New Mexico Tech (October 30, 2023)

Spectral Asymptotics

of Elliptic Operators

on Manifolds

Ivan G. Avramidi

Department of Mathematics

New Mexico Institute of Mining and Technology

Socorro, NM 87801, USA

E-mail<EMAIL_ADDRESS>

The study of spectral properties of natural geometric elliptic partial
differential operators acting on smooth sections of vector bundles over
Riemannian manifolds is a central theme in global analysis, differential
geometry and mathematical physics. Instead of studying the spectrum of a
differential operator $L$ directly one usually studies its spectral functions,
that is, spectral traces of some functions of the operator, such as the
spectral zeta function $\zeta(s)=\mathrm{Tr\,}L^{-s}$ and the heat trace
$\Theta(t)=\mathrm{Tr\,}\exp(-tL)$. The kernel $U(t;x,x^{\prime})$ of the heat
semigroup $\exp(-tL)$, called the heat kernel, plays a major role in quantum
field theory and quantum gravity, index theorems, non-commutative geometry,
integrable systems and financial mathematics. We review some recent progress
in the study of spectral asymptotics. We study more general spectral
functions, such as $\mathrm{Tr\,}f(tL)$, that we call quantum heat traces.
Also, we define new invariants of differential operators that depend not only
on the their eigenvalues but also on the eigenfunctions, and, therefore,
contain much more information about the geometry of the manifold. Furthermore,
we study some new invariants, such as
$\mathrm{Tr\,}\exp(-tL_{+})\exp(-sL_{-})$, that contain relative spectral
information of two differential operators. Finally we show how the convolution
of the semigroups of two different operators can be computed by using purely
algebraic methods.

##  1 Introduction

The study of spectral properties of natural geometric partial differential
operators is a central theme in global analysis, differential geometry and
mathematical physics. In particular, the basic question of spectral geometry
is: “To what extent does the spectrum of an elliptic partial differential
operator determine the geometry of the underlying manifold?”, or as M. Kac put
it in his famous paper [42]: “Can one hear the shape of a drum?” In general,
the answer to Kac’s question is “no” [54, 53]. Instead of studying the
spectrum of a differential operator directly one usually studies its spectral
functions, that is, spectral traces of some functions of the operator, such as
the spectral zeta function and the heat trace. The heat trace is the trace of
the heat kernel, which is the fundamental solution of the heat equation for an
elliptic partial differential operator with a positive leading symbol (see,
e.g. [35, 56, 24, 40, 43, 38, 30, 3, 10, 12]). The existence of non-isometric
isospectral manifolds demonstrates that the spectrum alone does not determine
the geometry. That is why, it makes sense to study more general invariants of
partial differential operators, maybe even such invariants that are not
spectral invariants, that is, invariants that depend not only on the
eigenvalues but also on the eigenfunctions, and, therefore, contain much more
information about the geometry of the manifold [19, 13, 14, 15].

Another motivation to study the heat kernel comes from quantum field theory
and statistical physics. The main objects of interest are the effective action
(or the partition function) and the Green functions (or the correlation
functions) [44, 31, 25, 57, 47]. All these objects are expressed in terms of
the functional determinants of some self-adjoint elliptic partial differential
operators, their heat traces and their resolvents. It turns out that all of
them can be expressed in terms of the heat kernels of those operators.

In financial mathematics one uses stochastic differential equations to model
the random behavior of some financial assets. Then the behavior of the
corresponding derivative securities is determined by deterministic parabolic
partial differential equation (such as the diffusion or heat equation) with an
elliptic partial differential operator of second order. The conditional
probability density is then nothing else but the heat kernel (for a brief
introduction see [12] with more references therein).

Many problems in mathematics and physics naturally lead to the presence of
boundaries and to the corresponding boundary value problems for partial
differential operators. The type of the boundary conditions is not limited to
the classical Dirichlet and Neumann ones. In some applications, such as
quantum gravity and applied mathematics, there appear mixed boundary
conditions on vector bundles (that mix the Dirichlet and the Neumann one),
oblique boundary conditions (that involve the tangential derivatives to the
boundary) [20, 21, 8], and even discontinuous boundary conditions, so called
Zaremba boundary conditions, (that jump from Dirichlet to Neumann across a co-
dimension $2$ submanifold in the boundary) [6, 52].

Most natural elliptic partial differential operators are second order
operators with scalar leading symbols, so called Laplace type operators, or
first order operators whose square is a Laplace type operator, so called Dirac
type operators. However, in some applications, such as in gauge field theories
and quantum gravity, there appear second-order elliptic partial differential
operators with non-scalar leading symbols, so called non-Laplace type
operators [18, 8]. Another motivation for studying such operators is non-
commutative geometry and matrix general relativity where the metric tensor,
that is, the inner product in the tangent bundle, is endomorphism valued in
some vector bundle [4, 5, 7, 8].

In the generic situation when it is impossible to compute the heat kernel
exactly, it becomes very important to study highly symmetric spaces such as
Lie groups and symmetric spaces [29, 40, 2, 16, 9, 17] or various asymptotic
regimes. Of special interest is the study of the short-time asymptotic
expansion of the heat kernel [34, 35, 24, 30, 1, 10]. This expansion is
closely related to the semi-classical expansion in quantum theory and the
high-temperature expansion in statistical physics [31, 25]. The coefficients
of this expansion, called the heat invariants, are spectral invariants
associated with the asymptotic properties of the spectrum. There are also non-
trivial links between spectral invariants and non-linear completely integrable
systems, such as the Korteweg-de Vries hierarchy [48, 46, 45, 32, 40, 41, 23,
11]. In many interesting cases such systems are, in fact, infinite-dimensional
Hamiltonian systems, and the infinite set of integrals of motion of these
systems is related to the spectral invariants of a linear elliptic partial
differential operator (see [32, 11]).

##  2 Heat Kernel

###  2.1 Elliptic Operators

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ equipped
with a positive definite Riemannian metric $g$. We denote the local
coordinates by $x^{\mu}$, with Greek indices running over $1,...,n$. The
Riemannian volume element is defined as usual by
$d\mathrm{vol\,}=dx\;g^{1/2}$, where $g=\det g_{\mu\nu}$ and
$dx=dx^{1}\wedge\dots\wedge dx^{n}$ is the standard Lebesgue measure.

Let ${\cal V}$ be a smooth vector bundle over $M$ with the typical fiber $V$
of dimension $N$. Let $C^{\infty}({\cal V})$ be the space of smooth sections
of the bundle ${\cal V}$. The completion of the space $C^{\infty}({\cal V})$
defines the Hilbert space $L^{2}({\cal V})$ of square integrable sections. Let

$\nabla^{\cal V}:C^{\infty}({\cal V})\to C^{\infty}(T^{*}M\otimes{\cal V})$
(2.1)

be a compatible connection on the vector bundle ${\cal V}$. By using the Levi-
Civita connection of the metric $g$ the connection is given its unique natural
extension to bundles in the tensor algebra over ${\cal V}$, its dual ${\cal
V}^{*}$ and the tangent and cotangent bundles, $TM$ and $T^{*}M$; the
resulting connection will usually be denoted just by $\nabla$. The fiber inner
product $\langle\;,\;\rangle$ defines a natural $L^{2}$ inner product,
$(\;,\;)$, on the bundle ${\cal V}$ and the $L^{2}$-trace, $\mathrm{Tr\,}$,
using the invariant Riemannian measure on the manifold $M$.

Let

$\nabla^{*}:C^{\infty}(TM\otimes{\cal V})\to C^{\infty}({\cal V})$ (2.2)

be the formal adjoint of the connection. Let $a$, $B$ and $Q$ be smooth maps

$\displaystyle a:T^{*}M\otimes{\cal V}\to TM\otimes{\cal V},$ (2.3)
$\displaystyle B:T^{*}M\otimes{\cal V}\to{\cal V},$ (2.4) $\displaystyle
Q:{\cal V}\to{\cal V}.$ (2.5)

defined by endomorphism-valued tensors satisfying

$a^{\mu\nu}=a^{\nu\mu},$ (2.6)
$(a^{\mu\nu})^{*}=a^{\mu\nu},\qquad(B^{\mu})^{*}=-B^{\mu},\qquad Q^{*}=Q$
(2.7)

Every formally self-adjoint second-order partial differential operator

$L:C^{\infty}({\cal V})\to C^{\infty}({\cal V})$ (2.8)

has the form

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\nabla_{\mu}a^{\mu\nu}\nabla_{\nu}+B^{\mu}\nabla_{\mu}+\nabla_{\mu}B^{\mu}+Q.$
(2.9)

Natural non-Laplace type operators can be constructed as follows. Let

$\Gamma:T^{*}M\otimes{\cal V}\to{\cal V}$ (2.10)

be a smooth map; this defines the Dirac type operator

$D=\Gamma\nabla:C^{\infty}({\cal V})\to C^{\infty}({\cal V})$ (2.11)

and the operator

$L=D^{*}D:C^{\infty}({\cal V})\to C^{\infty}({\cal V});$ (2.12)

in this case

$a^{\mu\nu}=\frac{1}{2}\left(\Gamma^{\mu\;*}\Gamma^{\nu}+\Gamma^{\nu\;*}\Gamma^{\mu}\right).$
(2.13)

More generally, let ${\cal V}_{j}$, $j=1,\dots,s$, be some vector bundles and

$P_{j}:T^{*}M\otimes{\cal V}\to{\cal V}_{j}$ (2.14)

be some smooth maps. This defines the gradients

$G_{j}=P_{j}\nabla:C^{\infty}({\cal V})\to C^{\infty}({\cal V}_{j}),$ (2.15)

and the operator

$L=\sum_{j=1}^{s}\alpha_{j}G_{j}^{*}G_{j}:C^{\infty}({\cal V})\to
C^{\infty}({\cal V}),$ (2.16)

where $\alpha_{j}$ are some real constants. In this case

$a^{\mu\nu}=\sum_{j=1}^{s}\alpha_{j}\frac{1}{2}\left(P_{j}^{\mu\;*}P_{j}^{\nu}+P_{j}^{\nu\;*}P_{j}^{\mu}\right).$
(2.17)

The leading symbol of the operator $L$ is given by the endomorphism

$H(x,\xi)=a^{\mu\nu}(x)\xi_{\mu}\xi_{\nu},$ (2.18)

with $x\in M$ and $\xi\in T_{x}^{*}M$. Since this matrix is self-adjoint all
its eigenvalues must be real. The operator $L$ is elliptic if all eigenvalues
are positive, in other words, the matrix $H(x,\xi)$ is positive. In the case
when the manifold $M$ is closed, that is, compact without boundary, the
operator $L$ is also self-adjoint; strictly speaking, it is essentially self-
adjoint, that is, it has a unique self-adjoint extension (from now on, we will
just say that the operator $L$ is self-adjoint).

