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

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