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Determinant of Laplacians on Heisenberg Manifolds

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the"radius"of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two dimensional flat torus and Kronecker's second limit formula.


Introduction
In the conformal field theory, especially in the string theory an infinite dimensional analogue of the determinant for elliptic operators appears and it is considered in the framework of the analytic continuation method of the spectral zeta-function of elliptic operators. In the string theory it seems to be most interesting to calculate the explicit values for compact Riemannian surfaces and also from the mathematical point of view the quantity relates with various aspects of special functions already in such two dimensional cases. As for the values for the spheres there are relations with the values of the Riemann ζ-functions at the negative integers, for the two dimensional flat torus it is expressed by using the famous formula, so called "Kroneker's second limit formula", and for the cases of compact surfaces with constant negative curvature it was calculated by using Selberg's trace formula and deep properties of modified Bessel functions ( [10], [15], [5], [4], [7]). Thus for all two dimensional cases we already know the values of the zeta-regularized determinant of the Laplacians or relations with other quantities. A purpose of this note is to give an integral representation of the zeta-regularized determinant for three dimensional Heisenberg manifolds and state the corresponding results for five dimensional Heisenberg manifolds. It will be possible to give similar expressions for higher dimensional cases, but we restrict ourselves to these two cases and also we restrict ourselves to deal with a certain kind of uniform discrete subgroups of the Heisenberg group, since even in these cases they contain all necessary features for determining the values for any cases and make us the calculations to be simple. Then in the last half of this paper we derive a general formula of the zeta-regularized determinant for manifolds of the product form of two Riemannian manifolds and the formulas for flat tori of two, three and four dimensions.
In §1 we gather up the basic data of the spectrum of the three dimensional Heisenberg group and Heisenberg manifolds. In §2 first, we explain the zeta-reguralized determinant of the Laplacian and give a calculation for the three dimensional Heisenberg manifolds based on an integral representation of the spectral zeta-function (= the Mellin transformation of the trace of heat kernel divided by a Gamma function). In §3 as an application of an integral representation of the spectral zeta-function given in §2 we give expressions of the all coefficients of the asymptotic expansion of the heat kernel for the three dimensional Heisenberg manifolds. In §4 we state the corresponding results in §2 and §3 for five dimensional Heisenberg manifolds. In §5 we give a general expression of the zeta-regularized determinant of the Laplacian on the product of two Riemannian manifolds. In §6 we give a precise form of the formula derived in §5 for a product type manifold with S 1 . Finally in §7 we give such formulas for two, three and four dimensional flat tori and explain a relation of the formula for two dimensional flat torus and the Kroneker's second limit formula.

