Abstract
We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray–Singer analytic torsion, up to a constant.
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1 Introduction
The Ray–Singer analytic torsion [77] of a closed smooth manifold is a zeta regularized graded determinant of its de Rham complex and computes the Reidemeister torsion [6, 32, 33, 72]. Rumin and Seshadri [85] have recently introduced an analytic torsion for the Rumin complex of contact manifolds [79, 80, 82] and showed that it coincides with the Ray–Singer torsion for 3-dimensional CR Seifert manifolds equipped with a unitary representation [85, Theorem 4.2]. In this paper we propose an analytic torsion of the Rumin complex associated with another filtered geometry [81, 83, 84] known as generic rank two distributions in dimension five [29].
A generic rank two distribution on a 5-manifold M is a smooth rank two subbundle \({\mathcal {D}}\) in the tangent bundle TM such that Lie brackets of sections of \({\mathcal {D}}\) span a rank three subbundle \([{\mathcal {D}},{\mathcal {D}}]\) in TM and triple brackets span all of the tangent bundle. In other words, \({\mathcal {D}}\) is a bracket generating distribution with growth vector (2, 3, 5). Clearly, this is a \(C^2\)-open condition on the 2-plane bundle \({\mathcal {D}}\), whence the name generic.
If \({\mathcal {D}}\) is a generic rank two distribution, then the Lie bracket of vector fields induces an algebraic (Levi) bracket on the associated graded bundle
where \({\mathfrak {t}}^{-1}M={\mathcal {D}}\), \({\mathfrak {t}}^{-2}M=[{\mathcal {D}},{\mathcal {D}}]/{\mathcal {D}}\) and \({\mathfrak {t}}^{-3}M=TM/[{\mathcal {D}},{\mathcal {D}}]\). This turns \({\mathfrak {t}}M\) into a bundle of graded nilpotent Lie algebras called the bundle of osculating algebras. Its fibers are all isomorphic to the graded nilpotent Lie algebra \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with graded basis \(X_1,X_2\in {\mathfrak {g}}_{-1}\), \(X_3\in {\mathfrak {g}}_{-2}\), \(X_4,X_5\in {\mathfrak {g}}_{-3}\) and brackets
A basic example of a generic rank two distribution is obtained by equipping the simply connected Lie group G corresponding to the Lie algebra \({\mathfrak {g}}\) with the G-invariant 2-plane bundle obtained by translating \({\mathfrak {g}}_{-1}\).
The study of generic rank two distributions reaches back to Cartan’s “five variables paper” [29] from 1910 where Cartan constructs a canonical connection and derives a non-trivial local invariant, the harmonic curvature tensor which is a section of \(S^4{\mathcal {D}}^*\). A generic rank two distribution is locally diffeomorphic to the invariant rank two distribution on the nilpotent Lie group G if and only if Cartan’s curvature tensor vanishes.
In modern terminology, generic rank two distributions can equivalently be described as regular normal parabolic geometries [27] of type \((G_2,P)\), where P is a particular parabolic subgroup in the split real form of the exceptional Lie group \(G_2\), see also [88]. Their geometry has many intriguing features. The most symmetric example, the flat model, is the homogeneous space \(G_2/P\) which is locally diffeomorphic to the nilpotent Lie group G and has an underlying manifold diffeomorphic to \(S^2\times S^3\), see [87]. In general, the symmetry group of a generic rank two distribution is a Lie group of dimension at most 14 and is subject to further restrictions [23, Theorem 7]. Nurowski [73] constructed a conformal metric of signature (2, 3) with conformal holonomy \(G_2\) which is naturally associated with a generic rank two distribution, see also [24, 54, 88,89,90]. Bryant and Hsu have shown that generic rank two distributions admit many rigid curves [18], i.e., curves which are isolated in the path space of curves tangent to \({\mathcal {D}}\) with fixed endpoints. In the same paper Bryant and Hsu also discuss the mechanical system of a surface rolling without slipping and twisting on another surface. This gives rise to a 5-dimensional configuration space equipped with a rank two distribution encoding the no slipping and twisting condition, which is generic iff the Gaussian curvatures are disjoint [18, Sect. 4.4]. If both surfaces are round spheres and the ratio of their radii is 3:1 then the universal covering of the configuration space is diffeomorphic to \(G_2/P\), see [8, 29]. Let us also mention the following recent studies of generic rank two distributions [2, 9, 50, 55, 64].
By virtue of Gromov’s h-principle for open manifolds [51, Sect. 2.2.2], it is well understood [37, Theorem 2] which open 5-manifolds admit globally defined generic rank two distributions. For closed manifolds the situation appears to be more subtle but the underlying topological problem has been settled, see [37, Theorem 1(b)]. The question to what extent generic rank two distributions abide by an h-principle [45, 51] has served as a motivation for us to study the analytic torsion of the associated Rumin complex, cf. [10, 30, 31, 41, 44, 48, 97].
The Rumin complex [81, 83, 84] associated with a generic rank two distribution on a 5-manifold M is a natural complex of higher order differential operators,
where \(E^q={\mathcal {H}}^q({\mathfrak {t}}M)\) denotes the vector bundle obtained by taking the fiberwise Lie algebra cohomology of the bundle of osculating algebras \({\mathfrak {t}}M\). The ranks of the bundles \(E^q\) are 1, 2, 3, 3, 2, 1 and the Heisenberg orders of the operators \(D_q\) are 1, 3, 2, 3, 1, see [19, Sect. 5] or [39, Example 4.24]. There exist injective differential operators \(L_q:\Gamma ^\infty (E^q)\rightarrow \Omega ^q(M)\), embedding the Rumin complex as a subcomplex in the de Rham complex, whose image can be characterized using differential operators. The operators \(L_q\) intertwine the Rumin differential with the de Rham differential, \(dL_q=L_{q+1}D_q\), and induce isomorphisms in cohomology. In particular, the Rumin complex computes the cohomology of M.
Let us emphasize that the Rumin differentials \(D_q\) are natural, i.e. independent of any further choices, and so is the map induced on cohomology by \(L_q\), see Lemma 3.2. The operators \(L_q\), however, depend on the choice of a sub-Riemannian metric \({\tilde{g}}\) on \({\mathcal {D}}\) and a splitting of the filtration, \(S:{\mathfrak {t}}M\rightarrow TM\). By the latter we mean a filtration preserving isomorphism inducing the identity on the associated graded, that is, \(S|_{{\mathfrak {t}}^{-1}M}={{\,\mathrm{id}\,}}\), \(S({\mathfrak {t}}^{-2}M)\subseteq [{\mathcal {D}},{\mathcal {D}}]\), \(S|_{{\mathfrak {t}}^{-2}M}={{\,\mathrm{id}\,}}\) mod \({\mathcal {D}}\), and \(S|_{{\mathfrak {t}}^{-3}M}={{\,\mathrm{id}\,}}\) mod \([{\mathcal {D}},{\mathcal {D}}]\). Using parabolic geometry, one can construct a variant of the Rumin complex \(D_q\) with splitting operators \(L_q\) which are entirely natural. Indeed, the Rumin complex is isomorphic to the curved BGG sequence [25] associated with the trivial representation of \(G_2\). Writing down the BGG operators explicitly, however, requires the determination of (a good portion of) the canonical regular normal Cartan connection associated with the rank two distribution [29], a quite involved procedure, see [88]. While this more natural perspective might prove to be insightful when studying the sub-Riemannian limit, it does not appear to be helpful for the purpose of this paper, and we will therefore not adopt it here.
Suppose F is a flat complex vector bundle over M and let \(D^F_q\) denote the differential complex obtained by twisting the Rumin complex with F. The definition of the analytic torsion requires a sub-Riemannian metric [52, 69], i.e., a fiberwise Euclidean inner product \({\tilde{g}}\) on \({\mathcal {D}}\), as well as a fiberwise Hermitian metric h on F. These choices give rise to fiberwise Hermitian metrics on the vector bundles \(E^q\) and permit to define formal adjoints of the Rumin differentials, denoted by \((D^F_q)^*\). The essential part of the analytic torsion is a zeta regularized graded determinant which may be expressed in the form
where the complex powers are understood to vanish on the kernels of the operators, see Remark 2.9. For our rigorous definition, however, we will proceed as in Ray–Singer [77] or Rumin–Seshadri [85] and rewrite this in terms of zeta functions associated with hypoelliptic Rumin–Seshadri type operators
see Sect. 2.3. For appropriate choices of the numbers \(a_q\), the operators \(\Delta _q^F\) are Rockland of Heisenberg order \(2\kappa \) and admit a parametrix in the Heisenberg calculus [39, 66, 96]. The zeta function \({{\,\mathrm{tr}\,}}\bigl ((\Delta _q^F)^{-s}\bigr )\) converges for \(\Re (s)>10/2\kappa \) and extends to a meromorphic function on the entire complex plane which is holomorphic at zero, see [38, Theorem 2]. Correspondingly, as \(t\rightarrow 0\), the heat kernel admits an asymptotic expansion [38, Theorem 1] of the form
where \(p^F_{q,j}\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F)\otimes |\Lambda |\bigr )\) are locally computable, and \(p_{q,j}^F=0\) for all odd j. Here \(|\Lambda |\) denotes the line bundle of 1-densities over M.
Following [6] we incorporate the zero eigenspaces and consider the analytic torsion as a norm \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(M;F))}_{{\mathcal {D}},{\tilde{g}},h}\) on the graded determinant line [62]
where \(H^q(M;F)\) denotes the de Rham cohomology of M with coefficients in the flat bundle F. This analytic torsion behaves as expected with respect to finite coverings and Poincaré duality, see Theorem 3.8. Moreover, this definition leads to a simple and local anomaly formula, describing the metric dependence, which is analogous to the corresponding formulas for the Ray–Singer torsion [6, Theorem 0.1] and the Rumin–Seshadri torsion [85, Corollary 3.7].
To state this formula, note that restriction provides natural isomorphisms
where \({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M)\) denotes the bundle of fiberwise graded Lie algebra automorphisms and \({{\,\mathrm{der}\,}}({\mathfrak {t}}M)\) denotes the bundle of fiberwise derivations of degree zero. Hence, each \(A\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathcal {D}}))\) induces \({\mathcal {H}}^q(A)\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathcal {H}}^q({\mathfrak {t}}M)))\) and correspondingly, each \(\dot{A}\in \Gamma ^\infty ({{\,\mathrm{end}\,}}({\mathcal {D}}))\) induces \({\mathcal {H}}^q(\dot{A})\in \Gamma ^\infty ({{\,\mathrm{end}\,}}({\mathcal {H}}^q({\mathfrak {t}}M)))\).
Theorem 1.1
Suppose \({\tilde{g}}_u\) is a family of sub-Riemannian metrics on \({\mathcal {D}}\) and \(h_u\) is a family of fiberwise Hermitian metrics on F, both depending smoothly on a real parameter u. Then
where \(\dot{g}_u:={\tilde{g}}^{-1}_u\frac{\partial }{\partial u}{\tilde{g}}_u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}({\mathcal {D}}))\), \(\dot{h}_u:=h^{-1}_u\frac{\partial }{\partial u}h_u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(F))\), and \(p^F_{q,u,10}\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F)\otimes |\Lambda |\bigr )\) denotes the constant term in the heat kernel expansion associated with the Rumin–Seshadri operator \(\Delta ^F_{u,q}\), cf. (1).
Of course, the Ray–Singer torsion of a 5-manifold is independent of the Riemannian metric and the Hermitian metric on F, for there are no constant terms in the small time asymptotic expansion of the heat trace of the Hodge Laplacians in odd dimensions. The relevant analogue in the heat trace asymptotics for Rockland operators on filtered manifolds is the homogeneous dimension, which happens to be even (ten) for generic rank two distributions in dimension five. Hence, there is no apparent analytical reason, why the corresponding analytic torsion should be independent of \({\tilde{g}}\) and h. The situation is similar for the Rumin–Seshadri analytic torsion of contact manifolds which have even homogeneous dimension too. On contact 3-manifolds, by adding (potentially vanishing) local correction terms, Rumin and Seshadri were able to turn their torsion into a CR-invariant [85, Corollary 3.8(3)] which coincides with the Ray–Singer torsion according to [1]. We will not compute the local quantities appearing in the anomaly formula in Theorem 1.1 more explicitly in this paper.
In view of Gray’s stability theorem [49, Theorem 2.2.2], a smooth deformation of a contact structure can always be absorbed by adjusting the sub-Riemannian metric. Hence, the infinitesimal change of the Rumin–Seshadri analytic torsion under a smooth deformation of the contact structure can be expressed as an integral over a local quantity. For generic rank two distributions the situation is fundamentally different due to the local invariant provided by Cartan’s curvature tensor. Even a small perturbation of the 2-plane bundle will in general result in a distribution which is not even locally diffeomorphic to the original one. Nevertheless, the torsion changes by a local quantity. More precisely, we have
Theorem 1.2
Suppose \(\Theta _u\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}(TM))\) depends smoothly on a real parameter u such that \(\Theta _0={{\,\mathrm{id}\,}}_{TM}\). Consider the 2-plane bundle \({\mathcal {D}}_u=\Theta _u({\mathcal {D}})\) equipped with the sub-Riemannian metric \({\tilde{g}}_u=(\Theta _u)_*{\tilde{g}}\). Then, provided u is sufficiently small, \({\mathcal {D}}_u\) is a generic rank two distribution and we have
where the density \(\alpha \) is locally computable. More precisely, \(\alpha (x)\) can be computed from the germ of \((M,{\mathcal {D}},{\tilde{g}},F,h,{\dot{\Theta }}|_{{\mathcal {D}}})\) at x, where \({\dot{\Theta }}=\frac{\partial }{\partial u}\big |_{u=0}\Theta _u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(TM))\).
A more explicit formula for the density \(\alpha \) will be provided in the proof presented in Sect. 4.1, see (149), (150), and (151).
We will compute the torsion for generic rank two distributions on some nilmanifolds, up to a constant. To formulate this result, let G denote the simply connected Lie group with Lie algebra \({\mathfrak {g}}\) and let \({\mathcal {D}}\) denote a right invariant generic rank two distribution on G. We equip \({\mathcal {D}}\) with a right invariant sub-Riemannian metric \({\tilde{g}}\). Since the structure constants of \({\mathfrak {g}}\) are rational, the group G admits lattices, i.e., discrete cocompact subgroups, [76, Theorem 2.12]. If \(\Gamma \) is a lattice in G, then \({\mathcal {D}}\) and \({\tilde{g}}\) descend to the nilmanifold \(G/\Gamma \). The induced structures on \(G/\Gamma \) will be denoted by \({\mathcal {D}}_\Gamma \) and \({\tilde{g}}_\Gamma \), respectively. We have the following comparison with the Ray–Singer torsion [6, 77] which will be denoted by \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(G/\Gamma ;F))}_{\mathrm{RS}}\).
Theorem 1.3
There exists a constant \(c>0\) such that
for every right invariant generic rank two distribution \({\mathcal {D}}\) on G, every lattice \(\Gamma \) in G which is generated by two elements, and every flat line bundle F over \(G/\Gamma \) with parallel fiberwise Hermitian metric h.
Actually, we will establish a slightly stronger result in Theorem 4.7 below.
Since the nilmanifold \(G/\Gamma \) admits a free circle action, its Ray–Singer torsion can be read off the corresponding Gysin sequence, see [65, Corollary 0.9].
To prove Theorem 1.3, we will consider the lattice \(\Gamma _0\) in G generated by two elements \(\exp (X_1)\) and \(\exp (X_2)\) where \(X_1,X_2\) is a basis of \({\mathfrak {g}}_{-1}\). The group of graded automorphisms \({{\,\mathrm{Aut}\,}}({\mathfrak {g}})\cong {{\,\mathrm{GL}\,}}(2,{\mathbb {R}})\) acts on G by diffeomorphisms preserving \({\mathcal {D}}_0\), the right invariant rank two distribution obtained by translating \({\mathfrak {g}}_{-1}\). The subgroup preserving the lattice, \({{\,\mathrm{Aut}\,}}({\mathfrak {g}},\log \Gamma _0)\), is a subgroup of finite index in \({{\,\mathrm{GL}\,}}(2,{\mathbb {Z}})\) which acts on the nilmanifold \(G/\Gamma _0\) by filtration preserving diffeomorphisms. Moreover, the space of unitary characters of \(\Gamma _0\), that is, the space of flat line bundles over \(G/\Gamma _0\) which admit a parallel Hermitian metric, can be covered by finitely many orbit closures of \({{\,\mathrm{Aut}\,}}({\mathfrak {g}},\log \Gamma _0)\). Exploiting this fact, and using several other results established here, including Theorems 1.1 and 1.2, we are able to give a proof of Theorem 1.3.
We expect that the statement in Theorem 1.3 remains true with \(c=1\) for arbitrary lattices \(\Gamma \) and all flat bundles F of arbitrary rank with parallel Hermitian metric h. This expectation is motivated by a recent result of Albin and Quan [1, Corollary 3], asserting that the quotient of the Rumin–Seshadri torsion and the Ray–Singer torsion can be expressed in terms of an integral over a local quantity. If their analysis of the sub-Riemannian limit can be generalized to generic rank two distributions, the aforementioned strengthening of Theorem 1.3 would follow at once.
The remaining part of this paper is organized as follows. In Sect. 2 we define an analytic torsion for Rockland differential complexes over general closed filtered manifolds. We establish a metric anomaly formula for this torsion in Theorem 2.11 and describe the variation of the torsion through a deformation of the underlying filtration in Theorem 2.17. Some basic properties including compatibility with finite coverings and duality are collected in Sect. 2.6.
In Sect. 3 we apply this general framework to the Rumin complex associated with certain filtered manifolds. The resulting anomaly formulas are contained in Theorems 3.11 and 3.15, compatibility with Poincaré duality is formulated in Theorem 3.8. In order for the Rumin complex to be Rockland, it is necessary to assume that the osculating algebras have pure cohomology. This apparently very restrictive assumption will be discussed in Sect. 3.7. We only know of three types of filtered manifolds with this property: trivially filtered manifolds giving rise to the Ray–Singer torsion, contact manifolds giving rise to the Rumin–Seshadri torsion, and generic rank two distributions in dimension five giving rise to the torsion proposed in this paper.
In Sect. 4 we specialize to generic rank two distributions in dimension five and prove the results stated in the introduction.
2 Analytic Torsion of Rockland Complexes
To every Rockland complex of differential operators over a closed filtered manifold one can associate an analytic torsion. For the de Rham complex this specializes to the classical Ray–Singer torsion [6, 77]. For the Rumin complex of a contact manifold it specializes to the Rumin–Seshadri analytic torsion [85].
The purpose of this section is to provide a rigorous definition of the analytic torsion of general Rockland complexes and to discuss some basic properties. We will show that this torsion is compatible with duality and finite coverings, see Sect. 2.6. Moreover, we will establish an anomaly formula which shows that the dependence on the metric is given by an integral over a local quantity, see Theorem 2.11. We will also study the variation of the torsion during a deformation of the underlying filtration, and show that in certain situations, this can be expressed in terms of local quantities also, cf. Theorem 2.17.
In Sects. 3 and 4 we will apply these results to Rumin complexes associated with certain filtered manifolds.
2.1 Differential Operators on Filtered Manifolds
Recall that a filtered manifold is a smooth manifold M together with a filtration of the tangent bundle TM by smooth subbundles,
which is compatible with the Lie bracket of vector fields in the following sense: If X is a smooth section of \(T^pM\) and Y is a smooth sections of \(T^qM\) then the Lie bracket [X, Y] is a section of \(T^{p+q}M\). Putting \({\mathfrak {t}}^pM:=T^pM/T^{p+1}M\), the Lie bracket induces a vector bundle homomorphism \({\mathfrak {t}}^pM\otimes {\mathfrak {t}}^qM\rightarrow {\mathfrak {t}}^{p+q}M\) referred to as (generalized) Levi bracket [27, Definition 3.1.7]. This turns the associated graded vector bundle \({\mathfrak {t}}M:=\bigoplus _p{\mathfrak {t}}^pM\) into a bundle of graded nilpotent Lie algebras called the bundle of osculating algebras. The Lie algebra structure on the fiber \({\mathfrak {t}}_xM=\bigoplus _p{\mathfrak {t}}^p_xM\) depends smoothly on the base point \(x\in M\), but will in general not be locally trivial. In particular, the Lie algebras \({\mathfrak {t}}_xM\) might be non-isomorphic for different \(x\in M\). The simply connected nilpotent Lie group with Lie algebra \({\mathfrak {t}}_xM\) is called osculating group at x and will be denoted by \(\mathcal T_xM\).
Using negative degrees, we are following a convention common in parabolic geometry, see [27, 70, 71] for instance. The other convention, where everything is concentrated in positive degrees, is the one that has been adopted in [95, 96].
The filtration on M gives rise to a Heisenberg filtration on differential operators which is compatible with composition and transposition. The basic idea is to consider differentiation along a vector field tangent to \(T^{-k}M\) as an operator of Heisenberg order at most k. Suppose E and F are two vector bundles over M. A differential operator \(A:\Gamma ^\infty (E)\rightarrow \Gamma ^\infty (F)\) of Heisenberg order at most k has a Heisenberg principal (co)symbol, encoding the highest order derivatives in the Heisenberg sense at \(x\in M\),
where \(\mathcal U_{-k}({\mathfrak {t}}_xM)\) denotes the degree \(-k\) part of the universal enveloping algebra of the graded nilpotent Lie algebra \({\mathfrak {t}}_xM=\bigoplus _p{\mathfrak {t}}^p_xM\). Equivalently, the Heisenberg principal symbol may be regarded as a left invariant differential operator
on the osculating group \(\mathcal T_xM\) which is homogeneous of degree k with respect to the grading automorphism. The Heisenberg principal symbol is compatible with composition and transposition of differential operators.
