Abstract
The quantum Batalian-Vilkovisky master action for closed string field theory consists of kinetic term and infinite number of interaction terms. The interaction strengths (coupling constants) are given by integrating the off-shell string measure over the distinct string diagrams describing the elementary interactions of the closed strings. In the first paper of this series, it was shown that the string diagrams describing the elementary interactions can be characterized using the Riemann surfaces endowed with the hyperbolic metric with constant curvature −1. In this paper, we construct the off-shell bosonic-string measure as a function of the Fenchel-Nielsen coordinates of the Teichmüller space of hyperbolic Riemann surfaces. We also describe an explicit procedure for integrating the off-shell string measure over the region inside the moduli space corresponding to the elementary interactions of the closed strings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Noncommutative Geometry and String Field Theory, Nucl. Phys.B 268 (1986) 253 [INSPIRE].
C.B. Thorn, String Field Theory, Phys. Rept.175 (1989) 1 [INSPIRE].
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys.B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
K. Strebel, Quadratic Differentials, Springer, Berlin Heidelberg, Germany (1984).
S.F. Moosavian and R. Pius, Hyperbolic Geometry of Superstring Perturbation Theory, arXiv:1703.10563 [INSPIRE].
S.F. Moosavian and R. Pius, Hyperbolic Geometry and Closed Bosonic String Field Theory I: The String Vertices Via Hyperbolic Riemann Surfaces, arXiv:1706.07366 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Gauge Algebra and Quantization, Phys. Lett.102B (1981) 27 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev.D 28 (1983) 2567 [Erratum ibid.D 30 (1984) 508] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys.174 (1995) 57 [hep-th/9405109] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. II. Application to Yang-Mills theory, Commun. Math. Phys.174 (1995) 93 [hep-th/9405194] [INSPIRE].
M. Henneaux, Lectures on the Antifield-BRST Formalism for Gauge Theories, Nucl. Phys. Proc. Suppl.A 18 (1990) 47.
M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, U.S.A. (1992).
J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept.259 (1995) 1 [hep-th/9412228] [INSPIRE].
J.M.L. Fisch and M. Henneaux, Homological Perturbation Theory and the Algebraic Structure of the Antifield-Antibracket Formalism for Gauge Theories, Commun. Math. Phys.128 (1990) 627 [INSPIRE].
M. Schnabl, Analytic solution for tachyon condensation in open string field theory, Adv. Theor. Math. Phys.10 (2006) 433 [hep-th/0511286] [INSPIRE].
H. Yang and B. Zwiebach, A closed string tachyon vacuum?, JHEP09 (2005) 054 [hep-th/0506077] [INSPIRE].
N. Moeller, Closed bosonic string field theory at quartic order, JHEP11 (2004) 018 [hep-th/0408067] [INSPIRE].
T. Erler, S. Konopka and I. Sachs, One Loop Tadpole in Heterotic String Field Theory, JHEP11 (2017) 056 [arXiv:1704.01210] [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys.B 271 (1986) 93 [INSPIRE].
R. Pius, A. Rudra and A. Sen, Mass Renormalization in String Theory: Special States, JHEP07 (2014) 058 [arXiv:1311.1257] [INSPIRE].
R. Pius, A. Rudra and A. Sen, Mass Renormalization in String Theory: General States, JHEP07 (2014) 062 [arXiv:1401.7014] [INSPIRE].
A. Sen, Off-shell Amplitudes in Superstring Theory, Fortsch. Phys.63 (2015) 149 [arXiv:1408.0571] [INSPIRE].
R. Pius, A. Rudra and A. Sen, String Perturbation Theory Around Dynamically Shifted Vacuum, JHEP10 (2014) 70 [arXiv:1404.6254] [INSPIRE].
A. Sen, Supersymmetry Restoration in Superstring Perturbation Theory, JHEP12 (2015) 075 [arXiv:1508.02481] [INSPIRE].
A. Sen, BV Master Action for Heterotic and Type II String Field Theories, JHEP02 (2016) 087 [arXiv:1508.05387] [INSPIRE].
R. Pius and A. Sen, Cutkosky rules for superstring field theory, JHEP10 (2016) 024 [Erratum ibid.09 (2018) 122] [arXiv:1604.01783] [INSPIRE].
R. Pius and A. Sen, Unitarity of the Box Diagram, JHEP11 (2018) 094 [arXiv:1805.00984] [INSPIRE].
A. Sen, Equivalence of Two Contour Prescriptions in Superstring Perturbation Theory, JHEP04 (2017) 025 [arXiv:1610.00443] [INSPIRE].
