Narrow Results

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\mathrm{\text{Sp}}(4,ℝ)$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\mathrm{\text{Sp}}(4,ℝ)$-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

The original Hilbert and Pólya conjecture is the assertion that the nontrivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However, the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by Number Theory. In this paper, we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularizations, is that the various Dirichlet series attached to such sequences — such as those which are sums over "primes" of (norm P)^-s have a natural boundary, i.e. they cannot be continued beyond the line Re(s) = 0. The main argument is that once the regularized determinant of a Laplacian is meromorphic in s, it follows that the series considered above cannot be a regularized determinant. In other words, we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron–Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\to ℝ$ and $G:X\to ℝ$. We let $\nu =e^{-U}\mu$ and $\mathrm{\Omega }=G^{-1}(-\infty ,0]$. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.