Graded partial differential

Question

Hi! Dealing with graded manifolds, I would like to define a graded coordinate of degree 1, denoted

$\xi_{a}$

and the associated partial derivative

$D_{a}$

of degree -1 (position of indices is unimportant for my purpose).

In particular, both

$\xi_{a}$

and

$D_{a}$ are self-anticommuting and anti-commuting with each other (being both of odd degree).

However, when I implement the situation in the following way:

{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::Indices(T, position=free, parent=double);

D{#}::PartialDerivative;

{\xi{#}}::Depends(D{#});

{D{#},\xi{#}}::AntiCommuting;

{D{#},\xi{#}}::SelfAntiCommuting;

Exp:=D_{a}{D_{b}{\xi_{c}}*\xi_{d}};

product_rule(_);

the above leads to

$D_{a b}{\xi_{c}} \xi_{d}-D_{b}{\xi_{c}} D_{a}{\xi_{d}}$

where the second sign is wrong, since

$D_{b}{\xi_{c}}$

should be of degree 0, and then commute with

$D_{a}$.

Is there a way to fix this?

Thank you for any information!

Answer 1

Assign the AntiCommuting property to the indices, not to the derivatives. So

{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::Indices(T, position=free, parent=double);
D{#}::PartialDerivative;
\xi{#}::Depends(D{#});
{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::AntiCommuting;

and then

Exp:=D_{a}{D_{b}{\xi_{c}}*\xi_{d}};
product_rule(_);

produces

$$D_{a b}{\xi_{c}} \xi_{d} + D_{b}{\xi_{c}} D_{a}{\xi_{d}}$$

Comments

Please let me know if you run into any other issues. I am reworking the code for this, so that it works properly if you have a combination of explicit and implicit indices, so now is the time to fix other issues (if any).

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Thank you for quick reply!

However, I'm not sure to quite get how Anticommuting works on coordinates yet.

For example,

{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::Indices(T, position=free, parent=double);
{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::AntiCommuting;

Exp:=\xi_{a}*\xi_{b}+\xi_{b}*\xi_{a};
sort_product(_);

renders $2\xi{a} \xi{b}$ where I was expecting zero.

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For that you do need the

\xi_{a}::SelfAntiCommuting;

property. The problem at the moment is that not all algorithms make use of the same property information (sort_product needs the SelfAntiCommuting, while product_rule looks at the property directly on the indices). This is being cleaned up, but still a bit confusing at the moment.

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Ok, now it works fine, thank you!

By the way, do you know if there is a way to make D_{a b} skewsymmetric in a and b, i.e. such that:

{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::Indices(T, position=free, parent=double);
{a, b, c, d, e, f, g, h, i, j, k, l, m, n ,o, r,s,t,u,v,w,x,y,z#}::AntiCommuting;
D{#}::PartialDerivative;
Exp:=D_{a b}{f}-D_{b a}{f};
canonicalise(_);

doesn't yield zero?

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Partial derivatives are always assumed to be commuting (which should not be, in case they are in anticommuting directions, but that's not handled right now). So you'll need to declare D{#}::Derivative. That does lead to slightly messier output unfortunately.

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Dear Kasper,

I'm still slightly confused about how the sort_product algorithm deals with Anticommuting indices. In the following example, it doesn't consider the indices a and b as anticommuting in the first example but seems to do so when one adds a partial derivative.

{\mu,\nu}::Indices; {a, b}::Indices; {a, b}::AntiCommuting;

\partial{#}::PartialDerivative;

Exp1:=\zeta_{b}*\xi_{a}; sort_product(_);

Exp2:=\zeta_{b}*\partial_{\mu}{\xi_{a}}; sort_product(_);

Is there a way to harmonise this so that to obtain a plus sign in the second output?

Thank you!