Category Archives: Research work

Contact structures and supersymmetric mechanics

Contact structures and supersymmetric mechanics

Andrew James Bruce

Abstract
We establish a relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with supersymmetric mechanics.

arXiv:1108.5291v1 [math-ph]

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In the preprint above I show that aspects of  d=1, N=2 supersymmetric quasi-classical mechanics in the superspace formulation can be understood in terms of  a contact structure on the supermanifold \(R^{1|2}\).

 

In particular if we pick local coordinates \((t, \theta, \bar{\theta})\) then the super contact structure is given by

 

\(\alpha = dt + i \left(  d \bar{\theta}\theta + \bar{\theta} d \theta  \right)\),
which is a Grassmann odd one form. One could motivate the study of such a one form as a “superisation” of the contact form on \(R^{3}\).

 

Associated with any odd one form that is nowhere vanishing is a hyperplane distribution of codimension (1|0). That is we have a subspace of the tangent bundle that contains one less even vector field in its (local) basis as compared to the  tangent bundle.  This is why we should refer to the above structure as an even (pre-)contact structure.

 

The hyperplane distribution associated with the super contact structure is spanned by two odd vector fields. These odd vector fields are exactly the SUSY covariant derivatives. More over we do have a genuine contact structure as the exterior derivative of the super contact form is non-degenerate on the hyperplane distribution. For more details see the preprint.

 

Generalising contact structures  on manifolds to  supermanifolds appears fairly straight forward. We have the non-classical case of odd contact structures to also handle, here the hyperplane distribution is of corank (0|1), i.e. one less odd vector field. There is also a subtly when defining kernels and contactomorphisms as we will have to take care with nilpotent objects.

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Comments on the preprint will be very much appreciated.

 

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Update A third revised version has now been submitted. 08/02/2012

Higher Lie-Schouten brackets

I thought it would be interesting to point out a geometric construction related to  \(L_{\infty}\)-algebras.  (See earlier post here) Recall that given a Lie algebra \((\mathfrak{g}, [,] )\) one can associate on the dual vector space a linear Poisson structure known as the Lie-Poisson bracket.  So, as a  manifold \((\mathfrak{g}^{*}, \{, \}) \) is a Poisson manifold.  It is convenient to  replace the “classical” language of linear and replace this with a graded condition. That is, if we associate weight one to the coordinates on \(\mathfrak{g}^{*} \) then the Lie-Poisson bracket is of weight minus one.

The Lie Poisson bracket is very important in deformation quantisation (both formal and C*-algebraic). There are some nice theorems and results that I should point to at some later date.

Now, it is also known that one has an odd version of this known as the Lie-Schouten brackets on \(\Pi \mathfrak{g}^{*}\). The key difference is the shift in the Grassmann parity of the “linear” coordinates.  Note that this all carries over to Lie super algebras with no problem.  I will drop the prefix super from now on…

 

So, let us look at the situation for \(L_{\infty}\)-algebras. We understand these either as a series of higher order brackets on a vector space  \(U\) that satisfies a higher order generalsiation of the Jacobi identities or more conveniently we can understand all this in terms of a homological vector field on the formal manifold \(\Pi U\).

Definition An \(L_{\infty}\)-algebra is a vector space \(V = \Pi U\) together with a homological vector field \(Q = (Q^{\delta} + \xi^{\alpha} Q_{\alpha}^{\delta} + \frac{1}{2!} \xi^{\alpha} \xi^{\beta} Q_{\beta \alpha}^{\delta} + \frac{1}{3!} \xi^{\alpha} \xi^{\beta} \xi^{\gamma} Q_{\gamma \beta \alpha}^{\delta} + \cdots) \frac{\partial}{\partial \xi^{\delta}}\),

where we have picked coordinates on \(\Pi U\)  \(\{  \xi^{\alpha}\}\). Note that these coordinates are odd as compared to the coordinates on \(U\). Thus we assign the Grassmann parity \(\widetilde{\xi^{\alpha}} = \widetilde{\alpha} + 1\)  Note that \(Q\) is odd and that if we restrict to the quadratic part then we are back to Lie algebras.

I will simply state the result, rather than derive it.

Proposition Let \((\Pi U, Q)\) be an \(L_{\infty}\)-algebra. Then the formal manifold \(\Pi U^{*}\) has a homotopy Schouten algebra structure.