If all eigenvalues of the leading symbol of the operator $L$ are equal, that
is, the operator $L$ has a scalar positive definite leading symbol defined by
the Riemannian metric

$H(x,\xi)=|\xi|^{2}I,$ (2.19)

with $|\xi|^{2}=g^{\mu\nu}(x)\xi_{\mu}\xi_{\nu}$, then the operator is called
of Laplace type. In this case the connection can be redefined to absorb the
vector $B^{\mu}$, so that every Laplace type operator has the form

$L=-\Delta+Q,$ (2.20)

where

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ (2.21) $\displaystyle=$ $\displaystyle
g^{-1/2}(\partial_{\mu}+{\cal A}_{\mu})g^{1/2}g^{\mu\nu}(\partial_{\nu}+{\cal
A}_{\nu})$

is the Laplacian. Here and below $\partial_{\mu}=\partial/\partial x^{\mu}$
denotes the partial derivative. A Laplace type operator is defined in terms of
three pieces of local information: the Riemannian metric $g$, the connection
one-form ${\cal A}$ and the potential $Q$.

###  2.2 Heat Kernel

Let $L$ be a self-adjoint elliptic positive partial differential operator of
second order on the Hilbert space $L^{2}({\cal V})$. For manifolds with
boundary the domain of the operator $L$ has to be supplemented with some
suitable elliptic boundary conditions. For compact manifolds the spectrum of
the operator $L$ is an increasing real sequence of eigenvalues
$\\{\lambda_{k}\\}_{k\in{\mathbb{Z}}_{+}}$ with the corresponding orthonormal
eigensections $\\{\varphi_{k}\\}_{k\in{\mathbb{Z}}_{+}}$ (counted with
multiplicities) defined by [35, 24]

$(L-\lambda_{k})\varphi_{k}=0,\qquad(\varphi_{i},\varphi_{j})_{L^{2}}=\delta_{ij}.$
(2.22)

For noncompact manifolds the spectrum of the operator $L$ is continuous, it
goes from a positive real constant $c$ to $\infty$. In general, it is
impossible to compute the spectrum exactly. That is why, it becomes of special
importance the study of the asymptotics of the eigenvalues (and the
eigensections) as $k\to\infty$. Rather than doing this directly it is more
convenient to study the asymptotics of some spectral functions and special
traces such as the heat trace and the zeta function [35, 24, 43].

Let ${\cal V}\boxtimes{\cal V}^{*}$ be the external tensor product of the
bundles ${\cal V}$ and ${\cal V}^{*}$ over the product manifold $M\times M$.
The heat kernel $U(t;x,x^{\prime})$ of the operator $L$ is a one-parameter
family of smooth sections of ${\cal V}\boxtimes{\cal V}^{*}$ defined by
requiring it to satisfy the heat equation

$(\partial_{t}+L)U(t;x,x^{\prime})=0$ (2.23)

for $t>0$ with the initial condition

$U(0^{+};x,x^{\prime})=\delta(x,x^{\prime})\,,$ (2.24)

that is,

$U(t;x,x^{\prime})=\exp(-tL)\delta(x,x^{\prime})\,,$ (2.25)

where $\delta(x,x^{\prime})$ is the covariant Dirac delta-distribution.

The heat kernel is regular at the diagonal with a well defined diagonal value
[35]

$U^{\rm diag}(t;x)=U(t;x,x),$ (2.26)

which is a section of the endomorphism bundle $\mathrm{End}({\cal V})$. The
heat kernel diagonal is well defined on both compact and noncompact manifolds.
For compact manifolds the heat kernel can be computed in terms of the spectral
data

$\displaystyle U(t;x,x^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}e^{-t\lambda_{k}}\varphi_{k}(x)\varphi^{*}_{k}(x^{\prime}).$
(2.27)

and has a well defined heat trace $\Theta(t)=\mathrm{Tr\,}\exp(-tL)$,

$\displaystyle\Theta(t)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}e^{-t\lambda_{k}}$ (2.28) $\displaystyle=$
$\displaystyle\int\limits_{M}d\mathrm{vol\,}\;\mathrm{tr\,}U^{\rm diag}(t);$

here and below $\mathrm{tr\,}$ denotes the fiber trace. This also enables one
to define the spectral zeta function by the Mellin-Laplace transform of the
heat trace

$\zeta(s,\lambda)=\sum_{k=1}^{\infty}\frac{1}{(\lambda_{k}-\lambda)^{s}}=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}t^{s-1}e^{t\lambda}\Theta(t),$
(2.29)

where $\lambda$ is a complex parameter with a sufficiently large negative real
part, and the regularized spectral determinant [35, 43]

$\mathrm{Det\,}(L-\lambda)=\exp\left\\{-\partial_{s}\zeta(s,\lambda)\big{|}_{s=0}\right\\}.$
(2.30)

The resolvent $G(\lambda;x,x^{\prime})$ of the operator $L$ is a section of
${\cal V}\boxtimes{\cal V}^{*}$ depending on a complex parameter $\lambda$
defined by

$(L-\lambda)G(\lambda;x,x^{\prime})=\delta(x,x^{\prime}).$ (2.31)

The resolvent is related to the heat kernel by the Laplace transform

$\displaystyle G(\lambda;x,x^{\prime})$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}dt\;e^{t\lambda}U(t;x,x^{\prime}).$
(2.32)

For compact manifolds there is the spectral representation of the resolvent

$\displaystyle G(\lambda;x,x^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{\lambda_{k}-\lambda}\varphi_{k}(x)\varphi^{*}_{k}(x^{\prime}).$
(2.33)

Off diagonal, that is, for $x\neq x^{\prime}$, the resolvent is an analytic
function of $\lambda$ for $\mathrm{Re\,}\lambda<c$ with sufficiently large
negative real constant $c$. It has a diagonal singularity as $x\to
x^{\prime}$.

On a general manifold the heat kernel cannot be computed exactly, and that is
why the short-time asymptotic expansion as $t\to 0$ of the heat kernel and its
heat trace is studied. For Laplace type operators it has a rather simple form.
Let $x^{\prime}$ be a fixed point in the interior of the manifold; we consider
a sufficiently small geodesic ball centered at $x^{\prime}$, so that each
point $x$ of the ball can be connected by a unique geodesic with the point
$x^{\prime}$. This can be always done if the size of the ball is smaller than
the injectivity radius of the manifold $r_{\rm inj}(M)$.

Let $\sigma=\sigma(x,x^{\prime})$ be the Ruse-Synge function defined by [50,
55]

$\sigma(x,x^{\prime})=\frac{1}{2}r^{2}(x,x^{\prime}),$ (2.34)

where $r(x,x^{\prime})$ is the geodesic distance between the points $x$ and
$x^{\prime}$, and $\Delta(x,x^{\prime})$ be the Van Vleck-Morette determinant

$\Delta(x,x^{\prime})=g^{-1/2}(x)\det\left\\{-\frac{\partial^{2}\sigma(x,x^{\prime})}{\partial
x^{\mu}\partial x^{\nu^{\prime}}}\right\\}g^{-1/2}(x^{\prime}).$ (2.35)

Near the diagonal of $M\times M$ these two-point functions are smooth single-
valued functions of the coordinates of the points $x$ and $x^{\prime}$ [55, 1,
3, 10].

In the interior of the manifold (on a finite distance from the boundary, if
present) there is a local short time asymptotic expansion of the heat kernel
of a Laplace type operator as $t\to 0^{+}$ [35, 31, 3]

$U(t;x,x^{\prime})\sim(4\pi
t)^{-n/2}\Delta^{1/2}(x,x^{\prime})\exp\left\\{-\frac{\sigma(x,x^{\prime})}{2t}\right\\}\sum_{k=0}^{\infty}t^{k}a_{2k}(x,x^{\prime}),$
(2.36)

where $a_{k}(x,x^{\prime})$ are the so-called local off-diagonal heat kernel
coefficients. Notice that there are no odd-order coefficients here, that is,
$a_{2k+1}(x,x^{\prime})=0$. The heat kernel coefficients can be computed in
form of a covariant Taylor series from the recurence relations obtained by
substituting this ansatz in the heat equation [1] with the initial condition

$a_{0}(x,x^{\prime})={\cal P}(x,x^{\prime}),$ (2.37)

where ${\cal P}(x,x^{\prime})$ is the operator of parallel transport of
sections along the geodesic from the point $x^{\prime}$ to the point $x$ (for
details, see [1, 3]).

This expansion immediately gives the asymptotic expansion as $t\to 0^{+}$ of
the heat kernel diagonal

$U^{\rm diag}(t;x)\sim(4\pi t)^{-n/2}\sum_{k=0}^{\infty}t^{k}a^{\rm
diag}_{2k}(x,x),$ (2.38)

where $a_{k}^{\rm diag}(x)=a_{k}(x,x)$ are the diagonal local heat kernel
coefficients.

For manifolds without boundary this also gives the asymptotic expansion of the
heat trace [37, 51, 35]

$\Theta(t)\sim(4\pi t)^{-n/2}\sum_{k=0}^{\infty}t^{k}A_{2k},$ (2.39)

where

$A_{2k}=\int\limits_{M}d\mathrm{vol\,}\;\mathrm{tr\,}a^{\rm diag}_{2k}$ (2.40)

are the global heat kernel coefficients. The heat trace, as well as the global
heat kernel coefficients, are obviously spectral invariants of the operator
$L$. These coefficients were computed up to $a_{8}^{\rm diag}$ (see [34, 1,
3]). The low-order asymptotics of the heat trace of a Laplace type operator
has the form

$\displaystyle\Theta(t)$ $\displaystyle=$ $\displaystyle(4\pi
t)^{-n/2}\Biggl{\\{}N\mathrm{vol\,}(M)+t\int\limits_{M}d\mathrm{vol\,}\left(\frac{N}{6}R-\mathrm{tr\,}Q\right)+O\left(t^{2}\right)\Biggr{\\}},$
(2.41)

where $R$ is the scalar curvature.

