Letters in Mathematical Physics

, Volume 81, Issue 1, pp 41–59 | Cite as

Semi-Classical Quantum Fields Theories and Frobenius Manifolds



We show that a semi-classical quantum field theory comes with a versal family with the property that the corresponding partition function generates all path integrals., satisfies a system of second order differential equations determined by algebras of classical observables. This versal family gives rise to a notion of special coordinates that is analogous to that in string theories. We also show that for a large class of semi-classical theories, their moduli space has the structure of a Frobenius super-manifold.

Mathematics Subject Classification (2000)

81T70 58H15 


Quantum fields theory BV quantization special coordinates Frobenius manifold 


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  1. 1.
    Barannikov, S., Kontsevich, M.: Frobenius manifolds., formality of Lie algebras of polyvector fields. Int. Math. Res. Not. 4, 201–215 (1998) [arXiv:alg-geom/9710032]Google Scholar
  2. 2.
    Batalin I.A.., Vilkovisky G.A. (1981). Gauge algebra., quantization. Phys. Lett. B 102: 27–31 CrossRefADSGoogle Scholar
  3. 3.
    Dijkraaf R., Verlinde E.., Verlinde H. (1991). Topological strings in d < 1. Nucl. Phys. B 352: 59–86 CrossRefADSGoogle Scholar
  4. 4.
    Dubrovin, B.A.: Geometry of 2D topological field theories. In: (Montecatini Terme 1993) Integrable systems., quantum groups. Lecture Notes in Math, vol. 1620, pp. 120–348. Springer, Berlin (1996), [arXiv:hep-th/9407018]Google Scholar
  5. 5.
    Huebschmann J.., Stasheff J. (2002). Formal solution of the master equation via HPT., deformation theory. Forum Mathematicum 14: 847–868. [arXiv:math.AG/9906036] MATHCrossRefGoogle Scholar
  6. 6.
    Manin, Yu. I.: Three constructions of Frobenius manifolds: a comparative study. In: Surv. Differ. Geom., vol. 7, pp. 497–554. International Press, Somerville (2000) [arXiv: math.QA/9801006]Google Scholar
  7. 7.
    Park, J.S.: Flat family of QFTs., quantization of d-algebras. [arXiv:hep-th/ 0308130]Google Scholar
  8. 8.
    Park, J.S.: Special coordinates in quantum fields theory I: affine classical structure.Google Scholar
  9. 9.
    Saito K. (1981). Primitive forms for an universal unfolding of a functions with isolated critical point. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28(3): 777–792 Google Scholar
  10. 10.
    Schwarz A. (1993). Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155: 249–260. [arXiv:hep-th/9205088] MATHCrossRefADSGoogle Scholar
  11. 11.
    Witten E. (1990). On the structure of the topological phase of two-dimensional gravity. Nucl. Phys. B 340: 281–332 CrossRefADSGoogle Scholar
  12. 12.
    Witten, E.: Mirror manifolds., topological field theory. [arXiv:hep-th/9112056]Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of MathematicsYonsei UniversitySeoulSouth Korea

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