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References

Published online by Cambridge University Press:  28 November 2024

Eric D'Hoker  and
Justin Kaidi
Eric D'Hoker
Affiliation:
University of California, Los Angeles
Justin Kaidi
Affiliation:
Kyushu University
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Modular Forms and String Theory , pp. 455 - 476
Publisher: Cambridge University Press
Print publication year: 2024

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References

Mohammad, Akhond, Guillermo, Arias-Tamargo, Alessandro, Mininno, Hao-Yu, Sun, Zhengdi, Sun, Yifan, Wang, and Fengjun, Xu, The hitchhiker’s guide to 4d = 2 superconformal field theories, SciPost Phys. Lect. Notes 64 (2022), 1.Google Scholar
Alday, Luis F., Agnese, Bissi, and Eric, Perlmutter, Genus-one string amplitudes from conformal field theory, J. High Energy Phys. 06 (2019), 010.Google Scholar
Vassilis, Anagiannis and Miranda, C. N. Cheng, TASI lectures on Moonshine, PoS TASI2017 (2018), 010.CrossRefGoogle Scholar
Enrico, Arbarello, Maurizio, Cornalba, Griffiths, Phillip A., and Joseph, Harris, Geometry of Algebraic Curves, Volume I, Grundlehren der mathematischen Wissenschaften, vol. 267, Springer-Verlag, 1985.Google Scholar
Thales, Azevedo, Marco, Chiodaroli, Henrik, Johansson, and Oliver, Schlotterer, Heterotic and bosonic string amplitudes via field theory, J. High Energy Phys. 10 (2018), 012.Google Scholar
Argyres, Philip C. and Douglas, Michael R., New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995), 93–126.CrossRefGoogle Scholar
Martin, Ammon and Johanna, Erdmenger, Gauge/Gravity Duality, Foundations and Applications, Cambridge University Press, 2015.Google Scholar
Aspman, Johannes, Elias, Furrer, and Jan, Manschot, Elliptic loci of SU(3) vacua, Ann. Henri Poincar´e 22 (2021), no. 8, 2775–2830.Google Scholar
Aspman, Johannes, Elias, Furrer, and Jan, Manschot, Cutting and gluing with running couplings in N=2 QCD, Phys. Rev. D 105 (2022), no. 2, 025021.Google Scholar
Aspman, Johannes, Elias, Furrer, and Jan, Manschot, Four flavors, triality, and bimodular forms, Phys. Rev. D 105 (2022), no. 2, 025017.Google Scholar
Angelantonj, Carlo, Ioannis, Florakis, and Boris, Pioline, A new look at one-loop integrals in string theory, Commun. Number Theory Phys. 6 (2012), 159–201.CrossRefGoogle Scholar
Angelantonj, Carlo, Ioannis, Florakis, and Boris, Pioline, One-loop BPS amplitudes as BPS-state sums, J. High Energy Phys. 6 (2012), 070.Google Scholar
Luis, Alvarez-Gaume, Jean-Benoît, Bost, Moore, Gregory W., Nelson, Philip C., and Cumrun, Vafa, Bosonization on higher genus riemann surfaces, Commun. Math. Phys. 112 (1987), 503.Google Scholar
Luis, Alvarez-Gaume, Daniel, Z. Freedman, and Sunil, Mukhi, The background field method and the ultraviolet structure of the supersymmetric nonlinear sigma model, Ann. Phys. 134 (1981), 85.Google Scholar
Luis, Alvarez-Gaume and Hassan, S. F., Introduction to S duality in N=2 supersymmetric gauge theories: A Pedagogical review of the work of Seiberg and Witten, Fortsch. Phys. 45 (1997), 159–236.Google Scholar
Ofer, Aharony, Steven, S. Gubser, Juan Martin Maldacena, Hirosi, Ooguri, and Yaron, Oz, Large N field theories, string theory and gravity, Phys. Rep. 323 (2000), 183–386.Google Scholar
Luis, Alvarez-Gaume, Gregory, W. Moore, and Cumrun, Vafa, Theta functions, modular invariance and strings, Commun. Math. Phys. 106 (1986), 1–40.Google Scholar
Luis, Alvarez-Gaume and Edward, Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984), 269.Google Scholar
Nima, Arkani-Hamed, Jacob, Bourjaily, Freddy, Cachazo, Alexander, Goncharov, Alexander Postnikov, and Jaroslav Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, 2016.Google Scholar
Nima, Arkani-Hamed, Lorenz, Eberhardt, Yu-tin, Huang, and Sebastian, Mizera, On unitarity of tree-level string amplitudes, J. High Energy Phys. 02 (2022), 197.Google Scholar
Olof, Ahl´én and Axel, Kleinschmidt, D6R4 curvature corrections, modular graph functions and Poincar´e series, J. High Energy Phys. 05 (2018), 194.Google Scholar
Albert, Jan, Justin, Kaidi, and Ying-Hsuan, Lin, Topological modularity of supermoonshine, Prog. Theor. Exp. Phys. 2023 (2023), no. 3, 033B06.CrossRefGoogle Scholar
Alvarez, Orlando, Theory of strings with boundaries: fluctuations, topology, and quantum geometry, Nucl. Phys. B 216 (1983), 125.CrossRefGoogle Scholar
Emil, Artin and Arthur, Norton Milgram, Galois Theory, vol. 2, Courier Corporation, 1998.Google Scholar
Apostol, Tom M., Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer Verlag, 1976.Google Scholar
Apostol, Tom M., Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, vol. 41, Springer Verlag, 1976.Google Scholar
Henry, Peter Francis Swinnerton-Dyer and Arthur, Oliver Lonsdale Atkin, Modular forms on non-congruence subgroups, Combinatorics 19 (1971), 1–25.CrossRefGoogle Scholar
Aharony, Ofer, Nathan, Seiberg, and Yuji, Tachikawa, Reading between the lines of four-dimensional gauge theories, J. High Energy Phys. 08 (2013), 115.Google Scholar
Ofer, Aharony and Yuji, Tachikawa, S-folds and 4d N=3 superconformal field theories, J. High Energy Phys. 06 (2016), 044.Google Scholar
Bantay, Peter, The Kernel of the modular representation and the Galois action in RCFT, Commun. Math. Phys. 233 (2003), 423–438.CrossRefGoogle Scholar
Basu, Anirban, Poisson equation for the Mercedes diagram in string theory at genus one, Class. Quant. Grav. 33 (2016), no. 5, 055005.CrossRefGoogle Scholar
Basu, Anirban, Poisson equation for the three loop ladder diagram in string theory at genus one, Int. J. Mod. Phys. A 31 (2016), no. 32, 1650169.Google Scholar
Basu, Anirban, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016), no. 23, 235011.CrossRefGoogle Scholar
Basu, Anirban, Eigenvalue equation for the modular graph Ca,b,c,d, J. High Energy Phys. 07 (2019), 126.Google Scholar
Becker, Katrin, Becker, Melanie, and Schwarz, John H., String Theory and M-theory: A Modern Introduction, Cambridge University Press, 2006.CrossRefGoogle Scholar
Zvi, Bern, John, Joseph Carrasco, Chiodaroli, Marco, Henrik, Johansson, and Radu, Roiban, The SAGEX review on scattering amplitudes Chapter 2: An invitation to color-kinematics duality and the double copy, J. Phys. A 55 (2022), no. 44, 443003.Google Scholar
Benjamin, Nathan, Scott, Collier, Fitzpatrick, A. Liam, Alexander, Maloney, and Eric, Perlmutter, Harmonic analysis of 2D CFT partition functions, J. High Energy Phys. 09 (2021), 174.Google Scholar
Bouchard, Vincent, Thomas, Creutzig, and Aniket, Joshi, Hecke operators on vector-valued modular forms, SIGMA 15 (2019), 041.Google Scholar
Eric, Braaten, Thomas, L. Curtright, and Cosmas, K. Zachos, Torsion and geometrostasis in nonlinear sigma models, Nucl. Phys. B 260 (1985), 630 [Erratum: Nucl. Phys. B 266, 748–748 (1986)].Google Scholar
Johannes, Broedel, Claude, Duhr, Falko, Dulat, and Lorenzo, Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, J. High Energy Phys. 05 (2018), 093.Google Scholar
Nathan, Berkovits, Eric, D’Hoker, Green, Michael B., Henrik, Johansson, and Oliver, Schlotterer, Snowmass white paper: String perturbation theory, 2022 Snowmass Summer Study, arXiv:2203.09099 (2022).Google Scholar
Jin-Beom Bae, Zhihao Duan, Lee, Kimyeong, Sungjay, Lee, and Matthieu, Sarkis, Fermionic rational conformal field theories and modular linear differential equations, Prog. Theor. Exp. Phys. 2021 (2021), no. 8, 08B104.Google Scholar
Jin-Beom Bae, Zhihao Duan, Lee, Kimyeong, Sungjay, Lee, and Matthieu, Sarkis, Bootstrapping fermionic rational CFTs with three characters, J. High Energy Phys. 01 (2022), 089.Google Scholar
Berera, Arjun, Unitary string amplitudes, Nucl. Phys. B 411 (1994), 157–180.CrossRefGoogle Scholar
Berkovits, Nathan, Super Poincare covariant quantization of the superstring, J. High Energy Phys. 04 (2000), 018.Google Scholar
