Let $K$ denote a finite-dimensional vector space over $\mathbb{C}$. If $K$ is paired with a map $(\cdot,\cdot):K\times K\to\mathbb{C}$ such that, for all $\varphi,\psi,\chi\in K$ and $z,w\in\mathbb{C}$ we have
- $(z \varphi+w\psi,\chi) = z(\varphi,\chi) + w (\psi,\chi)$,
- $(\varphi,\psi) = (\psi,\varphi)^*$,
- and $(\varphi,\psi) = 0$ for all $\psi$ implies $\varphi=0$,
then we say $(K,(\cdot,\cdot))$ is a complex indefinite inner product space (IIS).
I am interested in resources that study "IIS bundles". I would define an IIS bundle as a tuple $(E,M,\pi)$ with $E$ (the total space) and $M$ (the base space) both manifolds and $\pi:E\to M$ a continuous surjection such that $\pi^{-1}(\{m\})$ is an IIS for each $m\in M$. Moreover, I would require that the associated sesquilinear forms $(\cdot,\cdot)_m$ defined on $\pi^{-1}(\{m\})$ would vary smoothly with $m$.
Specifically, have people studied connections on these bundles in detail? The dream resource would be one akin to Barry Simon's paper but for IIS bundles.
I am somewhat versed in the case of Hilbert space bundles and figured it is likely that people have considered the above generalization. I am coming from the perspective of physics whereby the so-called "Berry connection" provides a connection on certain Hilbert space bundles defined over the parameter space of a given Hamiltonian.
I am interested in generalizing this concept to the case where the fibers are one-dimensional (maybe higher) eigenspaces of a given pseudo-Hermitian Hamiltonian dependent on some parameters $m\in M$. Here a linear operator $G:K\to K$ is pseudo-Hermitian (sometimes called para-Hermitian or quasi-Hermitian) if $(G\varphi,\psi) = (\varphi,G\psi)$ for each $\varphi,\psi\in G$. If we demand that the eigenstates of $G(m)$ obey certain orthonormality conditions with respect to $(\cdot,\cdot)$ then a connection similar to the Berry connection can be constructed albeit with the usual inner-product essentially just replaced with the indefinite-inner product. However, I such orthonormality conditions can breakdown at certain points in $M$ and I would guess something interesting can happen there. Specifically, these "singularities" can be much richer than their Hermitian counterparts whereby one encounters eigenvalue degeneracy.
For those who are curious about my physical motivation, this type of problem arises when considering parallel/adiabatic transport of single-particle states in free bosonic systems.
*Edit 1: elaborated on the definition of an IIS.
*Edit 2: One observation I've made is the following: parallel/adiabatic transport of eigenvectors of pseudo-Hermitian matrices can elicit complex "pseudo-Hermitian" Berry phases. This is in contrast to their Hermitian counterparts whereby the Berry phase is always real (equivalently, the connection is always real).