Asymptotic properties of the quantum representations of the modular group
We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the known asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of the torus bundles with hyperbolic monodromy.
Quantum Chern-Simons theory was introduced twenty years ago by Witten [Wi] and Reshetikhin-Turaev [ReTu]. It provides among other things invariants of three-dimensional manifold and representation of the mapping class group of surfaces, cf. [BaKi], [Tu] for an exposition of the theory and [Fr] for a survey on recent developments. This theory has a semi-classical limit, where the level, an integral parameter denoted by , plays the role of the inverse of the Planck constant. In this paper, we are concerned with the torus and its mapping class group, . We study the large behaviour of the quantum representation of the modular group.
The quantum representations may be equivalently defined with algebraic or geometrical methods. Geometrically, we consider a line bundle, called the Chern-Simon bundle, over the moduli space of flat -principal bundles on the torus. Here is a compact Lie group that we assume to be simple and simply connected. Then the modular group acts linearly on the space of holomorphic sections of the -th tensor power of the Chern-Simons bundle.
A part of this construction is standard in geometric quantization: to any compact Kähler manifold with an integral fundamental form one associates the space of holomorphic sections of a prequantum bundle. In this general context the usual tools of microlocal analysis have been introduced. In particular given a prequantum bundle automorphism, we define a class of operators similar to the Fourier integral operators [oim_qm], [oim_LS] quantizing it. Our main result, theorem 9.1, says that each element of acts as a Fourier integral operator on the quantum spaces, the underlying action on the Chern-Simons bundle being defined through gauge theory. This establishes a clear link between the quantum and classical Chern-Simons theories.
As a corollary, we can estimate the character of the quantum representations of the hyperbolic elements of . More generally, under a transversality assumption, one proves that the trace of a Fourier integral operators has an asymptotic expansion, which generalizes in some sense the Lefschetz fixed point formula [oim_LS]. The characters of the quantum representation of the modular group are the three-dimensional invariants of the torus bundles. In this way we recover the asymptotic expansion proved by Jeffrey [Je], whose leading term is given in terms of the Chern-Simons invariants and the torsion of some flat bundles over the torus bundle. The proof in [Je] is completely different and relies on the reciprocity formula for Gauss sum. Our result is slightly more general since we treat any hyperbolic element with any simple simply connected group . But, what is more important, we hope that our analytic method will work in other cases. In the companion paper [oim_MCG], we prove similar result for the mapping class group in genus .
Besides the semiclassical results, we also give a careful construction of the quantum representations, comparing the geometric and algebraic methods. Strictly speaking, we do not have representations of the modular group but only projective representations which lift to genuine representations of the appropriate extension of . The extensions appearing naturally are not the same in the geometric and the algebraic approach.
The paper is organised as follows. In section 1, we state our result about the asymptotic expansion of the trace of the quantum representations. In section 3, we introduce the phase space of the Chern-Simons theory for the torus, its symplectic structure and prequantum bundle. The relation with gauge theory is the content of section 4. In section 5, we introduce a complex structures on the phase space and the associated quantum Hilbert spaces. We exhibit basis in terms of theta functions. The modular group acts naturally on the previous datas but does not preserve the complex structure. In section 6 we identify the quantum spaces associated to the various complex structures. This leads to the definition of the quantum representations. In section 7 we compare these representations with the ones defined by algebraic methods. The next two sections are devoted to semi-classical results: section 8 on the identification of the quantum spaces and section 9 on the quantum representations. In a first appendix we prove basic facts on theta functions. In a second appendix we list some notations used in the paper.
1 Characters of the quantum representations
Let be a compact simple and simply connected Lie group. The phase space of the Chern-Simons theory for an oriented surface and group is the moduli space of flat -principal bundles over . For a torus, this moduli space identifies with the quotient , where is a maximal torus of and is the Weyl group acting diagonally. The modular group , being the mapping class group of the torus, acts on this moduli space. More explicitly, since , with the Lie algebra of and the integral lattice, we have a bijection . Identify with , then an element acts on the moduli space by sending the class of to the class of .
Applying geometric quantization, we obtain a family of projective representations of the modular group indexed by a positive integer . Since the construction is rather long, we only give in this introduction the representation of the generators
of the modular group. Let be the basic inner product of the Lie algebra of . We choose a set of positive roots and denote by the corresponding open fundamental Weyl alcove. We identify the weight lattice with a lattice of via the basic inner product. The -th representation has a particular basis indexed by the set . For any , let
where is the rank of and is the integral part of .
