# TULCZYJEW TRIPLES AND HIGHER POISSON/SCHOUTEN STRUCTURES ON LIE ALGEBROIDS

My paper “Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids” arXiv:0910.1243v4 [math-ph] is going to appear in Reports on Mathematical Physics, Vol. 66, No 2, (2010).

Abstract
We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.

# Lie infinity-Algebras

As $$L_{\infty}$$-algebras play a large role in my research, and more generally in mathematical physics, homotopy theory, modern geometry etc I thought it maybe useful to say a few words about them.

One should think of $$L_{\infty}$$-algebras as “homotopy relatives” of Lie algebras. In a sense I think of them as differential graded Lie algebras + “more”. I hope to make this a little clearer.

Definition: A supervector space $$V = V_{0} \oplus V_{1}$$ is said to be an $$L_{\infty}$$-algebra if it comes equipped with a series of parity odd $$n$$-linear operations ($$n \geq 0$$ ), which we denote as “brackets” $$(, \cdots , )$$ that

1) are symmetric $$( \bullet , \cdots, a, b , \cdots, \bullet) = (-1)^{\widetilde{a}\widetilde{b} }( \bullet , \cdots, b, a , \cdots, \bullet)$$, $$a,b \in V$$.

2) satisfy the homotopy Jacobi identities

$$\sum_{k+l=n-1} \sum_{(k,l)-\textnormal{unshuffels}}(-1)^{\epsilon}\left( (a_{\sigma(1)}, \cdots , a_{\sigma(k)}), a_{\sigma(k+1)}, \cdots, a_{\sigma(k+l)} \right)=0$$

hold for all $$n \geq 1$$. Here $$(-1)^{\epsilon}$$ is a sign that arises due to the exchange of homogenous elements $$a_{i} \in V$$. Recall that a $$(k,l)$$-unshuffle is a permutation of the indices $$1, 2, \cdots k+l$$ such that $$\sigma(1)$$ < $$\cdots$$ < $$\sigma(k)$$ and $$\sigma(k+1)$$ < $$\cdots$$ < $$\sigma(k+l)$$. The LHS of the above are referred to as Jacobiators.

So, we have a vector space with a series of brackets; $$(\emptyset)$$, $$(a,b)$$ , $$(a,b,c)$$ etc. If the zero bracket $$(\emptyset)$$ is zero then the $$L_{\infty}$$-algebra is said to be strict. Often the definition of $$L_{\infty}$$-algebra assumes this. With a non-vanishing zero bracket the algebra is often called “weak”, “with background” or “curved”.

Let us examine the first few Jacobi identities in order to make all this a little clearer. First let us assume a strict algebra and we will denote the one bracket as $$d$$ (this will become clear).

1) $$d^{2}a = 0$$.

That is we have a differential graded algebra.

2) $$d (a,b) + (da, b) + (-1)^{\widetilde{a} \widetilde{b}} (db, a) =0$$.

So the one bracket (the differential) satisfied a derivation rule over the 2-bracket.

3) $$d (a,b,c) + (da,b,c) + (-1)^{\widetilde{a} \widetilde{b}}(db, a, c) + (-1)^{\widetilde{c}(\widetilde{a} + \widetilde{b})} (dc, a, b)$$
$$+ ((a,b), c) + (-1)^{\widetilde{b}\widetilde{c}}((a,c), b) + (-1)^{\widetilde{a}(\widetilde{b}+ \widetilde{c})} ((b,c), a)= 0$$.

So we have the standard Jacobi identity up to something exact.

The higher Jacobi identities are not so easy to interpret in terms of things we all know. There are higher homotopy relations and thus the word “strong”. This should make it clearer what I mean by “differential graded Lie algebra + more”.

Note that the conventions here are not quite the same as originally used by Stasheff. In fact he used a $$\mathbb{Z}$$-grading where we use a $$\mathbb{Z}_{2}$$-grading. The brackets of Stasheff are skew-symmetric and (with superisation) they are even/odd parity for even/odd number of arguments. By employing the parity reversion function and including a few extra sign factors one can construct a series of brackets on $$\Pi V$$ that are closer to Stasheff’s conventions, of course “superised”. This series of brackets on $$\Pi V$$ then directly includes Lie superalgebras.

There are other “similarities” between Lie algebras and $$L_{\infty}$$-algebras. I may post more about some of these another time.

A few words about applications. $$L_{\infty}$$-algebras can be found behind the BV (BFV) formalism, deformation quantistion of Poisson manifolds and closed string field theory, for example.

# Lie-Infinity Algebroids?

I have been thinking a little bit recently about $$L_{\infty}$$-algebroids. So what is such an object?

Let us work with super-stuff from the start. I will be lax about signs etc, so this should not course any real confusion. First we need a little background.

Heuristically, a supermanifold is a “manifold” in which the coordinates have an underlying $$\mathbb{Z}_{2}$$-grading. Morphisms between charts are smooth and respect this grading. In more physical language, we have bosonic coordinates and even coordinates. The bosonic coordinates commute as where the fermionic coordinates anticommute. I will refer to bosonic coordinates as even parity and fermionic as odd parity. To set this up properly one needs the theory of locally ringed spaces. However, we will not need this here.

