Laplacian on forms and anomalies in closed hyperbolic manifolds

Abstract

The global multiplicative properties of the Laplacian on j — forms and related zeta functions are analyzed. The explicit form of the multiplicative and conformal anomalies in closed oriented hyperbolic manifolds gamma<img src="http:/img/fbpe/bjp/v30n3/16eq01.gif" border="0">d are derived.

Laplacian on forms and anomalies in closed hyperbolic manifolds

A.A. Bytsenko^* A. E. Gonçalves, and M. Simões

Departamento de Física, Universidade Estadual de Londrina,

Caixa Postal 6001, CEP 86051-970, Londrina, PR, Brazil

Received 1st June, 1999

The global multiplicative properties of the Laplacian on

j

— forms and related zeta functions are analyzed. The explicit form of the multiplicative and conformal anomalies in closed oriented hyperbolic manifolds G

^d

are derived.

I Introduction

The multiplicative properties of (pseudo-) differential operators as well as properties of their determinants have been studied actively during recent years in the mathematical and physical literature. The anomaly associated with product of regularized determinants of operators can be expressed by means of the non-commutative residue, the Wodzicki residue [1] (see also Refs. [2, 3]). The Wodzicki residue, which is the unique extension of the Dixmier trace to the wider class of (pseudo-) differential operators [4, 5], has been considered within the non-commutative geometrical approach to the standard model of the electroweak interactions [6, 7, 8] and the Yang-Mills action functional. The product of two (or more) differential operators of Laplace type can arise in higher derivative field theories (for example, in higher derivative quantum gravity).

Some recent papers along these lines can be found in Refs. [9, 10, 11, 12]. The zeta function associated to the product of Laplace type operators acting in irreducible rank 1 symmetric spaces and the explicit form of the multiplicative anomaly have been derived in [11].

Under such circumstances we should note that the conformal deformation of a metric and the corresponding conformal anomaly can also play an important role in quantum theories with higher derivatives. It is well known that evaluation of the conformal anomaly is actually possible only for even dimensional spaces and up to now its computation is extremely involved. The general structure of such an anomaly in curved d-dimensional spaces (d even) has been studied in [13]. We briefly mention here analysis related to this phenomenon for constant curvature spaces. The conformal anomaly calculation for the d- dimensional sphere can be found, for example, in Ref. [14]. The explicit computation of the anomaly (of the stress-energy tensor) in irreducible rank 1 symmetric spaces has been carried out in [15, 16, 17] using the zeta-function regularization and the Selberg trace formula.

The purpose of the present paper is to investigate the spectral zeta functions associated with a product of Laplacians on j- forms and to calculate in an explicit form the multiplicative and conformal anomalies for d- dimensional closed oriented hyperbolic manifolds G/^d.

II The spectral zeta function and the trace formula

We shall be working with irreducible rank 1 symmetric spaces X = G/K of non-compact type. Thus G will be a connected non-compact simple split rank 1 Lie group with finite center and K Ì G will be a maximal compact subgroup. Up to local isomorphism we choose X = SO₁(d,1)/SO(d). Thus the isotropy group K of the base point (1, 0, ...0) is SO(d); X can be identified with hyperbolic d- space ^d, d = dim X. It is possible to view ^d, for example, as one sheet of the hyperboloid of two sheets in ^d+1 given by q(x) = - + + ... + = -1, x₀ > 0 with the metric induced by the quadratic form q(x). Let G Ì G be a discrete, co-compact, torsion free subgroup, and let c(g) = trace(c(g)) be the character of a finite-dimensional unitary representation c of G for g Î G. Let L^(j)º D_G^(j) be the Laplacian on j- forms acting on the vector bundle V(X_G) over X_G = G\G/K induced by c. Note that the non-twisted j- forms on X_G are obtained by taking c = 1. One can define the heat kernel of the elliptic operator ^(j) = L^(j) + b^(j) by

where

₀ is an arc in the complex plane ; the b^(j) are endomorphisms of the vector bundle V(X_G). By standard results in operator theory there exist e,d > 0 such that for 0 < t < d the heat kernel expansion holds

where {l_l^(j)}_{l = 0}^¥ is the set of eigenvalues of operator L^(j) and n_l(c) denote the multiplicity of l_l^(j). Eventually we would like also to take b^(j) = 0, but for now we consider only non-zero modes: b^(j) + l_l^(j) > 0, "l : l₀^(j) = 0, b^(j) > 0.