Spectrum of three dimensional Heisenberg manifolds
Let H 3 be the three dimensional Heisenberg group: The Lie algebra is It is decomposed into a direct sum in the form of (1.3) We identify h 3 and H 3 through the exponential map. Then the group multiplication of two elements X = X (x, y, z) ∈ h 3 and X = X ( x, y, z) ∈ h 3 is given by (1.4) Y = X * X = X + X + 1 2 [X, X], that is, exp (Y ) = exp X · exp X.
Left invariant Riemannian metrics are determined by its restriction to the tangent space at the identity element ( ∼ = its Lie algebra), and among the left invariant Riemannian metrics we only consider such a metric that X 1 = where ℓ is a positive integer. These choices do not loose the essential features in treating with the spectrum of the Laplacian on the quotient space, H 3 /Γ ℓ , so called, Heisenberg manifolds, since the spectrum is given more or less in a similar form ( [6], we state them later).
Then the inverse image of Γ ℓ by the exponential map is which is a direct sum of two uniform lattices Γ B in g + ⊕ g − and Γ V (ℓ) in z such that The kernel K t (g, g) of e −t∆ , the heat kernel on the Heisenberg group H 3 , is given by where we regard η ∈ z * = RZ * ∼ = R and g = xX 1 + yY 1 + zZ, g = xX 1 + yY 1 + zZ ∈ h 3 ∼ = H 3 through the identification by the exponential map exp : h 3 → H 3 . Then the heat kernel k H 3 /Γ ℓ (t; [g], [ g]) on the Heisenberg manifold H 3 /Γ ℓ is expressed by making use of the heat kernel K t (g, g) on the whole group, because of the invariance K t (γ · g, γ · g) = K t (g, g) , γ ∈ H 3 : Its trace is calculated in the following form: Here F ℓ denotes a fundamental domain of the uniform discrete subgroup Γ ℓ and Γ * B and Γ V (ℓ) * are dual lattices, i.e., The spectrum of the Laplacian on H 3 /Γ ℓ is the union of eigenvalues of two types (a) and (b): we know that the Heisenberg manifold is the total space of a principal bundle with the structure group and a torus T 2 ∼ = (g + ⊕ g − ) /Γ B as the base manifold. We denote the projection map H 3 /Γ ℓ → T 2 by ρ.
Proposition 1.3. The fibers of the bundle ρ : H 3 /Γ ℓ → T 2 are totally geodesic and so the map ρ * : commutes with the action of Laplacians on the each space.
By this property the eigenvalues of the type (b) are those coming from the base space T 2 through the map ρ.

Zeta-regularized Determinant
Let M be an n-dimensional closed Riemannian manifold with the Laplacian ∆ M , and denote the heat kernel by k M (t; x, y): Then the Mellin transformation of the trace of the heat kernel, is meromorphically continued to the whole complex plane with only poles of order one (at most) at s = n/2, n/2 − 1, · · · , especially at s = 0 it is holomorphic. We put this function as ζ M (s) and call the spectral zeta-function of the Riemannian manifold M. Then we can regard the value as a "determinant" (= product of non-zero eigenvalues) of the Laplacian ∆ M acting on the space of functions orthogonal to the space of constant functions( [14], [13], [7]) and call it as zeta-regularized determinant of the Laplacian. We denote it by Det ∆ M . Of course it can be defined in a same way for more general elliptic operators.
In our case of the three dimensional Heisenberg manifold M(ℓ) = H 3 /Γ ℓ we put then the second term is the trace of the heat kernel of the flat torus T 2 ∼ = (g + ⊕ g − ) /Γ B , and so

Hence we have
The value ζ ′ T 2 (0) is given by the formula called " Kroneker's second limit formula": [7], [13]) We give an elementary proof of this formula and expressions of the zeta-reguralized determinant for the three and four dimensional flat tori in §7.

So in this section we only consider the value ζ
Since the functions g and h are rapidly decreasing on the positive real axis, we have, Next we consider the behavior of the functions ∂ k+l G ∂u k ∂v l , when u ↓ 0. Let Note that α 0 = 1, α 1 = −1/2, α 2 = 1/12, α 2i+1 = 0 for i = 1, 2, 3, · · · · · · and β 0 = 1, β 2 = 1/6, β 4 = 7/360. The coefficients are expressed in the following forms with For |u| < π and v ∈ [0, 1], the function G(u, v) is expanded as follows: where we denote the polynomial P n (v) Let g(v, s) be a sufficiently many times differentiable function defined on a domain in R × C including [0, 1] × D, where D = {s ∈ C | Re(s) > −ǫ} (ǫ > 0 and fixed) and g is holomorphic on the domain D for each fixed t ∈ [0, 1]. has the Laurent expansion at s = 0 as where R −1 and R 0 are given by the formulas: and By applying this to the function of the form g(v, s) 1 + v we have the Laurent expansion of the function When we restrict the variable s in the domain Re(s) > 3/2, we have Now by Corollary (2.6) when we put then Proposition 2.7.
Proof. First we show Next we calculate R 0 : Summing up we have and the zeta-regularized determinant of the Laplacian on the Heisenberg manifold H 3 /Γ ℓ is given by the formula: we only determine the sign of this integral. For this purpose we decompose the integral (2.8) in the form For this purpose, recall that h(u) = u sinh u is expanded for u ∈ (−π, π) as with the coefficients with positive coefficients γ 2k such that Hence From these estimates we see that the function h(u/2) 2 − 1 + 1 12 u 2 takes positive values on the interval [0, 2], since h(u/2) 2 − 1 + 1 12 is the form of the alternative sum consisting of decreasing positive sequences when u ∈ [0, 2]. Now we take r = 2, then we see that I < 0, hence we have the desired result.

