Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
What mathematicians call the Mellin transform relating a theta function to its (completed) zeta function
is precisely what physicists call the Schwinger parameter-formulation which takes the partition function of the worldline formalism to the zeta-regulated Feynman propagator
I have tried to briefly mention this relation in relevant entries and to cross-link a bit. But more should be done.
I created a stub on Sam Gitler who has recently died. He was very important not only for his contribution to Yang-Mills theory and the Brown-Gitler spectrum, but also for his creation, with Adem of the school of algebraic topology in Mexico. (I have changed all the mentions of Gitler to be ‘active’.)
I have not yet made this change – as a newbie, I want to get an opinion first.
associative unital algebra describes an -algebra , for commutative, as a “ring under ”. From under category, this is just an object in the coslice category : a map , where is another ring. However, I believe that such a map gives an -algebra only if its image is in the center of (for example, Wikipedia). I’m not sure how to fix that. Maybe we should just remove the “under ” item from that first page. Thoughts?
since the story of the various duals, compactifications and twists of gauge field theories which constitute “Witten’s grand story” (or whatever it should be called in total) gets a bit long, I thought it would be good to have a birds-eye view digest of it – and so I created a survey table
gauge theory from AdS-CFT – table
and included it into some relevant entries.
(There is clearly still room for expansion and further details, but maybe it’s a start).
finally created a minimum at Dirichlet theta function, cross-linked with Dirichlet character and Dirichlet L-function and added it to the table (bottom left entry)
(I have gotten a funny problem with my Opera browser having trouble loading nLab pages. Something makes it choke. For instance when I try to edit a page it tends to show me a blank screen, but when I then go to edit the same page with another browser, then that informs me that the page is locked, so Opera did get to that point, but then got stuck. This happens since the last few days. I tried clearing caches, but it didn’t seem to help. Hm. )
Someone set up a new page with title www.fuw.edu.pl/~slworono/PDF-y/OP.pdf. It seems that this is an attempt by Stanisław Lech Woronowicz to create a nlab entry. Should we just convert it to a usual format page for him? The pdf file is a copy of his paper:Operator theory in the -algebra framework., joint with K.Napi ́orkowski.
started some bare minimum at Spin Chern-Simons theory
had added some minimum at moduli space of framed manifolds.
Are there any general results characterizing moduli of -framings on -manifolds?
brief entry holomorphic line 2-bundle, just to have the link and to record the reference there
added at Beilinson regulator a section Geometric constructions such as to finally give a canonical home to the pointer to Brylinki’s article that David Roberts keeps highlighting (in other threads).
started something at Friedlander-Milnor isomorphism conjecture. But handle with care, I am only just watching the video linked to there.
Added a stub at Waldhausen K-theory of a dg-category. I call this the Waldhausen K-theory and not simply K-theory because I imagine that there should also be a more intrinsic definition not passing through Waldhausen categories or stable infinity-categories.
created classical anomaly
started some minimum at vacuum amplitude. Briefly mentioned relation to a) generating functionals for correlators and b) to zeta functions and c) to expected evanishing in supersymmetric theories
Remarked that in view of b) and c) one is tempted to expect some relation between 1-loop vacuum amplitudes of supersymmetric field/string theories with the Riemann hypothesis. Added a pointer to the article ACER 11 which seems to find just that.
If anyone has further pointers to literature relating vanishing of susy 1-loop vacuum amplitudes and (generalized) Riemann hypotheses, please drop me a note.
created a table-for-inclusion and included it into the relevant entries:
zeta-functions and eta-functions and theta-functions and L-functions – table
Presently it looks like this:
The main statement of this analogy is discussed for instance on the first pages of
I have added some paragraphs at eta invariant, accordingly.
started some minimum at vacuum energy, but running out of battery now.
started some minimum at Mellin transform
This here to collect resources on the observation that – in view of pertinent arithmetic/differential-geometry analogies – an Artin L-function of a Galois representation looks like the zeta function of a Laplace operator of a Dirac operator twisted by a flat bundle.
