Discussion:
unoriented Strings coupled to KR?
(too old to reply)
Urs Schreiber
2005-08-23 18:20:18 UTC
Permalink
An oriented string couples to the Kalb-Ramond field by means of gerbe
holonomy.

Under certain conditions there can be KR fields for unoriented strings, too.
Is there anything known about how these unoriented strings would couple to
the KR field in this sense?
Lubos Motl
2005-08-24 00:07:31 UTC
Permalink
Post by Urs Schreiber
An oriented string couples to the Kalb-Ramond field by means of gerbe
holonomy.
Under certain conditions there can be KR fields for unoriented strings, too.
Is there anything known about how these unoriented strings would couple to
the KR field in this sense?
Dear Urs,

you may have something more fancy in mind. But if you just consider
ordinary unoriented strings, it means that you include an orientifold. One
of the implications of an orientifold planes, as seen by its effects on
the closed string modes, is the constraint for the B-field

B_{mn} (x) = -B_{mhat nhat} (xhat)

where "xhat" is the point associated with "x" by the orientifold Z_2
symmetry, and "mhat, nhat" are the correspondingly related indices (i.e.
they include one minus sign for every index "m,n" that is transverse to
the orientifold plane).

For example, the spacetime-filling orientifold O9-plane is something that
creates type I theory out of type IIB (if you also add 32
spacetime-filling half-D9-branes). In this case, "xhat=x" and you simply
get
B_{mn} = -B_{mn} = 0.

This means that in type I, there is no B-field. The strings are thus not
charged under it and they can decay - e.g. open strings can split because
they don't carry any conserved charges.

However, for localized orientifold planes, the B-field can still be
nonzero away from the orientifold plane; that's not surprising, the local
physics is still physics of type II string theory. The fundamental strings
are always coupled to the B-field in the same way. It's just the B-field
itself that becomes constrained - or, equivalently, there is a
cancellation of the B-field contributions between different worldsheets.

All the best
Lubos
______________________________________________________________________________
E-mail: ***@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
2005-08-24 08:54:14 UTC
Permalink
you may have something more fancy in mind. But if you just consider ordinary
Hi Lubos, thanks for the reply.

I am not sure yet if I have anything fancy in mind. :-)

Recently I was confronted with the (general) question what the holonomy of
an abelian gerbe for an unoriented surface should be, if anything.

Thinking physicswise, I was wondering if in string theory we ever
want to integrate the KR field over an unoriented piece of string
worldsheet. As far as I can see the answer is "no". Is that correct?
Urs Schreiber
2005-08-24 09:07:30 UTC
Permalink
Thinking physicswise, I was wondering if in string theory we ever want to
integrate the KR field over an unoriented piece of string
worldsheet.
P.S.

Of course, the case I am interested in is the unorient_able_ case.
Lubos Motl
2005-08-24 15:13:23 UTC
Permalink
Thinking physicswise, I was wondering if in string theory we ever want to
integrate the KR field over an unoriented piece of string
worldsheet. As far as I can see the answer is "no". Is that correct?
Hi Urs,

this is a question that one faces in string backgrounds with orientifolds.
In that case, unoriented worldsheets definitely do contribute to the
scattering amplitudes. So what do we do?

Locally, the Lagrangian on the worldsheet is always the same thing. So if
there is any B-field (if the orientifold planes are localized), you
compute the induced B-field on the worldsheet and integrate it. But how do
you integrate it over unorientable worldsheets?

You may always imagine that the unorientable worldsheet is a Z_2 quotient
of an orientable one. For example, the Klein bottle is a Z_2 worldsheet
orbifold of a torus. Klein bottles are tori with configurations that
preserve a Z_2 constraint.

The Klein bottle itself is one half of the area of the torus, a
fundamental domain. So the only question is whether the two halves of the
torus contribute the same sign to the B-part of the action or not: in the
latter case the term cancels.

That's easy to answer for general orientifolds. The Z_2 action on the
torus configuration, for example, reverses the orientation of the
worldsheet coordinates - the 2-volume form - but it also reverts the sign
of the B-field, namely its components with the odd number of indices along
the orientifold directions. So I think two minus signs may combine to a
plus and that it does not cancel as long as the B-field varies. (It cannot
vary for spacetime filling O9-planes where it's zero everywhere.)

Of course, there is typically no constant mode of a B-field on these
orientifold compactifications, so there are never any natural "periods"
over "unorientable cycles" - something that does not really exist in
homology, except for torsion that can only carry discrete B-fields.

But I think that for non-constant B-fields, the B-field of the action does
not cancel and its sign can be defined consistently. It's only the
spacetime components B_{iM} where one index "i" is along the orientifold
plane and the other index "M" is transverse to it that contribute nonzero;
the other components contribute nothing, by a sign counting. The latter
ones that cancel are the same ones for which you would have problems to
define the sign globally.

There's a 10% probability that I made an odd number of sign errors in
which case the answer is just the opposite. ;-)

Best wishes
Lubos
______________________________________________________________________________
E-mail: ***@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Loading...