Heat kernel asymptotics
As an application of the formula (2.1) we calculate the heat kernel asymptotics for the three dimensional Heisenberg manifolds, which are given in terms of Bernoulli numbers.
We know that the heat kernel k M (t; x, y) has the asymptotic expansion: Of course the coefficients are given in terms of quantities coming from metric tensors, but here we calculate the values explicitly by the method of analytic continuation by making use of the formula (2.1), since we know that the Mellin transform of the heat kernel has poles of order one (at most) at the points n/2 − k, k = 0, 1, 2, · · · · · · , and the residue at the pole n/2 − k is given by the integral In our cases the trace of the heat kernel and the second term Z T 2 (t) is that corresponding to the two dimensional flat torus. So that it is enough to consider the Mellin transform of the first term Z V (t): Here, .

Since the function defined by the integral
is holomorphic for any s ∈ C and so the second term in the above expression (3.1) has poles of order at most one at the points s = 1, 0, −1, −2, · · · . Hence residues at these points must vanish. Consequently it is enough to consider the first term, for calculating the residues at the poles s = 3/2 − k, k = 0, 1, 2, 3 · · · . Then we have To calculate the residues at the poles s = 3/2 − k, k = 0, 1, 2, · · · · · · , again it is enough to consider the terms corresponding to m = even = 2k in the sum (3.3).
cally continued to the whole complex plane and we have Proof. Let i > 0, then So at the point s = 3/2 − (i + j) it takes the form Now by Lemma (3.1), to calculate the residues of the function (2.6) at the points s = 3/2 − k, it is enough to consider the function is given one by one by the following lemma where we put is given by the formula:

