Communications in Mathematical Physics manuscript No. (will be inserted by the editor)

Quantum Monodromy in Integrable Systems

San Vũ Ngo.c^{1}^{,}^{2}

1 Institut Fourier UMR5582, B.P. 74, 38402 Saint-Martin d’Hères, France.

E-mail: San.Vu-Ngoc@ujf-grenoble.fr 2 Mathematics Institute, P.O. Box 80010, 3508 TA Utrecht, The Netherlands

Received: 21 April 1998 / Accepted: 8 December 1998

Abstract: Let P_{1}(h), …, P_{n}(h) be a set of commuting self-adjoint h-pseudo-differential operators on an n-dimensional manifold. If the joint principal symbol p is proper, it is known from the work of Colin de Verdière [6] and Charbonnel [3] that in a neighbourhood of any regular value of p, the joint spectrum locally has the structure of an affine integral lattice. This leads to the construction of a natural invariant of the spectrum, called the quantum monodromy. We present this construction here, and show that this invariant is given by the classical monodromy of the underlying Liouville integrable system, as introduced by Duistermaat [9]. The most striking application of this result is that all two degree of freedom quantum integrable systems with a focus-focus singularity have the same non-trivial quantum monodromy. For instance, this proves a conjecture of Cushman and Duistermaat [7] concerning the quantum spherical pendulum.

1. Introduction

Obstructions to the existence of global action-angle coordinates for completely integrable systems are well known since Duistermaat’s article [9]. It was then natural to raise the question about the impact of these obstructions on quantum integrable systems, at least for the (semi)-classical pseudo-differential quantisation on cotangent bundles. The first attempts in this direction were [7] and [11], both of them concerning the monodromy invariant for the example of the spherical pendulum. This system is indeed one of the simplest (along with the Champagne bottle [1]) that exhibits a non-trivial monodromy. The first of these articles [7] proposed a particularly interesting way of detecting the monodromy by observing a shift in the lattice structure of the joint spectrum. It is the purpose of this article to state, prove and explain this idea.

2  S. Vũ Ngo.c

Surprisingly enough, this idea of quantum monodromy has been sleeping for ten years, before new interest resulted in its experimental discovery in the spectrum of excited water molecules [4,5].

Back to mathematics, it turns out that, in the framework of semi-classical microlocal analysis (developed for integrable systems in [3]), there is a natural way of defining an invariant of the joint spectrum away from singularities of the principal symbols, that precisely describes the obstruction to the existence of a global lattice structure for the spectrum. The organisation of this article is as follows: we first extract the relevant properties of joint spectra, and define the quantum monodromy invariant for any set that shares these properties (Sect. 2). Then we prove in Sect. 3 that, for spectra, the quantum monodromy is precisely given by the classical monodromy of the underlying classical Hamiltonian system. The result is applied in Sect. 4 to the particularly interesting case of systems admitting a focus-focus singularity. The last Sect. 5 finally shows how to read off the monodromy from a picture of the spectrum. As an example, we use the spectrum of the Champagne bottle computed by Child [4].

2. Construction of the Quantum Monodromy

Let U be an open subset of ℝ^{n}, let H be a set of positive real numbers accumulating at 0, and for any h in H let Σ(h) be a discrete subset of U.

If B is an open subset of U, a family (f(h))_{h∈H} of smooth functions on B with values in ℝ^{n} is called a symbol (of order zero) if it admits an asymptotic expansion of the form

f(h) = f_{0} + hf_{1} + h^{2}f_{2} + ⋯

for smooth functions f_{i} : B → ℝ^{n}. More precisely we require that for any ℓ ≥ 0, for any N ≥ 0, and for any compact K ⊂ B, there is a constant C_{ℓ,N,K} such that for all h ∈ H,

∥f(h) − ∑_{k=0}^{N} h^{k}f_{k}∥_{ℓ} ≤ C_{ℓ,N,K}h^{N+1},

where ∥.∥_{ℓ} denotes the C^{ℓ} norm in K. The symbol f(h) is elliptic if its principal part f_{0} is a local diffeomorphism of B into ℝ^{n}. The value of f(h) at a point c ∈ B will be denoted by f(h; c).

A family (r(h))_{h∈H} of elements of a finite dimensional vector space is said to be O(h^{∞}) if for any N ≥ 0 there is a constant C > 0 such that ∥r(h)∥ ≤ Ch^{N}, uniformly for all h ∈ H. If S(h) is any family of sets depending on h, then the notation f(h) ∈ S(h) + O(h^{∞}) means that the function dist(f(h), S(h)) is O(h^{∞}).

