Let \(\mathbb{T}^3\) be the unit torus parameterized by cyclic coordinates \(x^\alpha\), \(\alpha=1,2,3\), of period \(2\pi\) with Euclidean metric. The massless Dirac operator corresponding to the standard spin structure reads \[ W= -i \begin{pmatrix} \frac\partial{\partial x^3}&\frac\partial{\partial x^1}-i\frac\partial{\partial x^2} \\ \frac\partial{\partial x^1}+i\frac\partial{\partial x^2}&-\frac\partial{\partial x^3} \end{pmatrix}. \] Spectrum of the operator \(W\) is symmetric about zero and zero is an eigenvalue of multiplicity two.
The authors of the paper under review perturb the metric, i.e. consider a metric \(g_{\alpha\beta}(x;\varepsilon)\) the components of which are smooth functions of coordinates \(x^\alpha\), \(\alpha=1,2,3\), and small real parameter \(\varepsilon\), and which satisfies \(g_{\alpha\beta}(x;0)=\delta_{\alpha\beta}\). The main result is asymptotic formula for the eigenvalue with smallest modulus \(\lambda_0\) for arbitrary perturbations of the metric: \[ \lambda_0(\varepsilon)=c\,\varepsilon^2+O(\varepsilon^3) \quad\text{as}\quad\varepsilon\to 0, \] where constant \(c\) calculated explicitly. If the constant \(c\) is nonzero, then this formula tells us that for sufficiently small nonzero \(\varepsilon\) the spectrum of massless Dirac operator is asymmetric about zero.
Two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly are presented. A relation between asymptotic formula and the eta invariant is also established.