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I am a postdoc in maths with a degree in theoretical physics. I am interested in mathematical structures in quantum field theory and string theory, see here.
Jan 3 |
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Reference Request for TQFTs
For mathematical supersymmetry I recommend the articles by D. Freed listed here ncatlab.org/nlab/show/… particularly the "Five lectures on supersymmetry" ncatlab.org/nlab/show/Five+lectures+on+supersymmetry and his article with Deligne ncatlab.org/nlab/show/supersymmetry#DeligneFreed99 |
Jan 3 |
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Reference Request for TQFTs
I'd suggest Witten 91 ncatlab.org/nlab/show/topological+quantum+field+theory#Witten91 and Cordes-Moore-Ramgoolam 94 ncatlab.org/nlab/show/… |
Jan 3 |
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Reference Request for TQFTs
Did you look at the original articles by Witten here ncatlab.org/nlab/show/… ? |
Jan 3 |
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Reference Request for TQFTs
A commented list of references with further pointers is here: ncatlab.org/nlab/show/… |
Dec 28 |
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pullback of Lie algebra cocycles along Cartan connections
Literature on the cocycles that I am thinking of is listed here: ncatlab.org/nlab/show/… Discussion of the need to prolong these to closed forms on curved superspace is listed here: ncatlab.org/nlab/show/… I am preparing some notes on this extension problem. Will send you more once its ready. |
Dec 27 |
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pullback of Lie algebra cocycles along Cartan connections
Thanks for your comment. My main motivating class of examples is that where $\mathfrak{g}$ is an extended super-Poincare Lie algebra and $\mathfrak{h}$ is the Lie algebra of the Lorentz group. Then $\mathfrak{g}/\mathfrak{h}$ is extended Minkowski-spacetime regarded as a super-translation Lie algebra. There are a finite number of exceptional super-Lie algebra cocycles on these, and in supergravity theory it is of key interest to prolong these to closed forms over a curved superspacetime, hence over an $(\mathfrak{h} \hookrightarrow \mathfrak{g})$-Cartan geometry. |
Dec 25 |
awarded | Announcer |
Dec 18 |
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What is torsion in differential geometry intuitively?
Further putting this all together, it seems we should say that given an $(H \to G)$-Cartan connection ncatlab.org/nlab/show/Cartan+connection (which subsumes G-structures and soldering forms) then torsion is the projection of its curvature under $\mathfrak{g}\to \mathfrak{g}/\mathfrak{h}$. |
Dec 17 |
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What, precisely, does Klein's Erlangen Program state?
Riemannian geometry (and many other types of geometries) is subsumed by the globalization of Klein geometry known as "Cartan geometry". ncatlab.org/nlab/show/Cartan+geometry |
Dec 15 |
awarded | Announcer |
Dec 11 |
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Artin L-function and Zeta function of twisted Dirac operator
John Baez points out that discussion of more analogy along these lines is in D. Brown "Lifting properties of prime geodesics" ncatlab.org/nlab/show/Selberg+zeta+function#Brown09 . On p. 9 there is a comprehensive table of pertinent analogies. |
Dec 10 |
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Questions about analogy between Spec Z and 3-manifolds
I am wondering if it is not better to think of the prime fields as being not like chosen knots, but like all the prime geodesics in a hyperbolic 3-manifold. Then everything falls into place: we want the Selberg zeta function given as a product over determinants of holonomies over all these prime geodesics (as in Bunke-Olbrich 94) and so that's the product over all determinants of Frobenius, hence is the Artin L-function. |
Nov 28 |
awarded | Announcer |
Nov 13 |
awarded | Custodian |
Nov 13 |
reviewed | Close Websites for Math Shopping |
Nov 12 |
awarded | Announcer |
Nov 12 |
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Synthetic vs. classical differential geometry
Using modern developments in type theory, one may take this quite far by formalulating the axioms for SDG, incarnated in the guise of (differential) cohesion, in the new foundations given by homotopy type theory. Doing so leads to "cohesive homotopy type theory" ncatlab.org/nlab/show/cohesive+homotopy+type+theory . This connects to the question about applications of SDG in physics: in cohesive HoTT a large amount of cutting-edge modern physics may naturally be found and studied. That's the point of dcct ncatlab.org/schreiber/show/… |
Nov 12 |
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Synthetic vs. classical differential geometry
+QiaochuYuan makes a nice point above about how synthetic reasoning is about passing from axioms for objects to axioms for their categories. Another perspective on that is foundational: Everyone is familiar with how the ZFC axioms lay foundations for sets. Similarly the axioms of "intuitionistic type theory" lay foundations for sets that may have geometric structure (may be interpreted as sheaves of sets). In "synthetic" reasoning we are adding axioms to this intuitionistic type theory that further specify the nature of this geometry. This way SDG connects to the foundations of mathematics. |
Nov 12 |
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Synthetic vs. classical differential geometry
Consider the standard textbook on SDG by Anders Kock (ncatlab.org/nlab/show/synthetic+differential+geometry#KockBookA). The first chapter titled "The synthetic theory" develops differential geometry purely logically just by speaking intuitionistic logic+KL-axioms. Then chapter II "Categorical logic" recalls what it means to have a model for these axioms in a category, and finally chapter III "Models" discusses actual such models in certain categories. It may be useful to read these chapters in reverse order, because chapter III connects to classical theory, while I abstracts from that. |
Nov 12 |
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Synthetic vs. classical differential geometry
Yes, sorry, I should have used less jargon. As David Roberts and Qiaochu Yuan point out, the "models" that I was referring to are simply the categories (the toposes) which satisfy the axioms of SDG (or else those of differential cohesion). SDG in itself is the theory, in the formal sense of formal logic, obtained by starting with intuitionistic set theory and adding to it the Kock-Lawvere axiom. The act of "speaking in this logic" is what people refer to when they say that "in SDG every function is smooth". This logic now has "models" in terms of categories built in ordinary set theory. |