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May 10, 2008
E8 Quillen Superconnection
Posted by Urs Schreiber
A remark on the nature of Quillen superconnections with values in -graded Lie algebras, such as .
May 8, 2008
(Action) Lie Infinity-Algebroids
Posted by Urs Schreiber
Taking a day off at HIM (will report tomorrow on what Liang Kong has been teaching us about vertex operator algebras from Segal’s CFT axioms, using theorems by Huang), today I am giving a talk in Hamburg in our series on BRST-BV formalism (as you will have guessed), the goal being to illuminate the geometric -categorical meaning of the BRST complex regarded as a -algebroid:
The things I’ll say and draw to the board are those at the beginning of section 1 of
On action Lie -groupoids and action Lie -algebroids
(pdf)
Pernicious Symbolization
Posted by David Corfield
Gian-Carlo Rota upset a number of analytic philosophers when in The pernicious influence of mathematics upon philosophy he likened their use of symbolism to someone paying for groceries with Monopoly money. But it didn’t take an outsider to object to such practices. Gilbert Ryle, one of the so-called ‘ordinary language philosophers’, reviewing Rudolf Carnap’s Meaning and Necessity in Philosophy XXIV, 1949, remarks on Carnap’s
…growing willingness to present his views in quite generous rations of English prose. He still likes to construct artificial ‘languages’ (which are not languages but codes), and he still interlards his formulae with unhandy because, for English speakers, unsayable Gothic letters. But the expository importance of these encoded formulae seems to be dwindling. Indeed I cannot satisfy myself that they have more than a ritual value. They do not function as a sieve against vagueness, ambiguity or sheer confusion, and they are not used for the abbreviation or formalization of proofs. Calculi without calculations seem to be gratuitous algebra. Nor, where explicitness is the desideratum, is shorthand a good substitute.
(Interlard is literally ‘to intersperse with alternate layers of lard’.)
May 7, 2008
Integrability of Lie Brackets
Posted by Urs Schreiber
I would like to advertise the beautiful review
Marius Crainic, Rui Loja Fernandes
Lectures on Integrability of Lie Brackets
arXiv:math/0611259
on the integration of Lie algebroids to Lie groupoids .
Section 3.2 has a nice review of the method of integrating Lie algebras to Lie groups using equivalence classes of paths in the Lie algebra. Then in 3.3 it is discussed how this generalizes to Lie algebroids.
In section 5.3 of On action Lie -groups and action Lie -algebras (pdf) I describe how this integration method is secretly (well, it’s pretty obvious, but still deserves to be made explicit) nothing but forming the fundamental path groupoid of the smooth classifying space of -valued differential forms:
May 6, 2008
Ambimorphic?
Posted by Urs Schreiber
A question on the interpretation of the fundamental path -groupoid as an ambimorphic object: an -groupoid valued co-presheaf:
Quaternionic Analysis
Posted by David Corfield
Nobody else has mentioned it, but perhaps a few extracts from a paper by one of the founding fathers of categorification, Igor Frenkel, might be of interest, even if not on our topic:
Quaternionic Analysis, Representation Theory and Physics, Igor Frenkel and Matvei Libine.
Abstract
We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.
Theorems Into Coffee II
Posted by John Baez
Nobody instantly solved my first coffee challenge, but I hope that interest is brewing. Maybe some of you will perk up if I throw another $15 in the pot?
It’s a slight variation on the same theme: taking a nice category where the morphisms are matrices, interpreting it as a PROP, and asking what sort of algebraic gadget is defined by this PROP.
May 4, 2008
Theorems Into Coffee
Posted by John Baez
As the famous quote goes, “a mathematician is a machine for turning coffee into theorems”. But every chemical reaction is reversible, at least under the right conditions. So, there’s got to be some way turn theorems into coffee!
And now you can do it here.
FQXi
Posted by Urs Schreiber
I agreed to participate in the review panel of FQXi’s new round of grant competition.