Berkovits, Nathan, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, J. High Energy Phys. 09 (2004), 047.Google Scholar
Berkovits, Nathan, Super-Poincare covariant two-loop superstring amplitudes, J. High Energy Phys. 01 (2006), 005.Google Scholar
Berndt, Bruce C., Ramanujan’s Notebooks: Part III, Springer Science & Business Media, 2012.Google Scholar
Jan, Hendrik Bruinier and Jens, Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45–90.Google Scholar
Borisov, Lev A. and Gunnels, Paul E., Toric modular forms of higher weight, J. Reine Angew. Math. 560 (2003), 43–64.Google Scholar
Peter, Bantay and Terry, Gannon, Vector-valued modular functions for the modular group and the hypergeometric equation, arXiv:0705.2467 (2007).CrossRefGoogle Scholar
Brown, J. David and Marc, Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104 (1986), 207–226.CrossRefGoogle Scholar
Jean-Benoît, Bost and Thierry, Jolicoeur, A holomorphy property and critical dimension in string theory from an index theorem, Phys. Lett. B 174 (1986), 273–276.Google Scholar
Belavin, Alexander A. and Knizhnik, Vadim G., Algebraic geometry and the geometry of quantum strings, Phys. Lett. B 168 (1986), 201–206.CrossRefGoogle Scholar
Belavin, Alexander A., Knizhnik, Vadim, Alexei, Morozov, and Askold, Perelomov, Two and three loop amplitudes in the bosonic string theory, JETP Lett. 43 (1986), 411.Google Scholar
Nathan, Benjamin, Shamit, Kachru, Ken Ono, and Larry Rolen, Black holes and class groups, arXiv:1807.00797 (2018).CrossRefGoogle Scholar
Bossard, Guillaume, Axel, Kleinschmidt, and Boris, Pioline, 1/8-BPS couplings and exceptional automorphic functions, SciPost Phys. 8 (2020), no. 4, 054.CrossRefGoogle Scholar
Francis, Brown and Andrey, Levin, Multiple elliptic polylogarithms, arXiv:1110.6917 (2011).Google Scholar
Christopher, Beem, Madalena, Lemos, Pedro, Liendo, Wolfger, Peelaers, Rastelli, Leonardo, and Balt, C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015), no. 3, 1359–1433.Google Scholar
Blumenhagen, Ralph, Dieter Lu¨st, and Stefan Theisen, Basic Concepts of String Theory, Theoretical and Mathematical Physics, Springer, Heidelberg, Germany, 2013.Google Scholar
Nathan, Berkovits and Carlos, R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006), 011602.Google Scholar
Bost, Jean-Benoît, Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties, From Number Theory to Physics, Springer, 1992, pp. 64–211.Google Scholar
Belavin, A. A., Polyakov, Alexander M., and Zamolodchikov, A. B., Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984), 333–380.CrossRefGoogle Scholar
Christopher, Beem and Leonardo, Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, J. High Energy Phys. 08 (2018), 114.Google Scholar
Matthew, Buican and Rajath, Radhakrishnan, Galois conjugation and multiboundary entanglement entropy, J. High Energy Phys. 12 (2020), 045.Google Scholar
Matthew, Buican and Rajath, Radhakrishnan, Galois orbits of TQFTs: Symmetries and unitarity, J. High Energy Phys. 01 (2022), 004.Google Scholar
Brown, Francis, A class of non-holomorphic modular forms I, arXiv:1707.01230 (2017).CrossRefGoogle Scholar
Brown, Francis, A class of non-holomorphic modular forms III: Real analytic cusp forms for SL2(ℤ), arXiv preprint arXiv:1710.07912 (2017).Google Scholar
Brown, Francis, A class of nonholomorphic modular forms II: Equivariant iterated, Forum Math. Sigma 8 (2020), e31.CrossRefGoogle Scholar
Broedel, Johannes, Oliver, Schlotterer, and Stephan, Stieberger, Polylogarithms, multiple zeta values and superstring amplitudes, Fortsch. Phys. 61 (2013), 812–870.Google Scholar
Broedel, Johannes, Oliver, Schlotterer, and Federico, Zerbini, From elliptic multiple zeta values to modular graph functions: Open and closed strings at one loop, J. High Energy Phys. 01 (2019), 155.Google Scholar
Bump, Daniel, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, 1998.Google Scholar
Jan, Hendrik Bruinier, Gerard, Van der Geer, Gu¨nter, Harder, and Don, Zagier, The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway, Springer Science & Business Media, 2008.Google Scholar
Cartier, Pierre, Introduction to Zeta Functions, From Number Theory to Physics, Springer, 1992, pp. 1–63.Google Scholar
Cheng, Miranda C. N., Coman, Ioana, Davide, Passaro, and Gabriele, Sgroi, Quantum modular ZˆG-invariants, arXiv:2304.03934 (2023).Google Scholar
Cacciatori, Sergio L., Francesco, Dalla Piazza, and Bert, van Geemen, Genus four superstring measures, Lett. Math. Phys. 85 (2008), 185.CrossRefGoogle Scholar
Cacciatori, Sergio L., Francesco, Dalla Piazza, and Bert, van Geemen, Modular forms and three loop superstring amplitudes, Nucl. Phys. B 800 (2008), 565–590.CrossRefGoogle Scholar
Calegari, Frank, Vesselin Dimitrov, and Yunqing Tang, The un-bounded denominators conjecture, arXiv:2109.09040 (2021).Google Scholar
Henri, Cohen and Gerhard, Frey, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, 2006.Google Scholar
Coste, A. and Gannon, T., Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. B 323 (1994), 316–321.CrossRefGoogle Scholar
Antoine, Coste and Terry, Gannon, Congruence subgroups and rational conformal field theory, arXiv:math/9909080 (1999).Google Scholar
Chester, Shai M., Green, Michael B., Pufu, Silviu S., Yifan, Wang, and Congkao, Wen, New modular invariants in N = 4 Super-Yang-Mills theory, J. High Energy Phys. 04 (2021), 212.Google Scholar
Mirjam, Cvetic, James, Halverson, Gary Shiu, and Washington Taylor, Snowmass white paper: String theory and particle physics, arXiv:2204.01742 (2022).Google Scholar
Henry, Cohn, Abhinav, Kumar, Miller, Stephen, Danylo, Radchenko, and Maryna, Viazovska, The sphere packing problem in dimension 24, Ann. Math. 185 (2017), no. 3, 1017–1033.Google Scholar
Chandra, A. Ramesh and Sunil, Mukhi, Towards a classification of two-character rational conformal field theories, J. High Energy Phys. 04 (2019), 153.Google Scholar
Cox, David A., Galois Theory, vol. 61, John Wiley & Sons, 2011.Google Scholar
Cox, David A., Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication, vol. 34, John Wiley & Sons, 2011.Google Scholar
Scott, Collier and Eric, Perlmutter, Harnessing S-duality in N = 4 SYM & supergravity as SL(2, ℤ)-averaged strings, arXiv:2201.05093 (2022).CrossRefGoogle Scholar
Sarvadaman, Chowla and Atle, Selberg, On Epstein’s zeta function (I), Proc. Natl. Acad. Sci. USA 35 (1949), no. 7, 371–374.Google Scholar
Changha, Choi and Leon, A. Takhtajan, Supersymmetry and trace formulas I. Compact Lie groups, arXiv:2112.07942 (2021).Google Scholar
Changha, Choi and Leon, A, Supersymmetry and trace formulas II. Selberg trace formula, arXiv:2306.13636 (2023).Google Scholar
John, Coates and Shing-Tung, Yau (eds.), Elliptic Curves, Modular Forms and Fermat’s Last Theorem, International Press, 1997.Google Scholar
Jan, de Boer, Miranda, C. N. Cheng, Robbert, Dijkgraaf, Jan, Manschot, and Erik, Verlinde, A farey tail for attractor black holes, J. High Energy Phys. 11 (2006), 024.Google Scholar
Jan, De Boer and Jacob, Goeree, Markov traces and II1 factors in conformal field theory, Commun. Math. Phys. 139 (1991), 267–304.Google Scholar
Eric, D’Hoker and William, Duke, Fourier series of modular graph functions, J. Number Theory 192 (2018), 1–36.Google Scholar
Daniele, Dorigoni, Mehregan, Doroudiani, Joshua, Drewitt, Martijn, Hidding, Axel, Kleinschmidt, Nils, Matthes, Oliver, Schlotterer, and Bram, Verbeek, Modular graph forms from equivariant iterated Eisenstein integrals, J. High Energy Phys. 12 (2022), 162.Google Scholar
Arun, Debray, Markus, Dierigl, Heckman, Jonathan J., and Miguel Montero, The anomaly that was not meant IIB, arXiv:2107.14227 (2021).CrossRefGoogle Scholar
Developers, Sage, Sagemath, the Sage Mathematics Software System, www.sagemath.org.Google Scholar
