We introduce the topologically twisted index for four-dimensional
N
=
1
gauge theories quantized on
AdS
2
×
S
1
. We compute the index by applying supersymmetric localization to partition functions ...of vector and chiral multiplets on
AdS
2
×
T
2
, with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a
R
-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization, the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the
R
-charge.
We consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the ...nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compactness principle for weighted‐variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.
"In this paper, we mainly study the existence of radial solutions for a class of biharmonic equation with a convection term, involving two real parameters λ and ρ. We mainly use a combination of the ...fixed-point index theory and the Banach contraction theorem to prove that there are λ0 > 0 and ρ0 > 0 such the equation admits at least one radial solution for all (λ, ρ) ∈ −λ0, ∞ × 0, ρ0. Keywords: Radial solution, biharmonic equation, index theory, existence."
In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to ...infinity, then the coarse Baum–Connes assembly map is injective, but not surjective, for the associated metric space
X.
Expanders with this girth property are a necessary ingredient in the construction of the so-called ‘Gromov monster’ groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum–Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum–Connes assembly map with these coefficients is an isomorphism.
In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.
This article concerns topologically non-trivial interface Hamiltonians that find many applications in materials science and geophysical fluid flows. The non-trivial topology manifests itself in the ...existence of topologically protected, asymmetric edge states at the interface between two two-dimensional half spaces. The asymmetric transport is characterized by a quantized interface conductivity. The objective of this article is to compute such a conductivity and show its stability under perturbations. We present two methods. The first one computes the conductivity using the winding number of branches of absolutely continuous spectrum of the interface Hamiltonian. This calculation is independent of any bulk properties but requires a sufficient understanding of the spectral decomposition of the Hamiltonian. In the fluid flow setting, it also applies in cases where the so-called bulk-interface correspondence fails. The second method establishes a bulk-interface correspondence between the interface conductivity and a so-called bulk-difference invariant. We introduce the bulk-difference invariants characterizing pairs of half spaces. We then relate the interface conductivity to the bulk-difference invariant by means of a Fedosov-Hörmander formula, which computes the index of a related Fredholm operator and is obtained using semiclassical calculus. The two methods are used to compute invariants for representative 2 × 2 and 3 × 3 systems of equations that appear in the applications.
In this paper, we develop a fixed point index theory for decomposable multivalued maps, that is, compositions of two multivalued nonlinear upper semicontinuous maps. As an application, this fixed ...point index theory is combined with the method of lower and upper solutions in order to obtain new existence, localization and multiplicity results for ϕ-Laplacian problems with discontinuous nonlinearities and nonlinear functional boundary conditions.
Let T be a circle group, and LT be its loop group. We hope to establish an LT-equivariant index theory for Hamiltonian LT-spaces, from the viewpoint of KK-theory. We have already constructed several ...objects in the previous paper (Takata, arXiv:1701.06055), including a Hilbert space H which can be regarded as a “Hilbert space consisting of L2-sections of a Spinor bundle on the infinite-dimensional manifold”, an “LT-equivariant Dirac operator D” acting on H, and the “twisted group C∗-algebra of LT”, in an ad hoc way.
In this paper, we will construct more sophisticated objects for the index problem, in terms of KK-theory. Concretely, we will define analytic index of D as a K-theory element of the “twisted group C∗-algebra of LT” mentioned above. Then, we will also define the “latter half” of the assembly map at the “index element” (H,D). Finally, we will prove that the value of the “latter half” of the assembly map at the “index element”, coincides with the above analytic index of D. This is an infinite-dimensional version of a part of the index theorem for complete Riemannian manifolds with proper cocompact actions of Kasparov.
We are mainly interested in the LT-equivariant index problem of infinite-dimensional manifolds, and so we use the language of representation of LT. However, our theory can be easily generalized to the setting of infinite-dimensional abstract Heisenberg groups equipped with some additional structures. This is potentially useful for researchers of other areas.
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