Arcadian Functor

A building nor’west breeze means a fine winter’s day in Kaikoura, so I headed up the coast with three friendly astronomers. We did our best to avoid the seal colony, but there were over 200 seals, with many pups, blocking our path around the coast. Unable to detour over crumbling cliffs, we were forced to run the gauntlet of a few angry, but fortunately lazy, fellas. After a late burger and chips in town we stopped off for scallops and mussels (in garlic) at a caravan by the beach, before heading back home.

Tommaso Dorigo is being heavily criticised for his post about Lisa Randall. Although I understand the issue, the hypocrisy here is just running red down the walls! After two definitely-without-a-doubt on topic remarks (supporting Tommaso) on Asymptotia, I was put onto the moderation queue! So who is the worst offender here? The self righteous, politically correct heroes? Personally I’d rather discuss physics with a hot blooded Italian man who listens to what I say (and perhaps even gives constructive criticism) any day.

Sure, a world where one can have a decent conversation without worrying about anything would be nice, but let’s be realistic here: that just isn’t possible without some degree of equality.

Update: I’m told that moderation queues are automatic (sorry, Clifford) but if you read the comments on the relevant threads you might see my point.

Both Heisenberg’s honeycombs and Kapranov’s Fourier transform suggest the use of non-commutative polynomial invariants. Now discussions with Matti Pitkanen about knot invariants has led to the idea of introducing non-commutativity for twisted ribbons, as in the original Mulase and Waldron symplectic matrix ensemble.

The Jones polynomial, the two variable homflypt polynomial and the Kauffman polynomial are all commutative. The most interesting invariant in this context is the Kauffman polynomial, defined in terms of the (planar isotopy) Kauffman bracket, because the latter has always been associated to the idea of diagrams as numbers. Is there a way to associate letters to (a skein relation for) twists that would extend these invariants to a non-commutative braided ribbon invariant? There is certainly an analogy between the three basic crossing diagrams ($L_{+}$, $L_{-}$ and $L_{0}$) for the homflypt polynomial and the three possible twist elements: left handed half twist, right handed half twist and flat ribbon.

As it turns out, Bar-Natan seems to be thinking about such things! As he says on his knot wiki, “If you know what this is about, good. If not, bummer.”

Back in Lesson 21 we looked at how holography suggests turning pants diagrams into little disc diagrams. The reverse process would be an operad inclusion map taking little discs to tube-like trees. This is in fact what was considered by Getzler in the paper recently discussed at The Cafe. In this paper, Getzler shows that BV algebras come from a little disc operad which can be included in an operad of moduli of genus zero surfaces (the pants pictures).

But it is not the ordinary little discs operad, where discs may be mapped into a larger disc allowing for translations and a dilation of each little disc. The BV disc operad also allows for a rotation of each little disc. That is, we allow a $\mathbb{C}^{\times}$ action on the disc. This extension of the usual little discs operad may be viewed in terms of braid groups. In fact, the little disc and framed little disc operads are just homologies:

$LD(k) = H_{\ast} (\mathbb{P}_{k})$, $FLD(k) = H_{\ast} (\mathbb{Z}^{k} \times \mathbb{P}_{k})$

where $\mathbb{P}_{k}$ is the pure braid group on $k$ strands. There is a fibration of the framed operad over the usual disc operad with a torus fibre $T^{k}$ which represents all the phases of the framed little discs. The factor of $\mathbb{Z}^{k}$ actually comes from a ribbon braid group, with generators for twisting the ribbons. This associates the rotation of a little disc with a twisting of ribbons, as previously discussed. Recall also that the connection to the topological moduli operad imbues the index $k$ (number of strands) with a dimensional meaning, since the dimension of the moduli spaces increases with the number of punctures $k$.

In the 1960s Gian-Carlo Rota wrote a series of papers on the foundations of combinatorics, looking in particular at incidence algebras for locally finite posets. This is an algebra of functions $f(a,b)$ defined on closed intervals of the poset. Multiplication is given by convolution

$(f * g)(a,b) = \sum_{x \in [a,b]} f(a,x) g(x,b)$

This algebra includes the zeta functions $\zeta (a,b) = 1$ and their inverses the Mobius functions, as discussed in Leinster’s paper on Euler characteristics for categories.

In the fourth paper in the series [1] Rota considered the analogue for vector spaces over finite fields, which are of some interest here since a category of vector spaces may be seen as a simple quantum analogue to the topos Set. The paper begins with q-analogues of binomial coefficients, namely the Gaussian coefficients, which count the rank $k$ subspaces of an $n$ dimensional vector space over the field of order $q$. It is then observed that by allowing $q$ to take a wider range of values, the limit of $q \rightarrow 1$ reduces q-identities to the classical case. The Mobius function $\mu (V,W)$ for elements of the lattice of vector subspaces is given by

$\mu (V,W) = (-1)^{k} q^{(k;2)}$

where $(k;2)$ is the binomial coefficient and $k = \textrm{dim} W – \textrm{dim} V$. The zeta function is defined by $\zeta (V,W) = 1$ when $V$ is contained in $W$, and $0$ otherwise. It is the inverse of the Mobius function in the incidence algebra for the $n$ dimensional vector space.