Eric, D’Hoker and Daniel, Z. Freedman, Supersymmetric gauge theories and the AdS / CFT correspondence, Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2001): Strings, Branes and EXTRA Dimensions, 2002, pp. 3–158.Google Scholar
Philippe, Di Francesco, Pierre, Mathieu, and David S´en´echal, Conformal Field Theory, Springer, 1997.Google Scholar
Philippe, Di Francesco, Hubert, Saleur, and Jean-Bernard, Zuber, Modular invariance in nonminimal two-dimensional conformal theories, Nucl. Phys. B 285 (1987), 454.Google Scholar
Philippe, Di Francesco, Hubert, Saleur, and Jean-Bernard, Zuber, Generalized coulomb gas formalism for two-dimensional critical models based on SU(2) Coset Construction, Nucl. Phys. B 300 (1988), 393–432.Google Scholar
Eric, D’Hoker and Michael, B. Green, Zhang–Kawazumi invariants and superstring amplitudes, J. Number Theory 144 (2014), 111–150.Google Scholar
Eric, D’Hoker and Michael, B. Green, Identities between modular graph forms, J. Number Theory 189 (2018), 25–80.Google Scholar
Eric, D’Hoker and Michael, B. Green, Exploring transcendentality in superstring amplitudes, J. High Energy Phys. 07 (2019), 149.Google Scholar
Eric, D’Hoker and Nicholas, Geiser, Integrating three-loop modular graph functions and transcendentality of string amplitudes, J. High Energy Phys. 02 (2022), 019.Google Scholar
Eric, D’Hoker, Michael, B. Green, O¨mer, Gu¨rdogan, and Pierre, Vanhove, Modular graph functions, Commun. Number Theory Phys. 11 (2017), 165–218.Google Scholar
Eric, D’Hoker, Michael, Gutperle, and Phong, Duong H., Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005), 81–118.Google Scholar
Eric, D’Hoker, Michael, B. Green, and Boris, Pioline, Asymptotics of the D8R4 genus-two string invariant, Commun. Number Theory Phys. 13 (2019), 351–462.Google Scholar
Eric, D’Hoker, Michael, B. Green, and Boris, Pioline, Higher genus modular graph functions, string invariants, and their exact asymptotics, Commun. Math. Phys. 366 (2019), no. 3, 927–979.Google Scholar
Eric, D’Hoker, Michael, B. Green, Boris Pioline, and Rodolfo, Russo, Matching the D6R4 interaction at two-loops, J. High Energy Phys. 01 (2015), 031.Google Scholar
Eric, D’Hoker, Michael, B. Green, and Pierre, Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion, J. High Energy Phys. 08 (2015), 041.Google Scholar
Daniele, Dorigoni, Michael, B. Green, and Congkao, Wen, Exact expressions for n-point maximal U (1)Y-violating integrated correlators in SU (N ) = 4 SYM, J. High Energy Phys. 11 (2021), 132.Google Scholar
Daniele, Dorigoni, Michael, B. Green, and Congkao, Wen, Exact properties of an integrated correlator in N = 4 SU(N) SYM, J. High Energy Phys. 05 (2021), 089.Google Scholar
Daniele, Dorigoni, Michael, B. Green, and Congkao, Wen, The SAGEX review on scattering amplitudes. Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their supersymmetric Yang–Mills duals, J. Phys. A 55 (2022), no. 44, 443011.Google Scholar
Daniele, Dorigoni, Michael, B. Green, Congkao Wen, and Haitian, Xie, Modular-invariant large-N completion of an integrated correlator in N = 4 supersymmetric Yang-Mills theory, J. High Energy Phys. 04 (2023), 114.Google Scholar
Eric, D’Hoker, String theory, Quantum Fields and Strings: A Course for Mathematicians (P. Deligne and et al., eds.), vol. 2, American Mathematical Society and Institute for Advanced Study, 1999, pp. 807–1012.Google Scholar
Dixon, Lance J., Harvey, Jeffrey A., Vafa, C., and Edward, Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985), 678–686.CrossRefGoogle Scholar
Dixon, Lance J., Harvey, Jeffrey A., Vafa, C., and Edward, Witten, Strings on orbifolds. 2., Nucl. Phys. B 274 (1986), 285–314.CrossRefGoogle Scholar
Eric, D’Hoker and Justin, Kaidi, Hierarchy of modular graph identities, J. High Energy Phys. 11 (2016), 051.Google Scholar
Eric, D’Hoker and Justin, Kaidi, Modular graph functions and odd cuspidal functions. Fourier and Poincar´e series, J. High Energy Phys. 04 (2019), 136.Google Scholar
Daniele, Dorigoni and Axel, Kleinschmidt, Modular graph functions and asymptotic expansions of Poincar´e series, Commun. Number Theory Phys. 13 (2019), no. 3, 569–617.Google Scholar
Eric, D’Hoker, Axel, Kleinschmidt, and Oliver, Schlotterer, Elliptic modular graph forms. Part I. Identities and generating series, J. High Energy Phys. 03 (2021), 151.Google Scholar
Frederik, Denef, Shamit, Kachru, Zimo Sun, and Arnav Tripathy, Higher genus Siegel forms and multi-center black holes in N=4 supersymmetric string theory, arXiv:1712.01985 (2017).Google Scholar
Dong, Chongying, Xingjun, Lin, and Siu-Hung, Ng, Congruence property in conformal field theory, Algebra Number Theory 9 (2015), no. 9, 2121–2166.CrossRefGoogle Scholar
Douglas, Michael R. and Moore, Gregory W., D-branes, quivers, and ALE instantons, arXiv:9603167 (1996).Google Scholar
Robbert, Dijkgraaf, Juan, Martin Maldacena, Moore, Gregory W., and Verlinde, Erik P., A Black hole Farey tail, arXiv:hep-th/0005003 (2000).Google Scholar
Eric, D’Hoker, Carlos, R. Mafra, Boris Pioline, and Oliver, Schlotterer, Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors, J. High Energy Phys. 08 (2020), 135.Google Scholar
Eric, D’Hoker, Carlos, R. Mafra, Boris Pioline, and Oliver, Schlotterer, Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality, J. High Energy Phys. 02 (2021), 139.Google Scholar
Robbert, Dijkgraaf, Gregory, W. Moore, Erik P. Verlinde, and Herman, L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997), 197–209.Google Scholar
Dabholkar, Atish, Sameer Murthy, and Don Zagier, Quantum black holes, wall crossing, and mock modular forms, arXiv:1208.4074 (2012).Google Scholar
Eric, D’Hoker and Duong, H. Phong, Multiloop Amplitudes for the Bosonic Polyakov String, Nucl. Phys. B 269 (1986), 205–234.Google Scholar
Eric, D’Hoker and Duong, H. Phong, On determinants of laplacians on riemann surfaces, Commun. Math. Phys. 104 (1986), 537.Google Scholar
Eric, D’Hoker and Duong, H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988), no. 4, 917.Google Scholar
Eric, D’Hoker and Duong, H. Phong, Momentum analyticity and finiteness of the one loop superstring amplitude, Phys. Rev. Lett. 70 (1993), 3692–3695.Google Scholar
Eric, D’Hoker and Duong, H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995), 24–94.Google Scholar
Eric, D’Hoker and Duong, H. Phong, Calogero-Moser Lax pairs with spectral parameter for general Lie algebras, Nucl. Phys. B 530 (1998), 537–610.Google Scholar
Eric, D’Hoker and Duong, H. Phong, Calogero-Moser systems in SU(N) Seiberg-Witten theory, Nucl. Phys. B 513 (1998), 405–444.Google Scholar
Eric, D’Hoker and Duong, H. Phong, Spectral curves for superYang-Mills with adjoint hypermultiplet for general Lie algebras, Nucl. Phys. B 534 (1998), 697–719.Google Scholar
Eric, D’Hoker and Duong, H. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, 9th CRM Summer School: Theoretical Physics at the End of the 20th Century, 1999, pp. 1–125.CrossRefGoogle Scholar
Eric, D’Hoker and Duong, H. Phong, Two loop superstrings. 1. Main formulas, Phys. Lett. B 529 (2002), 241–255.CrossRefGoogle Scholar
Eric, D’Hoker and Duong, H. Phong, Two loop superstrings 4: The Cosmological constant and modular forms, Nucl. Phys. B 639 (2002), 129–181.Google Scholar
D’Hoker, Eric and Phong, D. H., Lectures on two-loop superstrings, Superstring Theory (Yau, S.-T., Liu, K. and Zhu, C., eds.), Advanced Lectures in Mathematics, Vol 1, International Press, 2002, pp. 85–123.Google Scholar
D’Hoker, Eric and Phong, D. H., Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005), 3–90.Google Scholar
Pierre, Deligne and Michael, Rapoport, Les sch´emas de modules de courbes elliptiques, Lecture Notes in Mathematics, vol. 349, Springer-Verlag, 1973.Google Scholar
Fred, Diamond and Jerry, Michael Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228, Springer, 2005.Google Scholar
