# Topological recursion in the Ramond sector

- 20 Downloads

## Abstract

We investigate supereigenvalue models in the Ramond sector and their recursive structure. We prove that the free energy truncates at quadratic order in Grassmann coupling constants, and consider super loop equations of the models with the assumption that the 1*/N* expansion makes sense. Subject to this assumption, we obtain the associated genus-zero algebraic curve with two ramification points (one regular and the other irregular) and also the supersymmetric partner polynomial equation. Starting with these polynomial equations, we present a recursive formalism that computes all the correlation functions of these models. Somewhat surprisingly, correlation functions obtained from the new recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction — the new recursion may lead us to a further development of supersymmetric generalizations of the Eynard-Orantin topological recursion.

## Keywords

1/N Expansion Matrix Models Supergravity Models## Notes

### **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

## References

- [1]J.E. Andersen, G. Borot, L.O. Chekhov and N. Orantin,
*The ABCD of topological recursion*, arXiv:1703.03307 [INSPIRE]. - [2]J.E. Andersen, G. Borot and N. Orantin,
*Geometric recursion*, arXiv:1711.04729. - [3]A.S. Alexandrov, A. Mironov and A. Morozov,
*Instantons and merons in matrix models*,*Physica***D 235**(2007) 126 [hep-th/0608228] [INSPIRE]. - [4]G. Akemann and J.C. Plefka,
*The Chiral superEigenvalue model*,*Mod. Phys. Lett.***A 12**(1997) 1745 [hep-th/9705114] [INSPIRE]. - [5]L. Álvarez-Gaumé, H. Itoyama, J.L. Manes and A. Zadra,
*Superloop equations and two-dimensional supergravity*,*Int. J. Mod. Phys.***A 7**(1992) 5337 [hep-th/9112018] [INSPIRE]. - [6]L. Álvarez-Gaumé, K. Becker, M. Becker, R. Emparan and J. Manes,
*Double scaling limit of the superVirasoro constraints*,*Int. J. Mod. Phys.***A 8**(1993) 2297 [hep-th/9207096] [INSPIRE]. - [7]K. Becker and M. Becker,
*Nonperturbative solution of the superVirasoro constraints*,*Mod. Phys. Lett.***A 8**(1993) 1205 [hep-th/9301017] [INSPIRE]. - [8]V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti,
*Remodeling the B-model*,*Commun. Math. Phys.***287**(2009) 117 [arXiv:0709.1453] [INSPIRE]. - [9]V. Bouchard and M. Marin˜o,
*Hurwitz numbers, matrix models and enumerative geometry*,*Proc. Symp. Pure Math.***78**(2008) 263 [arXiv:0709.1458] [INSPIRE]. - [10]V. Bouchard and K. Osuga,
*Supereigenvalue Models and Topological Recursion*,*JHEP***04**(2018) 138 [arXiv:1802.03536] [INSPIRE]. - [11]V. Bouchard and K. Osuga,
*Super Topological Recursion*, work in progress.Google Scholar - [12]V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba and P. Su-lkowski,
*Super Quantum Airy Structures*, arXiv:1907.08913 [INSPIRE]. - [13]A. Brini, B. Eynard and M. Mariño,
*Torus knots and mirror symmetry*,*Annales Henri Poincaré***13**(2012) 1873 [arXiv:1105.2012] [INSPIRE]. - [14]W.G. Brown,
*Enumeration of triangulations of the disk*,*Proc. Lond. Math. Soc.***s3-14**(1964) 746.Google Scholar - [15]L. Chekhov, B. Eynard and N. Orantin,
*Free energy topological expansion for the 2-matrix model*,*JHEP***12**(2006) 053 [math-ph/0603003] [INSPIRE]. - [16]P. Ciosmak, L. Hadasz, M. Manabe and P. Su-lkowski,
*Super-quantum curves from super-eigenvalue models*,*JHEP***10**(2016) 044 [arXiv:1608.02596] [INSPIRE]. - [17]P. Ciosmak, L. Hadasz, M. Manabe and P. Su-lkowski,