I currently see this in the literature in three steps:
the Selberg zeta function, which is originally defined as some Euler product, is specifially equal to an Euler product of characteristic polynomials (just as the Artin L-function). This turns out to be due to Gangolli77 and Fried86, and I have collected these references now at Selberg zeta function – Analogy with Artin L-function with a cross-linking paragraph also at Artin L-function itself
more specifically, those characteristic polynomials are those of the monodromies/holonomies of the given group representation, regarded as a flat connection. This is prop. 6.3 in Bunke-Olbrich 94.
finally, that product over characteristic polynomials of monodromies is indeed the zeta function of the bundle-twisted Laplace operator. This is the main point in Bunke-Olbrich 94, somehow, but I still need to fiddle with extracting a more explicit version of this statement.
I made a start on regular representation (via a stub from normalizer). My first thought was to made this a generic regular representation page so I put in definitions for groups and algebras.
Once I’d created the page I thought that it could be said to be an example of a more general thing whereby a monoid acts on itself. However, someone’s already editing the page (that was fast!) so I’ll have to wait to put that in.
(Unless the anonymous coward reads this and decides to put it in themselves!)
Updates to zeta function.
Some time ago I started a stub characteristic variety to record few references, mainly in D-module context. Regarding that the related notion of a characteristic ideal also appears in the treatment of Iwasawa polynomial and Alexander polynomial which Urs wants to understand from the point of view of connections and differential refinements of cohomology, maybe we should do some effort to make some pages which will connect various notions of characteristic ideals and their loci across various subjects. I just recorded
at characteristic ideal for the version in the context of Iwasawa theory.
In some thread here (which I seem to have lost) there was the open question of whether the Selberg zeta function is indeed the zeta function of the corresponding Laplace operator. The answer is of course Yes, I have added the following paragraph to zeta function of a Riemann surface:
That the Selberg zeta function is indeed proportional to the zeta function of a Laplace operator is due to (D’Hoker-Phong 86, Sarnak 87), and that it is similarly related to the eta function of a Dirac operator on the given Riemann surface/hyperbolic manifold goes back to (Milson 78), with further development including (Park 01). For review of the literature on this relation see also the beginning of (Friedman 06).
(the links will only work from within the entry)
gave Langlands correspondence an actual Idea-section.
(Am in a rush and on a horrible wifi connection. Need to proof-read and add more links later.)
added the following story to the Properties-section of Dedekind eta function and also to the Examples-section of functional determinant and zeta function of an elliptic differential operator:
For a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator is
The corresponding functional determinant is
where is the Dedekind eta function.
Recorded the basic idea at Artin L-function
Also: added the following paragraph to the Idea-section at Langlands correspondence (below the numbered two items stating the conjectured correspondence between Galois representations and automorphic representations):
Moreover, to each such automorphic representation is associated an L-function – the automorphic L-function – and in generalization of Artin reciprocity the conjecture is that the Artin L-function associated with the given Galois representation is equal to the automorphic L-function of the corresponding automorphic representation.
I have added the following sentence to class number formula and to all the other entries that the sentence links to:
Given a number field , the Dedekind zeta function of has a simple pole at . The class number formula says that its residue there is proportional the product of the regulator with the class number of
In particular I have also created regulator of a number field and cross-linked it with Beilinson regulator, which I have renamed to higher regulator.
gave zeta function regularization its own entry and expanded a bit, added more pertinent references
just to clean up entries, I have given torsion module its own entry (the keyword used to have non-overlapping discussion at torsion subgroup and at torsion approximation)
added to zeta function of an elliptic differential operator also some first minimum comments on the functional determinant and zeta-function regularization
added the definition of idele class group to group of ideles, with a brief pointer to its role in the moduli stack of line bundles. Added both then to the function field analogy – table
started a minimum at Euler product, the main point being for the moment to mention briefly how Euler product forms naturally appears from the point of view of adelic integration/Iwasawa-Tate theory.
I have renamed the entry pretriangulated dg-category to stable dg-category. I think this is a logical move that emphasizes the close relationship with the notions of stable (infinity,1)-category and stable model category. If anyone disagrees I would be interested to hear why. I have also edited the body of the page to give an exposition more in line with modern references like Cisinski-Tabuada. However I have adopted the term dg-presheaf for what is usually called (right) dg-module. This seems much more natural to me, but again I am open to hearing any arguments against it. When I get a minute I will also update the entry dg-category to match the conventions of this page.
Also, I never liked the term “quasi-equivalence”. I think that this should just be called equivalence of dg-categories, or maybe Dwyer-Kan equivalence if this is too ambiguous. Any thoughts?
@Urs: I do not quite agree with the sentence “This is unrelated to other notions of monads” in Beilinson monad.