We will say that Σ(h) has the structure of an “asymptotic affine lattice” whenever it can be described with a locally finite set of “asymptotic affine integral charts”, in the following sense:

Definition 1. (Σ(h), U) is an “asymptotic affine lattice” if for any c ∈ U, there exists a small open ball B ⊂ U around c, and an elliptic symbol f(h) : B → ℝ^{n} of order zero such that, for any family λ(h) ∈ B :

UB  f(h)  h hℤ^{n}

Fig. 1. An asymptotic affine lattice

– λ(h) ∈ Σ(h) ∩ B + O(h^{∞}) ⟺ f(h; λ(h)) ∈ hℤ^{n} + O(h^{∞})

– if λ(h) and λ′(h) are in Σ(h) ∩ B, then λ′(h) − λ(h) = O(h^{∞}) if and only if for small h, λ′(h) = λ(h).

Intuitively this means that zooming by a factor of \frac{1}{h} inside B makes Σ(h)∩B converge to the standard lattice as h tends to zero. The issue here is to see what prevents Σ(h) from globally converging to a lattice. Of course, the reason for this definition is that, under suitable hypothesis, the joint spectrum of a set of n commuting h-pseudo-differential operators on an n-dimensional manifold is indeed an “affine asymptotic lattice” (see the next section).

For short, a symbol f(h) satisfying Definition 1 will be referred to as an “affine chart” of Σ(h).

The main point is that the transition functions associated to these charts are elements of the affine group GA(n, ℤ) (following Berger [2], we denote by GA(n, ℝ) the group of invertible affine transformations of ℝ^{n}, which is the semidirect product of the linear group GL(n, ℝ) by the normal subgroup of translations. Some authors use the notation Aff_{n}(ℝ) instead. The subgroup GA(n, ℤ) consists then of elements A ∈ GA(n, ℝ) such that A and A^{−1} leave ℤ^{n} globally invariant).

Proposition 1. Let f(h) and g(h) be two affine charts of Σ(h), both defined on a ball B. Then there is a unique A ∈ GA(n, ℤ) ⊂ GA(n, ℝ) such that

(\frac{g(h)}{h}) ∘ (\frac{f(h)}{h})^{−1} = A_{↾f(h)(B)/h} + O(h^{∞}).

Suppose now that U is covered by a locally finite union of balls B_{α} on each of which is defined an affine chart f_{α}(h) of Σ(h). Proposition 1 yields a family of affine linear maps A_{αβ} such that on non-empty intersections B_{α} ∩ B_{β},

\frac{1}{h}f_{α}(h) = A_{αβ} (\frac{1}{h}f_{β}(h)) .

This in turn defines a 1-cocycle M in the Čech cohomology of U with values in the non-Abelian group GA(n, ℤ).

Definition 2. The class [M] ∈ Ȟ^{1}(U, GA(n, ℤ)) of the cocycle defined by A_{αβ} is called the quantum monodromy of (Σ(h), U).

Let L be the canonical homomorphism, whose kernel is the group of translations:

L : GA(n, ℝ) → GL(n, ℝ).

Let ι be the inclusion of GL(n, ℝ) into GA(n, ℝ) such that for any M ∈ GL(n, ℝ), ι(M) leaves the origin 0 ∈ ℝ^{n} invariant. Then ι is an injective homomorphism that depends on the choice of the origin 0, satisfying L ∘ ι = Id. Any A ∈ GA(n, ℝ) can be written in a unique way

A = τ(k) ∘ ι(M),

(which is usually written A = M + k), where M = L(A) ∈ GL(n, ℤ) and τ(k) is translation by the vector k ∈ ℤ^{n}.

The exact sequence of group homomorphisms

0 \overline{}ℤ^{n} ^{^{τ} \overline{}}GA(n, ℤ) ^{^{L} \overline{}}GL(n, ℤ) \overline{}1

gives rise to the following sequence of maps (which are not homomorphisms, since cohomology sets with values in a non-abelian group have no natural group structure – see [12, p. 38]):

Ȟ^{1}(U, ℤ^{n}) ^{^{τ_{∗}} \overline{}}Ȟ^{1}(U, GA(n, ℤ)) ^{^{L_{∗}} \overline{}}Ȟ^{1}(U, GL(n, ℤ)) \overline{}1.

This sequence is “exact” in the sense that L_{∗} is surjective, and if L_{∗}([M]) = 1, then there is an integer cocycle [ω] ∈ Ȟ^{1}(U, ℤ^{n}) such that [M] = τ_{∗}([ω]). The surjectivity of L_{∗} is due to the existence of the cross section ι, which gives rise to the map

Ȟ^{1}(U, GA(n, ℤ)) ^{\overline{}^{ι_{∗}}} Ȟ^{1}(U, GL(n, ℤ))

such that L_{∗}ι_{∗} = Id. For the second point, we remark that if the cocycle L(A_{αβ}) is a coboundary, then it can be written M_{α}M_{β}^{−1}. Therefore the cocycle ι(M_{α}^{−1})A_{αβ}ι(M_{β}) (which is equivalent to A_{αβ}) has a linear part equal to the identity, hence is a translation.

