Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 090, 30 pages      arXiv:2307.06355      https://doi.org/10.3842/SIGMA.2024.090

Moving NS Punctures on Super Spheres

Dimitri P. Skliros
Blackett Laboratory, Imperial College London, SW7 2AZ, UK

Received February 20, 2024, in final form September 18, 2024; Published online October 10, 2024

Abstract
One of the subtleties that has made superstring perturbation theory intricate at high string loop order is the fact that as shown by Donagi and Witten, supermoduli space is not holomorphically projected, nor is it holomorphically split. In recent years, Sen (further refined by Sen and Witten) has introduced the notion of vertical integration in moduli space. This enables one to build BRST-invariant and well-defined amplitudes by adding certain correction terms to the contributions associated to the traditional ''delta function'' gauge fixing for the worldsheet gravitino on local patches. The Sen and Witten approach is made possible due to there being no obstruction to a smooth splitting of supermoduli space, but it may not necessarily be the most convenient or natural solution to the problem. In particular, this approach does not determine what these corrections terms actually are from the outset. Instead, it shows that such correction terms in principle exist, and when included make all perturbative amplitudes well-defined. There may be situations however where one would like to instead have a well-defined and fully determined path integral at arbitrary string loop order from the outset. In this paper, I initiate an alternative (differential-geometric) approach that implements the fact that a smooth gauge slice for supermoduli space always exists. As a warmup, I focus specifically on super Riemann surfaces with the topology of a sphere in heterotic string theory, incorporating the corresponding super curvature locally, and introduce a new well-defined smooth gauge fixing that leads to a globally defined path integral measure that translates arbitrary fixed ($-1$) picture NS vertex operators (or handle operators) (that may or may not be offshell) to integrated (0) picture. I also provide some comments on the extension to arbitrary super Riemann surfaces.

Key words: integrated NS vertex operators; picture changing; super curvature; curved super Riemann surfaces; superconformal normal coordinates; deformations of supercomplex structures; superstring perturbation theory; heterotic strings.

