This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group $G$ with Lie algebra $\mathfrak{g}$, we choose a compact torus $T$ of $G$, with Lie algebra $\mathfrak{t}$. Each root of $\mathfrak{g}$ with respect to the complexification of $\mathfrak{t}$ restricts to an element in $\mathfrak{t}^*$.
Consider $\mathfrak{t} \otimes \mathbb{R}^3 \setminus \Delta$, where $$ \Delta = \{ \operatorname{ker}(\alpha \otimes \mathbf{1}) ; \alpha \in \Phi \} $$ where $\Phi$ is the set of roots of $\mathfrak{g}$ (with each root restricted to $\mathfrak{t}$) and $\mathbf{1}$ denotes the identity on $\mathbb{R}^3$. Thus each $\alpha \otimes \mathbf{1}$ is a linear map from $\mathfrak{t} \otimes \mathbb{R}^3$ to $\mathbb{R}^3$.
Let us introduce the notation $$ \mathcal{C} = \mathfrak{t} \otimes \mathbb{R}^3 \setminus \Delta. $$
Given $\mathbf{x} \in \mathcal{C}$ and $\alpha \in \Phi$, note that $(\alpha \otimes \mathbf{1})(\mathbf{x}) \in \mathbf{R}^3 \setminus \{\mathbf{0}\}$. But $\mathbf{R}^3$ is isomorphic to the Lie algebra $\mathfrak{su}(2)$ (or actually, to be closer to physicists, we consider $i\mathfrak{su}(2)$ instead, which consists of trace-free hermitian $2$ by $2$ matrices), via the map: $$ x^a \mapsto x^a \sigma_a, $$ for $a = 1, \ldots, 3$, where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are the Pauli matrices. Thus $(\alpha \otimes \mathbf{1})(\mathbf{x}).\mathbf{\sigma}$ is a trace-free hermitian $2$ by $2$ matrix, where $\mathbf{\sigma}$ is the $3$-vector of Pauli matrices. Let $\psi_{\alpha}(\mathbf{x})$ denote a normalized eigenvector of it in $S^3 \subset \mathbb{C}^2$ with respect to its positive eigenvalue. Note that $\psi_{\alpha}(\mathbf{x})$ is only well defined up to a phase factor. Nevertheless, the map from $\mathcal{C}$ to $\mathbb{C}P^1$ which maps $\mathbf{x}$ to $[\psi_{\alpha}(\mathbf{x})]$ is a well defined smooth map, which we will denote by $\hat{\psi}_\alpha$.
We can think of $\psi_\alpha(\mathbf{x})$ as a qubit. This allows us to do the following. If we choose a direction, say $\nu \in S^2$, which for simplicity, I will assume it is $(0, 0, 1)^T$ (the "up" direction), then at each $\mathbf{x} \in \mathcal{C}$, a root $\alpha$ will be positive with probability amplitude $u$ (and thus probability $|u|^2$) and negative with probability amplitude $v$ (and thus probability $|v|^2$), with respect to the chosen direction $\nu$, where $\psi_\alpha(\mathbf{x}) = (u, v)^T \in S^3 \subset \mathbb{C}^2$.
I will not provide a complete description here, but if $G = SU(n)$ and if we form the $|\Phi|$-qubit $$ \psi(\mathbf{x}) = \bigotimes_{\alpha \in \Phi} \psi_{\alpha}(\mathbf{x}), $$ then the Atiyah-Sutcliffe determinant (related to the Atiyah problem on configurations) can then be thought of as the probability amplitude associated to an irreducible $1$-dimensional summand of the above $|\Phi|$-qubit space, with the latter thought of as a representation space of $SU(2)$. Thus the absolute value squared of the Atiyah-Sutcliffe determinant (properly normalized) is the probability that the quantum state $\psi$ is in that $1$-dimensional subspace after "measurement". The Atiyah-Sutcliffe conjecture 2 gives thus a positive lower bound on this probability. The Atiyah-Sutcliffe conjecture 1 (non-vanishing of the Atiyah-Sutcliffe determinant) can then be interpreted as the following statement: there is no $\mathbf{x} \in \mathcal{C}$ for which the corresponding quantum state $\psi(\mathbf{x})$ has a vanishing component in this $1$-dimensional subspace. Stated this way, this seems quite plausible (in my opinion)!
