Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^{2}$-differential forms.

In the remainder of this section, we introduce the background on stochastic flows and spectral action, and delve into motivating ideas. In section 2, the structure matrix for connection Laplacian is computed following the standard prescription. The noise space turns out to be $L^{2}$ differential forms and the flow lives on the Fock space over the differential forms. The section concludes by writing the structure matrix and the qsde for the stochastic flow in the coordinate free quantum stochastic calculus notation; the relevant material from quantum stochastic differential equations an appendix reviewing quantum stochastic differential equations is included at the end (appendix B). Necessary bounds for controlling growth of Laplacian powers are established in section 3. The existence of flows for the derived qsde is established in section 4 by providing estimates that can be plugged into the standard theory.

The canonical spectral triple for Riemannian $(M,g)$ which carries a spin structure is the data $(C^{\infty}(M),L^{2}(S),D_{M})$, $L^{2}(S)$ being the Hilbert space of square integrable sections of a spinor bundle $S\to M$, and $D_{M}$ the Dirac operator associated to the lift of Levi-Civita connection to the spinor bundle[16, pg 67]. We will take $S$ to be any spinor bundle associated to $TM$ and $D_{M}$ to be the Atiyah-Singer operator $\not{D}$ (see [11, ex II.5.9]), then we have that $\not{D}^{2}=\mathop{}\!\mathbin{\bigtriangleup}+\kappa/4$ where $\mathop{}\!\mathbin{\bigtriangleup}$ is the connection Laplacian for the connection on $S$ and $\kappa$ is the scalar curvature[11, Thm II.8.8].

The bosonic spectral action is the linear funtional $S_{b}^{M}:=\operatorname{{Tr}}f(D_{M}/\Lambda)$ for a choice of test function $f$ which we take as $e^{-x^{2}}$ and $\Lambda$ a cutoff parameter\cites[§ 5.1]ncg_standard_model[§ 7.1]suijlekom_ncg, so that the parameter $t$ of the Dirac heat semigroup $e^{-tD_{M}^{2}}$ will be taken to satisfy $t=\Lambda^{-2}$. From the asymptotic expansion $\lim_{\Lambda\to\infty}S_{b}^{M}$ for a Riemannian spin $4$-manifold $M$, the Einstein-Hilbert action, $S_{EH}$, can be recovered\cites[§ 5.3]ncg_standard_model[§ 8.3]suijlekom_ncg.

Since $D_{M},D_{M}^{2}$ are self-adjoint on $L^{2}(S)$, suppose $(\phi_{i,j})$ is an basis of orthonormal eigensections with eigenvalues $\lambda_{i,j}^{2}$ for $D^{2}_{M}$ where $j\in[n_{i}]$ runs over the multiplicity $n_{i}$ of eigenvalue $\lambda^{2}_{i,j}:=\lambda^{2}_{i}$ Define the state $\Phi^{z}=1/N(z)\sum_{i:\lambda_{i}\leq z}\sum_{j\in[n_{i}]}\phi_{i}^{j}$ with normalization $N(z):=\sqrt{\sum_{i:\lambda_{i}\leq z}n_{i}}$ then

$\displaystyle\lim_{z\to\infty}{\langle\Phi^{z},e^{-tD_{M}^{2}}\Phi^{z}\rangle}N(z)$

$\displaystyle=\sum_{i:\lambda_{i}\leq z}\sum_{j\in n_{i}}{\langle\phi_{i,j},e^{-tD_{M}^{2}}\phi_{i,j}\rangle}=\lim_{z\to\infty}{\langle\Phi^{z},e^{-tD_{M}^{2}}\Phi^{z}\rangle}=\operatorname{{Tr}}e^{-tD_{M}^{2}}$