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References

  1. Ahmadain A., Wall A.C., Off-shell strings I: S-matrix and action, SciPost Phys. 17 (2024), 005, 61 pages, arXiv:2211.08607.
  2. Ahmadain A., Wall A.C., Off-shell strings II: Black hole entropy, SciPost Phys. 17 (2024), 006, 37 pages, arXiv:2211.16448.
  3. Angelantonj C., Florakis I., Pioline B., A new look at one-loop integrals in string theory, Commun. Number Theory Phys. 6 (2012), 159-201, arXiv:1110.5318.
  4. Anninos D., Bautista T., Mühlmann B., The two-sphere partition function in two-dimensional quantum gravity, J. High Energy Phys. 2021 (2021), no. 9, 116, 29 pages, arXiv:2106.01665.
  5. Atick J.J., Moore G., Sen A., Some global issues in string perturbation theory, Nuclear Phys. B 308 (1988), 1-101.
  6. Baranov M.A., Frolov I.V., Shvarts A.S., Geometry of two-dimensional superconformal field theories, Theoret. and Math. Phys. 70 (1987), 64-72.
  7. Belavin A.A., Knizhnik V.G., Algebraic geometry and the geometry of quantum strings, Phys. Lett. B 168 (1986), 201-206.
  8. Bergman O., Zwiebach B., The dilaton theorem and closed string backgrounds, Nuclear Phys. B 441 (1995), 76-118, arXiv:hep-th/9411047.
  9. Berkovits N., D'Hoker E., Green M.B., Johansson H., Schlotterer O., Snowmass white paper: string perturbation theory, arXiv:2203.09099.
  10. Bern Z., Carrasco J.J., Chiodaroli M., Johansson H., Roiban R., The duality between color and kinematics and its applications, J. Phys. A 57 (2024), 333002, 179 pages, arXiv:1909.01358.
  11. Crane L., Rabin J.M., Super Riemann surfaces: uniformization and Teichmüller theory, Comm. Math. Phys. 113 (1988), 601-623.
  12. de Lacroix C., Erbin H., Pratap Kashyap S., Sen A., Verma M., Closed superstring field theory and its applications, Internat. J. Modern Phys. A 32 (2017), 1730021, 115 pages, arXiv:1703.06410.
  13. D'Hoker E., Mafra C.R., Pioline B., Schlotterer O., Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors, J. High Energy Phys. 2020 (2020), no. 8, 135, 77 pages, arXiv:2006.05270.
  14. D'Hoker E., Phong D.H., The geometry of string perturbation theory, Rev. Modern Phys. 60 (1988), 917-1065.
  15. D'Hoker E., Phong D.H., Conformal scalar fields and chiral splitting on super Riemann surfaces, Comm. Math. Phys. 125 (1989), 469-513.
  16. D'Hoker E., Phong D.H., Two-loop superstrings. I. Main formulas, Phys. Lett. B 529 (2002), 241-255, arXiv:hep-th/0110247.
  17. D'Hoker E., Phong D.H., Higher order deformations of complex structures, SIGMA 11 (2015), 047, 14 pages, arXiv:1502.03673.
  18. Dijkgraaf R., Witten E., Developments in topological gravity, Internat. J. Modern Phys. A 33 (2018), 1830029, 63 pages.
  19. Distler J., Nelson P., Topological couplings and contact terms in 2d field theory, Comm. Math. Phys. 138 (1991), 273-290.
  20. Donagi R., Witten E., Supermoduli space is not projected, in String-Math 2012, Proc. Sympos. Pure Math., Vol. 90, American Mathematical Society, Providence, RI, 2015, 19-71, arXiv:1304.7798.
  21. Doyle M.D., The operator formalism and contact terms in string theory, Ph.D. Thesis, Princeton University, 1992.
  22. Eberhardt L., Partition functions of the tensionless string, J. High Energy Phys. 2021 (2021), no. 3, 176, 43 pages, arXiv:2008.07533.
  23. Eberhardt L., Pal S., The disk partition function in string theory, J. High Energy Phys. 2021 (2021), no. 3, 026, 28 pages, arXiv:2105.08726.
  24. Erbin H., Maldacena J., Skliros D., Two-point string amplitudes, J. High Energy Phys. 2019 (2019), no. 7, 139, 6 pages, arXiv:1906.06051.
  25. Erler T., The closed string field theory action vanishes, J. High Energy Phys. 2022 (2022), no. 5, 055, 8 pages, arXiv:2204.12863.
  26. Erler T., Konopka S., Vertical integration from the large Hilbert space, J. High Energy Phys. 2017 (2017), no. 12, 112, 36 pages, arXiv:1710.07232.
  27. Fischler W., Susskind L., Dilaton tadpoles, string condensates and scale invariance, Phys. Lett. B 171 (1986), 383-389.
  28. Fischler W., Susskind L., Dilaton tadpoles, string condensates and scale invariance. II, Phys. Lett. B 173 (1986), 262-264.
  29. Friedan D., Notes on string theory and two-dimensional conformal field theory, in Workshop on Unified String Theories (Santa Barbara, Calif., 1985), World Scientific Publishing, Singapore, 1986, 162-213.
  30. Friedan D., Martinec E., Shenker S., Conformal invariance, supersymmetry and string theory, Nuclear Phys. B 271 (1986), 93-165.
  31. Geyer Y., Monteiro R., Stark-Muchão R., Superstring loop amplitudes from the field theory limit, Phys. Rev. Lett. 127 (2021), 211603, 9 pages, arXiv:2106.03968.
