# Open questions in “Spin geometry”

This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics as a tool, but I don't see papers published on the development of Spin geometry by itself. Is this true? Loosely speaking, by "Spin geometry" I mean here the area in mathematics that would have the "Spin geometry" book of Lawson and Michelsohn as the basic reference.

Are there any open questions in Spin geometry?

Thanks.

• There are a number of things that you could mean by "spin geometry" for which Lawson - Michelsohn is still the basic reference. For instance, you might mean the representation theory of Clifford algebras, in which case I don't think there's much left to say over $\mathbb{R}$ or $\mathbb{C}$, but I don't know the state of the art over other fields. Or you might mean the geometry of spin manifolds, in which case a lot is not known. For instance, a lot of people are interested in the topology of the space of positive scalar curvature metrics on a compact spin manifold. – Paul Siegel Aug 16 '15 at 16:21
• @PaulSiegel: Thanks for the information. I was meaning also the geometry of spin manifolds. However, the topic that you mention seems to be only "accidentally" (please correct me if I am wrong) related with Spin geometry: the goal would be to study the space of positive scalar curvature metrics, not study something inherent of Spin geometry. – Bilateral Aug 16 '15 at 23:06
• Widening slightly the scope of what "spin geometry" might mean, I'd be very happy to know whether this question is an open one or not. – Igor Khavkine Aug 17 '15 at 5:18
• @Bilateral On the contrary, there are strong arguments that the theory of positive scalar curvature invariants should be considered part of spin geometry. For instance, it is a theorem of Gromov and Lawson that every compact simply connected non-spin manifold of dimension at least 5 admits a metric of positive scalar curvature. The underlying reason for the connection between spin geometry and positive scalar curvature is the Lichnerowicz formula for the square of the spinor Dirac operator. – Paul Siegel Aug 17 '15 at 5:51
• @PaulSiegel: Thanks for the explanation, I see the connection is deep than I expected. – Bilateral Aug 17 '15 at 21:26

As @314159 has explained, Spin geometry on "spin manifolds" is a very active field of research, but the subject goes way beyond the domain spin manifolds. The key point to notice is that every pseudo-Riemannian spin manifold $(M,g)$ admits a bundle of irreducible Clifford modules over the bundle of Clifford algebras $Cl(M,g)$, which is usually called the "spinor bundle", but remarkably enough, the opposite is not true. In other words, there exist manifolds which admit "spinor bundles" but do not admit any spin structure. This is already clear just by looking at the literature on $Spin^{c}$ manifolds, but it turns out the possibilities go beyond $Spin^{c}$ manifolds as well. In fact, so you can see a very recent paper on the subject, the following paper by C. Lazaroiu and Carlos Shahbazi:

https://arxiv.org/abs/1606.07894

classifies all the spinorial structures associated to a bundle of irreducible real Clifford modules, and gives the topological obstructions for the existence of such bundles on any pseudo-Riemannian manifold. You can check that these obstructions are in general different from that required by a spin structure, and correspond to what they call a "Lipschitz structure". They even have some application of his formalism to physics. This proves that "spin geometry" is not always associated to a spin structure and indeed it is a very rich and subtle construction. I think spin geometry in this more general setting is very little explored, and I believe is going to attract a lot of attention in the future.

Spin geometry is an active field and of course is not exhausted in the book of Lawson and Michelson. In fact, nowadays, there are new books on the topic, including more recent results.

In the introduction of N.Ginoux's book, one finds,

''...one of the most famous achievements of spin geometry was the discovery of a topological obstruction to positive scalar curvature, as a relatively straightforward application of Atiyah-Singer Index theorem''

and below

''...the non commutative geometry made the Dirac operator one of its keystones, as it allows to reconstruct the given Riemannian spin manifold from its canonical spectral triple''

or

''...special eigenvectors of the Dirac operator, called Killing spinors, have become some of the physicists' main tools in the investigation of supersymmetric models in string theory...''

These are just a few reasons that make spin geometry so important.

Nowadays, spin geometry (and all these that it includes), is still very active in several different directions, especially in differential geometry, representation theory, functional analysis, etc. For example, computing the spectrum of the Dirac operator on certain manifolds is a widely open problem (there are a few spaces that we have a complete picture and most of them are homogeneous).

Moreover, (and since the question focus on developments of the spin geometry itself): The last decades there is an enormous progress on Dirac operators related to metric connections different than the Levi-Civita connection and their relationship with the field of non-integrable $G$-structures and special geometries. Such connections are not any more torsion-free (I mean metric connections with skew-torsion, vectorial torsion, etc) and under specific conditions, they become nice replacements of the L-C connection, in the sense that they preserve the special geometric structure (as the L-C connection does in the integrable case). On the other hand, there is a plethora of special structures carrying such connections, e.g. Sasakian, almost Hermitian (e.g. nearly-K\"ahler), or $G_2$-geometries.

In fact, this part of research, i.e. Dirac operators associated to metric connections with skew-torsion, like the characteristic connection $$\nabla^{c}=\nabla^{g}+\frac{1}{2}T$$ associated to a special $G$-structure, where $0\neq T\in\Lambda^{3}T^{*}M$ denotes the torsion of $\nabla^{c}$,
became a subject of interest in theoretical and mathematical physics as well, e.g. in string theory of Type II, where the basic model consists of a Riemannian spin manifold endowed with a 3-form, a dilation function and a spinor field, which is actually parallel under the characteristic connection $\nabla^{c}$ (interpreted as supersymmetric transformation, while the torsion $T$ plays the role of B-field).

From the mathematical point of view, the most famous of such type Dirac operator is the ''cubic Dirac operator'' (with applications both in representation theory and differential geometry). Notice finally that nowadays, there are even generalizations of Killing spinor fields, etc.