A differential operator \(A:\Gamma ^\infty (E)\rightarrow \Gamma ^\infty (F)\) of Heisenberg order at most k is said to satisfy the Rockland condition [78] if
is injective for every \(x\in M\) and every non-trivial irreducible unitary Hilbert space representation \(\pi \) of the osculating group \(\mathcal T_xM\), where \({\mathcal {H}}_{x,\infty }\) denotes the subspace of smooth vectors, cf. [61]. The Rockland theorem asserts that in this situation the operator admits a left parametrix which is of order \(-k\) in an appropriate Heisenberg calculus of pseudodifferential operators adapted to the filtration. In particular, Rockland operators are hypoelliptic.
This result has a long history. For trivially filtered manifolds, that is if \(TM=T^{-1}M\), it reduces to the classical, elliptic case. In this situation all irreducible unitary representations of the (abelian) osculating group are one dimensional, and the Rockland condition at \(x\in M\) becomes the familiar condition that the principal symbol of the operator is invertible at every \(0\ne \xi \in T_x^*M\), provided \({{\,\mathrm{rk}\,}}(E)={{\,\mathrm{rk}\,}}(F)\). Helffer–Nourrigat [56,57,58] proved maximal hypoellipticity for left invariant scalar Rockland differential operators on graded nilpotent Lie groups, thus confirming a conjecture due to Rockland [78]. For contact and (more generally) Heisenberg manifolds, a pseudodifferential calculus has been developed independently by Beals–Greiner [3] and Taylor [93], see also [74]. Rockland theorems for Heisenberg manifolds can be found in [3, Theorem 8.4] or [74, Theorem 5.4.1]. These investigations can be traced back to the work of Kohn [63], Boutet de Monvel [11], and Folland–Stein [47] on CR manifolds, cf. the introduction of [3] for further historical comments. A pseudodifferential calculus for general filtered manifolds was first described by Melin [66]. In his unpublished manuscript Melin shows [66, Theorem 7.2] that a scalar Rockland differential operator admits a parametrix in his calculus. More recent constructions of pseudodifferential calculi on filtered manifolds are based on the Heisenberg tangent groupoid [34, 53, 68, 95, 96] and the idea of essential homogeneity introduced in [40]. In [35, Theorem 2.5(d)] one finds a scalar Rockland theorem for graded nilpotent Lie groups which suffices to study the flat models in parabolic geometry as well as topologically stable [75] structures like contact and Engel manifolds. For general systems of (pseudo)differential operators on arbitrary filtered manifolds the Rockland theorem may be found in [39, Theorem A].
Using the parametrix of the heat operator and the Heisenberg pseudodifferential calculus, one can extend the results on the structure of complex powers [4, 67, 74, 91] to (formally) selfadjoint positive Rockland differential operators over general filtered manifolds, see [37, Corollary 2]. In particular, the zeta function of such an operator admits a meromorphic extension to the entire complex plane which is holomorphic at zero. Hence, its derivative at zero may be used to define regularized determinants for these operators.
2.2 Rockland Complexes
Consider a finite complex of differential operators over a closed filtered manifold M,
which are of Heisenberg order \(k_q\ge 1\), respectively. Here \(E^q\) are smooth vector bundles over M, only finitely many of these vector bundles are non-zero, and \(D_qD_{q-1}=0\) for all q. We assume the sequence is Rockland [39, Definition 2.14], i.e., the Heisenberg principal symbol sequence
becomes exact in every non-trivial irreducible unitary representation of \(\mathcal T_xM\).
Fix a volume density \(\mu \) on M and fiber wise Hermitian inner products \(h_q\) on the bundles \(E^q\), and consider the associated \(L^2\) inner product on \(\Gamma ^\infty (E^q)\),
Let \(D_q^*:\Gamma ^\infty (E^{q+1})\rightarrow \Gamma ^\infty (E^q)\) denote the formal adjoint of \(D_q\), that is
for all \(\phi \in \Gamma ^\infty (E^{q+1})\) and \(\psi \in \Gamma ^\infty (E^q)\). Clearly, \(D_{q-1}^*D_q^*=0\).
Fix natural numbers \(a_q\in \mathbb {N}\) such that
for all q, and put
Then the Rumin–Seshadri operators,
are all of Heisenberg order \(2\kappa \). These operators generalize the classical Hodge Laplacians as well as the Laplacians associated with the Rumin complex in [85, Sect. 2.3] which are of Heisenberg order two or four.
It will be convenient to consider the operators \(\Delta :=\bigoplus _q\Delta _q\), \(D=\bigoplus _qD_q\) and \(D^*=\bigoplus _qD_q^*\) acting on sections of the graded vector bundle \(E:=\bigoplus _qE^q\).
Lemma 2.1
The Rumin–Seshadri operator \(\Delta \) is formally selfadjoint and commutes with D and \(D^*\), i.e., we have the graded commutator relations
Moreover, defining \(\sigma _q:\Gamma ^\infty (E^q)\rightarrow \Gamma ^\infty (E^{q-1})\) by
and \(s_q:\Gamma ^\infty (E^q)\rightarrow \Gamma ^\infty (E^{q+1})\) by
we have the graded commutator relations
Proof
Clearly, \(\Delta _q^*=\Delta _q\). Using \(D_qD_{q-1}=0\), we obtain \(D_q\Delta _q=\Delta _{q+1}D_q\) and then \(\Delta _qD_q^*=D_q^*\Delta _{q+1}\) by passing to the formal adjoint, whence (8). Moreover, \(\Delta _q=D_{q-1}\sigma _q+\sigma _{q+1}D_q\) and thus \(\Delta _q=s_{q-1}D_{q-1}^*+D_q^*s_q\), whence (11). \(\square \)
Lemma 2.2
([39, Lemma 2.18]) The Rumin–Seshadri operator \(\Delta \) is Rockland of Heisenberg order \(2\kappa \).
Lemma 2.3
([38, Lemma 1]) The Rumin–Seshadri operator \(\Delta \) is essentially self-adjoint with compact resolvent on \(L^2(E)\). Moreover, \(\Delta \ge 0\).
Using the spectral theorem [60, Sect. VI§5.3] we obtain a strongly differentiable semi-group \(e^{-t\Delta }\) for \(t\ge 0\). Let
denote the homogeneous dimension of M. Note that \(n\ge 0\) since with our convention the grading of \({\mathfrak {t}}M=\bigoplus _p{\mathfrak {t}}^pM\) is concentrated in negative degrees. We will denote the line bundle of 1-densities on M by \(|\Lambda |\).
Lemma 2.4
([38, Theorem 1]) In this situation \(e^{-t\Delta }\) is a smoothing operator for each \(t>0\), and the corresponding heat kernel \(k_t\in \Gamma ^\infty (E\boxtimes (E^*\otimes |\Lambda |))\) depends smoothly on \(t>0\). Furthermore, as \(t\rightarrow 0\), we have an asymptotic expansion
where \(p_j\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E)\otimes |\Lambda |)\). Moreover, \(p_j(x)=0\) for all odd j.
For \(A\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E))\) we thus obtain an asymptotic expansion as \(t\rightarrow 0\),
where \({{\,\mathrm{str}\,}}\) denotes the graded trace. In particular,
where \({{\,\mathrm{LIM}\,}}\) denotes the constant term in the asymptotic expansion.
Lemma 2.5
The graded heat trace \({{\,\mathrm{str}\,}}\bigl (e^{-t\Delta }\bigr )\) is constant in t and
Here \(\chi (E,D)\) denotes the Euler characteristics of the complex in (2).
Proof
We adapt the classical argument. Combining the graded commutator relation \(\Delta =[D,\sigma ]\) from Lemma 2.1 with \([D,e^{-t\Delta }]=0\) and the fact that the graded trace vanishes on graded commutators, we see that the graded heat trace is constant in t:
Moreover, with respect to the trace norm we have [38, Lemma 5]
where P denotes the spectral (orthogonal) projection onto the kernel of \(\Delta \). In view of Lemma 2.3, this kernel is finite dimensional. By Hodge theory [39, Corollary 2.20], the inclusion \({{\,\mathrm{img}\,}}(P)\subseteq \Gamma ^\infty (E)\) induces an isomorphism in cohomology. Hence,
Putting \(A={{\,\mathrm{id}\,}}\) in (14) and combining the resulting equation with (16) and (18) we obtain (15). \(\square \)
2.3 Analytic Torsion
For \(\lambda \ge 0\), let \(P_\lambda \) denote the spectral projection of \(\Delta \) corresponding to the eigenvalues at most \(\lambda \), and let \(Q_\lambda :={{\,\mathrm{id}\,}}-P_\lambda \) denote the complementary projection. Fix numbers \(N_q\) such that
for all q. We let \(N\in \Gamma ({{\,\mathrm{end}\,}}(E))\) denote the operator given by multiplication with \(N_q\) on \(E^q\). For \(\lambda \ge 0\) and \(\Re (s)>n/2\kappa \), we put
According to [38, Corollary 2] this zeta function admits a meromorphic continuation to the entire complex plane which is holomorphic at \(s=0\). Moreover,
where \({{\,\mathrm{LIM}\,}}\) denotes the constant term in the asymptotic expansion, and
Putting \(A=N\) in (14) and using (21) we obtain
Recall that \({{\,\mathrm{img}\,}}(P_\lambda )\) is a finite dimensional subcomplex of \((\Gamma ^\infty (E),D)\). Moreover, the inclusion induces a canonical isomorphism in cohomology,
The torsion of finite dimensional complexes, see [7, Sect. a] or [6, Sect. Ia], provides a canonical isomorphism of graded determinant lines [62]
Recall here that the graded determinant line of a finite dimensional graded vector space \(V=\bigoplus _qV_q\) is the line \({{\,\mathrm{sdet}\,}}(V):=\bigotimes _q(\det V_q)^{(-1)^q}\) where \(\det W:=\Lambda ^{\dim W}W\) and the exponent \(-1\) is used to denote the dual line. Combining the latter two identifications, we obtain a canonical isomorphism
The \(L^2\) inner product on \(\Gamma ^\infty (E)\) restricts to a graded inner product on \({{\,\mathrm{img}\,}}(P_\lambda )\) and induces an inner product on the graded determinant line \({{\,\mathrm{sdet}\,}}({{\,\mathrm{img}\,}}(P_\lambda ))\). Via (23) this corresponds to an inner product on the line \({{\,\mathrm{sdet}\,}}(H^*(E,D))\). We will denote the corresponding norm by \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(E,D))}_{[0,\lambda ],h,\mu }\). Following [6, 7], we define the analytic torsion of the Rockland complex (E, D) to be the norm
on the graded determinant line
A priori, the analytic torsion of a Rockland complex (E, D) depends on the graded fiber wise Hermitian inner product h on E, the volume density \(\mu \) on M, the choice of numbers \(a_q\) satisfying (5), the choice of numbers \(N_q\) satisfying (19), and the choice of a spectral cutoff \(\lambda \ge 0\). It is fairly easy to see that the analytic torsion is actually independent of \(a_q\), \(N_q\) and \(\lambda \), see Lemmas 2.6, 2.7 and 2.8 below. The dependence on \(h_q\) and \(\mu \) will be discussed in Sect. 2.4, see Theorem 2.11 below.
Lemma 2.6
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of the spectral cutoff \(\lambda \ge 0\).
Proof
Suppose \(0\le \lambda _1\le \lambda _2\) and consider the finite rank spectral projection \(P_{(\lambda _1,\lambda _2]}:=P_{\lambda _2}-P_{\lambda _1}\). Clearly, \(C:={{\,\mathrm{img}\,}}(P_{(\lambda _1,\lambda _2]})\) is a finite dimensional graded vector space invariant under D, \(D^*\), and \(\Delta \). Moreover, \(\ker \bigl (D|_{C^q}\bigr )={{\,\mathrm{img}\,}}(D|_{C^{q-1}}\bigr )\) and \(\ker \bigl (D^*|_{C^q}\bigr )={{\,\mathrm{img}\,}}(D^*|_{C^{q+1}}\bigr )\). From the decomposition of complexes
we obtain
where the right hand side denotes the torsion of the finite dimensional acyclic complex (C, D), see [6, Proposition 1.5].
Consider the decomposition of graded vector spaces
where \(K^q:={{\,\mathrm{img}\,}}\bigl (P_{(\lambda _1,\lambda _2]}D_{q-1}\bigr )\) and \(L^q:={{\,\mathrm{img}\,}}\bigl (P_{(\lambda _1,\lambda _2]}D_q^*\bigr )\). A well known expression for the torsion of an acyclic complex, see [85, Proposition 2.7], gives
Clearly, \(\det \bigl (DD^*|_{K^q}\bigr )=\det \bigl (D^*D|_{L^{q-1}}\bigr )\) and thus
A telescoping argument yields
Using \(Q_{\lambda _2}-Q_{\lambda _1}=-P_{(\lambda _1,\lambda _2]}\) we obtain, see (20),
Hence,
and
Consequently,
Combining (25), (26), (27), and (28) we obtain
whence the lemma. \(\square \)
Lemma 2.7
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of numbers \(N_q\), see (19).
Proof
If \({\tilde{N}}_q\) is another choice such that \({\tilde{N}}_{q+1}-{\tilde{N}}_q=k_q\), then there exists a constant c such that \({\tilde{N}}_q-N_q=c\) for all q. Recall that the graded commutator relations \(\Delta =[D,\sigma ]\), see (11), and \([D,e^{-t\Delta }]=0=[Q_\lambda ,D]\) yield
Since \(\lim _{t\rightarrow \infty }{{\,\mathrm{tr}\,}}\bigl (Q_{\lambda ,q}e^{-t\Delta _q}\bigr )=0\), see (17), we conclude \({{\,\mathrm{str}\,}}\bigl (Q_\lambda e^{-t\Delta }\bigr )=0\) for all \(t>0\) and all \(\lambda \ge 0\). Hence, for \(\Re (s)>n/2\kappa \) we obtain
By analytic continuation, this remains true for all s, whence the lemma. \(\square \)
Lemma 2.8
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of numbers \(a_q\), see (5).
Proof
Suppose \({\tilde{a}}_q\in \mathbb {N}\) is another choice of positive integers satisfying (5), that is, \(k_{q-1}{\tilde{a}}_{q-1}=k_q{\tilde{a}}_q\) for all q. Then there exist positive integers \(r,{\tilde{r}}\in \mathbb {N}\) such that \(ra_q={\tilde{r}}{\tilde{a}}_q\) for all q. Indeed, we may use \(r:={\tilde{a}}_{q_0}\) and \({\tilde{r}}:=a_{q_0}\) for some fixed \(q_0\). W.l.o.g. we may assume \({\tilde{r}}=1\) and thus
for all q. Writing \({\tilde{\kappa }}:=k_q{\tilde{a}}_q\) we find, see (6),
Denoting the corresponding Laplacian by \({\tilde{\Delta }}_q:=(D_{q-1}D^*_{q-1})^{{\tilde{a}}_{q-1}}+(D^*_qD_q)^{{\tilde{a}}_q}\), see (4), and using \((D^*)^2=0=D^2\), we find
Using the spectral cutoff
and denoting the spectral projections of \({\tilde{\Delta }}\) by \({\tilde{Q}}\), we have
Using \((\Delta ^r)^{-s}=\Delta ^{-rs}\), (30) and (31) we find, see (20),
for all s. Differentiating and using (29) we find
In view of (31) we have \({{\,\mathrm{img}\,}}({\tilde{P}}_{{\tilde{\lambda }}})={{\,\mathrm{img}\,}}(P_\lambda )\) and thus
Combining (32) with (33) and Lemma 2.6 we see that the analytic torsion with the choices \(a_q\) and \({\tilde{a}}_q\) coincide, cf. (24). \(\square \)
Remark 2.9
The analytic torsion in (24) may be expressed in a way which does not require the choice of numbers \(a_q\) as in (5) and \(N_q\) as in (19). More precisely, for \(\lambda \ge 0\) we have
Here \({{\,\mathrm{str}\,}}_\lambda \bigl ((D^*D)^{-s}\bigr )\) denotes the graded trace, disregarding all eigenvalues \(\le \lambda \), in particular, disregarding the eigenvalue zero which will have infinite multiplicity in general. To see this, note that \(D^2=0\) gives, see (7)
Moreover, since \(D_{q-1}D_{q-1}^*\) and \(D^*_{q-1}D_{q-1}\) have the same spectrum
Using the latter two displayed equations, and proceeding by induction on q, one readily concludes, from the corresponding fact for \({{\,\mathrm{tr}\,}}(Q_\lambda \Delta _q^{-s})\), that the functions \({{\,\mathrm{tr}\,}}_\lambda \bigl ((D^*_qD_q)^{-a_qs}\bigr )\) and \({{\,\mathrm{tr}\,}}_\lambda \bigl ((D_{q-1}D^*_{q-1})^{-a_qs}\bigr )\) admit meromorphic extensions which are holomorphic at \(s=0\). Using (19) we obtain
Similarly,
Combining the latter two equations with the obvious equality
obtained via analytic continuation, we arrive at the formula in (34).
Remark 2.10
The analytic torsion of a Rockland complex remains unchanged if the differentials \(D_q\) are replaced with \(c_qD_q\) where \(c_q\in \mathbb C\) and \(|c_q|=1\).
2.4 Dependence on the Metric
Suppose the volume density \(\mu _u\) on M and the graded Hermitian inner product \(h_u\) on E depend smoothly on a real parameter u. Let \(D^*_u\) and \(\Delta _u\) denote the corresponding family of operators. For real u and v, we let \(G_{v,u}\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}(E))\),
denote the unique vector bundle automorphism such that, cf. (3),
for all \(\phi ,\psi \in \Gamma ^\infty (E)\). Clearly, \(G_{v,u}\) preserves the decomposition \(E=\bigoplus _qE^q\). Using (4) we immediately see that \(D^*_u\) varies by conjugation, i.e.,
The aim of this section is to establish the following variational formula generalizing well known results for the Ray–Singer torsion [6, 77] and the Rumin–Seshadri torsion, see [85, Corollary 3.7].
Theorem 2.11
In this situation we have
where \(\dot{h}_u:=h_u^{-1}\frac{\partial }{\partial u}h_u\), \({\dot{\mu }}_u:=\mu _u^{-1}\frac{\partial }{\partial u}\mu _u\), and \(p_{u,n}\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E)\otimes |\Lambda |)\) denotes the constant term in the heat kernel expansion associated with the Rumin–Seshadri operator \(\Delta _u\), see (13).
Remark 2.12
If the homogeneous dimension n is odd, then Theorem 2.11 implies that the analytic torsion of a Rockland complex does not depend on the choice of \(\mu \) and h, for in this case \(p_{u,n}=0\) according to Lemma 2.4. In the classical case this has already been observed by Ray and Singer [77].
Let \(\dot{G}_u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E))\) be defined by
where the middle equality is a consequence of the cocycle relation \(G_{w,v}G_{v,u}=G_{w,u}\), and the last equality follows by differentiating (35).
The following can be traced back to the original work of Ray–Singer [77, Theorem 2.1], see also [6, Theorem 5.6] or [85, Sect. 3.2].
Lemma 2.13
In this situation we have
Proof
Let us define \(\sigma _{u,q}:\Gamma ^\infty (E^q)\rightarrow \Gamma ^\infty (E^{q-1})\) by
and \(s_{u,q}:\Gamma ^\infty (E^q)\rightarrow \Gamma ^\infty (E^{q+1})\) by
Hence, from Lemma 2.1 we have graded commutator relations
Using Duhamel’s formula one shows
where \({\dot{\Delta }}_u:=\frac{\partial }{\partial u}\Delta _u\), see [85, Lemma 3.5] or [5, Corollary 2.50]. Consequently,
Put \({\dot{\sigma }}_u:=\frac{\partial }{\partial u}\sigma _u\) and let k denote the operator given by multiplication with \(k_q\) on \(\Gamma ^\infty (E^q)\). Then we have graded commutator relations \({\dot{\Delta }}_u=[D,{\dot{\sigma }}_u]\) and \([N,D]=Dk\), see (42) and (19), respectively. Combining this with \([D,e^{-t\Delta _u}]=0\) and the fact that the graded trace vanishes on graded commutators we obtain
Put \(\dot{D}^*_u:=\frac{\partial }{\partial u}D^*_u\). From (40) we get
Since \([D_{q-1}D^*_{u,q-1},e^{-t\Delta _{u,q}}]=0\) and the trace vanishes on commutators this yields
see also (6) and (41). Consequently,
Differentiating (37) and using (38), we obtain \(\dot{D}^*_u=[D^*_u,\dot{G}_u]\). Combining this with \([D^*_u,e^{-t\Delta _u}]=0\) and \([s_u,D^*_u]=\Delta _u\), see (42), and using the fact that the graded trace vanishes on graded commutators we obtain
Clearly,
Combining equations (44), (45), (46), (47) and (48), we obtain (39). \(\square \)
The following generalizes [85, Theorem 3.4].