Remark 1. The lack of injectivity for τ_{∗} is measured by Ȟ^{0}(U, GL(n, ℤ)) : one can check that two cocycles [k] and [k′] in Ȟ^{1}(U, ℤ^{n}) yield the same element of Ȟ^{1}(U, GA(n, ℤ)) if and only if there is an M ∈ Ȟ^{0}(U, GL(n, ℤ)) such that [k′] = [M · k].

Let us now give various interpretations of the quantum monodromy M.

The action of GA(n, ℤ) on ℤ^{n} being effective, it is a standard fact that the cohomology set Ȟ^{1}(U, GA(n, ℤ)) classifies the isomorphism classes of fibre bundles over U with structure group GA(n, ℤ) and fibre ℤ^{n} (see for instance [12, pp.40–41]). Let L be such a lattice bundle associated to M. The elements A_{αβ} just define the transition functions between two adjacent trivialisations of L.

Since these trivialisation functions are locally constant, there is a naturally defined parallel transport γ.p of a point p ∈ L_{c} along a path γ in the base U.

This defines the holonomy of L, as a map from π_{1}(U, c) into GA(L_{c}). We will always identify the latter with GA(n, ℤ) by choosing an affine basis of L_{c}.

The choice of such a basis is equivalent to that of a trivialisation f of L above c that sends this basis to the canonical basis of ℤ^{n}; the holonomy μ_{f} is then defined by :

f(γ.p) = μ_{f}(γ)f(p).  (1)

Finally, this is also equivalent to the choice of an affine chart f(h) of Σ(h) around c. If M is any cocycle associated to this trivialisation, then

μ_{f}(γ) = A_{1,ℓ} ∘ ⋯ ∘ A_{3,2} ∘ A_{2,1},  (2)

where A_{i,j} denotes the transition element corresponding to a pair of intersecting open balls (B_{i}, B_{j}), and B_{1}, …, B_{ℓ} enumerate elements of a cover of U encountered by γ(t) when t runs from 0 to 1.

We shall always assume that U is connected, so that μ_{f} does not depend on the base point c. Note that since (γ′γ).p = γ.(γ′.p), we have

μ_{f}(γ′γ) = μ_{f}(γ)μ_{f}(γ′).

It should be noticed that the bundles considered here have discrete fibres, so that we could reduce the discussion to the theory of coverings. The fibre bundle formulation seems however to be more natural when it comes to comparing them with objects arising in Hamiltonian systems. Nevertheless, the covering approach will be used in Sect. 5.

Other geometric interpretations of M will also be discussed in Sect. 5. For the moment just notice that the non-triviality of [M] is equivalent to the nontriviality of the lattice bundle L and to the fact that there is no globally defined symbol f(h) on U sending Σ(h) to the straight lattice hℤ^{n}.

Proof of Proposition 1. There are no surprises in this quite elementary proof. Let c ∈ U, and f(h), g(h) be two affine charts of Σ defined on a ball B around c. Because of Definition 1, any open ball around c contains, for h small enough, at least one element of Σ(h). Therefore, there exists a family λ(h) ∈ Σ(h) ∩ B such that

lim_{h→0} λ(h) = c.

Let k ∈ ℤ^{n} and let λ′(h) be a family of elements of Σ(h) ∩ B such that

f(h; λ(h)) = f(h; λ′(h)) + hk + O(h^{∞}).

Then, as h tends to zero, \frac{λ′(h)−λ(h)}{h} tends towards a limit v ∈ ℝ^{n} satisfying

k = df_{0}(c)v

(recall that f_{0} denotes the principal part of f(h)).

Since λ(h) and λ′(h) are in Σ(h), there is a family k′(h) ∈ ℤ^{n} such that

(\frac{g(h; λ′(h)) − g(h; λ(h))}{h}) = k′(h) + O(h^{∞}).

The left-hand side of the above equation has limit dg_{0}(c)v as h → 0. Therefore k′(h) is equal to a constant integer k′ for small h, and we have

k′ = dg_{0}(c)(df_{0}(c))^{−1}k,

which implies that dg_{0}(c)(df_{0}(c))^{−1} ∈ GL(n, ℤ). Since GL(n, ℤ) is discrete, there is a constant matrix M ∈ GL(n, ℤ) such that for all c ∈ B, dg_{0}(c) = M ·(df_{0}(c)); this in turn implies the existence of a constant k ∈ ℤ^{n} such that, on B,

g_{0} = M · f_{0} + k.