Let a₀, n₀ denote the Lie algebras of A, N in an Iwasawa decomposition G = KAN. Since the rank of G is 1, dima₀ = 1 by definition, say a₀ = H₀ for a suitable basis vector H₀. One can normalize the choice of H₀ by b(H₀) = 1, where b: a₀ ® is the positive root which defines n₀; for more detail see Ref. [18]. Since G is torsion free, each g Î G-{1} can be represented uniquely as some power of a primitive element d : g = d^j(g) where j(g) ³ 1 is an integer and d cannot be written as g₁^j for g₁Î G, j > 1 an integer. Taking g Î G, g ¹ 1, one can find t_g > 0 and m_gÎ M {m_gÎ K | m_ga = am_g, "a Î A} such that g is G conjugate to m_gexp(t_gH₀), namely for some g Î G, ggg^-1 = m_gexp(t_gH₀). Besides let c_s(m) = trace(s(m)) be the character of s, for s a finite-dimensional representation of M.

II.1 Fried's trace formula [19]

For 0 £ j £ d-1,

where

r₀ = (d - 1)/2, and the function C(g), g Î G, defined on G-{1} by

For Ad denoting the adjoint representation of G on its complexified Lie algebra, one can compute t_g as follows [20]

Here C_G is a complete set of representatives in G of its conjugacy classes; Haar measure on G is suitably normalized. In our case K SO(d), M SO(d - 1). For j = 0 (i.e. for smooth functions or smooth vector bundle sections) the measure m₀(r) corresponds to the trivial representation of M. For j ³ 1 there is a measure m_s(r) corresponding to a general irreducible representation s of M. Let s_j is the standard representation of M = SO(d-1) on L ^j(d-1). If d = 2n is even then s_j (0 £ j £ d -1) is always irreducible; if d = 2n + 1 the every s_j is irreducible except for j = (d-1)/2 = n, in which case s_n is the direct sum of two (1/2)- spin representations s^± : s_n = s⁺ Ås^-. For j = n the representation t_n of K = SO(2n) on Lⁿ²ⁿ is not irreducible, t_n = Å is the direct sum of (1/2)- spin representations.

II.2 The Harish-Chandra Plancherel measure

Let the group G = SO₁(2n, 1). Then

and ms_j(r) = ms_2n-j-1(r).

For the group G = SO₁(2n + 1, 1) one has

We should note that the reason for the pair of terms {I ^(j), I ^(j-1)}, {H ^(j), H ^(j-1)} in the trace formula Eq. (2.3) is that t_j satisfies t_j|_M = s_j Å s_j-1.

Finally using Eqs. (2.8)-(2.11) we have

where the P(r, d) are even polynomials (with suitable coefficients (d)) of degree d-1 for G ¹ SO(2n+1, 1), and of degree d = 2n + 1 for G = SO₁(2n + 1, 1) [21, 18].

II.3 Case of the trivial representation

For j = 0 we take I ^(-1) = H ^(-1) = 0. Since s₀ is the trivial representation cs₀(m_g) = 1. In this case Fried's formula Eq. (2.3) reduces exactly to the trace formula for j = 0 [20, 22]:

where r₀ is associated with the positive restricted (real) roots of G (with multiplicity) with respect to a nilpotent factor N of G in an Iwasawa decomposition G = KAN. The function H⁽⁰⁾(t, b⁽⁰⁾) has the form

III Case of zero modes.

It can be shown [23] that the Mellin transform of H⁽⁰⁾(t, 0) (b⁽⁰⁾ = 0, i.e. the zero modes case)

is a holomorphic function on the domain Res < 0. Then using the result of Refs. [21, 18] one can obtain on Res < 0,

where K_n(s) is the modified Bessel function, and finally

Here y_G(s; c) º d(logZ_G(s; c))/ds, and Z_G(s; c) is a meromorphic suitably normalized Selberg zeta function [24-29, 22, 30, 21].

III The multiplicative anomaly

In this section the product of the operators on j- forms Ä , = L^(j) + , p = 1,2 will be considered. We are interested in multiplicative properties of determinants, the multiplicative anomaly [2, 3]. The multiplicative anomaly F(, ) reads

where we assume a zeta-regularization of determinants, i.e.

Generally speaking, if the anomaly related to elliptic operators is nonvanishing then the relation logdet(Ä ) = Trlog(Ä ) does not hold.

II.1 The zeta function of the product of Laplacians

The spectral zeta function associated with the product Ä

has the form

We shall always assume that ¹ , say > . If = then z(s| Ä ) = z(2s|^(j)) is a well-known function. For , Î , set b₊ ( + )/2, b_- = ( - )/2, thus = b₊ + b_- and = b₊ - b_-.

