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Super Riemann surfaces: Uniformization and Teichmüller theory. (English) Zbl 0659.30039
In view of applications in the superstring theory the global structure of the space of super moduli is studied. By employing the Rogers theory of supermanifolds and adopting Friedans global definition of a super Riemann surface the Teichmüller theory of Riemann surfaces is generalized.
The construction of super Teichmüller space follows closely that of the ordinary Teichmüller space. For the cases relevant for superstring applications the results characterizing the structure of super Teichmüller space are proven. It appears to be a complex super-orbifold whose body is the ordinary Teichmüller space of the associated Riemann surfaces with spin structure and has, for $$g>1$$, 3g-3 complex even and 2g-2 complex odd dimensions. The important result is that the super modular group is simply the ordinary modular group.
The uniformization theorem for super Riemann surfaces is proved.
The generalization of the representation of Riemann surfaces in terms of Beltrami differentials is given. The super Beltrami equations are derived and their solution is discussed. It is then shown that metrizable super Riemann surfaces can be represented by Schottky supergroups. The notion of universal super Teichmüller space is defined.
Some remarks about the uniformization theorem are super Riemann surfaces with arbitrary topology are also included.
Reviewer: P.Maślanka

##### MSC:
 30F99 Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 83E99 Unified, higher-dimensional and super field theories 81Q99 General mathematical topics and methods in quantum theory 81V99 Applications of quantum theory to specific physical systems
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##### References:
 [1] Polyakov, A. M.: Quantum geometry of bosonic strings. Phys. Lett.103B, 207-210 (1981) [2] D’Hoker, E., Phong, D. H.: Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys.B269, 205-234 (1986) [3] Moore, G., Nelson, P.: Measure for moduli. Nucl. Phys.B266, 58-74 (1986) [4] Belavin, A. A., Knizhnik, V. G.: Algebraic geometry and the geometry of quantum strings. Phys. Lett.168B, 201-206 (1986) · Zbl 0693.58043 [5] D’Hoker, E., Phong, D. H.: Loop amplitudes for the fermionic string. Nucl. Phys.B278, 225-241 (1986) [6] Moore, G., Nelson, P., Polchinski, J.: Strings and supermoduli. Phys. Lett.169B, 47-53 (1986) [7] Friedan, D.: Notes on string theory and two-dimensional conformal field theory. In: The proceedings of the workshop on unified string theories. Gross, D., Green, M. (eds.), Singapore: World Press 1986 · Zbl 0648.53057 [8] Rogers, A.: A global theory of supermanifolds. J. Math. Phys.21, 1352-1365 (1980) · Zbl 0447.58003 [9] Rabin, J. M., Crane, L.: Global properties of supermanifolds. Commun. Math. Phys.100, 141-160 (1985) · Zbl 0576.58004 [10] Rabin, J. M., Crane, L.: How different are the supermanifolds of Rogers and DeWitt? Commun. Math. Phys.102, 123-137 (1985) · Zbl 0576.58003 [11] Baranov, M. A., Shvarts, A. S.: Multiloop contribution to string theory. JETP Lett.42, 419-421 (1986) [12] Martinec, E.: Conformal field theory on a (super-)Riemann surface. Nucl. Phys.B281, 157-210 (1987) [13] Robers, A.: Graded manifolds, supermanifolds, and infinite-dimensional Grassmann algebras. Commun. Math. Phys.105, 375-384 (1986) · Zbl 0602.58003 [14] Arvis, J. F.: Classical dynamics of the supersymmetric Liouville theory. Nucl. Phys.B212, 151-172 (1983) [15] Berkovits, N.: Calculation of scattering amplitudes for the Neveu-Schwarz model using supersheet functional integration. Berkeley preprint UCB-PTH-85/55 (1985) [16] DeWitt, B. S.: Supermanifolds. Cambridge: Cambridge University Press 1984 [17] Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys.106, 1-40 (1986) · Zbl 0605.58049 [18] Farkas, H. M., Kra, I.: Riemann surfaces. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0475.30001 [19] Forster, O.: Lectures on Riemann surfaces. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0475.30002 [20] Nelson, P., Moore, G.: Heterotic geometry. Nucl. Phys.B274, 509-519 (1986) [21] Rothstein, M.: Deformations of complex supermanifolds. Proc. AMS95, 255-260 (1985) · Zbl 0584.32041 [22] Hodgkin, L.: A direct calculation of super Teichmüller space. King’s College Mathematics Department preprint, January 1987 · Zbl 0627.58004 [23] Kodaira, K.: Complex manifolds and deformation of complex structures. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0581.32012 [24] Bers, L.: Uniformization, moduli and Kleinian groups. Bull. London Math. Soc.4, 257-300 (1972) · Zbl 0257.32012 [25] Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math.72, 385-404 (1960) · Zbl 0104.29902 [26] Earle, C. J.: Teichmüller theory. In: Discrete groups and automorphic functions. Harvey, W. J. (ed.). London: Academic Press 1977 [27] Batchelor, M.: The structure of supermanifolds. Trans. AMS253, 329-338 (1979) · Zbl 0413.58002 [28] Batchelor, M.: Two approaches to supermanifolds. Trans. AMS258, 257-270 (1980) · Zbl 0426.58003 [29] Rogers, A.: On the existence of global integral forms on supermanifolds. J. Math. Phys.26, 2749-2753 (1985) · Zbl 0582.53055 [30] Bers, L.: Uniformization by Beltrami equations. Commun. Pure Appl. Math.14, 215-228 (1961) · Zbl 0138.06101 [31] Friedan, D., Shenker, S.: The integrable analytic geometry of quantum string. Phys. Lett.175B, 287-296 (1986) [32] Lawson, H. B.: Foliations. Bull. AMS80, 369-418 (1974) · Zbl 0293.57014 [33] Bott, R., Tu, L. W.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0496.55001 [34] Leites, D.: Introduction to the theory of supermanifolds. Russ. Math. Surv.35, 1-64 (1980) · Zbl 0462.58002 [35] Manin, Yu. I.: Quantum strings and algebraic curves, talk presented at the International Congress of Mathematicians, Berkeley 1986 [36] Le Brun, C., Rothstein, M.: Moduli of super Riemann surfaces. Princeton preprint 1987
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