Therefore, the bosonic spectral action can be approximated by expectation of small time expectation of $e^{-t(D_{M})^{2}}$ in state $\Phi^{z}$ for $z$ large:

$$S_{EH}\approx\lim_{\sqrt{t}\to 0}\operatorname{{Tr}}e^{-tD_{M}^{2}}=\lim_{\sqrt{t}\to 0}\lim_{z\to\infty}{\langle\Phi^{z},e^{-tD_{M}^{2}}\Phi^{z}\rangle}N(z)$$

This motivates the interest in evaluating ${\langle\Phi^{z},e^{-tD_{M}^{2}}\Phi^{z}\rangle}$. The approach we take is that of a quantum stochastic dilation associated to the heat semigroup, $e^{-tD^{2}_{M}}$, which is a quantum dynamical semigroup by [9].

The issue is complicated by the fact that for the existence of Evans-Husdon flow, the generator must annihilate identity, that is, the semigroup must be conservative. The Dirac laplacian acting on endomorphisms by composition does not satisfy this. We instead have to work with the endomorphism connection and the associated endomorphism connection laplacian and endomorphism Dirac laplacian and then extract the spinor bundle Dirac laplacian from it (see [9] more detailed discussion on this). Because of Ricci flatness, $\not{D}^{2}=\mathop{}\!\mathbin{\bigtriangleup}$, the semigroup of interest will be $e^{-t\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}}$ for the endomorphism connection laplacian $\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}$.

A quantum stochastic dilation of the Evans-Hudson type²²2see appendix B for a brief review of quantum stochastic processes and integrals, and [14, 13] for detailed standard theory for the quantum dynamical semigroup $e^{-t{\mathcal{L}}}$ on the C^*-algebra ${\mathcal{A}}:={\mathcal{C}}(M)\subset\operatorname{{End}}(L^{2}(S))$ is a family of $*$-homomorphisms $j_{t}:{\mathcal{A}}\to{\mathcal{A}}^{\prime\prime}\otimes{\mathcal{B}}(\operatorname{{\Gamma}}(\mathds{R}_{+},k_{0}))$ where $k_{0}$ is a Hilbert space, called the noise (or multiplicity) Hilbert space that is constructed from the generator $-{\mathcal{L}}$, such that the following holds

• •

  There exists an ultra-weak dense $*$-subalgebra ${\mathcal{A}}_{0}\subset{\mathcal{A}}$ such that the map-valued process $J_{t}:{\mathcal{A}}\otimes\textsc{E}(L^{2}(\mathds{R}_{+},k_{0}))\to{\mathcal{A}}^{\prime\prime}\otimes\operatorname{{\Gamma}}(L^{2}(\mathds{R}_{+},k_{0}))$ with $J_{t}(x\otimes\textsc{E}(f))u:=j_{t}(x)(u\textsc{E}(f))$ for $x\in{\mathcal{A}},u\in{\textsf{H}}:=L^{2}(S)$, $f\in L^{2}(\mathds{R}_{+},k_{0})$, satisfying the qsde:

  $\displaystyle dJ_{t}=J_{t}\circ(a_{\delta}(dt)+a_{\delta}^{\dagger}(dt)+\Lambda_{\sigma}(dt)+I_{\mathcal{L}}(dt)),J_{0}={\textbf{1}}$

  (1)

  on ${\mathcal{A}}_{0}\otimes\textsc{E}(L^{2}(\mathds{R}_{+},k_{0}))$ where $\delta:{\mathcal{A}}_{0}\to{\mathcal{A}}\otimes k_{0},\sigma:{\mathcal{A}}_{0}\to{\mathcal{A}}\otimes{\mathcal{B}}(k_{0})$ are linear maps, called the structure maps for the qsde, derived from the generator ${\mathcal{L}}$ for $e^{-tD_{M}^{2}}$, ${\mathcal{A}}_{0}\subset\operatorname{{Dom}}({\mathcal{L}})$, and $a_{\delta},a^{\dagger}_{\delta},\Lambda_{\sigma},I_{\mathcal{L}}$ the fundamental processes with respect to which stochastic integral is defined.