Lemma 2.14
For \(\lambda \ge 0\) not in the spectrum of \(\Delta _u\) and \(\Re (s)>n/2\kappa \) we have
Moreover,
and
Proof
As \(\lambda \) is not in the spectrum of \(\Delta _{u,q}\), the spectral projector \(Q_{u,\lambda ,q}\) is smooth in u. Since \(\Delta _{u,q}\) commutes with \(Q_{u,\lambda ,q}\), Duhamel’s formula gives
where \(\dot{Q}_{u,\lambda ,q}:=\frac{\partial }{\partial u}Q_{u,\lambda ,q}\). Differentiating \(Q_{u,\lambda ,q}=Q_{u,\lambda ,q}^2\), we get \(\dot{Q}_{u,\lambda ,q}=\dot{Q}_{u,\lambda ,q}Q_{u,\lambda ,q}+Q_{u,\lambda ,q}\dot{Q}_{u,\lambda ,q}\) and then \(Q_{u,\lambda ,q}\dot{Q}_{u,\lambda ,q}Q_{u,\lambda ,q}=0\). Hence, \(\dot{Q}_{u,\lambda ,q}=Q_{u,\lambda ,q}\dot{Q}_{u,\lambda ,q}P_{u,\lambda ,q}+P_{u,\lambda ,q}\dot{Q}_{u,\lambda ,q}Q_{u,\lambda ,q}\) and thus
Combining this with (52), we obtain the following analogue of (43)
Proceeding exactly as in the proof of Lemma 2.13 and using the fact that \(Q_{u,\lambda }\) commutes with \(\Delta _u\), D and \(D^*_u\), one obtains
Using (20) and (54) it is straight forward to derive (49). Using the fact that
is holomorphic at \(s=0\), the equation (50) follows at once. Furthermore, we get
Combining this with (14) we obtain (51). \(\square \)
The following can be proved as in the classical case, see for instance [20, p 56].
Lemma 2.15
If \(\lambda \ge 0\) is not contained in the spectrum of \(\Delta _u\), then
Combining Lemma 2.15 with Eqs. (51) and (38), we obtain Theorem 2.11, see (24).
Remark 2.16
Twisting a Rockland complex as in (2) with a flat vector bundle F we obtain a new Rockland complex:
Suppose \(h^F\) is a parallel fiber wise Hermitian inner product on F and equip \(E^q\otimes F\) with the Hermitian metric \(h_q\otimes h^F\). Locally, the flat bundle F and the parallel Hermitian metric \(h^F\) can be trivialized simultaneously. Hence, locally, we have
Using locality of the heat kernel expansion, we conclude
for all \(j\in \mathbb N_0\), where \(p_j^F\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}(E\otimes F)\otimes |\Lambda |\bigr )\) and \(p_j\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}(E)\otimes |\Lambda |\bigr )\) denote the local quantities in the heat kernel expansions of the Rumin–Seshadri operators \(\Delta ^F\) and \(\Delta \), respectively, see (13).
2.5 Deformation of the Filtration
In this section we consider a family of closed filtered manifolds \(M_u\) smoothly depending on a real parameter \(u\in {\mathbb {R}}\). More explicitly, we assume the underlying manifold M remains fixed, and the filtration subbundles \(T^pM_u\subseteq TM\) depend smoothly on u. Suppose we have a smooth family of differential complexes \(D_u\) acting between sections of vector bundles \(E^q_u\) which depend smoothly on u,
We assume that each \(D_u\) is a Rockland complex with respect to the filtration on \(M_u\) and that the Heisenberg order \(k_q\) of \(D_{q,u}\) is independent of u. We consider graded Hermitian metrics \(h_u\) on \(E_u=\bigoplus _qE^q_u\) and volume densities \(\mu _u\) on M depending smoothly on u, and aim at describing the dependence of the torsion \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(E_u,D_u))}_{h_u,\mu _u}\) on the parameter u, cf. (24).
We will add the subscript u to the notation used in the preceding sections to indicate the dependence on this parameter. In particular, \(\Delta _u\) denotes the Rumin–Seshadri operator associated with \(E_u,D_u,h_u,\mu _u\), see (7), and \(k_{t,u}(x,y)\) will denote the (smooth) kernel of \(e^{-t\Delta _u}\). We take the numbers \(a_q\) satisfying (5) and the numbers \(N_q\) as in (19) to be constant in u.
Suppose we have differential operators \(A_{q,v,u}:\Gamma ^\infty (E^q_u)\rightarrow \Gamma ^\infty (E^q_v)\) which depend smoothly on real parameters u and v such that
We assume that \(A_{v,u}\) induces an isomorphism in cohomology,
and that \(A_{u,u}={{\,\mathrm{id}\,}}\). In this situation we have
Theorem 2.17
With respect to the isomorphism of determinant lines
induced by (56), we have
where \({\tilde{p}}_{n,u}\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E_u)\otimes |\Lambda |)\) is the constant term in the asymptotic expansion
Here \({\dot{\mu }}_u=\mu _u^{-1}\frac{\partial }{\partial u}\mu _u\), \(\dot{h}_u=h_u^{-1}\nabla _{\frac{\partial }{\partial u}}h_u\), and \(\dot{A}_u=\nabla _{\frac{\partial }{\partial v}}|_{v=u}A_{v,u}\) with respect to an auxiliary linear connection on the bundle \(\bigsqcup _uE_u\) and \(r_u\) denotes the Heisenberg order of \(\dot{A}_u\) with respect to the filtration on \(M_u\). In particular, \({\tilde{p}}_{n,u}\) is locally computable. More precisely, \({\tilde{p}}_{n,u}(x)\) can be computed from \((\dot{A}_u,\dot{h}_u,{\dot{\mu }}_u)\) at x and the germ of \((M_u,E_u,D_u,h_u,\mu _u)\) at x. Furthermore, \({\tilde{p}}_{n,u}=0\) if the homogeneous dimension n is odd.
In the remaining part of this section we will give a proof of this theorem. We begin by addressing the existence of the asymptotic expansion in (58).
Lemma 2.18
Suppose B is a differential operator, acting on sections of \(E_u\), which is of Heisenberg order at most \(r_u\) with respect to the filtration on \(M_u\), and let \((Bk_{t,u})(x,y)\) denote the (smooth) kernel of \(Be^{-t\Delta _u}\). Then we have an asymptotic expansion, as \(t\rightarrow 0\),
where \({\tilde{p}}^B_{j,u}\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(E_u)\otimes |\Lambda |)\) are locally computable. More precisely, \({\tilde{p}}^B_{j,u}(x)\) can be computed from the coefficients of B at x and the germ of \((M_u,E_u,D_u,h_u,\mu _u)\) at x. Furthermore, \({\tilde{p}}_{j,u}^B(x)=0\) for all odd j.
Proof
According to [38, Sect. 4], the heat operator \(\Delta _u+\frac{\partial }{\partial t}\) admits a parametrix Q in the Heisenberg calculus on \(M\times {\mathbb {R}}\) whose kernel \(k_Q\) satisfies
Here \(M\times {\mathbb {R}}\) is considered as a filtered manifold such that the vector field \(\frac{\partial }{\partial t}\) has Heisenberg order \(2\kappa \). Hence, BQ is a pseudodifferential operator of Heisenberg order \(r_u-2\kappa \) in the Heisenberg calculus on \(M\times {\mathbb {R}}\). For its kernel \(k_{BQ}\) we find
Since BQ is in the Heisenberg calculus, its kernel \(k_{BQ}\) admits an asymptotic expansion along the diagonal. Proceeding exactly as in [38, Sect. 4], this leads to the asymptotic expansion in (59). \(\square \)
Using parallel transport with respect to the auxiliary connection to identify the bundles \(E_u\) with a single vector bundle, we may w.l.o.g. assume \(E_u=E\) for all u. Then \(\dot{h}_u=h_u^{-1}\frac{\partial }{\partial u}h_u\) and \(\dot{A}_u=\frac{\partial }{\partial v}|_{v=u}A_{v,u}\). Differentiating (55), we obtain
where \(\dot{D}_u:=\tfrac{\partial }{\partial u}D_u\). By the chain rule,
In view of Theorem 2.11 we may thus assume w.l.o.g. that the Hermitian metrics and the volume densities are independent of u, that is, \(h_u=h\) and \(\mu _u=\mu \).
Lemma 2.19
If \(\lambda \ge 0\) is not in the spectrum of \(\Delta _u\) then
Proof
From Lemma 2.1 we have the graded commutator relation
Using Duhamel’s formula one shows as in (53)
where \({\dot{\Delta }}_u:=\frac{\partial }{\partial u}\Delta _u\). Consequently,
Put \({\dot{\sigma }}_u:=\frac{\partial }{\partial u}\sigma _u\) and let k denote the operator given by multiplication with \(k_q\) on \(\Gamma ^\infty (E^q)\). Then we have graded commutator relations \({\dot{\Delta }}_u=[\dot{D}_u,\sigma _u]+[D_u,{\dot{\sigma }}_u]\), \([N,D_u]=D_uk\) and \([N,\sigma _u]=-k\sigma _u\), see (62), (19) and (9), respectively. Combining this with \([D_u,e^{-t\Delta _u}]=0=[\sigma _u,e^{-t\Delta _u}]\) and \([D_u,Q_{\lambda ,u}]=0=[\sigma _u,Q_{\lambda ,u}]\) and the fact that the graded trace vanishes on graded commutators, we obtain
Clearly,
As \([D_{u,q}^*D_{u,q},e^{-t\Delta _{u,q}}]=0=[D_{u,q}^*D_{u,q},Q_{\lambda ,u,q}]\), this yields
see also (6). Consequently,
Using (60), \([D_u,e^{-t\Delta _u}]=0=[D_u,Q_{\lambda ,u}]\), \([\sigma _u,D_u]=\Delta _u\) and the fact that the graded trace vanishes on graded commutators, we obtain \({{\,\mathrm{str}\,}}(Q_{\lambda ,u}\sigma _u\dot{D}_ue^{-t\Delta _u})=-{{\,\mathrm{str}\,}}(\dot{A}_uQ_{\lambda ,u}\Delta _ue^{-t\Delta _u})\). Similarly, \({{\,\mathrm{str}\,}}(Q_{\lambda ,u}\dot{D}_u^*\sigma _u^*e^{-t\Delta _u})=-{{\,\mathrm{str}\,}}(\dot{A}_u^*Q_{\lambda ,u}\Delta _ue^{-t\Delta _u})\) and thus
Clearly,
Combining equations (63), (64), (65), (66) and (67), we obtain (61). \(\square \)
Lemma 2.20
For \(\lambda \ge 0\) not in the spectrum of \(\Delta _u\) and \(\Re (s)>(n+r_u)/2\kappa \) we have
Moreover,
and
where \({\tilde{p}}_{n,u}\) is the constant term in the asymptotic expansion (58).
Proof
Integrating (58), we obtain the asymptotic expansion
as \(t\rightarrow 0\). Using Eq. (61) and proceeding exactly as in the second half of the proof of Lemma 2.14, we obtain the lemma. \(\square \)
Lemma 2.21
If \(\lambda \ge 0\) is not contained in the spectrum of \(\Delta _u\), then
Proof
We consider the linear map
This is a map of finite dimensional complexes inducing (56) on cohomology. Since \(\alpha _{u,u}={{\,\mathrm{id}\,}}\), the map \(\alpha _{v,u}\) is an isomorphism of complexes, for v sufficiently close to u. Hence, for these v, we have
Differentiating and using \(P_{\lambda ,u}\dot{P}_{\lambda ,u}P_{\lambda ,u}=0\), we obtain the lemma. \(\square \)
Combining Lemma 2.21 with Eq. (70), we obtain Theorem 2.17, cf. (24).
2.6 Elementary Properties of the Analytic Torsion
In this section we collect a few properties of the analytic torsion of Rockland complexes generalizing well known facts about the Ray–Singer and the Rumin–Seshadri analytic torsion.
Throughout this section M denotes a closed filtered manifold, (E, D) is a Rockland complex over M, \(\mu \) is a volume density on M and h is a graded fiberwise Hermitian inner product on E.
Proposition 2.22
Let \(({\tilde{E}},{\tilde{D}})\) denote the Rockland complex obtained by shifting the grading by one. More explicitly, \({\tilde{E}}^q=E^{q+1}\), \({\tilde{D}}_q=D_{q+1}\), and \({\tilde{h}}_q=h_{q+1}\). Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q({\tilde{E}},{\tilde{D}})=H^{q+1}(E,D)\), we have
where the right hand side denotes the induced norm on the dual (inverse) line.
Proof
Denoting the zeta function associated with the Rockland complex \(({\tilde{E}},{\tilde{D}})\) by \({\tilde{\zeta }}_\lambda \), see (20), we clearly have \({\tilde{\zeta }}_\lambda (s)=-\zeta _\lambda (s)\), for any \(\lambda \ge 0\). Moreover, up to the canonical identification in (72) we have
see [94, Remark 1.4(4)] for the acyclic case. Combining these facts, the proposition follows at once, cf. (24). \(\square \)
For the torsion of a sum of two Rockland complexes we have:
Proposition 2.23
Suppose (E, D) and \(({\tilde{E}},{\tilde{D}})\) are two Rockland complexes over M such that the Heisenberg orders of \(D_q\) and \({\tilde{D}}_q\) coincide for all q. Moreover, let h and \({\tilde{h}}\) be graded, fiber wise Hermitian inner products on E and \({\tilde{E}}\), respectively. Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q(E\oplus {\tilde{E}},D\oplus {\tilde{D}})=H^q(E,D)\oplus H^q({\tilde{E}},{\tilde{D}})\), we have
Proof
Since the Heisenberg orders of \(D_q\) and \({\tilde{D}}_q\) coincide, \(D\oplus {\tilde{D}}\) is a Rockland complex and we may use the same numbers \(a_q\) and \(N_q\) to compute the analytic torsion of all three complexes. With theses choices, we clearly have
In particular,
for each \(\lambda \ge 0\), and we obtain
From (74) we get a canonical isomorphism of finite dimensional complexes,
It is well known that up to the canonical identification in (73) we have
see [94, Theorem 1.5] for the acyclic case. Combining this with (75), the proposition follows at once, cf. (24). \(\square \)
Recall that the transposed of a differential operator \(A:\Gamma ^\infty (E)\rightarrow \Gamma ^\infty (F)\) is the differential operator \(A^t:\Gamma ^\infty (F')\rightarrow \Gamma ^\infty (E')\) characterized by
where the parentheses \((-,-)_E\) denote the canonical pairing between sections of E and \(E':=E^*\otimes |\Lambda |\). The following observation will be useful in the discussion of Poincaré duality in Sect. 3.3 below.
Proposition 2.24
Let \((E',D^t)\) denote the transposed complex, i.e., \((E')^q=(E^{-q})^*\otimes |\Lambda |\) and \((D^t)_q=(D_{-q-1})^t\). Equip \((E')^q\) with the induced fiberwise Hermitian inner product, \(h'_q=(h_{-q})^{-1}\otimes \mu ^{-2}\). Then, up to the canonical identification
induced by the canonical isomorphism \(H^q(E',D^t)=H^{-q}(E,D)^*\), we have
where the right hand side denotes the induced norm on the dual (inverse) line.
Proof
Since \((D^t)_q=(D_{-q-1})^t\) we have
Dualizing, we obtain
Note here that the vector bundle isomorphism \(h_{-q}\otimes \mu :{\bar{E}}^{-q}\rightarrow (E^{-q})^*\otimes |\Lambda |=(E')^q\) intertwines the Hermitian metric \({\bar{h}}\) on \({\bar{E}}\) with the Hermitian metric \(h'\) on \(E'\), and thus, with respect to these Hermitian metrics, we have
We obtain
Denoting the Heisenberg order of \((D^t)_q\) by \(k'_q\), we have
Hence, we may use the numbers \(a'_q=a_{-q-1}\) to compute the analytic torsion of \((E',D^t)\), cf. (5) and Lemma 2.8. With this choice, we find
In particular,
In view of (77), the numbers \(N'_q:=-N_{-q}\) also satisfy the relation (19), that is, \(N'_{q+1}-N'_q=k'_q\). Clearly,
Combining (78), (79) and (80), we obtain
Recall here that according to Lemma 2.7, we may use any sequence of numbers satisfying (19) to compute this zeta function.
Clearly, the adjoints of D with respect to h and \({\bar{h}}\) coincide. Hence, \(\Delta ^{({\bar{E}},D)}_{{\bar{h}},\mu }=\Delta ^{(E,D)}_{h,\mu }\) and \(Q^{({\bar{E}},D)}_{\lambda ,{\bar{h}},\mu }=Q^{(E,D)}_{\lambda ,h,\mu }\). Since the spectrum of \(\Delta ^{(E,D)}_{h,\mu }\) is real, we thus get
Note that from (78) and (79) we also obtain
Hence, the canonical weakly non-degenerate pairing \(\Gamma ^\infty ((E')^q)\times \Gamma ^\infty (E^{-q})\rightarrow \mathbb C\) restricts to a (weakly) non-degenerate pairing
The latter pairing induces an isomorphism between finite dimensional complexes,
where the right hand side denotes the dual of the complex \(\bigl ({{\,\mathrm{img}\,}}\bigl (P^{(E,D)}_\lambda \bigr ),D\bigr )\) with the grading reversed. Via this isomorphism, the inner product on the left hand side induced by the \(L^2\) inner product on \(\Gamma (E')\) coincides with the inner product on the right hand side induced by the \(L^2\) inner product on \(\Gamma (E)\). Moreover, by the universal coefficient theorem, this identification induces isomorphisms in cohomology, \(H^q(E',D^t)=H^{-q}(E,D)^*\), which in turn induce the canonical identification (76). Hence, via (76), we have
see [94, Theorem 1.9] for the acyclic case. Combining this with (81) and (82), the proposition follows at once, see (24). \(\square \)
With respect to finite coverings we have:
Proposition 2.25
Suppose \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds, let \(({\tilde{E}},{\tilde{D}})\) be a Rockland complex over \({\tilde{M}}\), and consider the Rockland complex \((\pi _*{\tilde{E}},\pi _*{\tilde{D}})\) over M, where \(\pi _*{\tilde{E}}\) denotes the vector bundle over M with fibers \((\pi _*{\tilde{E}})_x=\Gamma ({\tilde{E}}|_{\pi ^{-1}(x)}) =\bigoplus _{{\tilde{x}}\in \pi ^{-1}(x)}{\tilde{E}}_{{\tilde{x}}}\) and \(\pi _*{\tilde{D}}\) denotes the induced differential operator. Moreover, suppose \({\tilde{h}}\) is a graded Hermitian inner product on \({\tilde{E}}\) and let \((\pi _*{\tilde{h}})_x=\bigoplus _{{\tilde{x}}\in \pi ^{-1}(x)}h_{{\tilde{x}}}\) denote the induced Hermitian inner product on \(\pi _*{\tilde{E}}\). Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q(\pi _*{\tilde{E}},\pi _*{\tilde{D}})=H^q({\tilde{E}},{\tilde{D}})\), we have
Proof
Restricting a section \(\phi \) of \({\tilde{E}}\) to fibers of \(\pi :{\tilde{M}}\rightarrow M\), we obtain a section \(\pi _*\phi \) of \(\pi _*{\tilde{E}}\), \((\pi _*\phi )(x)=\phi |_{\pi ^{-1}(x)}\), \(x\in M\). This provides a canonical isomorphism
which intertwines \(\pi _*{\tilde{D}}\) with \({\tilde{D}}\) and induces the canonical identification in cohomology. Moreover, this isomorphism intertwines the \(L^2\) inner product induced by \(\pi _*{\tilde{h}}\) and \(\mu \) on \(\Gamma ^\infty (\pi _*{\tilde{E}})\) with the \(L^2\) inner product induced by \({\tilde{h}}\) and \(\pi ^*\mu \) on \(\Gamma ^\infty ({\tilde{E}})\). The proposition follows at once. \(\square \)
Proposition 2.26
Suppose \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds and consider the pulled back Rockland complex \((\pi ^*E,\pi ^*D)\) over \({\tilde{M}}\). Let \(\mathcal V\) denote the flat vector bundle over M with fibers
and let \(h^{\mathcal V}\) denote the canonical parallel fiber wise Hermitian inner product on \(\mathcal V\). Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q(\pi ^*E,\pi ^*D)=H^q(E\otimes \mathcal V,D^{\mathcal V})\), we have
cf. Remark 2.16.
Proof
This follows from Proposition 2.25. Indeed, via the canonical isomorphism of vector bundles \(\pi _*\pi ^*E=E\otimes \mathcal V\) we have \(\pi _*\pi ^*D=D^{\mathcal V}\) and \(\pi _*\pi ^*h=h\otimes h^{\mathcal V}\). \(\square \)
3 Analytic Torsion of Rumin Complexes
The Rumin complex [81, 83, 84] associated with a filtered manifold is a complex of higher order differential operators which is conjugate to a subcomplex of the de Rham complex and computes the cohomology of the underlying manifold. Rumin has shown that these are Rockland complexes in a graded sense. Assuming that the cohomology of the osculating algebras is pure, we obtain a Rockland complex in the sense of Sect. 2. The aim of this section is to establish basic properties of the analytic torsion associated with these complexes. In particular, we will address Poincaré duality in Theorem 3.8, metric dependence in Theorem 3.11, and the dependence on the filtration in Theorem 3.15.
For trivially filtered manifolds the Rumin complex coincides with the de Rham complex and gives rise to the classical Ray–Singer torsion [6, 77]. For contact manifolds the Rumin complex [79, 80, 82] has been used by Rumin and Seshadri [85] to define an analytic torsion. Although these two classical torsions appear as special cases of our general construction, we will say nothing new about them. Cartan’s (2, 3, 5) geometries [29] constitute another class of 5-dimensional filtered manifolds whose osculating algebras have pure cohomology. Their analytic torsion will be discussed in Sect. 4 below.
