The c = 1 string theory S-matrix revisited
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Springer
Received: December 15, Revised: April 8, Accepted: April 17, Published: April 24,
2018 2019 2019 2019
Bruno Balthazar, Victor A. Rodriguez and Xi Yin Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We revisit the perturbative S-matrix of c = 1 string theory from the worldsheet perspective. We clarify the origin of the leg pole factors, the non-analyticity of the string amplitudes, and the validity as well as limitations of earlier computations based on resonance momenta. We compute the tree level 4-point amplitude and the genus one 2point reflection amplitude by numerically integrating Virasoro conformal blocks with DOZZ structure constants on the sphere and on the torus, with sufficiently generic complex Liouville momenta, and find agreement with known answers from the c = 1 matrix model. Keywords: Bosonic Strings, M(atrix) Theories, Matrix Models, String Duality ArXiv ePrint: 1705.07151
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)145
JHEP04(2019)145
The c = 1 string theory S-matrix revisited
Contents 1 Introduction
1
2 The structure of the perturbative S-matrix in c = 1 string theory 2.1 The worldsheet theory 2.2 The dual matrix model
4 4 6 6 6 7 8 10 11 11
4 Genus one 2-point reflection amplitude 4.1 Limits of the moduli integral 4.2 Torus conformal blocks 4.3 Torus 2-point amplitude at resonance momenta 4.4 Numerical evaluation at generic imaginary momenta
13 15 16 16 18
5 Discussion
19
A Torus 2-point conformal blocks
21
B Further details of the numerics B.1 On the sphere 4-point amplitude B.2 On the torus 2-point amplitude
22 22 23
1
Introduction
The two-dimensional noncritical “c = 1” string theory has been an invaluable source of inspiration in the exploration of string dualities and quantum gravity for nearly three decades (for reviews, see [1–5]). The perturbative excitations of the c = 1 string theory are massless bosons in 1 + 1 dimensions that interact and scatter off a Liouville wall. The full perturbative S-matrix (as well as its non-perturbative completion) was solved from the conjectured dual matrix quantum mechanics (the “c = 1 matrix model”) in [6]. The string theory side of the story has been much murkier. The tree level string S-matrix was computed by an analytic continuation from “resonance momenta” in [7, 8], where the relevant Liouville correlators were computed based on Coulomb gas integrals, and the result for the 1 → n
–1–
JHEP04(2019)145
3 Tree level 4-point string amplitude from Liouville correlator 3.1 The 1 → 3 amplitude 3.1.1 Analytic continuation of Liouville correlator 3.1.2 Regularization of moduli integral 3.2 The 2 → 2 amplitude and crossing symmetry 3.2.1 Resonance momenta 3.3 Numerical results
i=1
where ω is the energy of the incoming massless particle, ω1 , · · · , ωn are the energies of the outgoing particles (after reflection from the Liouville wall). It is subject
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