From classical theta functions to topological quantum field theory
Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrödinger representation is obtained as an action on curves in a handlebody. A careful analysis of the discrete Fourier transform yields the Murakami-Ohtsuki-Okada formula for invariants of 3-dimensional manifolds. In this context, we give an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.
2010 Mathematics Subject Classification:14K25, 57R56, 57M25, 81S10, 81T45
In this paper we construct the abelian Chern-Simons topological quantum field theory directly from the theory of classical theta functions, without the insights of quantum field theory.
It has been known for years, within abelian Chern-Simons theory, that classical theta functions relate to low dimensional topology , . Abelian Chern-Simons theory is considerably simpler than its non-abelian counterparts, and has been studied thoroughly (see e.g. , ). Here we do not start with abelian Chern-Simons theory, but instead give a direct construction of the associated topological quantum field theory based on the theory of theta functions, and arrive at skein modules from representation theoretical considerations.
We consider theta functions in the representation theoretic point of view of André Weil . As such, the space of theta functions is endowed with an action of a finite Heisenberg group (the Schrödinger representation), which induces, via a Stone-von Neumann theorem, the Hermite-Jacobi action of the modular group. All this structure is what we shall mean by the theory of theta functions.
We show how the group algebra of the finite Heisenberg group and its Schrödinger representation on the space of theta functions, lead to algebras of curves on surfaces and their actions on spaces of curves in handlebodies. These notions are formalized using skein modules.
The Hermite-Jacobi representation of the modular group on theta functions is a discrete analogue of the metaplectic representation. The modular group acts by automorphisms that can be interpreted as discrete Fourier transforms. We show that these discrete Fourier transforms can be expressed as linear combinations of curves. A careful analysis of their structure and of their relationship to the Schrödinger representation yields the Murakami-Ohtsuki-Okada formula  of invariants of 3-manifolds.
As a corollary of our discussion we obtain an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds obey the same formula.
The paper uses results and terminology from the theory of theta functions, quantum mechanics, and low dimensional topology. To make it accessible to different audiences we include a fair amount of detail. A more detailed discussion of these ideas can be found in .
Section 2 reviews the theory of theta functions on the Jacobian variety of a surface. The action of the finite Heisenberg group on theta functions is defined via Weyl quantization of the Jacobian variety in a Kähler polarization. In fact it has been found recently that Chern-Simons theory is related to Weyl quantization , , and this was the starting point of our paper. The next section exhibits the representation theoretical model for theta functions. In Section 4 we show that this model for theta functions is topological in nature, and reformulate it using algebras of curves on surfaces, together with their action on skeins of curves in handlebodies which are associated to the linking number.
In Section 5 we derive a formula for the discrete Fourier transform as a skein. This formula is interpreted in terms of surgery in the cylinder over the surface. Section 6 analizes the exact Egorov identity which relates the Hermite-Jacobi action to the Schrödinger representation. This analysis shows that the topological operation of handle slides is allowed over the skeins that represent discrete Fourier transforms, and this yields in the next section the abelian Chern-Simons invariants of 3-manifolds defined by Murakami, Ohtsuki, and Okada. We point out that the above-mentioned formula was introduced in an ad-hoc manner by its authors , our paper derives it naturally.
Section 8 shows how to associate to the discrete Fourier transform a 4-dimensional manifold, and explains why the cocycle of the Hermite-Jacobi action is related to that governing the non-additivity of the signature of 4-manifolds . Section 9 should be taken as a conclusion; it puts everything in the context of Chern-Simons theory.
2. Theta functions
We start with a closed genus Riemann surface , and consider a canonical basis of , like the one in Figure 1. To it we associate a basis in the space of holomorphic differential -forms , defined by the conditions , . The matrix with entries , is symmetric with positive definite imaginary part. This means that if , then , and . The matrix is called the period matrix of , its columns , called periods, generate a lattice in . The complex torus
is the Jacobian variety of . The map
induces a homeomorphism .
The complex coordinates on are inherited from . We introduce real coordinates by imposing . A fundamental domain for the period lattice in terms of the coordinates is . has the canonical symplectic form
with the complex structure and symplectic form is a Kähler manifold. The symplectic form induces a Poisson bracket on , given by , where is the Hamiltonian vector field defined by .
Classical theta functions arise when quantizing in a Kähler polarization in the direction of this Poisson bracket. In this paper we perform the quantization in the case where Planck’s constant is the reciprocal of an even positive integer: where , . The Hilbert space of the quantization consists of the holomorphic sections of a line bundle obtained as the tensor product of a line bundle with curvature and the square root of the canonical line bundle. The latter is trivial for the complex torus and we ignore it. The line bundle with curvature is the tensor product of a flat line bundle and the line bundle defined by the cocycle ,
. (See e.g. §4.1.2 of  for a discussion of how this cocycle gives rise to a line bundle with curvature .) We choose the trivial flat bundle to tensor with. Then the Hilbert space can be identified with the space of entire functions on satisfying the periodicity conditions
We denote this space by ; its elements are called classical theta functions.111The precise terminology is canonical theta functions, classical theta functions being defined by a slight alteration of the periodicity condition. We use the name classical theta functions to emphasize the distinction with the non-abelian theta functions. A basis of consists of the theta series
The definition of theta series will be extended for convenience to all , by for any . Hence the index is taken in .