Our assumption that the osculating algebras have pure cohomology appears to be very restrictive. The three cases mentioned previously are the only types of filtered manifolds we know of which have this property. In Sect. 3.7 we present a necessary condition which restricts the dimensions of such Lie algebras tremendously.
3.1 Rumin Complexes
We continue to consider a closed filtered manifold M. Recall that \({\mathfrak {t}}M\) denotes the bundle of osculating algebras and let \(\partial _q:\Lambda ^q{\mathfrak {t}}^*M\rightarrow \Lambda ^{q+1}{\mathfrak {t}}^*M\) denote the fiberwise Chevalley–Eilenberg codifferential. We assume that the dimension of the Lie algebra cohomology \({\mathcal {H}}^q({\mathfrak {t}}_xM)=\ker \partial _{q,x}/{{\,\mathrm{img}\,}}\partial _{q,x}\) is locally constant in \(x\in M\), cf. [83, Definition 2.4]. Hence, \({\mathcal {H}}^q({\mathfrak {t}}M)\) is a smooth vector bundle over M, for each q. The filtration on TM induces a filtration on \(\Lambda ^*T^*M\). More explicitly, \(\alpha \in \Lambda ^qT^*_xM\) is contained in filtration degree p iff \(\alpha (X_1,\dotsc ,X_q)=0\) for all \(X_i\in T_x^{p_i}M\) with \(-p<\sum _ip_i\), see [27, Sect. 3.1.1]. Since the filtration on M is compatible with Lie brackets, the de Rham differential d on \(\Omega ^*(M)\) is filtration preserving and induces \({{\,\mathrm{gr}\,}}(d)=\partial \) on the associated graded, \({{\,\mathrm{gr}\,}}(\Lambda ^*T^*M)=\Lambda ^*{\mathfrak {t}}^*M\).
Let \({\tilde{g}}\) be a graded fiberwise Euclidean inner product on \({\mathfrak {t}}M=\bigoplus _p{\mathfrak {t}}^pM\). We will denote the induced fiberwise Euclidean inner product on \(\Lambda ^q{\mathfrak {t}}^*M\) by \({\tilde{g}}^{-1}\). Let \(\partial ^*_q:\Lambda ^{q+1}{\mathfrak {t}}^*M\rightarrow \Lambda ^q{\mathfrak {t}}^*M\) denote the corresponding fiberwise adjoint of \(\partial _q\). Fiberwise finite dimensional Hodge theory provides an orthogonal decomposition of vector bundles
where \({\mathcal {H}}^q({\mathfrak {t}}M)=\ker \partial _q/{{\,\mathrm{img}\,}}\partial _{q-1}=\ker \partial _q\cap \ker \partial ^*_{q-1} =\ker \partial _{q-1}^*/{{\,\mathrm{img}\,}}\partial _q^*\).
We fix a splitting of the filtration \(S:{\mathfrak {t}}M\rightarrow TM\), i.e., a filtration preserving vector bundle isomorphism inducing the identity on the associated graded. Then \((S^t)^{-1}:{\mathfrak {t}}^*M\rightarrow T^*M\) is a splitting for the dual filtration which will be denoted by S too. We extend it to a splitting \(S:\Lambda ^*{\mathfrak {t}}^*M\rightarrow \Lambda ^*T^*M\) characterized by
for \(\alpha ,\beta \in \Lambda ^*{\mathfrak {t}}^*_xM\). Hence \(\delta _q:\Lambda ^qT^*M\rightarrow \Lambda ^{q-1}T^*M\),
is a filtration preserving vector bundle homomorphism inducing \({{\,\mathrm{gr}\,}}(\delta )=\partial ^*\) on the associated graded. Actually, \(\delta \) is a Kostant type codifferential [39, Definition 4.8 and Remark 4.14] for the de Rham complex \(\Omega ^*(M)\).
Kostant’s box operator \(\Box =\delta d+d\delta \) is filtration preserving and induces \({{\,\mathrm{gr}\,}}(\Box )={\tilde{\Box }}:=\partial ^*\partial +\partial \partial ^*\) on the associated graded. If \(0\ne z\in \mathbb C\) is sufficiently close to zero, then the vector bundle map \(z-{\tilde{\Box }}\) is invertible. This implies that \(z-\Box \) is invertible too and its inverse is again a filtration preserving differential operator. Indeed, the inverse can be expressed using a finite geometric series, cf. [81, Lemma 1], [83, Lemma 2.5], [25, 22, Theorem 5.2], or [39, Lemma 4.3]. Hence, for sufficiently small \(\varepsilon >0\),
is a filtration preserving differential projector, \(\Pi ^2=\Pi \), inducing \({{\,\mathrm{gr}\,}}(\Pi )={\tilde{\Pi }}\) on the associated graded, where \({\tilde{\Pi }}\) denotes the orthogonal projection onto the subbundle \(\ker {\tilde{\Box }}=\ker \partial ^*\cap \ker \partial \) of \(\Lambda ^*{\mathfrak {t}}^*M\). Since \(\Box \) commutes with d and \(\delta \), the same is true for \(\Pi \), that is, \(d\Pi =\Pi d\), \(\delta \Pi =\Pi \delta \). The operator \(\Pi \) coincides with Rumin’s projector \(\Pi _E\) in [81, Theorem 1] and [83, Theorem 2.6]. For more details on the construction presented here we refer to [39, Lemma 4.4(b)].
The differential operator \(\mathbf{L}:\Gamma ^\infty (\Lambda ^*{\mathfrak {t}}^*M)\rightarrow \Omega ^*(M)\),
is filtration preserving and induces the identity on the associated graded, \({{\,\mathrm{gr}\,}}(\mathbf{L})={{\,\mathrm{id}\,}}\). Hence, \(\mathbf{L}\) is invertible and its inverse is a filtration preserving differential operator. Since \(\Pi \mathbf{L}=\mathbf{L}{\tilde{\Pi }}\) and \(\Pi d=d\Pi \), the differential operator \(\mathbf{L}^{-1}d\mathbf{L}\) commutes with \({\tilde{\Pi }}\). Hence, \(\mathbf{L}^{-1}d\mathbf{L}\) decouples into a sum of two differential operators,
with respect to the decomposition \(\Lambda ^*{\mathfrak {t}}^*M={{\,\mathrm{img}\,}}({\tilde{\Pi }})\oplus \ker ({\tilde{\Pi }})\), where
From \(d^2=0\) we obtain \(\mathbf{D}^2=0\) and \(\mathbf{B}^2=0\). The complex \(\mathbf{D}\) is Rumin’s complex denoted \((E_0,d_c)\) in [81, Theorem 1] and [83, Theorem 2.6]. The complex \(\mathbf{B}\) is acyclic and the restriction of \(\mathbf{L}\) induces an isomorphism between the cohomology of \(\mathbf{D}\) and the de Rham cohomology [81, 83]. Indeed, \(\partial ^*\mathbf{B}+\mathbf{B}\partial ^*\) is invertible for it is filtration preserving and the induced homomorphism on the associated graded, \({{\,\mathrm{gr}\,}}(\partial ^*\mathbf{B}+\mathbf{B}\partial ^*)=\partial ^*\partial +\partial \partial ^*\), is invertible on \(\ker {\tilde{\Pi }}\). One can show that the complex \(\mathbf{B}\) is actually conjugate, via an invertible filtration preserving differential operator, to the acyclic tensorial complex \(\partial |_{\Gamma ^\infty (\ker {\tilde{\Pi }})}\), see [39, Proposition 4.5(c)].
We use the decomposition in (83) to identify \({\mathcal {H}}({\mathfrak {t}}M)=\ker \partial /{{\,\mathrm{img}\,}}\partial \) with \({{\,\mathrm{img}\,}}{\tilde{\Pi }}=\ker \partial ^*\cap \ker \partial \). More precisely, we let \(\iota :{\mathcal {H}}({\mathfrak {t}}M)\rightarrow {{\,\mathrm{img}\,}}{\tilde{\Pi }}\subseteq \ker \partial \) denote the corresponding isomorphism of vector bundles, splitting the canonical projection \(\ker \partial \rightarrow \ker \partial /{{\,\mathrm{img}\,}}\partial ={\mathcal {H}}({\mathfrak {t}}M)\). Putting \(D:=\iota ^{-1}\mathbf{D}\iota \), we obtain a complex of differential operators we will refer to as Rumin complex:
Moreover, \(L:\Gamma ^\infty \bigl ({\mathcal {H}}({\mathfrak {t}}M)\bigr )\rightarrow \Omega (M)\), \(L:=\mathbf{L}\iota \) is a chain map,
inducing an isomorphism in cohomology. The inverse is induced by the differential operator \(\iota ^{-1}{\tilde{\Pi }}\mathbf{L}^{-1}:\Omega (M)\rightarrow \Gamma ^\infty \bigl ({\mathcal {H}}({\mathfrak {t}}M)\bigr )\) which satisfies \((\iota ^{-1}{\tilde{\Pi }}\mathbf{L}^{-1})L={{\,\mathrm{id}\,}}\), \(L(\iota ^{-1}{\tilde{\Pi }}\mathbf{L}^{-1})=\Pi \), and
The latter three equations follow from \(L=\mathbf{L}\iota \), \(\mathbf{L}{\tilde{\Pi }}=\Pi \mathbf{L}\), (87), and \(D=\iota ^{-1}\mathbf{D}\iota \). The construction of the Rumin complex presented here is motivated by the construction of natural differential operators [22, 25, 28, 39] in parabolic geometry.
Remark 3.1
The operator L can be characterized as the unique differential operator \(L:\Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M))\rightarrow \Omega (M)\) for which
where \(\pi :\ker \delta \rightarrow {\mathcal {H}}({\mathfrak {t}}M)\) denotes the composition of \(S^{-1}:\ker \delta \rightarrow \ker \partial ^*\) with the projection \(\ker \partial ^*\rightarrow \ker \partial ^*/{{\,\mathrm{img}\,}}\partial ^*=\ker \partial ^*\cap \ker \partial ={\mathcal {H}}({\mathfrak {t}}M)\). This follows readily from
and
see [81, 83] or [39, Lemma 4.9]. Moreover, we have \(D=\pi dL\) and, cf. (90),
Rumin has shown that the sequence (88) becomes exact in every non-trivial unitary representation of the osculating group, see [81, Theorem 3] or [83, Theorem 5.2]. Hence, the operators in (88) form a graded Rockland complex which has graded Heisenberg order zero, see also [39, Corollary 4.20(b)]. In the flat case, where M is locally diffeomorphic to a graded nilpotent Lie group, Rumin [81, 83] proved that this complex is C-C elliptic.
To obtain a Rockland complex, we assume from now on that the osculating algebras have pure cohomology [81, 83], that is, \({\mathcal {H}}^q({\mathfrak {t}}_xM)=\bigoplus _p{\mathcal {H}}^q({\mathfrak {t}}_xM)_p\) is concentrated in a single degree for each q. In other words, we assume that the grading automorphism \(\phi _t\) given by multiplication with \(t^p\) on \({\mathfrak {t}}^pM\), \(t>0\), acts as a scalar in each cohomology group. Hence, there exist numbers \(p_q\) such that
It follows from Lemma 3.21 below that \({\mathcal {H}}^q({\mathfrak {t}}M)\ne 0\) for all \(0\le q\le \dim M\) and that the numbers \(p_q\) are mutually different. In fact, the Betti numbers of any nilpotent Lie algebra are all nonzero [43, Théorème 2]. Clearly, \(p_0=0\). The Rumin differential \(D_q\) has Heisenberg order
We expect \(p_{q+1}>p_q\), but are unable to prove this in general. If this inequality failed, then \(D_q\) would have to vanish. Subsequently, we will assume \(p_{q+1}>p_q\).
Lemma 3.2
If the osculating algebras have pure cohomology, then the Rumin complex (88) does not depend on the choice of a graded fiberwise Euclidean inner product \({\tilde{g}}\) on \({\mathfrak {t}}M\) or the choice of a splitting \(S:{\mathfrak {t}}M\rightarrow TM\). Moreover, the isomorphism in cohomology induced by L is independent of these choices too.
Proof
Suppose the Euclidean inner product \({\tilde{g}}_u\) and the splitting \(S_u\) depend on a (discrete) parameter u. Let \({\tilde{\Pi }}_u\), \(\mathbf{L}_u\), \(L_u\), \(\iota _u\), and \(D_u\) denote the associated operators. Combining (89) and (90), we obtain
It remains to show that \(\iota _v^{-1}{\tilde{\Pi }}_v\mathbf{L}_v^{-1}L_u={{\,\mathrm{id}\,}}\), for any two parameters u and v. Note that each factor is filtration preserving and the composition induces
on the associated graded. Indeed, \({{\,\mathrm{gr}\,}}(\mathbf{L})={{\,\mathrm{id}\,}}\), \(\iota \) takes values in \(\ker \partial \), and \(\iota ^{-1}{\tilde{\Pi }}|_{\ker \partial }\) coincides with the canonical projection \(\ker \partial \rightarrow \ker \partial /{{\,\mathrm{img}\,}}\partial ={\mathcal {H}}({\mathfrak {t}}M)\). Since the filtration on \({\mathcal {H}}^q({\mathfrak {t}}M)\) was assumed to be trivial, this implies \(\iota _v^{-1}{\tilde{\Pi }}_v\mathbf{L}_v^{-1}L_u={{\,\mathrm{id}\,}}\) on \(\Gamma ^\infty ({\mathcal {H}}^q({\mathfrak {t}}M))\), for each q. Composing with \(L_v\) from the left, the latter equation also yields \(\Pi _vL_u=L_v\). Hence, as \(\Pi _v\) induces the identity on cohomology, \(L_u\) and \(L_v\) induced the same isomorphism in cohomology. \(\square \)
3.2 Analytic Torsion
We continue to consider a closed filtered manifold M whose osculating algebras have pure cohomology of locally constant dimension. Twisting the Rumin complex in (88) with a flat vector bundle F, we obtain a Rockland complex
computing the cohomology of M with coefficients in the flat bundle F. More explicitly, the twisted operator \(L^F:\Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M)\otimes F)\rightarrow \Omega (M;F)\) provides a chain map,
which induces an isomorphism on cohomology,
Let \({\tilde{g}}\) be a graded fiberwise Euclidean inner product on \({\mathfrak {t}}M\) and let \({\tilde{g}}^{-1}\) denote the induced fiberwise Euclidean inner product on \(\Lambda ^q{\mathfrak {t}}^*M\). Via the orthogonal decomposition in (83) we obtain a fiberwise Euclidean inner product on \({\mathcal {H}}^q({\mathfrak {t}}M)\) that will be denoted by \({\tilde{g}}^{-1}\) too. If h is a fiberwise Hermitian inner product on F, we obtain a fiberwise Hermitian inner product on \({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F\) we will denote by \({\tilde{g}}^{-1}\otimes h\). Using the volume density \(\mu _{{\tilde{g}}}\) on M induced from the Euclidean inner product \({\tilde{g}}^{-1}\) via the canonical identification \(\Lambda ^m{\mathfrak {t}}^*M=\Lambda ^mT^*M\) where \(m=\dim M\), we obtain formal adjoints \((D^F)^*\) and Rumin–Seshadri operators, see (7), which will be denoted by
These are Rockland operators of Heisenberg order \(2\kappa \), see (6), which also depend on the choice of numbers \(a_q\) as in (5), but they are independent of the splitting S according to Lemma 3.2. According to Lemma 2.5, the graded heat trace of \(\Delta _{{\tilde{g}},h}^F\) is constant in t and
where \(\chi (M;F)=\chi (M){{\,\mathrm{rk}\,}}(F)\) denotes the Euler characteristics of M and \(p_n^F\) denotes the constant term in the heat kernel expansion associated with \(\Delta ^F_{{\tilde{g}},h}\).
Using the identification in (97), we obtain an analytic torsion, i.e., a norm
on the graded determinant line,
cf. (24). The subscript \(\mathcal F\) indicates the dependence on the filtration on M. We will refer to this as the analytic torsion of the filtered manifold M with coefficients in the flat bundle F. It is defined for closed filtered manifolds whose osculating algebras have pure cohomology of locally constant dimension.
The analytic torsion defined above is a common generalization of the Ray–Singer torsion [6, 77] and the Rumin–Seshadri [85] analytic torsion. In Sect. 4 we will discuss another instance of this torsion associated with a certain five dimensional geometry. We are not aware of further filtered manifolds whose osculating algebras have pure cohomology, cf. Sect. 3.7 below.
Let us spell out the following immediate consequence of Proposition 2.23:
Proposition 3.3
Suppose \(F_1\) and \(F_2\) are two flat vector bundles with Hermitian inner products \(h_1\) and \(h_2\), respectively. Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q(M;F_1\oplus F_2)=H^q(M;F_1)\oplus H^q(M;F_2)\), we have
With respect to finite coverings we have
Proposition 3.4
Suppose \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds, let \({\tilde{F}}\) be a flat vector bundle over \({\tilde{M}}\), and let \(\pi _*{\tilde{F}}\) denote the flat vector bundle over M with fibers \((\pi _*{\tilde{F}})_x=\bigoplus _{{\tilde{x}}\in \pi ^{-1}(x)}{\tilde{F}}_{{\tilde{x}}}\). Moreover, let \({\tilde{h}}\) be a Hermitian inner product on \({\tilde{F}}\) and let \((\pi _*{\tilde{h}})_x=\bigoplus _{{\tilde{x}}\in \pi ^{-1}(x)}{\tilde{h}}_{{\tilde{x}}}\) denote the induced Hermitian inner product on \(\pi _*{\tilde{F}}\). Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q(M;\pi _*{\tilde{F}})=H^q({\tilde{M}};{\tilde{F}})\), we have
Proof
This follows from Proposition 2.25. Indeed, since the covering map is a local diffeomorphism of filtered manifolds, we have canonical isomorphisms of vector bundles \(\pi _*({\mathcal {H}}^q({\mathfrak {t}}{\tilde{M}})\otimes {\tilde{F}})={\mathcal {H}}^q({\mathfrak {t}}M)\otimes \pi _*{\tilde{F}}\) intertwining \(\pi _*D_q^{{\tilde{F}}}\) with \(D^{\pi _*{\tilde{F}}}_q\), cf. (95), and intertwining the Hermitian inner product \(\pi _*(\pi ^*{\tilde{g}}^{-1}\otimes {\tilde{h}})\) with the Hermitian inner product \({\tilde{g}}^{-1}\otimes \pi _*{\tilde{h}}\). Moreover, \(\pi ^*\mu _{{\tilde{g}}}=\mu _{\pi ^*{\tilde{g}}}\). \(\square \)
Proposition 3.5
Suppose \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds, let F be a flat vector bundle over M, and let h be a fiberwise Hermitian inner product on F. Moreover, let \(\mathcal V\) denote the flat vector bundle over M with fibers \(\mathcal V_x=H_0(\pi ^{-1}(x);\mathbb {C})=\mathbb {C}[\pi ^{-1}(x)]\), \(x\in M\), and let \(h^{\mathcal V}\) denote the canonical parallel Hermitian inner product on \(\mathcal V\). Then, up to the canonical identification of determinant lines
induced by the canonical isomorphisms \(H^q({\tilde{M}};\pi ^* F)=H^q(M;F\otimes \mathcal V)\), we have
Proof
This follows from Proposition 3.4. Indeed, via the canonical isomorphism of flat vector bundles \(\pi _*\pi ^*F=F\otimes \mathcal V\) we have \(\pi _*\pi ^*h=h\otimes h^{\mathcal V}\). \(\square \)
3.3 Poincaré Duality
Rumin observed that his complex is Hodge \(\star \) self-dual, see [81, Sect. 2] or [83, Proposition 2.8]. In this section we recall this duality and discuss implications for the analytic torsion.
Let us denote the dimension of M by
Recall that the wedge product provides a fiberwise non-degenerate bilinear pairing
The corresponding vector bundle isomorphism
is an isometry with respect to the Euclidean metric \({\tilde{g}}^{-1}\) on the left hand side and the induced Euclidean metric \({\tilde{g}}\otimes {\tilde{g}}^{-1}\) on the right hand side. Hence,
Since the Chevalley–Eilenberg codifferential \(\partial :\Lambda ^*{\mathfrak {t}}_x^*M\rightarrow \Lambda ^{*+1}{\mathfrak {t}}_x^*M\) is a graded derivation, the wedge product induces a fiberwise bilinear pairing
where we regard \({\mathcal {H}}({\mathfrak {t}}M)=\ker \partial /{{\,\mathrm{img}\,}}\partial \). By Poincaré duality for the nilpotent Lie algebra \({\mathfrak {t}}_xM\), this pairing is fiberwise non-degenerate. In particular, the codifferential in top degree, \(\partial :\Lambda ^{m-1}{\mathfrak {t}}^*M\rightarrow \Lambda ^m{\mathfrak {t}}^*M\) vanishes. Hence, we have canonical identifications
Note that the vector bundle map \(\pi :\ker \delta \rightarrow {\mathcal {H}}({\mathfrak {t}}M)\) from Remark 3.1 coincides with the canonical identification (104) in top degree. Moreover, in top degree L is algebraic, inducing the canonical identification in (104) inverse to \(\pi \), cf. (91).