But k is necessarily zero : indeed, applying the above equality to λ(h) gives a sequence k′(h) ∈ ℤ^{n} such that

hk′(h) =^{def} g(h; λ(h)) − M · f(h; λ(h)) = k + O(h).

Therefore k′(h) must tend to zero, and hence must equal zero for small h, implying that k = 0.

We have proved the existence of a smooth symbol F (h) such that

M · f(h) − g(h) = hF (h).

Because F (h; λ(h)) ∈ ℤ^{n} +O(h^{∞}) and lim_{h→0} F (h; λ(h)) = F_{0}(c), we must have F_{0}(c) ∈ ℤ^{n}. So

F_{0} = const ∈ ℤ^{n} in B.

This easily implies that all lower order terms in F (h) must vanish on B, so we are left with

F (h) = k + O(h^{∞}), for a k ∈ ℤ^{n}.

This gives g(h) = M · f(h) − hk + O(h^{∞}), which reads

\frac{1}{h}g(h) = A(\frac{1}{h}f(h)) + O(h^{∞}),

with A ∈ GA(n, ℤ) defined by A(p) = M · p − k, p ∈ ℤ^{n}. ⊓ ⊔

Remark 2. Because of the discreteness of GA(n, ℤ), Proposition 1 implies that there is an h_{0} > 0 such that the transition element A is uniquely defined by (g(h_{0})/h_{0}) (f(h_{0})/h_{0})^{−1} acting on a finite subset of ℤ^{n}. Therefore, when restricted to any open subset of U with compact closure in U, the cocycle [M] is really a quantum object, in the sense that “you don’t need to let h tend to zero” to define it.

3. Link with the Classical Monodromy

Let P_{1}(h), …, P_{n}(h) be a set of commuting self-adjoint h-pseudo-differential operators on an n-dimensional manifold X. They will be assumed to be classical and of order zero, in the sense that in any coordinate chart their Weyl symbols p_{j}(h) have an asymptotic expansion of the form

p_{j}(h; x, ξ) = p_{0}^{j}(x, ξ) + hp_{1}^{j}(x, ξ) + h^{2}p_{2}^{j}(x, ξ) + · · · .

Because the principal symbols p_{0}^{1}, …, p_{0}^{n} commute with respect to the symplectic Poisson bracket on T^{∗}X, the map

T^{∗}X ∋ (x, ξ) ^{^{p}\overline{}}(p_{0}^{1}(x, ξ), …, p_{0}^{n}(x, ξ)) ∈ ℝ^{n}

is a momentum map for the local Hamiltonian action of ℝ^{n} on T^{∗}X defined by the Hamiltonian flows of the p_{0}^{j}. We will always assume that p is proper, so that the level sets

Λ_{c} = p^{−1}(c)

are compact. Moreover, we ask that these level sets be connected. Conclusions for non-connected Λ_{c} can be obtained by separately studying the different connected components.

Let U_{r} be the open subset of regular values of the momentum map p, and let U be an open subset of U_{r} with compact closure.

It follows from the Arnold-Liouville theorem that p_{↾U} is a smooth fibration whose fibres are Lagrangian tori. The structure of this fibration is semi-globally (i.e. in a neighbourhood of a fibre) described with the help of action-angle coordinates. However, the flat fibre bundle H_{1}(Λ_{c}, ℤ) → c ∈ U (with fibre ℤ^{n}) may have non-trivial monodromy, preventing the construction of global action variables on p^{−1}(U) (see Duistermaat [9]). We will denote by [M_{cl}] (classical monodromy) the cocycle in Ȟ^{1}(U, GL(n, ℤ)) associated to this lattice bundle.

On the other hand, let Σ(h) be the intersection with U of the joint spectrum of the operators P_{1}(h), …, P_{n}(h). It is known from [3] that this spectrum is discrete and for small h is composed of simple eigenvalues. Moreover, the following result holds:

Proposition 2 ([3]). Σ(h) is an asymptotic affine lattice on U.

We denote by [M_{qu}] ∈ Ȟ^{1}(U, GA(n, ℤ)) the quantum monodromy of the spectrum on U, given by Definition 2.

Recall that ι denotes the inclusion of GL(n, ℝ) into GA(n, ℝ) such that for any M ∈ GL(n, ℝ), ι(M) leaves the origin 0 ∈ ℝ^{n} invariant.

The relation between [M_{qu}] and the classical monodromy [M_{cl}] is then given by the following theorem :

Theorem 1. The quantum monodromy is “dual” to the classical monodromy in the following sense:

[M_{qu}] = ι_{∗}(^{t}[M_{cl}]^{−1}).

In other words, for any c ∈ U there exists a choice of basis of H_{1}(Λ_{c}, ℤ) and of an affine chart of Σ(h) such that the monodromy representations

μ^{cl} : π_{1}(U, c) → GL(n, ℤ)

and

μ^{qu} : π_{1}(U, c) → GA(n, ℤ)

defined by [M_{cl}] and [M_{qu}] satisfy :

μ^{qu} = ι ∘ (^{t}μ^{cl})^{−1}.

