In this series, I will study T-branes as they relate to F-theory in the context of GUT-model-construction. Aside: the book on the cover is a great read. Note that D-branes are crucial since they allow us to build various 4-D string theory vacua and dictate the phenomenology of compactifications as well. T-branes are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Without loss of generality, I will work in type IIB string theory compactified on a Calabi-Yau threefold quotiented by an orientifold action stacks of D3-branes and D7-branes. The BPS conditions are functionals:
and in 4-D are interpreted as a superpotential and D-term for each D7-brane.
and generally, the second quantized integral of the D-brane partition function for closed strings is given by:
with a non-Abelian D-term:
is the first Pontryagin class-term, and is the flat space Kähler form:
is given by:
In this, part one, I will show that the 7-brane superpotential for F-theory and the corresponding F-term conditions are purely topological and in no need for -corrections.
Recalling that a central class of T-brane configurations feature a Higgs field and a set of non-commuting generators as well as a non-primitive worldvolume flux of the form:
which is a solution to the D-brane equation, and is the Cartan generator of the gauge group and is a function of the 7-brane coordinates that solves the D-brane equation. The central problem of this series of posts is that this Abelian profile for needs the -correction for 4-D compactificational Calabi-Yau consistency conditions.
First, note that the Kähler form and:
a holomorphic (3,0)-form in , satisfy:
with the worldvolume flux and . Hence,
the pull-back on the D7-brane worldvolume is
where is a coordinate on .
The non-Abelian profiles for and must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:
are built up as follows. Write locally:
and localize the integral in:
the non-Abelian generalisation of and have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion
So effectively, we have a Kähler-equivalence of the derivatives in the pull-back with gauge-covariant ones, yielding:
with the inclusion of the complex Higgs field , and represents the symmetrization over gauge indices.
In this local description, the Higgs field is given by:
where is a matrix in the complexified adjoint representation of and its Hermitian conjugate. Thus, locally, we have:
a Kähler coordiante expansion of and gives us, after inserting it in: