Atiyah Patodi Singer Index Theorem Research Articles

Deformation theory of asymptotically conical Spin(7)-instantons

We develop the deformation theory of instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. By relating the deformation complex with spinors, we identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold.Finally we apply this theory to describe the deformations of Fairlie–Nuyts–Fubini–Nicolai Spin(7)-instantons on R8, where R8 is considered to be an asymptotically conical Spin(7)-manifold asymptotic to the cone over S7. We calculate the virtual dimension of the moduli space using Atiyah–Patodi–Singer index theorem and the spectrum of the twisted Dirac operator.

Curved domain-wall fermion and its anomaly inflow

AbstractWe investigate the effect of a U(1) gauge field on lattice fermion systems with a curved domain-wall mass term. In the same way as the conventional flat domain-wall fermion, the chiral edge modes appear localized at the wall, whose Dirac operator contains the induced gravitational potential as well as the U(1) vector potential. In the case of anS1 domain-wall fermion on a two-dimensional flat lattice, we find a competition between the Aharonov–Bohm(AB) effect and a gravitational gap in the Dirac eigenvalue spectrum, which leads to an anomaly inthe time-reversal (T) symmetry. Our numerical result shows a good consistency with the Atiyah–Patodi–Singer index theorem on a disk inside the S1 domain wall, which describes the cancellation of the T anomaly between the bulk and edge. When the U(1) flux is squeezed inside one plaquette, and the AB phase takes a quantized value π mod $2\pi \mathbb {Z}$, the anomaly inflow drastically changes: the strong flux creates another domain wall around the flux to make the two zero modes coexist. This phenomenon is also observed in the S2 domain-wall fermion in the presence of a magnetic monopole. We find that the domain-wall creation around the monopole microscopically explains the Witten effect.

Secondary higher invariants and cyclic cohomology for groups of polynomial growth

We prove that if $\Gamma$ is a group of polynomial growth, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocycle, we thus define a higher analogue of Lott’s delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if $\Gamma$ is of polynomial growth, then there is a well-defined pairing between delocalized cyclic cocycles and $K$-theory classes of $C^\*$-algebraic secondary higher invariants. When this $K$-theory class is that of a higher rho invariant of an invertible differential operator, we show this pairing is precisely the aforementioned higher analogue of Lott’s delocalized eta invariant. As an application of this equivalence, we provide a delocalized higher Atiyah–Patodi–Singer index theorem, given that $M$ is a compact spin manifold with boundary, equipped with a positive scalar metric $g$ and having fundamental group $\Gamma=\pi\_1(M)$ which is finitely generated and of polynomial growth.

An Equivariant Atiyah–Patodi–Singer Index Theorem for Proper Actions I: The Index Formula

AbstractConsider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index ${\operatorname {index}}_g(D)$, in terms of a parametrix for $D$ and a trace associated to $g$. We prove an equivariant Atiyah–Patodi–Singer index theorem for this index. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if $g=e$ is the identity element; if $G$ is a finitely generated, discrete group, and the conjugacy class of $g$ has polynomial growth; and if $G$ is a connected, linear, real semisimple Lie group, and $g$ is a semisimple element. In the classical case, where $M$ is compact and $G$ is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah–Patodi–Singer index theorem. In part II of this series, we prove that, under certain conditions, ${\operatorname {index}}_g(D)$ can be recovered from a $K$-theoretic index of $D$ via a trace defined by the orbital integral over the conjugacy class of $g$.

Anomaly and superconnection

Abstract We study anomalies of fermions with spacetime-dependent mass. Using Fujikawa’s method, it is found that the anomalies associated with the U(N)+ × U(N)− chiral symmetry and U(N) flavor symmetry for even and odd dimensions, respectively, can be written in terms of superconnections. In particular, the anomaly for a vector-like U(1) symmetry is given by the Chern character of the superconnection in both even- and odd-dimensional cases. It is also argued that the non-Abelian anomaly for a system in D-dimensional spacetime is characterized by a (D + 2)-form part of the Chern character of the superconnection which generalizes the usual anomaly polynomial for the massless case. These results enable us to analyze anomalies in the systems with interfaces and spacetime boundaries in a unified way. Applications to index theorems, including the Atiyah–Patodi–Singer index theorem and a Callias-type index theorem, are also discussed. In addition, we give a natural string theory interpretation of these results.

Atiyah–Patodi–Singer index theorem for domain walls

We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the Atiyah–Patodi–Singer theorem that relates the index to the bulk integral of Pontryagin density and η-invariants of auxiliary Dirac operators on the domain wall. Thus the index is expressed through the global chiral anomaly in the volume and the parity anomaly on the wall.