The close analogy with poset incidence algebras suggests an extension of Leinster’s weightings for a category, which are defined in terms of the Mobius function on the object set incidence algebra with $\zeta (A,B)$ counting the arrows in Hom($A,B$) (a simple extension of the single arrow counted for posets). For a category enriched in Vect the Hom space is a vector space, and it is natural to extend $\zeta (V,W)$ to the dimension of the Hom space Hom($V,W$). Euler characteristics that count dimensions of linear spaces, rather than elements of a set, hopefully bring us a little closer to the knot invariants in their categorified homological guise.

[1] J. Goldman and G-C. Rota, Stud. Appl. Math. 49 (1970) 239-258

Recall that knots have primes. For example, all (m,n) torus knots are prime. Mike Sullivan has shown that although the universal Ghrist template contains all knots, there exist simpler templates (such as the Lorenz one) for which all the knots are prime. Zeta functions for templates have been defined.

Is there a zeta function for knots? The question is what to use in place of the factor $p$. Presumably the correct choice is the Jones (or homflypt) polynomial, or categorified versions thereof (a polynomial is really a kind of number, anyway). Nothing prevents one from simply defining a zeta function

$\zeta (z) = \prod_{K} (1 – J(K)^{-z})^{-1}$

over prime knots $K$, and the Jones polynomial is functorial with respect to knot composition. Then if the Jones polynomial were itself a zeta function for the knot diagram, this would be an iterated zeta function. This looks horrible, but at least the number of prime knots of $n$ crossings is bounded above by $e^n$ (!!) and convergence might be reasonable since, for torus knots, the invariant $J(K)$ looks like an ordinary ordinal when normalised to the simplest knot in the series. That is, for negative integers $z$ the corresponding Euler series would be a summation of counting numbers, just as for the Riemann zeta function.

There have been attempts to observe time lags in gamma flares and in gamma-ray bursts, but we have never seen something like this….

said Daniel Ferenc of U.C. Davis, discussing the new MAGIC result.

The observation of this group of galaxies that is almost devoid of dark matter flies in the face of our current understanding of the cosmos

said Arif Babul of the University of Victoria, discussing the Abell 520 cluster.

Not only has no one ever found a void this big, but we never even expected to find one this size

said Lawrence Rudnick of U. Minnesota regarding the void that corresponds to a cold spot on the WMAP map.

The Concorde cosmology is ready to crash

said Louise Riofrio on her blog.

The Old Man in a Cave reports on the latest news regarding arctic ice levels. As of August 22 there is only 3.22 million square kilometres of sea ice in the north: a new record low. It is said that the (supposedly alarmist) UN Panel on Climate Change predicted this level of ice for the year 2050, not 2007.

On a lighter note: Happy 8th Birthday, Blogger!

How does one discuss a category of operads when the intention was to define categories using operads in the first place? Such self referential questions appear to lurk behind the mystery of weak n-categories. We want an $\omega$-category of $\omega$-operads such that the categories defined as algebras are precisely what we get when we take the algebras of a special Koszul monad. Gee, that’s already way too much mathematics. And duality won’t do for all (categorical) dimensions: in logos theory there are n-alities, so we need a concept of Koszul n-ality. Fortunately, the importance of 2 to topos theory (a la locales and schizophrenic objects) is just the place we thought about extending dualities. So we want an $\omega$-category of $\omega$-operads such that an $\omega$-monad of Koszul n-alities gives weak n-categories as n-algebras. Sigh. Maybe we should return to pictures of trees and discs.

Aside: I don’t like inventing new words for things, since there are so many definitions in mathematics as it is, so I will ignore all objections to the term schizophrenic object. After all, do dwarfs object to having stars named after them? If a term is descriptive, it is a good term.

In an interesting exchange on the Cafe, Schreiber made the remark, “we are understanding that all these differential graded algebras are really just the Koszul dual incarnation of the Lie version of n-groupoids of physical configurations and/or states.” In physical M theory we like to avoid phrasing everything in terms of complicated algebras, because the physics does not require this and the algebras arise naturally from operads anyway. The point is that Koszul duality is a fundamental process for operads themselves, playing the role of 2-ality for the n-logos Machian correspondence.

In a short paper on operads, Kapranov (who introduced modular operads with Getzler) discusses Koszul duals. Examples are the duals of the operads for associative, commutative and Lie algebras, which are respectively the associative, Lie and commutative operads. That is, associativity is self-dual but commutativity and Lie structures are interchanged. These operads are linear. More generally there is a functor $K$ on the category of operads (this is just about 1-operads) which has the property that $K^{2}$ is a kind of isomorphism. Sounds like a monad, doesn’t it? Indeed, Kapranov says that one should think of $F$ as a type of cohomology theory on the category of operads. It’s a bit like the power set monad for the topos Set, which is why the comment of the enigmatic David Corfield regarding duality based on the number 2 is particularly relevant here.

Another example of Koszul dual operads are the homologies of (a) uncompactified moduli spaces (actually varieties) of $n$ punctured Riemann surfaces of genus $g$ and (b) the compactified spaces.

So it would seem that the absolutely brilliant Schreiber is fast converging on the right mathematics, and he’s doing it the hard way, without really thinking about the physics at all. Somebody (respectable) needs to tell him that there is no Dark Energy or no SUSY partners or no KK modes.