Viewing message <3C91DC20.67AB2EFB@uni-essen.de>
 
 

 Von:Urs Schreiber (Urs.Schreiber@uni-essen.de)
 Betrifft:Re: Higher analogues of Maxwell's equations
 Newsgroups:sci.physics.research
 Datum:2002-03-15 10:03:34 PST
 

John Baez wrote:

[...]

> Unless I'm confused, you're talking here about "p-form electromagnetism",
> which is important in string theory, and has also been the topic of
> my quantum gravity seminar this quarter, where I've been talking about
> its relation to n-categories.

I have looked at the quantum gravity seminar notes by Miguel
Carrión Álvarez, but p-form electromagnetism seems not to have
been included yet. Will there be notes available online?
(Maybe I have not looked close enough?) I'd be interested.

> In differential form notation, p-form electromagnetism goes like this:
>
> You start with a "vector potential" A which is a p-form.  You
> form the "electromagnetic field"
>
> F = dA
>
> which is a (p+1)-form.  This automatically satisfies
>
> dF = 0
>
> but you also impose the equations
>
> *d*F = J
>
> where J is a (p-1)-form called the "current".
>
> For p = 1 this is just Maxwell's equations.  For higher p you
> get analogues of Maxwell's equations where the current is naturally
> produced not by point particles but by (p-1)-branes, that is,
> by (p-1)-dimensional surfaces which trace out p-dimensional
> worldsheets in spacetime.

For vanishing current J I look at all p-form electromagnetism
as a special case of the massless Kaehler equation, i.e. of
the equation

 (1)  (d + *d*) F = 0

for F any section of the exterior bundle (i.e. an
inhomogeneous form).

Source-free p-form electromagnetism results from this general
equation  by imposing a further restrictions on F, namely

 (2) N F = p  (where N is the number operator) .

But imposing this condition on the original Kaehler equation
destroys many of its nice features. For example, the Kaehler
equation on unrestricted form sectors allows a reformulation
as a time evolution equation, which may be straigtforwardly
integrated for given boundary conditions. This is not possible
when restricting it to p-forms. But there is a way out, which
solves this problem and at the same time has a powerful
generalization:

Instead of (2) impose
 

 (3) (d - *d*) F = 0 .

in addition to (1). By adding and substracting (1) and (3) it
is immediate that

 (4) d F = 0

 (5) d*F = 0 .

Since d and *d* strictly map p-form sectors of different p
into each other, these two equations must hold *in every
p-form sector seperately*. Hence solving (1) and (3) gives
vacuum solutions to *all* p-form electromagnetism theories at
once! Before you complain that I have just reformulated the
problem of solving (4) and (5) in each sector, without
achieving any simplification to this task, let me point out
that there is indeed a major simplification:

Observation: Consider any theory defined by N operators D_i
that satisfy the relation

  (6) {D_i, D_j} = 2 delta_ij L

where {.,.} is the anticommutator and (by definition)

 L = (D_i)^2 , for i in {1,...,N},

and where physical states F are defined by

 D_i F = 0 ,  for i in {1,...,N}.

>From any solution G to the *single* equation

 D_1 G = 0

one obtains a solution to all N constraints by the following
algorithm:

 (a) Choose any i in {1,...,N} such that D_i G <> 0. If there
     is no such i we are done.

 (b) Replace G by its image under D_i, i.e. set

       G <-  D_i G,

     and continue with (a).

It is easlily seen, by using (6), that the resulting G of this
algorithm solves all the equations

 D_i G = 0,  i in {1,...,N} .
 
 

Now for p-form electromagnetism set

 (7) D_1 = d + *d*

 (8) D_2 = i(d - *d*),

which satisfies (6) for N=2. For simplicity assume that there
is a coordinate patch in which the connection coefficients are
time independent. Then one can write

      (d + *d*) F = 0

<=>   yn (@n + 1/4 omega_nab (ya yb - Ya Yb) ) F = 0

<=>   @0 F = - y0 (ym @n - 1/4 omega_nab (ya yb - Ya Yb)) F

<=>   @0 F = H F

<=>   F = exp(x0 H) F(x0 = 0)

with

 yn = (e/\)n - (e->)n
 Yn = (e/\)n + (e->)n
 omega_nab = the Levi-Civita connection components
 H  = - y0 (ym @n - 1/4 omega_nab (ya yb - Ya Yb)) .

Hence D_1 F = 0 is (formally) emmediatly integrated. It may
happen that the result

 exp(x0 H) F(x0 = 0)

by accident also satisfies

 D_2 exp(x0 H) F(x0 = 0).

If it does we are done. If not choose instead

 G = D_2 exp(x0 H) F(x0 = 0) .

This will do since

 D_2 G = (D_2)^2 F = L F = (D_1)^2 F = 0 .

Thus G solves (1) and (3) and, by the above consideration,
every p-form sector of G is a solution to the respective
(source free) p-form electromagnetism.