A. Sen, Reality of Superstring Field Theory Action, JHEP11 (2016) 014 [arXiv:1606.03455] [INSPIRE].
A. Sen, Unitarity of Superstring Field Theory, JHEP12 (2016) 115 [arXiv:1607.08244] [INSPIRE].
A. Sen, Gauge Invariant 1PI Effective Action for Superstring Field Theory, JHEP06 (2015) 022 [arXiv:1411.7478] [INSPIRE].
A. Sen, Gauge Invariant 1PI Effective Superstring Field Theory: Inclusion of the Ramond Sector, JHEP08 (2015) 025 [arXiv:1501.00988] [INSPIRE].
A. Sen, Wilsonian Effective Action of Superstring Theory, JHEP01 (2017) 108 [arXiv:1609.00459] [INSPIRE].
C. de Lacroix, H. Erbin, S.P. Kashyap, A. Sen and M. Verma, Closed Superstring Field Theory and its Applications, Int. J. Mod. Phys.A 32 (2017) 1730021 [arXiv:1703.06410] [INSPIRE].
H.M. Farkas and M.I. Kra, Riemann surfaces, Springer, New York, U.S.A. (1992).
A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology19 (1980) 221.
A. Hatcher, Pants decompositions of surfaces, math/9906084.
G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math.132 (1998) 607.
M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math.167 (2006) 179.
S.A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Diff. Geom.31 (1990) 417.
K. Obitsu and S.A. Wolpert, Grafting hyperbolic metrics and Eisenstein series, Math. Ann.341 (2008) 685.
R. Melrose and X. Zhu, Boundary Behaviour of Weil-Petersson and Fiber Metrics for Riemann Moduli Spaces, arXiv:1606.01158.
S.A. Wolpert, Families of Riemann surfaces and Weil-Petersson geometry, No. 113, American Mathematical Society, (2010).
S.P. Kerckhoff, The Nielsen realization problem, Annals Math.117 (1983) 235.
S.A. Wolpert, The Fenchel-Nielsen deformation, Annals Math.115 (1982) 501.
S.A. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Annals Math.117 (1983) 207.
S.A. Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Am. J. MAth.107 (1985) 969.
W. Fenchel and J. Nielsen, Discontinuous groups of non-Euclidean motions, Walter De Gruyter Inc (2002).
Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller spaces, Springer Science & Business Media (2012).
W. Abikoff, The Uniformization theorem, Am. Math. Mon.88 (1981) 574.
F. Benson and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press (2011).
F. Luo and S.P. Tan, A dilogarithm identity on moduli spaces of curves, J. Diff. Geom.97 (2014) 255.
H. Hengnan and S.P. Tan, New identities for small hyperbolic surfaces, Bull. London Math. Soc.46 (2014) 1021.
R.C. Penner, Decorated Teichmüller theory, European Mathematical Society (2012).
R.C. Penner, Weil-Petersson volumes, J. Diff. Geom.35 (1992) 559.
S.A. Wolpert, Lectures and notes: Mirzakhani’s volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers, arXiv:1108.0174 [INSPIRE].
S.A. Wolpert, On the Kähler form of the moduli space of once punctured tori, Comment. Math. Helv.58 (1983) 246.
R. Fricke and F. Klein, Vorlesungen über die Theorie der automorphen Functionen, Volume 1, BG Teubner (1897).
L. Keen, On fundamental domains and the Teichmüller modular group, Contributions to Analysis, Academic Press, (1974) pp. 185-194.
P. Sarnak, Determinants of Laplacians, Commun. Math. Phys.110 (1987) 113.
E. D’Hoker and D.H. Phong, On Determinants of Laplacians on Riemann Surfaces, Commun. Math. Phys.104 (1986) 537.
J. Bolte and F. Steiner, Determinants of Laplace-like operators on Riemann surfaces, Commun. Math. Phys.130 (1990) 581.
D.A. Hejhal, The Selberg trace formula for PSL(2, R), Volume 2, Springer, (2006).
E. D’Hoker and D.H. Phong, Multiloop Amplitudes for the Bosonic Polyakov String, Nucl. Phys.B 269 (1986) 205 [INSPIRE].
L.P. Teo, Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points, arXiv:1901.07898. [arXiv:1901.07898].
B. Maskit, Matrices for Fenchel-Nielsen coordinates, RECON no. 20010088230, Annales Academiae Scientiarum Fennicae: Mathmatica26 (2001) 267.
D. Stanford and E. Witten, JT Gravity and the Ensembles of Random Matrix Theory, arXiv:1907.03363 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1708.04977
We dedicate this paper to the memory of Maryam Mirzakhani who tragically passed away recently, and whose seminal ideas about the space of hyperbolic Riemann surfaces form some of the basic tools that we use in this work.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Moosavian, S.F., Pius, R. Hyperbolic geometry and closed bosonic string field theory. Part II. The rules for evaluating the quantum BV master action. J. High Energ. Phys. 2019, 177 (2019). https://doi.org/10.1007/JHEP08(2019)177
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2019)177