*Singular vector structure of quantum curves*, 2017, arXiv:1711.08031 [INSPIRE]. - [18]P. Ciosmak, L. Hadasz, Z. Jaskólski, M. Manabe and P. Sulkowski,
*From CFT to Ramond super-quantum curves*,*JHEP***05**(2018) 133 [arXiv:1712.07354] [INSPIRE]. - [19]N. Do and P. Norbury,
*Topological recursion on the Bessel curve*,*Commun. Num. Theor. Phys.***12**(2018) 53 [arXiv:1608.02781] [INSPIRE]. - [20]R. Donagi and E. Witten,
*Supermoduli Space Is Not Projected*,*Proc. Symp. Pure Math.***90**(2015) 19 [arXiv:1304.7798] [INSPIRE]. - [21]P. Dunin-Barkowski, N. Orantin, S. Shadrin and L. Spitz,
*Identification of the Givental formula with the spectral curve topological recursion procedure*,*Commun. Math. Phys.***328**(2014) 669 [arXiv:1211.4021] [INSPIRE]. - [22]B. Eynard,
*Counting Surfaces*, Birkhäuser (2016).Google Scholar - [23]B. Eynard, M. Mulase and B. Safnuk,
*The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers*,*Publ. Res. Inst. Math. Sci. Kyoto***47**(2011) 629 [arXiv:0907.5224] [INSPIRE]. - [24]B. Eynard and N. Orantin,
*Invariants of algebraic curves and topological expansion*,*Commun. Num. Theor. Phys.***1**(2007) 347 [math-ph/0702045] [INSPIRE]. - [25]B. Eynard and N. Orantin,
*Algebraic methods in random matrices and enumerative geometry*, arXiv:0811.3531 [INSPIRE]. - [26]B. Eynard and N. Orantin,
*Computation of Open Gromov-Witten Invariants for Toric Calabi-Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture*,*Commun. Math. Phys.***337**(2015) 483 [arXiv:1205.1103] [INSPIRE]. - [27]B. Fang, C.-C.M. Liu and Z. Zong,
*On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds*, arXiv:1604.07123 [INSPIRE]. - [28]B. Fang, C.C.M. Liu and Z. Zong,
*The SYZ mirror symmetry and the BKMP remodeling conjecture*, arXiv:1607.06935. - [29]J. Gu, H. Jockers, A. Klemm and M. Soroush,
*Knot Invariants from Topological Recursion on Augmentation Varieties*,*Commun. Math. Phys.***336**(2015) 987 [arXiv:1401.5095] [INSPIRE]. - [30]P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2
*-D Gravity and random matrices*,*Phys. Rept.***254**(1995) 1 [hep-th/9306153] [INSPIRE]. - [31]M. Kontsevich and Y. Soibelman,
*Airy structures and symplectic geometry of topological recursion*, arXiv:1701.09137 [INSPIRE]. - [32]J.P. Kroll,
*Topological Recursion and the Supereigenvalue Model*, MSc Thesis in Mathematical Physics, University of Alberta (2012).Google Scholar - [33]M. Mariño,
*Open string amplitudes and large order behavior in topological string theory*,*JHEP***03**(2008) 060 [hep-th/0612127] [INSPIRE]. - [34]I.N. McArthur,
*The Partition function for the supersymmetric Eigenvalue model*,*Mod. Phys. Lett.***A 8**(1993) 3355 [INSPIRE]. - [35]J.C. Plefka,
*Iterative solution of the supereigenvalue model*,*Nucl. Phys.***B 444**(1995) 333 [hep-th/9501120] [INSPIRE]. - [36]J.C. Plefka,
*The Supereigenvalue model in the double scaling limit*,*Nucl. Phys.***B 448**(1995) 355 [hep-th/9504089] [INSPIRE]. - [37]
- [38]J.M. Rabin and P.G.O. Freund,
*Supertori are algebraic curves*,*Commun. Math. Phys.***114**(1988) 131 [INSPIRE]. - [39]D. Stanford and E. Witten,
*JT Gravity and the Ensembles of Random Matrix Theory*, arXiv:1907.03363 [INSPIRE]. - [40]
- [41]
- [42]
- [43]E. Witten,
*More On Superstring Perturbation Theory: An Overview Of Superstring Perturbation Theory Via Super Riemann Surfaces*, arXiv:1304.2832 [INSPIRE].