Five dimensional Heisenberg manifolds
So far we only considered three dimensional cases and illustrated the procedure to calculate the determinants and heat kernel asymptotics somehow precisely. These indicate that our method for calculating the determinants and heat kernel asymptotics for higher dimensional Heisenberg manifolds would also be valid. Even so the calculations and the results are so complicated, here we state the results for the cases of 5-dimensional Heisenberg manifolds.
Let h 5 be the 5-dimensional Heisenberg Lie algebra: where z is the center and [g + , g − ] = z. The corresponding Lie group H 5 is realized as As in the three dimensional cases we only consider a left invariant Riemannian metric defined from such an inner product on the Lie algebra h 5 that Also as in the three dimensional cases let us take uniform discrete subgroups Γ ℓ (ℓ ∈ N) of the form We identify H 5 and h 5 by the exponential map then the uniform discrete subgroup Γ ℓ ⊂ H 5 is identified with a direct sum of two lattices Γ B ⊂ g + ⊕ g − and Γ V (ℓ) ⊂ z: The heat kernel on the whole group H 5 is given as the following form: K t (g, g) = K t (x, y, z; x, y, z) where we regarded η ∈ z * = RZ * 1 ∼ = R. and g = (x, y, z) = x 1 X 1 + x 2 X 2 + yY 1 + y 2 Y 2 + zZ 1 , and similar to g through the exponential map.
Then the heat kernel k H 5 /Γ ℓ (t; [g], [ g]) on the Heisenberg manifold M ℓ = H 5 /Γ ℓ , is expressed as : and its trace is calculated in the following form: Here F ℓ denotes a fundamental domain of the uniform discrete subgroup Γ ℓ and Γ * B and Γ V (ℓ) * are dual lattices, as before. The second term Z T 4 (t) is that corresponding to 4-dimensional flat torus (g Let ζ M ℓ (s) be the function then Theorem 4.2.
Corresponding to the formula (2.1) we have an integral representation of the function ζ V,ℓ (s):   Proof. Put I(ℓ) it is enough to see the sign of this integral as we did before in Corollary (2.9) to determine the behavior of the determinant Det ∆ H 5 /Γ ℓ when ℓ → ∞. Then, by a similar calculation in Corollary 2.9 we have an expression of the integral It is clear that the second integrand takes always positive values on the positive real axis. We can also prove that the integrand in the first integral takes positive values on the positive real axis. Since the coefficients of the Taylor expansion of the function a n x 2n are given as and all take positive values, which we can see from the expression of a n for n ≥ 4, a n = 1 20 · (2n + 3)! 3 2n−2 − 1 (2n + 3)(2n + 2)(2n + 1)(2n) − 15 · 3 4 + 20 · (2n + 3)(2n + 2)(2n + 1) − 15 · (3 4 − 1) , a 4 = 1 20 · (11)! 3 6 − 1 11 · 10 · 9 · 8 − 15 · 3 4 + 20 · 11 · 10 · 9 − 15 · (3 4 − 1) and for n = 3, a 3 = 4 189 . From these facts we can prove the desired result. Finally we list the heat asymptotics for five dimensional Heisenberg manifolds M ℓ . Let c k be the coefficients of the asymptotic expansion of the heat kernel k H 5 /Γ ℓ (t; x, y) of the five dimensional Heisenberg manifolds M ℓ = H 5 /Γ ℓ : Let us denote the Taylor expansion of the function h(x) 2 as then δ k is given by Proposition 4.6. (4.11) where W k is given in (3.5).
In this section we give a formula of the zeta-regularized determinant of the Laplacian on the product Riemannian manifold M × N in terms of the each value of the zeta-regularized determinant and heat invariants.
The spectral zeta-function ζ M×N (s) for the product Riemannian manifold M × N is given by

We express this as
where N = dim N, M = dim M and {b i } ∞ i=0 are the coefficients of asymptotic expansion of the trace of the heat kernel k N (t; x, y) on N: We also denote by {a i } ∞ i=0 the coefficients of the asymptotic expansion of trace of the heat kernel k M (t; x, y) on the manifold M: , the first term Q 0 (s) is holomorphic on the domain {s ∈ C | Re(s) > −1/2}, at least under this expression for any case of the dimensions of the two manifolds. So we can put s = 0 in (5.1) and we have Next we put the second term as By the remark above we can describe the derivative at s = 0 of the each term in (5.6) in terms of the spectral zeta-function ζ M (s). For this purpose we consider four cases separately. Hence, Remark 5.6. −Γ ′ (1) = − ∞ 0 e −t log tdt = C e = 0.57721 · · · · · · = Euler's constant. We can now write down an expression of the value Det ∆ M×N corresponding to the each case(5.2, 5.3, 5.4 and 5.5) above. Here we only state for the case that both of dim M and dim N are even. In the next section we state a special case of N = S 1 .
Theorem 5.7. Let dim N and dim M be both even, then The formula we gave in the last section says that even in the product manifold case the zeta-regularized determinant is not expressed in a simple way in terms of each zeta-regularized determinant. In the paper [7] a formula of the zeta-regularized determinant for a manifold of the type M × S 1 is given, in fact a formula is derived for a higher order operator of the form with the variable in S 1 being separated, by reducing the problem to a boundary value problem on the manifold M×[0, 1] In this section we give a direct proof of this formula by restricting ourselves to the case of Laplacians on M × S 1 by following the same line as we did in the last section. Here in this special case the Poisson summation formula allows us to make an integration of the term (5.5) and we arrive at a precise formula.