Lemma 3.6
We have
for \(\alpha \in \Lambda ^q{\mathfrak {t}}_x^*M\) and \(\beta \in \Lambda ^{m-q-1}{\mathfrak {t}}_x^*M\). Moreover,
for \(\alpha \in \Lambda ^{q+1}{\mathfrak {t}}_x^*M\) and \(\beta \in \Lambda ^{m-q}{\mathfrak {t}}_x^*M\). Moreover,
for \(\phi \in \Lambda ^{q+1}T_x^*M\) and \(\psi \in \Lambda ^{m-q}T^*_xM\). Moreover,
for \(\phi \in \ker \delta _{q,x}\) and \(\psi \in \ker \delta _{m-q,x}\). Moreover,
for \(\alpha \in \Gamma ^\infty ({\mathcal {H}}^q({\mathfrak {t}}M))\) and \(\beta \in \Gamma ^\infty ({\mathcal {H}}^{m-q}({\mathfrak {t}}M))\).
Proof
Since \(\partial \) is a graded derivation that vanishes in top degree, we have (105). Using the vector bundle isomorphism in (101), this equation may be written in the form \(\wedge _{q+1}\circ \partial _q=-(-1)^q(\partial ^t_{m-q-1}\otimes {{\,\mathrm{id}\,}})\circ \wedge _q\). Dualizing, we obtain \(\partial _q^*\circ \wedge _{q+1}^*=-(-1)^q\wedge _q^*\circ ((\partial ^t_{m-q-1})^*\otimes {{\,\mathrm{id}\,}})\). Using \((\partial ^t)^*=(\partial ^*)^t\) and (102), this yields \(\wedge _q\circ \partial _q^*=-(-1)^q\circ ((\partial ^*_{m-q-1})^t\otimes {{\,\mathrm{id}\,}})\circ \wedge _{q+1}\), whence (106). Combining (106) with (85) and (84), we obtain (107).
Note that (108) is obviously true for \(\phi \in S_q(\ker \partial ^*_{q-1,x}\cap \ker \partial _{q,x})\) and \(\psi \in S_{m-q}(\ker \partial ^*_{m-q-1,x}\cap \ker \partial _{m-q,x})\), see (84) and (85). The general case follows from (107) since \(\pi \) induces an isomorphism \(\ker \delta /{{\,\mathrm{img}\,}}\delta \rightarrow \ker \partial ^*/{{\,\mathrm{img}\,}}\partial ^*=\ker \partial ^*\cap \ker \partial ={\mathcal {H}}({\mathfrak {t}}M)\). Combining (108) with the relations \(\delta L=0\) and \(\pi L={{\,\mathrm{id}\,}}\) from (91), we obtain \(\pi (L\alpha \wedge L\beta )=\alpha \wedge \beta \) and then (109). \(\square \)
For every flat vector bundle F, the wedge product in (103) induces a fiberwise non-degenerate bilinear pairing
Here \({\mathcal {O}}\) denotes the orientation bundle of M and we are using the canonical identification of line bundles
From (109) we obtain
for all \(\alpha \in \Gamma ^\infty \bigl ({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F^*\otimes {\mathcal {O}}\bigr )\) and \(\beta \in \Gamma ^\infty \bigl ({\mathcal {H}}^{m-q}({\mathfrak {t}}M)\otimes F\bigr )\).
Variants of the following statement can be found in [81, Sect. 2], [83, Proposition 2.8], and [22, Sect. 7].
Lemma 3.7
We have
for \(\alpha \in \Gamma ^\infty \bigl ({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F^*\otimes {\mathcal {O}}\bigr )\) and \(\beta \in \Gamma ^\infty \bigl ({\mathcal {H}}^{m-q-1}({\mathfrak {t}}M)\otimes F\bigr )\).
Proof
By the Leibniz rule for the de Rham differential, (96) and (112) we find:
\(\square \)
By Lemma 3.7 and Stokes’ theorem, the pairing in (110) induces a pairing
In view of (112), the isomorphisms induced on cohomology by \(L^{F^*\otimes {\mathcal {O}}}\) and \(L^F\), see (97), intertwine this pairing with the Poincaré duality pairing
In particular, the pairing in (113) is non-degenerate.
Theorem 3.8
Via the canonical isomorphism of determinant lines
induced by Poincaré duality \(H^q(M;F^*\otimes {\mathcal {O}})=H^{m-q}(M;F)^*\), we have
Here \(h^{-1}\) denotes the fiberwise Hermitian inner product on \(F^*\otimes {\mathcal {O}}\) induced by the fiberwise Hermitian inner product h on F.
Proof
From the preceding lemma and Stokes’ theorem we obtain the following commutative diagram:
Here the vertical arrows denote the vector bundle isomorphisms
corresponding to the pairing in (110). Via the latter vector bundle isomorphism, the Hermitian metric \({\tilde{g}}^{-1}\otimes h^{-1}\) on \({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F^*\otimes {\mathcal {O}}\) corresponds to the Hermitian metric \(h':=({\tilde{g}}^{-1}\otimes h)^{-1}\otimes \mu _{{\tilde{g}}}^{-2}\) on the right hand side, cf. (102). Hence, by Proposition 2.22 and Remark 2.10,
Moreover, from Proposition 2.24 we obtain
Combining the preceding to equations, we obtain the theorem. \(\square \)
Duality can also be understood in terms of an analogue of the Hodge star operator, cf. [81, Sect. 2] or [83, Proposition 2.8]. To this end, we let \(\star :\Lambda ^q{\mathfrak {t}}^*M\rightarrow \Lambda ^{m-q}{\mathfrak {t}}^*M\otimes {\mathcal {O}}\) denote the star operator characterized by
for \(\alpha ,\beta \in \Lambda ^q{\mathfrak {t}}^*_xM\) where we are using the canonical identification given by the second and third equality in (111). Recall that the star operator is isometric, i.e.,
Furthermore, (105) and (114) yield [81, 83]
for \(\alpha \in \Lambda ^q{\mathfrak {t}}_x^*M\) where \(\partial ^{{\mathcal {O}}}:=\partial \otimes {{\,\mathrm{id}\,}}_{{\mathcal {O}}}\).
The star restricts to vector bundle isomorphisms
cf. (83). From Lemma 3.7 we obtain [81, 83]
Using (115) this also yields
Hence,
Since
we have \(a_q=a_{m-q-1}\), see (6). Hence, we may use the same numbers \(a_q\) for the Rumin–Seshadri operator associated with flat bundle \(F^*\otimes {\mathcal {O}}\), and obtain (7)
In particular,
In view of (116), the numbers \({\tilde{N}}_q:=-N_{m-q}\) also satisfy the relation (19), that is, \({\tilde{N}}_{q+1}-{\tilde{N}}_q=k_q\). Clearly,
Combining (117), (118) and (119), we obtain
cf. (20) and (82). Recall here that according to Lemma 2.7, we may use any sequence of numbers satisfying (19) to compute this zeta function.
The relation (120) immediately leads to a slightly different proof of Theorem 3.8. Let us spell out two more consequences.
Proposition 3.9
If M is an orientable manifold of even dimension and h is a parallel Hermitian metric on F, then \(\zeta _{\lambda ,{\tilde{g}},h}^F(s)=0\).
Proof
Since h is parallel, it provides, together with an orientation of M, an isomorphism of flat vector bundles \(h:{\bar{F}}\rightarrow F^*\otimes {\mathcal {O}}\) which maps the Hermitian metric \({\bar{h}}\) on \({\bar{F}}\) to the Hermitian metric \(h^{-1}\) on \(F^*\otimes {\mathcal {O}}\). Hence, \(\zeta ^{{\bar{F}}}_{\lambda ,{\tilde{g}},{\bar{h}}}(s)=\zeta ^{F^*\otimes {\mathcal {O}}}_{\lambda ,{\tilde{g}},h^{-1}}(s)\). The proposition now follows from (120). \(\square \)
From (117) we also obtain
where \(p_{q,j}^F\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}^q({\mathfrak {t}}M)\otimes F)\otimes |\Lambda |\bigr )\) denote the degree q part of the local quantities in the heat kernel asymptotics associated with the Rumin–Seshadri operator \(\Delta ^F_{{\tilde{g}},h}\), cf. (13). Using locality, this gives
Proposition 3.10
If the dimension of M is odd and if h is a parallel Hermitian metric on F, then \({{\,\mathrm{str}\,}}(p_j^F)=0\) for all \(j\in \mathbb N_0\).
Proof
Since h is parallel, it provides an isomorphism of flat vector bundles \(h:{\bar{F}}\rightarrow F^*\) which maps the Hermitian metric \({\bar{h}}\) on \({\bar{F}}\) to the Hermitian metric \(h^{-1}\) on \(F^*\). Hence, \(p^{{\bar{F}}}_{q,j}=p^{F^*}_{q,j}\). Combining this with (121), we see that the summands in \({{\,\mathrm{str}\,}}(p^F_j)\) cancel pairwise if the dimension is odd. \(\square \)
3.4 Variation of the Metrics
In this section we apply Theorem 2.11 to determine how the analytic torsion of a Rumin complex depends on the fiberwise inner products \({\tilde{g}}\) on \({\mathfrak {t}}M\) and h on F.
If \(A\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\) is a fiberwise automorphism of graded (nilpotent) Lie algebras, we let \({\mathcal {H}}(A)\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathcal {H}}({\mathfrak {t}}M)))\) denote the automorphism in fiberwise cohomology induced by the transpose inverse \((A^{-1})^t\). Hence, we have covariant functoriality \({\mathcal {H}}(AB)={\mathcal {H}}(A){\mathcal {H}}(B)\) just like for the dual representation of a group, \(A,B\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\). If \(\dot{A}\in \Gamma ^\infty ({{\,\mathrm{der}\,}}({\mathfrak {t}}M))\) is a fiberwise derivation of graded (nilpotent) Lie algebras, we define \({\mathcal {H}}(\dot{A})\in \Gamma ^\infty ({{\,\mathrm{end}\,}}({\mathcal {H}}({\mathfrak {t}}M)))\) by \({\mathcal {H}}(\dot{A}):=\frac{\partial }{\partial t}|_{t=0}{\mathcal {H}}(\exp (t\dot{A}))\) so that \({\mathcal {H}}([\dot{A},\dot{B}])=[{\mathcal {H}}(\dot{A}),{\mathcal {H}}(\dot{B})]\) for \(\dot{A},\dot{B}\in \Gamma ^\infty ({{\,\mathrm{der}\,}}({\mathfrak {t}}M))\).
The following generalizes anomaly formulas for the Ray–Singer [77] torsion [6, Theorem 0.1] and the Rumin–Seshadri analytic torsion, see [85, Corollary 3.7].
Theorem 3.11
Suppose the graded fiberwise Euclidean inner product \({\tilde{g}}_u\) on \({\mathfrak {t}}M\) and the fiberwise Hermitian inner product \(h_u\) on F depend smoothly on a real parameter u such that \({\tilde{g}}_v^{-1}{\tilde{g}}_u\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\) is a fiberwise automorphism of graded (nilpotent) Lie algebras, for all u and v. Then
where \(\dot{g}_u:={\tilde{g}}_u^{-1}\frac{\partial }{\partial u}{\tilde{g}}_u\in \Gamma ^\infty \bigl ({{\,\mathrm{der}\,}}({\mathfrak {t}}M)\bigr )\), \(\dot{h}_u:=h_u^{-1}\frac{\partial }{\partial u}h_u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(F))\), and \(p_{u,n}^F\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}({\mathfrak {t}}M)\otimes F)\otimes |\Lambda |\bigr )\) denotes the constant term in the heat kernel expansion associated with the Rumin–Seshadri operator \(\Delta ^F_{{\tilde{g}}_u,h_u}\).
Proof
Recall that we are using the fiberwise Hermitian inner products \({\tilde{g}}_u^{-1}\otimes h_u\) on the Rumin complex \({\mathcal {H}}({\mathfrak {t}}M)\otimes F\), where \({\tilde{g}}_u^{-1}\) denotes the fiberwise Euclidean inner product on \({\mathcal {H}}({\mathfrak {t}}M)\) obtained by restriction via the fiberwise Hodge decomposition (83) from the induced Euclidean inner product on \(\Lambda ^*{\mathfrak {t}}^*M\) which is also denoted by \({\tilde{g}}^{-1}_u\). Put \(A_{v,u}={\tilde{g}}_v^{-1}{\tilde{g}}_u\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\) and let \(A_{v,u}^t={\tilde{g}}_u{\tilde{g}}_v^{-1}\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}^*M))\) denote the fiberwise dual automorphism. Extend this to an automorphism of the exterior bundle, \(A_{v,u}^t\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}(\Lambda ^*{\mathfrak {t}}^*M))\) characterized by
for \(\alpha ,\beta \in \Lambda ^*{\mathfrak {t}}^*_xM\). Note that the equality
holds on \(\Lambda ^*{\mathfrak {t}}^*M\) for we are using the induced Euclidean inner products on the exterior bundle. Since \(A_{v,u}\) is a fiberwise automorphism of Lie algebras, \(A_{v,u}^t\) commutes with fiberwise Chevalley–Eilenberg codifferential \(\partial \), the fiberwise adjoint \(\partial ^*\) does not depend on u, and \(A_{v,u}^t\) commutes with \(\partial ^*\) too. Hence, the decomposition in (83) is independent of u and invariant under \(A_{v,u}^t\). Hence,
Combining (122) and (123), we conclude
Differentiating, we find
Furthermore, we clearly have \(\mu _{{\tilde{g}}_u}={\det }^{1/2}({\tilde{g}}_v^{-1}{\tilde{g}}_u)\mu _{{\tilde{g}}_v}\) and thus
The statement thus follows from Theorem 2.11. \(\square \)
Recall that \(\phi _t\) denotes the grading automorphism of \({\mathfrak {t}}_xM=\bigoplus _p{\mathfrak {t}}^p_xM\) given by multiplication with \(t^p\) on the summand \({\mathfrak {t}}_x^pM\), \(t>0\). For a positive smooth function f on M, we let \(\phi _f\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\) denote the fiberwise Lie algebra automorphism acting by \(\phi _{f(x)}\) on the fiber \({\mathfrak {t}}_xM\).
Corollary 3.12
Suppose f is a real valued smooth function on M and consider a family of Euclidean inner products of the form \({\tilde{g}}_u={\tilde{g}}\phi _{\exp (uf)}\) on \({\mathfrak {t}}M\). Then
where \(p_{u,n}^F\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}({\mathfrak {t}}M)\otimes F)\otimes |\Lambda |\bigr )\) denotes the constant term in the heat kernel expansion associated with the Rumin–Seshadri operator \(\Delta ^F_{{\tilde{g}}_u,h}\).
Proof
Since \(\dot{g}_u={\tilde{g}}_u^{-1}\tfrac{\partial }{\partial u}{\tilde{g}}_u\) acts by pf on \({\mathfrak {t}}^pM\), we have cf. (12)
Using (19), (94) and \(p_0=0\), we find \(N_q-N_0=p_q\). Combining this with (93), we obtain \({\mathcal {H}}^q(\phi _{\exp (uf)})=(e^{uf})^{p_q}=e^{u(N_q-N_0)f}\). Hence, \({\mathcal {H}}^q(\dot{g}_u)=(N_q-N_0)f\) and
The statement thus follows from Theorem 3.11. \(\square \)
Remark 3.13
Conformal invariance of the analytic torsion, that is, independence under scaling of the Euclidean inner product on \({\mathfrak {t}}M\) as in Corollary 3.12, thus is equivalent to the pointwise vanishing of the local quantity \({{\,\mathrm{str}\,}}\bigl ((N-N_0-\frac{n}{2})p_n^F\bigr )\). Using Lemma 2.5 and (22), we find
where \(\chi (M;F)=\sum _q(-1)^q\dim H^q(M;F)={{\,\mathrm{rk}\,}}(F)\chi (M)\) denotes the Euler characteristics and \(\chi '(M;F)=\sum _q(-1)^qN_q\dim H^q(M;F)\). In view of (50) the integral in (124) is independent of \({\tilde{g}}\) and h, hence this is a smooth invariant of the filtered manifold M and the flat bundle F. Actually, this invariant only depends on F and the homotopy class of the underlying filtration, see Remark 3.17 below for a more precise statement. In the trivially filtered case, this vanishes according to [6, Theorem 7.10]. A discussion of the contact case can be found at the end of Sect. 3.2 in [85]. Whether (124) vanishes in general is unclear. If m is odd and h is parallel then \({{\,\mathrm{str}\,}}(p_n^F)=0\) according to Proposition 3.10.
Remark 3.14
If \(h_u\) are parallel Hermitian inner products on F, then the integrands in the preceding statements can be simplified somewhat:
Here \(p_{u,n}\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}({\mathfrak {t}}M))\otimes |\Lambda |\bigr )\) denotes the constant term in the heat kernel expansion for the corresponding Rumin–Seshadri operator associated with the trivial flat line bundle over M equipped with the canonical Hermitian metric. This follows from Remark 2.16.
3.5 Deformation of the Filtration
In this section we consider a family of closed filtered manifolds \(M_u\) smoothly depending on a real parameter u. More explicitly, we assume the underlying manifold M remains fixed, and the filtration subbundles \(T^pM_u\subseteq TM\) depend smoothly on u. As before, we assume that the osculating algebras of \(M_u\) have pure cohomology of locally constant dimension. Moreover, suppose \({\tilde{g}}_u\) is a smooth family of graded fiberwise Euclidean metrics on \({\mathfrak {t}}M_u\). For any flat bundle F with Hermitian metric h, we obtain an analytic torsion \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(M;F))}_{\mathcal F_u,{\tilde{g}}_u,h}\) on the determinant line \({{\,\mathrm{sdet}\,}}(H^*(M;F))\). Below, we will describe qualitatively how this torsion depends on the parameter u.
Clearly, the bundle of osculating algebras \({\mathfrak {t}}M_u\) depends smoothly on u also. For simplicity, we will assume that these are isomorphic as bundles of graded Lie algebras. Hence, there exist isomorphisms of graded vector bundles \({\tilde{\psi }}_{v,u}:{\mathfrak {t}}M_u\rightarrow {\mathfrak {t}}M_v\) which intertwine the fiberwise Lie algebra structures and depend smoothly on the parameters u and v. W.l.o.g. we may assume \({\tilde{\psi }}_{w,v}{\tilde{\psi }}_{v,u}={\tilde{\psi }}_{w,u}\) and in particular \({\tilde{\psi }}_{u,u}={{\,\mathrm{id}\,}}\). Let \(S_u:{\mathfrak {t}}M_u\rightarrow TM_u\) be splittings of the filtration which depend smoothly on u and define \(\psi _{v,u}\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}(TM))\) by the equation
Then \(\psi _{v,u}:TM_u\rightarrow TM_v\) is a filtration preserving isomorphism, \(\psi _{v,u}(T^pM_u)=T^pM_v\), which induces \({\tilde{\psi }}_{v,u}:{\mathfrak {t}}M_u\rightarrow {\mathfrak {t}}M_v\) on the associated graded. Moreover, \(\psi _{w,v}\psi _{v,u}=\psi _{w,u}\) and \(\psi _{u,u}={{\,\mathrm{id}\,}}\). Since we have already discussed the dependence on the metric before, we will consider a family of fiberwise Euclidean metrics \({\tilde{g}}_u\) on \({\mathfrak {t}}M_u\) such that
Theorem 3.15
In this situation we have
where \({\tilde{p}}^F_{n,u}\in \Gamma ^\infty \bigl ({{\,\mathrm{end}\,}}({\mathcal {H}}({\mathfrak {t}}M_u)\otimes F)\otimes |\Lambda |\bigr )\) is the constant term in the asymptotic expansion, as \(t\rightarrow 0\),
Here \(\dot{A}^F_u\) is the differential operator given by the expression (139) below, \({\dot{\psi }}_u:=\frac{\partial }{\partial v}|_{v=u}\psi _{v,u}\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(TM))\), and \(k^F_{t,u}\) denotes the heat kernel of the Rumin–Seshadri operator \(\Delta ^F_{{\tilde{g}}_u,h}\). In particular, \({\tilde{p}}^F_{n,u}\) is locally computable. More precisely, \({\tilde{p}}^F_{n,u}(x)\) can be computed from the germ of \((M,\mathcal F_u,{\tilde{g}}_u,F,h,S_u,{\dot{\psi }}_u)\) at x. Furthermore, \({\tilde{p}}^F_{n,u}=0\) if the homogeneous dimension n is odd.