I would like to pursue this line of thought, bringing in qubits and QM to Lie algebras, with the aim of applying such ideas to the Atiyah problem on configurations. I guess I wanted to share my idea, and wonder if similar ideas were considered before. I know there exist quantum groups, but from what I understand, it is the Lie bracket which gets deformed, so it is not the same as what I am considering here.
Thoughts, remarks and comments are welcome. References to related ideas in the literature are welcome too.
Edit 1: I had swept this under the rug, but it is possible to take care of the phase ambiguity of $\psi(\mathbf{x})$ by dividing it with a non-zero scalar-valued quantity with the same kind of homogeneity. So $\psi$ is really a smooth function on $\mathcal{C}$ with values in the vector space $$ \bigotimes_{\alpha \in \Phi} \mathbb{C}^2. $$
Edit 2: given an irreducible representation $V$ of $\mathfrak{g}$ corresponding to a dominant integral weight $\lambda$, we can choose a vector $0 \neq v \in V$ corresponding to the weight $\lambda$. If we then consider the principal $\mathfrak{su}(2)$ in $\mathfrak{g}$ with special element $H$ given by $2 \delta^\vee$, where $\delta^\vee$ is half the sum of the positive coroots (I forgot the corresponding formulas for the $E$ and $F$ elements), then under that representation, the $\mathfrak{su}(2)$ orbit of $v$ defines a representation of $\mathfrak{su}(2)$ which I think is irreducible (is this always the case, please?). Let us denote it by $W_\lambda$.
We could have started with an element say $\lambda'$ in the Weyl orbit $W.\lambda$ of $\lambda$. A construction of mine, generalizing the Atiyah problem on configurations, assigns, to each choice of $\lambda' \in W.\lambda$ a smooth function from $\mathcal{C}$ to $\mathbb{P}(W_\lambda)$, where $\mathbb{P}(.)$ denotes the projectivization operation.
Moreover, using the previously mentioned smooth functions defined on $\mathcal{C}$, one can construct a smooth complex-valued function defined on $\mathcal{C}$, which generalizes the Atiyah-Sutcliffe determinant, starting from a choice of $V$. Such a function is Weyl-invariant and is also invariant under an action of $SO(3)$ on $\mathcal{C}$. I feel like such a function is naturally associated to each choice of irreducible representation of $\mathfrak{g}$ (this vaguely reminds me of the geometric Langlands program, which I know very little about, but I am getting too speculative now, so I will stop).
What kind of applications may such concepts have? I don't know. I just feel these are natural objects to consider. These generalizations of the Atiyah-Sutcliffe (AS) determinant remind me a bit of characters, except they are more like determinants rather than traces, or actually, they can be defined essentially as successive contractions of a multi-qubit using a complex symplectic form on $\mathbb{C}^2$, until we get a scalar.
From this point of view it makes sense to ask the following questions. Can one recover $V_\lambda$, or equivalently $\lambda$, by just knowing the corresponding generalized AS function? How is information about $V_\lambda$ encoded by such generalized AS functions?
I guess it is hard to get help this way since this work is unpublished and I haven't given all the details (in particular, the precise definition of the generalized AS functions), yet (almost) all of the building blocks were already there in my article in the Journal of Exp. Math. ("Weights, Weyl-Equivariant Maps and a Rank Conjecture").
Note: writing my thoughts in this post made me realize I now have a research project.
Motivation: the generalized AS functions are natural and they allow one to generalize the Atiyah problem on configurations.
Questions to try to answer in the research project:
- Can one recover $V_\lambda$, or equivalently $\lambda$, by just knowing the corresponding generalized AS function?
- How is information about $V_\lambda$ encoded by such generalized AS functions?
Thank you MO!