Eric, D’Hoker and Oliver, Schlotterer, Two-loop superstring five-point amplitudes. Part III. Construction via the RNS formulation: Even spin structures, J. High Energy Phys. 12 (2021), 063.Google Scholar
Eric, D’Hoker and Oliver, Schlotterer, Identities among higher genus modular graph tensors, Commun. Number Theory Phys. 16 (2022), no. 1, 35–74.Google Scholar
Anatol’evich Dubrovin, Boris, Theta functions and non-linear equations, Russ. Math. Surv. 36 (1981), no. 2, 11.CrossRefGoogle Scholar
Robbert, Dijkgraaf, Erik, P. Verlinde, and Herman, L. Verlinde, C = 1 conformal field theories on Riemann surfaces, Commun. Math. Phys. 115 (1988), 649–690.Google Scholar
Robbert, Dijkgraaf, Erik, P. Verlinde, and Herman, L. Verlinde, Counting dyons in N=4 string theory, Nucl. Phys. B 484 (1997), 543–561.CrossRefGoogle Scholar
Robbert, Dijkgraaf, Cumrun, Vafa, Verlinde, Erik P., and Verlinde, Herman L., The operator algebra of orbifold models, Commun. Math. Phys. 123 (1989), 485.Google Scholar
Ron, Donagi and Edward, Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996), 299–334.Google Scholar
Tohru, Eguchi and Kazuhiro, Hikami, Superconformal algebras and mock theta functions 2. Rademacher expansion for K3 surface, Commun. Number Theory Phys. 3 (2009), 531–554.Google Scholar
Henrietta, Elvang and Yu-Tin, Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University press, 2015.Google Scholar
Lorenz, Eberhardt and Sebastian, Mizera, Evaluating one-loop string amplitudes, SciPost Phys. 15 (2023), no. 3, 119.Google Scholar
Enriquez, Benjamin, Analogues elliptiques des nombres multiz´etas, Bull. Soc. Math. France 144 (2016), no. 3, 395–427. MR 3558428.CrossRefGoogle Scholar
Eguchi, Tohru, Hirosi, Ooguri, and Yuji, Tachikawa, Notes on the K3 surface and the Mathieu group M24, Exp. Math. 20 (2011), 91–96.CrossRefGoogle Scholar
Tohru, Eguchi, Hirosi, Ooguri, Anne, Taormina, and Sung-Kil, Yang, Superconformal algebras and string compactification on manifolds with SU(N) Holonomy, Nucl. Phys. B 315 (1989), 193–221.Google Scholar
Erdelyi, A. (ed.), Higher Transcendental Functions, The Bateman Manuscript Project, vol. 2, Krieger Publishing Company, 1981.Google Scholar
Erdelyi, A. (ed.), Higher Transcendental Functions, The Bateman Manuscript Project, vol. 1, Krieger Publishing Company, 1981.Google Scholar
Erdelyi, A. (ed.), Higher Transcendental Functions, The Bateman Manuscript Project, vol. 3, Krieger Publishing Company, 1981.Google Scholar
Escofier, Jean-Pierre, Galois Theory, Graduate Texts in Mathematics, vol. 204, Springer, 2001.Google Scholar
Evtikhiev, Mikhail, N = 3 SCFTs in 4 dimensions and non-simply laced groups, J. High Energy Phys. 06 (2020), 125.Google Scholar
Martin, Eichler and Don, Zagier, The theory of jacobi forms, progress in Math. Vol 55 (1985), Birkh¨auser-Verlag, Progress in Math. 55 (1985).Google Scholar
Benjamin, Enriquez and Federico, Zerbini, Analogues of hyperlogarithm functions on affine complex curves, arXiv:2212.03119 (2023).Google Scholar
Fay, John D., Theta Functions on Riemann Surfaces, Graduate Texts in Mathematics, vol. 352, Springer-Verlag, 2006.Google Scholar
Philipp, Fleig, Henrik, P. A. Gustafsson, Axel Kleinschmidt, and Daniel Persson, Eisenstein Series and Automorphic Representations, Cambridge University Press, 2018.Google Scholar
Sergio, Fubini, Andrew, J. Hanson, and Roman, Jackiw, New approach to field theory, Phys. Rev. D 7 (1973), 1732–1760.Google Scholar
Hershel, M. Farkas and Irwin, Kra, Riemann Surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, 1980.Google Scholar
Cameron, Franc and Geoffrey, Mason, Fourier coefficients of vector-valued modular forms of dimension 2, Canadian Mathematical Bulletin 57 (2014), no. 3, 485–494.Google Scholar
Cameron, Franc and Geoffrey, Mason, Hypergeometric series, modular linear differential equations and vector-valued modular forms, The Ramanujan Journal 41 (2016), no. 1, 233–267.Google Scholar
Cameron, Franc and Geoffrey, Mason, Three-dimensional imprimitive representations of the modular group and their associated modular forms, J. Number Theory 160 (2016), 186–214.Google Scholar
Daniel, Friedan, Emil, J. Martinec, and Stephen, H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B 271 (1986), 93–165.Google Scholar
Ioannis, Florakis and Boris, Pioline, On the Rankin–Selberg method for higher genus string amplitudes, Commun. Number Theory Phys. 11 (2017), 337–404.Google Scholar
Daniel, Harry Friedan, Nonlinear models in Two + epsilon dimensions, Annals Phys. 163 (1985), 318.Google Scholar
David, Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (1986), no. 3, 523–540.Google Scholar
Daniel, Z. Freedman and Antoine, Van Proeyen, Supergravity, Cambridge University Press, Cambridge, UK, 2012.Google Scholar
Gaiotto, Davide, N=2 dualities, J. High Energy Phys. 08 (2012), 034.Google Scholar
Terry, Gannon, Monstrous moonshine: The First twenty five years, arXiv:math/0402345 (2004).Google Scholar
Terry, Gannon, Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, 2006.Google Scholar
Terry, Gannon, The theory of vector-modular forms for the modular group, arXiv preprint arXiv:1310.4458 (2013).CrossRefGoogle Scholar
Carl, Friedrich Gauss, Disquisitiones Arithmeticae, Lipsiae, 1801.Google Scholar
Krzysztof, Gawedzki, Conformal field theory and strings, Quantum Fields and Strings: A Course for Mathematicians (Deligne, P. et al., eds.), vol. 2, American Mathematical Society and Institute for Advanced Study, 1999, pp. 727–806.Google Scholar
In˜aki, Garc´ıa-Etxebarria and Diego, Regalado, N = 3 four dimensional field theories, J. High Energy Phys. 03 (2016), 083.Google Scholar
In˜aki, Garc´ıa-Etxebarria and Diego, Regalado, Exceptional N = 3 theories, J. High Energy Phys. 12 (2017), 042.Google Scholar
Jan, Erik Gerken, Modular graph forms and scattering amplitudes in String theory, Ph.D. thesis, Humboldt U., Berlin, Humboldt U., Berlin, 2020.Google Scholar
Gerken, Jan E., Basis decompositions and a mathematica package for modular graph forms, J. Phys. A 54 (2021), no. 19, 195401.CrossRefGoogle Scholar
Michael, B. Green and Michael, Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997), 195–227.Google Scholar
Gaberdiel, Matthias R. and Green, Michael B., An SL(2, Z) anomaly in IIB supergravity and its F theory interpretation, J. High Energy Phys. 11 (1998), 026.Google Scholar
Gel’land, I. M., Graev, M. I., and Pyatetskii-Shapiro, II., Representation theory and automorphic functions, (Transl. from Russian) Philadelphia: Saunders, 1969.Google Scholar
Green, Michael B., Michael, Gutperle, and Pierre, Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997), 177–184.CrossRefGoogle Scholar
Gross, David J., Harvey, Jeffrey A., Martinec, Emil J., and Ryan, Rohm, Heterotic String theory. 1. The free heterotic string, Nucl. Phys. B 256 (1985), 253.CrossRefGoogle Scholar
Gross, David J., Harvey, Jeffrey A., Martinec, Emil J., and Ryan, Rohm, Heterotic String theory. 2. The interacting heterotic string, Nucl. Phys. B 267 (1986), 75–124.CrossRefGoogle Scholar
Ginsparg, Paul H., Applied conformal field theory, Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, 1988.Google Scholar
Jan, E. Gerken and Justin, Kaidi, Holomorphic subgraph reduction of higher-point modular graph forms, J. High Energy Phys. 01 (2019), 131.Google Scholar
Alexander, Gorsky, Igor, Krichever, Marshakov, Andrei, Andrei, Mironov, and Alexei, Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995), 466–474.Google Scholar
Gerken, Jan E., Axel, Kleinschmidt, and Oliver, Schlotterer, Heteroticstring amplitudes at one loop: Modular graph forms and relations to open strings, J. High Energy Phys. 01 (2019), 052.Google Scholar
Gerken, Jan E., Axel, Kleinschmidt, and Oliver, Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, J. High Energy Phys. 01 (2020), 064.Google Scholar
Gerken, Jan E., Axel, Kleinschmidt, and Oliver, Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, J. High Energy Phys. 07 (2020), no. 07, 190.CrossRefGoogle Scholar