A graded manifold is a supermanifold, in which the coordinates are assigned an additional weight in $$\mathbb{Z}^{n}$$ and the changes of coordinates respect the parity as well as the additional weight. In general the parity and weight are completely independent.

A Q-manifold is a supermanifold (or a graded manifold ) that comes equipped with a homological vector field, usually denoted by Q. That is we have an odd parity vector field that “self-commutes” under the Lie bracket,

$$[Q,Q] = 0$$.

Note that as the homological vector field is odd, this is a non-trivial condition. Sometimes, if the supermanifold is a graded manifold then conditions on the weight of Q can be imposed.

Now we can describe $$L_{\infty}$$-algebroids. The best way to describe them is as follows:

Definition:
A vector bundle $$E \rightarrow M$$ is said to have an $$L_{\infty}$$-algebroid structure if there exists a homological vector field, denote as $$Q$$ on the total space of $$\Pi E$$, thought of as a graded manifold.

That is the pair $$(\Pi E, Q)$$ is a Q-manifold. We call this pair an $$L_{\infty}$$-algebroid.

The weight, in this case just in $$\mathbb{Z}$$ is assigned by equipping the base coordinates of $$E$$ with weight zero and the fibre coordinates with weight one (or some other integer). The “$$\Pi$$” is the parity reversion functor. It shifts the parity of the fibre coordinates. So, a coordinate that is originally even\odd get replaced by a coordinate that is odd\even. It does nothing to the weight. Note that this shift is fundamental here and not just for convenience.

It is very easy to see that the original vector bundle $$E \rightarrow M$$ is equivalent to the graded manifold $$\Pi E$$. Stronger than this, the equivalence is functorial. That is we have equivalent categories.

Further note that there is no condition on the weight of the homological vector bundle in this definition, nor is there any “compatibility condition” with the vector bundle (or graded) structure.

Definition:
An $$L_{\infty}$$-algebroid is said to be strict if the restriction of Q to the “base manifold” $$M \subset \Pi E$$ is a genuine homological vector field on $$M$$.

This does not sound very invariant at first, but simply put restriction of Q to the weight zero “part” of $$\Pi E$$ should still be homological.

For those that know Lie algebroids and $$L_{\infty}$$-algebras, the question is why call them $$L_{\infty}$$-algebroids? An $$L_{\infty}$$-algebroid is to an $$L_{\infty}$$-algebra what a Lie algebroid is to a Lie algebra.

It also turns out that some of the main constructions relating to Lie algebroids carry over to $$L_{\infty}$$-algeboids, see . (I may say more another time.) This may also be of use for $$L_{\infty}$$-algebras. I am currently also pondering this.

So maybe I should end for now on a little motivation as to why such things are interesting. First, if we insist on the homological vector field being of weight one we recover Lie algebroids. If we insist on the vector bundle being over a point we recover $$L_{\infty}$$-algebras. (I should post on these later) Also, very similar things appear in quantum field theory via the BV and BFV formalisms (again I should post on these another time). However, at the moment it is not exactly clear how $$L_{\infty}$$-algebroids fit in here. One “barrier” is that Q is inhomogenous in weight, in the BV and BFV formulations the homological vector field is homogeneous in “ghost number”. It would also be interesting to see if these structures can be used in the BV-AKSZ formalism.

—————————–
References

 Andrew James Bruce, From $$L_{\infty}$$-algebroids to higher Schouten\Poisson structures. Submitted for publication, available as arXiv:1007.1389v2 [math-ph]

# Is "three" the new "two"?

The number “two” may appear to be very special in theoretical physics, but maybe it has had it’s day…

By this we mean that much of physics is described in terms of “binary objects”: Lie brackets, commutators, metrics, rank two curvature tensors, quadratic Lagrangians, two dimensional world sheets of strings, Poisson structures, symplectic two forms, Laplacians and I am sure the list goes on. However, it has become increasingly clear in recent years that “higher objects” play an important role in theoretical physics as well as modern geometry.

For example, it has become increasingly clear that n-aray generalisations of Lie algebras play a role in physics. The Sh Lie algebras of Stasheff and the (not completely unrelated) n-Lie algebras of Filipov are great examples here. In one form or another, they can be found behind the BV-antifield formalism, Zwiebach’s closed string field theory, Kontsevich’s deformation quantisation of Poisson manifolds, Nambu’s generalised mechanics and the Bagger–Lambert–Gustavsson (BLG) description of multiple
stacked M2 branes
.

The last one has been of interest to me lately.

So, M-theory was introduced by Witten in 1995 as a non-perturbative unification of the various superstring theories. Here, the fundamental objects are not strings but extended membranes of dimension 2 and 5, the so called M2 and M5 branes. Since then progress has been slow. No-one really knows what M-theory is and there is no proper understanding of the dynamics of interacting branes.

Then in Bagger & Lambert  in 2006 and independently Gustavsson  in 2007 construct and effective action for the low energy description of a stack of two M2 branes. The novel feature here is that 3-Lie algebras play a role here. A 3-Lie algebra should be thought of as a “Lie algebra” but with a tribracket not a bibracket. Details should not worry us.

