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How and why did mathematicians develop spin-manifolds in differential geometry?

+ 12 like - 0 dislike
156 views

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that this question is not appropriate for math.SE.

As far as I have searched in mathematical physics literature, historically, physicists such as Pauli and Dirac pioneered the concept of spin for a particle. Dirac developed his theory for spin-1/2 electrons by factorizing the Klein-Gordon equation to find a linear relativistic equation that is compatible with Schrodinger's wave equation but doesn't give negative probability density. He factorized the Klein-Gordon equation and found algebraic constraints that gave the Clifford algebra $Cl(1,3)$.

Now that's the algebraic part of the story. What I don't understand is how the concepts in Spin manifolds and Spin geometry were developed from the point of view of differential geometry and topology.

Apparently, Elie Cartan was one of the pioneers and he has written a book about it (I have read its first few chapters). But his language is very different from the language of differential geometry that we use today. He doesn't talk about covering spaces, vector bundles or connections the way that has become common in today's math literature. So, I find it very difficult to trace the chain of thoughts that has led physicists and mathematicians to develop spin geometry in its current language.

I'd like to know how our today's mathematical physics literature has been developed historically and what is the aim of using and further developing this sophisticated language.


This post imported from StackExchange MathOverflow at 2015-12-14 21:43 (UTC), posted by SE-user H. Z.

asked Oct 14, 2015 in Mathematics by H. Z. (60 points) [ revision history ]
recategorized Dec 14, 2015 by Dilaton
Most voted comments show all comments
@AndréHenriques: I am a math undergraduate. I said I wasn't a "mathematician" because mathoverflow is for research level math and my current knowledge of mathematics is definitely not enough to call myself a mathematician. To answer your second question, my interest in spin geometry arises from my interest in Quantum Field Theory.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user H. Z.
The invention of the notion of a spin structure on a manifold is usually credited to André Haefliger (1956), building on work of his Ph.D. advisor Charles Ehresmann.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Carlo Beenakker
Seems like a perfect question for hsm.stackexchange.com

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Willie Wong
I highly recommend reading: TOMONAGA, S. (1974) 1997. The story of spin. Translated by Oka, T. Chicago, University of Chicago Press. The thing to realize, reading later authors such as the DeWitts, is that fiber bundles etc are ways of concisely stating problems in physics. The mathematics appeared in the 1950s. When in 1960s relativity and mechanics was restated into the language of algebraic topology, spin was also naturally translated, along with most of physics. I have a list of reference in historical order to mid 1950s somewhere. This is better as comment until I find it.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Guido Jorg
While I'm not a spin expert (by far) I need to advertise a vastly underappreciated book: W.A.Poor, Differential Geometric Structures. The last section seems to me the best primer on spin geometry available. If you can afford your own math books, don't let it be missing in your library, it's one of the cheapest and best differential geometry books.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Martin Gisser
Most recent comments show all comments
The concept of spinor goes back to work of Elie Cartan en.wikipedia.org/wiki/Spinor

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Liviu Nicolaescu
@GuidoJorg: "reading later authors such as the DeWitts, is that fiber bundles etc are ways of concisely stating problems in physics". If you could refer me to some of Dewitt's works that explains this, that would be wonderful. I am looking forward to seeing the reference list.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user H. Z.

1 Answer

+ 9 like - 0 dislike

I can't give a comprehensive history (if you don't get that here, you might try [hsm.se]---a lot of mathematicians are active on that site), nor can I explain how or why the theory of spin manifolds first emerged. But I think I can say something about how and why spin manifolds became important.

The pre-history is an observation due to some combination of Atiyah and Hirzebruch that a certain characteristic number, the $\hat{A}$-genus, happens to take integer values on spin manifolds (a priori it is a rational number). I believe they were able to prove it using spin cobordism theory (though I'm not sure), but it still called for a convincing conceptual explanation.

This was certainly on Atiyah and Singer's minds when they were working on the index theorem, which computes the Fredholm index of an elliptic (pseudo)differential operator in terms of topological data. This is undoubtedly why in their Index of Elliptic Operators III they introduced the spinor Dirac operator associated to a spin manifold. The index of this operator is obviously an integer on one hand, and on the other hand they calculated that the index is precisely the $\hat{A}$-genus, beautifully explaining Atiyah and Hirzebruch's observation.

Moreover, Atiyah and Singer realized that many of the other operators used to give applications of their index theorem (including the de Rham operator, the signature operator, and the Dolbeault operator) can be constructed in a uniform way using the representation theory of Clifford algebras, so in a certain sense the spinor Dirac operator is the fundamental example in index theory and therefore has its tentacles in many different parts of geometry and topology. This observation explains why spin geometry is so ubiquitous in the theory of positive scalar curvature obstructions, for instance.

The significance of this operator, and therefore spin geometry, was elevated and clarified by the development of K-homology (the homology theory corresponding to the K-theory spectrum). Perhaps the most fundamental mathematical explanation of the importance of spin manifolds is that the spin condition corresponds to orientability for the KO-theory spectrum, and spin manifolds equipped with spinor Dirac operators correspond to the fundamental classes. (The counterpart for traditional K-theory is the spin$^c$ condition.) This was inspired by Atiyah, who argued that there should be a model of K-homology in which the generators are elliptic pseudo-differential operators, and sorted out by Baum and Douglas. The result is that spin geometry infiltrates many problems that involve topological K-theory.

The next (and current) chapter in the story involves the recent interest in loop spaces of manifolds, inspired by physics. It turns out that a spin structure on a manifold is in some sense the same thing as an orientation on its loop space, an observation which Witten used to sketch a proof of the Atiyah-Singer index theorem. There is another kind of structure on a manifold - a string structure - which corresponds to a spin structure on the loop space, and Witten constructed an invariant (the Witten genus) which he argued ought to be the index of a loop space Dirac operator. So far as I know nobody knows how to construct such an operator, but the Witten genus provided motivation for a lot of exciting modern geometry and topology, including topological modular forms and Stolz-Teichner functorial field theories.

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Paul Siegel
answered Oct 14, 2015 by Paul Siegel (90 points) [ no revision ]
H.Z. seems to more like to know how E.Cartan discovered spinors. Do you have more detail (reference) on this generalistically fascinating bit: "It turns out that a spin structure on a manifold is in some sense the same thing as an orientation on its loop space."

This post imported from StackExchange MathOverflow at 2015-12-14 21:44 (UTC), posted by SE-user Martin Gisser

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