C 0 ; (d) the subspaceỸ is S k -invariant.
For example, we can chooseỸ = Y O(p) . Let prỸ : Sing W →Ỹ be the orthogonal projection with respect to the contravariant form.
Important def inition. Denote by P S k the algebra of polynomials with complex coefficients in n variables a j /f j , j ∈ J, such that for any F (a j /f j , j ∈ J) ∈ P S k the function F (a j /f j (z 0 , ·), j ∈ J) ∈ U (A(z 0 )) is S k -invariant.
Lemma 10.13. The natural homomorphism P S k → A p,Φ is an epimorphism.
Let F (a j /f j (z 0 , ·), j ∈ J) ∈ P S k . Replace in F each variable a j /f j with the operator prỸ K 1 j (z 0 ) * . Denote the resulting operator on Sing W by F (prỸ K 1 j (z 0 ) * , j ∈ J). This operator preserves Y σ(p) for any σ ∈ S k and its restriction to Y σ(p) does not depend on the choice ofỸ . The operator F (prỸ K 1 j (z 0 ) * , j ∈ J) is S k -invariant by Corollary 10.4. Hence, F (prỸ K 1 j (z 0 ) * , j ∈ J) preserves Y − O(p) . Theorem 10.14.
(i) For F ∈ P S k , the operators commute and are symmetric with respect to the contravariant form.
induces an algebra monomorphism , is a maximal commutative subalgebra .
(v) The maps α − p , β − p define an isomorphism of the regular representation of A p,Φ and the . The linear map α − p identifies the contravariant form on Y − O(p) and the residue bilinear form on A p,Φ multiplied by (−1) k k 1 ! · · · k r !.
, F ∈ P S k , we have  (c) for any j ∈ J, the subspaceỸ lies in the kernel of L * C , C ∈ C 0 ; (d) the subspaceỸ is S k -invariant.
For example, we can chooseỸ = ⊕ d s=1 Y O(ps) . Let prỸ : Sing W →Ỹ be the orthogonal projection. Denote do not depend on the choice ofỸ . They commute and are symmetric with respect to the contravariant form.
Consider the isomorphisms . Then (c) the isomorphisms α − , β − identify the regular representation of the algebra ⊕ d s=1 A ps,Φ and the A Y − -module Y − ; the linear map α − identifies the contravariant form on Y − and the residue bilinear form ( , ) = ⊕ d s=1 ( , ) ps on ⊕ d s=1 A ps,Φ multiplied by (−1) k k 1 ! · · · k r !; (d) in particular, if the rank of the contravariant form S (a) | Sing W − (z 0 ) equals the sum of Milnor numbers of the points p 1 , . . . , p d , then the module Sing W − (z 0 ) = Y − over the unital subalgebra of End(Sing W − (z 0 )) generated by geometric Hamiltonians F (prỸ K 1 j (z 0 ) * , j ∈ J)| Y − , F ∈ P S k , is isomorphic to the regular representation of the algebra ⊕ d s=1 A ps,Φ .
Proof . In this case Y − = Sing W − (z 0 ) and the corollary follows from Theorem 10.17.
11 Applications to the Bethe ansatz of the Gaudin model