##  3 Boundary Value Problems

###  3.1 Mixed Boundary Value Problem

Now, let $M$ be a compact Riemannian manifold with smooth boundary $\partial
M$. We denote the local coordinates on the boundary $\partial M$ by $x^{i}$,
with Latin indices running over $1,...,(n-1)$. The Riemannian volume element
is defined as usual by $d\mathrm{vol\,}=d\hat{x}\;\hat{g}^{1/2}$, where
$d\hat{x}=d\hat{x}^{1}\wedge\dots\wedge d\hat{x}^{n-1}$ is the standard
Lebesgue measure and $\hat{g}=\det\hat{g}_{ij}$, with $\hat{g}_{ij}$ the
induced metric on the boundary.

Let $N$ be the inward-pointing unit normal vector field to the boundary
$\partial M$. To make the operator $L$ elliptic we need to impose some
boundary conditions on the boundary data. The classical boundary conditions
are

$\displaystyle\varphi\big{|}_{\partial M}$ $\displaystyle=$ $\displaystyle
0,\qquad\mbox{(Dirichlet)},$ (3.1)
$\displaystyle\nabla_{N}\varphi\big{|}_{\partial M}$ $\displaystyle=$
$\displaystyle 0,\qquad\mbox{(Neumann)}.$ (3.2)

The Robin boundary condition is a slight generalization of the Neumann one

$\left(\nabla_{N}+S\right)\varphi\Big{|}_{\partial M}=0,\qquad\mbox{(Robin)}.$
(3.3)

where $S$ is a smooth self-adjoint endomorphism. One can go further and mix
these boundary conditions as follows

$\displaystyle(I-\Pi)\varphi\big{|}_{\partial M}$ $\displaystyle=$
$\displaystyle 0\,,\qquad$ (3.4)
$\displaystyle\Pi\left(\nabla_{N}+S\right)\Pi\varphi\big{|}_{\partial M}$
$\displaystyle=$ $\displaystyle 0,$ (3.5)

where $\Pi$ is a self-adjoint projection. A Laplace type operator $L$ equipped
with mixed boundary conditions is essentially self-adjoint and elliptic [35,
28, 21, 6].

The heat kernel asymptotics as $t\to 0$ does not depend on the boundary
conditions in the interior of the manifold and has the same form as the heat
kernel for manifolds without boundary. Near the boundary there is a narrow
strip of the width of order $t^{1/2}$ where one should add an additional term,
whose role is to satisfy the boundary conditions. In a narrow strip near the
boundary this compensating term in the heat kernel diagonal behaves as a
distribution near the boundary as $t\to 0$ [6]. It is precisely this feature
that leads to the presence of boundary terms in the global heat kernel
coefficients. The heat trace asymptotic expansion as $t\to 0^{+}$ has the
half-integer powers of $t$, [37, 51, 27, 35, 56, 6]

$\Theta(t)\sim(4\pi t)^{-n/2}\sum_{k=0}^{\infty}t^{k/2}A_{k};$ (3.6)

here

$A_{k}=\int\limits_{M}d\mathrm{vol\,}\;\mathrm{tr\,}\,a_{k}^{\rm
diag}+\int\limits_{\partial M}d\mathrm{vol\,}b_{k}\,,$ (3.7)

where the boundary heat kernel coefficients $b_{k}$ are local invariants
constructed polynomially from the jets of the symbols of both the operator $L$
and the boundary operator.

The low-order asymptotics have the form [27, 28, 6]

$\displaystyle\Theta(t)$ $\displaystyle=$ $\displaystyle(4\pi
t)^{-n/2}\Biggl{\\{}N\mathrm{vol\,}(M)+t^{1/2}\,\int\limits_{\partial
M}d\mathrm{vol\,}\frac{\sqrt{\pi}}{2}\left(2\mathrm{tr\,}\Pi-N\right)$
$\displaystyle+t\Biggl{[}\int\limits_{M}d\mathrm{vol\,}\left(\frac{N}{6}R-\mathrm{tr\,}Q\right)+\int\limits_{\partial
M}d\mathrm{vol\,}\left(\frac{N}{3}K+2\mathrm{tr\,}\Pi
S\right)\Biggr{]}+O\left(t^{3/2}\right)\Biggr{\\}},$

where $K$ is the trace of the extrinsic curvature of the boundary.

###  3.2 Grubb-Gilkey-Smith Boundary Value Problem

The mixed boundary conditions described above contain only normal derivative.
More general boundary conditions contain also tangential derivatives along the
boundary [36, 52, 20, 21]. Let $\hat{\nabla}_{i}$ be the tangential covariant
derivative, $\Gamma^{i}$ be a vector-valued anti-self-adjoint endomorphism and
$\Lambda$ be a first-order formally self-adjoint tangential differential
operator defined by

$\Lambda=\frac{1}{2}\left(\Gamma^{i}\hat{\nabla}_{i}+\hat{\nabla}_{i}\Gamma^{i}\right)+S\,.$
(3.9)

The Grubb-Gilkey-Smith boundary conditions (also called oblique boundary
conditions) then read [36]

$\displaystyle(I-\Pi)\varphi\Big{|}_{\partial M}$ $\displaystyle=$
$\displaystyle 0\,,\qquad$ (3.10)
$\displaystyle\Pi\left(\nabla_{N}+\Lambda\right)\Pi\varphi\Big{|}_{\partial
M}$ $\displaystyle=$ $\displaystyle 0.$ (3.11)

A Laplace type operator $L$ equipped with such boundary conditions is
essentially self-adjoint but not necessarily elliptic. To be elliptic the
operator $\Lambda$ has to satisfy the strong ellipticity condition [36, 26,
39, 35]. Let $T$ be a matrix defined by the leading symbol of the operator
$\Lambda$,

$T(\hat{\xi})=\Gamma^{j}\hat{\xi}_{j},$ (3.12)

where $\hat{\xi}\in T^{*}\partial M$ is a covector on the boundary. Since the
matrices $\Gamma^{i}$ are anti-self-adjoint, the matrix $T^{2}(\xi)$ is self-
adjoint and negative, $T^{2}(\hat{\xi})<0$, for any $\hat{\xi}\neq 0$. Then
the oblique boundary value problem is elliptic if the matrix

$I|\hat{\xi}|^{2}+T^{2}(\hat{\xi})>0$ (3.13)

is positive for any $\hat{\xi}\neq 0$. Here
$|\hat{\xi}|^{2}=\hat{g}^{ij}\hat{\xi}_{i}\hat{\xi}_{j}$ is defined with the
metric $\hat{g}^{ij}$ on the boundary.

When the boundary value problem is elliptic the heat trace asymptotic
expansion has the canonical form (3.6). Contrary to the classical boundary
value problems (Dirichlet, Neumann, mixed), because of the non-commutativity
of the matrices $\Gamma^{i}$, the explicit form of the coefficients $b_{k}$ is
unknown, in general; the low-order asymptotic expansion has the form [21, 20]

$\Theta(t)=(4\pi
t)^{-n/2}\left\\{N\mathrm{vol\,}(M)+t^{1/2}\,\int\limits_{\partial
M}d\mathrm{vol\,}{\sqrt{\pi}\over
2}\Bigl{(}2\mathrm{tr\,}\Pi-3N+2\gamma\Bigr{)}+O(t)\right\\},$ (3.14)

where

$\gamma=\int\limits_{{{\mathbb{R}}}^{n-1}}{d\hat{\xi}\,\over\pi^{(n-1)/2}}\,\mathrm{tr\,}\exp\left(-|\hat{\xi}|^{2}-T^{2}(\xi)\right).$
(3.15)

This can be computed explicitly in special cases. If the matrices $\Gamma^{i}$
commute then

$\gamma=\mathrm{tr\,}\left(I+\Gamma^{2}\right)^{-1/2},$ (3.16)

where $\Gamma^{2}=\hat{g}_{ij}\Gamma^{i}\Gamma^{j}$. One can also compute the
coefficient $\gamma$ explicitly in the non-commutative case when the matrices
$\Gamma^{i}$ form a Clifford algebra

$\Gamma^{i}\Gamma^{j}+\Gamma^{j}\Gamma^{i}=-2\varkappa\Pi\hat{g}^{ij}$ (3.17)

with a real parameter $\varkappa$. This problem is elliptic if $\varkappa<1$
and we obtain

$\gamma=(1-\varkappa)^{-(n-1)/2}\mathrm{tr\,}\Pi.$ (3.18)

###  3.3 Zaremba Boundary Value Problem

Let the boundary $\partial M$ be decomposed as a disjoint union $\partial
M=\Sigma_{1}\cup\Sigma_{2}\cup\Sigma_{0}$, where $\Sigma_{1}$ and $\Sigma_{2}$
are smooth compact submanifolds of $\partial M$ of dimension $(n-1)$, with the
boundary $\Sigma_{0}$ which is a smooth compact manifold without boundary of
dimension $(n-2)$. Zaremba boundary value problem is defined by the following
boundary conditions [6, 52]

$\displaystyle\varphi\big{|}_{\Sigma_{1}}$ $\displaystyle=$ $\displaystyle
0,\qquad$ (3.19) $\displaystyle(\nabla_{N}+S)\varphi\big{|}_{\Sigma_{2}}$
$\displaystyle=$ $\displaystyle 0\,,$ (3.20)

where $S$ is a smooth self-adjoint endomorphism.

