This post is on the String-String Duality. In particular, the D=6 string-string duality, which is crucial since it allows interchanging the roles of 4-D spacetime and string-world-sheet loop expansion, and this is mathematically essential for phenomenologically adequate string-compactifications. Here I will prove an equivalence between K3 membrane action and orbifold action and show how it entails D=6 string-string duality. Working in the bosonic sector, the membrane action is:
which is deep since Clifford algebras are a quantization of exterior algebras, and applying to the ‘Einstein-Minkowski’ tangent bundle, we get via Gaussian matrix elimination, an expansion of via Green-functions, that yields M-Theory’s action:
with the kappa symmetry term. With the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. The worldvolume is:
The bosonic sector lives on the boundary of the open membrane: two copies of , which naturally couple to the U(1) connections .
Now, double dimensional reduction of the twisted supermembrane on:
entails that the bosonic sector is that of the heterotic string:
with gauge group indices I = 1, …, 16.
It gets interesting when we consider:
since the worldsheet action:
is now just a sum of three terms:
and the index I = 1, …, 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:
Now, something fundamentally deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, and not a three-dimensional theory
This is precisely the long wavelength limit behavior of the open membrane:
the gauge fields are defined in terms of fields which live on 10-dimensional boundaries of M-theory
In the closed membrane case:
the gauge fields are defined in terms of 11-dimensional fields
Hence, the gauge fields of the closed membrane to be defined over M3 and not over its boundary, unlike the closed membrane, whose action on is:
where is with the spacetime being .
Hence, the closed membrane action on reduces to:
and since surfaces have no one-cycles, it follows that the three-form potential that appears in of the action:
can be expanded in terms of the cocycles of .
For the 22 2-cocycles of , one can decompose in a similar way for the two-form potential:
with I = 1, …, 22 labeling the two-cycles of . So after insertion into , we can derive:
Applying reparametrization invariance, one can set:
where is a worldvolume coordinate, and now one performs a dimensional reduction of:
Here are the key propositions relevant to the membrane/string duality of the low energy theory in D=7.
- the kinetic terms for the gauge fields in D=7 supergravity are:
derived by a split of the 4-4 field strength , of the 11-dimensional supergravity action:
from the the following term:
- Membrane/string duality in D=7 requires the existence of a point in the moduli space of where all the 22 gauge fields are enhanced via