Now in this case the Laplacian ∆ M×S 1 is given by ∆
x ∈ R, ℓ > 0 and the spectrum are are the spectrum of M, and k ∈ Z .
where the constant C M is given by (a) when dim M = even = 2m, then Here a (M +1)/2 denotes (M + 1)/2-th coefficient of the asymptotic expansion of the heat kernel Z M (t) on M.
Proof. We express the spectral zeta-function ζ M×S 1 (s) as Then by using the Poisson's summation formula this equals to where we put We denote the second term Since the sum To calculate the value P ′ 0 (0) recall a formula of a modified Bessel function K 1/2 (z) = K −1/2 (z) (see [1]):

Now we have
Hence from Propositions 5.2 and 5.4 the determinant Det ∆ M×S 1 is of the form where the constant C M is given by (a) when dim M is even, then Example 6.2. As an application of our formula (6.1) we give an expression of Det ∆ S 2 ×S 1 .
For the standard 2-dimensional sphere S 2 the spectral zeta-function is 2k + 1 k s (k + 1) s , which converges for Re(s) > 1. We rewrite this as where we used the expansion (1 + z) α = d m (α)z m , for |z| < 1. Note that for α > 0 this series converges for −1 ≤ z ≤ 1. Then for Re(s) > (2 − n)/2, by the estimate and the functional relation for the Riemann ζ-function the expression (6.4) gives us the analytic continuation of ζ S 2 (s) to the complex plane of Re(s) > (2 − n)/2 for each n > 0. So we can put s = −1/2 in (6.4) and we have an expression of ζ S 2 (−1/2): This is also expressed as So, finally we have (6.6) Again by applying Propositions 5.4 and 5.3 we have an alternative representation of the determinant Det ∆ M×S 1 .

Corollary 6.3. When dim M is odd, then
Here we used the formula ζ(s) = 1/(s − 1) + C e s + O((s − 1) 2 ). When dim M is even, then Note that ζ(−2k) = 0 for k = 1, 2, · · · . Remark 6.4. Our formula (6.1) is of course a special case of formulas given in the paper [7] for more general elliptic operators on the product manifolds M × S 1 . However here we gave an expression of the constant C M , although the formula itself is not a computable form, especially for M being odd dimensional. To obtain a further information we must specify the manifolds M. So in the next section we give a more precise form of this factor C M for some flat tori.