The theory has the fields take their values in a 3-Lie algebra and their is a novel gauge symmetry. However, the original BLG-model can be recast as a conventional gauge theory, the ABJM theory . So it starts to look that maybe 3-Lie algebras are some weird artificial artefact of M2 branes.

(There are many, many papers on the arXiv dealing with modifying the original BLG model. This usually can be understood in terms of 3-Lie algebra. I won’t say any more right now.)

But then very recently, Lambert & Papageorgakis  provided evidence that the effective description on M5 branes would also require 3-Lie algebras. However, they have not yet produced an action, which would be essential if the more or less standard methods of quantisation were to be applied.

This is fascinating. M-theory seems to be deeply tied to the theory of n-aray algebras, and in particular 3-aray algebras. There are may open questions here, both from a physics and mathematics point of view. In all it looks like n-aray algebra are here to stay.

——————-
REFERENCES

 Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP, 10:091, 2008.

 Jonathan Bagger and Neil Lambert. Modeling multiple M2’s. Phys. Rev., D75:045020, 2007.

 Andreas Gustavsson. Algebraic structures on parallel M2-branes. Nucl. Phys., B811:66–76, 2009.

 Neil Lambert and Constantinos Papageorgakis. Nonabelian (2,0) Tensor Multiplets and 3-algebras. 2010,
arXiv:1007.2982 [hep-th].

# Can an "amateur" today make useful contributions to theoretical physics or mathematics?

This was a question I posed to a friend of mine. We decided to define an “amateur” as someone without a PhD in physics, mathematics or something close.

We came to the conclusion that it is very unlikely that some one with out a PhD could in fact make a real contribution. This is despite the fact that things are far more open today than they ever have been. I mean, we have the arXiv and open access journals online. Almost everyone has the internet at home these days, and if not the local libraries do.

From time to time undergraduate students can contribute, but these are with close supervision. The supervisor will guide the student and nurture the natural ability. This was really the closest to an “amateur” that could contribute we agreed on.

So, why can’t “amateurs” contribute? Here are my thoughts…

1) Without spending some time in academia, “amateurs” are not aware of the culture and what is expected of anyone wishing to contribute to mathematical science. They do not know how to do research.

2) “Amateurs”, although interested and very keen at times do not often realise just how much of a prerequisite can be required to conduct research. They can often lack the mathematical skills to contribute. Claims like “I can solve the Riemann hypothesis using high school mathematics” only suggests that they don’t understand the hypothesis correctly in the first place. Trying to rewrite particle theory using high school maths is also redundant. We have a great construct for doing particle physics, it is called the standard model.

3) Theoretical physics, mathematical physics and mathematics as a whole is split up into smaller sections. One can only hope to get acquainted properly with a small subset of what is out there. Without specialising to a large extent, it is unlikely that one can discover something new and interesting. Trying to find smaller, specialised problems to work on is usually the way forward: unless you are a genius and can discover a whole new branch of mathematics! “Amateurs” seem to be focused on very well-known and published open questions. In number theory the Riemann hypothesis is a great example of this. In physics, a theory of quantum gravity is an example.

4) Because the individual does not understand it, it must be wrong. “Amateurs” fall into this mind set quite often. Finding a simpler more elegant approach to things is a large part of the mathematical sciences. However, trying to show that special relativity or quantum mechanics are mathematically inconsistent or do not agree with nature is futile. This also includes the desire to use nothing but high school maths to explain all of physics.

Not that I want to discourage anyone from thinking about mathematics and physics. I encourage it, but with a caveat: reading Wikepedia and popular accounts of science are not enough for one to start to do research.

__________________________________________________________
UPDATE (15th May 2014)

Please do not post about your pet theories in the comments here. If you have something to say related to this post about the role of amateurs in science then please by all means share it here. Thank you.

# Workshop on Operator Algebras and Physics

There is a workshop on operator algebras and physics at the University of Cardiff, 21-25 June inclusive.

Operator algebras have their conception in quantum physics. The states of a quantum particle are understood as vectors in a Hilbert space. As all vector space of a given dimension are essentially the same one cannot really extract information about the system from the vector space structure. The important thing here is the algebra of operators on the Hilbert space. These do contain useful information.

For physics and noncommutative geometry C^{*} and von Neumann algebras are important, as are more general Banach algebras.

I am no expert on operator algebras, but I am interested in there use in physics and geometry. I need to brush up on them!

Constantin Teleman will be delivering a series of lectures on the five days on Two Dimensional Topological Quantum Field Theories and Gauge theories as part of the workshop. The interest in two dimensional theories comes largely from string theory and statistical physics. Also, systems in two dimensions can be thought of “toy models” for more realistic 4-d theories. There are some nice mathematical properties of low dimensional systems that allow one to get at features of quantum field theory that are hidden behind the difficulties encountered in higher dimensions.

A good place to start reading about these things is the preprint by Teleman “Topological field theories in 2 dimensions”; it can be found here as a pdf.

There are other good speakers giving talks.

I will write some kind of overview after the event.