Application of supersymmetric vertex algebras to the investigation of supermoduli spaces

• Applicant: Dr. Christoph Sachse
• Subject Area: Mathematics
• Term: from 2008 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 73745670

• Final Report Year: 2011

Final Report Abstract

The goal of this project is to develop a geometric understanding of two-dimensional superconformal field theories and supersymmetric field theories in general. Due to the wide variety of difficulties encountered I focused on particularly promising topics. Progress was achieved along several lines. The first focus was the study of supergeometric tools, in particular diffeomorphism supergroups, methods of infinite-dimensional supergeometry and possible analytic approaches to the construction of moduli spaces of superconformal structures. Secondly I started a collaboration with C. Wockel and T. Nikolaus to study the representation theory of the string group and associated vector bundles on manifolds. The goal here is to achieve a better (super-)geometric understanding of the relation between supersymmetric sigma models, elliptic cohomology and higher categorical structures in geometry. Finally, in a collaboration with C. Devchand and P. Guha, I could make a contribution to the study of supersymmetric integrable systems, more precisley, supersymmetric analogues of the KdV equation. All these projects can be seen as steps towards my long-term goal of understanding the geometry of supersymmetric sigma models, in particular their relation to cohomology theories.

Publications

• Fuzzy Scalar Field Theory as Matrix Quantum Mechanics
  Matthias Ihl, Christoph Sachse, Christian Sämann
  
• On critical dimensions of string theories. Proc. of the XII. Regional Conference on Mathematical Physics, Islamabad, Pakistan, eds.: M.J. Aslam, F. Hussain, A. Qadir, Riazuddin and H. Saleem, World Scientific
  D. A. Leites, C. Sachse
  
• Superconformal Algebras and Supersymmetric Integrable Flows
  C. Devchand, P. Guha, C. Sachse
  
• Sylvester-t’Hooft generators and relations between them for sl(n) and gl(n|n). Theor. Math. Phys. 149(1): 1299-1311, 2006
  C. Sachse
  
• A Categorical Formulation of Superalgebra and Supergeometry. 02/2008
  C. Sachse
  
• The diffeomorphism supergroup of a finitedimensional supermanifold. 04/2009, Adv.Math.Theor.Phys.
  C. Sachse, C. Wockel