Definition of $k^0$ in Getzler's proof of the Atiyah-Singer index theorem I have read Getzler's proof of the Atiyah-Singer index theorem. Unfortunately, I don't understand the most important definition.
The definition
For all $(t,x)\in(0,\infty)\times\mathbf{R}^n$, Getzler makes the following definition:
\begin{equation}\tag{1}
k^0_t(x)=(4\pi t)^{-n/2}\det\left(\frac{t\Omega/2}{\sinh{t\Omega/2}}\right)^{1/2}\exp\left[tF-\frac{1}{4t}\left(\frac{t\Omega/2}{\tanh{t\Omega/2}}\right)_{ij}x^ix^j\right]
\end{equation}
Here's what Getzler says regarding the RHS:

The symbol $\Omega$ denotes the Riemannian curvature at the point $0$ thought of as
an antisymmetric $n\times n$ matrix of $2$-forms and $F$ is the curvature of the connection at $0$ thought of as a hermitian $m\times m$ matrix of $2$-forms.

Why $(1)$ might be different from what you think ...
If $K$ is the curvature of a covariant derivative (e.g. the Riemannian curvature), a local frame for the tangent bundle on $U\subset M$ defines a matrix ${K^i}_j\in\Omega^2(U)$ of $2$-forms. However, that's not what Getzler is talking about:
Let $\Delta$ be the vector space of spinors and let $\Lambda(n)\cong\mathrm{End}(\Delta)$ be the complexification of $\Lambda^*\mathbf{R}^n$.
As mentioned in the beginning of the proof of the theorem on page $3$,
\begin{equation}
k^0\colon(0,\infty)\to C^\infty(\mathbf{R}^n)\otimes\mathrm{End}(\Delta\otimes\mathbf{C}^m)
\end{equation}
is the heat kernel associated to
\begin{equation}
D^0:=\sum_i\left(\partial_i+\frac{1}{4}\Omega_{ij}(0)x^j\right)^2+F(0).
\end{equation}
I think that Getzler is referring to elements of $\Lambda^2\mathbf{R}^n$ as $2$-forms - for example, I'm pretty sure that the $F$ in $(1)$ is actually defined by
\begin{equation}
F=\sum_{\mu<\nu}F_{\mu\nu}(0)e^\mu\wedge e^\nu\in\mathrm{End}(\mathbf{C}^m)\otimes \Lambda(n).
\end{equation}
Question
How is the expression
\begin{equation}
\det\left(\frac{t\Omega/2}{\sinh{t\Omega/2}}\right)^{1/2}
\end{equation}
in $(1)$ defined? In case this helps, I also found two recent proofs of the local index theorem that are based on Getzler's rescaling method and that contain the expression on the RHS of $(1)$:

*

*Proposition 25.10 on page 114 of Walpuski's notes.