Gaudin model
Let g be a simple Lie algebra over C with Cartan matrix (a i,j ) r i,j=1 . Let h ⊂ g be a Cartan subalgebra. Fix simple roots α 1 , . . . , α r in h * and a nondegenerate g-invariant bilinear form ( , ) on g. The form identifies g and g * and defines a bilinear form on g * . Let H 1 , . . . , H r ∈ h be the corresponding coroots, λ, H i = 2(λ, α i )/(α i , α i ) for λ ∈ h * . In particular, α j , H i = a i,j .
Let E 1 , . . . , E r ∈ n + , H 1 , . . . , H r ∈ h, F 1 , . . . , F r ∈ n − be the Chevalley generators of g, and (ad E i ) 1−a i,j E j = 0, (ad F i ) 1−a i,j F j = 0, for all i = j. Let (J i ) i∈I be an orthonormal basis of g, Ω = i∈I J i ⊗ J i ∈ g ⊗ g the Casimir element.
For a g-module V and µ ∈ h * denote by V [µ] the weight subspace of V of weight µ and by Sing V [µ] the subspace of singular vectors of weight µ, Let N > 1 be an integer and Λ = (Λ 1 , . . . , Λ N ), Λ b ∈ h * , a set of weights. For µ ∈ h * let V µ be the irreducible g-module with highest weight µ. Denote For X ∈ End(V Λ i ), denote by the operator acting nontrivially on the i-th factor only. For The operators are called the Gaudin Hamiltonians. The Hamiltonians commute, The Hamiltonians commute with the g-action on V Λ . Hence they preserve the subspaces Sing V Λ [µ] ⊂ V Λ of singular vectors of a given weight µ. Let τ : g → g be the anti-involution sending E i , H i , F i , to F i , H i , E i , respectively, for all i. Let W be a highest weight g-module with a highest weight vector w. The Shapovalov form S on W is the unique symmetric bilinear form such that for all u, v ∈ W and g ∈ g.
Fix highest weight vectors v 1 , . . . , v N of V Λ 1 , . . . , V Λ N , respectively. Define a symmetric bilinear form on the tensor product V Λ by the formula where S b is the Shapovalov form on V Λ b . The form S Λ is called the tensor Shapovalov form.
For any µ ∈ h * , the Gaudin model on Sing V Λ [µ] ⊂ V Λ is the collection see [7,8]. The main problem for the Gaudin model is to find common eigenvectors and eigenvalues of the Gaudin Hamiltonians.
11.2 Master function and weight function, [23] The eigenvectors of the Gaudin Hamiltonians are constructed by the Bethe ansatz method. We remind that construction in this section. Fix a collection of nonnegative integers k = (k 1 , . . . , k r ). Denote k = k 1 + · · · + k r , Consider C k with coordinates kr . Define the master function We consider Φ as a function of t depending on parameters x 0 . Denote by U the set of all points p ∈ C k such that for any log h entering (11.1) (with a nonzero coefficient) we have h(p) = 0. The set U is the complement in C k to the union of hyperplanes. The coefficients of the logarithms in (11.1) define weights of the hyperplanes. This weighted arrangement is called discriminantal. Discriminantal arrangements were considered in Section 10.2.
Let us construct the weight function ω : ). To each b ∈ B and σ ∈ Σ(b) we assign the vector and the rational function and ω e σ,b (x 0 e ) = 1 if b e = 0. Then the weight function is given by the formula The weight function is a function of t depending on parameters x 0 .
Lemma 11.1 (Lemma 2.1 in [16]). The weight function is regular on U .
The vector ω(x 0 , p, k) is called the Bethe vector corresponding to the critical point p.
Theorem 11.2. If p is an isolated critical point of the master function, then In particular, if the critical point is nondegenerate, then the Bethe vector is nonzero.
This theorem is proved for g = sl r+1 in [16] and for any simple Lie algebra in [27].

Identification of Gaudin and naive geometric Hamiltonians
Let us identify constructions and statements of Theorems 11.1, 11.2 and of Theorem 10.14.
From statements (i)-(iv) and Theorem 10.14 we get the following improvement of Theorem 11.2. Theorem 11.3. For any simple Lie algebra g, if p is an isolated critical point of the master function Φ(x 0 , ·, Λ, k), then the corresponding Bethe vector ω(x 0 , p, k) is nonzero.

