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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 $(1)$:

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  • $\begingroup$ This expression seems to come from regularizing (taking the determinant after diagonalizing) the $A^{\hat}$-roof genus operator see for example page 8, equation 1.1.13 in Freed's notes on Dirac operators. I haven't read the paper you are interested in; but the physical viewpoint is that this term is, roughly, the index of the Dirac operator. $\endgroup$ May 11 at 17:33
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    $\begingroup$ A nice starting point of view would be to think on how your equation (1) look in the case of a Riemann surface (see equation (1.1) in Anomalies in String Theory with branes) The mathematical intuition on the other hand, is that the $A-$genus realizes the idea that the $K$-theory of a submanifold $S$ depends on the spin structure one chooses for the normal bundle of $S$ in its ambient space. $\endgroup$ May 11 at 17:43
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    $\begingroup$ @RamiroHum-Sah Thank you for the links! I am only an undergraduate and I haven't had an introduction to cohomology, but I would be surprised if that's what Getzler means (if I haven't misunderstood your comment). I've added two recent references at the bottom of my question that prove the local index theorem based on Getzler's rescaling and that contain the same expression, maybe that helps. $\endgroup$
    – Filippo
    May 12 at 12:13
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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.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Chris
    May 16 at 21:11
  • $\begingroup$ I don't see why the matrix $\Omega$ in equation (10) is antisymmetric. Of course, $R(e_k,e_l)=-R(e_l,e_k)$, but I don't see why $e^iR(e_k,e_l)e_j=-e^jR(e_k,e_l)e_i$. But this is very important, since otherwise I can't apply the part "Application to an antisymmetric matrix". Could you please explain? $\endgroup$
    – Filippo
    Aug 8 at 14:03
  • $\begingroup$ @Filippo It's not obvious. It comes from the definition of the spin connection. Here's a little more context: In (10), the indices $a,b$ are the raised/lowered using $\delta_{ab}$, because they refer to a non-coordinate orthonormal basis $e^a_\mu$ in which the Riemannina metric tensor is $g_{\mu\nu}=\delta_{ab}e^a_\mu e^b_\nu$. Equation (10) can be written $\Omega_{ab} = d\omega_{ab}+\omega_{ac}\wedge\omega^c{}_b$ (equation (11.6) in Bilal) where $\omega$ is the spin connection, related to the Levi-Civita connection using $e^a_\mu$. $\endgroup$ Aug 8 at 23:11
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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}

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