Proof. Let α be the Liouville 1-form on T^{∗}X. Let c_{0} ∈ U and for c near c_{0} let (γ_{1}(c), …, γ_{n}(c)) be a smooth family of loops on Λ_{c} whose homology classes form a basis of H_{1}(Λ_{c}, ℤ). It is known from [3,6] (see also [14] for a viewpoint closer to this article) that one can find an affine chart f(h) for Σ(h) around c such that the principal part f_{0} is equal to the action integral associated to γ_{1}, …, γ_{n}:

f_{0}(c) = (\frac{1}{2π} ∫_{γ_{1}(c)} α, …, \frac{1}{2π} ∫_{γ_{n}(c)} α).

Because of Proposition 1, any other affine chart around c having the same principal part must equal f(h) (modulo O(h^{∞})). In this way, the choice of a local smooth basis of H_{1}(Λ_{c}, ℤ) determines an affine chart of Σ(h). If (γ′_{1}(c), …, γ′_{n}(c)) is another basis of H_{1}(Λ_{c}, ℤ) such that

(γ′(c)) = M(c) · (γ(c)),  (3)

for a matrix M(c) ∈ GL(n, ℤ) depending smoothly on c, then the corresponding affine charts f(h) and f′(h) of Σ(h) satisfy :

f′(h; c) = M(c) · f(h; c) + O(h^{∞}).

Recall that the notation “M·” here means matrix multiplication by M, which is of course the same as affine composition by ι(M).

But formula (3) says that if k and k′ are trivialisation functions of the bundle H_{1}(Λ_{c}, ℤ) → c associated to the basis γ and γ′, then k′ = ^{t}M^{−1}k. Therefore, if tM_{αβ}^{−1} are transition elements for the lattice bundle H_{1}(Λ_{c}, ℤ) → c, then ι(M_{αβ}) define a monodromy cocycle for Σ(h). ⊓ ⊔

Remark 3. The fact that the affine nature of quantum monodromy is here naturally reduced to an action of the linear group GL(n, ℤ) is due the the global existence of a primitive of the symplectic form on T^{∗}X, namely the Liouville 1-form α.

4. Monodromy of a Focus-Focus Singularity

It is probably not worth discussing monodromy in arbitrary degrees of freedom, for it is a typical phenomenon of 4-dimensional symplectic manifolds (see [13]).

More precisely, let X be a 2-dimensional manifold, and let P_{1}(h), P_{2}(h) be two commuting self-adjoint h-pseudo-differential operators on X. As before, suppose that the momentum map p = (p_{0}^{1}, p_{0}^{2}) defined by the principal symbols is proper with connected level sets.

We shall make the following hypothesis. There exists a critical point m ∈ T^{∗}X of p of maximal corank (i.e. both p_{0}^{1} and p_{0}^{2} are critical at m) such that, in some local symplectic coordinates (x, y, ξ, η), the Hessians (p_{0}^{1})′′(m) and (p_{0}^{2})′′(m) (thereafter denoted by H(p_{0}^{1}) and H(p_{0}^{2})) generate a 2-dimensional subalgebra of the algebra Q(4) of quadratic forms in (x, y, ξ, η) under Poisson bracket that admits the following basis (q_{1}, q_{2}):

q_{1} = xξ + yη,
q_{2} = xη − yξ.

Such a singularity m is called a focus-focus singularity. The point m is then isolated amongst critical points of p. Therefore, we can choose U ⊂ U_{r} to be a small punctured disc around o = p(m). Finally, we shall always assume that m is the only critical point of the critical level set Λ_{0} = p^{−1}(o).

It is known (probably since [15]; see for instance [14] or [8] for discussions and more references on this topic) that the fibration p_{↾U} has non-trivial monodromy, and can be described in the following way:

Near m, we know from [10] that the integrable Hamiltonian system (p_{0}^{1}, p_{0}^{2}) can be brought into a normal form given by (q_{1}, q_{2}). In other words there exists a local diffeomorphism F : (ℝ^{2}, 0) → (ℝ^{2}, o) such that

(p_{0}^{1}, p_{0}^{2}) = F (q_{1}, q_{2}).

This allows one to define transversal vector fields X_{1} and X_{2} tangent to the fibres Λ_{c} that are equal to the Hamiltonian vector fields X_{q_{1}} and X_{q_{2}} near m. Note that X_{2} is periodic of period 2π.

Around each c ∈ U, we can now define the following smooth basis (γ_{1}(c), γ_{2}(c)) of H_{1}(Λ_{c}, ℤ) ≃ π_{1}(Λ_{c}):

– γ_{2}(c) is a simple integral loop of X_{2}.

