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Inspired by Geometric interpretation of the determinant of a complex matrix, I ask the following related questions.

The complex determinant is determined up to a global scaling factor by the fact that it is multilinear and skew-symmetric. Let us try to understand the phase of the determinant. For that purpose, let us restrict our attention to $G = U(n)$. Let $E = G \times U(1)$ be the trivial $U(1)$ bundle over $G$. Let $T$ denote a maximal torus of $G$; to fix the ideas, let us assume it is the one consisting of all diagonal matrices in $G$. Let $N(T)$ denote the normalizer of $T$ in $G$. $N(T)$ is the semi-direct product of $T$ with the symmetric group $S_n$ ($S_n$ is the full permutation group on $n$ elements). We can now say that the determinant is a global smooth section $s$ of $E$ over $G$ which is further $N(T)$-equivariant.

Question 1: do these properties determine $s$ up to a global phase factor?

If we let $\nabla$ denote the unique flat connection on $E$ which makes the actual determinant function parallel (i.e. covariantly constant) everywhere, then if we require that the section $s$ be covariantly constant with respect to $\nabla$, then we recover the determinant function uniquely up to a global phase factor, and the symmetry properties under $N(T)$ are automatically satisfied.

What remains to be done, is to describe this flat connection $\nabla$ on $E$ in a way that does not involve the determinant function, so as to avoid circularity.

What we know about $\nabla$ is that it is a flat connection on the trivial $U(1)$ bundle $E$ over $G = U(n)$, such that there exists a global parallel section $s$, unique up to a global phase factor, which is equivariant under $N(T)$.

Question 2: how can $\nabla$ be the defined directly (say using Maurer-Cartan 1-forms and geometric things like that)?

Edit: The whole point of my post, is to be able to define the determinant function on $U(n)$, simply as a parallel section (parallel with respect to the connection $\nabla$) which has the value 1 at the identity.

Malkoun
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1 Answers1

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The answer to your Question 1 is 'no', there are other global smooth sections of $E$ that are invariant under $N(T)$. For example, the section $s:G\to E = G\times \mathrm{U}(1)$ defined by $s(A) = (A,1)$ is also a global smooth section that is invariant under $N(T)$. In fact, if $f:G\to\mathbb{R}$ is any $N(T)$-invariant function (such as, for example, $f(A) = |\mathrm{tr}(A)|^2$), then $s(A) = (A,\mathrm{e}^{i\,f(A)})$ is a smooth $N(T)$-invariant section of $E$.

The answer to your Question 2 is as follows: You need to specify the connection $1$-form on $E$. If $p:E\to \mathrm{U}(1)$ is the projection onto the second factor, then you take the connection $1$-form to be $$ \pi = p^{-1}\,\mathrm{d}p - \mathrm{tr}(g^{-1}\,\mathrm{d}g), $$ where $g:E\to G$ is the projection onto the first factor. Then the section $s:G\to E$ defined by $s(A) = \bigl(A,\det(A)\bigr)$ satisfies $$ s^*\pi = (\det(A))^{-1}\mathrm{d}(\det(A)) - \mathrm{tr}(A^{-1}\,\mathrm{d}A) = 0, $$ so $s$ is a $\nabla$-parallel section of $E$, where $\nabla$ is the connection whose canonical $1$-form is $\pi$.

Robert Bryant
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  • Thank you Prof. Bryant. I was thinking about the exact same formula for the connection 1-form, but did not check the details. Thank you! – Malkoun Jul 01 '17 at 16:31
  • I know my question was perhaps trivial but, I wonder whether there could be any links with the Berry phase, say, of which I know little about. I wonder whether such a connection 1-form has any physical meanings, say. – Malkoun Jul 01 '17 at 16:37
  • Just a minor comment, but for question 1, I really meant $N(T)$ equivariance, rather than invariance, where the $T^n$ part of $N(T)$ acts on $U(1)$ by first multiplying all the factors together, and then multiplying by that result, and the $S_n$ part acts via the signature homomorphism. But you are right that the answer to Question 1 is no. Once you have one such smooth section, one can very simply create many others using small local perturbations, while preserving $N(T)$-equivariance. – Malkoun Jul 01 '17 at 17:37