Bethe algebra
The subalgebra of End(Sing V Λ [µ]) generated by the Gaudin Hamiltonians can be extended to a larger commutative subalgebra called the Bethe algebra. A construction of the Bethe algebra for any simple Lie algebra g is given in [4]. As a result of that construction, for any x 0 one obtains a commutative subalgebra B(x 0 ) ⊂ (U g) ⊗N which commutes with the diagonal subalgebra U g ⊂ (U g) ⊗N  A more straightforward construction of the Bethe algebra is known for the Gaudin model of gl r+1 , see [24]. Below we give its description.
Let gl r+1 [s] = gl r+1 ⊗ C[s] be the Lie algebra of gl r+1 -valued polynomials with the pointwise commutator. For g ∈ gl r+1 , we set g(u) = ∞ i=0 (g ⊗ s i )u −i−1 . We identify gl r+1 with the subalgebra gl r+1 ⊗ 1 of constant polynomials in gl r+1 [s]. Hence, any gl r+1 [s]-module has a canonical structure of a gl r+1 -module. For each a ∈ C, there exists an automorphism ρ a of gl r+1 [s], ρ a : g(u) → g(u − a). Given a gl r+1 [s]-module W , we denote by W (a) the pull-back of W through the automorphism ρ a . As gl r+1 -modules, W and W (a) are isomorphic by the identity map.
We have the evaluation homomorphism, gl r+1 [s] → gl r+1 , g(u) → gu −1 . Its restriction to the subalgebra gl r+1 ⊂ gl r+1 [s] is the identity map. For any gl r+1 -module W , we denote by the same letter the gl r+1 [s]-module, obtained by pulling W back through the evaluation homomorphism.
As a subalgebra of U gl r+1 [s], the algebra B acts on any gl r+1 [s]-module W . Since B commutes with U gl r+1 , it preserves the gl r+1 weight subspaces of W and the subspace Sing W of gl r+1singular vectors.
If W is a B-module, then the image of B in End(W ) is called the Bethe algebra of W . Let V Λ = ⊗ n b=1 V Λ b be a tensor product of irreducible highest weight gl r+1 -modules. For given x 0 = (x 0 1 , . . . , x 0 N ), consider V Λ as the gl r+1 [s]-module ⊗ n b=1 V Λ b (x 0 b ). This gl r+1 [s]-module structure on V Λ provides V Λ with a Bethe algebra, a commutative subalgebra of End(V Λ ). It is known that this Bethe algebra of V Λ contains the Gaudin Hamiltonians K b (x 0 ), b = 1, . . . , N . In fact, the Gaudin Hamiltonians are suitably normalized residues of the generating function B 2 (u), see Appendix B in [10]. Theorem 11.4 ([12]). Consider V Λ as the gl r+1 [s]-module ⊗ n b=1 V Λ b (x 0 b ). Then any element B ∈ B acts on V Λ as a symmetric operator with respect to the tensor Shapovalov form, S Λ (Bu, v) = S Λ (u, Bv) for any u, v ∈ V Λ . 11.6 gl r+1 Bethe algebra and critical points of the master function In [15] the following generalization of Theorem 11.1 for g = gl r+1 was obtained.
Let u be a variable. Define polynomials T 1 , . . . , T r ∈ C[u], Q 1 , . . . , Q r ∈ C[u, t], j , and the differential operator where ∂ u = d/du and log ′ f denotes (df /du)/f . We have Let p ∈ U be an isolated critical point of the master function Φ(x, , · , Λ, k) with Milnor number µ. Let A p,Φ be its local algebra. For f ∈ C(t) U denote by [f ] the image of f in A p,Φ . Denote be the element induced by the weight function.
Let g 1 , . . . , g µ be a basis of A p,Φ considered as a C-vector space. Write [ω] p = i g i ⊗ w i , with w i ∈ Sing V Λ [Λ ∞ ]. Denote by Y p ⊂ Sing V Λ [Λ ∞ ] the vector subspace spanned by w 1 , . . . , w µ . Let ( , ) p be the bilinear form on A p,Φ . Define a linear map (ii) The map α p : A p,Φ → Y p is an isomorphism of vector spaces.
(iii) The map [G ij ] p →B ij extends uniquely to an algebra isomorphism β p : A p,Φ → A Yp .
(iv) The isomorphisms α p and β p identify the regular representation of A p,Φ and the B-module Y p , that is, for any f, g ∈ A p,Φ we have α p (f g) = β p (f )α p (g).  Then (a) A Y = ⊕ d s=1 A Yp s ; (b) A Y is a maximal commutative subalgebra of End(Y); (c) the isomorphisms α, β identify the regular representation of the algebra ⊕ d s=1 A ps,Φ and the A Y -module Y.
Let us identify constructions and statements of Theorem 11.6 and of Theorems 10.14, 10.17. Consider the discriminantal arrangement A(z 0 ), z 0 ∈ U (X), defined in Section 11.4 and the isomorphism γ| Additional information that is given by Theorems 10.14, 10.17 is the following theorem.
Theorem 11.7. The monomorphism α p of Theorem 11.6 identifies the tensor Shapovalov form on Y p and the residue bilinear form on A p,Φ .
Proof . The theorem is a corollary of properties (iii), (iv) in Section 11.4 and Lemma 10.12.

Expectations
One may expect that for the Gaudin models of any simple Lie algebras the associated Bethe algebra defined in [4] coincides with the algebra of geometric Hamiltonians associated with the arrangement of the corresponding master function. That topic will be discussed in a forthcoming paper.
Below we consider three examples in which we have a well-defined algebra of geometric Hamiltonians on Sing V Λ [Λ ∞ ].

Example
Consider the Gaudin model corresponding to the following data: the Lie algebra gl r+1 , the collection of dominant integral weights Λ = (Λ 1 , . . . , Λ N ) with Λ b = (1, 0, . . . , 0) for all b, a vector of nonnegative integers k = (k 1 , . . . , k r ) such that Λ ∞ is a partition, a collection of generic distinct complex numbers x 0 = (x 0 1 , . . . , x 0 N ). By [16], for generic x 0 the associated master function has a collection of critical points p 1 , . . . , p d such that the S k -orbits of these points do not intersect and the sum of Milnor numbers of these points equals the dimension of Sing V Λ [Λ ∞ ]. In this case Theorem 10.17 defines a maximal commutative subalgebra A Sing (W − (z 0 )) ⊂ End(Sing (W − (z 0 ))) containing naive geometric Hamiltonians. The isomorphism γ sends A Sing (W − (z 0 )) to a maximal commutative subalgebra of End(Sing V Λ [Λ ∞ ]) containing the Gaudin Hamiltonians. By [12,13], the Bethe algebra of Sing V Λ [Λ ∞ ] is a maximal commutative subalgebra of End(Sing V Λ [Λ ∞ ]) and the Bethe algebra of Sing V Λ [Λ ∞ ] is generated by the Gaudin Hamiltonians. Hence, in this case, γ establishes an isomorphism of the algebra of geometric Hamiltonians A Sing (W − (z 0 )) and the Bethe algebra of Sing V Λ [Λ ∞ ].