– Take a point on γ_{2}(c); let it evolve under the flow of X_{1}. After a finite time, it goes back on γ_{2}(c). Close it up on γ_{2}(c). This defines γ_{1}(c).

Proposition 3 ([15]). Let c ∈ U. With respect to the basis (γ_{1}(c), γ_{2}(c)), the action of the classical monodromy map μ^{cl} on a simple loop δ ∈ π_{1}(U, c) enclosing o is given by the matrix

μ^{cl}(δ) = ( \begin{matrix}1 & 0 \\ ϵ & 1 \end{matrix} ) .

Here ϵ is the sign of det M, where M ∈ GL(2, ℝ) is the unique matrix such that :

(H(p_{0}^{1}), H(p_{0}^{2})) = M · (H(q_{1}), H(q_{2})).

Λ_{c} γ_{1}(c)  γ_{2}(c)
Fig. 2. The basis (γ_{1}(c), γ_{2}(c))

Note also that M = dF (0).

This, together with Theorem 1, proves the following result:

Theorem 2. Let P_{1}(h), P_{2}(h) be a quantum integrable system with a focus-focus singularity. Then there exists a small punctured neighbourhood U of the critical value o such that for any c ∈ U, if f(h) is an affine chart of the joint spectrum Σ(h) around c having principal part

(\frac{1}{2π} ∫_{γ_{1}(c)} α, \frac{1}{2π} ∫_{γ_{2}(c)} α) ,

then the value of the quantum monodromy map μ_{f}^{qu} ∈ GA(2, ℤ) at a simple loop δ ∈ π_{1}(U, c) enclosing o is given by the matrix

μ_{f}^{qu}(δ) = ι ( \begin{matrix}1 & −ϵ \\ 0 & 1 \end{matrix} ) .

Here ϵ is the sign of det M, where M ∈ GL(2, ℝ) is the unique matrix such that :

(H(p_{0}^{1}), H(p_{0}^{2})) = M · (H(q_{1}), H(q_{2})).

5. How to Detect Quantum Monodromy

5.1. Introduction. Theorem 1 wouldn’t be of much interest if one could not “read off” the quantum monodromy from a picture of the joint spectrum.

This is actually easy to do, at least in a heuristic way. The rigorous mathematical formulation may however look slightly awkward.

The first idea is the following. Given a straight lattice ℤ^{n}, and any two points A and B in ℤ^{n}, there is a natural parallel translation from A to B acting on ℤ^{n}, namely the translation by the integral vector ^{→} AB.

Now, the joint spectrum Σ(h) locally around any point c ∈ U looks like a lattice. If the points A and B in Σ(h) are close enough to c and h is small enough, one can still define a parallel translation from A to B, taking points of Σ(h) near A to points in Σ(h) near B. This allows us to pass from one chart to another, and hence to define the notion of parallel transport along any loop through c. This yields a map from π_{1}(U, c) to GL(n, ℤ) which is precisely the linear part of the quantum monodromy μ^{qu}.

Fig. 3. Parallel transport on Σ(h)

This idea is made precise in Sect. 5.2.

The problem can also be viewed the other way round. Roughly speaking, (Σ(h), U) is an affine manifold, and hence can be defined by the data of a local diffeomorphism f(h) from the universal cover Ũ of U to hℝ^{n} sending Σ(h) to hℤ^{n}, and of the holonomy ν associated to it :

f(h; γ.c̃) = νc̃(γ)f(h; c̃), ∀γ ∈ π_{1}(U), ∀c̃ ∈ Ũ.

Of course, ν should be related to the quantum monodromy μ_{f}. The diffeomorphism f(h) can be seen as an “unwinding” of Σ(h) onto ℝ^{n}. This viewpoint is developed in Sect. 5.3.

5.2. Parallel transport on Σ(h). We discuss here the notion of parallel transport on any asymptotic affine lattice (Σ(h), U).

1. First suppose that there exists an affine chart f(h) of Σ(h) defined globally on U. Since f(h) is elliptic and sends elements of Σ(h) into hℤ^{n} + O(h^{∞}), there is an h_{0} > 0 such that for any h < h_{0}, there is an injective map f̃(h) sending elements of Σ(h) exactly into hℤ^{n} and such that f̃(h) − f(h) = O(h^{∞}).

Because f(h) is of order zero, there is a fixed open ball B̃′ ⊂ f(h; U) such that B̃′ ∩ (hℤ^{n}) is contained in f̃(h; Σ(h)).

