scholarly journals Manifolds with Many Rarita–Schwinger Fields

2021 ◽  
Author(s):  
Christian Bär ◽  
Rafe Mazzeo
Keyword(s):  
Dirac Operator ◽  
Compact Manifolds ◽  
Overdetermined Problem ◽  
Complex Dimension ◽  
Linearly Independent ◽  
Twisted Dirac Operator ◽  
Flat Tori ◽  
Divergence Free ◽  
Simply Connected ◽  
Einstein Constant

AbstractThe Rarita–Schwinger operator is the twisted Dirac operator restricted to $$\nicefrac 32$$ 3 2 -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.

scholarly journals Special Types of Locally Conformal Closed G2-Structures

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 90 ◽  
Author(s):  
Giovanni Bazzoni ◽  
Alberto Raffero
Keyword(s):  
Lie Groups ◽  
Symplectic Geometry ◽  
Complete Characterization ◽  
Compact Manifolds ◽  
Different Types ◽  
Simply Connected ◽  
Locally Conformal

Motivated by known results in locally conformal symplectic geometry, we study different classes of G 2 -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G 2 -structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G 2 -structures.

scholarly journals THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

2011 ◽  
Vol 90 (1) ◽  
pp. 53-80 ◽  
Author(s):  
JOUKO MICKELSSON ◽  
SYLVIE PAYCHA
Keyword(s):  
Dirac Operator ◽  
Differential Form ◽  
Flat Space ◽  
Invariant Polynomial ◽  
Four Dimensions ◽  
Special Cases ◽  
Twisted Dirac Operator ◽  
Finite Sum ◽  
Logarithmic Residue ◽  
Generalized Laplacian

AbstractWe show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

A super-twisted Dirac operator and Novikov inequalities

2000 ◽  
Vol 43 (5) ◽  
pp. 470-480 ◽  
Author(s):  
Huitao Feng ◽  
Enli Guo
Keyword(s):  
Dirac Operator ◽  
Twisted Dirac Operator

Tropical Geometric Compactification of Moduli, II: A g Case and Holomorphic Limits

2018 ◽  
Vol 2019 (21) ◽  
pp. 6614-6660 ◽  
Author(s):  
Yuji Odaka
Keyword(s):  
Algebraic Curves ◽  
Abelian Varieties ◽  
Moduli Of Curves ◽  
Complex Dimension ◽  
Crucial Difference ◽  
Flat Tori ◽  
Hausdorff Limits

Abstract We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties.

scholarly journals Twistor spaces and the adiabatic limits of Dirac operators

2001 ◽  
Vol 164 ◽  
pp. 53-73 ◽  
Author(s):  
Masayoshi Nagase
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Twistor Space ◽  
Dirac Operators ◽  
Twistor Spaces ◽  
Adiabatic Limits ◽  
Twisted Dirac Operator

We show that a (Spinq-style) twistor space admits a canonical Spin structure. The adiabatic limits of η-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

scholarly journals DETERMINANT LINE BUNDLES AND TOPOLOGICAL INVARIANTS OF HYPERBOLIC GEOMETRY

2002 ◽  
Vol 17 (06n07) ◽  
pp. 951-955
Author(s):  
ANDREI A. BYTSENKO
Keyword(s):  
Dirac Operator ◽  
Hyperbolic Geometry ◽  
Hyperbolic Manifolds ◽  
Line Bundles ◽  
Topological Invariants ◽  
Twisted Dirac Operator ◽  
Determinant Line Bundles

Some remarks on determinant line bundles and topological invariants are given. The index of the twisted Dirac operator acting on real hyperbolic manifolds is computed. We briefly discuss physical applications of results.

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