The geometry and representation theory of superconformal quantum mechanics

We study aspects of the quantum mechanics of nonlinear σ-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\mathfrak{su}(1,1|1),~\mathfrak{u}(1,1|2)$ and \mathfrak{osp}(4^*|4) explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $\mathcal{N}=4$ SUSY Yang-Mills. In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra \mathfrak{osp}(4^*|4). We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for \mathfrak{osp}(4^*|4). States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the L² Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of σ-model with $\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\mathfrak{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. Consideration of the spectral problem for the dilatation operator in these models leads to a deformation which we interpret, via an extension of the moduli space approximation, as an anti-self-dual spacetime magnetic field coupling to the topological instanton current.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2016

Author dc:creator

• Singleton, Andrew John

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9171c46a-d3c2-4d02-8dd6-f91a2f5f5874/download
• https://www.rioxx.net/licenses/all-rights-reserved/
• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9866eafd-05af-4ac4-ad16-a412e39fb02c/download

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.275

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/260821