• •

  For all $u,v\in{\textsf{H}},x\in{\mathcal{A}}$,

  $\displaystyle{\langle v\textsc{E}(0),j_{t}(x)\textsc{E}(0)\rangle}={\langle v,e^{-t{\mathcal{L}}}u\rangle}$

  (2)

Therefore, ${\langle\Phi^{z},e^{-t{\mathcal{L}}}\Phi_{z}\rangle}$ is realized as operator algebraic expectation of $j_{t}$ with respect vacuum state ${|\Phi^{z}\textsc{E}(0)\rangle}\in L^{2}(S)\otimes\operatorname{{\Gamma}}(L^{2}(\mathds{R}_{+},k_{0}))$, where $j_{t}$ is obtained from the flow $J_{t}$ for the Evans-Hudson qsde (equation 1). Schemes for solving for Evans-Hudson qsde, for example, using Picard iterates, provide a way to algorithmically construct the flow.

To start we note the following sign conventions of the Laplacians. Primarily the signs are fixed so the Laplace-Beltrami operator has non-negative spectrum, and signs on all other Laplacians cascade from there. On Riemannian $(M,g)$, $M$ compact, without boundary, $\operatorname{{Tr}}_{g}$ denotes the trace of a covariant tensor taken after identifying with a contravariant tensor via the metric $g$, $\operatorname{{Tr}}_{g}(h):=g^{ij}h_{ij}$. Note that trace on contravariant tensor, e.g. vector fields, is simply the sum. For $X\in\Gamma(TM)$, $\text{div}(X)=\operatorname{{Tr}}(\nabla X)$. The Laplace-Beltrami operator is taken as the operator with non-negative spectrum, that is, $-\text{div}(\nabla)=-\operatorname{{Tr}}(\nabla_{\cdot,\cdot})$, where $\nabla_{\cdot,\cdot}$ is the second invariant derivative $\nabla^{2}_{V,W}:=\nabla_{V}\nabla_{W}-\nabla_{\nabla_{V}W}:\Gamma(E)\to\Gamma(E)$. The connection Laplacian is $\nabla^{*}\nabla$ where $\nabla^{*}$ is adjoint of the connection $\nabla:\Gamma(E)\to\Gamma(E)\otimes T^{*}M$. Equivalently, $\nabla^{*}\nabla=-\operatorname{{Tr}}(\nabla_{\cdot,\cdot})$. Further, $\mathop{}\!\mathbin{\bigtriangleup}=-g^{ij}\nabla_{i}\nabla_{j}$.

Let $\nabla$ be any connection on the vector bundle $E\to M$, $(M,g)$ a compact Riemannian manifold. The connection Laplacian at $p\in M$, $\mathop{}\!\mathbin{\bigtriangleup}=\nabla^{*}\nabla$ in local coordinates $(e_{i})$ is given by $\mathop{}\!\mathbin{\bigtriangleup}=-(\sum_{i}\nabla_{i}\nabla_{i}-\nabla_{\nabla_{i}e_{i}}$). To evaluate $\mathop{}\!\mathbin{\bigtriangleup}\phi$ at any $p\in M$ and $\phi\in\Gamma(E)$, we will use Riemann normal coordinates centered at $p$ so $\nabla_{i}e_{j}$ vanish, yielding $\mathop{}\!\mathbin{\bigtriangleup}\phi(p)=-\sum_{i}\nabla_{i}\nabla_{i}\phi(p)$.

