What is the Jacobian of a Riemann Surface with Boundary?

  • Thomas M. Fiore
  • Igor Kriz


We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of “open abelian varieties” which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of “conformal field theory” to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties.


Modulus Space Riemann Surface Boundary Component Abelian Variety Mapping Class Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T.M. Fiore: Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Mem. Amer. Math. Soc. 182 (2006).Google Scholar
  2. 2.
    T.M. Fiore: On the cobordism and commutative monoid with cancellation approaches to conformal field theory, J. Pure Appl. Algebra 209 (2007) 583–620.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    T.M. Fiore, P. Hu, I. Kriz: Laplaza sets, or how to select coherence diagrams for pseudo algebras, to appear in the Advances of Mathematics.Google Scholar
  4. 4.
    P. Hu, I. Kriz: Conformal field theory and elliptic cohomology, Adv. Math. 189 (2004) 325–412.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    P. Hu, I. Kriz: Closed and open conformal field theories and their anomalies, Comm. Math. Phys. 254 (2005) 221–253.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Mumford: Abelian varieties, Tata Inst. of Fund. Res. Studies in Math. 5, London 1970.Google Scholar
  7. 7.
    A. Pressley, G. Segal: Loop groups, Oxford Math. Monographs, Oxford University Press 1986.Google Scholar
  8. 8.
    B. Riemann: Theorie der Abelschen Functionen, J. Reine und Angew. Mathematik 54 (1857) 101–155.MATHCrossRefGoogle Scholar
  9. 9.
    G. Segal: The definition of conformal field theory, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 308, Cambridge University Press, 2004, 421–577.Google Scholar
  10. 10.
    G. Segal: Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981). 301–342.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2010

Authors and Affiliations

  • Thomas M. Fiore
  • Igor Kriz

There are no affiliations available

Personalised recommendations