*Lemma 2.10 on page 9 of Ponge's paper.
 A: The question asks how the quantity
$$
 \det\left(\frac{t\Omega/2}{\sinh t\Omega/2}\right)^{1/2}
\tag{2}
$$
is defined. I'll build up the definition one piece at a time, and then I'll relate it to Getzler's statement about the nature of the quantity $k^0_t(x)$ shown as equation (1) in the question.
 The power series 
Consider the function defined by
$$
 f(x)\equiv \frac{x/2}{\sinh(x/2)}
 = \frac{x/2}{(e^{x/2}-e^{-x/2})/2}
\tag{3}
$$
where $x$ is an abstract variable. We can expand this function in powers of $x$. If $x$ is a real variable, then the series never terminates. However, in equation (2), the abstract variable $x$ is replaced by a different kind of object, and we'll see that the series does terminate in that case. Therefore, the expansion in powers of $x$ is all we need, and we can think of (3) as a convenient abbreviation for the resulting series. The first few nonzero terms are
\begin{align*}
 f(x) &= 1 - \frac{1}{3!}\left(\frac{x}{2}\right)^2
  +\left(\frac{1}{(3!)^2}-\frac{1}{5!}\right)
  \left(\frac{x}{2}\right)^4
  - O(x^6)
\\
 &=1 - \frac{x^2}{24} + \frac{7x^4}{5760} - O(x^6).
\tag{4}
\end{align*}
This involves only even powers of $x$, so we can define
$$
 g(x^2)\equiv f(x).
\tag{5}
$$
 Application to an antisymmetric matrix 
Consider any antisymmetric matrix, $M=-M^T$, and use a basis in which it has the form
$$
 M = \left(
 \begin{array}{cc|cc|cc|c}
  0 & m_1  & 0 & 0 & 0&0&\cdots \\
  -m_1 & 0 & 0 & 0 & 0&0& \\
\hline
  0  & 0 & 0 & m_2 & 0&0&\\
  0  & 0 & -m_2 & 0 & 0&0&\\
\hline
  0  & 0 & 0  & 0 & 0 & m_3 & \\
  0  & 0 & 0  & 0 & -m_3 & 0 & \\
\hline
  \vdots &  & & &&& \ddots
 \end{array}
 \right).
\tag{6}
$$
The square of $M$ is
$$
 M^2 = \left(
 \begin{array}{cc|cc|cc|c}
  -m_1^2 & 0  & 0 & 0 & 0&0&\cdots \\
  0 & -m_1^2 & 0 & 0 & 0&0& \\
\hline
  0  & 0 & -m_2^2 & 0 & 0&0&\\
  0  & 0 & 0 & -m_2^2 & 0&0&\\
\hline
  0  & 0 & 0  & 0 & -m_3^2 & 0 & \\
  0  & 0 & 0  & 0 & 0 & -m_3^2 & \\
\hline
  \vdots &  & & &&& \ddots
 \end{array}
 \right).
\tag{7}
$$
Since this is diagonal, we have no trouble using it as the input $x$ in the power series defined by equations (3)-(5):
$$
 f(M) = g(M^2) = \mathrm{diag}\big(
  g(-m_1^2),
  g(-m_1^2),
  g(-m_2^2),
  g(-m_2^2),
  g(-m_3^2),
  g(-m_3^2),...\big)
\tag{8}
$$
The square root of $f(M)$ is given by replacing each diagonal element $g(-m_k^2)$ with its square root. Use this to deduce
$$
 \det \big(f(M)\big)^{1/2}=\prod_k g(-m_k^2).
\tag{9}
$$
 The curvature $2$-form 
In equation (2), $\Omega$ is the curvature $2$-form associated with the Levi-Civita connection familiar from general relativity. The curvature $2$-form is really a matrix's worth of $2$-forms, with components
$$
 \Omega^a{}_b\propto \frac{1}{2}
 \sum_{\mu,\nu}R^a{}_{b\mu\nu}dx^\mu\wedge dx^\nu,
\tag{10}
$$
where $R^a{}_{b\mu\nu}$ is the Riemann curvature tensor. I'm using the notation from section 11.2 in Bilal's Lectures on Anomalies. The matrix $\Omega$ is antisymmetric, so we can use a basis in which it has the form (3), where now the quantities $m_k$ are individual $2$-forms.
Let $t$ be an ordinary real variable. When the abstract variable $x$ in (3) is replaced by $t\Omega$, we get a series of powers of $t\Omega$. These powers are defined by a product that includes both the exterior product of differential forms and also the usual matrix product, because $\Omega$ is a matrix's worth of differential forms. For example, $\Omega^2$ has components
$$
 (\Omega^2)^a{}_c \propto 
 \frac{1}{4}
 \sum_b\left(\sum_{\mu,\nu}R^a{}_{b\mu\nu}dx^\mu\wedge dx^\nu\right)
 \wedge
 \left(\sum_{\rho,\sigma}R^{b}{}_{c\rho\sigma}
 dx^{\rho}\wedge dx^{\sigma}\right).
\tag{11}
$$
The sum over $b$ implements the matrix-product aspect. The series (4) terminates at a finite order in this case, because on an $n$-dimensional manifold, there are no differential forms of degree higher than $n$.
 Putting the pieces together 
Altogether, equation (2) is defined by taking the quantities $m_k$ in (9) to be the individual $2$-forms (10) when $\Omega$ is written a basis like (6). The power series (4) is automatically truncated at a finite order, because there are no differential forms of degree higher than $n$ on an $n$-dimensional manifold. The quantity $t$ is an ordinary real variable.
 Consistency with Getzler's paper 
In Getzler's paper, the quantity $k_t^0(x)$ shown in the question as equation (1) is said to be an element of this frightening-looking space:
$$
 C^\infty((0,\infty)\times\mathbb{R}^n)\otimes \Lambda(n)\otimes
 \mathrm{End}(\mathbb{C}^m).
\tag{12}
$$
Let's break it down:

*

*The $(0,\infty)$ part specifies the range of the variable $t$.