Humberto, Gomez and Carlos, R. Mafra, The overall coefficient of the two-loop superstring amplitude using pure spinors, J. High Energy Phys. 05 (2010), 017.Google Scholar
Humberto, Gomez and Carlos, R. Mafra, The closed-string 3-loop amplitude and S-duality, J. High Energy Phys. 10 (2013), 217.Google Scholar
Alexander, Gorsky, Andrei, Marshakov, Andrei, Mironov, and Alexei, Morozov, RG equations from Whitham hierarchy, Nucl. Phys. B 527 (1998), 690–716.Google Scholar
Green, Michael B., Miller, Stephen D., Russo, Jorge G., and Pierre, Vanhove, Eisenstein series for higher-rank groups and string theory amplitudes, Commun. Number Theory Phys. 4 (2010), 551–596.CrossRefGoogle Scholar
Geyer, Yvonne, Monteiro, Ricardo, and Ricardo, Stark-Much˜ao, Superstring loop amplitudes from the field theory limit, Phys. Rev. Lett. 127 (2021), no. 21, 211603.CrossRefGoogle ScholarPubMed
Green, Michael B., Miller, Stephen D., and Pierre, Vanhove, SL(2, Z)-invariance and D-instanton contributions to the D6R4 interaction, Commun. Number Theory Phys. 09 (2015), 307–344.CrossRefGoogle Scholar
Green, Michael B., Miller, Stephen D., and Pierre, Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, J. Number Theory 146 (2015), 187–309.CrossRefGoogle Scholar
Goddard, Peter, Nuyts, Jean, and Olive, David I., Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977), 1–28.CrossRefGoogle Scholar
Goldfeld, Dorian, Automorphic forms and L-functions for the group GL(n, R), Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, 2006.Google Scholar
Gottschling, Erhard, U¨ber die fixpunkte der Siegelschen modulgruppe, Math. Ann. 143 (1961), no. 2, 111–149.Google Scholar
Gottschling, Erhard, U¨ber die fixpunktuntergruppen der siegelschen modulgruppe, Math. Ann. 143 (1961), no. 5, 399–430.CrossRefGoogle Scholar
Gottschling, Erhard, Die Uniformisierbarkeit der Fixpunkte eigentlich diskontinuierlicher Gruppen von biholomorphen Abbildungen, Math. Ann. 169 (1967), no. 1, 26–54.CrossRefGoogle Scholar
David, J. Gross and Vipul, Periwal, String perturbation theory diverges, Phys. Rev. Lett. 60 (1988), 2105.Google Scholar
Giveon, Amit, Massimo, Porrati, and Eliezer, Rabinovici, Target space duality in string theory, Phys. Rep. 244 (1994), 77–202.CrossRefGoogle Scholar
Gradshteyn, I. S., Ryzhik, I. M., and Romer, Robert H., Tables of integrals, series, and products, 1988.CrossRefGoogle Scholar
Grushevsky, Samuel, Superstring scattering amplitudes in higher genus, Commun. Math. Phys. 287 (2009), 749–767.CrossRefGoogle Scholar
Green, Michael B., Russo, Jorge G., and Pierre, Vanhove, Low energy expansion of the four-particle genus-one amplitude in type II superstring theory, J. High Energy Phys. 02 (2008), 020.Google Scholar
Green, Michael B., Russo, Jorge G., and Pierre, Vanhove, Automorphic properties of low energy string amplitudes in various dimensions, Phys. Rev. D 81 (2010), 086008.CrossRefGoogle Scholar
Green, Michael B. and Schwarz, John H., Supersymmetrical String theories, Phys. Lett. B 109 (1982), 444–448.CrossRefGoogle Scholar
Green, Michael B. and Schwarz, John H., Anomaly cancellation in supersymmetric D=10 Gauge theory and superstring theory, Phys. Lett. B 149 (1984), 117–122.CrossRefGoogle Scholar
Gross, David J. and Sloan, John H., The quartic effective action for the heterotic string, Nucl. Phys. B 291 (1987), 41–89.CrossRefGoogle Scholar
Michael, B. Green and Savdeep, Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999), 046006.Google Scholar
Green, Michael B., Schwarz, John H., and Lars, Brink, N=4 Yang-Mills and N=8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982), 474–492.CrossRefGoogle Scholar
Gliozzi, Ferdinando, Joel, Scherk, and David, Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys. B 122 (1977), no. 2, 253–290.CrossRefGoogle Scholar
Greene, Brian R., Shapere, Alfred D., Cumrun, Vafa, and Shing-Tung, Yau, Stringy cosmic strings and noncompact Calabi–Yau manifolds, Nucl. Phys. B 337 (1990), 1–36.CrossRefGoogle Scholar
Green, Michael B., Schwarz, John H., and Edward, Witten, Superstring theory Vol. 1: 25th anniversary edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2012.Google Scholar
Green, Michael B., Schwarz, John H., and Edward, Witten, Superstring theory Vol. 2: 25th anniversary edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2012.Google Scholar
Gunning, R. C., The Eichler cohomology groups and automorphic forms, Trans. Am. Math. Soc. 100 (1961), no. 1, 44–62.CrossRefGoogle Scholar
Gunning, Robert C., Lectures on Vector Bundles over Riemann Surfaces, vol. 6, Princeton University Press, 1967.Google Scholar
Gunning, Robert C., Lectures on Riemann Surfaces, Lectures on Riemann Surfaces, Princeton University Press, 2015.CrossRefGoogle Scholar
Michael, B. Green and Pierre, Vanhove, The low-energy expansion of the one loop type II superstring amplitude, Phys. Rev. D 61 (2000), 104011.Google Scholar
Sergei, Gukov and Cumrun, Vafa, Rational conformal field theories and complex multiplication, Commun. Math. Phys. 246 (2004), 181–210.Google Scholar
Michael, B. Green and Pierre, Vanhove, Duality and higher derivative terms in M theory, J. High Energy Phys. 01 (2006), 093.Google Scholar
Grisaru, Marcus T., Ven, A. E. M. van de, and Daniela, Zanon, Four loop beta function for the N=1 and N=2 supersymmetric nonlinear sigma model in two-dimensions, Phys. Lett. B 173 (1986), 423–428.CrossRefGoogle Scholar
Gukov, Sergei, Cumrun, Vafa, and Edward, Witten, CFT’s from Calabi–Yau four folds, Nucl. Phys. B 584 (2000), 69–108.CrossRefGoogle Scholar
David, J. Gross and Edward, Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986), 1.Google Scholar
Sergei, Gukov and Edward, Witten, Gauge theory, ramification, and the geometric Langlands program, arXiv:0612073 (2006).CrossRefGoogle Scholar
Michael, B. Green and Congkao, Wen, Modular Forms and SL(2, ℤ)-covariance of type IIB superstring theory, J. High Energy Phys. 06 (2019), 087.Google Scholar
Benedict, Gross and Don, Zagier, Singular moduli, J. Reine Angew. Math. 355 (1985), 191–220.Google Scholar
Hamilton, S., Richard, , Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom. 17 (1982), 255–306.Google Scholar
Harvey, Jeffrey A., TASI 2003 lectures on anomalies, TASI 2003, 2005.Google Scholar
Hecke, Erich, U¨ber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, Mathematische Annalen 114 (1937), no. 1, 1–28.CrossRefGoogle Scholar
Hecke, Erich, U¨ber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II, Mathematische Annalen 114 (1937), no. 1, 316–351.Google Scholar
Hecke, Erich, Lectures on the Theory of Algebraic Numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, 1981.Google Scholar
Hejhal, Dennis A., The Selberg Trace Formula for SL (2, R): Volumes 1 and 2, vol. 1001, Springer, 1983.CrossRefGoogle Scholar
Harrison, Sarah M., Harvey, Jeffrey A., and Paquette, Natalie M., Snowmass White Paper: Moonshine, arXiv:2201.13321 (2022).Google Scholar
Jeffrey, A. Harvey, Yichen, Hu, and Yuxiao, Wu, Galois symmetry induced by Hecke relations in rational conformal field theory and as-sociated modular tensor categories, J. Phys. A 53 (2020), no. 33, 334003.Google Scholar
Harsha, R. Hampapura and Sunil, Mukhi, On 2d conformal field theories with two characters, J. High Energy Phys. 01 (2016), 005.Google Scholar
Harsha, R. Hampapura and Sunil, Mukhi, Two-dimensional RCFT’s without Kac–Moody symmetry, J. High Energy Phys. 07 (2016), 138.Google Scholar
Hopkins, Michael J., Algebraic topology and modular forms, arXiv preprint math/0212397 (2002).Google Scholar
Hidding, Martijn, Oliver Schlotterer and Bram Verbeek, Elliptic modular graph forms II: Iterated integrals, arXiv:2208.11116 (2022).Google Scholar
Hull, Chris M. and Townsend, Paul K., Unity of superstring dualities, Nucl. Phys. B 438 (1995), 109–137.CrossRefGoogle Scholar
