Lie derivatives and N=1 SUSY

[ajb]

I don't know if this is well-known, but I have a nice geometric realisation of the N=1 supersymmetric algebra in terms of the Lie derivative along a vector field of differential forms over a (super)manifold.

 

Let [math]M[/math] be a (smooth) supermanifold and let us work in local coordinates [math]\{x^{A}\}[/math]. Here we have [math]x^{A}x^{B} = (-1)^{\widetilde{A}\widetilde{B}}x^{B}x^{A}[/math] where the parity is [math]\widetilde{A} = 0/1[/math], i.e. bosonic and fermionic coordinates. From now on by manifold I mean supermanifold.

 

Over the manifold [math]M[/math] we can build the two natural bundles [math]\Pi TM[/math] and [math]\Pi T^{*}M[/math]. Let us equip these bundles with natural local coordinates [math]\{x^{A}, dx^{A} \}[/math] and [math]\{ x^{A}, x^{*}_{A} \}[/math] respectively. Here we have [math]\widetilde{dx}^{A} = \widetilde{x}^{*}_{A}=\widetilde{A}+1[/math]. The bundle automorphisms induced by (local) diffeomorphisms of [math]M[/math] are

 

[math]\overline{x}^{A} = \overline{x}^{A}(x)[/math],

[math]\overline{dx}^{A} = dx^{B}\frac{\partial \overline{x}^{A}}{\partial x^{B}}[/math],

[math]\overline{x}^{*}_{A} = \frac{\partial x^{B}}{\partial \overline{x}^{A}} x^{*}_{B}[/math].

 

 

Now, I define differential forms over [math]M[/math] to be functions on [math]\Pi TM[/math] and similarly multivector fields to be functions on [math]\Pi T^{*}M[/math]. The Schouten-Nijenhuis bracket on multivector fields is just the canonical odd Poisson bracket on [math]\Pi T^{*}M[/math]. You should think of it as the extension of the Lie bracket on vector fields.

 

In particular, we have the odd isomorphism which take vectors [math]X^{A}\frac{\partial}{\partial x^{A}}[/math] to one-vectors [math]X^{A}x^{*}_{A}[/math]. Note that this shifts the parity of the object. The Schouten--Nijenhuist bracket is then up-to parity and sign conventions the Lie bracket. As I won't use anything other than one-vectors just think of the Lie bracket.

 

 

Now for the fun bit. Let us define an odd vector field on [math]\Pi TM[/math] (derivation acting on differential forms) as

 

[math]Q_{X} = d + i_{X}[/math]

 

where [math]d= dx^{A}\frac{\partial}{\partial x^{A}} [/math] is the de Rham differential and [math]i_{X} = (-1)^{\widetilde{X}}X^{A}\frac{\partial }{\partial dx^{A}}[/math] where [math]X^{A}[/math] is the component of an odd one-vector (even vector field). By my conventions we have [math]\widetilde{X}=1[/math].

 

Result

 

[math] [Q_{X},Q_{X}] =2 [d,i_{X}] = 2 L_{X}[/math],

[math] [Q_{X},L_{X}] = i_{[X,X]} =0[/math].

 

Here the bracket on vector fields is standard Lie bracket (graded commutator) where the bracket on one-vectors is the SN bracket. [math]L_{X}[/math] is the Lie derivative of a differential form along a one-vector (turns out in my conventions to be minus the classical Lie derivative along a vector field.)

 

Thus we have the N=1 SUSY algebra realised as the geometric variation of differential forms.

 

Via direct inspection you see the (infinitesimal) "point transformations are"

 

[math]\overline{x}^{A} = x^{A} + \epsilon dx^{A}[/math],

[math]\overline{dx}^{A} =dx^{A} - \epsilon X^{A}[/math],

 

but note that these are not vector bundle automorphisms of [math]\Pi TM[/math] but rather more general diffeomrphisms (in the category of supermanifolds).

 

 

Any comments? Is this a coincidence or does it point to a deeper relation between SUSY and differential geometry?

Edited July 28, 2009 by ajb
added point transformations explicity

[hewj11]

i wish i understood this better haha

[Severian]

I am afraid that is a little too mathematical for me.

 

But isn't it a bit of a fix? Since you started with a supermanifold and have fermionic coordinates, isn't supersymmetry quite natural (I can see how it isn't necessary).

 

I am also confused by your notation. Your commutators under 'Result' are a little bit reminiscent of the susy algebra, but not quite. Could you explain to me how the susy algebra comes about from this?

[ajb]

You can take the original manifold [math]M[/math] to be a classical manifold. By thinking of differential forms as functions on [math]\Pi TM[/math], which is a supermanifold the Lie derivative along a vector field has this interesting "square root". It is this square root that made me think of SUSY.

 

In particular, applying the odd operator [math]Q_{X} = d + i_{X}[/math] twice results in the Lie derivative, that is the "geometric variation" or an "infinitesimal spacial translation".

 

This formally reminds me of supersymmetry. (Square root of operators are always going to look like SUSY.)

 

 

You work with SUSY more than I do, how is this not quite SUSY as you put it? (all commutators are graded)

 

I am thinking that the [math]Q_{X}[/math] plays the role of the supercharge and the Lie derivative of the Hamiltonian. The second commutator tells us "the Hamiltonian (Lie derivative) commutes with the supercharge (my Q)".

Edited August 1, 2009 by ajb

[ajb]

I should also say this generalises to multivector fields, that is arbitrary functions on [math]\Pi T^{*}M[/math] (may need odd constants so we can include everything).

 

To my original question "Is this supersymmetry" there appears to be (at lest) two answers.

 

1) Yes. I have a sub Lie algebra isomorphic to the 1 dimensional SUSY algebra.

 

2) Possibly as there could be a (0+1) dimensional sigma model whose supersymmetry algebra is given by my construction (maybe with some extra conditions on the vector/multivector and/or the supermanifold).

 

The second half answer is by far more interesting that the first. However, I do not know what the model is nor even if it actually exists. So, if I find the time I think I will have to search the literature for sigma models with supermanifolds as the target spaces. I know there is plenty of work out their.