*The $\mathbb{R}^n$ part refers to the coordinates $x^k$.


*The notation $C^\infty((0,\infty)\times\mathbb{R}^n)$ means that $k_t^0(x)$ is a function of the $n+1$ variables $t$ and $x$.


*The factor $\Lambda(n)$ refers to the exterior algebra of differential forms on $\mathbb{R}^n$, as explained in the second paragraph on page 112 in Getzler's paper. The components of $\Omega$ belong to $C^\infty(\mathbb{R}^n)\times \Lambda(n)$, so the quantity (2) belongs to $C^\infty((0,\infty)\times\mathbb{R}^n)\times \Lambda(n)$.


*The factor $\mathrm{End}(\mathbb{C}^m)$ is a fancy way of denoting the set of $m\times m$ matrices over $\mathbb{C}$. That accounts for the matrix nature of the $F$-dependent factor, because $F$ is an $m\times m$ matrix's worth of $2$-forms representing the curvature of a non-abelian gauge field, as explained in the text surrounding equation (3) in Getzler's paper. The quantity $F(x=0)$ belongs to $\Lambda(n)\otimes\mathrm{End}(\mathbb{C}^m)$, as suggested in the question.
This is all consistent with the definition described above.
A: This answer is a warning. It seems like the definition of $k^0$ paper can easily be misunderstood if one doesn't understand the entire proof:
By putting multiple equations together$^1$, one obtains
\begin{equation}\tag{1}
\lim_{t\to 0}k_t(0,0)=\mathrm{Str}\,\Big[(4\pi t)^{-n/2}\hat{A}(t\Omega)\wedge\exp(tF)\Big](0)=(2\pi i)^{-n/2}\Big[\hat{A}(\Omega)\wedge\mathrm{ch}(F)\Big]_n(0)
\end{equation}
Since the first equation in $(1)$ contains the limit $t\to 0$, you would probably expect the second equation in $(1)$ to hold in the limit $t\to 0$, too. And, depending on how $F$ is  interpreted, this does actually work, at least in the case $\Omega=0$:
\begin{equation}
\lim_{t\to 0}(4\pi t)^{-n/2}\cdot\mathrm{Str}\,\exp\left(t\sum_{k>l}F_{kl}(x)\gamma^k\gamma^l\right)=(2\pi i)^{-n/2}\mathrm{ch}_n(F)(x)
\end{equation}
However, I think that Getzler didn't include the limit $t\to 0$ for a good reason:
$k_t^0(0)$ is evaluated at $t=1$:
Getzler's $k^0_t(x)$ seems to correspond to the $r^{x_0}(0,t,x)$ in Reutter's essay and the equation
\begin{equation}
\lim_{t\to 0}\mathrm{Str}\,k_t(x_0,x_0)\,\mathrm{d}x=(2/i)^{n/2}\mathrm{tr}\,\Big[r^{x_0}(0,1,0)\Big]_n
\end{equation}
is one of the main results (Reutter writes $p_t$ instead of $k_t$ to denote the heat kernel of $D^2$). In addition, this equation suggests that Getzler omitted the volume form in $(1)$.
I recommend reading Reutter's essay (in particular, section $2.3.5$ on the Getzler scaling) to everyone who wants to understand Getzler's paper in detail.

$^1$ Firstly,
\begin{equation}
\lim_{t\to 0}k_t(0,0)=\lim_{\epsilon\to 0}(2/i)^{n/2}\int k^\epsilon_t(0)=(2\pi i)^{-n/2}\Big[\hat{A}(\Omega)\wedge\mathrm{Tr}\,\exp(F)\Big]_n(0)
\end{equation}
(page $115$). Secondly,
\begin{equation}
\mathrm{Str}\,a=(2/i)^{n/2}\int a
\end{equation}
for all $a\in\Lambda(n)$ and
\begin{equation}
\lim_{\epsilon\to 0}k^\epsilon_t(0)=k^0_t(0)\in\Lambda(n)\otimes\mathrm{End}(\mathbf{C}^m).
\end{equation}