(You can find useful discussion of the nature of FQXi in blog posts over at Cosmic Variance: The Foundational Questions Institute (Anthony Aguirre) and Foundational Questioners announced.)
The review meeting is June 3 to June 5 in Santa Cruz, California.
I am just mentioning this in case anyone feels like taking advantage of me being in the US with flight tickets having been taken care of :-) I just need to check how many days the HIM allows me be be away. On June 10 I give a talk in Hamburg, so I need to be back at least by then.
April 30, 2008
Questions on 2-covers
Posted by David Corfield
The following may well have been talked about by John in his lectures but I didn’t see it explicitly there, so I’ll ask.
Since at level 1, we have
A Galois connection between subgroups of the fundamental group and path-connected covering spaces of X for path-connected . A universal covering space is simply connected.
should we not expect a level 0 analogue:
A ‘connection’ between subsets of the set of connected components and covering spaces of whose locally constant fibres are either empty or ?
This puts these latter ‘covering spaces’ into correspondence with the poset of subsets of , so that the universal covering space with truth valued fibres is the empty set.
So do we have then:
A Galois 2-connection between sub-2-groups of the fundamental 2-group and path-connected 2-covering spaces (with groupoid fibres) of X for path-connected , a universal 2-covering space being 2-connected?
So, if we think of a nice space with nontrivial first and second homotopy, say the loop space of the 2-sphere, do we have a correspondence between 2-covering spaces and sub-2-groups of its fundamental 2-group? And is there a universal 2-connected 2-cover with 1- and 2-homotopy killed off?
April 29, 2008
Returning to Lautman
Posted by David Corfield
I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le réel physique, Vrin, 2006, a copy of which arrived through the post the other day. It’s remarkable how much contemporary mathematics Lautman covers – class field theory, algebraic topology, analytic number theory, etc.
At the same time as I was reading Lautman I became excited by category theory, via Colin McLarty’s Uses and Abuses of the History of Topos Theory, British Journal for the Philosophy of Science 1990 41(3):351-375, and Saunders Mac Lane’s Mathematics: Function and Form, and noticed an affinity with Lautman’s thinking, supported by a remark made by Jean Dieudonné in his 1977 Preface:
La “montée vers l’absolu” qu’il y discerne, et où il voit une tendance générale, a pris en effet, grâce au langage des catégories, une forme applicable à toutes les parties des mathématiques: c’est la notion de ‘foncteur représentable’ qui joue aujourd’hui un rôle considérable, tant dans la découverte que dans la structuration d’une théorie. (p. 36)
That Lautman worked with Claude Chevalley and Charles Ehresmann may not be unconnected.
April 28, 2008
Dual Formulation of String Theory and Fivebrane Structures
Posted by Urs Schreiber
We would like to share the following:
Hisham Sati, U.S. and Jim Stasheff
Dual Formulation of String Theory and Fivebrane Structures
(pdf)
Update: now available as arXiv:0805.0564
Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual” version of the anomaly cancellation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under electric-magnetic duality.
This expands on some of the material announced in section 3 of
H. S., U.S., J. S.
-connections and application to String- and Chern-Simons transport
(arXiv, blog pdf)
but so far concentrates on the topological aspects of Fivebrane structures. A discussion of the differential geometry of Fivebrane 6-bundles with connection – which are nonabelian differential cocycles that are to super 5-branes as String 2-bundles with connection are to superstrings and as ordinary Spin bundles with connection are to spinning particles – as well as of the Chern-Simons 7-bundles with connection obstructing their existence, will be given elsewhere, following the general approach described in
On nonabelian differential cohomology
(pdf).
I’d be grateful for comments, but should add that I’ll be travelling in Ireland until 3rd of May, which will reduce my responsiveness here for that period.
April 27, 2008
Charges and Twisted Bundles, IV: Anomaly Canellation
Posted by Urs Schreiber
Last time # I had talked about how the presence of electric and magnetic charges makes the would-be action functional of (bosonic, abelian, possibly higher) gauge theory a section of a potentially nontrivial line bundle with connection on the space of fields, here called . This time I talk about how this “anomaly cancels” against another anomly caused by spinorial fields: the Pfaffian line bundle.