The endomorphism connection $\nabla^{\operatorname{{End}}}$ on the bundle $\operatorname{{End}}(E)=E\otimes E^{*}$ associated to a connection $\nabla$ on the Hermitian vector bundle $E$ over the Riemannian manifold $M$ is such that for $X\in TM,\nabla_{X}^{\operatorname{{End}}}=\nabla_{X}\otimes 1+1\otimes\widetilde{\nabla}_{X}$, where $\widetilde{\nabla}$ is the dual connection on $E^{*}$. The endomorphism Laplacian is defined as usual: at $p\in M$ in Riemann normal coordinates centered at $p$ (denoting $\widetilde{\nabla}^{*}\widetilde{\nabla},\widetilde{\nabla}$ by $\mathop{}\!\mathbin{\bigtriangleup},\nabla$ again),

$$\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}=-\sum_{i}\nabla_{i}^{\operatorname{{End}}}\nabla_{i}^{\operatorname{{End}}}=-\left(\mathop{}\!\mathbin{\bigtriangleup}\otimes 1+2\sum_{i}\nabla_{i}\otimes\nabla_{i}+1\otimes\mathop{}\!\mathbin{\bigtriangleup}\right)$$

Note that as $E\otimes E^{*}$ is balanced over ${\mathcal{C}}(M)$, the action of ${\mathcal{C}}(M)$ on $\operatorname{{End}}(E)$ can be written as $f\cdot 1_{\operatorname{{End}}}=\sum_{i}(f\cdot h_{i})\otimes h_{i}^{*}$; this convention is used for all computation with Laplacian expressed in this tensor form. It’s also very useful to note that in any local coordinates , $\nabla^{\operatorname{{End}}}$ acts by commutator: if over chart $U$, the connection has potential $A$, $\nabla=d+A$, then for a local orthonormal frame $(\mu_{i})$ and dual frame $(\mu^{j})$, $\nabla^{\operatorname{{End}}}\sum_{ij}\sigma^{i}_{j}\mu_{i}\otimes\mu^{j}=\sum_{ij}(d\sigma_{j}^{i})\mu_{i}\otimes\mu^{j}+\sum_{jk}[\sigma A-A\sigma]_{jk}\mu_{k}\otimes\mu^{j}$. In particular, since ${\textbf{1}}_{\operatorname{{End}}}$ is given by the identity matrix locally, it follows (see [9]) that $\nabla^{\operatorname{{End}}}({\textbf{1}}_{\operatorname{{End}}})=0$. This implies that again in normal Riemann coordinates centered at $p$ yields that for any $f\in{\mathcal{C}}^{\infty}(M)$,

$\displaystyle\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}(f{\textbf{1}}_{\operatorname{{End}}})(p)$

$\displaystyle=-\sum_{i}\nabla^{\operatorname{{End}}}_{i}\nabla^{\operatorname{{End}}}_{i}(f\cdot{\textbf{1}}_{\operatorname{{End}}})(p)=\sum_{i}\nabla_{i}^{\operatorname{{End}}}\left(\partial_{i}f{\textbf{1}}_{\operatorname{{End}}}\right)=-\sum_{i}\partial_{i}\partial_{i}f\cdot{\textbf{1}}_{\operatorname{{End}}}(p)$

(3)

Now for any constant $c$, $\operatorname{{Tr}}(e^{-t(\mathop{}\!\mathbin{\bigtriangleup}+c)})$ is just $e^{-ct}\operatorname{{Tr}}(e^{-t\mathop{}\!\mathbin{\bigtriangleup}})$. If there exists a parallel section $\phi$ for $\nabla$ (equivalently for the connection on the dual bundle), then with $H_{\phi}=\overline{\{s\otimes\phi:s\in\Gamma(E)\}}$, and appropriate normalization on $\phi$,

$\displaystyle\operatorname{{Tr}}(e^{-t\mathop{}\!\mathbin{\bigtriangleup}})=\operatorname{{Tr}}|_{H_{\phi}}(e^{-t\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}})$

(4)

The parallel section, therefore, allows one to extract the heat kernel trace $\operatorname{{Tr}}(e^{-t\mathop{}\!\mathbin{\bigtriangleup}})$ from the $\operatorname{{Tr}}(e^{-t\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}})$.

