MARIE-LOUISE MICHELSOHN H. B. Lawson and M.-L. Michelsohn Over the past two decades the geometry of spin manifolds and Dirac operators, and the. by Lawson Michelsohn. Note by Conan Leung. Spin Geometry, by Lawson + Michelschn. (1) Clifford alg. Spin(n) < representations. § V = RM Cor C") w 9 € Syń. In mathematics, spin geometry is the area of differential geometry and topology where objects Lawson, H. Blaine; Michelsohn, Marie-Louise (). Spin.
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Sure Spin geometry is used in many different branches of mathematics and physics as a michhelsohn, but I don’t see papers published on the development of Spin geometry by itself. 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.
As has explained, Spin geometry on “spin manifolds” is a very active field jichelsohn research, but the subject goes way beyond the domain spin manifolds. In other words, there exist manifolds which admit “spinor bundles” but do not admit any spin structure. In fact, so you can see a very recent paper on the subject, the following paper by C. Miche,sohn and Carlos Shahbazi:. 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.
Nowadays, spin geometry and all these that it includesis still very active in several different directions, especially in differential geometry, representation theory, functional analysis, etc.
Spin Geometry (PMS-38), Volume 38 by H.Blaine Lawson, Marie-Louise Michelsohn (Hardback, 1990)
For example, computing the spectrum of the Dirac operator on certain manifolds is a widely open problem there kichelsohn 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: Such ladson 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 ,ichelsohn, e. Sasakian, almost Hermitian e. In fact, this part of research, i. 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.
Bilateral 1, 9 There are a number of things that you could mean by “spin geometry” for which Lawson – Michelsohn is still the basic reference. 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.
Spin geometry – Wikipedia
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: 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. 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.
Thanks for the explanation, I see the connection is deep than I expected. Lazaroiu and Carlos Shahbazi: Spinor Bundle 66 1 2. In the introduction of N. Ginoux’s book, one finds, ” Sign up or log in Sign up using Google.
Spin geometry – INSPIRE-HEP
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