Abstract
We present a complete computation of superstring scattering amplitudes at tree level, for the case of Neveu-Schwarz insertions. Mathematically, this is to say that we determine explicitly the superstring measure on the moduli space \( {\mathcal{M}}_{0,n,0} \) of super Riemann surfaces of genus zero with n ≥ 3 Neveu-Schwarz punctures. While, of course, an expression for the measure was previously known, we do this from first principles, using the canonically defined super Mumford isomorphism [1]. We thus determine the scattering amplitudes, explicitly in the global coordinates on \( {\mathcal{M}}_{0,n,0} \), without the need for picture changing operators or ghosts, and are also able to determine canonically the value of the coupling constant. Our computation should be viewed as a step towards performing similar analysis on \( {\mathcal{M}}_{0,0,n} \), to derive explicit tree-level scattering amplitudes with Ramond insertions.
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References
A. A. Voronov, A formula for the Mumford measure in superstring theory, Funk. Anal. Prilozh. 22 (1988) 67 [Funct. Anal. Appl. 22 (1988) 139].
A.A. Belavin and V.G. Knizhnik, Algebraic Geometry and the Geometry of Quantum Strings, Phys. Lett. B 168 (1986) 201 [INSPIRE].
E. D’Hoker and D.H. Phong, Multiloop Amplitudes for the Bosonic Polyakov String, Nucl. Phys. B 269 (1986) 205 [INSPIRE].
J. Jost, Bosonic Strings: A Mathematical Treatment, AMS (2001) [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory Vol. 1: 25th Anniversary Edition, Cambridge University Press (2012) [https://doi.org/10.1017/CBO9781139248563] [INSPIRE].
J. Polchinski, String theory. Volume 1: An introduction to the bosonic string, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511816079] [INSPIRE].
E. Witten, Perturbative superstring theory revisited, Pure Appl. Math. Quart. 15 (2019) 213 [INSPIRE].
G. Felder, D. Kazhdan and A. Polishchuk, Regularity of the superstring supermeasure and the superperiod map, Selecta Math. 28 (2022) 17 [arXiv:1905.12805] [INSPIRE].
G. Felder, D. Kazhdan and A. Polishchuk, The moduli space of stable supercurves and its canonical line bundle, Am. J. Math. 145 (2023) 1777 [arXiv:2006.13271] [INSPIRE].
U. Bruzzo, D.H. Ruiperez and A. Polishchuk, Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes, Adv. Math. 415 (2023) 108890 [arXiv:2008.00700] [INSPIRE].
Yu. I. Manin, Critical dimensions of the string theories and the dualizing sheaf on the moduli space of (super) curves, Funk. Anal. Prilozh. 20 (1986) 88 [Funct. Anal. Appl. 20 (1986) 244].
D. Diroff, On the super Mumford form in the presence of Ramond and Neveu-Schwarz punctures, J. Geom. Phys. 144 (2019) 273 [arXiv:1802.07865] [INSPIRE].
N. Ott and A.A. Voronov, The supermoduli space of genus zero super Riemann surfaces with Ramond punctures, J. Geom. Phys. 185 (2023) 104726 [arXiv:1910.05655] [INSPIRE].
R. Donagi and N. Ott, Supermoduli Space with Ramond punctures is not projected, arXiv:2308.07957 [INSPIRE].
R. Donagi and E. Witten, Supermoduli Space Is Not Projected, Proc. Symp. Pure Math. 90 (2015) 19 [arXiv:1304.7798] [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory Volume 2: 25th Anniversary Edition, Cambridge University Press (2012) [https://doi.org/10.1017/CBO9781139248570] [INSPIRE].
J. Polchinski, String theory. Volume 2: Superstring theory and beyond, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511618123] [INSPIRE].
E. D’Hoker and D.H. Phong, Two loop superstrings. I. Main formulas, Phys. Lett. B 529 (2002) 241 [hep-th/0110247] [INSPIRE].
E. D’Hoker and D.H. Phong, Two loop superstrings. II. The chiral measure on moduli space, Nucl. Phys. B 636 (2002) 3 [hep-th/0110283] [INSPIRE].
E. D’Hoker and D.H. Phong, Two loop superstrings. III. Slice independence and absence of ambiguities, Nucl. Phys. B 636 (2002) 61 [hep-th/0111016] [INSPIRE].
E. D’Hoker and D.H. Phong, Two loop superstrings IV. The cosmological constant and modular forms, Nucl. Phys. B 639 (2002) 129 [hep-th/0111040] [INSPIRE].