In the setting of the canonical spectral triple, we specialize to the spinor bundle. The existence of a parallel spinor constrains the holonomy of Levi-Civita connection of the manifold. In particular, this implies for Riemannian $(M,g)$ that the Ricci tensor vanishes[8, § 6.3], therefore, $D^{2}=\mathop{}\!\mathbin{\bigtriangleup}$.

Following Goswami-Sinha construction of the flow generator, we compute the Kolmogorov decomposition for the kernel for ${\mathcal{L}}=-\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}{}}/2$ (to avoid factors of $-2$), the data we need will be derived from the structure theory of this kernel,

$\displaystyle K_{{\mathcal{L}}}((a_{1},a_{2}),(b_{1},b_{2}))=\left(\sum_{k}da_{1}(e_{k})db_{1}(e_{k})\right)a_{2}b_{2}$

(6)

As in [14, Thm 2.2.7], the Kolmogorov decomposition is taken to be the reproducing kernel Hilbert space

$$R_{{\mathcal{L}}}=\overline{\operatorname{{LinSpan}}\{K_{{\mathcal{L}}}(\cdot,b)u:b\in C^{\infty}(M),u\in E\}}$$

with map $V:{\mathcal{C}}^{\infty}(M)\to{\mathcal{B}}(\mathrm{E},R_{\mathcal{L}})$ for $x\in{\mathcal{C}}^{\infty}(M)$ given by

$$V(x):E\to R_{\mathcal{L}},E\ni u\to K_{\mathcal{L}}(\cdot,x)u\in R_{\mathcal{L}}$$

By definition, $K_{\mathcal{L}}(\cdot,x)u$ is total in $R_{{\mathcal{L}}}$ making the decomposition minimal.

The structure matrix is the map $\Theta:{\mathcal{A}}_{0}\to{\mathcal{B}}(H\otimes(\mathbb{C}\oplus k_{0}))$ for a dense subalgebra ${\mathcal{A}}_{0}\subset{\mathcal{A}}$, and Hilbert spaces $H,k_{0}$ (alternatively, a map $\Theta:{\mathcal{A}}_{0}\to{\mathcal{A}}_{0}\otimes{\mathcal{B}}(\mathbb{C}\oplus k_{0}),{\mathcal{A}}_{0}\subset{\mathcal{B}}(H)$) given by

$\displaystyle\Theta(x)=\begin{pmatrix}{\mathcal{L}}(x)&\delta^{\dagger}(x)\\ \delta(x)&\sigma(x)\end{pmatrix}$

(7)

where ${\mathcal{L}}:{\mathcal{A}}_{0}\to{\mathcal{A}}_{0}$ is a $*$-linear map such that ${\mathcal{L}}(xy)-{\mathcal{L}}(x)y-x{\mathcal{L}}(y)=\delta^{\dagger}(x)\delta(y)$, $\delta:{\mathcal{A}}_{0}\to{\mathcal{A}}_{0}\otimes k_{0}$ is a $\pi$-derivation, $\Theta(1)=0$. Additionally, $\sigma(x)=\pi(x)-x\otimes 1$. The maps $\mathop{}\!\mathbin{\bigtriangleup},\delta,\sigma$ satisfy these conditions iff the first order Ito product formula holds[1, lemma 2.2].

For the structure matrix ${\mathcal{L}}$ is the semigroup generator $-\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}$. Following [14], $\delta,\sigma$ are extracted from the minimal Kolmogorov decomposition: the decomposition $(R_{{\mathcal{L}}},V)$, induces the following maps on ${\mathcal{A}}_{\infty}:={\mathcal{C}}^{\infty}(M)\subset{\mathcal{C}}(M)={\mathcal{A}}$,

	$\displaystyle\rho:{\mathcal{A}}_{\infty}\to{\mathcal{B}}(R_{\mathcal{L}}),\rho(x)(V(\cdot,b)u)=V(\cdot,xb)u$
	$\displaystyle\alpha:{\mathcal{A}}_{\infty}\to{\mathcal{B}}(E,R_{{\mathcal{L}}}),\alpha(x)=V(x,1)$