Proof
The differential operators \(A_{v,u}:\Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M_u))\rightarrow \Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M_v))\)
satisfy \(A_{u,u}={{\,\mathrm{id}\,}}\) and \(A_{v,u}D_u=D_vA_{v,u}\), see (89) and (90). Twisting with the flat bundle F, we get differential operators \(A^F_{v,u}:\Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M_u)\otimes F)\rightarrow \Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M_v)\otimes F)\) such that \(A_{v,u}^FD_u^F=D_v^FA_{v,u}^F\). Clearly, \(A_{v,u}^F\) induces the identity on \(H^*(M;F)\), up to the canonical identifications in (97). In order to apply Theorem 2.17, we have to analyze \(\dot{A}^F_u\). It will be convenient to rewrite \(A_{v,u}\) in the form, see (92),
We let \(\psi _{v,u}:\Lambda ^*T^*M_u\rightarrow \Lambda ^*T^*M_v\) and \({\tilde{\psi }}_{v,u}:\Lambda ^*{\mathfrak {t}}^*M_u\rightarrow \Lambda ^*{\mathfrak {t}}^*M_v\) also denote the isomorphisms induced by \(\psi _{v,u}:TM_u\rightarrow TM_v\) and \({\tilde{\psi }}_{v,u}:{\mathfrak {t}}M_u\rightarrow {\mathfrak {t}}M_v\), respectively. Then Eq. (125) remains true on \(\Lambda ^*{\mathfrak {t}}^*M_u\) if \(S_u:\Lambda ^*{\mathfrak {t}}^*M_u\rightarrow \Lambda ^*T^*M_u\) denotes the splitting induced by \(S_u:{\mathfrak {t}}M_u\rightarrow TM_u\) as in (84). Furthermore, (126) yields
where \({\tilde{g}}_u^{-1}\) denotes the induced Euclidean metric on \(\Lambda ^*{\mathfrak {t}}^*M_u\) as before. We have
since \({\tilde{\psi }}_{v,u}\) is an isomorphism of graded Lie algebra bundles. By (129) this gives
and combination with (125) also yields
see (85). Moreover,
where \({\mathcal {H}}({\tilde{\psi }}_{v,u}):{\mathcal {H}}({\mathfrak {t}}M_u)\rightarrow {\mathcal {H}}({\mathfrak {t}}M_v)\) denotes the induced isomorphism, cf. (130). From (125), (128) and (133) we obtain
When applying Theorem 2.17 we use an auxiliary linear connection on the bundle \(\bigsqcup _u{\mathcal {H}}({\mathfrak {t}}M_u)\otimes F\) such that \(\dot{A}^F_u=\bigl (\frac{\partial }{\partial v}|_{v=u}{\mathcal {H}}({\tilde{\psi }}_{v,u})^{-1}A_{v,u}\bigr )^F\). Differentiating (134) yields
where \({\dot{\psi }}_u=\frac{\partial }{\partial v}|_{v=u}\psi _{v,u}\in \Gamma ^\infty ({{\,\mathrm{der}\,}}(\Lambda ^*T^*M))\) denotes the fiberwise graded derivation given by contraction with \({\dot{\psi }}_u\in \Gamma ^\infty ({{\,\mathrm{end}\,}}(TM))\). Since \(\Box _v=[\delta _v,d]\) we obtain from (132)
and thus
Combining this with (86), we find
Combining this with \([\delta _u,\Box _u]=0\), \(\delta _uL_u=0\) and \({\tilde{\Pi }}_uS_u^{-1}\delta _u=0\), we conclude that the first summand in (135) vanishes. Hence, using (92) we obtain
In particular, the coefficients of the differential operator \(\dot{A}_u^F\) at x can be expressed in terms of the germ of \((M,\mathcal F_u,{\tilde{g}}_u,F,h,S_u,{\dot{\psi }}_u)\) at x.
Clearly, \(\mu _{{\tilde{g}}_v}=\psi _{v,u}(\mu _{{\tilde{g}}_u})=\det _{TM}(\psi _{v,u}^{-1}) \mu _{{\tilde{g}}_u}\) and thus
Moreover, (129) and (133) give \({\mathcal {H}}({\tilde{\psi }}_{v,u})^*({\tilde{g}}_v^{-1}|_{{\mathcal {H}}({\mathfrak {t}}M_v)})={\tilde{g}}_u^{-1}|_{{\mathcal {H}}({\mathfrak {t}}M_u)}\) and therefore
The statement thus follows from Theorem 2.17. \(\square \)
Remark 3.16
Note that \(\dot{A}^F_u\), \({{\,\mathrm{tr}\,}}({\dot{\psi }}_u)\) and \({\tilde{p}}_{j,u}^F\) in Theorem 3.15 remain unchanged if \(\psi _{v,u}\) is replaced by any other filtration preserving family \(\psi _{v,u}':TM_u\rightarrow TM_v\) satisfying \(\psi '_{w,v}\psi '_{v,u}=\psi '_{w,u}\) and inducing the same automorphism \({\mathfrak {t}}M_u\rightarrow {\mathfrak {t}}M_v\) on the associated graded, that is, \({{\,\mathrm{gr}\,}}(\psi '_{v,u})={{\,\mathrm{gr}\,}}(\psi _{v,u})={\tilde{\psi }}_{v,u}\). Indeed, in this case there exists a smooth family of filtration preserving automorphism \(B_{v,u}\) of \(TM_u\) such that \(\psi _{v,u}'=\psi _{v,u}B_{v,u}\) and \({{\,\mathrm{gr}\,}}(B_{v,u})={{\,\mathrm{id}\,}}\). Differentiating, we obtain \({\dot{\psi }}'_u={\dot{\psi }}_u+\dot{B}_u\) where \(\dot{B}_u:=\frac{\partial }{\partial v}|_{v=u}B_{v,u}\) is a filtration preserving endomorphism of \(TM_u\) inducing \({{\,\mathrm{gr}\,}}(\dot{B}_u)=0\) on \({\mathfrak {t}}M_u\). In particular, \({{\,\mathrm{tr}\,}}(\dot{B}_u)=0\) and thus \({{\,\mathrm{tr}\,}}({\dot{\psi }}'_u)={{\,\mathrm{tr}\,}}({\dot{\psi }}_u)\). Moreover, \({{\,\mathrm{gr}\,}}(\pi _u\Pi _u\dot{B}_uL_u)={{\,\mathrm{gr}\,}}(\pi _u\Pi _u){{\,\mathrm{gr}\,}}(\dot{B}_u){{\,\mathrm{gr}\,}}(L_u)=0\) since each factor is filtration preserving. As \({\mathfrak {t}}M_u\) has pure cohomology, we conclude \(\pi _u\Pi _u\dot{B}_uL_u=0\), cf. the proof of Lemma 3.2. We obtain \(\pi _u\Pi _u{\dot{\psi }}'_uL_u=\pi _u\Pi _u{\dot{\psi }}_uL_u\) and thus \(\dot{A}^F_u\) remains unchanged, see (139). Clearly this implies that \({\tilde{p}}_{j,u}^F\) remains unchanged as well, see (127).
Remark 3.17
Using (69), we conclude that the quantity in (124) remains unchanged if the filtration on M is smoothly deformed as above, i.e., in such a way that the bundles of osculating algebras remain isomorphic. Hence, this quantity only depends on F and the corresponding homotopy class of the filtration.
3.6 Comparison with the Ray–Singer Torsion
We continue to use the notation set up above. Moreover, we consider a Riemannian metric g on M. Comparing the torsion of the Rumin complex constructed above, see (99), with the Ray–Singer torsion [6, 77], we obtain a positive real number
For trivially filtered manifolds the analytic torsion constructed above coincides, by definition, with the Ray–Singer torsion. We will add the decoration RS to our notation if an object is associated to the underlying trivially filtered manifold.
For contact manifolds, Albin and Quan [1, Corollary 3] have shown that the quotient (140) can be expressed as an integral over local quantities. Possibly, their analysis of the sub-Riemannian limit can be extended to filtered manifolds whose osculating algebras have pure cohomology. We will not pursue this here though. Returning to the general case we have.
Proposition 3.18
Suppose \(F_u\) is a complex vector bundle over M with a flat connection that depends smoothly on a real parameter u. Then \(R_{{\tilde{g}};g}^{F_u,h}(M,\mathcal F)\) is smooth in u.
Proof
Let \(\Delta _u=\Delta _{{\tilde{g}},h}^{F_u}\) denote the Rumin–Seshadri operator (98) and let \(\Delta _{{\mathrm{RS}},u}=\Delta ^{F_u}_{{\mathrm{RS}},g,h}\) denote the Hodge Laplacian. Given \(u_0\), we choose \(\lambda >0\) such that all non-zero eigenvalues of \(\Delta _{u_0}\) and all non-zero eigenvalues of \(\Delta _{{\mathrm{RS}},u_0}\) are strictly larger than \(\lambda \). By definition, see (24) and (99), we have
If u is sufficiently close to \(u_0\), then the spectra of \(\Delta _u\) and \(\Delta _{{\mathrm{RS}},u}\) do not contain \(\lambda \). For these u, the spectral projectors \(P_{\lambda ,u}\) and \(P_{{\mathrm{RS}},\lambda ,u}\) depend smoothly on u, and so do \(\zeta '_{\lambda ,u}(0)\) and \(\zeta '_{{\mathrm{RS}},\lambda ,u}(0)\), cf. (20). Hence, numerator and denominator of the first factor in (141) depend smoothly on u, provided u remains sufficiently close to \(u_0\). To analyze the second factor in (141), we consider the chain map, cf. (96),
Clearly, \(\ell _u\) induces the isomorphism (97) in cohomology. By positivity, and since \(\Delta _{u_0}\) vanishes on \({{\,\mathrm{img}\,}}(P_{\lambda ,u_0})\), the differential \(D^{F_{u_0}}\) vanishes on \({{\,\mathrm{img}\,}}(P_{\lambda ,u_0})\), cf. (7). Similarly, \(d^{F_{u_0}}\) vanishes on \({{\,\mathrm{img}\,}}(P_{{\mathrm{RS}},\lambda ,u_0})\). Hence, \(\ell _{u_0}\) is an isomorphism of (trivial) chain complexes. We conclude that \(\ell _u\) is an isomorphism of chain complexes, provided u is sufficiently close to \(u_0\). In this situation, we have the well-known formula
where for these u, the right hand side clearly depends smoothly on u. \(\square \)
From Propositions 3.3, 3.4, 3.5, and Theorem 3.8 we immediately obtain
Proposition 3.19
In this situation we have:
-
(a)
If \(F_1\) and \(F_2\) are two flat vector bundles with fiberwise Hermitian inner products \(h_1\) and \(h_2\), respectively, then
$$\begin{aligned} R_{{\tilde{g}};g}^{F_1\oplus F_2,h_1\oplus h_2}(M,\mathcal F)=R_{{\tilde{g}};g}^{F_1,h_1}(M,\mathcal F)\cdot R_{{\tilde{g}};g}^{F_2,h_2}(M,\mathcal F). \end{aligned}$$ -
(b)
If \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds and \({\tilde{F}}\) is a flat vector bundle over \({\tilde{M}}\) with fiberwise Hermitian inner product \({\tilde{h}}\), then
$$\begin{aligned} R_{{\tilde{g}};g}^{\pi _*{\tilde{F}},\pi _*{\tilde{h}}}(M,\mathcal F)=R_{\pi ^*{\tilde{g}};\pi ^*g}^{{\tilde{F}},{\tilde{h}}}({\tilde{M}},\pi ^*\mathcal F). \end{aligned}$$ -
(c)
If \(\pi :{\tilde{M}}\rightarrow M\) is a finite covering of filtered manifolds, then
$$\begin{aligned} R_{\pi ^*{\tilde{g}};\pi ^*g}^{\pi ^*F,\pi ^*h}({\tilde{M}},\pi ^*\mathcal F)=R_{{\tilde{g}};g}^{F\otimes \mathcal V,h\otimes h^{\mathcal V}}(M,\mathcal F). \end{aligned}$$ -
(d)
With respect to Poincaré duality, we have
$$\begin{aligned} R_{{\tilde{g}};g}^{F^*\otimes {\mathcal {O}},h^{-1}}(M,\mathcal F) =\left( R_{{\tilde{g}};g}^{{\bar{F}},{\bar{h}}}(M,\mathcal F)\right) ^{(-1)^{m+1}} =\left( R_{{\tilde{g}};g}^{F,h}(M,\mathcal F)\right) ^{(-1)^{m+1}}.\nonumber \\ \end{aligned}$$(142)
Remark 3.20
For even dimensional orientable M and parallel Hermitian metrics h, Equation (142) yields \(R_{{\tilde{g}};g}^{F,h}(M,\mathcal F)=1\), for in this case, h together with an orientation of M provides an isomorphism of flat bundles \({\bar{F}}\cong F^*\otimes {\mathcal {O}}\) that intertwines the Hermitian metric \({\bar{h}}\) with \(h^{-1}\), cf. Proposition 3.9. Hence, in this situation
and, in particular, the analytic torsion on the left hand side is independent of the metric \({\tilde{g}}\). However, besides the trivially filtered case, we do not know of any even dimensional filtered manifold whose osculating algebras have pure cohomology: contact manifolds are odd dimensional and so are the five dimensional geometries discussed in Sect. 4 below.
We expect that the analytic torsion proposed in this paper can be refined to a complex bilinear form on the determinant line \({{\,\mathrm{sdet}\,}}(H^*(M;F))\) such that the comparison with the refined complex valued Ray–Singer torsion [12,13,14,15,16, 20, 21, 92] analogous to (140) yields a ratio that depends holomorphically on the flat connection.
3.7 Nilpotent Lie Algebras with Pure Cohomology
The analytic torsion discussed above is only defined for filtered manifolds which have osculating algebras with pure cohomology. This assumption appears to be very restrictive. The only examples we are aware of are: trivially filtered manifolds corresponding to abelian osculating algebras giving rise to the classical Ray–Singer torsion; contact manifolds corresponding to Heisenberg algebras giving rise to the Rumin–Seshadri torsion; and the five dimensional geometries discussed in Sect. 4 below.
The aim of this section is to present a necessary condition which the dimensions of the grading components of a finite dimensional graded nilpotent Lie algebra
have to satisfy, if the cohomology of \({\mathfrak {g}}\) is pure.
To this end let \(\phi _t\) denote the grading automorphism acting by multiplication with \(t^p\) on the summand \({\mathfrak {g}}_p\) and consider the polynomial [42]
If \({\mathfrak {g}}\) has pure cohomology, then there exist numbers \(p_q\in \mathbb N_0\) such that \(\phi _t\) acts by the scalar \(t^{p_q}\) on \(H^q({\mathfrak {g}})\). Hence, in this case the polynomial
has at most \(d+1\) non-trivial coefficients, where \(d:=\dim {\mathfrak {g}}\).
Clearly, \({{\,\mathrm{str}\,}}(\phi _t|H^*({\mathfrak {g}}))={{\,\mathrm{str}\,}}(\phi _t|\Lambda ^*{\mathfrak {g}}^*)=\prod _p{{\,\mathrm{str}\,}}(\phi _t| \Lambda ^*{\mathfrak {g}}^*_p)\) and thus
where \(n_p:=\dim {\mathfrak {g}}_{-p}\). This is a polynomial of degree \(n:=\sum _ppn_p\). Moreover,
where the (positive, integral) coefficients \(a_l\) are defined by
and we are using the convention \(\left( {\begin{array}{c}d\\ m\end{array}}\right) =0\) if \(m<0\) or \(m>d\). Using
which is valid for all integers \(0\le i\le n\) and \(0\le l\le n-d\), we obtain
where
and
are polynomials of degree \(n-d\) in i. Hence, the polynomial P(t) has at least \(d+1\) non-trivial coefficients. Moreover, we obtain
Lemma 3.21
If \({\mathfrak {g}}\) has pure cohomology, then P(t) has precisely \(d+1\) non-trivial coefficients and c(i) has \(n-d\) mutually different integral zeros located in the range \(\{0,\dotsc ,n\}\). More explicitly,
where \(\mathcal P:=\{p_q:q=0,\dotsc ,d\}\) and \(\mathcal P':=\{0,\dotsc ,n\}\setminus \mathcal P\).
Example 3.22
If \({\mathfrak {g}}={\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=2\), then \(a_l=1,2,1\) and
with zeros \(i=\frac{n\pm \sqrt{n}}{2}\). Hence, if \({\mathfrak {g}}\) has pure cohomology, then n must be a square according to Lemma 3.21. The case \(n=4\) corresponds to \(n_1=0\) and is realized by an abelian Lie algebra concentrated in degree 2. The case \(n=9\) corresponds to \(n_1=5\). It is not hard to see that this cannot be realized by a nilpotent Lie algebra though. We do no know if larger squares n can be realized by nilpotent Lie algebras.
Example 3.23
If \({\mathfrak {g}}={\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=3\), then \(a_l=1,3,3,1\) and
Hence, if \({\mathfrak {g}}\) has pure cohomology, then n must be even and \(3n-2\) must be a square according to Lemma 3.21. The case \(n=6\) corresponds to \(n_1=0\) and is realized by an abelian Lie algebra concentrated in degree 2. The general n consistent with Lemma 3.21 is of the form \(n=(m^2+2)/3\) where \(m\ge 4\) is an integer congruent to 2 or 4 mod 6. Which of these can be realized by nilpotent Lie algebras is unclear.
Example 3.24
If \({\mathfrak {g}}={\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=4\), then \(a_l=1,4,6,4,1\) and
Hence, if \({\mathfrak {g}}\) has pure cohomology, then the four roots of c(r),
must all be integral according to Lemma 3.21. The case \(n=8\) corresponds to \(n_1=0\) and is realized by an abelian Lie algebra concentrated in degree 2. We do not know of any other n for which this happens. Using a computer, we ruled out all \(8<n\le 10000\). For some n, e.g. \(n=17\), \(n=66\), or \(n=1521\), two roots are integral, but never all four, in this range.
Example 3.25
If \({\mathfrak {g}}={\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=5\), then \(a_l=1,5,10,10,5,1\) and
Hence, if \({\mathfrak {g}}\) has pure cohomology, then the root \(i=n/2\) must be integral, hence n must be even and the other four roots,
must all be integral according to Lemma 3.21. The case \(n=10\) corresponds to \(n_1=0\) and is realized by an abelian Lie algebra concentrated in degree 2. We do not know of any other n for which this happens. Using a computer, we ruled out all \(10<n\le 10000\). For some n, e.g. \(n=17\), \(n=36\), or \(n=289\), two roots are integral, but never all five. For \(n=67\) four roots are integral, but this one is odd.
Using computer algebra, we also ruled out Lie algebras of the form \({\mathfrak {g}}={\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) in the range \(1\le n_1\le 1000\) and \(6\le n_2\le 11\) — non of these dimensions are consistent with the condition in Lemma 3.21.
Example 3.26
If \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=n_3=1\), then \(a_l=1,2,2,1\) and
Hence, if \({\mathfrak {g}}\) has pure cohomology, then the root \(i=n/2\) must be integral, hence n must be even and the other two roots,
must be integral according to Lemma 3.21. All \(n\ge 12\) can be ruled out since in this case these roots are complex. Hence, \(n=10\) is the only case consistent with Lemma 3.21. However, one readily shows that this can not be realized by a nilpotent Lie algebra.
Example 3.27
If \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=2\) and \(n_3=1\), then we have \(a_l=1,3,4,3,1\) and
If this polynomial has four real zeros, then it must take a positive value at \(i=n/2\), hence we must have \(n^2-14n+24<0\). This rules out any \(n\ge 12\). The remaining finitely many n can easily be excluded. Hence, no nilpotent Lie algebra of this form is consistent with the condition in Lemma 3.21.
Example 3.28
If \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) with \(n_2=1\) and \(n_3=2\), then we have \(a_l=1,3,5,5,3,1\) and
If \({\mathfrak {g}}\) has pure cohomology, then the zero \(i=n/2\) must be integral, hence n must be even. Moreover, in order to obtain five real zeros, the polynomial c(i) must have a local maximum at \(i=n/2\). Hence we must have \(6n^2-132n+252<0\), which rules out any \(n\ge 20\). Checking the remaining n by computer, we see that the only case consistent with Lemma 3.21 is \(n=10\) corresponding to \(n_1=2\). The latter can indeed be realized by a nilpotent Lie algebra, see Sect. 4 below.
Using computer algebra, we also searched \({\mathfrak {g}}={\mathfrak {g}}_{-5}\oplus {\mathfrak {g}}_{-4}\oplus {\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) in the range \(0\le n_1\le 100\), \(0\le n_2,n_3,n_4,n_5\le 5\). The only cases consistent with Lemma 3.21 are the ones mentioned above, perhaps with a different grading (abelian concentrated in any degree, or contact in even degrees). We also searched \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) in the range \(0\le n_1\le 200\), \(0\le n_2\le 50\), \(0\le n_3\le 20\) and did not find anything new which is consistent with Lemma 3.21. The algorithm we used expands the polynomial \(P(t)=(1-t)^{n_1}(1-t^2)^{n_2}\cdots (1-t^r)^{n_r}\) and counts the number of non-zero coefficients.
4 Generic Rank Two Distributions in Dimension Five
Recall that a generic rank two distribution on a 5-manifold M is a smooth rank two subbundle \({\mathcal {D}}\) in the tangent bundle TM such that Lie brackets of sections of \({\mathcal {D}}\) span a rank three subbundle \([{\mathcal {D}},{\mathcal {D}}]\) in TM and triple brackets span all of the tangent bundle. Hence, putting \(T^{-1}M:={\mathcal {D}}\), \(T^{-2}M:=[{\mathcal {D}},{\mathcal {D}}]\) and \(T^{-3}M:=TM\), a generic rank two distribution turns M into a filtered manifold,
We will denote this filtration by \(\mathcal F_{{\mathcal {D}}}\).
The osculating algebras \({\mathfrak {t}}_xM={\mathfrak {t}}^{-3}_xM\oplus {\mathfrak {t}}^{-2}_xM\oplus {\mathfrak {t}}_x^{-1}M\) of such a filtration are all isomorphic, as graded Lie algebras, to the five dimensional graded Lie algebra
with graded basis \(X_1,X_2\in {\mathfrak {g}}_{-1}\), \(X_3\in {\mathfrak {g}}_{-2}\), \(X_4,X_5\in {\mathfrak {g}}_{-3}\) and non-trivial brackets
see [29, 27, Sect. 4.3.2], [86, Example 1], [87, Sect. 4] or [88, Sect. 2.3.1]. Note that these geometries have even homogeneous dimension
Lemma 4.1
Suppose \(Y_1,Y_2\in {\mathfrak {g}}\) project to a basis of \({\mathfrak {g}}/[{\mathfrak {g}},{\mathfrak {g}}]\). Then there exists a unique (in general ungraded) Lie algebra automorphism \(\varphi :{\mathfrak {g}}\rightarrow {\mathfrak {g}}\) such that \(\varphi (X_1)=Y_1\) and \(\varphi (X_2)=Y_2\).