One can indeed view the Beilinson monad as the monad of an adjoint equivalence between (interpreted as the heart of and some category of linear complexes over an exterior algebra (the Koszul dual of the Cox ring of ).
created a minimum at adelic integration, for the moment this is just a glorified pointer to Fesenko 08, section 3
At differential cohesion there used to be the statement that every object canonically has a “spectrum” given by , but the (simple) argument that indeed satisfies the axioms of a structure sheaf used to be missing. I have now added it here.
added references.
Any book that develops a bit of algebraic geometry of non-unital commutative rings or one that discusses what would be hte major things that break?
started a wee bit at nonunital Ek-algebra
started augmented A-infinity algebra
I added a clarifying clause to infinity-field so it now reads
The Morava K-theory A-∞ rings are essentially the only -fields. See at Morava K-theory – As infinity-Fields, where and we define as .
This is from Lurie’s lectures. What precisely does he mean? He says in lecture 24 that for any field that its E-M spectrum is an infinity-field, so the “essentially” is doing some work. Is the idea that all infinity-fields are -modules (cor 10, lecture 25), so the essentially cover things?
On another point, would there be a higher form of the rational/p-adic fracturing of , involving the ?
collected some introductions and surveys
started completion of a module, for the moment mainly so as to record a bunch of basic definitions and facts about completion of -modules from DAG12
I have given the notion of canonical transformation as used in Hamiltonian mechanics its own brief page.
So in particular I removed the redirect of that term to canonical morphism and instead added disambiguation lines on the top of both entries. I think this is justified: the term “canonical transformation” has been standard since ancient times in Hamiltonian mechanics and is in each and every textbook on the matter. On the other hand the same term as referring to canonical morphisms was mainly the proposal of one single person in category theory, and never caught up much, I think. (Also I find the term ill-motivated in category theory in the first place).
Therefore, while the disambiguation redirects ensure that both notions still can be found, I think it is clear that the default meaning must be that in Hamiltonian mechanics.
started a table-for-inclusion arithmetic cohesion – table and included it into relevant entries.
In the course of this I started a minimum at adic residual.
created a minimum at higher local field and higher arithmetic geometry and added disambiguation with “E-∞ arithmetic geometry”.
I have worked on the entry synthetic differential infinity-groupoid;
added a brief remark in the Idea section;
spelled out statement and proof that is totally -connected over ;
began some discussion on how the induced relative fundamental -groupoid functor is : the infinitesimal path -groupoid functor, such that is the de Rham space of and a morphism an -stack of D-modules on . But this deserves more discussion.
Concerning the writeup of the second point I had myself confused about the direction of one of the arrows for a while. Hope I got it right now.
Look at hausdorff locally convex space.
created an entry for torsion approximation, mainly to be able to refer to this concept by a link.
Unfortunately, I am lacking chocolat medals (as well as the authority to award them), but thanks to the author (presumably Todd) who graced Karoubi envelope with the proof that smooth manifolds result from open sets by idempotent splitting.
I have added a reference to Lawvere’s Perugia notes where this appeared as an exercise.
Entre parenthèses: it appears to me that it’d be better to have the proof at the page for smooth manifolds and to mention the result only at Karoubi envelope as I think this is kind of a butterfly at Karoubi though a beautiful one but an important result for manifolds.
polished and expanded somewhat the entry groupoid object in an (infinity,1)-category
created a bare minimum at Fermat quotient.
I created generalized uniform structures - table in the style of all of those tables that Urs makes and included it on most of the relevant pages. (I left the pages on the simplest concepts, the binary relations.)
I hope that the headers “monad on an object” and “monad on a pro-object” are accurate. These should be objects in an equipment, I think. Perhaps Mike can help me figure out what equipments are relevant here.
I have greatly expanded the basic definition at prometric space to show other ways to look at the concept.
Started adjoint lifting theorem. For now, it only includes a version for lifting left adjoints (I still haven’t read Johnstone’s 1975 paper for the case of right adjoints). I hope there is no substantial error in the appliaction for cocompleteness.
New entry movable singularity (for ODE’s with complex time) redirecting also fixed singularity.
started stub on Tannaka duality for geometric stacks, but need to interrupt now.
The theorem there can be read as justifying the point of view of derived noncommutative geometry to regard the 2-algebra as a valid replacement for the 1-algebra .
I edited group scheme and scheme a bit.