E. D’Hoker and D.H. Phong, Two-loop superstrings. V. Gauge slice independence of the N-point function, Nucl. Phys. B 715 (2005) 91 [hep-th/0501196] [INSPIRE].
E. D’Hoker and D.H. Phong, Two-loop superstrings VI. Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].
E. D’Hoker and D.H. Phong, Two-Loop Superstrings. VII. Cohomology of Chiral Amplitudes, Nucl. Phys. B 804 (2008) 421 [arXiv:0711.4314] [INSPIRE].
Y. Manin, Topics in Non-Commutative Geometry, Princeton University Press (1991) [https://doi.org/10.1515/9781400862511].
E. Witten, Notes On Super Riemann Surfaces And Their Moduli, Pure Appl. Math. Quart. 15 (2019) 57 [arXiv:1209.2459] [INSPIRE].
D. Friedan, S.H. Shenker and E.J. Martinec, Covariant Quantization of Superstrings, Phys. Lett. B 160 (1985) 55 [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys. B 271 (1986) 93 [INSPIRE].
V.G. Knizhnik, Covariant Superstring Fermion Amplitudes From the Sum Over Fermionic Surfaces, Phys. Lett. B 178 (1986) 21 [INSPIRE].
J.H. Schwarz, Superstring Theory, Phys. Rept. 89 (1982) 223 [INSPIRE].
M.D. Freeman and P.C. West, Ramond String Scattering in the Group Theoretic Approach to String Theory, Phys. Lett. B 217 (1989) 259 [INSPIRE].
F. Knudsen and D. Mumford, The Projectivity of the Moduli Space of Stable Curves. I: Preliminaries on ‘Det’ and ‘Div.’ , Math. Scand. 39 (1976) 19.
P. Deligne, Le déterminant de la cohomologie, in Current Trends in Arithmetical Algebraic Geometry, American Mathematical Society (1987), p. 93 [https://doi.org/10.1090/conm/067/902592].
P. Deligne, Letter to Y.I. Manin, (1988).
C. Lebrun and M. Rothstein, Moduli of Superriemann Surfaces, Commun. Math. Phys. 117 (1988) 159 [INSPIRE].
K.A. Maxwell, The super Mumford form and Sato Grassmannian, J. Geom. Phys. 180 (2022) 104604 [arXiv:2002.06625] [INSPIRE].
A.A. Beilinson, Y.I. Manin and V.V. Schechtman, Sheaves of the Virasoro and Neveu-Schwarz algebras, in K-Theory, Arithmetic and Geometry, Y.I. Manin eds., Springer Berlin Heidelberg (1987), p. 52–66 [https://doi.org/10.1007/bfb0078367].
Y.I. Manin, Neveu-Schwarz sheaves and differential equations for Mumford superforms, J. Geom. Phys. 5 (1988) 161 [INSPIRE].
A.A. Voronov, A.A. Roslyi and A.S. Schwarz, Geometry of Superconformal Manifolds, Commun. Math. Phys. 119 (1988) 129 [INSPIRE].
R. Hartshorne, Algebraic Geometry, Springer New York (1977) [https://doi.org/10.1007/978-1-4757-3849-0].
K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer Berlin Heidelberg (2005) [https://doi.org/10.1007/b138372].
M. Manetti, Lectures on deformations of complex manifolds (deformations from differential graded viewpoint), Rend. Mat. Appl. 7 (2004) 1.
E. Witten, Notes On Holomorphic String And Superstring Theory Measures Of Low Genus, Contemp. Math. 644 307 [arXiv:1306.3621] [INSPIRE].
Acknowledgments
The second author is grateful to Luis Álvarez-Gaumé for sharing his knowledge of the subject and for literature references. The third author thanks Mikhail Kapranov and Katherine Maxwell for useful discussions and patience during the work on parallel projects. The first and third authors would like to thank the Simons Center for Geometry and Physics for hospitality in Spring 2023 during the “Supergeometry and supermoduli” program, where all the authors started their attempt to reach a complete mathematical understanding of genus zero superstring scattering amplitudes.
Research of the second author is supported in part by NSF grant DMS-21-01631. Research of the third author is supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and Collaboration grant #585720 from the Simons Foundation.
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Cacciatori, S.L., Grushevsky, S. & Voronov, A.A. Tree-level superstring amplitudes: the Neveu-Schwarz sector. J. High Energ. Phys. 2024, 8 (2024). https://doi.org/10.1007/JHEP09(2024)008
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DOI: https://doi.org/10.1007/JHEP09(2024)008