# Long string scattering in *c* = 1 string theory

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## Abstract

We study the scattering of long strings in *c* = 1 string theory, both in the worldsheet description and in the non-singlet sector of the dual matrix quantum mechanics. From the worldsheet perspective, the scattering amplitudes of long strings are obtained from a decoupling limit of open strings amplitudes on FZZT branes, which we compute by integrating Virasoro conformal blocks along with structure constants of boundary Liouville theory. In particular, we study the tree level amplitudes of (1) a long string decaying by emitting a closed string, and (2) the scattering of a pair of long strings. We show that they are indeed well defined as limits of open string amplitudes, and that our results are in striking numerical agreement with computations in the adjoint and bi-adjoint sectors of the dual matrix model (based on proposals of Maldacena and solutions due to Fidkowski), thereby providing strong evidence of the duality.

## Keywords

Bosonic Strings D-branes Long strings M(atrix) Theories## Notes

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## References

- [1]I.R. Klebanov,
*String theory in two-dimensions*, talk given at the*Spring School on String Theory and Quantum Gravity*, April 15-23, Trieste, Italy (1991), hep-th/9108019 [INSPIRE]. - [2]P.H. Ginsparg and G.W. Moore,
*Lectures on*2*D gravity and*2*D string theory*, in the proceedings of the*Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles*, June 1-26, Boulder, U.S.A. (1992), hep-th/9304011 [INSPIRE]. - [3]A. Jevicki,
*Development in*2−*string theory*, talk given at the*Workshop on String Theory, Gauge Theory and Quantum Gravity*, April 28-29, Trieste, Italy (1993), hep-th/9309115 [INSPIRE]. - [4]J. Polchinski,
*What is string theory?*, talk given at the*NATO Advanced Study Institute: Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory*, August 2-September 9, Les Houches, France (1994), hep-th/9411028 [INSPIRE]. - [5]E.J. Martinec,
*Matrix models and*2*D string theory*, talk given at the 9^{th}*Frontiers of Mathematical Physics Summer School on Strings, Gravity and Cosmology*, August 2-13, Vancouver, Canada (2004), hep-th/0410136 [INSPIRE]. - [6]H. Dorn and H.J. Otto,
*Two and three point functions in Liouville theory*,*Nucl. Phys.***B 429**(1994) 375 [hep-th/9403141] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [7]A.B. Zamolodchikov and A.B. Zamolodchikov,
*Structure constants and conformal bootstrap in Liouville field theory*,*Nucl. Phys.***B 477**(1996) 577 [hep-th/9506136] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [8]J. Teschner,
*On the Liouville three point function*,*Phys. Lett.***B 363**(1995) 65 [hep-th/9507109] [INSPIRE]. - [9]J. Teschner,
*Liouville theory revisited*,*Class. Quant. Grav.***18**(2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [10]B. Balthazar, V.A. Rodriguez and X. Yin,
*The c*= 1*string theory S-matrix revisited*, arXiv:1705.07151 [INSPIRE]. - [11]A.B. Zamolodchikov and A.B. Zamolodchikov,
*Liouville field theory on a pseudosphere*, hep-th/0101152 [INSPIRE]. - [12]V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov,
*Boundary Liouville field theory. 1. Boundary state and boundary two point function*, hep-th/0001012 [INSPIRE]. - [13]J. Teschner,
*Remarks on Liouville theory with boundary*,*PoS***tmr2000**(2000) 041 [hep-th/0009138] [INSPIRE]. - [14]J. McGreevy and H.L. Verlinde,
*Strings from tachyons: the c*= 1*matrix reloaded*,*JHEP***12**(2003) 054 [hep-th/0304224] [INSPIRE]. - [15]
- [16]J.M. Maldacena,
*Long strings in two dimensional string theory and non-singlets in the matrix model*,*JHEP***09**(2005) 078 [hep-th/0503112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [17]L. Fidkowski,
*Solving the eigenvalue problem arising from the adjoint sector of the c*= 1*matrix model*, hep-th/0506132 [INSPIRE]. - [18]B. Ponsot and J. Teschner,
*Boundary Liouville field theory: Boundary three point function*,*Nucl. Phys.***B 622**(2002) 309 [hep-th/0110244] [INSPIRE]. - [19]B. Ponsot and J. Teschner,
*Structure constant of boundary operators in Liouville field theory*, in*From integrable models to gauge theories: a volume in honor of Sergei Matinyan*, V.G. Gurzadyan and A.G. Sedrakian eds., World Scientific, Singapore (2002).Google Scholar - [20]K. Hosomichi,
*Bulk boundary propagator in Liouville theory on a disc*,*JHEP***11**(2001) 044 [hep-th/0108093] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [21]C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,
*Little string amplitudes (and the unreasonable effectiveness of*6*D SYM)*,*JHEP***12**(2014) 176 [arXiv:1407.7511] [INSPIRE]. - [22]A.B. Zamolodchikov,
*Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude*,*Commun. Math. Phys.***96**(1984) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [23]P. Di Francesco and D. Kutasov,
*Correlation functions in*2*D string theory*,*Phys. Lett.***B 261**(1991) 385 [INSPIRE]. - [24]P. Di Francesco and D. Kutasov,
*World sheet and space-time physics in two-dimensional (Super)string theory*,*Nucl. Phys.***B 375**(1992) 119 [hep-th/9109005] [INSPIRE]. - [25]J.L. Karczmarek,
*Scattering in the adjoint sector of the c*= 1*matrix model*,*JHEP***02**(2009) 011 [arXiv:0809.3543] [INSPIRE]. - [26]J.E. Bourgine, K. Hosomichi, I. Kostov and Y. Matsuo,
*Scattering of long folded strings and mixed correlators in the two-matrix model*,*Nucl. Phys.***B 795**(2008) 243 [arXiv:0709.3912] [INSPIRE]. - [27]K. Demeterfi, A. Jevicki and J.P. Rodrigues,
*Scattering amplitudes and loop corrections in collective string field theory*,*Nucl. Phys.***B 362**(1991) 173 [INSPIRE]. - [28]K. Demeterfi, A. Jevicki and J.P. Rodrigues,
*Scattering amplitudes and loop corrections in collective string field theory. 2.*,*Nucl. Phys.***B 365**(1991) 499 [INSPIRE]. - [29]G. Marchesini and E. Onofri,
*Planar limit for*SU(*N*)*symmetric quantum dynamical systems*,*J. Math. Phys.***21**(1980) 1103 [INSPIRE]. - [30]A. Donos, A. Jevicki and J.P. Rodrigues,
*Matrix model maps in AdS/CFT*,*Phys. Rev.***D 72**(2005) 125009 [hep-th/0507124] [INSPIRE]. - [31]P. Betzios and O. Papadoulaki,
*FZZT branes and non-singlets of matrix quantum mechanics*, arXiv:1711.04369 [INSPIRE].