Hungerford, Thomas W., Abstract Algebra: An Introduction, Cengage Learning, 2012.Google Scholar
Jeffrey, A. Harvey and Yuxiao, Wu, Hecke relations in rational conformal field theory, J. High Energy Phys. 09 (2018), 032.Google Scholar
Igusa, Jun-ichi, Theta Functions, Grundlehren der mathematischen Wissenschaften, vol. 194, Springer Science & Business Media, 2012.Google Scholar
Henryk, Iwaniec and Emmanuel, Kowalski, Analytic Number Theory, vol. 53, American Mathematical Soc., 2021.Google Scholar
Kenneth, A. Intriligator and Nathan, Seiberg, Lectures on supersymmetric gauge theories and electric–magnetic duality, Nucl. Phys. B Proc. Suppl. 45BC (1996), 1–28.Google Scholar
Iwaniec, Henryk, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, 1997.Google Scholar
Iwaniec, Henryk, Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, vol. 53, American Mathematical Society, 2002.Google Scholar
Jay, Jorgenson and Serge, Lang, Basic Analysis of Regularized Series and Products, Lecture Notes in Mathematics 1564 (1993).Google Scholar
Johnson, Clifford V., D-branes, Cambridge University Press, 2006.Google Scholar
, Nariya Kawazumi, Johnson’s homomorphisms and the Arakelov-Green function, arXiv:0801.4218 (2008).Google Scholar
Johnson, Clifford V., Some tensor field on the Teichmu¨ller space, Lecture at MCM2016 (2016).Google Scholar
Kiritsis, Elias B., Fuchsian differential equations for characters on the torus: A classification, Nucl. Phys. B 324 (1989), 475–494.CrossRefGoogle Scholar
Kiritsis, Elias, String Theory in a Nutshell, vol. 21, Princeton University Press, 2019.Google Scholar
Masanobu, Kaneko and Masao, Koike, On modular forms arising from a differential equation of hypergeometric type, Ramanujan J. 7 (2003), no. 1, 145–164.Google Scholar
Justin, Kaidi, Zohar, Komargodski, Ohmori, Kantaro, Sahand, Seifnashri, and Shu-Heng, Shao, Higher central charges and topological boundaries in 2+1-dimensional TQFTs, SciPost Phys. 13 (2022), no. 3, 067.Google Scholar
Klingen, Helmut, Introductory lectures on Siegel modular forms, Camb. Stud. Adv. Math. 20 (1990).Google Scholar
Kaidi, Justin, Ying-Hsuan, Lin, and Julio, Parra-Martinez, Holomorphic modular bootstrap revisited, J. High Energy Phys. 12 (2021), 151.Google Scholar
Khuri-Makdisi, Kamal, Moduli interpretation of Eisenstein series, arXiv:0903.1439 (2011).Google Scholar
Justin, Kaidi, Mario, Martone, Leonardo, Rastelli, and Mitch, Weaver, Needles in a haystack. An algorithmic approach to the classification of 4d = 2 SCFTs, J. High Energy Phys. 03 (2022), 210.Google Scholar
Kaidi, Justin, Mario, Martone, and Gabi, Zafrir, Exceptional moduli spaces for exceptional N = 3 theories, J. High Energy Phys. 08 (2022), 264.Google Scholar
Knapp, Anthony W., Elliptic Curves, Mathematical Notes, Princeton University Press, 1992.Google Scholar
Koblitz, Neal I, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics, vol. 114, Springer Science & Business Media, 1994.Google Scholar
Koblitz, Neal I, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, vol. 97, Springer Science & Business Media, 2012.Google Scholar
Kodaira, Kunihiko, Complex Manifolds and Deformation of Complex Structures, Grundlehren der mathematischen Wissenschaften, vol. 283, Springer, 1986.Google Scholar
Gu¨nter, K¨ohler, Eta Products and Theta Series Identities, Springer Monographs in Mathematics, Springer Science & Business Media, 2011.Google Scholar
Shamit, Kachru and Arnav, Tripathy, Black holes and Hurwitz class numbers, Int. J. Mod. Phys. D 26 (2017), no. 12, 1742003.Google Scholar
Kubota, Tomio, Elementary Theory of Eisenstein Series, Halsted Press, 1973.Google Scholar
Axel, Kleinschmidt and Valentin, Verschinin, Tetrahedral modular graph functions, J. High Energy Phys. 09 (2017), 155.Google Scholar
Anton, Kapustin and Edward, Witten, Electric–magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1–236.Google Scholar
Lang, Serge, Sl(2,r), Addison-Wesley Publishing Company, 1975.Google Scholar
Lang, Serge, Introduction to Modular Forms, Grundlehren der mathematischen Wissenschaften, vol. 222, Springer-Verlag, 1976.Google Scholar
Lang, Serge, Elliptic Functions, Graduate Texts in Mathematics, vol. 112, Springer-Verlag, 1987.Google Scholar
Lin, Ying-Hsuan, Topological modularity of Monstrous Moonshine, arXiv:2207.14076 (2022).Google Scholar
Ying-Hsuan, Lin and Du, Pei, Holomorphic CFTs and topological modular forms, Commun. Math. Phys. 401 (2023), no. 1, 325–332.Google Scholar
Ruth, Lawrence and Don, Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93–108.Google Scholar
Marcus, Daniel A., Number Fields, Springer Universitext, 1977.CrossRefGoogle Scholar
Emil, J. Martinec, Nonrenormalization theorems and Fermionic string finiteness, Phys. Lett. B 171 (1986), 189.Google Scholar
Emil, J. Martinec, Integrable structures in supersymmetric gauge and string theory, Phys. Lett. B 367 (1996), 91–96.Google Scholar
Marks, Christopher, Fourier coefficients of three-dimensional vectorvalued modular forms, arXiv preprint arXiv:1201.5165 (2012).Google Scholar
Martone, Mario, The constraining power of Coulomb Branch Geometry: Lectures on Seiberg-Witten theory, Young Researchers Integrability School and Workshop 2020: A modern primer for superconformal field theories, 6 2020.Google Scholar
Mason, Geoffrey, On the Fourier coefficients of 2-dimensional vectorvalued modular forms, Proc. Am. Math. Soc. 140 (2012), no. 6, 1921–1930.CrossRefGoogle Scholar
McKean, H., Selberg’s trace formula for PSL(2,R), Commun. Pure Appl. Math. 25 (1972), 223.Google Scholar
Miyake, Toshitsune, Modular Forms, Springer Science & Business Media, 2006.Google Scholar
Jan, Manschot and Gregory, W. Moore, A Modern Farey Tail, Commun. Number Theory Phys. 4 (2010), 103–159.Google Scholar
Jan, Manschot and Gregory, W. Moore, Topological correlators of SU (2) N = 2∗ SYM on four-manifolds, arXiv:2104.06492 (2021).Google Scholar
Mathur, Samir D., Sunil, Mukhi, and Ashoke, Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988), 303–308.CrossRefGoogle Scholar
Mathur, Samir D., Sunil, Mukhi, and Ashoke, Sen, Reconstruction of conformal field theories from modular geometry on the torus, Nucl. Phys. B 318 (1989), 483–540.CrossRefGoogle Scholar
Jan, Manschot, Gregory, W. Moore, and Xinyu, Zhang, Effective gravitational couplings of four-dimensional N = 2 supersymmetric gauge theories, J. High Energy Phys. 06 (2020), 150.Google Scholar
Joseph, A. Minahan and Dennis, Nemeschansky, An N=2 superconformal fixed point with E(6) global symmetry, Nucl. Phys. B 482 (1996), 142–152.Google Scholar
Joseph, A. Minahan and Dennis, Nemeschansky, Superconformal fixed points with E(n) global symmetry, Nucl. Phys. B 489 (1997), 24–46.Google Scholar
Minahan, Joseph A., Nemeschansky, Dennis, and Warner, Nicholas P., Instanton expansions for mass deformed N=4 superYang-Mills theories, Nucl. Phys. B 528 (1998), 109–132.CrossRefGoogle Scholar
Claus, Montonen and David, I. Olive, Magnetic monopoles as Gauge particles?, Phys. Lett. B 72 (1977), 117–120.Google Scholar
Moore, Gregory W., Modular forms and two loop string physics, Phys. Lett. B 176 (1986), 369–379.CrossRefGoogle Scholar
Louis, Joel. Mordell, On Mr. Ramanujan’s empirical expansion of modular functions, Proc. Camb. Philos. Soc., vol. 19, 1917, pp. 117–124.Google Scholar
Mukhi, Sunil, Rahul, Poddar, and Palash, Singh, Rational CFT with three characters: The quasi-character approach, J. High Energy Phys. 05 (2020), 003.Google Scholar
Gregory, W. Moore and Nathan, Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988), 451–460.Google Scholar
Gregory, W. Moore and Nathan, Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989), 177.Google Scholar
Gregory, W. Moore and Nathan, Seiberg, Lectures on RCFT, 1989 Banff NATO ASI: Physics, Geometry and Topology, 1989.Google Scholar