But one can do more:

Suppose there are more D_i around than just (7) and (8), so
that (6) is satisfied for N > 2. This happens if the
underlying geometry admits complex structures on the tangent
bundle that satisfy one of the normed division algebras (i.e.
Kaehler, Hyperkaehler,... geometry). Then, acting with any one
of these additional operators on the above solution G gives
yet another solution in every p-form sector.

In fact, one can do even more:

Suppose you have used up all the complex structures and want
still more solutions. Then look for Killing-Yano tensors on
spacetime that are not complex structures. These give even
more D_i operators, but now the algebra will no longer close
on L = (D_1)^2 but will have "central charges":
 

 {D_i, D_j} = 2 delta_ij Z_i

with

 Z_1 = L

and all Z_i are Casimir operators of the algebra. (The Z_i
will generically have the form c1 L + c2 L', where L' is the
exterior Laplace operator of a metric "dual" to the original
metric.)

Just like D_1 + D_2 = 2d gives a nilpotent operator one may
generically construct two nilpotent operators by lineasr
combinations of the above D_i. This allows to put the above
diagonal algebra in "polar" form, with nontrivial bracket

 {d_i, del_j} = delta _ij Z_i.

Find simultaneous eigenstates

 F = |z1,z2,...zN>

of the Casimirs. One can assume these to be "vacuua" with
respect to all del_i:

 del_i |z1,z2,...zN> = 0,

because if they are not, one applies del_i to them until they
are.

This then gives us entire "supermultiplets" (I am still
talking about ordinary EM (well, ordinary p-form EM in
arbitrary dimensions)) of states by acting on these vacuua
with the "creation operators"

 d_i1 di_2 ... |z1,z2,...zN> .

All of these with z1 = 0 are (again in every p-form sector)
solutions to p-form electromagnetism.
 
 
 

I realize that a possible objection to the above is, that I
merrily generate further and further solutions, but without
proper control on the boundary conditions. This deserves
further thinking on my part. But it is nevertheless
remarkable, I think, that all p-form electromagnetisms (on a
given spacetime) are intimitely related and that one can map
solutions from one to the other in many ways. This is why I,
personally, do not think in terms of different types of EM,
but simply in terms of the massless Kaehler equation and its
higher symmetric solutions, which subsumes all of it.
 
 

> There is a lot of cool stuff about how these higher analogues
> of Maxwell's equations relate to n-category theory, and how they
> generalize to higher analogues of Yang-Mills equations, but this
> is top secret.

I'd be interested to know if and how the Kaehler-equation
point of view carries over to n-Category theory.

--
Urs.Schreiber@uni-essen.de
 
 
 

Viewing message <a7o174$120a$1@rs04.hrz.uni-essen.de>
 
 

 Von:Urs Schreiber (Urs.Schreiber@uni-essen.de)
 Betrifft:Re: Higher analogues of Maxwell's equations
 Newsgroups:sci.physics.research
 Datum:2002-03-26
 09:23:42 PST
 

"Aaron Bergman" <abergman@princeton.edu> schrieb im Newsbeitrag
news:abergman-6C4F00.00431823032002@news.bellatlantic.net...
> In article <3C9B2912.461E2127@uni-essen.de>,
>  Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

[...]

> > The 3-form potential A couples to 2-branes with worldvolume
> > V via L = \int_V A. How do the 5-branes couple to A? Via \int A^A ?
>
> Let F = dA and define (locally a set of) *F = dA'. A' is a 6 form and
> couples to five branes.

Interesting. I rewrite that as

  F = dA = *dA' = *d* *A' = del *A'

(up to possibly a sign). The potentials A and *A' exist because F is closed
and coclosed and hence locally exact and coexact. Adding both potentials
gives a single inhomogeneous-form potential B:

 B = A - *A'

which satisfies the Kaehler equation

 (d+del) B = 0

if A is given in Lorentz gauge

 del A = 0

and *A' in "co-Lorentz" gauge.

 d *A' = 0.

(Other gauges may be chosen while preserving the Kaehler equation
by adding suitable terms to B. These terms will then serve as potentials
for other p-form sectors.)

So to every p-form potential (p<d-1) there is a "dual" (d-(p+2))-form
potential. To every p'-brane (p'<d-3) coupled to p=(p'+1)-form EM there
is a  "dual" (d-(p+2))-1=(d-((p'+1)+2))-1=(d-(p'+4))-brane. (Here "dual"
refers to the above relation.)

Is there a "dual" brane for ordinary EM in 4 dimensions? In ordinary EM the
brane is a point particle, a (p'=0)-brane, coupled to (p=1)-form
vectorpotential.
The "dual" degrees are (d-(p'+4)) = 0, (d-(p+2)) = 1. Hence ordinary EM
is "self-dual" in the above sense, there is only one kind of brane here.

--
Urs.Schreiber@uni-essen.de