Let us pick local coordinates \(\{ \eta_{\alpha}\}\) on \(\Pi U^{*}\). Furthermore, we consider this as a graded manifold and attach a weight of one to each coordinate.  A general function,  a  “multivector” has the form

\(X = \stackrel{0}{X} + X^{\alpha} \eta_{\alpha} + \frac{1}{2!}X^{\alpha \beta}\eta_{\beta} \eta_{\alpha} + \cdots \)

The higher Lie-Schouten brackets are given by

\((X_{1}, X_{2}, \cdots, X_{r}) = \pm Q_{\alpha_{r}\cdots  \alpha_{1} }^{\beta}\eta_{\beta}\frac{\partial X_{1}}{\partial \eta_{\alpha_{1}}} \cdots   \frac{\partial X_{1}}{\partial \eta_{\alpha_{r}}}\),

being slack with an overall sign.  Note that with respect to the natural weight the n-bracket has weight (1-n). Thus not unexpectedly, restricting to n=2 gives an odd bracket of weight minus one: up to conventions this is the Lie-Schouten bracket of a Lie algebra.

The above collection of brackets forms an \(L_{\infty}\)-algebra in the “odd super” conventions that satisfies a derivation rule of the product of “multivectors”. Thus the nomenclature homotopy Schouten algebra and higher Lie-Schouten bracket.

A similar statement holds in terms of a  homotopy Poisson algebra on \(U^{*}\). Here the brackets as skewsymmetric and of  even/odd Grassmann parity for even/odd number of arguments.  (I rather the odd conventions overall).

Now this is quite a new construction and the technical exploration of this nice geometric construction awaits to be explored. How much of the geometric theory associated with Lie algebras and Lie groups carries over to \(L_{\infty}\)-algebras and \(\infty\)-groups is an open question.

Details can be found in Andrew James Bruce ” From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures”, Reports on Mathematical Physics Vol. 67, (2011), No. 2  (also on the arXiv).

 

Also see earlier post here on Lie infinity algebroids.

29th North British Mathematical Physics Seminar

I have been invited to give a talk at the 29th North British Mathematical Physics Seminar (NBMPS) in Edinburgh on the 16th February 2011.

The NBMPS has been running since 2001 and is a forum for mathematical physicists in North Britain to meet up. They organise four one-day meetings that are held in rotation every year in Durham, Edinburgh, York and Nottingham.

The 29th meeting is in Edinburgh and I am listed a the first speaker!

The topic of my talk will be my preprint on Odd Jacobi manifolds and classical BV-gauge systems. This paper is also discussed on my blog here.

I will place a link to the slides in due course.

I will also place an update of the event at some later date.

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Update:

The talk I felt went well. I had a few questions and comments, but nothing off putting or massively critical.

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Update: The slides for the talk can be found here.

Odd Jacobi structures and BV-gauge systems

Abstract
In this paper we define Grassmann odd analogues of Jacobi structures on supermanifolds. We then examine their potential use in the Batalin-Vilkovisky formalism of classical gauge theories.

arXiv:1101.1844v1 [math-ph]

In my latest preprint I construct a Grassmann odd analogue of Jacobi structures on supermanifolds.

Without any details (being slack with signs) an odd Jacobi structure on a supermanifold is an ” almost Schouten structure”, \(S\) that is an odd function on the total space of the cotangent bundle of the supermanifold quadratic in fibre coordinates and a homological vector field \(Q\) on the supermanifold together with the compatibility conditions

\(L_{Q}S = \{\mathcal{Q}, S \} = 0\),
\( \{S, S \} = 2 S \mathcal{Q}\),

where \(\mathcal{Q} \in C^{\infty}(T^{*}M)\) is the “Hamiltonian” or principle symbol of the Homological vector field. The brackets here are the canonical Poisson brackets on the cotangent bundle.

An odd Lie bracket can then be constructed on \(C^{\infty}(M)\)

\([f,g] = \pm \{ \{ S,f \},g\} \pm \{ \mathcal{Q},fg \} \).

So, this odd bracket satisfies all the properties of a Schouten bracket i.e. symmetry and the Jacobi identity, but the Leibniz rule is not identically satisfied. There is an “anomaly” to the Leibniz rule of the form

\( [f,gh] = \pm [f,g]h \pm g [f,h] \pm [f,1] gh\)

In the preprint I examine the basic properties of odd Jacobi manifolds. The definition and study of odd Jacobi manifolds appears to be missing from the previous literature despite the wide interest in Schouten manifolds and Q-manifolds in mathematical physics.

One should note that for classical or even Jacobi structures (if you know what these are) the Reeb vector field has no constrain on it like being homological. For odd structures the homological condition is essential.

I also consider if the classical BV-antifield formalism can be generalised to odd Jacobi manifolds. In short, does one require the antibracket to be a Schouten bracket or can one weaken the Leibniz rule? I show that it looks possible to extend the BV formalism, classically anyway to odd Jacobi manifolds with the extra condition that the extended classical action not just be a Maurer-Cartan element,

\([s,s] = 0\),

but in addition should be Q-closed,

\(Qs =0\).