Then, one can find a smaller ball B̃ ⊂ B̃′ such that for any two points P̃, Q̃

in B̃ ∩ (hℤ^{n}), the translation by the vector ^{→} P̃Q̃ takes any point of B̃ ∩ (hℤ^{n}) into B̃′ ∩ (hℤ^{n}) (Fig. 4). Let us denote by B an open ball in ℝ^{n} such that f(h; B) ⊂ B̃. Pulling back by f̃(h), one thus defines the “parallel transport”

h hℤ^{n}  B̃′ B̃ P̃  Q̃ Ã  Ã′

Fig. 4. Parallel translation

τ_{→_{P Q}}(A) of a point A ∈ Σ(h) ∩ B along the direction given by two points P and Q in Σ(h) ∩ B. When the composition is defined, we have

τ_{→_{QR}} ∘ τ_{→_{P Q}} = τ_{→_{P R}}.  (4)

Moreover, because translation in ℤ^{n} is an isometry, there exists a constant C > 0, independent of h, such that for any A ∈ Σ(h) ∩ B

||  ^{→} Qτ_{→_{P Q}}(A)|| < C|| ^{→} P A||.  (5)

Because of Proposition 1, any other choice of affine chart f(h) gives the same parallel transport.

2. Now, let (Σ(h), U) be a general asymptotic affine lattice. If γ is any path in U, one can cover its image by open balls B_{i} on which parallel transport is well defined for h less than some h_{i} > 0. If \overline{U} is compact, as we shall always assume, this can be done with a finite number of such balls B_{1}, …, B_{ℓ}, ordered in a way that for each 1 ≤ i < ℓ, B_{i} ∩ B_{i+1} ≠ ∅.

In the following, take h to be less than min_{i} h_{i}. Let P ∈ Σ(h) ∩ B_{0} and Q ∈ Σ(h)∩B_{ℓ}. For each i = 1, …, ℓ−1, pick up a point P_{i} ∈ Σ(h)∩(B_{i} ∩B_{i+1}). For h small enough, this set is not empty. Because of the estimate (5), the mapping

τ_{γ,P,Q} =^{def} τ_{→_{P_{ℓ−1}Q}} ∘ ⋯ ∘ τ_{→_{P_{1}P_{2}}} ∘ τ_{→_{P P_{1}}}

is well-defined when restricted to a sufficiently small ball B_{0} around P (here again, Σ(h) ∩ B_{0} won’t be empty if h is small enough). Equation (4) shows that this map does not depend on the choice of the intermediate points P_{i}. Therefore it depends only on P, Q, and on the homotopy class of γ (as a path from a point in B_{1} to a point in B_{ℓ}).

If Q = P, and γ is a loop (B_{ℓ} ∩ B_{1} ≠ ∅ and B_{0} ⊂ B_{1}) then τ_{γ,P,P} is a map from Σ(h) ∩ B_{0} to Σ(h) ∩ B_{1} leaving P invariant. If f(h) is an affine chart for Σ(h) on B_{1}, then f̃(h) ∘ τ_{γ,P,P} ∘ f̃(h)^{−1} is a locally defined map τ̃_{γ,f(h),P} from hℤ^{n} to itself leaving f̃(h; P) invariant.

We know from Sect. 2 (formula (1)) that the choice of such an affine chart allows the quantum monodromy map μ_{f} to take its values in GA(n, ℤ). Remember that L denotes the natural homomorphism from GA(n, ℝ) to GL(n, ℝ).

Proposition 4. The map τ̃_{γ,f(h),P} is equal to the linearisation at P̃ = f̃(h; P) of the quantum monodromy along γ :

∀R̃ ∈ hℤ^{n}, ^{→} P̃τ̃_{γ,f(h),P} (R̃) = L(μ_{f}(γ)) ^{→} P̃R̃,

whenever the left-hand side of the above is defined.

Proof. If we choose affine charts f_{i}(h) for Σ(h) on each of the B_{i}’s with f_{1} = f, and let A_{i,i+1} be the transition elements of the monodromy cocycle

f_{i}(h)/h = A_{i,i+1}(f_{i+1}(h)/h) + O(h^{∞}) (convention ℓ + 1 ≡ 1),

then it is easy to check that

→ P̃τ̃_{γ,f(h),P} (R̃) = L(A_{1,ℓ}) ⋯ L(A_{3,2})L(A_{2,1}) · ^{→} P̃R̃,

whenever the composition is defined. Using (2) finishes the proof. □

As an application, one can easily “read off” from the spectrum of the quantum Champagne bottle (Fig. 5) that the linear part of the quantum monodromy is

conjugate to the matrix ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ).

5.3. Unwinding the spectrum. We keep here the notation of the previous paragraph. In particular, Σ(h) is any asymptotic affine lattice on U, γ is a path in U whose image is covered by balls B_{i} on which local parallel translation is defined. We choose points P ∈ B_{1} ∩ Σ(h), Q ∈ B_{ℓ} ∩ Σ(h) and P_{1}, P_{2}, …, P_{ℓ−1}, P_{ℓ} = Q such that for i = 1, …, ℓ − 1, P_{i} ∈ B_{i} ∩ B_{i+1} ∩ Σ(h).