Since the boundary operator is discontinuous, this problem is a singular
boundary value problem. It is well known that in this case the heat trace
asymptotic expansion as $t\to 0^{+}$ contains, in general, logarithmic terms
[39, 33]

$\Theta(t)\sim(4\pi t)^{-n/2}\sum_{k=0}^{\infty}t^{k/2}A_{k}+\log
t\sum_{k=0}^{\infty}t^{k/2}H_{k}\,.$ (3.21)

However, Seeley [52] has shown that for Zaremba problem the logarithmic terms
do not appear, that is, all

$H_{k}=0.$ (3.22)

Therefore, the heat trace asymptotics has the canonical form (3.6). However,
the global heat kernel coefficients get a contribution from the codimension 2
submanifold $\Sigma_{0}$ [6, 52]

$A_{k}=\int\limits_{M}d\mathrm{vol\,}\mathrm{tr\,}\,a_{k}^{\rm
diag}+\int\limits_{\Sigma_{1}}d\mathrm{vol\,}b_{k}^{(1)}+\int\limits_{\Sigma_{2}}d\mathrm{vol\,}b_{k}^{(2)}+\int\limits_{\Sigma_{0}}d\mathrm{vol\,}c_{k}\,.$
(3.23)

It turns out [6, 52] that the boundary conditions on the open sets
$\Sigma_{1}$ and $\Sigma_{2}$ are not enough to fix the problem, and an
additional boundary condition along the singular set $\Sigma_{0}$ is needed.
This additional boundary condition can be considered formally as an extension
of Dirichlet conditions from $\Sigma_{1}$ to $\Sigma_{0}$, (regular boundary
condition)

$\left(\sqrt{\rho}\varphi\right)\Big{|}_{\Sigma_{0}}=0,$ (3.24)

where $\rho$ is the normal geodesic distance to the singular set $\Sigma_{0}$,
or an extension of Neumann (or Robin) conditions from $\Sigma_{2}$ to
$\Sigma_{0}$,

$(\partial_{\rho}-h)\left(\sqrt{\rho}\varphi\right)\Big{|}_{\Sigma_{0}}=0\,,$
(3.25)

where $h$ is a real parameter. However, strictly speaking the boundary
condition on $\Sigma_{0}$ does not follow from the boundary conditions on
$\Sigma_{1}$ and $\Sigma_{2}$ and can be chosen rather arbitrarily.

The coefficients of the asymptotic expansion can be computed by constructing
asymptotic solutions [6]:

  1. 1.

in the interior,

  2. 2.

in a thin shell near the $\Sigma_{1}$ and $\Sigma_{2}$, and

  3. 3.

in a thin strip close to the singular submanifold $\Sigma_{0}$.

The trace of the heat kernel of the Zaremba boundary value problem has the
following asymptotic expansion as $t\to 0^{+}$

$\displaystyle\Theta(t)$ $\displaystyle=$ $\displaystyle(4\pi
t)^{-n/2}\Biggl{\\{}N\mathrm{vol\,}(M)+t^{1/2}{\sqrt{\pi}\over
2}N\left[\mathrm{vol\,}(\Sigma_{2})-\mathrm{vol\,}(\Sigma_{1})\right]$ (3.26)
$\displaystyle+t\Biggl{[}\int\limits_{M}d\mathrm{vol\,}\left({N\over
6}R-\mathrm{tr\,}Q\right)+{N\over
3}\int\limits_{\Sigma_{1}}d\mathrm{vol\,}K+\int\limits_{\Sigma_{2}}d\mathrm{vol\,}\left({N\over
3}K+2\mathrm{tr\,}S\right)\Biggr{]}$ $\displaystyle+\alpha\,{\pi\over
4}N\mathrm{vol\,}(\Sigma_{0})+O\left(t^{3/2}\right)\Biggr{\\}}\,,$

where $\alpha=-1$ for the boundary condition (3.24) and $\alpha=7$ for the
boundary condition (3.25) (for the details of the calculation see [6]).

##  4 Non-Laplace Type Operators

We consider a non-Laplace type operator of the form

$L=-\nabla_{\mu}a^{\mu\nu}\nabla_{\nu}+Q,$ (4.1)

where $a^{\mu\nu}$ is an endomorphism-valued symmetric tensor. If we assume,
in addition, that the leading symbol of the operator $L$ is positive then the
operator $L$ is self-adjoint and elliptic and, therefore, has the same
canonical heat trace expansion of the form (3.6)

$\Theta(t)\sim(4\pi t)^{-n/2}\sum_{k=0}^{\infty}t^{k/2}A_{k}.$ (4.2)

For manifolds without boundary one can get the asymptotic expansion of the
heat kernel as follows [18]. Let $\xi\in T^{*}M$ be a covector and
$\langle\xi,x\rangle=\xi_{\mu}x^{\mu}$. It is easy to see that

$\exp\left(-i\langle\xi,x\rangle\right)L\exp\left(i\langle\xi,x\rangle\right)=H+K+L,$
(4.3)

where

$H(x,\xi)=a^{\mu\nu}(x)\xi_{\mu}\xi_{\nu}$ (4.4)

is the leading symbol of the operator $L$ and $K$ is the first order operator
defined by

$K=-i\xi_{\mu}\left(a^{\mu\nu}\nabla_{\nu}+\nabla_{\nu}a^{\mu\nu}\right).$
(4.5)

Then the asymptotics of the heat trace as $t\to 0$ are given by

$\Theta(t)\sim(4\pi
t)^{-n/2}\int\limits_{M}dx\int\limits_{{\mathbb{R}}^{n}}{d\xi\over\pi^{n/2}}\mathrm{tr\,}\exp\left(-H-\sqrt{t}K-tL\right)\cdot
I\,,$ (4.6)

By using the Volterra series for the heat semigroup we get

$\displaystyle\exp\left(-H-\sqrt{t}K-tL\right)=e^{-H}-t^{1/2}\int\limits_{0}^{1}d\tau_{1}e^{-(1-\tau_{1})H}Ke^{-\tau_{1}H}$
$\displaystyle\qquad+\,t\Biggl{[}\int\limits_{0}^{1}d\tau_{2}\int\limits_{0}^{\tau_{2}}d\tau_{1}e^{-(1-\tau_{2})H}Ke^{-(\tau_{2}-\tau_{1})H}Ke^{-\tau_{1}H}-$
$\displaystyle\qquad\hphantom{+\,t\Biggl{[}}-\int\limits_{0}^{1}d\tau_{1}e^{-(1-\tau_{1})H}Le^{-\tau_{1}H}\Biggr{]}+O(t^{2})\,.$
(4.7)

Since $K$ is linear in $\xi$, the term proportional to $t^{1/2}$ vanishes
after integration over $\xi$ and we obtain the first two coefficients of the
asymptotic expansion of the heat kernel diagonal in the form [18]

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\int\limits_{M}dx\int\limits_{{\mathbb{R}}^{n}}\frac{d\xi}{\pi^{n/2}}\mathrm{tr\,}\exp\left[-H(x,\xi)\right],$
(4.8) $\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 0,$ (4.9)
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\int\limits_{M}dx\,\int\limits_{{\mathbb{R}}^{n}}{d\xi\over\pi^{n/2}}\,\mathrm{tr\,}\,\Biggl{[}\int\limits_{0}^{1}d\tau_{2}\int\limits_{0}^{\tau_{2}}d\tau_{1}e^{-(1-\tau_{2})H}Ke^{-(\tau_{2}-\tau_{1})H}Ke^{-\tau_{1}H}-$
(4.10)
$\displaystyle\hphantom{\int\limits_{M}dx\,\int\limits_{{\mathbb{R}}^{n}}{d\xi\over\pi^{n/2}}\,\mathrm{tr\,}_{V}\,\Biggl{[}}-\int\limits_{0}^{1}d\tau_{1}e^{-(1-\tau_{1})H}Le^{-\tau_{1}H}\Biggr{]}\,.$

Since there is no Riemannian metric, the spectral invariants of a non-Laplace
type operator are not expressed in terms of the invariants of the curvature.
It is very important to develop the corresponding differential-geometric
language based on the non-scalar leading symbol of a non-Laplace type
operator.

Boundary value problems for non-Laplace type operators are even more
complicated. For the Dirichlet boundary value problem the asymptotics are
computed as follows (for details see [8]). Let $x=(r,\hat{x})$ be the local
coordinates near the boundary where $r$ is the normal geodesic distance to the
boundary and $\hat{x}^{i}$ are local coordinates on the boundary. We decompose
a covector $\xi\in T^{*}M$ as $\xi=(\omega,\hat{\xi})$ where $\omega$ is a
real number and $\hat{\xi}\in T^{*}\partial M$ is a covector on the boundary.
Let $\Phi(\lambda;\hat{x},\hat{\xi})$ be a function defined by

$\Phi(\lambda;\hat{x},\hat{\xi})=\int\limits_{\mathbb{R}}\frac{d\omega}{2\pi}\left\\{H(0,\hat{x},\omega,\hat{\xi})-\lambda
I\right\\}^{-1}$ (4.11)

and $\Psi(x,\xi)$ be a function defined by

$\Psi(x,\xi)=\int\limits_{c-i\infty}^{c+i\infty}\frac{d\lambda}{2\pi
i}e^{-\lambda}\frac{\partial}{\partial\lambda}\log\det\Phi(\lambda;\hat{x},\hat{\xi}),$
(4.12)

where $c$ is a sufficiently large positive real constant. Then the heat trace
boundary coefficient $A_{1}$ for the Dirichlet boundary conditions has the
form [8]

$A_{1}=-\sqrt{\pi}\int\limits_{\partial
M}d\hat{x}\int\limits_{{\mathbb{R}}^{n-1}}\frac{d\hat{\xi}}{\pi^{(n-1)/2}}\Psi(\hat{x},\hat{\xi}).$
(4.13)

##  5 Non-perturbative Spectral Asymptotics

Let $M$ be a compact Riemannian manifold without boundary and ${\cal V}$ be a
complex vector bundle over $M$ realizing a representation of the group
$G\times U(1)$. Let $\varphi$ be a section of the bundle ${\cal V}$ and
$\nabla$ be the total connection on the bundle $S$ (including the
$G$-connection as well as the $U(1)$-connection). Then the commutator of
covariant derivatives defines the curvatures

$[\nabla_{\mu},\nabla_{\nu}]\varphi=({\cal R}_{\mu\nu}+iF_{\mu\nu})\varphi,$
(5.1)

where ${\cal R}_{\mu\nu}$ is the curvature of the $G$-connection and
$F_{\mu\nu}$ is the curvature of the $U(1)$-connection.

We assume that the $U(1)$-connection is parallel, that is,

$\nabla_{\mu}F_{\alpha\beta}=0.$ (5.2)

This equation puts severe algebraic restriction on the curvature tensor

$R^{\lambda}{}_{\alpha\mu\nu}F_{\lambda\beta}-R^{\lambda}{}_{\beta\mu\nu}F_{\lambda\alpha}=0$
(5.3)

and, therefore, gives powerful restriction on the holonomy group of the
manifold. For example, $F$ could be a simplectic form of a Kähler manifold.

