Abstract
In [1] we have studied the single-particle free energy of a class of Little String Theories of A-type, which are engineered by N parallel M5-branes on a circle. To leading instanton order (from the perspective of the low energy U(N) gauge theory) and partially also to higher order, a decomposition was observed, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the N = 1 BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions were written as infinite series and similarities to modular graph functions were remarked. In the current work we continue and extend this study: working with the full non-perturbative BPS free energy, we analyse in detail the cases N = 2, 3 and 4. We argue that in these cases to leading instanton order all coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. We provide closed form expressions, which we conjecture to hold for generic N. To higher instanton order, we observe that a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. We nevertheless provide evidence that tentative coupling functions are still combinations of scalar Greens functions, which are decorated with derivatives or multiplied with holomorphic Eisenstein series. We interpret these decorations as corrections of the leading order effective couplings and in particular link the latter to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs.
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References
S. Hohenegger, From little string free energies towards modular graph functions, JHEP 03 (2020) 077 [arXiv:1911.08172] [INSPIRE].
E. Witten, Some comments on string dynamics, in Strings ′95: future perspectives in string theory, (1995) [hep-th/9507121] [INSPIRE].
P.S. Aspinwall, Point-like instantons and the Spin(32)/Z2 heterotic string, Nucl. Phys. B 496 (1997) 149 [hep-th/9612108] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
N. Seiberg, New theories in six-dimensions and matrix description of M-theory on T5 and T5/Z2, Phys. Lett. B 408 (1997) 98 [hep-th/9705221] [INSPIRE].
K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].
L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the classification of little strings, Phys. Rev. D 93 (2016) 086002 [Erratum ibid. 100 (2019) 029901] [arXiv:1511.05565] [INSPIRE].
L. Bhardwaj, Revisiting the classifications of 6d SCFTs and LSTs, JHEP 03 (2020) 171 [arXiv:1903.10503] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
D. Xie and S.-T. Yau, 4d N = 2 SCFT and singularity theory. Part I. Classification, arXiv:1510.01324 [INSPIRE].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [INSPIRE].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
M. Caorsi and S. Cecotti, Geometric classification of 4d N = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE].
L. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE].
M. Martone, Towards the classification of rank-r N = 2 SCFTs. Part I. Twisted partition function and central charge formulae, JHEP 12 (2020) 021 [arXiv:2006.16255] [INSPIRE].
P.C. Argyres and M. Martone, Towards a classification of rank-r N = 2 SCFTs. Part II. Special Kähler stratification of the Coulomb branch, JHEP 12 (2020) 022 [arXiv:2007.00012] [INSPIRE].
S. Hohenegger, A. Iqbal and S.-J. Rey, M-strings, monopole strings, and modular forms, Phys. Rev. D 92 (2015) 066005 [arXiv:1503.06983] [INSPIRE].
S. Hohenegger, A. Iqbal and S.-J. Rey, Instanton-monopole correspondence from M-branes on S1 and little string theory, Phys. Rev. D 93 (2016) 066016 [arXiv:1511.02787] [INSPIRE].
S. Hohenegger, A. Iqbal and S.-J. Rey, Self-duality and self-similarity of little string orbifolds, Phys. Rev. D 94 (2016) 046006 [arXiv:1605.02591] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Triality in little string theories, Phys. Rev. D 97 (2018) 046004 [arXiv:1711.07921] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Beyond triality: dual quiver gauge theories and little string theories, JHEP 11 (2018) 016 [arXiv:1807.00186] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Five-dimensional gauge theories from shifted web diagrams, Phys. Rev. D 99 (2019) 046012 [arXiv:1810.05109] [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
S. Hohenegger and A. Iqbal, M-strings, elliptic genera and N = 4 string amplitudes, Fortsch. Phys. 62 (2014) 155 [arXiv:1310.1325] [INSPIRE].
V. Gritsenko, Complex vector bundles and Jacobi forms, math.AG/9906191 [INSPIRE].
A. Kanazawa and S.-C. Lau, Local Calabi-Yau manifolds of type \( \tilde{A} \) via SYZ mirror symmetry, J. Geom. Phys. 139 (2019) 103 [arXiv:1605.00342] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
B. Bastian and S. Hohenegger, Five-brane webs and highest weight representations, JHEP 12 (2017) 020 [arXiv:1706.08750] [INSPIRE].
S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings from F-theory and flop transitions, JHEP 07 (2017) 112 [arXiv:1610.07916] [INSPIRE].
A. Ahmed, S. Hohenegger, A. Iqbal and S.-J. Rey, Bound states of little strings and symmetric orbifold conformal field theories, Phys. Rev. D 96 (2017) 081901 [arXiv:1706.04425] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings and their partition functions, Phys. Rev. D 97 (2018) 106004 [arXiv:1710.02455] [INSPIRE].
B. Bastian and S. Hohenegger, Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds, Phys. Rev. D 99 (2019) 066013 [arXiv:1811.03387] [INSPIRE].