Gregory, W. Moore and Nathan, Seiberg, Taming the conformal zoo, Phys. Lett. B 220 (1989), 422–430.Google Scholar
Carlos, R. Mafra and Oliver, Schlotterer, Towards the n-point one-loop superstring amplitude. Part I. Pure spinors and superfield kinematics, J. High Energy Phys. 08 (2019), 090.Google Scholar
Carlos, R. Mafra and Oliver, Schlotterer, Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics, J. High Energy Phys. 08 (2019), 091.Google Scholar
Carlos, R. Mafra and Oliver, Schlotterer, Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure, J. High Energy Phys. 08 (2019), 092.Google Scholar
Carlos, R. Mafra and Oliver, Schlotterer, Tree-level amplitudes from the pure spinor superstring, Phys. Rep. 1020 (2023), 1–162.Google Scholar
Mafra, Carlos R., Oliver, Schlotterer, and Stephan, Stieberger, Complete N-Point superstring disk amplitude I. Pure spinor computation, Nucl. Phys. B 873 (2013), 419–460.Google Scholar
Minasian, Ruben, Soumya, Sasmal, and Raffaele, Savelli, Discrete anomalies in supergravity and consistency of string backgrounds, J. High Energy Phys. 02 (2017), 025.Google Scholar
Mukhi, Sunil, Dualities and the SL(2,Z) anomaly, J. High Energy Phys. 12 (1998), 006.Google Scholar
Mukhi, Sunil, Classification of RCFT from Holomorphic Modular Boot-strap: A Status Report, Pollica Summer Workshop 2019: Mathematical and Geometric Tools for Conformal Field Theories, 10 2019.Google Scholar
Mumford, David, Tata Lectures on Theta. I (Modern Birkh¨auser Classics), Birkh¨auser Boston Incorporated, 2007.Google Scholar
Naculich, Stephen G., Differential equations for rational conformal characters, Nucl. Phys. B 323 (1989), 423–440.CrossRefGoogle Scholar
Narain, Kumar S., New heterotic string theories in uncompactified dimensions < 10, Phys. Lett. B 169 (1986), 41–46.CrossRefGoogle Scholar
Nekrasov, Nikita A., Seiberg-Witten prepotential from instanton counting, Adv. Theory Math. Phys. 7 (2003), no. 5, 831–864.CrossRefGoogle Scholar
Hans, Peter Nilles, Saul, Ramos-Sanchez, Trautner, Andreas, and Vaudrevange, Patrick K. S., Orbifolds from Sp(4,Z) and their modular symmetries, Nucl. Phys. B 971 (2021), 115534.Google Scholar
Andr´e, Neveu and John, H Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971), no. 1, 86–112.Google Scholar
Narain, Kumar S., Sarmadi, M. H., and Edward, Witten, A note on toroidal compactification of heterotic string theory, Nucl. Phys. B 279 (1987), 369–379.CrossRefGoogle Scholar
Fritz, Oberhettinger and Wilhelm, Magnus, Anwendung der elliptischen Funktionen in Physik und Technik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 55, Springer-Verlag, 2013.Google Scholar
Ono, Ken, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS, vol. 102, American Mathematical Society, 2004.Google Scholar
Ono, Ken, Harmonic Maass forms, mock modular forms, and quantum modular forms, Notes (2013), 347–454.Google Scholar
Niels, A. Obers and Boris, Pioline, U duality and M theory, Phys. Rep. 318 (1999), 113–225.Google Scholar
Niels, A. Obers and Boris, Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000), 275–324.Google Scholar
Osborn, Hugh, Topological charges for N=4 supersymmetric Gauge theories and monopoles of Spin 1, Phys. Lett. B 83 (1979), 321–326.CrossRefGoogle Scholar
Osborn, Hugh, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991), 486–526.CrossRefGoogle Scholar
Pestun, Vasily et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017), no. 44, 440301.CrossRefGoogle Scholar
Perelman, Giorgi, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG 0211159 (2002).Google Scholar
Perelman, Giorgi, Ricci flow with surgery on three-manifolds, arXiv:math.DG 0303109 (2003).Google Scholar
Pestun, Vasily, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012), 71–129.CrossRefGoogle Scholar
Petersson, Hans, U¨ber die Entwicklungskoeffizienten der automorphen Formen, Acta Math. 58 (1932), 169–215.Google Scholar
Petersson, Hans, U¨ber eine Metrisierung der ganzen Modulformen., Jahres-bericht der Deutschen Mathematiker-Vereinigung 49 (1939), 49–75.Google Scholar
Pioline, Boris, A Note on nonperturbative R**4 couplings, Phys. Lett. B 431 (1998), 73–76.CrossRefGoogle Scholar
Pioline, Boris, Lectures on black holes, topological strings and quantum attractors, Class. Quantum Grav. 23 (2006), S981.CrossRefGoogle Scholar
Pioline, Boris, D6R4 amplitudes in various dimensions, J. High Energy Phys. 04 (2015), 057.Google Scholar
Pioline, Boris, A Theta lift representation for the Kawazumi–Zhang and Faltings invariants of genus-two Riemann surfaces, J. Number Theory 163 (2016), 520–541.CrossRefGoogle Scholar
Polchinski, Joseph, Evaluation of the one loop string path integral, Commun. Math. Phys. 104 (1986), 37.CrossRefGoogle Scholar
Aleksandr, Michajloviˇc Polyakov, Gauge Fields and Strings, Taylor & Francis, 1987.Google Scholar
Polchinski, Joseph, String Theory. Vol. 1: An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007.Google Scholar
Polchinski, Joseph, String theory. Vol. 2: Superstring theory and beyond, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007.Google Scholar
Paul, Hynek, Eric, Perlmutter, and Himanshu, Raj, Integrated correlators in N = 4 SYM via SL(2, ℤ) spectral theory, J. High Energy Phys. 01 (2023), 149.Google Scholar
Joseph, Polchinski and Edward, Witten, Evidence for heterotic-type I string duality, Nucl. Phys. B 460 (1996), 525–540.Google Scholar
Rademacher, Hans, The Fourier coefficients of the modular invariant J (τ), Am. J. Math. 60 (1938), no. 2, 501–512.CrossRefGoogle Scholar
Rademacher, Hans, Topics in Analytic Number Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 169, Springer, 1970.Google Scholar
Ramond, Pierre, Dual theory for free fermions, Phys. Rev. D 3 (1971), no. 10, 2415.CrossRefGoogle Scholar
Rankin, R., Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetic functions, Proc. Camb. Philol. Soc. 35 (1939), 351.Google Scholar
Rankin, Robert A., Modular Forms and Functions, Cambridge University Press, 1977.CrossRefGoogle Scholar
Ray, Daniel B. and Singer, Isadore M., R-torsion and the Laplacian on Riemannian manifolds, Advances in Mathematics 7 (1971), no. 2, 145–210.CrossRefGoogle Scholar
Shlomo, Razamat, Evyatar, Sabag, Orr Sela, and Gabi Zafrir, Aspects of 4d supersymmetric dynamics and geometry, arXiv:2203.06880 (2022).Google Scholar
Daniel, Robbins and Thomas, Vandermeulen, Modular orbits at higher genus, J. High Energy Phys. 02 (2020), 113.Google Scholar
Sarnak, Peter, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120.CrossRefGoogle Scholar
Atle, Selberg and Sarvadaman, Chowla, On Epstein’s Zeta-function., Walter de Gruyter, 1967.Google Scholar
Schwarz, John H., An SL(2,Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995), 13–18.CrossRefGoogle Scholar
Schwarz, John H., Lectures on superstring and M theory dualities: Given at ICTP Spring School and at TASI Summer School, Nucl. Phys. B Proc. Suppl. 55 (1997), 1–32.CrossRefGoogle Scholar
Selberg, Atle, Bemerkungen u¨ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47.Google Scholar
Selberg, Atle, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. of the Indian Mathematical Society 20 (1956), no. 7, 47–87.Google Scholar
Selberg, Atle, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math., vol. 8, AMS, 1965, pp. 1–15.Google Scholar
Serre, Jean-Pierre, A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, 1973.Google Scholar
Shimura, Goro, Introduction to the Arithmetic Theory of Automorphic Functions, vol. 1, Princeton University Press, 1971.Google Scholar
Shifman, Mikhail, Advanced Topics in Quantum Field Theory, Cambridge University Press, 2022.CrossRefGoogle Scholar
Carl, Ludwig Siegel, On Advanced Analytic Number Theory, Tata Institute of Fundamental Research, 1961.Google Scholar
Carl, Ludwig Siegel, Topics in Complex Function Theory, Volume 2: Automorphic Functions and Abelian Integrals, vol. 16, John Wiley & Sons, 1989.Google Scholar
