A high-resolution version for printing - CP3-Origins

The geometry group at SDU provides the mathematical soul of the centre and is involved in
providing a strong training in mathematics and complementary expertise for the high energy
component. SM extensions have a significant mathematical content, particularly in the form of
differential geometry, Lie group theory and topology. We report below some of their scientific
output and ongoing research.
Hyperkähler Modifications
We investigated a general framework for cutting constructions and reinterpret in this setting the
work on non-Abelian symplectic cuts by Weitsman. We then introduced two analogous non-
Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly,
the other gives metric deformations. We highlighted ways in which the geometry of
moment maps for non-Abelian hyperkähler actions differs from the Abelian case and from the
non-Abelian symplectic case.
[Non-Abelian Cut Constructions and Hyperkähler Modifications. Andrew Dancer (Oxford),
Andrew Swann (IMADA & CP3-Origins), arXiv:1002.1837. Submitted for Publication]
Harmonic Maps
We used filtrations of the Grassmannian model to produce explicit algebraic formulae for harmonic
maps of finite uniton number from a Riemann surface to the unitary group for a general
class of factorizations by unitons. We showed how these specialize to give explicit formulae for
harmonic maps into the special orthogonal and symplectic groups, real, complex and quaternionic
Grassmannians, and the spaces SO(2m)/U(m) and Sp(n)/U(n), i.e., all the classical compact
Lie groups and their inner symmetric spaces. Our methods also give explicit J_2-
holomorphic lifts for harmonic maps into Grassmannians and an explicit Iwasawa decomposition.
[Filtrations, factorizations and explicit formulae for harmonic maps. Martin Svensson
(IMADA & CP 3 -Origins), John C. Wood (Leeds), arXiv:1002.1837. Submitted for Publication]
Strong KT geometry
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is
closed. We classified the invariant SKT structures on four-dimensional solvable Lie groups. The
classification includes solutions on groups that do not admit compact four-dimensional quotients.
It also shows that there are solvable groups in dimension four that admit invariant complex
structures but have no invariant SKT structure.
[Invariant strong KT geometry on four-dimensional solvable Lie groups. Thomas Bruun Madsen
(IMADA and CP 3 -Origins) and Andrew Swann (IMADA and CP 3 -Origins), IMADA-PP-2009-16, CP3-
ORIGINS: 2009-22,arXiv:0911.0535v1 [math.DG]. Submitted for Publication]
We will continue investigating the geometric structure of the Standard Model and its extensions.
One interesting example is the topological structure of generic models of dynamical electroweak
symmetry breaking. The physics of these topological terms can be investigated at the
LHC.
CP³-Black book 18