Proof
Define a linear map \(\varphi :{\mathfrak {g}}\rightarrow {\mathfrak {g}}\) by \(\varphi (X_1):=Y_1\), \(\varphi (X_2):=Y_2\), \(\varphi (X_3):=[Y_1,Y_2]\), \(\varphi (X_4):=[Y_1,[Y_1,Y_2]]\), \(\varphi (X_5)=[Y_2,[Y_1,Y_2]]\). Using the relations in (145) one readily checks that \(\varphi \) is a homomorphism of Lie algebras. With respect to the decomposition \({\mathfrak {g}}={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\oplus {\mathfrak {g}}_{-1}\) the linear map \(\varphi \) has upper triangular block form with invertible diagonal blocks. Hence, \(\varphi \) is invertible. \(\square \)
In particular, this lemma shows that restriction provides an isomorphism
where the left hand side denotes the group of graded Lie algebra automorphisms.
For the Betti numbers \(b^q:=\dim H^q({\mathfrak {g}})\) we find \(b^0=b^5=1\), \(b^1=b^4=2\), \(b^2=b^3=3\). Moreover, \({\mathfrak {g}}\) has pure cohomology. Indeed, the grading automorphism acts by multiplication \(t^{p_q}\) on \(H^q({\mathfrak {g}})\) where \(p_0=0\), \(p_1=1\), \(p_2=4\), \(p_3=6\), \(p_4=9\), \(p_5=10\). Hence, the Rumin complex is a Rockland complex of the form
with ranks \({{\,\mathrm{rk}\,}}{\mathcal {H}}^0({\mathfrak {t}}M))={{\,\mathrm{rk}\,}}{\mathcal {H}}^5({\mathfrak {t}}M))=1\), \({{\,\mathrm{rk}\,}}{\mathcal {H}}^1({\mathfrak {t}}M))={{\,\mathrm{rk}\,}}{\mathcal {H}}^4({\mathfrak {t}}M))=2\), and \({{\,\mathrm{rk}\,}}{\mathcal {H}}^2({\mathfrak {t}}M))={{\,\mathrm{rk}\,}}{\mathcal {H}}^3({\mathfrak {t}}M))=3\). Moreover, the differential operators \(D_0\) and \(D_4\) have Heisenberg order 1; the operators \(D_1\) and \(D_3\) have Heisenberg order 3; and \(D_2\) has Heisenberg order 2, see [19, Sect. 5] or [39, Example 4.24].
To obtain an analytic torsion, it suffices to choose a sub-Riemannian metric \({\tilde{g}}\) on the rank two bundle \({\mathcal {D}} =T^{-1}M={\mathfrak {t}}^{-1}M\). We can use the fiberwise Lie algebra structure to extend this to a graded Euclidean inner product on \({\mathfrak {t}}M\), which will be denoted by \({\tilde{g}}\) too. Indeed, in view of (145), the fiberwise (Levi) bracket on \({\mathfrak {t}}M\) induces canonical isomorphisms of vector bundles
More explicitly, using a graded basis as in (145), we extend \({\tilde{g}}\) such that
This choice of peculiar constants 4 and 3 is motivated by parabolic geometry, see [27, Sect. 3.3.1] and [88, Equations (3.21) and (3.39)]. Hence, a sub-Riemannian metric \({\tilde{g}}\) on the rank two bundle \({\mathcal {D}}\) and a Hermitian metric h on a flat bundle F over M give rise to an analytic torsion
i.e. a norm on the graded determinant line, cf. (99),
For the purpose of this paper, any other pair of positive constants in (147) and (148) would work equally well. We will not discuss here to what extent the torsion depends on the choice of these constants. The following will be crucial though: If \({\tilde{g}}_u\) is a family of sub-Riemannian metrics on \({\mathcal {D}}\), then their extensions differ by a graded automorphism of the bundle of osculating algebras \({\mathfrak {t}}M\), that is, \({\tilde{g}}_v^{-1}{\tilde{g}}_u\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}({\mathfrak {t}}M))\), see (146). Moreover, \({{\,\mathrm{tr}\,}}_{{\mathfrak {t}}M}(\dot{g}_u)=5{{\,\mathrm{tr}\,}}_{{\mathcal {D}}}(\dot{g}_u)\). Hence, Theorem 3.11 is applicable and specializes Theorem 1.1.
4.1 Proof of Theorem 1.2
We begin by noting that a sub-Riemannian metric \({\tilde{g}}\) on a generic rank two distribution \({\mathcal {D}}\) induces a splitting of the filtration \(S_{{\mathcal {D}},{\tilde{g}}}:{\mathfrak {t}}M\rightarrow TM\). Indeed, the sub-Riemannian metric induces a nowhere vanishing section of the line bundle \(|\Lambda ^2{\mathcal {D}}^*|\) that provides a scale which is closely related to the generalized contact forms considered in [88, Sect. 3.1]. It induces [27, Corollary 5.1.6] an (exact) Weyl structure [27, Definition 5.1.7] for the canonical (regular and normal) Cartan connection [29], see also [26]. In particular, it provides a splitting of the filtration such that \(S_{{\mathcal {D}},{\tilde{g}}}(x)\) only depends on the germ of \((M,{\mathcal {D}},{\tilde{g}})\) at x. Using these splittings, the coefficients at x of the differential operators \(L:\Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M))\rightarrow \Omega (M)\) and \(\pi \Pi :\Omega (M)\rightarrow \Gamma ^\infty ({\mathcal {H}}({\mathfrak {t}}M))\) only depend on the germ of \((M,{\mathcal {D}},{\tilde{g}})\) at x.
We denote the filtered manifold corresponding to the generic rank two distribution \({\mathcal {D}}_u\) by \(M_u\). Let \({\tilde{\psi }}_u:{\mathfrak {t}}M\rightarrow {\mathfrak {t}}M_u\) denote the unique isomorphism of graded Lie algebra bundles which restricts to \(\Theta _u|_{{\mathcal {D}}}:{\mathcal {D}}\rightarrow {\mathcal {D}}_u\). Clearly, \(({\tilde{\psi }}_u)_*{\tilde{g}}={\tilde{g}}_u\), cf. (126), where \({\tilde{g}}\) and \({\tilde{g}}_u\) denote the graded Euclidean metrics on \({\mathfrak {t}}M\) and \({\mathfrak {t}}M_u\) which are induced from the sub-Riemannian metrics on \({\mathcal {D}}\) and \({\mathcal {D}}_u\) as indicated in (147) and (148). We use the splittings discussed above to define \(\psi _u\in \Gamma ^\infty ({{\,\mathrm{Aut}\,}}(TM))\) by \(\psi _uS_{{\mathcal {D}},{\tilde{g}}}=S_{{\mathcal {D}}_u,{\tilde{g}}_u}{\tilde{\psi }}_u\), cf. (125). Put \({\dot{\psi }}=\frac{\partial }{\partial u}|_{u=0}\psi _u\) and consider the differential operator, cf. (139),
We will now show that \({{\,\mathrm{tr}\,}}({\dot{\psi }})(x)\) and the coefficients of \(\dot{A}\) at x only depend on the germ of \((M,{\mathcal {D}},{\tilde{g}},{\dot{\Theta }}|_{{\mathcal {D}}})\) at x. To see this let X, Y be a local frame of \({\mathcal {D}}\) defined on an open subset \(U\subseteq M\). Then X, Y, [X, Y], [X, [X, Y]], [Y, [X, Y]] is a frame of \(TM|_U\) and we may define a smooth family of filtration preserving automorphisms \(\psi _u':TM|_U\rightarrow TM_u|_U\) by:
Clearly \({{\,\mathrm{gr}\,}}(\psi _u')={\tilde{\psi }}_u\) and \(\psi '_0={{\,\mathrm{id}\,}}\). Differentiating, we see that \({\dot{\psi }}'(x)=\frac{\partial }{\partial u}|_{u=0}\psi _u'(x)\) only depends on the germ of \((M,{\mathcal {D}},{\dot{\Theta }}|_{{\mathcal {D}}},X,Y)\) at x. According to Remark 3.16 we have \({{\,\mathrm{tr}\,}}({\dot{\psi }})|_U={{\,\mathrm{tr}\,}}({\dot{\psi }}')\) and \(\dot{A}|_U=-\pi \Pi {\dot{\psi }}'L|_U\). We conclude that \({{\,\mathrm{tr}\,}}({\dot{\psi }})(x)\) and the coefficients of \(\dot{A}\) at x only depend on the germ of \((M,{\mathcal {D}},{\tilde{g}},{\dot{\Theta }}|_{{\mathcal {D}}})\) at x.
From the preceding paragraph we see that the coefficients at x of the differential operator \(\dot{A}^F+(\dot{A}^F)^*-{{\,\mathrm{tr}\,}}({\dot{\psi }})\) only depend on the germ of \((M,{\mathcal {D}},{\tilde{g}},F,h,{\dot{\Theta }}|_{{\mathcal {D}}})\) at x. Hence, in the asymptotic expansion
the terms \({\tilde{p}}_j^F(x)\) can be computed from the germ of \((M,{\mathcal {D}},{\tilde{g}},F,h,{\dot{\Theta }}|_{{\mathcal {D}}})\) at x. Hence, in view of Theorem 3.15, the density
has the desired property.
4.2 Computation for Nilmanifolds
Let G denote the simply connected nilpotent Lie group with Lie algebra (144). Since the structure constants are rational, this group admits lattices, i.e., discrete cocompact subgroups, [76, Theorem 2.12]. Suppose \(\Gamma \subseteq G\) is a lattice and let \(\rho :\Gamma \rightarrow U(k)\) be a finite dimensional unitary representation of \(\Gamma \). Then \(F_\rho :=G\times _\Gamma \mathbb {C}^k\) is a flat vector bundle over the nilmanifold \(G/\Gamma \) which comes with a parallel Hermitian metric \(h_\rho \). Let \({\mathcal {D}}\) be a right G-invariant generic rank two distribution on G and let \({\mathcal {D}}_\Gamma \) denote the induced generic rank two distribution on \(G/\Gamma \). Note that a 2-dimensional subspace in \({\mathfrak {g}}\) gives rise to a right invariant generic rank two distribution on G iff it intersects \([{\mathfrak {g}},{\mathfrak {g}}]={\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\) trivially, cf. Lemma 4.1. We consider a right invariant sub-Riemannian metric \({\tilde{g}}\) on \({\mathcal {D}}\) and let \({\tilde{g}}_\Gamma \) denote the induced sub-Riemannian metric on \({\mathcal {D}}_\Gamma \).
Lemma 4.2
The torsion \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}H^*(G/\Gamma ;F_\rho )}_{{\mathcal {D}}_\Gamma ,{\tilde{g}}_\Gamma ,h_\rho }\) does not depend on the right invariant generic rank two distribution \({\mathcal {D}}\) on G nor does it depend on the right invariant sub-Riemannian metric \({\tilde{g}}\) on \({\mathcal {D}}\).
Proof
We start by showing independence of \({\tilde{g}}\). By convexity, any two right invariant sub-Riemannian metrics on \({\mathcal {D}}\) can be connected by a straight path of right invariant sub-Riemannian metrics on \({\mathcal {D}}\). In view of Theorem 1.1, it therefore suffices to show that the constant term in the asymptotic expansion of the heat kernel of the Rumin–Seshadri operator \(\Delta _{{\mathcal {D}}_\Gamma ,{\tilde{g}}_\Gamma ,h_\rho }^{F_\rho }\) vanishes. Since this asymptotic expansion can be computed locally, we may work on the universal covering \(p:G\rightarrow G/\Gamma \). Over G, the flat bundle \(F_\rho \) may be (canonically) trivialized and the Hermitian metric \(h_\rho \) becomes constant in this trivialization. In view of Lemma 4.1 there exists a group isomorphism \(\phi :G\rightarrow G\) such that \(\phi ^*{\mathcal {D}}={\mathcal {D}}_0\), the right invariant standard distribution obtained by translating \({\mathfrak {g}}_{-1}\). Moreover, \({\tilde{g}}_0:=\phi ^*{\tilde{g}}\) is a right invariant sub-Riemannian metric on \({\mathcal {D}}_0\). Hence, by naturality,
where \(h_0\) denotes the standard (constant) Hermitian metric on the trivial flat bundle \(\phi ^*p^*F_\rho =G\times \mathbb C^{{{\,\mathrm{rk}\,}}(\rho )}\). Note that \({\mathcal {D}}_0\) and \({\tilde{g}}_0\) are also homogeneous with respect to the grading automorphism of G. Hence, \(\Delta _{{\mathcal {D}}_0,{\tilde{g}}_0,h_0}^{\mathbb C^{{{\,\mathrm{rk}\,}}(\rho )}}\) is right invariant and homogeneous of degree \(2\kappa \). The same is true for the heat operator \(\partial _t+\Delta _{{\mathcal {D}}_0,{\tilde{g}}_0,h_0}^{\mathbb C^{{{\,\mathrm{rk}\,}}(\rho )}}\) on \({\mathbb {R}}\times G\), after assigning degree \(2\kappa \) to the time direction \({\mathbb {R}}\). Consequently, this heat operator admits a parametrix which is homogeneous of degree \(-2\kappa \). Combining this with (152), we see that all but the leading term in the asymptotic expansion of the heat kernel of \(\Delta _{{\mathcal {D}}_\Gamma ,{\tilde{g}}_\Gamma ,h_\rho }^{F_\rho }\) vanish, cf. the proof of Lemma 2.18 or [46, Theorem 1.1].
To show independence of \({\mathcal {D}}\), we proceed analogously using Theorem 1.2. Since the space of 2-dimensional subspaces in \({\mathfrak {g}}\) intersecting \({\mathfrak {g}}_{-3}\oplus {\mathfrak {g}}_{-2}\) trivially is connected, any two right invariant generic rank two distributions on G can be connected by a smooth family of rank two distributions of the same type. According to Theorem 1.2 and the formula in (151) it thus suffices to show that for every differential operator B of Heisenberg order at most r on \(G/\Gamma \) we have an asymptotic expansion of the form, as \(t\rightarrow 0\),
and, in particular, the constant term \({\tilde{p}}_{10}\) vanishes. This in turn follows from (152). Indeed, since \(\Delta _{{\mathcal {D}}_0,{\tilde{g}}_0,h_0}^{\mathbb C^{{{\,\mathrm{rk}\,}}(\rho )}}\) is homogeneous and right invariant, \((\phi ^*p^*B)e^{-\Delta _{{\mathcal {D}}_0,{\tilde{g}}_0,h_0}^{\mathbb C^{{{\,\mathrm{rk}\,}}(\rho )}}}\) has an asymptotic expansion of the same form. \(\square \)
Since \(G/\Gamma \) is odd dimensional, the Ray–Singer torsion \(\Vert -\Vert ^{{{\,\mathrm{sdet}\,}}(H^*(G/\Gamma ;F_\rho ))}_{{\mathrm{RS}},g,h_\rho }\) is independent of the Riemannian metric g and the fiberwise Hermitian inner product h on \(F_\rho \), see [77] and [6, Theorem 0.1]. Combining this with Lemma 4.2, we see that the relative torsion in (140),
only depends on the lattice \(\Gamma \) in G and the unitary representation \(\rho :\Gamma \rightarrow U(k)\).
Lemma 4.3
Let \(\rho \), \(\rho _1\) and \(\rho _2\) be finite dimensional unitary representations of a lattice \(\Gamma \) in G. Moreover, let \({\tilde{\rho }}\) be a finite dimensional unitary representations of a sublattice \({\tilde{\Gamma }}\subseteq \Gamma \). Then the following hold true:
-
(a)
\(R_\Gamma (\rho _1\oplus \rho _2)=R_\Gamma (\rho _1)\cdot R_\Gamma (\rho _2)\).
-
(b)
\(R_\Gamma (\rho )=R_{{\tilde{\Gamma }}}({\tilde{\rho }})\) if \(\rho ={{\,\mathrm{Ind}\,}}_{{\tilde{\Gamma }}}^\Gamma ({\tilde{\rho }})\) is the induced representation of \(\Gamma \).
-
(c)
\(R_{{\tilde{\Gamma }}}(\rho |_{{\tilde{\Gamma }}})=R_\Gamma (\rho \otimes \mathbb C[\Gamma /{\tilde{\Gamma }}])\).
-
(d)
\(R_\Gamma (\rho ^*)=R_\Gamma ({\bar{\rho }})=R_\Gamma (\rho )\).
-
(e)
\(R_{\phi ^{-1}(\Gamma )}(\rho \phi )=R_\Gamma (\rho )\) for every Lie group isomorphism \(\phi :G\rightarrow G\).
Proof
With respect to the canonical identification \(F_{\rho _1\oplus \rho _2}=F_{\rho _1}\oplus F_{\rho _2}\) we have \(h_{\rho _1\oplus \rho _2}=h_{\rho _1}\oplus h_{\rho _2}\). Hence (a) follows from Proposition 3.19(a).
The canonical projection \(\pi :G/{\tilde{\Gamma }}\rightarrow G/\Gamma \) is a finite covering of filtered manifolds, where we use the generic rank two distributions \({\mathcal {D}}_{{\tilde{\Gamma }}}\) and \({\mathcal {D}}_\Gamma \) on \(G/{\tilde{\Gamma }}\) and \(G/\Gamma \), respectively. If \(\rho ={{\,\mathrm{Ind}\,}}_{{\tilde{\Gamma }}}^\Gamma ({\tilde{\rho }})\), then with respect to the canonical identification \(\pi _*F_{{\tilde{\rho }}}=F_\rho \) we have \(\pi _*h_{{\tilde{\rho }}}=h_\rho \). Moreover, \(\pi ^*{\tilde{g}}_\Gamma ={\tilde{g}}_{{\tilde{\Gamma }}}\). Hence (b) follows from Proposition 3.19(b).
(c) follows from (b) for we have \({{\,\mathrm{Ind}\,}}_{{\tilde{\Gamma }}}^\Gamma (\rho |_{{\tilde{\Gamma }}})=\rho \otimes \mathbb {C}[\Gamma /{\tilde{\Gamma }}]\).
(d) follows from Proposition 3.19(d).
To see (e), note that \({\mathcal {D}}':=\phi ^{-1}({\mathcal {D}})\) is a right invariant generic rank two distribution on G and \({\tilde{g}}':=\phi ^*{\tilde{g}}\) is a right invariant sub-Riemannian metric on \(\phi ^{-1}({\mathcal {D}})\). Hence, \(\phi \) induces a diffeomorphism \({\bar{\phi }}:G/\Gamma '\rightarrow G/\Gamma \) where we consider the lattice \(\Gamma ':=\phi ^{-1}(\Gamma )\). Clearly, \({\bar{\phi }}({\mathcal {D}}'_{\Gamma '})={\mathcal {D}}_\Gamma \) and \({\bar{\phi }}^*{\tilde{g}}_\Gamma ={\tilde{g}}'_{\Gamma '}\). Moreover, via the canonical identification \({\bar{\phi }}^*F_\rho =F_{\rho '}\) we have \({\bar{\phi }}^*h_\rho =h_{\rho '}\), where we abbreviate \(\rho ':=\rho \phi \). Hence, by naturality
via the identification \({{\,\mathrm{sdet}\,}}(H^*(G/\Gamma ';F_{\rho '}))\cong {{\,\mathrm{sdet}\,}}(H^*(G/\Gamma ;F_\rho ))\) induced by \({\bar{\phi }}\). Combining this with a corresponding identity for the Ray–Singer torsion, we obtain \(R_{\Gamma '}(\rho ')=R_\Gamma (\rho )\), see (153) and Lemma 4.2. \(\square \)
The exponential map provides a diffeomorphism \(\exp :{\mathfrak {g}}\rightarrow G\). Using the Baker–Campbell–Hausdorff formula we find
where
We let \(\Gamma _0\) denote the subgroup generated by \(\gamma _1:=\exp (X_1)\) and \(\gamma _2:=\exp (X_2)\).
