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New page: monadic decomposition.
this page needs attention. For the moment I have at least added these original articles:
Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS, Tome 37 (1969) 5-26 numdam:PMIHES_1969__37__5_0
Max Karoubi, Espaces Classifiants en K-Théorie, Transactions of the American Mathematical Society 147 1 (Jan., 1970) 75-115 doi:10.2307/1995218
added pointers:
For superconformal field theory, such as D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM, D=6 N=(1,0) SCFT, D=6 N=(2,0) SCFT:
Christopher Beem, Madalena Lemos, Pedro Liendo, Leonardo Rastelli, Balt C. van Rees, The superconformal bootstrap (arXiv:1412.7541)
Christopher Beem, Madalena Lemos, Leonardo Rastelli, Balt C. van Rees, The superconformal bootstrap (arXiv:1507.05637)
Christopher Beem, Leonardo Rastelli, Balt C. van Rees, More superconformal bootstrap (arXiv:1612.02363)
Wrote a section on the associated monad at operad, in terms of the framework introduced under the section titled Preparation.
I am splitting off Zariski topology from Zariski site, in order to have a page for just the concept in topological spaces.
So far I have spelled out the details of the old definition of the Zariski topology on (here).
added to Lie algebra a brief paragraph general abstract perspective to go along with this MO reply
Made some edits and additions at simplicial complex.
just to make links work, I have started a minimum at gravitational wave.
tried to bring the entry orientation into a bit of shape
I expanded Maxwell’s equations by adding the integral form in SI system and then a shorter version of discussion from electromagnetism for the differential form of the equations, both in 3d and 4d formulations. Note also that Ampère’s law is about producing magnetic field from current; while it is Maxwell’s equation, or Ampère-Maxwell which adds the term with the change of electric field, the main discovery of Maxwell. Some people nowdays say generalized Ampère’s law what I wrote, but I am not happy about it as the general form does not generalize it in the straightforward manner, but adds new physics what needs a separate attribution.
I noticed only now that the entry bimodule is in bad shape and needs some attention. For the moment I have added here a mentioning of the 2-category of algebras, bimodules and intertwiners and a pointer to the Eilenberg-Watts theorem.
Todd, thanks for this interesting entry (colimits in categories of algebras)! I left a query box describing a possible simple proof for the last theorem using the adjoint lifting theorem (I hope there are no substantial errors).
at additive functor there was a typo in the diagram that shows the preservation of biproducts. I have fixed it.
Also formatted a bit more.
a stub, for the moment just so as to record pointer to Simpson 12 where “resolution of the paradox” is claimed to be achieved simply by passing from topological spaces to locales
I have added a comment and collected some references on the renormalization freedom in the cosmological constant: here
I have cross-linked this with related entries: renormalization, perturbative quantum gravity and stress-energy tensor
I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.
in order to satisfy links, but maybe really in procrastination of other duties, I wrote something at quantum gravity
I am trying to collect citable/authorative references that amplify the analog of the mass gap problem in particle phenomenology, where it tramslates into the open problem of computing hadron masses and spins from first principles (due to the open problem of showing existence of hadrons in the first place!).
This is all well and widely known, but there is no culture as in mathematics of succinctly highlighting open problems such that one could refer to them easily.
I have now created a section References – Phenomenology to eventually collect references that come at least close to making this nicely explicit. (Also checked with the PF community here)
Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.
Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.
I am giving this bare list of references its own entry, so that it may be !include-ed into related entries (such as topological quantum computation, anyon and Chern-Simons theory but maybe also elsewhere) for ease of updating and synchronizing
I’ll be working a bit on supersymmetry.
Zoran, you had once left two query boxes there with complaints. The second one is after this bit of the original entry (this will change any minute now)
The theory of supergravity is, as a classical field theory, an action functional on functions on a supermanifold which is invariant under the super-diffeomorphism group of .
where you say
Zoran: action functional is on paths, even paths in infinitedimensional space, but not on point-functions.
I think you got something mixed up here. If is spacetime, a field on is the “path” that you want to see. The statement as given is correct, but I’ll try to expand on it.
The second complaint is after where the original entry said
many models that suggest that the familiar symmetry of various action functionals should be enhanced to a supersymmetry in order to more properly describe fundamental physics.
You wrote:
This is doubtful and speculative. There are many models which have supersymmetry which is useful in their theoretical analysis, but the same models can be treated in formalisms not knowing about supersymmetry. Wheather the fundamental physics needs a model which has nontrivial supersymmetry is a speculative statement, and I disagree with equating theoretical physics with one direction in “fundamental physics”. I do not understand how can a model suggest supersymmetry; it is rather experimental evidence or problems with nonsupersymmetric models. Also one should distinguish the supersymmetry at the level of Lagrangean and the supersymmetry which holds only for each solution of the equation of motion.
I’ll rephrase the original statement to something less optimistic, but i do think that supersymmetry is suggsted more by looking at the formal nature of models than by lookin at the nature of nature. If you have a gauge theory for some Lie algebra (gravity, Poincaré Lie algebra) and the super extension of the Lie algebra has an interesting classification theory (the super Poincar´ algebra) then it is more th formalist in us who tends to feel compelled to investigate this than the phenomenologist. Supersymmetry is studied so much because it looks compelling on paper. Not because we have compelling phenomenological evidence. On the contrary.
So, if you don’t mind, I will remove both your query boxes and slightly polish the entry. Let’s have any further discussion here.
stub for quantum computation
Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.
Corrected typo (“Wikiedpia” to “Wikipedia”). Also added pentagon decagon hexagon identity as related entry.
added the full definition to factorization algebra
I gave Seiberg-Witten theory an Idea-paragraph, added the orinal reference and cross-linked with N=2 D=4 super Yang-Mills theory and with electric-magnetic duality.
starting a category:reference-entry.
Just a single item so far, but this entry should incrementally grow as more preprints appear (similar to what we have been doing at Handbook of Quantum Gravity and similar entries).
I know that a soft deadline for submissions of at least one of the sections is this December, so I am guessing this is planned to appear in 2024.
Expansion of references section at differential topology.
Explained why this definition is equivalent to the one at van Kampen colimit.
a bare list of references, to be !include-ed into relevant entries, such as at elliptic cohomology, but also at equivariant elliptic cohomology, elliptic genus, Witten genus etc.
(in an attempt to clean up and harmonize the referencing across all these entries – still some way to go towards that goal, but it should be a start)
have added a minimum on the level decompositon of the first fundamental rep of here.
I have half-heartedly started adding something to Kac-Moody algebra. Mostly refrences so far. But I don’t have the time right now to do any more.
There was a section about W-suspensions titled “Higher inductive types generated by graphs” in the article coequalizer type, so I moved the section into its own page at W-suspension.
Created this article on graph quotients using material split from W-suspension. The “W-suspensions in the first sense” in that article are actually called graph quotients in