With remark 2.1 and equation 6 in mind, $\rho(f)$, $f\in{\mathcal{C}}^{\infty}(M)$, is multiplication by $f$ on $R_{{\mathcal{L}}}$ while $\alpha(f)$ acts by contraction with $1$-form $\sum_{k}df(e_{k})de_{k}$. The representation $\rho$ is the identity map: $C^{\infty}(M)$ is interpreted as acting by multiplication on $R_{{\mathcal{L}}}$, and $\alpha$ is a derivation.

The construction of the structure maps proceeds as in [14, Thm 6.6.1]. To start define the Hilbert ${\mathcal{A}}$-module $E=\overline{\{\alpha(x)y:x,y\in{\mathcal{A}}_{\infty}\}}$ where the closure is with respect to operator norm for ${\mathcal{B}}({\textsf{H}}_{E},R_{{\mathcal{L}}})$. ${\mathcal{A}}$ has right action on $E$ by multiplication (where the norm density of ${\mathcal{A}}_{\infty}\subset{\mathcal{A}}$ is utilized) and the ${\mathcal{A}}$-valued inner product is $E\times E\ni(a,b)\to{\langle a,b\rangle}=a^{*}b$.

Now $E$ can be identified $\alpha(f)g\in E$ can be identified with $g\otimes df\in{\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}W\subset{\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}L^{2}(\Omega^{1}(M))$ where $W=\overline{\{df:f\in{\mathcal{C}}^{\infty}(M)\}}$ (note $M$ is compact) is a closed subspace of $L^{2}(\Omega^{1}(M))=W\oplus W^{\perp}$ (the ${\mathcal{A}}$ valued innerproduct is pointwise tensor contraction). However, ${\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}L^{2}(\Omega^{1}(M))=L^{2}(\Omega^{1}(M))$ since ${\mathcal{A}}={\mathcal{C}}(M)$ the tensor product is ${\mathcal{C}}(M)$-balanced: $a\otimes\omega\in{\mathcal{A}}\otimes_{\emph{C}}\textsuperscript{*}L^{2}(\Omega^{1}(M))$ is simply $1\otimes a\omega$.

This choice of the Hilbert space $L^{2}(\Omega^{1}(M))$ simplifies the application of Kasparov stabilization theorem[14, Thm 4.1.10]. Now the Hilbert-C^*-module ${\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}L^{2}(\Omega^{1}(M))={\mathcal{A}}\otimes L^{2}(\Omega^{1}(M))=L^{2}(\Omega^{1}(M))$, and by Kasparov stabilization theorem, there’s a unitary map $t^{\prime}:E\oplus{\mathcal{A}}\otimes L^{2}(\Omega^{1}(M))\to{\mathcal{A}}\otimes L^{2}(\Omega^{1}(M))$. In this case it can be explicitly computed, but it turns out not to matter.

The $t^{\prime}$ yields a unitary embedding $t:E\to{\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}L^{2}(\Omega^{1}(M)),t=t^{\prime}|_{E}$. Note that since the C^*-module ${\mathcal{A}}\otimes_{\emph{C}\textsuperscript{*}}L^{2}(\Omega^{1}(M))$ has a basis, one does not need Kasparov’s stabilization theorem and an abstract embedding, everything can be done explicitly.

Define $\delta(x)=t(\alpha(x))$. Now $\rho$ induces a left action $\hat{\rho}$ on $E$, $\hat{\rho}(x)(\alpha(y))=(\alpha(xy)-\alpha(x)y)$. But as $\alpha$ is a $\rho$-derivation, $\hat{\rho}(x)=x\alpha(y)$, so $\hat{\rho}(x)$ is multiplication by $x$ and is again identity representation of ${\mathcal{A}}$ acting by multiplication on sections on the bundle. Set $\pi(x)=t\hat{\rho}(x)t^{*}$, again $\pi={\textbf{1}}$ (so the explicit form of $t$ does not come into play). Therefore, with equation 7 in reference,

• •

  The multiplicity space $k_{0}=L^{2}(\Omega^{1}(M))$

• •

  $\delta:{\mathcal{A}}\to{\mathcal{A}}\otimes k_{0},\delta(f)=1\otimes df$.