Let $U(t;x,x^{\prime})$ be the heat kernel of the Laplacian
$\Delta=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$. We rescale the curvature by

$F\mapsto\frac{1}{\varepsilon^{2}}F,$ (5.4)

where $\varepsilon>0$ is a small positive parameter; let
$\Delta_{\varepsilon}$ be the rescaled Laplacian and
$U_{\varepsilon}(\varepsilon t;x,x^{\prime})$ be the rescaled heat kernel.
Then as $\varepsilon\to 0$ there is the new asymptotic expansion of the off-
diagonal heat kernel [22]

$U_{\varepsilon}(\varepsilon^{2}t;x,x^{\prime})\sim
U_{0}(t;x,x^{\prime})\sum_{k=0}^{\infty}\varepsilon^{k-n}t^{k/2}b_{k}(t;x,x^{\prime})\,,$
(5.5)

where

$U_{0}(t;x,x^{\prime})=(4\pi
t)^{-n/2}\Delta^{1/2}(x,x^{\prime})\det\left(\frac{tiF}{\sinh(tiF)}\right)^{1/2}\exp\left(-\frac{1}{4t}\left<u,tiF\coth(tiF)u\right>\right),$
(5.6)

$F$ is the matrix $F=(F_{\mu\nu})$ and $u^{\mu}$ are normal coordinates with
origin at $x^{\prime}$. Here the coefficients $b_{k}(t;x,x^{\prime})$ are
analytic functions of $t$ that depend on $F$ only in the dimensionless
combination $tF$. Of course, for $t=0$ they are equal to the standard heat
kernel coefficients, that is,

$b_{k}(0;x,x)=a_{k}(x,x^{\prime}),$ (5.7)

and, therefore, the odd-order coefficients at $t=0$ vanish,
$b_{2k+1}(0;x,x^{\prime})=0.$ Moreover, the odd-order coefficients vanish also
for any $t$ on the diagonal $x=x^{\prime}$ [22], that is,

$b^{\rm diag}_{2k+1}(t)=0$ (5.8)

Then the asymptotic expansion of the heat kernel diagonal and the heat trace
are

$\mathrm{Tr\,}\exp(\varepsilon^{2}t\Delta_{\varepsilon})\sim(4\pi
t)^{-n/2}\sum_{k=0}^{\infty}\varepsilon^{k-n}t^{k/2}B_{k}(t)\,,$ (5.9)

where

$B_{k}(t)=\int\limits_{M}d\mathrm{vol\,}\det\left(\frac{tiF}{\sinh(tiF)}\right)^{1/2}\mathrm{tr\,}\;b^{\rm
diag}_{k}(t)\,.$ (5.10)

The coefficients $B_{k}(t)$ are new spectral invariants of the Laplacian. They
are differential polynomials in the Riemann curvature tensor (and the
curvature of the $G$-connection) and its derivatives with universal
coefficients depending in a non-polynomial but analytic way on the curvature
$F$, more precisely, on $tF$.

We explicitly computed the coefficients $b_{k}(t)$ (both off diagonal and the
diagonal values) for $k=0,1,2,3,4$. These functions generate all terms
quadratic and linear in the Riemann curvature and of arbitrary order in $F$ in
the standard heat kernel coefficients $a_{k}^{\rm diag}$. In that sense, we
effectively sum up the usual short time heat kernel asymptotic expansion to
all orders of the curvature $F$. The first two non-zero coefficients have the
form [22]

$\displaystyle b_{0}^{\rm diag}(t)$ $\displaystyle=$ $\displaystyle I,$ (5.11)
$\displaystyle b_{2}^{\rm diag}(t)$ $\displaystyle=$ $\displaystyle
J_{\alpha\beta\mu\nu}(t)R^{\alpha\beta\mu\nu}I+\frac{1}{2}H^{\mu\nu}(t){\cal
R}_{\mu\nu},$ (5.12)

where

$H(t)=\coth(tiF)-\frac{1}{itF}$ (5.13)

and $J_{\alpha\beta\mu\nu}(t)$ is a more complicated tensor constructed from
the matrix $F$ that is analytic in $t$ (for more details see [22])

$J^{\alpha\beta}{}_{\mu\nu}(t)=\frac{1}{6}\delta^{\alpha}{}_{[\mu}\delta^{\beta}{}_{\nu]}+O(t^{2})$
(5.14)

##  6 Heat Determinant

The existence of non-isometric isospectral manifolds demonstrates that the
spectrum alone does not determine the geometry. That is why it is worth
studying new invariants that depend not only on the eigenvalues but also on
the eigenfunctions, and, therefore, contain much more information about the
geometry of the manifold.

Let

$L=-\Delta+Q$ (6.1)

be a Laplace type operator acting on sections of a $N$-dimensional vector
bundle ${\cal V}$ over a $n$-dimensional compact Riemannian manifold $M$
without boundary. We define a new invariant (called the heat determinant) by
(the detailed context and motivation for introducing this invariant is
outlined in [19])

$K(t)=\int\limits_{M\times
M}dx\,dx^{\prime}\;\det\left\\{\mathrm{tr\,}\left(U^{*}(t;x,x^{\prime})\nabla_{\mu}\nabla_{\nu^{\prime}}U(t;x,x^{\prime})\right)\right\\}\,,$
(6.2)

where $U(t;x,x^{\prime})$ is the heat kernel of the operator $L$. This
invariant has an interesting spectral representation. Let
$\\{\lambda_{k},\varphi_{k}\\}_{k=1}^{\infty}$ be the eigenvalues and the
eigensections of the operator $L$ and $\Phi^{k}_{l}$ be the one-forms defined
by

$\Phi^{k}_{l}=\left<\varphi_{k},D\varphi_{l}\right>=\left<\varphi_{k},\nabla_{\mu}\varphi_{l}\right>dx^{\mu}\,,$
(6.3)

where $\left<\cdot,\cdot\right>$ is the fiber inner product, and $D=d+{\cal
A}$ is the covariant exterior derivative. Then the coefficients

$\displaystyle\Psi^{k_{1}\dots k_{n}}_{l_{1}\dots l_{n}}$ $\displaystyle=$
$\displaystyle\int\limits_{M}\Phi^{k_{1}}_{l_{1}}\wedge\cdots\wedge\Phi^{k_{n}}_{l_{n}}\,$
(6.4)

measure the correlations between the eigensections. Then the invariant $K(t)$
takes the form [19]

$K(t)=\frac{1}{n!}\sum_{k_{1},l_{1},\dots,k_{n},l_{n}=1}^{\infty}\exp\left\\{-t(\lambda_{k_{1}}+\lambda_{l_{1}}+\cdots+\lambda_{k_{n}}+\lambda_{l_{n}})\right\\}\left|\Psi_{l_{1}\dots
l_{n}}^{k_{1}\dots k_{n}}\right|^{2}\,.$ (6.5)

We prove in [19] that there is an asymptotic expansion as $t\to 0$

$K(t)\sim\frac{1}{2}N^{n}(4\pi)^{-n^{2}}\left(\frac{\pi}{2n}\right)^{n/2}t^{-n\left(n+\frac{1}{2}\right)}\sum_{k=0}^{\infty}t^{k/2}C_{k},$
(6.6)

where

$C_{k}=\int\limits_{M}\;d\mathrm{vol\,}c_{k},$ (6.7)

and $c_{k}$ are differential polynomials in the Riemann curvature, the
curvature of the bundle connection and the potential $Q$ with some universal
numerical coefficients that depend only on the dimensions $n$ and $N$. On
manifolds without boundary all odd-order coefficients vanish [19]

$C_{2k+1}=0.$ (6.8)

In particular,

$C_{0}=\mathrm{vol\,}(M)$ (6.9)

and the coefficients $c_{2}$ and $c_{4}$ are computed explicitly in our paper
[19].

##  7 Quantum Heat Traces

We initiate the study of new invariants of second-order elliptic partial
differential operators acting on sections of vector bundles over compact
Riemannian manifolds without boundary. We draw a deep analogy between the
spectral invariants of elliptic operators and the (classical and quantum)
statistical physics [44].

In statistical physics one considers an ensemble of identical particles in
thermodynamic equilibrium at the temperature $T$ and the chemical potential
$\mu$. The average number of classical particles $n_{k}$ in the state with the
energy $E_{k}$ is determined by the Boltzman distribution

$n_{k}=\exp[-\beta(E_{k}-\mu)],$ (7.1)

where $\beta=1/T$ is the inverse temperature. The quantum particles are
indistinguishable [44]; as the result, there are two types of quantum
particles, bosons and fermions. The average number of bosons with energy
$E_{k}$ is given by the Bose-Einstein distribution

$n_{(b),k}=\frac{1}{\exp[\beta(E_{k}-\mu)]-1},$ (7.2)

while the average number of fermions with energy $E_{k}$ is given by the
Fermi-Dirac distribution

$n_{(f),k}=\frac{1}{\exp[\beta(E_{k}-\mu)]+1}.$ (7.3)

The energy spectrum $E_{k}$ is described by the Hamiltonian $H$. The total
number of particles determines the partition function

$Z(\beta,\mu)=\mathrm{Tr\,}\exp[-\beta(H-\mu)],$ (7.4)

in the classical case, and

$Z_{b,f}(\beta,\mu)=\mathrm{Tr\,}\frac{1}{\exp[\beta(H-\mu)]+1}.$ (7.5)

in the quantum case. Recall that the Hamiltonian of a free particle of mass
$m$ and momentum $p$ is

$H=\frac{p^{2}}{2m}$ (7.6)

in the nonrelativistic case and

$H=(p^{2}+m^{2})^{1/2}$ (7.7)

in the relativistic case (in the units with the speed of light $c=1$).

This motivates the study of generalized heat traces defined as follows [13].
Let $L$ be an elliptic self-adjoint positive partial differential operator of
second order acting on smooth section of a vector bundle ${\cal V}$ over a
closed manifold $M$. Its square root,

$H=L^{1/2},$ (7.8)

is an elliptic self-adjoint positive pseudo-differential operator of first
order that plays the role of the Hamiltonian. We interpret the classical heat
trace

$\Theta(\beta)=\mathrm{Tr\,}\exp(-\beta L)$ (7.9)

as the partition function for the Boltzman distribution. By analogy, we define
the relativistic heat trace

$\displaystyle\Theta_{r}(\beta)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\exp(-\beta H)$ (7.10)

and the quantum heat traces

$\displaystyle\Theta_{b}(\beta,\mu)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\frac{1}{\exp[\beta(H-\mu)]-1},$ (7.11)
$\displaystyle\Theta_{f}(\beta,\mu)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\frac{1}{\exp[\beta(H-\mu)]+1},$ (7.12)

where $\mu$ is a parameter that plays the role of the chemical potential. We
show that these new invariants can be reduced to some integrals of the
classical heat trace and compute the high-temperature asymptotics of these
invariants as $\beta\to 0$.