The spectral zeta function can be written as follows [11]:

where the integral converges absolutely for Re s > d/4. This formula is a main starting point to study the zeta function. It expresses z(s| Ä ) in terms of the Bessel function I_s-(b_-t) and (t, b₊), where the trace formula applies to (t, b₊). Let B_p(j) = (r₀(p) - j)²+ and A c(1) vol(G\G)C^(j)(d)/4.

For Res > d/4 the explicit meromorphic continuation holds:

where

which is an entire function of s,

and F(a, b; g; z) is the hypergeometric function.

The goal now is to compute the zeta function and its derivative at s = 0. Thus we have

where

_2n are the Bernoulli numbers.

A preliminary form of the zeta function z(s|Ä_p ) at s = 0 is

The derivative of the zeta function at s = 0 has the form:

where

and s_nå_{k = 1}ⁿk^-1.

III.2 The one-loop effective action

After a standard integration, the contribution to the Euclidean one-loop effective action can be written as follows:

where m² is a normalization parameter. As a result we have

where

_l^(j)(0), E_l^(j)(0) and are given by the formulae (3.9), (3.10) and (3.14)-(3.17) respectively.

III.3 The residue formula and the multiplicative anomaly

The value of F(₁, ₂) can be expressed by means of the non-commutative Wodzicki residue [1]. Let _p, p = 1, 2, be invertible elliptic (pseudo-) differential operators of real non-zero orders a and b such that a + b ¹ 0. Even if the zeta functions for operators ₁, ₂ and ₁Ä ₂ are well defined and if their principal symbols satisfy the Agmon-Nirenberg condition (with appropriate spectra cuts) one has in general that F(₁, ₂) ¹ 1. For such invertible elliptic operators the formula for the anomaly of commuting operators holds:

More general formulae have been derived in Refs. [2, 3]. Furthermore the anomaly can be iterated consistently. Indeed, using Eq. (3.20) we have

III.4 The explicit formula for the multiplicative anomaly

In particular, for n = 2 and _pº the anomaly is given by the following formula

where

We note that for the four-dimensional space with G = SO₁(4, 1), one derives from Eq. (3.22) the result

which also follows from Wodzicki's formula (3.20), where we should set = (4)/4.

IV The conformal anomaly and associated operator products.

In this section we start with a conformal deformation of a metric and the conformal anomaly of the energy stress tensor. It is well known that (pseudo-) Riemannian metrics g_mn(x) and _mn(x) on a manifold X are (pointwise) conformal if _mn(x) = exp(2f)g_mn(x), f Î C^¥(). For constant conformal deformations the variation of the connected vacuum functional (the effective action) can be expressed in terms of the generalized zeta function related to an elliptic self-adjoint operator [31]:

where < T_mn(x) > means that all connected vacuum graphs of the stress-energy tensor T_mn(x) are to be included. Therefore Eq. (4.1) leads to

The formulae (3.5), (3.9), (3.10) and (3.11) give an explicit result for the conformal anomaly, namely

where d is even. For B_1,2(j) = B(j), B_1,2(j-1) = B(j-1) the anomaly (4.3) has the form

Note that for a minimally coupled scalar field of mass m, B(0) = + m². The simplest case is, for example, G = SO₁(2, 1) SL(2, ); besides X = ² is a two-dimensional real hyperbolic space. Then we have = 1/4, = 1, and finally

For real d-dimensional hyperbolic space the scalar curvature is R(x) = -d(d-1). In the case of the conformally invariant scalar field we have B(0) = + R(x)(d - 2)/[4(d - 1)]. As a consequence, B(0) = 1/4 and

Thus in conformally invariant scalar theory the anomaly of the stress tensor coincides with one associated with operator product. This statement holds not only for hyperbolic spaces considered above but for all constant curvature manifolds as well [16].

V Concluding remarks

In this paper the one-loop contribution to the effective action (3.19), the multiplicative anomaly (3.22) and the conformal anomaly of the stress-energy momentum tensor (4.4), related to the operator product, have been evaluated explicitly. In addition we have considered the product Ä_p of Laplace operators acting in closed real hyperbolic manifolds. Note that the multiplicative anomaly is equal to zero for d = 2 and for the odd dimensional cases. It seems to us that the explicit results for the anomalies are not only interesting as mathematical results but are of physical interest. We hope that proposed discussion will be interesting in view of future applications to concrete problems in quantum field theory.

Acknowledgments

We thank F. L. Williams for useful discussion. First author partially supported by a CNPq grant (Brazil), RFFI grant (Russia) No 98-02-18380-a, and by GRACENAS grant (Russia) No 6-18-1997.

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*

On leave from Sankt-Petersburg State Technical University, Russia

Publication Dates

• Publication in this collection
  11 Jan 2002
• Date of issue
  2000

History

• Received
  01 June 1999