If the rank of is even, the map sending and to the matrices and extends to a unitary representation of the modular group. If the rank of is odd, we obtain a representation of an extension of the modular group by . Recall that the metaplectic group is the connected two-cover of . Then is defined as the subgroup of the metaplectic group consisting of the elements which project onto the modular group. The representation of the elements projecting onto and is given by the matrices and .
Assume the rank of is even, then for any hyperbolic element , we have
if the trace of is bigger than and otherwise.
if is the class of and .
If the rank of is odd, let projecting onto an hyperbolic element of the modular group. Then the same result holds for the trace of except that the equivalent has to be multiplied by , where modulo .
It is a general property of topological quantum field theories that the trace of the quantum representation of an element of the modular group is the invariant of the mapping torus
Here we ignore the the complications due to the framing of 3-dimensional manifold and the related fact that we only have a projective representation. For any and such that , consider the flat -principal bundle whose holonomies along the paths , and are respectively , and . Then is the Chern-Simons invariant of . This is in agreement with the formula obtained by Witten using the Feynman path integral (heuristic) definition of the three-dimensional invariants. We refer the reader to Jeffrey’s paper [Je] for more details. In particular the factor appears as an integral of the torsion of the adjoint bundles over the moduli space of flat -principal bundle over .
2 Lie group notations
Let be a compact simple Lie algebra, and the corresponding compact connected and simply-connected Lie group. Choose a maximal torus of and denote by its Lie algebra. The integral lattice of is defined as the kernel of the exponential map . Since is simply-connected, is the lattice of generated by the coroots for the (real) roots .
Let the basic inner product be the unique invariant inner product on such that for each long root , . Through the paper, we will use to identify with . The basic inner product has the important property that it restricts to an integer-valued -bilinear form on which takes even values on the diagonal.
We fix a set of positive roots and let be the corresponding positive open Weyl chamber. Let be the highest root and be the open fundamental Weyl alcove
We denote by the Weyl group of . Let be the alternating character of .
3 The symplectic data
In this section we endow with a symplectic form and a prequantum bundle , that is a complex Hermitian line bundle together with a connection of curvature . Furthermore we introduce commuting actions of the Weyl group and the modular group on .
3.1 A prequantum bundle on
Denote by and the projections on the first and second factor respectively. Let be the symplectic form on given by
Consider the trivial complex line bundle over with fiber and connection
Its curvature is , so it is a prequantum bundle.
3.2 Heisenberg group and reduction to
Introduce the (reduced) Heisenberg group with multiplication
The same formula defines an action of the Heisenberg group on . This action preserves the trivial metric and the connection. The lattice embeds into the Heisenberg group
Using that takes integral values on , we prove that this map is a group morphism. Hence we get an action of on by automorphisms of prequantum bundle.
By quotienting, we obtain a symplectic form on with a prequantum bundle over .
3.3 Weyl group
Consider the diagonal action of the Weyl group on and lift this action trivially on the bundle . Since acts on by isometries, acts on by linear symplectomorphisms and on by isomorphisms of prequantum bundle.
The Weyl group preserves the integral lattice . Consider the semi-direct product where acts diagonally on . Is is easily checked that the actions of and on the prequantum bundle over generate an action of . Then, quotienting by , we obtain an action of on . Since the action of the Weyl group on the base is not free, we will not consider the orbifold quotient and its prequantum bundle.
3.4 Modular group
Let us consider the symplectic action of the modular group on given by
The trivial lift to the prequantum bundle preserves the metric and the connection. Furthermore this action together with the action of define an action of the semi-direct product . To prove this, one has to use that is integral when and even if furthermore . Consequently we get an action of the modular group on by prequantum bundle isomorphisms. Observe that the Weyl group action on commute with the modular action.
4 Chern-Simons theory
We explain how the definitions of the previous section can be deduced from gauge theory. Our aim is only to motivate the constructions. No proof in the paper relies on the gauge theoretic considerations.
The phase space of the Chern-Simons theory for an oriented surface is the moduli space of representations of the fundamental group of in . When is a torus, the fundamental group is the free Abelian group with two generators, so each representation is given by a pair of commuting elements of unique up to conjugation. In the same way that , one shows that these representations are conjugate to a representation in the maximal torus , uniquely up to the action of the Weyl group. So the moduli space of representation for the torus is .
4.1 Gauge theory presentation
Consider the space of connections of the trivial -principal bundle with base . It is a symplectic vector space with symplectic product given by
The gauge group acts on by symplectic affine isomorphisms:
where is the right-invariant Maurer-Cartan form. Each gauge class of flat connections is determined by its holonomy representation. The quotient of the space of flat connections by the gauge group may be viewed as a symplectic quotient which defines a symplectic structure on the moduli space of representations.