Atiyah–Patodi–Singer Index Theorem―Domain-Wall Junction among Particle Theory, Condensed Matter Physics, and Mathematics

Atiyah–Patodi–Singer Index Theorem―Domain-Wall Junction among Particle Theory, Condensed Matter Physics, and Mathematics

Euler flows and singular geometric structures

Tichler proved (Tischler D. 1970 Topology 9 , 153–154. ( doi:10.1016/0040-9383(70)90037-6 )) that a manifold admitting a smooth non-vanishing and closed one-form fibres over a circle. More generally, a manifold admitting k -independent closed one-form fibres over a torus T k . In this article, we explain a version of this construction for manifolds with boundary using the techniques of b -calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem . Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. ( N. Y. ) 264 , 864–896. ( doi:10.1016/j.aim.2014.07.032 )). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b -symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638 ) and what we call b -Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352 , 5781–5794. ( doi:10.1090/S0002-9947-00-02651-9 )). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Eta and rho invariants on manifolds with edges

We establish existence of eta-invariants as well as of the Atiyah–Patodi–Singer and the Cheeger–Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature and the spin Dirac operator. We derive an analogue of the Atiyah–Patodi–Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments are based on the microlocal analysis of the heat kernel asymptotics associated to the Dirac laplacian of an incomplete edge metric. As an application, we discuss stability results for the two rho-invariants we have defined.

Two-cocycle twists and Atiyah–Patodi–Singer index theory

AbstractWe construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Index theorems and domain walls

The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an η invariant on the boundary (which defines the parity anomaly). We show that the APS index theorem holds for configurations with domain walls that are defined as surfaces where background gauge fields have discontinuities.

Harmonic spinors on a family of Einstein manifolds

The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub–NUT, Eguchi–Hanson and with the Fubini–Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of and extend to as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah–Patodi–Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.

Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.

Higher (odd) dimensional quantum Hall effect and extended dimensional hierarchy

We demonstrate dimensional ladder of higher dimensional quantum Hall effects by exploiting quantum Hall effects on arbitrary odd dimensional spheres. Non-relativistic and relativistic Landau models are analyzed on S2k−1 in the SO(2k−1) monopole background. The total sub-band degeneracy of the odd dimensional lowest Landau level is shown to be equal to the winding number from the base-manifold S2k−1 to the one-dimension higher SO(2k) gauge group. Based on the chiral Hopf maps, we clarify the underlying quantum Nambu geometry for odd dimensional quantum Hall effect and the resulting quantum geometry is naturally embedded also in one-dimension higher quantum geometry. An origin of such dimensional ladder connecting even and odd dimensional quantum Hall effects is illuminated from a viewpoint of the spectral flow of Atiyah–Patodi–Singer index theorem in differential topology. We also present a BF topological field theory as an effective field theory in which membranes with different dimensions undergo non-trivial linking in odd dimensional space. Finally, an extended version of the dimensional hierarchy for higher dimensional quantum Hall liquids is proposed, and its relationship to quantum anomaly and D-brane physics is discussed.

The noncommutative family Atiyah–Patodi–Singer index theorem

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fibration with boundary as a pairing of the family Chern–Connes character for a manifold with boundary with the Chern character of an idempotent matrix. We define the family b -Chern–Connes character and then we prove that it is entire and give its variation formula. By this variation formula, we prove another noncommutative family Atiyah–Patodi–Singer index theorem. Thus, we extend the results of Getzler and Wu to the family case.

A note on the higher Atiyah–Patodi–Singer index theorem on Galois coverings

Let $\Gamma$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $\Gamma$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (\Gamma;\mathbb{C})$ which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way.

A cohomological formula for the Atiyah–Patodi–Singer index on manifolds with boundary

The main result of this paper is a new Atiyah–Singer type cohomological formula for the index of Fredholm pseudodifferential operators on a manifold with boundary. The nonlocality of the chosen boundary condition prevents us to apply directly the methods used by Atiyah and Singer in [4, 5]. However, by using the K-theory of C*-algebras associated to some groupoids, which generalizes the classical K-theory of spaces, we are able to understand the computation of the APS index using classic algebraic topology methods (K-theory and cohomology). As in the classic case of Atiyah–Singer ([4, 5]), we use an embedding into a Euclidean space to express the index as the integral of a true form on a true space, the integral being over a C∞-manifold called the singular normal bundle associated to the embedding. Our formula is based on a K-theoretical Atiyah–Patodi–Singer theorem for manifolds with boundary that is inspired by Connes' tangent groupoid approach, it is not a groupoid interpretation of the famous Atiyah–Patodi–Singer index theorem.

Corners in M-theory

M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: the lift to bounding 12 dimensions of M-theory on anti-de Sitter spaces, ten-dimensional heterotic string theory in relation to 12 dimensions, and the two M-branes within M-theory in the presence of a boundary. The M2-brane is taken with (or as) a boundary and the worldvolume of the M5-brane is viewed as a tubular neighborhood. We then concentrate on the (variant) of the heterotic theory as a corner and explore analytical and geometric consequences. In particular, we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner(s). The analysis involves considering M-theory on disconnected manifolds and makes use of the extension of the Atiyah–Patodi–Singer index theorem to manifolds with corners and the b-calculus of Melrose.

Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s = 0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523–540], to a noncompact case.

A renormalized index theorem for some complete asymptotically regular metrics: The Gauss–Bonnet theorem

We study the Gauss–Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré–Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah–Patodi–Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L 2 -cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m , we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function.