We introduce a function $A_{q}$ of a complex variable $q$ defined by the
Mellin transform of the heat trace

$\displaystyle A_{q}$ $\displaystyle=$
$\displaystyle(4\pi)^{n/2}\frac{1}{\Gamma(-q)}\int_{0}^{\infty}dt\;t^{-q-1+n/2}\Theta(t).$
(7.13)

Then we show (for details see [1, 13]) that for a positive operator $L$:

  1. 1.

the function $A_{q}$ is an entire function of $q$,

  2. 2.

its values at non-negative integer points $A_{k}$, with
$k\in{\mathbb{Z}}_{+}$, are equal (up to a normalization factor) to the
standard heat trace coefficients (2.40), which are locally computable,

  3. 3.

while the values of the function $A_{k+1/2}$ at the half-integer points
$k+1/2$, with $k$ a positive integer, as well as the values of its derivative
$A^{\prime}_{k}=\partial_{q}A_{q}|_{q=k}$ at the positive integer points
$k\in{\mathbb{Z}}_{+}$ are new global invariants that are not locally
computable.

We use the integral [49]

$\exp(-x)=(4\pi)^{-1/2}\int_{0}^{\infty}dt\;t^{-3/2}\exp\left(-\frac{1}{4t}-tx^{2}\right),$
(7.14)

valid for $x\geq 0$, to reduce the relativistic heat trace to the classical
heat trace (7.9), that is,

$\Theta_{r}(\beta)=(4\pi)^{-1/2}\int_{0}^{\infty}dt\;t^{-3/2}\exp\left(-\frac{1}{4t}\right)\Theta(t\beta^{2}).$
(7.15)

Next, by using eqs. (7.15) and (7.13) we express the relativistic heat trace
in terms of the function $A_{q}$ via a Mellin-Barnes integral

$\Theta_{r}(\beta)=2(4\pi)^{-(n+1)/2}\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}dq\,\Gamma(-q)\Gamma\left[-q+(n+1)/2\right]\left(\frac{\beta}{2}\right)^{2q-n}A_{q}.$
(7.16)

We compute the asymptotics of the relativistic heat trace $\Theta_{r}(\beta)$
as $\beta\to 0$ (for details of the calculation see [13]). We obtained in even
dimension $n=2m$,

$\displaystyle\Theta_{r}(\beta)$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{\infty}\beta^{2k-2m}b^{(1)}_{k}A_{k}+\sum_{k=0}^{\infty}\beta^{2k+1}b^{(2)}_{k}A_{k+m+1/2},$
(7.17)

and in odd dimension $n=2m+1$,

$\displaystyle\Theta_{r}(\beta)$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{\infty}\beta^{2k-2m-1}b^{(3)}_{k}A_{k}+\log\beta\sum_{k=0}^{\infty}\beta^{2k+1}b^{(4)}_{k}A_{k+m+1}+\sum_{k=0}^{\infty}\beta^{2k+1}b^{(5)}_{k}A^{\prime}_{k+m+1},$

and computed all numerical coefficients $b^{(i)}_{k}$. Notice that the
coefficients of the singular part containing the inverse powers of $\beta$ and
the logarithm are locally computable.

Similarly, we express the quantum heat traces in terms of the classical one
[13]

$\displaystyle\Theta_{b,f}(\beta,\mu)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}dt\;h_{b,f}\left(t,\beta\mu\right)\Theta\left(t\beta^{2}\right),$
(7.19)

where

$\displaystyle h_{f}(t,\beta\mu)$ $\displaystyle=$
$\displaystyle(4\pi)^{-1/2}t^{-3/2}\sum_{k=1}^{\infty}(-1)^{k+1}k\exp\left(-\frac{k^{2}}{4t}+k\beta\mu\right),$
(7.20) $\displaystyle h_{b}(t,\beta\mu)$ $\displaystyle=$
$\displaystyle(4\pi)^{-1/2}t^{-3/2}\sum_{k=1}^{\infty}k\exp\left(-\frac{k^{2}}{4t}+k\beta\mu\right),$
(7.21)

which reduces the calculation of the quantum heat traces to the calculation of
the classical one (7.9). This gives the Mellin-Barnes representation of the
quantum heat traces [13]

$\displaystyle\Theta_{b,f}(\beta,\mu)$ $\displaystyle=$ $\displaystyle
2(4\pi)^{-(n+1)/2}\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}dq\;\Gamma(-q)\Gamma\left[-q+(n+1)/2\right]\left(\frac{\beta}{2}\right)^{2q-n}$
(7.22) $\displaystyle\times F_{b,f}(n-2q,\beta\mu)A_{q},$

where $c<0$ and

$\displaystyle F_{b}(s,{\beta\mu})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{e^{k{\beta\mu}}}{k^{s}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\;\frac{t^{s-1}}{e^{t-{\beta\mu}}-1},$
(7.23) $\displaystyle F_{f}(s,{\beta\mu})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}\frac{e^{k{\beta\mu}}}{k^{s}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\;\frac{t^{s-1}}{e^{t-{\beta\mu}}+1}.$
(7.24)

We compute their asymptotics as $\beta\to 0$ [13]. For $\mu=0$ we obtain an
asymptotic expansion as $\beta\to 0$: in even dimension $n=2m$,

$\displaystyle\Theta_{f}(\beta,0)$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{m}\beta^{2k-2m}c^{(1)}_{k}A_{k}+\sum_{k=0}^{\infty}\beta^{2k+1}c^{(2)}_{k}A_{k+m+1/2},$
(7.25) $\displaystyle\Theta_{b}(\beta,0)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m}\beta^{2k-2m}c^{(3)}_{k}A_{k}+\sum_{k=-1}^{\infty}\beta^{2k+1}c^{(4)}_{k}A_{k+m+1/2},$
(7.26)

and in odd dimension $n=2m+1$,

$\displaystyle\Theta_{f}(\beta,0)$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{\infty}\beta^{2k-2m-1}c^{(5)}_{k}A_{k}+\log\beta\sum_{k=0}^{\infty}\beta^{2k+1}c^{(6)}_{k}A_{k+m+1}+\sum_{k=0}^{\infty}\beta^{2k+1}c^{(7)}_{k}A^{\prime}_{k+m+1},$
$\displaystyle\Theta_{b}(\beta,0)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m-1}\beta^{2k-2m-1}c^{(8)}_{k}A_{k}+\log\beta\sum_{k=-1}^{\infty}\beta^{2k+1}c^{(9)}_{k}A_{k+m+1}+\sum_{k=-1}^{\infty}\beta^{2k+1}c^{(10)}_{k}A^{\prime}_{k+m+1}.$

and computed all numerical coefficients $c^{(i)}_{k}$ in [13].

##  8 Relative Spectral Invariants

As we already mentioned above the spectrum of a single elliptic partial
differential operator does not determine the geometry of a manifold. One could
ask the natural question whether the spectral information of two elliptic
differential operators determines the geometry. Our goal is to define new
relative spectral invariants that depend both on the eigenvalues and the
eigensections of both operators and contain much more information about
geometry. Such relative spectral invariants appear naturally, in particular,
in the study of particle creation in quantum field theory and quantum gravity
[31, 25, 14, 15].

Let $L_{\pm}$ be two self-adjoint elliptic second-order partial differential
operators acting on smooth sections of the vector bundle ${\cal V}$ over a
closed manifold $M$ with a positive definite scalar leading symbols of Laplace
type,

$L_{\pm}=-g_{\pm}^{-1/4}(\partial_{i}+{\cal
A}^{\pm}_{i})g_{\pm}^{1/2}g_{\pm}^{ij}(\partial_{j}+{\cal
A}^{\pm}_{j})g_{\pm}^{-1/4}+Q_{\pm}.$ (8.1)

Here $g_{ij}^{\pm}$ are two metrics, ${\cal A}^{\pm}_{i}$ are two connection
one-forms and $Q_{\pm}$ are two endomorphisms; also $g^{ij}_{\pm}$ are the
inverse metrics and $g_{\pm}=\det g^{\pm}_{ij}$.

We assume that ${\cal V}$ is a Clifford bundle. Let $D_{\pm}$ be two self-
adjoint elliptic first-order partial differential operators acting on smooth
sections of the vector bundle ${\cal V}$ of Dirac type,

$D_{\pm}=g_{\pm}^{1/4}i\gamma^{a}e^{j}_{\pm a}(\partial_{j}+{\cal
A}_{j}^{\pm})g_{\pm}^{-1/4}+S_{\pm},$ (8.2)

where $\gamma^{a}$ are the Dirac matrices satisfying

$\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a}=2\delta_{ab}I,$ (8.3)

$e^{i}_{\pm a}$ are two orthonormal frames for the metrics $g_{\pm}^{ij}$
satisfying

$g^{ij}_{\pm}=\delta^{ab}e^{i}_{\pm a}e^{i}_{\pm b},$ (8.4)

${\cal A}_{j}^{\pm}$ are two connections and $S_{\pm}$ are two endomorphisms.
We require that the endomorphisms $S_{\pm}$ commute with the Dirac matrices so
that the square of the Dirac type operators, $D_{\pm}^{2}$, are operators of
Laplace type.