  32. Gross D.J., Harvey J.A., Martinec E., Rohm R., Heterotic string theory. I. The free heterotic string, Nuclear Phys. B 256 (1985), 253-284.
  33. Gross D.J., Harvey J.A., Martinec E., Rohm R., Heterotic string theory. II. The interacting heterotic string, Nuclear Phys. B 267 (1986), 75-124.
  34. Kawai H., Lewellen D.C., Tye S.H.H., A relation between tree amplitudes of closed and open strings, Nuclear Phys. B 269 (1986), 1-23.
  35. La H., Nelson P., Effective field equations for fermionic strings, Nuclear Phys. B 332 (1990), 83-130.
  36. Mahajan R., Stanford D., Yan C., Sphere and disk partition functions in Liouville and in matrix integrals, J. High Energy Phys. 2022 (2022), no. 7, 132, 26 pages, arXiv:2107.01172.
  37. Mirzakhani M., Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179-222.
  38. Mirzakhani M., Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1-23.
  39. Mizera S., Combinatorics and topology of Kawai-Lewellen-Tye relations, J. High Energy Phys. 2017 (2017), no. 8, 097, 53 pages, arXiv:1706.08527.
  40. Nelson P., Covariant insertion of general vertex operators, Phys. Rev. Lett. 62 (1989), 993-996.
  41. Nelson P., Lectures on supermanifolds and strings, in Particles, Strings and Supernovae, Vol. 1, 2 (Providence, RI, 1988), World Scientific Publishing, Teaneck, NJ, 1989, 997-1073.
  42. Nelson P., Operator formalism and holomorphic factorization in supermoduli, in Superstrings '88 (Trieste, 1988), World Scientific Publishing, Teaneck, NJ, 1989, 276-287.
  43. O'Brien K.H., Tan C.I., Modular invariance of thermopartition function and global phase structure of heterotic string, Phys. Rev. D 36 (1987), 1184-1192.
  44. Polchinski J., Vertex operators in the Polyakov path integral, Nuclear Phys. B 289 (1987), 465-483.
  45. Polchinski J., Factorization of bosonic string amplitudes, Nuclear Phys. B 307 (1988), 61-92.
  46. Polchinski J., String theory. Vol. I. An introduction to the bosonic string, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1998.
  47. Polchinski J., String theory. Vol. II. Superstring theory and beyond, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1998.
  48. Seiberg N., Witten E., Spin structures in string theory, Nuclear Phys. B 276 (1986), 272-290.
  49. Sen A., Off-shell amplitudes in superstring theory, Fortschr. Phys. 63 (2015), 149-188, arXiv:1408.0571.
  50. Sen A., Witten E., Filling the gaps with PCO's, J. High Energy Phys. 2015 (2015), no. 9, 004, 34 pages, arXiv:1504.00609.
  51. Sen A., Zwiebach B., String field theory: a review, arXiv:2405.19421.
  52. Shenker S.H., The strength of nonperturbative effects in string theory, in Random Surfaces and Quantum Gravity, NATO Adv. Sci. Inst. Ser. B: Phys., Vol. 262, Plenum, New York, 1991, 191-200.
  53. Skliros D., Lüst D., Handle operators in string theory, Phys. Rep. 897 (2021), 1-180, arXiv:1912.01055.
  54. Skliros D.P., Copeland E.J., Saffin P.M., Highly excited strings I: generating function, Nuclear Phys. B 916 (2017), 143-207, arXiv:1611.06498.
  55. Stieberger S., A Relation between one-loop amplitudes of closed and open strings (one-loop KLT relation), arXiv:2212.06816.
  56. Stieberger S., One-loop double copy relation in string theory, Phys. Rev. Lett. 132 (2024), 191602, 8 pages, arXiv:2310.07755.
  57. Tseytlin A.A., On the ''macroscopic string'' approximation in string theory, Phys. Lett. B 251 (1990), 530-534.
  58. Verlinde E., Verlinde H., Multiloop calculations in covariant superstring theory, Phys. Lett. B 192 (1987), 95-102.
  59. Verlinde H.L., Path-integral formulation of supersymmetric string theory, Ph.D. Thesis, Rijksuniversiteit Utrecht (Netherlands), 1988, https://inis.iaea.org/search/search.aspx?orig_q=RN:20024923.
  60. Wang C., Yin X., On the equivalence between SRS and PCO formulations of superstring perturbation theory, J. High Energy Phys. 2023 (2023), no. 3, 139, 26 pages, arXiv:2205.01106.
  61. Witten E., The Feynman $i\epsilon$ in string theory, J. High Energy Phys. 2015 (2015), no. 4, 055, 24 pages, arXiv:1307.5124.
  62. Witten E., Notes on super Riemann surfaces and their moduli, Pure Appl. Math. Q. 15 (2019), 57-211, arXiv:1209.2459.
  63. Witten E., Notes on supermanifolds and integration, Pure Appl. Math. Q. 15 (2019), 3-56, arXiv:1209.2199.
  64. Witten E., Perturbative superstring theory revisited, Pure Appl. Math. Q. 15 (2019), 213-516, arXiv:1209.5461v3.
  65. Zwiebach B., Closed string field theory: quantum action and the Batalin-Vilkovisky master equation, Nuclear Phys. B 390 (1993), 33-152, arXiv:hep-th/9206084.

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