The inner product that makes the theta series into an orthonormal basis is
That the theta series form an orthonormal basis is a corollary of the proof of Proposition 2.1 below.
To define the operators of the quantization, we use the Weyl quantization method. This quantization method can be defined only on complex vector spaces, the Jacobian variety is the quotient of such a space by a discrete group, and the quantization method goes through. As such, the operator associated to a function on is the Toeplitz operator with symbol ( Proposition 2.97)222The variable of is not conjugated because we work in the momentum representation., where is the Laplacian on functions,
On a general Riemannian manifold this operator is given in local coordinates by the formula
where is the metric and . In the Kähler case, if the Kähler form is given in holomorphic coordinates by
where . In our situation, in the coordinates , , one computes that and therefore (recall that is the imaginary part of the matrix ). For Weyl quantization one introduces a factor of in front of the operator. As such, the Laplace (or rather Laplace-Beltrami) operator is equal to
(A word about the notation being used: represents the usual (column) vector of partial derivatives in the indicated variables, so that each object in the square brackets is a column vector of partial derivatives. The subindices are the corresponding components of those vectors.) A tedious calculation that we omit results in the following formula for the Laplacian in the coordinates:
We will only need to apply explicitly to exponentials, as part of the proof of the following basic proposition. Note that the exponential function
defines a function on the Jacobian provided .
The Weyl quantization of the exponentials is given by
Let us introduce some useful notation local to the proof. Note that and are fixed throughout.
For and , .
With these notations, in the coordinates
We first compute the matrix coefficients of the Toeplitz operator with symbol , namely , which is
Then a calculation shows that
The integral over of the term will be non-zero iff
in which case the integral will be equal to one. Therefore unless , where the brackets represent equivalence classes in . This shows that the Toeplitz operator with multiplier maps to a scalar times . We now compute the scalar.
Taking in the fundamental domain for , there is a unique representative, , of in the same domain. This is of the form
for a unique . With respect to the previous notation, .
It follows that
where in the th term. Using that , one obtains
and so we can write
Making the change of variables in the summand , the argument of the function can be seen to be equal to
Since and are integer vectors,
The dependence on of the integrand is a common factor that comes out of the summation sign. The series now is of integral over the translates of that tile the whole space. Therefore is equal to
A calculation of the integral333 yields that it is equal to
The exponent on the right-hand side is times
On the other hand, it is easy to check that and therefore
so that, by (2.2)
as desired. ∎
Let us focus on the group of quantized exponentials. First note that the symplectic form induces a nondegenerate bilinear form on , which we denote also by , given by
As a corollary of Proposition 2.1 we obtain the following result.
Quantized exponentials satisfy the multiplication rule
This prompts us to define the Heisenberg group
This group is a -extension of , with the standard inclusion of into it given by
defines a representation of on theta functions. To make this representation faithful, we factor it by its kernel.
The set of elements in that act on theta functions as identity operators is the normal subgroup consisting of the th powers of elements of the form with even. The quotient group is isomorphic to a finite Heisenberg group.
Recall (cf. ) that a finite Heisenberg group is a central extension
where is a finite abelian group such that the commutator pairing , (, and being arbitrary lifts of and to ) identifies with the group of homomorphisms from to .
By Proposition 2.1,
For to act as the identity operator, we should have
for all . Consequently, should be in . Then is a multiple of , so the coefficient equals . This coefficient should be equal to . For this implies that should be an even multiple of . But then by varying we conclude that is a multiple of . Because is even, it follows that is an even multiple of , and consequently is an even multiple of . Thus any element in the kernel of the representation must belong to . It is easy to see that any element of this form is in the kernel. These are precisely the elements of the form with even.
The quotient of by the kernel of the representation is a -extension of the finite abelian group , thus is a finite Heisenberg group. This group is isomorphic to
with the multiplication rule
The isomorphism is induced by the map ,
We denote by this finite Heisenberg group and by the image of in it. The representation of on the space of theta functions is called the Schrödinger representation. It is an analogue, for the case of the -dimensional torus, of the standard Schrödinger representation of the Heisenberg group with real entries on . In particular we have
(Stone-von Neumann) The Schrödinger representation of is the unique (up to an isomorphism) irreducible unitary representation of this group with the property that acts as for all .