Much work needs to be done to generalise the BV formalism to odd Jacobi manifolds including adding the required gradings of ghost number, antifield number etc as well as understanding the quantum aspects.

UPDATE: 22 March 2011. I have found a mistake in one of the examples I suggest. This is corrected and an updated version of the preprint will appear in due course. The mistake does not really effect the rest of the preprint.

Lie-Infinity Algebroids? II

This post should be considered as part two of the earlier post Lie-Infinity Algebroids?

The term \(L_{\infty} \) -algebroid seems not to be very well established in the literature. A nice discussion of this can be found at the nLab.

To quickly recall, the definition I use is that the Q-manifold \((\Pi E, Q)\) is an \(L_{\infty} \) -algebroid, where \(E \rightarrow M \) is a vector bundle and \(Q \) is an arbitrary weight homological vector field. The weight is provided by the assignment of zero to the base coordinates and one to the fibre coordinates. If the homological vector field is of weight one, then we have a Lie algebroid.

It is by now quite well established that a Lie algebroid, as above is equivalently described by

i) A weight minus one Schouten structure on the total space \(\Pi E^{*}\).
ii) A weight minus one Poisson structure on the total space of \(E^{*}\).

In other words, Lie algebroids are equivalent to certain graded Schouten or Poisson algebras. Recall, a Schouten algebra is an odd version of a Poisson algebra. The point is ignoring all gradings and parity, we have a Lie algebra such that the Lie bracket satisfies a Leibniz rule over the product of elements of the Lie algebra. We need a notion of multiplication, in this case it is just the “point-wise” product of functions.

Thus, there is a close relation between Poisson/Schouten algebras (or manifolds) and Lie algebroids.

The natural question now is “does something similar happen for \(L_{\infty}\)-algebroids?”

The answer is “yes”, but we now have to consider homotopy versions of Schouten and Poisson algebras.

Definition: A homotopy Schouten/Poisson algebra is a suitably “superised” \(L_{\infty}\)-algebra (see here) such that the n-linear operations (“brackets”) satisfy a Leibniz over the supercommutative product of elements.

This definition requires that we don’t have just an underlying vector space structure, but that of a supercommutative algebra. I will assume we also have a unit. Though, I think that noncommutative and non-unital algebras are no problem. The point is, I have in mind (at least for now) algebras of functions over (graded) supermanifolds.

Theorem: Given an \(L_{\infty}\)-algebroid \((\Pi E, Q)\) one can canonically construct
i) A total weight one higher Schouten structure on the total space of \(\Pi E^{*}\).
ii) A total weight one higher Poisson structure on the total space of \(E^{*}\).

Proof and details of the assignment of weights can be found in [1].

So, the point is that there is a close relation between homotopy versions of Poisson/Schouten algebras \(L_{\infty}\)-algebroids. To my knowledge, this has not appeared in the literature before. The specific case of \(L_{\infty}\)-algebras (algebroids over a “point”) also seems not to be discussed in the literature before.

The way we interpret this is interesting. We think of a Lie algebroid as a generalisation of the tangent bundle and a Lie algabra. The homological vector field \(Q \) “mixes” the de Rham differential over a manifold and the Chevalley-Eilenberg differential of a Lie algebra \(\mathfrak{g}\). Furthermore, we have a Poisson bracket on \(C^{\infty}( E^{*})\) which “mixes” the canonical Poisson on \(T^{*}M\) with the Lie-Poisson bracket on \(\mathfrak{g}^{*}\). Similar statements hold for the Schouten bracket.

For \(L_{\infty}\)-algebroids the homological vector field again generalises the de Rham and Chevalley-Eilenberg differentials, but it is now inhomogeneous. It resembles a “mix or higher order BRST-like” operator [3]. A homotopy version of the Maurer-Cartan equation naturally appears here. It is clear that we can consider the homotopy Schouten/Poisson algabras associated with an \(L_{\infty}\)-algebra as playing the role of the Lie-Poisson algebras, however there is no obvious higher brackets to consider on the cotangent bundle. It is not clear to me what should replace the tangent bundle here, if anything.

Exactly what technical use the theorem above is awaits to be explored. There are some interesting related notion in Mehta [2], I have yet to fully assimilate them. Maybe more on that another time.

References
[1] From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures. Andrew James Bruce, arXiv:1007.1389 [math-ph]

[2]On homotopy Poisson actions and reduction of symplectic Q-manifolds. Rajan Amit Mehta, arXiv:1009.1280v1 [math.SG]

[3] Higher order BRST and anti-BRST operators and cohomology for compact Lie algebras. C. Chryssomalakos, J. A. de Azcarraga, A. J. Macfarlane, J. C. Perez Bueno, arXiv:hep-th/9810212v2

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 [1]. (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.

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References

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