B. Bastian and S. Hohenegger, Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups, JHEP 03 (2020) 062 [arXiv:1911.07276] [INSPIRE].
B. Bastian and S. Hohenegger, Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums, JHEP 03 (2020) 016 [arXiv:1911.07280] [INSPIRE].
S. Hohenegger and A. Iqbal, Symmetric orbifold theories from little string residues, Phys. Rev. D 103 (2021) 066004 [arXiv:2009.00797] [INSPIRE].
B. Haghighat and R. Sun, M5 branes and theta functions, JHEP 10 (2019) 192 [arXiv:1811.04938] [INSPIRE].
H. Bachmann and U. Kühn, The algebra of generating functions for multiple divisor sums and applications to multiple zeta values, arXiv:1309.3920 [INSPIRE].
A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer, Berlin, Heidelberg, Germany (1976).
J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].
E. D’Hoker, M.B. Green, Ö. Gürdogan and P. Vanhove, Modular graph functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].
E. D’Hoker and M.B. Green, Identities between modular graph forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants, and their exact asymptotics, Commun. Math. Phys. 366 (2019) 927 [arXiv:1712.06135] [INSPIRE].
F. Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes, Ph.D. thesis, Bonn U., Bonn, Germany (2017) [arXiv:1804.07989] [INSPIRE].
F. Zerbini, Modular and holomorphic graph functions from superstring amplitudes, in KMPB conference: elliptic integrals, elliptic functions and modular forms in quantum field theory, Springer, Cham, Switzerland (2018) [arXiv:1807.04506] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE].
C.R. Mafra and O. Schlotterer, All order α′ expansion of one-loop open-string integrals, Phys. Rev. Lett. 124 (2020) 101603 [arXiv:1908.09848] [INSPIRE].
C.R. Mafra and O. Schlotterer, One-loop open-string integrals from differential equations: all-order α′-expansions at n points, JHEP 03 (2020) 007 [arXiv:1908.10830] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, JHEP 07 (2020) 190 [arXiv:2004.05156] [INSPIRE].
R. Gopakumar and C. Vafa, M-theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M-theory and topological strings. 2, hep-th/9812127 [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
I. Antoniadis, S. Hohenegger, K.S. Narain and T.R. Taylor, Deformed topological partition function and Nekrasov backgrounds, Nucl. Phys. B 838 (2010) 253 [arXiv:1003.2832] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Worldsheet realization of the refined topological string, Nucl. Phys. B 875 (2013) 101 [arXiv:1302.6993] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Non-perturbative Nekrasov partition function from string theory, Nucl. Phys. B 880 (2014) 87 [arXiv:1309.6688] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Probing the moduli dependence of refined topological amplitudes, Nucl. Phys. B 901 (2015) 252 [arXiv:1508.01477] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, NATO Sci. Ser. C 520 (1999) 359 [hep-th/9801061] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
M.R. Douglas, Conformal field theory techniques in large N Yang-Mills theory, in NATO advanced research workshop on new developments in string theory, conformal models and topological field theory, (1993) [hep-th/9311130] [INSPIRE].
R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B 493 (1997) 588 [hep-th/9609022] [INSPIRE].
R. Dijkgraaf, Mirror symmetry and elliptic curves, in The moduli space of curves, Birkhäuser, Boston, MA, U.S.A. (1995), pg. 149.
M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, MA, U.S.A. (1985).
S. Lang, Introduction to modular forms, Springer, Berlin, Heidelberg, Germany (1987).
W. Stein, Modular forms, a computational approach, American Mathematical Society, Providence, RI, U.S.A. (2007).
A. Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, arXiv:0904.1026.
T. Eguchi and H. Ooguri, Conformal and current algebras on general Riemann surface, Nucl. Phys. B 282 (1987) 308 [INSPIRE].
S.M. Kuzenko and O.A. Solovev, Equations for two point correlation functions on compact Riemann surfaces, Theor. Math. Phys. 88 (1991) 901 [Teor. Mat. Fiz. 88 (1991) 323] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α′-expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, multiple zeta values and superstring amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].
J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].
N. Matthes, Elliptic multiple zeta value, Ph.D. thesis, Universität Hamburg, Hamburg, Germany (2016).
F. Brown, A class of non-holomorphic modular forms I, Res. Math. Sci. 5 (2018) 7 [arXiv:1707.01230] [INSPIRE].
F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, arXiv:1708.03354.
J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, JHEP 01 (2019) 155 [arXiv:1803.00527] [INSPIRE].
E. D’Hoker and M.B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, Commun. Num. Theor. Phys. 14 (2020) 315 [arXiv:1904.06603] [INSPIRE].
D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, Commun. Num. Theor. Phys. 14 (2020) 413 [arXiv:1906.12339] [INSPIRE].
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Hohenegger, S. Diagrammatic expansion of non-perturbative little string free energies. J. High Energ. Phys. 2021, 275 (2021). https://doi.org/10.1007/JHEP04(2021)275
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DOI: https://doi.org/10.1007/JHEP04(2021)275