The spectral information about the operators $L_{\pm}$ and $D_{\pm}$ is
contained in the classical heat traces

$\displaystyle\Theta_{\pm}(t)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\exp(-tL_{\pm}),$ (8.5) $\displaystyle H_{\pm}(t)$
$\displaystyle=$ $\displaystyle\mathrm{Tr\,}D_{\pm}\exp(-tD_{\pm}^{2}).$ (8.6)

The relative spectral information is contained in the relative spectral
invariants [15]

$\displaystyle\Psi(t,s)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\left\\{\exp(-tL_{+})-\exp(-tL_{-})\right\\}\left\\{\exp(-sL_{+})-\exp(-sL_{-})\right\\},$
(8.7) $\displaystyle\Phi(t,s)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\left\\{D_{+}\exp(-tD^{2}_{+})-D_{-}\exp(-tD^{2}_{-})\right\\}\left\\{D_{+}\exp(-sD^{2}_{+})-D_{-}\exp(-sD^{2}_{-})\right\\},$

which can be expressed in terms of the combined heat traces

$\displaystyle X(t,s)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\left\\{\exp(-tL_{+})\exp(-sL_{-})\right\\}$ (8.9)
$\displaystyle=$ $\displaystyle\int\limits_{M\times
M}dx\;dx^{\prime}\;\mathrm{tr\,}\left\\{U_{+}(t;x,x^{\prime})U_{-}(s;x^{\prime},x)\right\\},$
$\displaystyle Y(t,s)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\left\\{D_{+}\exp(-tD^{2}_{+})D_{-}\exp(-sD^{2}_{-})\right\\}$
(8.10) $\displaystyle=$ $\displaystyle\int\limits_{M\times
M}dx\;dx^{\prime}\;\mathrm{tr\,}\left\\{D_{+}U_{+}(t;x,x^{\prime})D_{-}U_{-}(s;x^{\prime},x)\right\\}.$

We study the asymptotics of the combined heat traces (8.9) and (8.10). We
define the time-dependent metric $g_{ij}=g_{ij}(t,s)$ as the inverse of the
matrix

$g^{ij}=tg_{+}^{ij}+sg_{-}^{ij},$ (8.11)

with $t,s>0$; throughout the paper we use the notation $g=\det g_{ij}$ for the
determinant of the metric. Also, we define the time-dependent connection
${\cal A}_{i}={\cal A}_{i}(t,s)$ by

${\cal A}_{i}=g_{ij}\left(tg^{jk}_{+}{\cal A}^{+}_{k}+sg^{jk}_{-}{\cal
A}^{-}_{k}\right).$ (8.12)

and the vectors

${\cal C}^{\pm}_{i}={\cal A}^{\pm}_{i}-{\cal A}_{i}.$ (8.13)

###### Theorem 1

There are asymptotic expansions as $\varepsilon\to 0$

$\displaystyle X(\varepsilon{}t,\varepsilon{}s)$ $\displaystyle\sim$
$\displaystyle(4\pi\varepsilon)^{-n/2}\sum_{k=0}^{\infty}\varepsilon^{k}B_{k}(t,s),$
(8.14) $\displaystyle Y(\varepsilon{}t,\varepsilon{}s)$ $\displaystyle\sim$
$\displaystyle(4\pi\varepsilon)^{-n/2}\sum_{k=0}^{\infty}\varepsilon^{k-1}C_{k}(t,s),$
(8.15)

where

$\displaystyle B_{k}(t,s)$ $\displaystyle=$
$\displaystyle\int\limits_{M}dx\;g^{1/2}(t,s)b_{k}(t,s),$ (8.16)
$\displaystyle C_{k}(t,s)$ $\displaystyle=$
$\displaystyle\int\limits_{M}dx\;g^{1/2}(t,s)c_{k}(t,s).$ (8.17)

  1. 1.

The coefficients $b_{k}(t,s)$ and $c_{k}(t,s)$ are scalar invariants built
polynomially from the covariant derivatives (defined with respect to the
metric $g_{ij}$ and the connection ${\cal A}_{i}$) of the metrics
$g^{\pm}_{ij}$, the vectors ${\cal C}^{\pm}_{i}$ and the potentials $Q_{\pm}$
and $S_{\pm}$.

  2. 2.

The coefficients $b_{k}(t,s)$ are homogeneous functions of $t$ and $s$ of
degree $k$ and the coefficients $c_{k}(t,s)$ are homogeneous functions of $t$
and $s$ of degree $(k-1)$.

Proof: This theorem is proved in [15].

In particular, the first coefficients of the asymptotic expansion of the
combined heat traces are [15]

$\displaystyle b_{0}(t,s)$ $\displaystyle=$ $\displaystyle N,$ (8.18)
$\displaystyle c_{0}(t,s)$ $\displaystyle=$ $\displaystyle
N\frac{1}{2}\delta^{ab}e_{+a}^{i}g_{ij}(t,s)e_{-b}^{j},$ (8.19)

where $N=\mathrm{dim\,}{\cal V}$.

##  9 Bogolyubov Invariant

To motivate the definition of the Bogolyubov invariant we consider Klein-
Gordon and Dirac fields on a globally hyperbolic manifold ${\cal M}$ with
compact Cauchy surfaces $M_{t}$, $t\in{\mathbb{R}}$, satisfying an assumption
on good asymptotic behaviour as $t\to\pm\infty$. The time-dependence of the
operators leads in quantum field theory to the phenomenon of particle
creation. We introduce below an invariant, called a Bogolyubov invariant,
that, in a relevant asymptotic regime, describes the renormalized number of
created particles. This invariant is closely related to the relative spectral
invariants discussed above. A much more detailed discussion is presented in
[14].

Let ${\cal V}$ be a twisted spin-tensor bundle. Let $\xi$ and $\eta$ be two
self-adjoint anti-commuting involutive endomorphisms of the bundle ${\cal V}$,

$\xi^{2}=\eta^{2}=I,\qquad\xi^{*}=\xi,\qquad\eta^{*}=\eta,\qquad\xi\eta=-\eta\xi.$
(9.1)

Let $D_{\pm}$ be a self-adjoint elliptic first-order partial differential
operators of Dirac type acting on smooth sections of the bundle ${\cal V}$
that anti-commute with $\eta$ and commute with $\xi$,

$D_{\pm}\eta=-\eta D_{\pm},\qquad D_{\pm}\xi=\xi D_{\pm}.$ (9.2)

Suppose that the square of the operators $D_{\pm}$ are Laplace type operators

$D^{2}_{\pm}=H_{\pm}.$ (9.3)

Then the operators $(D_{\pm}+m\eta)$ with a mass parameter $m$ are Dirac type
operators whose square are Laplace type operators

$(D_{\pm}+m\eta)^{2}=H_{\pm}+m^{2}I.$ (9.4)

The bosonic and fermionic Bogolyubov invariants are defined by the following
traces [14]

$\displaystyle B_{b}(\beta)$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,}\left\\{\frac{1}{\exp(\beta\omega_{+})+1}-\frac{1}{\exp(\beta\omega_{-})+1}\right\\}\left\\{\frac{1}{\exp(\beta\omega_{+})-1}-\frac{1}{\exp(\beta\omega_{-})-1}\right\\},$
$\displaystyle B_{f}(\beta)$ $\displaystyle=$
$\displaystyle\beta^{2}\mathrm{Tr\,}\left\\{\frac{(D_{+}+m\eta)}{\sinh(\beta\omega_{+})}-\frac{(D_{-}+m\eta)}{\sinh(\beta\omega_{-})}\right\\}^{2}.$
(9.6)

where

$\omega_{\pm}=(H_{\pm}+m^{2})^{1/2},$ (9.7)

The Bogolyubov invariants can be expressed in terms of the the relative
spectral invariants $\Psi(t,s)$ and $\Phi(t,s)$ defined in (8.7) and
(LABEL:17via). The heat trace representation for the Bogolyubov invariant can
be obtained as follows [14]. Let $h_{b,f,0}$ be the functions defined by

$\displaystyle h_{f}(t)$ $\displaystyle=$
$\displaystyle(4\pi)^{-1/2}t^{-3/2}\sum_{k=1}^{\infty}(-1)^{k+1}k\exp\left(-\frac{k^{2}}{4t}\right),$
(9.8) $\displaystyle h_{b}(t)$ $\displaystyle=$
$\displaystyle(4\pi)^{-1/2}t^{-3/2}\sum_{k=1}^{\infty}k\exp\left(-\frac{k^{2}}{4t}\right),$
(9.9) $\displaystyle h_{0}(t)$ $\displaystyle=$
$\displaystyle(4\pi)^{-1/2}t^{-3/2}\sum_{k=0}^{\infty}\left(2k+1\right)\exp\left(-\frac{\left(2k+1\right)^{2}}{4t}\right).$
(9.10)

Then, for the bosonic case we get from (LABEL:311xxa) [14]

$B_{b}(\beta)=\int\limits_{0}^{\infty}dt\int\limits_{0}^{\infty}ds\;h_{f}\left(t\right)h_{b}\left(s\right)\exp\left(-m^{2}\beta^{2}(s+t)\right)\Psi\left(\beta^{2}t,\beta^{2}s\right),$
(9.11)

where

$\Psi(t,s)=\mathrm{Tr\,}\left[\exp(-tH_{+})-\exp(-tH_{-})\right]\left[\exp(-sH_{+})-\exp(-sH_{-})\right].$
(9.12)

For the fermionic case we get from (9.6) [14]

$\displaystyle B_{f}(\beta)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}dt\int\limits_{0}^{\infty}ds\;\,h_{0}\left(s\right)h_{0}\left(t\right)\exp\left(-m^{2}\beta^{2}(s+t)\right)$
(9.13)
$\displaystyle\times\beta^{2}\left\\{\Phi\left(\beta^{2}t,\beta^{2}s\right)+m^{2}\Psi\left(\beta^{2}t,\beta^{2}s\right)\right\\},$

where

$\Phi(t,s)=\mathrm{Tr\,}\left[D_{+}\exp(-tH_{+})-D_{-}\exp(-tH_{-})\right]\left[D_{+}\exp(-sH_{+})-D_{-}\exp(-sH_{-})\right].$
(9.14)

It is easy to see that the integrals for the Bogolyubov invariant converge
both as $t,s\to 0$ and (for sufficiently large $m^{2}$) also as
$t,s\to\infty$.