Given an affine chart f(h) on B_{1}, for h small there is a unique k_{1} ∈ ℤ^{n} such that the map f̃(h) ∘ τ_{→_{P P_{1}}} ∘ f̃(h)^{−1} is just translation by hk_{1}. If B_{1}, …, B_{ℓ} are endowed with affine charts f_{1}(h) = f(h), f_{2}(h), …, f_{ℓ}(h), in the same way we define k_{i} ∈ ℤ^{n} such that

f̃_{i}(h) ∘ τ_{→_{P_{i−1}P_{i}}} ∘ f̃_{i}(h)^{−1}

is translation by the vector hk_{i}. We unwind the points P, P_{1}, …, P_{ℓ} onto hℤ^{n} using the following procedure (see Fig. 6):

– P̃ = f̃(h; P);

– P̃_{1} = P̃ + hk_{1} = f̃(h, P_{1});

– P̃_{2} = P̃_{1} + hL(A_{2,1}) · k_{2};

– …

γ  E_{2} = hn R′ P R  E_{1}

Fig. 5. Spectrum of the Champagne bottle. The gray disc encloses the focus-focus critical value. R′ = τ_{γ,P,P} (R)

h  P̃_{11} P̃_{5}  P̃_{8} P̃_{7} P̃_{2} P̃_{1}  Q̃ P̃ f(h) P_{9} P_{10} P_{11} Q P_{8} P_{7} P_{6} P_{5} P_{4} P_{3} P_{2}  P_{1}  P  E_{1} E_{2} = hn

Fig. 6. Unwinding of the points P_{i}. We deduce that yP̃ = 4, which allows us to locate the horizontal line through the origin 0 ∈ hℤ^{2} (the dotted one)

– Q̃ = P̃_{ℓ} = P̃_{ℓ−1} + hL(A_{ℓ,ℓ−1}) ⋯ L(A_{2,1}) · k_{ℓ}.

Then one easily checks that

P̃_{i} = hA_{1,2} ∘ A_{2,3} ∘ ⋯ ∘ A_{i−1,i}(f̃_{i}(h; P_{i})/h).

In particular, applying this procedure to a loop γ (P = Q) proves the following :

Proposition 5. For h small enough, the quantum monodromy μ_{f} gives the end point Q̃ of the unwinding of any loop γ on U through a point P ∈ Σ(h) around which we are given an affine chart f(h) by the following formula :

Q̃ = h(μ_{f}(γ))^{−1}(f̃(h; P)/h).

Remark 4. There is a unique symbol g(h) defined on the universal cover Ũ of U that is an affine chart for Σ(h) and that coincides with f(h) above B_{0}. Then Q can be seen as the lift γ.P ∈ Ũ. The point is now that

g(h; Q) = Q̃ + O(h^{∞}).

For any P ∈ Ũ, and for any γ ∈ π_{1}(U), there is a unique ν_{P} (γ) ∈ GA(n, ℤ) such that

g(h; γ.P)/h = ν_{P} (γ)(g(h; P)/h) + O(h^{∞}).

By definition, we have ν_{P} (γγ′) = ν_{γ.P} (γ′)ν_{P} (γ). But one can show that for any loop γ such that γ.P = Q, then

ν_{Q}(γ′) = ν_{P} (γ)ν_{P} (γ′)ν_{P} (γ)^{−1}.

Therefore, ν_{P} is actually a homomorphism. Proposition 5 just says that

ν_{P} = μ_{f}^{−1}.

Applying this proposition together with Theorem 2 to a focus-focus singularity, we see that if the principal part of f(h) is given by the action integrals \frac{1}{2π} ∫_{γ_{1}} α and \frac{1}{2π} ∫_{γ_{2}} α then, for a small loop δ enclosing the critical value o,

ν(δ) = ι ( \begin{matrix}1 & ϵ \\ 0 & 1 \end{matrix} ) .

In particular, the whole horizontal line through the origin consists of fixed points. Of course, locating the origin on a diagram like Fig. 6 may require the computation of the action at one point. However, given P̃ and its image Q̃, it is easy to find the horizontal line through the origin, for

ϵyP̃ = xQ̃ − xP̃.

Acknowledgements. One of the reasons for having written this article is the enthusiasm of R. Cushman for the subject; I would like to thank him for this. I would also like to thank my adviser Y. Colin de Verdière, and J. J. Duistermaat, for stimulating discussions.

My research is supported by a Marie Curie Fellowship Nr. ERBFMBICT961572.

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Communicated by H. Araki