Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Factorization of log-corrections in AdS4/CFT3 from supergravity localization

We use the Atiyah-Singer index theorem to derive the general form of the one-loop corrections to observables in asymptotically anti-de Sitter (AdS4) supersymmetric backgrounds of abelian gauged supergravity. Using the method of supergravity localization combined with the factorization of the supergravity action on fixed points (NUTs) and fixed two-manifolds (Bolts) we show that an analogous factorization takes place for the one-loop determinants of supergravity fields. This allows us to propose a general fixed-point formula for the logarithmic corrections to a large class of supersymmetric partition functions in the large N expansion of a given 3d dual theory. The corrections are uniquely fixed by some simple topological data pertaining to a particular background in the form of its regularized Euler characteristic χ, together with a single dynamical coefficient that counts the underlying degrees of freedom of the theory.

Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

It is known that the Atiyah-Patodi-Singer index can be reformulated as the eta invariant of the Dirac operators with a domain wall mass which plays a key role in the anomaly inflow of the topological insulator with boundary. In this paper, we give a conjecture that the reformulated version of the Atiyah-Patodi-Singer index can be given simply from the Berry phase associated with domain wall Dirac operators when adiabatic approximation is valid. We explicitly confirm this conjecture for a special case in two dimensions where an analytic calculation is possible. The Berry phase is divided into the bulk and the boundary contributions, each of which gives the bulk integration of the Chern character and the eta-invariant.

On the quantisation and anomalies of antisymmetric tensor-spinors

We perform the quantisation of antisymmetric tensor-spinors (fermionic p-forms) $$ {\psi}_{\mu_1\dots {\mu}_p}^{\alpha } $$ ψ μ 1 … μ p α using the Batalin-Vilkovisky field-antifield formalism. Just as for the gravitino (p = 1), an extra propagating Nielsen-Kallosh ghost appears in quadratic gauges containing a differential operator. The appearance of this ‘third ghost’ is described within the BV formalism for arbitrary reducible gauge theories. We then use the resulting spectrum of ghosts and the Atiyah-Singer index theorem to compute gravitational anomalies.

Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

Modular symmetry and zeros in magnetic compactifications

We discuss the modular symmetry and zeros of zero-mode wave functions on two-dimensional torus T2 and toroidal orbifolds T2/ℤN (N = 2, 3, 4, 6) with a background homogeneous magnetic field. As is well-known, magnetic flux contributes to the index in the Atiyah-Singer index theorem. The zeros in magnetic compactifications therefore play an important role, as investigated in a series of recent papers. Focusing on the zeros and their positions, we study what type of boundary conditions must be satisfied by the zero modes after the modular transformation. The consideration in this paper justifies that the boundary conditions are common before and after the modular transformation.

Understanding the index theorems with massive fermions

The index theorems relate the gauge field and metric on a manifold to the solution of the Dirac equation on it. In the standard approach, the Dirac operator must be massless to make the chirality operator well defined. In physics, however, the index theorem appears as a consequence of chiral anomaly, which is an explicit breaking of the symmetry. It is then natural to ask if we can understand the index theorems in a massive fermion system which does not have chiral symmetry. In this review, we discuss how to reformulate the chiral anomaly and index theorems with massive Dirac operators, where we find nontrivial mathematical relations between massless and massive fermions. A special focus is placed on the Atiyah–Patodi–Singer index, whose original formulation requires a physicist-unfriendly boundary condition, while the corresponding massive domain-wall fermion reformulation does not. The massive formulation provides a natural understanding of the anomaly inflow between the bulk and edge in particle and condensed matter physics.