Flat tori
In the last two sections we considered the zeta-regularized determinant for manifolds of a product form as a Riemannian manifold. In this section we deal with the case that the manifolds are two, three and four dimensional flat tori, which are not always of a product form of lower dimensional tori as Riemannian manifolds.
We know by a similar calculation as we showed in §2 and §4 that the zetaregularized determinant of (2n + 1)-dimensional Heisenberg manifolds are always of the product form with a factor which is the zeta-regularized determinant of a 2n-dimensional torus. So of course it is required to determine the zeta-regularized determinant of flat tori to complete the calculation for Heisenberg manifolds. In this section we give an expression of it for two, three and four dimensional flat tori. Although our expressions are not of a computable form within a finite step, the expression for two dimensional cases are given by the famous limit formula of Kroneker as we cited in §2, and higher cases correspond to a generalization of this limit formula. The structure of a generalization was already stated and discussed focusing in their functional relations in the papers [2] and [3] for more general Dirichlet series than Epstein zeta-functions which are of our cases. Here we treat with the typical Epstein zeta-functions of two, three and four variables. Since it is enough for our purpose to give an explicit analytic continuation of the functions from a left half region in the complex plane to a region including zero, we give them based on the Jacobi identity and the Mellin transformation in a quite elementary way. For this purpose we fix the flat tori in the following way.
Let e 1 , e 2 , e 3 , e 4 be the standard orthonormal basis on R 4 and we fix a basis {u 1 , u 2 , u 3 , u 4 } of the following form u 1 = e 1 , u 2 = a 1,2 e 1 + a 2,2 e 2 , we put this = Ae 1 + Be 2 , u 3 = a 1,3 e 1 + a 2,3 e 2 + a 3,3 e 3 u 4 = a 1,4 e 1 + a 2,4 e 2 + a 3,4 e 3 + a 4,4 e 4 (a 2,2 , a 3,3 , a 4,4 > 0). I. Kroneker's second limit formula and two dimensional tori Since the dual lattice L * of L is given by non-zero eigenvalues of the Laplacian on T 2 L are By using the Poisson's summation formula we rewrite the first term as follows: is of the following form: Proposition 7.1. ( [11], [3]) where we put We know the integrand of H 0 (s) satisfies the asymptotics: Hence the first term H 0 (s) is a holomorphic function of s on the whole complex plane, and Then again by making use of (6.2) Then the zeta-regularized determinant Det ∆ T 2 L is given by the formula: Theorem 7.3.
Corollary 7.4. From the expression (7.4) we can see easily that Det ∆ T 2 L is periodic with respect to the parameter A, and when A = 0 we have both of Now we explain the relation of (7.4) with the Kroneker's second limit formula(see [7] where another explanation is given.). By using the integral representation of the modified Bessel function K α (z) ( [1]): the function H 0 (s) is expressed as Then from this we have a functional relation of the function H 0 (s): Proposition 7.5.
This relation (7.5) together with the functional relation of Riemann ζ-function gives us a very simple functional relation of the spectral zeta-function ζ T 2 L (s) for two dimensional flat torus T 2 L .
From the formula (7.1) we can easily see that the function ζ T 2 L (s) has (only)a pole of order one at s = 1, which comes from that of the second term in (7.1) and the Kroneker's second limit formula gives the constant term at this pole, that is, by the above relation (7.7) of the first term H 0 (s) we have Proposition 7.7. (Kroneker's second limit formula) This gives Corollary 7.8. log II. Three dimensional flat torus.
We put this as A 0 (s) + A 1 (s) + A 2 (s), and calculate each A ′ i (0). (a) where ζ T 2 L (s) is the spectral zeta-function of two dimensional torus T 2 L corresponding to the lattice L = L A,B , A = − g 2,1 g 2,2 , B = g 1,1 g 2,2 . Then Next we express the value ζ T 2 L (−1/2) in terms of a modified Bessel function K α (z) for α = 1: So we return to the expression (7.1): Then (c) Finally, for the lattice we have an expression of the zeta-regularized determinant Remark 7.10. As a special case of T 3 , let assume a 1,3 = a 2,3 = 0. Then the formula of Det ∆ T 3 for this case coincides with the formula (6.1) for T 2 × S 1 .

III. Four dimensional flat torus
Here we only state a formula for a four dimensional torus and a special case of them. As an application of the formula (5.7) we state a formula for a torus defined by the latter one, that is, the torus is a direct product of two 2-dimensional tori as Riemannian manifold. g 2,2 , b 0 = volume of N = a 3,3 a 4,4 . Note that the second term is obtained by making use of the Jacobi identity: Finally, we note that in the most special case, that is, let the matrix A be the identity matrix, then we have a formula for the spectral zeta-function ζ T 4 (s) It is possible to simplify this formula by using several formulas of Riemann zetafunction ζ(s) and to compare with our formula.
Remark 7.14. Similar to the case of ζ T 2 L (s), the function ζ T 3 L (s) (respectively ζ T 4 L (s)) has only a pole at s = 3/2 (resp. s = 2) of order one coming from the second term A 1 (s) (respectively B 1 (s)) and the term A 0 (s) (resp. B 0 (s)) will correspond to the term H 0 (s) in the two dimensional cases and there are similar functional relations like (7.5) also in these cases which are derived from the Jacobi identity.