Now we can compute the asymptotics of Bogolyubov invariant as $\beta\to 0$.
(We list below the results only; for the details of the calculation see [14].)
We obtain om odd dimension $n=2m+1$

$B_{b}(\beta)\sim\sum_{k=0}^{\infty}\beta^{2k-n}c_{k}^{(1)}+\sum_{k=0}^{\infty}\beta^{2k}c_{k}^{(2)},$
(9.15)

and in the even dimension $n=2m$

$B_{b}(\beta)\sim\sum_{k=0}^{m-1}\beta^{2k-n}c_{k}^{(3)}+\sum_{k=0}^{\infty}\beta^{2k}c_{k}^{(4)}+\log\beta^{2}\;\sum_{k=0}^{\infty}\beta^{2k}c_{k}^{(5)}.$
(9.16)

Here $c_{k}^{(i)}$ are some coefficients. Notice that the coefficients
$c^{(1)}_{k}$ of the all odd powers of $\beta$ as well as the coefficients
$c^{(3)}_{k}$ and $c^{(5)}_{k}$ of the singular part and the logarithmic part
are locally computable invariants whereas the coefficients $c^{(2)}_{k}$ of
the even non-negative powers of $\beta$ as well as the coefficients
$c^{(4)}_{k}$ of the regular part are non-locally computable global
invariants. The leading asymptotics have the form

$B_{b}(\beta)=\beta^{-n}c_{0}^{(1)}+\beta^{-n+2}c^{(1)}_{1}+O(\beta^{-n+4})+O(\log\beta).$
(9.17)

Similarly, for the Dirac operator we obtain: in the odd dimension $n=2m+1$:

$B_{f}(\beta)\sim\sum_{k=0}^{\infty}\beta^{2k-n}d_{k}^{(1)}+\sum_{k=0}^{\infty}\beta^{2k}d_{k}^{(2)},$
(9.18)

and in the even dimension $n=2m$

$B_{f}(\beta)\sim\sum_{k=0}^{m-1}\beta^{2k-n}d_{k}^{(3)}+\sum_{k=0}^{\infty}\beta^{2k}d_{k}^{(4)}+\log\beta^{2}\;\sum_{k=0}^{\infty}\beta^{2k}d_{k}^{(5)},$
(9.19)

Here the coefficients $d^{(1)}_{k}$ of the all odd powers of $\beta$ as well
as the coefficients $d^{(3)}_{k}$ and $d^{(5)}_{k}$ of the singular part and
the logarithmic part are locally computable invariants whereas the
coefficients $d^{(2)}_{k}$ of the even non-negative powers of $\beta$ as well
as the coefficients $d^{(4)}_{k}$ of the regular part are non-locally
computable global invariants. The leading asymptotics have the form

$B_{f}(\beta)=\beta^{-n}d_{0}^{(1)}+\beta^{-n+2}d^{(1)}_{1}+O(\beta^{-n+4})+O(\log\beta).$
(9.20)

##  10 Heat Semigroups on Weyl Algebra

Below we present a particular model when the combined heat traces can be
computed exactly based only on the algebraic properties of the operators
involved [16].

Let $x^{i}$ be the coordinates of the Euclidean space ${\mathbb{R}}^{n}$ and
$\partial_{i}$ be the corresponding partial derivatives. More precisely, we
consider the partial derivative operators acting on the space
$C^{\infty}_{0}({\mathbb{R}}^{n})$ of smooth functions of compact support.
Recall that this space is dense in the Hilbert space
$L^{2}({\mathbb{R}}^{n})$, which defines the extension of these operators to
the whole Hilbert space $L^{2}({\mathbb{R}}^{n})$. Moreover, the partial
derivatives are unbounded (essentially) anti-self-adjoint operators in this
Hilbert space. Then the Weyl algebra (the universal enveloping algebra of the
Heisenberg algebra) is simply the ring of all differential operators with
polynomial coefficients.

We define the anti-self-adjoint operators $\nabla_{k}$ of the form

$\nabla_{k}=\partial_{k}-\frac{1}{2}i{\cal R}_{kj}x^{j}.$ (10.1)

Given a positive matrix $g$ we introduce an operator called the Laplacian by

$\Delta_{g}=g^{ij}\nabla_{i}\nabla_{j},$ (10.2)

where $g^{ij}$ is the inverse matrix.

We consider two Laplacians $\Delta_{\pm}$ defined with two different matrices
${\cal R}^{\pm}_{ij}$ and two different matrices $g^{\pm}_{ij}$ and study the
convolution of the heat semigroups
$\exp(t\Delta_{+})\exp(s\Delta_{-}),$ more precisely, the integral kernel of
the product of the corresponding heat semigroups

$\displaystyle U(t,s;x,x^{\prime})$ $\displaystyle=$
$\displaystyle\exp(t\Delta_{+})\exp(s\Delta_{-})\delta(x-x^{\prime}).$ (10.3)

The $(2n+1)$ operators $(\nabla_{1},\dots,\nabla_{n},x^{1},\dots,x^{n},i)$
form the Lie algebra ${\mathfrak{h}}_{n}$ of the Heisenberg group $H_{2n+1}$
with the commutation relations

$\displaystyle[\nabla_{k},x^{j}]$ $\displaystyle=$
$\displaystyle\delta^{j}_{k},$ (10.4) $\displaystyle{}[\nabla_{k},\nabla_{j}]$
$\displaystyle=$ $\displaystyle i{\cal R}_{kj}\,,$ (10.5)
$\displaystyle{}[\nabla_{k},i]$ $\displaystyle=$ $\displaystyle[x^{k},i]=0.$
(10.6)

The universal enveloping algebra $U({\mathfrak{h}}_{n})$ of the Heisenberg
algebra (called the Weyl algebra) is the set of all polynomials in these
operators subject to these commutation relations.

The operator $\exp\left<\xi,\nabla\right>:C^{\infty}_{0}({\mathbb{R}}^{n})\to
C^{\infty}_{0}({\mathbb{R}}^{n})$ acts by

$\displaystyle\left(\exp\left<\xi,\nabla\right>f\right)(x)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\left<\xi,i{\cal
R}x\right>\right)f(x+\xi)\,.$ (10.7)

and is an isometry [16].

First, we show that the heat semigroup can be represented as a non-commutative
Gaussian integral [16]

$\exp(t\Delta_{g})=(4\pi)^{-n/2}\Omega(t)\int\limits_{{\mathbb{R}}^{n}}d\xi\exp\left\\{-\frac{1}{4}\left<\xi,D(t)\xi\right>\right\\}\exp\left<\xi,\nabla\right>\,,$
(10.8)

where $D(t)=(D_{ij})$ is a symmetric matrix defined by

$D(t)=i{\cal R}\coth\left(tg^{-1}i{\cal R}\right)\,,$ (10.9)

and

$\displaystyle\Omega(t)$ $\displaystyle=$
$\displaystyle\det\left(g^{-1}\frac{\sinh(tg^{-1}i{\cal R})}{g^{-1}i{\cal
R}}\right)^{-1/2}.$ (10.10)

Next, we consider two sets of such operators $\nabla_{i}^{+}$ and
$\nabla_{j}^{-}$ forming the Lie algebra

$\displaystyle[\nabla^{+}_{i},\nabla^{+}_{j}]$ $\displaystyle=$ $\displaystyle
i{\cal R}^{+}_{ij},$ (10.11) $\displaystyle{}[\nabla^{-}_{i},\nabla^{-}_{j}]$
$\displaystyle=$ $\displaystyle i{\cal R}^{-}_{ij},$ (10.12)
$\displaystyle{}[\nabla^{+}_{i},\nabla^{-}_{j}]$ $\displaystyle=$
$\displaystyle i{\cal R}_{ij}.$ (10.13)

where

${\cal R}_{ij}=\frac{1}{2}\left({\cal R}^{+}_{ij}+{\cal R}^{-}_{ij}\right).$
(10.14)

Then we compute the product of the semigroups
$\exp(t\Delta_{+})\exp(s\Delta_{-})$. We define the operators $\nabla_{i}$ and
$X_{i}$ by

$\displaystyle\nabla_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\nabla^{+}_{i}+\nabla^{-}_{i}),$ (10.15)
$\displaystyle X_{i}$ $\displaystyle=$
$\displaystyle\nabla^{+}_{i}-\nabla^{-}_{i}.$ (10.16)

Then we prove a highly non-trivial representation (for details of the proof
see [16])

$\displaystyle\exp(t\Delta_{+})\exp(s\Delta_{-})=(4\pi)^{-n/2}\Omega(t,s)\exp\left<X,D^{-1}(t,s)X\right>$
(10.17)
$\displaystyle\qquad\times\int\limits_{{\mathbb{R}}^{n}}d\xi\,\exp\left\\{-\frac{1}{4}\left<\xi,H(t,s)\xi\right>-\frac{1}{2}\left<\xi,Z^{T}(t,s)D^{-1}(t,s)X\right>\right\\}\exp\left<\xi,\nabla\right>,$

where $T_{\pm}(t)$, $D(t,s)$, $Z(t,s)$ and $H(t,s)$ are the matrices defined
by

$\displaystyle T_{\pm}(t)$ $\displaystyle=$ $\displaystyle D_{\pm}(t)+i{\cal
R},$ (10.18) $\displaystyle D(t,s)$ $\displaystyle=$ $\displaystyle
D_{+}(t)+D_{-}(s),$ (10.19) $\displaystyle Z(t,s)$ $\displaystyle=$
$\displaystyle D_{+}(t)-D_{-}(s)-2i{\cal R}_{-},$ (10.20) $\displaystyle
H(t,s)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(D(t,s)-Z^{T}(t,s)D^{-1}(t,s)Z(t,s)\right),$
(10.21)

and

$\Omega(t,s)=\left(\frac{\det T_{+}(t)\det T_{-}(s)}{\det
D(t,s)}\right)^{1/2}.$ (10.22)

This enables us to compute the convolution of the heat kernels [16]

$\displaystyle U(t,s;x,x^{\prime})$ $\displaystyle=$
$\displaystyle(4\pi)^{-n/2}\left(\det B(t,s)\right)^{1/2}$
$\displaystyle\times\exp\left\\{-\frac{1}{4}\left<x,A_{+}(t,s)x\right>-\frac{1}{4}\left<x^{\prime},A_{-}(t,s)x^{\prime}\right>+\frac{1}{2}\left<x,B(t,s)x^{\prime}\right>\right\\},$

where $A_{\pm}(t,s)$ and $B(t,s)$ be matrices defined by

$\displaystyle A_{+}(t,s)$ $\displaystyle=$ $\displaystyle
D_{+}(t)-T_{+}^{T}(t)D^{-1}(t,s)T_{+}(t),$ (10.24) $\displaystyle A_{-}(t,s)$
$\displaystyle=$ $\displaystyle D_{-}(s)-T_{-}^{T}(s)D^{-1}(t,s)T_{-}(s),$
(10.25) $\displaystyle B(t,s)$ $\displaystyle=$ $\displaystyle
T_{+}(t)D^{-1}(t,s)T_{-}(s).$ (10.26)

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