Twisted Dirac Operator Research Articles

Twisted Dirac Operators and General Kastler–Kalau–Walze Type Theorems for Manifolds with Boundary

In this paper, we establish some general Kastler–Kalau–Walze type theorems on any dimensional manifolds with boundary for twisted Dirac operators.

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.

Twisted Dirac operators and the Kastler-Kalau-Walze type theorem for five dimensional manifolds with boundary

Twisted Dirac operators and the Kastler-Kalau-Walze type theorem for five dimensional manifolds with boundary

On the spectral sets of Inoue surfaces

The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin$^c$ structure, and the corresponding chiral Dirac operators twisted by a flat $\mathbb C^*$--connection. The twisting connection is determined by $z \in \mathbb C^*$, and the points for which the twisted Dirac operators $\mathcal D^{\pm}_z$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus $\alpha^{-1/4} < |z| < \alpha^{1/4}$, where $\alpha >1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$.

Higher localised [formula omitted]-genera for proper actions and applications

For a finitely generated discrete group Γ acting properly on a spin manifold M, we formulate new topological obstructions to Γ-invariant metrics of positive scalar curvature on M that take into account the cohomology of the classifying space B_Γ for proper actions.In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher Aˆ-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of Γ-invariant positive scalar curvature on M. For classes arising from the subring of H⁎(B_Γ,R) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted L2-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of Wang-Wang [24] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of Mathai [17], which provided a partial answer to a conjecture of Gromov-Lawson on higher Aˆ-genera.If M is non-cocompact, we obtain obstructions to M being a partitioning hypersurface inside a non-cocompact Γ-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [12], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.

A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

We give a rigorous construction of the path integral in {mathcal {N}}=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten.

The Index Theorem of Lattice Wilson–Dirac Operators via Higher Index Theory

We give a proof of the index theorem of lattice Wilson–Dirac operators, which states that the index of a twisted Dirac operator on the standard torus is described in terms of the corresponding lattice Wilson–Dirac operator. Our proof is based on the higher index theory of almost flat vector bundles.

Representation theoretic embedding of twisted Dirac operators

Let G G be a non-compact connected semisimple real Lie group with finite center. Suppose L L is a non-compact connected closed subgroup of G G acting transitively on a symmetric space G / H G/H such that L ∩ H L\cap H is compact. We study the action on L / L ∩ H L/L\cap H of a Dirac operator D G / H ( E ) D_{G/H}(E) acting on sections of an E E -twist of the spin bundle over G / H G/H . As a byproduct, in the case of ( G , H , L ) = ( S L ( 2 , R ) × S L ( 2 , R ) , Δ ( S L ( 2 , R ) × S L ( 2 , R ) ) , S L ( 2 , R ) × S O ( 2 ) ) (G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\mathbb R})\times SL(2,{\mathbb R})),SL(2,{\mathbb R})\times SO(2)) , we identify certain representations of L L which lie in the kernel of D G / H ( E ) D_{G/H}(E) .

Manifolds with Many Rarita\u2013Schwinger Fields

The Rarita–Schwinger operator is the twisted Dirac operator restricted to nicefrac 32-spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spinc Dirac operator

We record an answer to the question “In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?”. In the process of doing so, we are naturally led to ask “For which values of ℓ is the connected sum of ℓ closed almost complex manifolds necessarily an almost complex manifold?”. We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n−1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted spinc Dirac operator.

Local index theory and the Riemann–Roch–Grothendieck theorem for complex flat vector bundles

The purpose of this paper is to give a proof of the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

A note on twisted Dirac operators on closed surfaces

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spinc Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results.

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.

Relative Chern character number and super-connection

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd-dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure, we give a formula to express the index of a special twisted Dirac operator.

A mod 2 index theorem for pin− manifolds

We establish a mod 2 index theorem for real vector bundles over 8k + 2 dimensional compact pin− manifolds. The analytic index is the reduced η invariant of (twisted) Dirac operators and the topological index is defined through KO-theory. Our main result extends the mod 2 index theorem of Atiyah and Singer (1971) to non-orientable manifolds.

Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac

Feynman’s time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N = 1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

Chern–Simons invariants on hyperbolic manifolds and topological quantum field theories

We derive formulas for the classical Chern-Simons invariant of irreducible $SU(n)$-flat connections on negatively curved locally symmetric three-manifolds. We determine the condition for which the theory remains consistent (with basic physical principles). We show that a connection between holomorphic values of Selberg-type functions at point zero, associated with R-torsion of the flat bundle, and twisted Dirac operators acting on negatively curved manifolds, can be interpreted by means of the Chern-Simons invariant. On the basis of Labastida-Marino-Ooguri-Vafa conjecture we analyze a representation of the Chern-Simons quantum partition function (as a generating series of quantum group invariants) in the form of an infinite product weighted by S-functions and Selberg-type functions. We consider the case of links and a knot and use the Rogers approach to discover certain symmetry and modular form identities.

Twisted Dirac operators and generalized gradients

On Riemannian or spin manifolds, there are geometric first order differential operators called generalized gradients. In this article, we prove that the Dirac operator twisted with an associated bundle is a linear combination of some generalized gradients. This observation allows us to find all the homomorphism type Weitzenbock formulas. We also give some applications.