Lemma 4.4
\(\Gamma _0\) is a lattice in G and
Moreover, \(H_1(\Gamma _0)=\Gamma _0/[\Gamma _0,\Gamma _0]\cong \mathbb {Z}^2\) is generated by the images of \(\gamma _1\) and \(\gamma _2\). Via the isomorphism in (146), the group of graded automorphisms preserving the lattice \(\Gamma _0\) corresponds to a (congruence) subgroup of finite index in \({{\,\mathrm{GL}\,}}(2,\mathbb {Z})\),
Proof
With z as in (154) we have:
Using these relations, one readily checks that the right hand side in (155) defines a lattice in G that contains \(\gamma _1\) and \(\gamma _2\). Using the computations
it is easy to see that this lattice is generated by \(\gamma _1\) and \(\gamma _2\). Here the formula
for commutators is helpful. We conclude that \(H_1(\Gamma _0)=\Gamma _0/[\Gamma _0,\Gamma _0]\) is generated by the images of \(\gamma _1\) and \(\gamma _2\). Since \(\Gamma _0\) maps onto a lattice in \(G/[G,G]\cong \mathbb {R}^2\), see [76, proof of Theorem 2.21], we must have \({{\,\mathrm{rk}\,}}H_1(\Gamma _0)=2\), whence \(H_1(\Gamma _0)\cong \mathbb {Z}^2\). Suppose \(\varphi \in {{\,\mathrm{Aut}\,}}({\mathfrak {g}})\) is a graded automorphism. Using the description in (155), we see that \(\varphi (X_1)=(a,c,0,0,0)^t\) and \(\varphi (X_2)=(b,d,0,0,0)^t\) are contained in \(\log \Gamma _0\) if and only if the numbers a, c, \(\frac{ac}{2}\), \(\frac{ac}{12}(2a+3)\), \(\frac{ac}{12}(2c+3)\) and b, d, \(\frac{bd}{2}\), \(\frac{bd}{12}(2b+3)\), \(\frac{bd}{12}(2d+3)\) are all integral. If \(\varphi ^{-1}\) has the same property, then a, c are coprime integers, b, d are coprime integers, and the latter integrality conditions hold if and only if \(ac\equiv bd\equiv 0\mod 12\). \(\square \)
Lemma 4.5
If \(\Gamma \) is a lattice in G, then there exists a, not necessarily graded Lie group automorphism \(\phi :G\rightarrow G\) such that \(\phi (\Gamma )\subseteq \Gamma _0\).
Proof
Every lattice \(\Gamma \) in G can be generated by five elements \(\omega _1,\dotsc ,\omega _5\). These generators may be chosen such that \(\omega _1,\omega _2\in \Gamma \), \(\omega _3\in \Gamma \cap [G,G]\) and \(\omega _4,\omega _5\in \Gamma \cap [G,[G,G]]\), see [76, Theorem 2.21]. According to Lemma 4.1 there exists a unique Lie algebra automorphism \(\varphi :{\mathfrak {g}}\rightarrow {\mathfrak {g}}\) such that \(\varphi (X_1)=\log \omega _1\) and \(\varphi (X_2)=\log \omega _2\). Then \(\exp \circ \varphi \circ \log :G\rightarrow G\) is a Lie group automorphism mapping \(\gamma _1\) to \(\omega _1\) and \(\gamma _2\) to \(\omega _2\). W.l.o.g. we may thus assume \(\omega _1=\gamma _1\) and \(\omega _2=\gamma _2\). Then \(\Gamma _0\) is a subgroup of finite index in \(\Gamma \). Hence, there exists a positive integer k such that \(\omega _3^k,\omega _4^k,\omega _5^k\) are all contained in \(\Gamma _0\). Writing
we see from (155) that the numbers \(ka,kb-\frac{ka}{2},kc+\frac{ka}{2},kd,ke,kf,kg\) must all be integral. For the action by the grading automorphism we find
Taking \(r=2k\), the numbers \(r,r^2a,r^3b-\frac{r^2a}{2},r^3c+\frac{r^2a}{2},r^3d,r^3e,r^3f,r^3g\) are all integral. Hence, using (155) again, we obtain \(\phi _r(\omega _i)\in \Gamma _0\) for \(i=1,\dotsc ,5\). As \(\Gamma \) is generated by \(\omega _1,\dotsc ,\omega _5\), this implies \(\phi _r(\Gamma )\subseteq \Gamma _0\), whence the lemma. \(\square \)
Lemma 4.6
If \(\rho _0\) is a finite dimensional unitary representation of \(\Gamma _0\), then
for all unitary characters \(\chi _0:\Gamma _0\rightarrow U(1)\).
Proof
W.l.o.g. we may assume \(\rho _0\) irreducible, see Lemma 4.3(a). According to [59, Proposition 1] there exists a normal subgroup \(\Gamma _1\subseteq \Gamma _0\) of finite index, a unitary representation \(\tau _0\) of \(\Gamma _0/\Gamma _1\), and a unitary character \(\sigma _0:\Gamma _0\rightarrow U(1)\) such that
We consider the group \({{\,\mathrm{Aut}\,}}(G,\Gamma _0)\) of graded automorphisms preserving \(\Gamma _0\). Since \(\Gamma _0\) only has finitely many subgroups of given finite index, see [59, Sect. II] or [17, Lemma 3], the subgroup
has finite index in \({{\,\mathrm{Aut}\,}}(G,\Gamma _0)\). For \(\phi \in A\) we have \(\tau _0\phi =\tau _0\) and Lemma 4.3(e) gives
for all unitary characters \(\sigma :\Gamma _0\rightarrow U(1)\). As \(H_1(\Gamma _0)\cong \mathbb {Z}^2\), cf. Lemma 4.4, the dual group \(\hom (\Gamma _0,U(1))\cong \hom (\mathbb {Z}^2,U(1))=U(1)\times U(1)\) is connected. Recall that the action of \({{\,\mathrm{GL}\,}}(2,\mathbb {Z})\) on \(\hom (\mathbb {Z}^2,U(1))\) has dense orbits [36]. Since A is a finite index subgroup of \({{\,\mathrm{GL}\,}}(2,\mathbb {Z})\), see Lemma 4.4, there exist finitely many A-orbits whose closures cover all of \(U(1)\times U(1)\). By (157) and Proposition 3.18 the function \(\sigma \mapsto R_{\Gamma _0}(\tau _0\otimes \sigma )\) is constant on each of these orbit closures. As \(U(1)\times U(1)\) is connected, \(R_{\Gamma _0}(\tau _0\otimes \sigma )\) has to be independent of \(\sigma \). We conclude
Combining this with (156), we obtain the lemma. \(\square \)
Combining these observations we will now show
Theorem 4.7
Suppose \(\Gamma \) is a lattice in G and let \(\rho \) denote a finite dimensional unitary representation of \(\Gamma \). Then
for all unitary characters \(\chi :\Gamma \rightarrow U(1)\) that vanish on \({{\,\mathrm{Tor}\,}}H_1(\Gamma )\).
Proof
In view of Lemmas 4.5 and 4.3(e) we may, w.l.o.g. assume \(\Gamma \subseteq \Gamma _0\). Since \(\chi \) vanishes on the torsion part of \(H_1(\Gamma )\), it can be extended to a unitary representation of G and, thus, extends to a unitary character \(\chi _0:\Gamma _0\rightarrow U(1)\), see [59, p. 284 above Proposition 1] for instance. Let \(\rho _0={{\,\mathrm{Ind}\,}}_\Gamma ^{\Gamma _0}(\rho )\) denote the induced representation of \(\Gamma _0\). Using Lemma 4.3(b) we find
The theorem thus follows form Lemma 4.6. \(\square \)
Let us finally give a proof of Theorem 1.3 formulated in the introduction. To this end, recall that every flat line bundle F with parallel Hermitian metric h over \(G/\Gamma \) is of the form \(F=F_\rho \) with \(h=h_\rho \) for some unitary character \(\rho :\Gamma \rightarrow U(1)\). Hence, it suffices to show that \(R_\Gamma (\rho )\) is constant in \(\Gamma \) and \(\rho \), cf. (153). Since \(\Gamma \) was assumed to be generated by two elements, there exists an automorphism \(\phi \) of G mapping \(\Gamma _0\) onto \(\Gamma \), cf. Lemma 4.1. Using Lemma 4.3(e) and applying Lemma 4.6 with the trivial character \(\rho _0=1\) and \(\chi _0=\rho \phi \), we obtain
Hence, the constant \(c=R_{\Gamma _0}(1)\) has the desired property.
References
Albin, P., Quan, H.: Sub-Riemannian limit of the differential form heat kernels of contact manifolds. Int. Math. Res. Not. IMRN 8, 5818–5881 (2022)
An, D., Nurowski, P.: Paweł, Twistor space for rolling bodies. Commun. Math. Phys. 326, 393–414 (2014)
Beals, R., Greiner, P.: Calculus on Heisenberg Manifolds. Annals of Mathematics Studies, vol. 119. Princeton University Press, Princeton (1988)
Beals, R., Greiner, P.C., Stanton, N.K.: The heat equation on a CR manifold. J. Differ. Geom. 20, 343–387 (1984)
Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Grundlehren der Mathematischen Wissenschaften, vol. 298. Springer, Berlin (1992)
Bismut, J.-M., Zhang, W.: An Extension of a Theorem by Cheeger and Müller. Astérisque 205. Société Mathématique de France (1992)
Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion. Commun. Math. Phys. 115, 49–78 (1988)
Bor, G., Montgomery, R.: \(G_2\) and the rolling distribution. Enseign. Math. (2) 55(1–2), 157–196 (2009)
Bor, G., Hernández, L., Nurowski, P.: The dancing metric, G2-symmetry and projective rolling. Trans. Am. Math. Soc. 370, 4433–4481 (2018)
Borman, M.S., Eliashberg, Y., Murphy, E.: Existence and classification of overtwisted contact structures in all dimensions. Acta Math. 215, 281–361 (2015)
Boutet de Monvel, L.: Hypoelliptic operators with double characteristics and related pseudo-differential operators. Commun. Pure Appl. Math. 27, 585–639 (1974)
Braverman, M., Kappeler, T.: A refinement of the Ray-Singer torsion. C. R. Math. Acad. Sci. Paris 341, 497–502 (2005)
Braverman, M., Kappeler, T.: Refined analytic torsion as an element of the determinant line. Geom. Topol. 11, 139–213 (2007)
Braverman, M., Kappeler, T.: Comparison of the refined analytic and the Burghelea-Haller torsions. Festival Yves Colin de Verdière. Ann. Inst. Fourier (Grenoble) 57, 2361–2387 (2007)
Braverman, M., Kappeler, T.: Ray-Singer type theorem for the refined analytic torsion. J. Funct. Anal. 243, 232–256 (2007)
Braverman, M., Kappeler, T.: Refined analytic torsion. J. Differ. Geom. 78, 193–267 (2008)
Brown, I.D.: Representation of finitely generated nilpotent groups. Pac. J. Math. 45, 13–26 (1973)
Bryant, R.L., Hsu, L.: Rigidity of integral curves of rank 2 distributions. Invent. Math. 114, 435–461 (1993)
Bryant, R.L., Eastwood, M., Neusser, K., and Gover, R.: Some differential complexes within and beyond parabolic geometry. In: Differential Geometry and Tanaka Theory (in honour of Professors Reiko Miyaoka and Keizo Yamaguchi). Adv. Stud. Pure Math., vol. 82, pp. 13–40 (2019). arXiv:1112.2142
Burghelea, D., Haller, S.: Complex valued Ray-Singer torsion. J. Funct. Anal. 248, 27–78 (2007)
Burghelea, D., Haller, S.: Complex valued Ray-Singer torsion II. Math. Nachr. 283, 1372–1402 (2010)
Calderbank, D.M., Diemer, T.: Differential invariants and curved Bernstein-Gelfand-Gelfand sequences. J. Reine Angew. Math. 537, 67–103 (2001)
Čap, A., Neusser, K.: On automorphism groups of some types of generic distributions. Differ. Geom. Appl. 27, 769–779 (2009)
Čap, A., Sagerschnig, K.: On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five. J. Geom. Phys. 59, 901–912 (2009)
Čap, A., Slovák, J., Souček, V.: Bernstein-Gelfand-Gelfand sequences. Ann. Math. 154, 97–113 (2001)
Čap, A., Slovák, J.: Weyl structures for parabolic geometries. Math. Scand. 93, 53–90 (2003)
Čap, A., Slovák, J.: Parabolic geometries. I. Background and general theory. In: Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence, RI (2009)
Čap, A., Souček, V.: Subcomplexes in curved BGG-sequences. Math. Ann. 354, 111–136 (2012)
Cartan, E.: Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. École Norm. Sup. 27, 109–192 (1910)
Casals, R., Pérez, J.L., del Pino, Á., Presas, F.: Existence h-principle for Engel structures. Invent. Math. 210, 417–451 (2017)
Casals, R., del Pino, Á., Presas, F.: Loose Engel structures. Compos. Math. 156, 412–434 (2020)
Cheeger, J.: Analytic torsion and Reidemeister torsion. Proc. Natl. Acad. Sci. USA 74, 2651–2654 (1977)
Cheeger, J.: Analytic torsion and the heat equation. Ann. Math. 109, 259–322 (1979)
Choi, W., Ponge, R.: Tangent maps and tangent groupoid for Carnot manifolds. Differ. Geom. Appl. 62, 136–183 (2019)
Christ, M., Geller, D., Głowacki, P., Polin, L.: Pseudodifferential operators on groups with dilations. Duke Math. J. 68, 31–65 (1992)
Dani, J.S.: Density properties of orbits under discrete groups. J. Indian Math. Soc. (N.S.) 39, 189–217 (1975)
Dave, S., Haller, S.: On \(5\)-manifolds admitting rank two distributions of Cartan type. Trans. Am. Math. Soc. 371, 4911–4929 (2019)
Dave, S., Haller, S.: The heat asymptotics on filtered manifolds. J. Geom. Anal. 30, 337–389 (2020)
Dave, S., Haller, S.: Graded hypoellipticity of BGG sequences. arXiv:1705.01659v2
Debord, C., Skandalis, G.: Adiabatic groupoid, crossed product by \({\mathbb{R}}^*_+\) and pseudodifferential calculus. Adv. Math. 257, 66–91 (2014)
del Pino, Á., Vogel, T.: The Engel-Lutz twist and overtwisted Engel structures. Geom. Topol. 24, 2471–2546 (2020)
Deninger, Ch., Singhof, W.: On the cohomology of nilpotent Lie algebras. Bull. Soc. Math. France 116, 3–14 (1988)
Dixmier, J.: Cohomologie des algèbres de Lie nilpotentes. Acta Sci. Math. (Szeged) 16, 246–250 (1955)
Eliashberg, Y.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98, 623–637 (1989)
Eliashberg, Y., Mishachev, N.: Introduction to the h-principle. In: Graduate Studies in Mathematics 48, American Mathematical Society, Providence, RI (2002)
Fischer, V.: Asymptotics and zeta functions on compact nilmanifolds. J. Math. Pures Appl. (9) 160, 1–28 (2022)
Folland, G.B., Stein, E.M.: Estimates for the \({\bar{\partial }}_b\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)
Geiges, H.: Contact structures on 1-connected 5-manifolds. Mathematika 38, 303–311 (1991)
Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)
Gover, A.R., Panai, R., Willse, T.: Nearly Kähler geometry and (2,3,5)-distributions via projective holonomy. Indiana Univ. Math. J. 66, 1351–1416 (2017)
Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 9. Springer, Berlin (1986)
Gromov, M.: Carnot-Carathéodory spaces seen from within. In: Sub-Riemannian Geometry. Progress in Mathematics, vol. 144, pp. 79–323. Birkhäuser, Basel (1996)
Haj Saeedi Sadegh, A.R., Higson, N.: Euler-like vector fields, deformation spaces and manifolds with filtered structure. Doc. Math. 23, 293–325 (2018)
Hammerl, M., Sagerschnig, K.: Conformal structures associated to generic rank 2 distributions on 5-manifolds—characterization and Killing-field decomposition. In: SIGMA Symmetry Integrability and Geometry: Methods and Application, vol. 5, Paper 081 (2009)
Hammerl, M., Sagerschnig, K.: The twistor spinors of generic 2- and 3-distributions. Ann. Global Anal. Geom. 39, 403–425 (2011)
Helffer, B., Nourrigat, F.: Hypoellipticité pour des groupes nilpotents de rang de nilpotence 3. Commun. Partial Differ. Equ. 3, 643–743 (1978)
Helffer, B., Nourrigat, J.: Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué. Commun. Partial Differ. Equ. 4, 899–958 (1979)
Helffer, B., Nourrigat, J.: Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Progress in Mathematics, vol. 58. Birkhäuser, Boston (1985)
Howe, R.E.: On representations of discrete, finitely generated, torsion-free, nilpotent groups. Pac. J. Math. 73, 281–305 (1977)
Kato, T., Perturbation Theory for Linear Operators. Reprint of the: edition, p. 1995. Classics in Mathematics. Springer, Berlin (1980)
Kirillov, A.A.: Lectures on the Orbit Method. Graduate Studies in Mathematics, vol. 64. American Mathematical Society, Providence, RI (2004)
Knudson, F.F., Mumford, D.: The projectivity of the moduli space of stable curves, I: Preliminaries on “det’’ and “div’’. Math. Scand. 39, 19–55 (1976)
Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of the Conference Complex Analysis (Minneapolis, 1964), pp. 81–94. Springer, Berlin (1965)
Leistner, T., Nurowski, P., Sagerschnig, K.: New relations between G2 geometries in dimensions 5 and 7. Int. J. Math. 28, 1750094 (2017)
Lück, W., Schick, T., Thielmann, T.: Torsion and fibrations. J. Reine Angew. Math. 498, 1–33 (1998)
Melin, A.: Lie filtrations and pseudo-differential operators. Preprint (1982)
Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)
Mohsen, O.: On the deformation groupoid of the inhomogeneous pseudo-differential Calculus. Bull. Lond. Math. Soc. 53, 575–592 (2021)
Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. In: Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence, RI (2002)
Morimoto, T.: Geometric structures on filtered manifolds. Hokkaido Math. J. 22, 263–347 (1993)
Morimoto, T.: Lie algebras, geometric structures and differential equations of filtered manifolds. In Lie groups, geometric structures and differential equations–one hundred years after Sophus Lie (Kyoto, Nara,: Adv. Stud. Pure Math. 37(2002), 205–252 (1999)
Müller, W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28, 233–305 (1978)
Nurowski, P.: Differential equations and conformal structures. J. Geom. Phys. 55, 19–49 (2005)
Ponge, R.S.: Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds. Mem. Am. Math. Soc. 194, 2008 (2008)
Presas, F.: Non-integrable distributions and the h-principle. Eur. Math. Soc. Newsl. 99, 18–26 (2016)
Raghunathan, M.S.: Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York (1972)
Ray, D.B., Singer, I.M.: \(R\)-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)
Rockland, C.: Hypoellipticity on the Heisenberg group-representation-theoretic criteria. Trans. Am. Math. Soc. 240, 1–52 (1978)
Rumin, M.: Un complexe de formes différentielles sur les variétés de contact. C. R. Acad. Sci. Paris Sér. I Math. 310, 401–404 (1990)
Rumin, M.: Formes différentielles sur les variétés de contact. J. Differ. Geom. 39, 281–330 (1994)
Rumin, M.: Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups. C. R. Acad. Sci. Paris Sér. I Math. 329, 985–990 (1999)
Rumin, M.: Sub-Riemannian limit of the differential form spectrum of contact manifolds. Geom. Funct. Anal. 10, 407–452 (2000)
Rumin, M.: Around heat decay on forms and relations of nilpotent Lie groups. Séminaire de Théorie Spectrale et Géométrie 19, Année 2000–2001, 123–164, Sémin. Théor. Spectr. Géom. 19(2001), Univ. Grenoble I, Saint-Martin-d’Hères, (2001)
Rumin, M.: An introduction to spectral and differential geometry in Carnot-Carathéodory spaces. Rend. Circ. Mat. Palermo (2) Suppl. No. 75, 139–196 (2005)
Rumin, M., Seshadri, N.: Analytic torsions on contact manifolds. Ann. Inst. Fourier 62, 727–782 (2012)
Sagerschnig, K.: Parabolic geometries determined by filtrations of the tangent bundle. Rend. Circ. Mat. Palermo (2) Suppl. No. 79, 175–181 (2006)
Sagerschnig, K.: Split octonions and generic rank two distributions in dimension five. Arch. Math. (Brno) 42(suppl), 329–339 (2006)
Sagerschnig, K.: Weyl structures for generic rank two distributions in dimension five. PhD thesis, University of Vienna, Vienna, Austria (2008). http://othes.univie.ac.at/2186/
Sagerschnig, K., Willse, T.: The almost Einstein operator for (2,3,5) distributions. Arch. Math. (Brno) 53, 347–370 (2017)
Sagerschnig, K., Willse, T., The geometry of almost Einstein (2,3,5) distributions. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017)
Seeley, R.T.: Complex powers of an elliptic operator. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307. American Mathematical Society, Providence (1967)
Su, G., Zhang, W.: A Cheeger-Müller theorem for symmetric bilinear torsions. Chin. Ann. Math. Ser. B 29, 385–424 (2008)
Taylor, M.E.: Noncommutative microlocal analysis I. Mem. Am. Math. Soc. 52, 1984 (1984)
Turaev, V.: Introduction to combinatorial torsions. In: Notes taken by Felix Schlenk. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2001)
van Erp, E., Yuncken, R.: On the tangent groupoid of a filtered manifold. Bull. Lond. Math. Soc. 49, 1000–1012 (2017)
van Erp, E., Yuncken, R.: A groupoid approach to pseudodifferential calculi. J. Reine Angew. Math. 756, 151–182 (2019)
Vogel, T.: Existence of Engel structures. Ann. Math. 169, 79–137 (2009)
Acknowledgements
The author is indebted to Shantanu Dave for countless encouraging discussions. He thanks an anonymous referee for numerous helpful comments. The author gratefully acknowledges the support of the Austrian Science Fund (FWF): P31663-N35 and Y963-N35.
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Haller, S. Analytic Torsion of Generic Rank Two Distributions in Dimension Five. J Geom Anal 32, 248 (2022). https://doi.org/10.1007/s12220-022-00987-z
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DOI: https://doi.org/10.1007/s12220-022-00987-z
Keywords
- Analytic torsion
- Rumin complex
- Rockland complex
- Generic rank two distribution
- (2,3,5) distribution
- Sub-Riemannian geometry