• •

  The requirement that $\delta(x)^{*}\delta(y)={\mathcal{L}}(x^{*}y)-x^{*}{\mathcal{L}}(y)-{\mathcal{L}}(x)^{*}y$ forces $\delta^{\dagger}(f):=\delta(f^{*})^{*}$ to act by $\delta(f)^{\dagger}(1\otimes\omega)={\langle df,\omega\rangle}=\sum_{i}df(e_{i})\omega(e_{i})\in{\mathcal{C}}(M)$ (using $f=f^{*}$); so $\delta(f)^{\dagger}(a\otimes\omega)=a{\langle df,\omega\rangle}$ works.

• •

  $\sigma=0$ since $\sigma(x)=\pi(x)-x\otimes{\textbf{1}}_{k_{0}}$ since $\pi:{\mathcal{A}}_{0}\to{\mathcal{A}}_{0}\otimes{\mathcal{B}}(k_{0})$ is identity

• •

  ${\mathcal{L}}$ is the generator $-\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}/2$ for the semigroup $e^{-t\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}/2}$.

Therefore, the structure matrix for the Laplacian generated flow is the map

$\displaystyle{\mathcal{A}}_{0}\ni f\to\Theta(f)=\begin{pmatrix}-\mathop{}\!\mathbin{\bigtriangleup}^{\operatorname{{End}}}(f\cdot{\textbf{1}})/2&\delta^{\dagger}(f)\\ 1\otimes df&0\end{pmatrix}\in{\mathcal{B}}(H\otimes(\mathbb{C}\oplus k_{0}))$

(8)

For $x\in M$ fix a small neighborhood of $x$, $V$, with $U\subset V$ open, such that $\bar{U}\subset V$, with coordinates $(x^{i})$ and $(\partial_{x^{i}})\equiv\partial_{i}$ the coordinate vector fields. For any multi-index $\alpha$ we denote $\partial^{\alpha}=\partial_{\alpha_{k}}\dots\partial_{\alpha_{1}}$, and same for $\nabla_{\alpha}$ with $\nabla_{i}\equiv\nabla_{\partial_{i}}$. Ricci-flatness implies for all $f\in C^{\infty}(M)$ (see, for instance, [5, lemma 1.36]),

$\displaystyle[\mathop{}\!\mathbin{\bigtriangleup},\nabla_{i}](f)=\text{Ric}_{i}^{k}\nabla_{k}(f)=0$

(9)

On $(p,q)$-tensors $s,s^{\prime}$ there’s a natural innerproduct by contraction with $g^{ij},g_{ij},{\langle s,s^{\prime}\rangle}=g^{i_{1}j_{1}}\dots g_{m_{1}n_{1}}{s^{\prime}}^{m_{1}\dots}_{i_{1}\dots}s^{n_{1}\dots}_{j_{1}\dots}$. The Levi-Civita connection has a lift to the tensor bundle and an associated connection Laplacian, both also denoted $\nabla,\mathop{}\!\mathbin{\bigtriangleup}$. Denote by $\nabla^{k}u$ the $k^{th}$-covariant derivative and define the point-wise length with the innerproduct[10, § 2.2.1]:

$\displaystyle\ell(\nabla^{k}u)^{2}=g^{i_{1}j_{1}}\dots g^{i_{k}j_{k}}(\nabla^{k}u)_{i_{1}\dots i_{k}}(\nabla^{k}u)_{j_{1}\dots j_{k}}={\langle\nabla^{k}u,\nabla^{k}u\rangle}$

(10)