Regarding how to type
A^1 (or any other math expression) in a section heading, ..., it depends.
For traditional publishers of pure mathematics, the answer is "enter it as ordinary math", or
$A^1$. The American Mathematical Society's style guide says this (section 3.1):
Math is always permissible in section heads, but displays are not. Any math in any head appears in math mode; it is not fonted to match the section head.
Translated out of jargon, this means that the font of a math expression is not changed in a bold (or small caps) heading; it should appear exactly as it does in text.
The reason for this is that, rather often, mathematicians use both a lightface and a bold version of the same letter in the same document to mean two different things. (Sometimes, even more variations on a single letter are used!) To embolden a lightface letter (as used in text) could therefore cause confusion, especially among less experienced readers.
That said, the practice is sometimes different in disciplines other than pure math, e.g., physics. In such a case,
\boldmath would serve to embolden the expression:
\boldmath ... $A^1$.
A comment to the question states that this is the reason why
\boldmath was created. Not so. Some fonts used for math are available only in lightface versions, and if a letter or other symbol in such a font is needed in bold form to make a necessary mathematical distinction,
\boldmath (or better,
\bm applied to only that character`) is a way to make the bold form available.