In the case is a torus, we can avoid this infinite dimensional quotient proceeding as follows. Represent as the quotient with coordinates . Then the map
is a symplectic embedding where the symplectic product of is the one of section 3.1. This embedding is equivariant with respect to the action of on and the morphism from to the gauge group sending an element to the constant gauge transform and to . Furthermore each gauge class of flat connections intersects the image of the embedding.
4.2 Prequantum bundle
Consider the trivial line bundle with base and connection , where is the primitive of given by
This is a prequantum bundle and the gauge group actions lifts to it in such a way that it preserves the trivial metric and the connection. Explicitly, the action is given by
where is the left invariant Maurer-Cartan one-form and is the Wess-Zumino-Witten term
Here is any three-dimensional compact oriented manifold with boundary , any extension of and is the Cartan three-form defined in terms of the left or right-invariant Maurer Cartan forms by
Since is the basic inner product, the cohomology class of is integral.
4.3 Mapping class group
The group of orientation preserving diffeomorphisms of acts symplectically on . The trivial lift to the prequantum bundle preserves the connection and the trivial metric. After quotienting by the gauge group, this defines an action of the mapping class group on the moduli space of representation and its prequantum bundle.
When is a torus, we recover the action of introduced in section 3.4. For any , define the diffeomorphism of the torus
where , , and are the coefficients of . On one hand, we recover the usual formula by considering the basis and . Indeed and . On the other hand,
which corresponds to the action of section 3.4.
Let us begin with a brief description of the general set-up. Consider a symplectic manifold with a prequantum bundle . Assume is endowed with a compatible positive complex structure, so is a form and for any non vanishing tangent vector of type . Then the prequantum bundle has a unique holomorphic structure such that the local holomorphic sections satisfy the Cauchy-Riemann equations:
The quantum space associated to these data is the space of holomorphic sections of . It has a natural scalar product obtained by integrating the punctual scalar product of sections against the Liouville measure .
In a first subsection we introduce complex structures on and define a basis of the quantum space by using theta functions. We also describe the action of the matrices and of in these basis. These results are standard. We provide proofs in appendix. Next we we move on to a subspace of equivariant sections with respect to the Weyl group action, the so called alternating sections. We compute the actions of and in this space.
5.1 Complex structure and theta functions
Denote by the Poincaré upper half-plane
Let . Identify with by the isomorphism sending to . Hence becomes a complex vector space and inherits a complex structure. One may compute the symplectic form in terms of the complex coordinate
It is a positive real form of type . So the prequantum bundle has a unique holomorphic structure compatible with the connection. We denote by its space of holomorphic sections.
Consider the section of
Its covariant derivative is . Since this form is of type , is holomorphic. Furthermore doesn’t vanish anywhere. So the holomorphic sections of identify with the sections over of the form such that is holomorphic and is -invariant.
As previously, we embed in via the identification given by the basic inner product. Recall that . For any , consider the theta function
This series converges uniformly on compact sets to a holomorphic function, it depends only on mod .
For any integer , the sections , where runs over mod , are -invariant and form an orthonormal basis of . Furthermore,
where is the Riemannian volume determined by .
Recall that the modular group acts on and its prequantum bundle. The induced action on the sections of doesn’t preserve the space , because of the complex structure. Actually, acts as a holomorphic map from to with
So for any , acts as an isomorphism
One may compute explicitly this isomorphism in the basis of theta functions when is the matrix or . We make explicit the dependence in in our notations to avoid any ambiguity.
For any and , one has
Here is the determination continuous with respect to and equal to when .
5.2 Alternating sections
The action of the Weyl group on is holomorphic with respect to the complex structure defined by any . Let us consider the alternating sections of , i.e. the sections satisfying
For any , let
Recall that we denote by the fundamental open Weyl alcove.
The family is a basis of the space of alternating holomorphic sections of .
Using that the Weyl group action preserves and , we check that for any ,
So is alternating.
For any root and integer , the orthogonal reflexion with respect to the hyperplane belongs to the affine Weyl group . So for any , there exists with such that modulo . Hence , so vanishes.
Recall that the affine Weyl group acts simply transitively on the set of components of and that is one of these components. The result follows from theorem 5.1. ∎
The modular action and the action of the Weyl group on and commute. So the representation of the modular group preserves the subspace of alternating sections.