Carl, Ludwig Siegel, Topics in Complex Function Theory, Volume 3: Abelian Functions and Modular Functions of Several Variables, vol. 16, John Wiley & Sons, 1989.Google Scholar
Silverman, Joseph H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, 1985.Google Scholar
Sinha, Aninda, The G(hat)**4 lambda**16 term in IIB supergravity, J. High Energy Phys. 08 (2002), 017.Google Scholar
Srednicki, Marc, Quantum Field Theory, Cambridge University Press, 2007.CrossRefGoogle Scholar
Joel, Scherk and John, H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979), 61–88.Google Scholar
Oliver, Schlotterer and Stephan, Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013), 475401.Google Scholar
Menahem, Schiffer and Donald, C. Spencer, Functionals of Finite Riemann Surfaces, Courier Corporation, 2014.Google Scholar
Joseph, H. Silverman and John, Tate, Rational Points on Elliptic Curves, vol. Undergraduate Texts in Mathematics, Springer-Verlag, 1992.Google Scholar
Stieberger, Stephan, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014), 155401.Google Scholar
Seiberg, N. and Edward, Witten, Spin structures in String theory, Nucl. Phys. B 276 (1986), 272.CrossRefGoogle Scholar
Seiberg, N. and Edward, Witten, Electric magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994), 19–52. [Erratum: Nucl. Phys. B 430, 485{486 (1994)].CrossRefGoogle Scholar
Seiberg, N. and Edward, Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B 431 (1994), 484–550.CrossRefGoogle Scholar
Ashoke, Sen and Edward, Witten, Filling the gaps with PCOs, J. High Energy Phys. 09 (2015), 004.Google Scholar
Stienstra, J. and Don, Zagier, Bimodular forms and holomorphic anomaly equation, Workshop on Modular Forms and String Duality, Banff International Research Centre, 2006 (2006).Google Scholar
Travaglini, Gabriele et al., The SAGEX review on scattering amplitudes, J. Phys. A 55 (2022), no. 44, 443001.CrossRefGoogle Scholar
Tachikawa, Yuji, N=2 Supersymmetric Dynamics for Pedestrians, Springer Hindustan Book Agency, 2013.Google Scholar
Tachikawa, Yuji, Topological modular forms and the absence of a heterotic global anomaly, Prog. Theor. Exp. Phys. 2022 (2022), no. 4, 04A107.CrossRefGoogle Scholar
Terras, Audrey, Harmonic Analysis on Symmetric Spaces and Applications I, Springer Science & Business Media, 1985.CrossRefGoogle Scholar
Terras, Audrey, Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag, 1988.CrossRefGoogle Scholar
Yuji, Tachikawa and Kazuya, Yonekura, Why are fractional charges of orientifolds compatible with Dirac quantization?, SciPost Phys. 7 (2019), no. 5, 058.Google Scholar
Yuji, Tachikawa and Mayuko, Yamashita, Topological modular forms and the absence of all heterotic global anomalies, Commun. Math. Phys. 402 (2023), no. 2, 1585–1620 [Erratum: Commun. Math. Phys. 402, 2131 (2023)].Google Scholar
Vafa, Cumrun, Evidence for F theory, Nucl. Phys. B 469 (1996), 403.CrossRefGoogle Scholar
Vafa, Cumrun, Geometric origin of Montonen-Olive duality, Adv. Theor. Math. Phys. 1 (1998), 158–166.Google Scholar
Gerard, Van Der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms, Springer, 2008, pp. 181–245.Google Scholar
Bartel, Leendert Van der Waerden, Algebra, vol. 2, Springer, 1991.Google Scholar
Verlinde, Erik P., Global aspects of electric – magnetic duality, Nucl. Phys. B 455 (1995), 211–228.CrossRefGoogle Scholar
Viazovska, Maryna, The sphere packing problem in dimension 8, Annals of Mathematics 185 (2017), no. 3, 991–1015.CrossRefGoogle Scholar
Verlinde, Erik P. and Verlinde, Herman L., Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987), 357.CrossRefGoogle Scholar
Cumrun, Vafa and Edward, Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994), 3–77.Google Scholar
Washington, Lawrence C., Elliptic Curves, Number Theory, and Cryptography, Chapman & Hall/CRC, 2003.Google Scholar
Julius, Wess and Jonathan, Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992.Google Scholar
Andr´e, Weil, Elliptic Functions According to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 88, Springer-Verlag, 1976.Google Scholar
Weinberg, Steven, Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.CrossRefGoogle Scholar
Weinberg, Steven, Quantum Theory of Fields, Volume II, Cambridge University Press, 1996.CrossRefGoogle Scholar
Weinberg, Steven, The Quantum Theory of Fields. Vol. 3: Supersymmetry, Cambridge University Press, 2013.Google Scholar
Witten, Edward, Dyons of charge e theta/2 pi, Phys. Lett. B 86 (1979), 283–287.CrossRefGoogle Scholar
Witten, Edward, On S duality in Abelian gauge theory, Selecta Math. 1 (1995), 383.Google Scholar
Witten, Edward, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995), 85–126.CrossRefGoogle Scholar
Witten, Edward, Solutions of four-dimensional field theories via M theory, Nucl. Phys. B 500 (1997), 3–42.CrossRefGoogle Scholar
Witten, Edward, Dynamics of quantum field theory, Quantum Fields and Strings: A Course for Mathematicians (Deligne, P. et al., eds.), vol. 2, American Mathematical Society and Institute for Advanced Study, 1999, pp. 1119–1424.Google Scholar
Witten, Edward, Three-dimensional gravity revisited, arXiv:0706.3359 (2007).Google Scholar
Witten, Edward, Superstring perturbation theory revisited, arXiv:1209.5461 (2012).Google Scholar
Witten, Edward, Notes on holomorphic string and superstring theory measures of low genus, arXiv:1306.3621 (2013).Google Scholar
Witten, Edward, The Feynman iϵ in String theory, J. High Energy Phys. 04 (2015), 055.Google Scholar
Witten, Edward, The super period matrix With Ramond punctures, J. Geom. Phys. 92 (2015), 210–239.Google Scholar
Witten, Edward, Notes on super Riemann surfaces and their moduli, Pure Appl. Math. Q. 15 (2019), no. 1, 57–211.Google Scholar
Edward, Witten and David, I. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978), 97–101.Google Scholar
Edmund, Taylor Whittaker and George, Neville Watson, A Course of Modern Analysis, Cambridge University Press, 1969.Google Scholar
Yi, Jinhee, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math. Anal. Appl. 292 (2004), no. 2, 381–400.CrossRefGoogle Scholar
Zagier, Don, Eisenstein Series and the Selberg Trace Formula, Automorphic Forms, Representation Theory and Arithmetic, Springer-Verlag, 1981, pp. 303–355.Google Scholar
Zagier, Don, Zetafunktionen und quadratische K¨orper: eine Einfu¨rhrung in die h¨ohere Zahlentheorie, Springer, 1981.CrossRefGoogle Scholar
Zagier, Don, The Rankin–Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 28 (1982), 415.Google Scholar
Zagier, Don, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990), 613–624.CrossRefGoogle Scholar
Zagier, Don, Introduction to Modular Forms, From Number Theory to Physics, Springer, 1992, pp. 238–291.Google Scholar
Zagier, Don, Modular forms and differential operators, Proc. Ind. Acad. Sci. 104 (1994), 57–75.Google Scholar
Zagier, Don, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, Elsevier 40 (2001), no. 5, 945–960.Google Scholar
Zagier, Don, Ramanujan’s mock theta functions and their applications, S´eminaire BOURBAKI (2007).Google Scholar
Zagier, Don, Elliptic Modular Forms and Their Applications, The 1-2-3 of modular forms, Springer, 2008, pp. 1–103.CrossRefGoogle Scholar
Zagier, Don, Quantum modular forms, Quanta of maths, American Mathematical Society Providence 11 (2010), 659–675.Google Scholar
Zerbini, Federico, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Number Theory Phys. 10 (2016), 703–737.CrossRefGoogle Scholar
Zhang, Shou-Wu, Gross–Schoen cycles and dualizing sheaves, arXiv preprint arXiv:0812.0371 (2008).Google Scholar
Zinn-Justin, J., Quantum Field Theory and Critical Phenomena, Oxford University Press, 1989.Google Scholar
Zwiebach, Barton, A First Course in String Theory, Cambridge University Press, 2004.CrossRefGoogle Scholar

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