April 25, 2008
Charges and Twisted Bundles, III: Anomalies
Posted by Urs Schreiber
In quantum physics a phenomenon called “(quantum) anomalies” plays a big role.
There are several different phenomena which go by this name, I think, and in the literature they don’t always tell you which one is which.
But generally, anomalies have to do with “global topological twists” (notably nontrivial fiber bundles) related to the configuration space of a field theory.
These twists are called “quantum” because they tend to become visible and/or relevant only when a classical theory is quantized.
They are called “anomalies”, I’d say, because to a large extent in physics the approach is to pretend that working locally is fine – until one happens to run head-on into global issues. A mathematician might say at this point: “We made a mistake at the beginning in assuming that everything is globally well defined, instead there may be obstructions to doing so”. The physicist says: “My naive approach of working locally is fine, but since it fails to work in this situation, it is the situation which is not normal: it is anomalous.”
A matter of perspective.
In any case, when you see the word “(quantum) anomaly” you should think obstruction to some global trivializability problem.
There is one particular kind of anomaly which arises in gauge theory and in higher gauge theory in the presence of electric and magnetic charges. This one is fully understood technically, to a large extent under control in concrete examples, and is the source of some very beautiful deep connections between physics on the one hand and index theory and differential cohomology on the other.
A good and rather exhaustive description, both as far as physical examples and as far as the mathematical machinery goes, of this phenomenon is given in
D. Freed
Dirac Charge Quantization and Generalized Differential Cohomology
arXiv:hep-th/0011220
This is one of the deepest articles on physics that I know of. The insights described there will rank one day with the central conceptual insights in physics of past centuries, I think. After differential equations in the 19th century and then later differential geometry in the 20th century, this identifies differential cohomology as the mathematical concept at the heart of physics.
The idea is simple: the action functional of gauge theory, in the presence of electric and magnetic charges, is, when you look closely, not really, in general, a function, the way they teach you in school. Rather, it is a multivalued function: a section of a line bundle over configuration space.
But whatever path integral quantization really is, it requires you to integrate the action against a measure. For that to be meaningful, the bundle that it is a section of must be trivializable.
The nontriviality of the bundle on configuration space that the action “functional” is a section of is “the” local anomaly: a measure for the failure of the starting point of the quantization procedure to be well defined.
But in fact more is true: the bundle on configuration space here is not just a bundle, but a bundle with connection: a differential cocycle. In order for everything to be well defined we need this bundle not only to be trivializable and have a flat connection, it also needs to have trivial connection. If not, we say we have a global anomaly.
So this kind of “anomaly” appearing in (higher) gauge theory in the presence of electric and magnetic charges is an obstruction which is measured by a class in differential cohomology.
As far as I know this was first realized in the study of the higher gauge theories that appear as effective target space field theories in string theory, notably in Witten’s discussion of the “5-brane anomaly”. But this is just where it was first realized. Remarkably, as nicely discussed at the beginning of section 2 the phenomenon is entirely visible in the ordinary 1.5 centuries old electromagnetism. And all the more complicated cases follow from this one simply by replacing line bundles with connection everywhere by higher differential cocycles (higher line bundles with connection).
Despite its crucial relevance, there is surprisingly little literature on this – which is however certainly due to the fact that the required differential cohomology theory is not widely familiar, and in fact in the process of being worked out more fully.
A big step in the direction of discussing the general theory of differential cohomology is the article
M.J. Hopkins, I.M. Singer
Quadratic functions in geometry, topology,and M-theory
arXiv:math/0211216.
Various aspects of its application to higher (abelian) quantum gauge theory have been discussed in
Daniel S. Freed, Gregory W. Moore, Graeme Segal
The Uncertainty of Fluxes
arXiv:hep-th/0605198