For any and , one has
with defined as in theorem 5.2 and
In the last line, we used that the affine Weyl group acts simply transitively on the set of connected components of and that vanishes if . Finally,
since the ’s are alternating. ∎
6 Geometric quantum representation
We introduce a representation of the modular group on the quantum spaces. To do this we identify the various spaces via the sections . Unfortunately the norm of these sections and the action of the modular group depend on as it appears in theorems 5.1 and 5.2. We introduce half-form bundle to correct this.
6.1 Half-form bundles
Let us begin with some definitions. Consider a symplectic manifold with a prequantum bundle and a positive compatible complex structure. Then a half-form bundle is a complex line bundle over with an isomorphism from to the canonical bundle of . A half-form bundle admits a natural metric and a natural holomorphic structure making the isomorphism with the canonical bundle a morphism of Hermitian holomorphic bundle. The quantization of with metaplectic correction is then the space of holomorphic sections of tensored with . The scalar product is defined by integrating the punctual norm of sections against the Liouville measure.
Let us return to our particular situation. Consider such that for a basis of , one has
is uniquely defined up to a plus or minus sign. For any , we defined a complex structure on via the isomorphism
So the holomorphic tangent bundle of is naturally isomorphic to the trivial bundle with fiber . Consequently the canonical bundle is naturally isomorphic to the trivial bundle with fiber . Denote by the section of the canonical bundle which is sent into the constant section equal to by this trivialization.
The section is holomorphic and has a constant punctual norm equal to .
Introduce an orthonormal basis of and denote by and the associated linear coordinates of . Let be the complex coordinate . Then if is a basis of , one has
where the plus or minus sign depends on the orientation of the various basis. Consequently
Now, by definition the Hermitian product of two tangent vectors of type is given by . Since
the vectors are mutually orthogonal and
So the are mutually orthogonal and
and concludes the proof. ∎
Let be a complex vector line and be an isomorphism . Let be such that
For any , the trivial bundle with base and fiber is a half-form bundle, with the squaring map sending into . By the previous lemma, the constant section equal to is a holomorphic section of with constant punctual norm equal to .
Instead of sections of , consider now sections of . The multiplication by is an isomorphism
For any , the sections
form a basis of . They are mutually orthogonal and
For any and in , let be the isomorphism from to defined by
By the previous theorem, is a unitary map.
6.2 Modular action
Let be the matrix with coefficients , , and . The map is a holomorphic map from to . Using the coordinates introduced in the proof of lemma 6.1, we show that
We will lift the action of to the half-form bundles in such a way that it squares to . Since doesn’t admit a preferred square root, we have to pass to an extension of the modular group.
Let be the set of pairs where and is a continuous function from to satisfying
is a group with the product given by where . acts on the product
Restricting to a particular value , we obtain a morphism from to lifting the action of on . The important point is that the square of this isomorphism is the natural map between the canonical bundles of and . Since the Hermitian and holomorphic structures of a half-form bundle are determined by the corresponding canonical bundle, the morphism that we consider is a unitary holomorphic isomorphism.
As previously we lift trivially the action of to the prequantum bundle. Then we obtain for any an isomorphism
By trivial reason, it is a unitary map. Miraculously, these isomorphisms all fit together.
For any and any , the diagram
It is sufficient to prove it for , or since these elements generate . The actions of and in the basis are given by the same formulas as in theorem 5.2 except that the constant has to be replaced by
This proves the result because does not depend on . ∎
For any , the map
is a unitary representation of on .
Consider now the diagonal action of the Weyl group on . Its trivial lift to the half-form bundle acts holomorphically and preserves the metric. So we have a unitary representation of on . It is easy to see that this representation commutes with the one of . This gives a representation of on the subspace of alternating sections that we denote by . By theorem 5.3, we have
The coefficients of the matrix of and in the basis , are respectively
7 Algebraic projective representation
Consider the category of representations of the quantum group for being the root of unity. This category has a subquotient, called the fusion category, which is a modular tensor category. Following [Tu], we can define a natural projective representation of the mapping class group of any 2-dimensional surface with marked points on appropriate spaces of morphisms in this category. In particular, for the torus, we get a projective representation of the modular group that we define in this section.
7.1 The representation
Denote by the half sum of positive roots and by the dual Coxeter number. Let be an integer bigger than . Then the set of admissible weights at level is
For any , let
Consider the action of the modular group on the real projective line induced by the standard action on . Choose a base point . Let be the set of pairs where and is a homotopy class (with fixed endpoint) of a path in from to . is an extension by of the modular group, the product being given by