Let , , , . Then , if , , , , , for all , and for all . Because commute pairwise, they have a common eigenvector . And because for all , the eigenvalues of with respect to the are roots of unity. The equalities
show that by applying ’s repeatedly we can produce an eigenvector of the commuting system whose eigenvalues are all equal to . The irreducible representation is spanned by the vectors , . Any such vector is an eigenvector of the system , with eigenvalues respectively . So these vectors are linearly independent and form a basis of the irreducible representation. It is not hard to see that the action of on the vector space spanned by these vectors is the Schrödinger representation. ∎
The operators , form a basis of the space of linear operators on .
For simplicity, we show that the operators
form a basis. Denote by the respective matrices of these operators in the basis . For a fixed , the nonzero entries of the matrices , are precisely those in the slots , with (here is taken modulo ). If we vary and and arrange these nonzero entries in a matrix, we obtain the th power of a Vandermonde matrix, which is nonsingular. We conclude that for fixed , the matrices , form a basis for the vector space of matrices with nonzero entries in the slots of the form . Varying , we obtain the desired conclusion. ∎
The algebra of linear operators on the space of theta functions is isomorphic to the algebra obtained by factoring by the relation .
Let us now recall the action of the modular group on theta functions. The modular group, known also as the mapping class group, of a simple closed surface is the quotient of the group of homemorphisms of by the subgroup of homeomorphisms that are isotopic to the identity map. It is at this point where it is essential that is even.
The mapping class group acts on the Jacobian in the following way. An element of this group induces a linear automorphism of . The matrix of has integer entries, determinant , and satisfies , where is the intersection form in . As such, is a symplectic linear automorphism of , where the symplectic form is the intersection form. Identifying with , we see that induces a symplectomorphism of . The map induces an action of the mapping class group of on the Jacobian variety. This action can be described explicitly as follows. Decompose into blocks as
Then maps the complex torus defined by the lattice and complex variable to the complex torus defined by the lattice and complex variable , where and .
This action of the mapping class group of the surface on the Jacobian induces an action of the mapping class group on the finite Heisenberg group by
The nature of this action is as follows: Since induces a diffeomorphism on the Jacobian, we can compose with an exponential and then quantize; the resulting operator is as above. We point out that if were not even, this action would be defined only for in the subgroup of the symplectic group (this is because only for even is the kernel of the map defined in Proposition 2.3 preserved under the action of ).
As a corollary of Theorem 2.4, the representation of the finite Heisenberg group on theta functions given by is equivalent to the Schrödinger representation, hence there is an automorphism of that satisfies the exact Egorov identity:
(Compare with , Theorem 2.15, which is the analogous statement in quantum mechanics in Euclidean space.) Moreover, by Schur’s lemma, is unique up to multiplication by a constant. We thus have a projective representation of the mapping class group of the surface on the space of classical theta functions that statisfies with the action of the finite Heisenberg group the exact Egorov identity from (2.5). This is the finite dimensional counterpart of the metaplectic representation, called the Hermite-Jacobi action.
We emphasize that the action of the mapping class group of on theta functions factors through an action of the symplectic group .
Up to multiplication by a constant,
(cf. (5.6.3) in ). When the Riemann surface is the complex torus obtained as the quotient of the complex plane by the integer lattice, and is the map induced by a rotation around the origin, then is the discrete Fourier transform. In general, like for the metaplectic representation (see ), can be written as a composition of partial discrete Fourier transforms. For this reason, we will refer to as a discrete Fourier transform.
3. Theta functions in the abstract setting
In this section we apply to the finite Heisenberg group the standard construction which identifies the Schrödinger representation as a representation induced by an irreducible representation (i.e. character) of a maximal abelian subgroup (see for example ).
Start with a Lagrangian subspace of with respect to the intersection form, which for our purpose is spanned by the elements of the canonical basis. Let be the intersection of this space with . Under the standard inclusion , becomes an abelian subgroup of the Heisenberg group with integer entries. This factors to an abelian subgroup of . Let be the subgroup of containing both and the scalars . Then is a maximal abelian subgroup. Being abelian, it has only 1-dimensional irreducible representations, which are its characters.
In view of the Stone-von Neumann Theorem, we consider the induced representation defined by the character , . This representation is
with acting on the left in the first factor of the tensor product. Explicitly, the vector space of the representation is the quotient of the group algebra by the vector subspace spanned by all elements of the form
with and . We denote this quotient by , and let be the quotient map. Let also the inner product be defined such that has norm , where is an element of the finite Heisenberg group seen as an element of its group algebra.
The left regular action of the Heisenberg group on its group algebra descends to an action on .
The map , defines a unitary map between the space of theta functions and , which intertwines the Schrödinger representation and the left action of the finite Heisenberg group.
It is not hard to see that and have the same dimension. Also, for , , hence the map from the statement is an isomorphism of finite dimensional spaces. The norm of is one, hence this map is unitary. We have