Integral formulas in Riemannian geometry

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress was made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, the classification of defects in uniaxial nematic liquid crystals was reviewed and expounded upon. Particular attention was paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied $XY$ model or the Heisenberg ferromagnet.

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Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals

We use homotopy theory to extend the notion of strong and weak topological insulators to the non-stable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly d-dimensional", i.e. not realizable by stacking lower-dimensional insulators, a more restrictive definition of "strong" is required. However, this does not exclude weak topological insulators from being "truly d-dimensional", which we demonstrate by an example. Additionally, we prove some useful technical results, including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime, as well as the rigorous justification of replacing $T^d$ by $S^d$ and $T^{d_k}\times S^{d_x}$ by $S^{d_k+d_x}$ as is common in the current literature.

Homotopy theory of strong and weak topological insulators

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss–Bonnet theorem using heat equation methods, to give a new proof of a result of Kuzʼmina and Labbi concerning the Euler–Lagrange equations of the Gauss–Bonnet integral, and to give a new derivation of the Euh–Park–Sekigawa identity.

Universal curvature identities

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spinc case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

GEOMETRY OF SPIN AND SPIN c STRUCTURES IN THE M-THEORY PARTITION FUNCTION

https://archive.maths.nuim.ie/staff/stefan/PP/indeta.pdf

The Computation of η-Invariants on Manifolds with Free Circle Action

We derive variational formulae for natural first order energy function-
als and obtain criteria for the stability of isometric immersions. This
generalizes known results for the classical energy, the p-energy and the
exponential energy

/pdf/tension-field-and-index-form-of-energy-type-functionals-3jfr1jg3qe.pdf

Tension field and index form of energy-type functionals

We show how far the local defect index determines the behaviour of an ordered medium in the vicinity of a defect.

https://archive.maths.nuim.ie/staff/stefan/PP/g.pdf

The Global Defect Index

We classify compact asystatic G-manifolds with fixed point singular orbits in cohomogeneity ⩽3 up to equivariant diffeomorphism. From this we derive existence results for invariant metrics of positive Ricci curvature on such objects. We also develop non-existence results for invariant metrics of positive Ricci curvature in cohomogeneity four.

https://archive.maths.nuim.ie/staff/stefan/PP/co.pdf

Manifolds of low cohomogeneity and positive Ricci curvature

We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mn Vk between compact connected Riemannian manifolds. If M contains a sub-manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

/pdf/infima-of-universal-energy-functionals-on-homotopy-classes-20scxkew9f.pdf

Stefan Bechtluft-Sachs

Papers

The Computation of η-Invariants on Manifolds with Free Circle Action

Tension field and index form of energy-type functionals

The Global Defect Index

Manifolds of low cohomogeneity and positive Ricci curvature

Infima of universal energy functionals on homotopy classes