Abstract
There are two alternative approaches to the minimal gravity — direct Liouville approach and matrix models. Recently there has been a certain progress in the matrix model approach, growing out of presence of a Frobenius manifold (FM) structure embedded in the theory. The previous studies were mainly focused on the spherical topology. Essentially, it was shown that the action principle of Douglas equation allows to define the free energy and to compute the correlation numbers if the resonance transformations are properly incorporated. The FM structure allows to find the explicit form of the resonance transformation as well as the closed expression for the partition function. In this paper we elaborate on the case of gravitating disk. We focus on the bulk correlators and show that in the similar way as in the closed topology the generating function can be formulated using the set of flat coordinates on the corresponding FM. Moreover, the resonance transformations, which follow from the spherical topology consideration, are exactly those needed to reproduce FZZ result of the Liouville gravity approach.
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References
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
H. Dorn and H.-J. Otto, On correlation functions for noncritical strings with c ≤ 1 but d ≥ 1, Phys. Lett. B 291 (1992) 39 [hep-th/9206053] [INSPIRE].
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].
F. David, A. Kupiainen, R. Rhodes and V. Vargas, Liouville Quantum Gravity on the Riemann sphere, Commun. Math. Phys. 342 (2016) 869 [arXiv:1410.7318] [INSPIRE].
A. Kupiainen, R. Rhodes and V. Vargas, Integrability of Liouville theory: proof of the DOZZ Formula, arXiv:1707.08785 [INSPIRE].
A.A. Belavin and A.B. Zamolodchikov, Integrals over moduli spaces, ground ring and four-point function in minimal Liouville gravity, Theor. Math. Phys. 147 (2006) 729 [INSPIRE].
K. Aleshkin and V. Belavin, On the construction of the correlation numbers in Minimal Liouville Gravity, JHEP 11 (2016) 142 [arXiv:1610.01558] [INSPIRE].
V.A. Kazakov, A.A. Migdal and I.K. Kostov, Critical Properties of Randomly Triangulated Planar Random Surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE].
V.A. Kazakov, Ising model on a dynamical planar random lattice: Exact solution, Phys. Lett. A 119 (1986) 140 [INSPIRE].
V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum Gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
M.R. Douglas, Strings in Less Than One-dimension and the Generalized KdV Hierarchies, Phys. Lett. B 238 (1990) 176 [INSPIRE].
E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom. 1 (1991) 243 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity, Nucl. Phys. B 348 (1991) 435 [INSPIRE].
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [INSPIRE].
V.A. Kazakov, The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity, Mod. Phys. Lett. A 4 (1989) 2125 [INSPIRE].
M. Staudacher, The Yang-Lee Edge Singularity on a Dynamical Planar Random Surface, Nucl. Phys. B 336 (1990) 349 [INSPIRE].
M.R. Douglas and S.H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys. B 335 (1990) 635 [INSPIRE].
D.J. Gross and A.A. Migdal, Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 64 (1990) 127 [INSPIRE].
E. Brézin and V.A. Kazakov, Exactly Solvable Field Theories of Closed Strings, Phys. Lett. B 236 (1990) 144 [INSPIRE].
G.W. Moore, N. Seiberg and M. Staudacher, From loops to states in two-dimensional quantum gravity, Nucl. Phys. B 362 (1991) 665 [INSPIRE].
A.B. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
A.A. Belavin and A.B. Zamolodchikov, Integrals over moduli spaces, ground ring and four-point function in minimal Liouville gravity, Theor. Math. Phys. 147 (2006) 729 [INSPIRE].
A.A. Belavin and A.B. Zamolodchikov, On Correlation Numbers in 2D Minimal Gravity and Matrix Models, J. Phys. A 42 (2009) 304004 [arXiv:0811.0450] [INSPIRE].
A. Belavin, B. Dubrovin and B. Mukhametzhanov, Minimal Liouville Gravity correlation numbers from Douglas string equation, JHEP 01 (2014) 156 [arXiv:1310.5659] [INSPIRE].
V. Belavin, Unitary Minimal Liouville Gravity and Frobenius Manifolds, JHEP 07 (2014) 129 [arXiv:1405.4468] [INSPIRE].
A.A. Belavin and V.A. Belavin, Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity, JHEP 09 (2014) 151 [arXiv:1406.6661] [INSPIRE].
V. Belavin, Correlation Functions in Unitary Minimal Liouville Gravity and Frobenius Manifolds, JHEP 02 (2015) 052 [arXiv:1412.4245] [INSPIRE].
V. Belavin and Y. Rud, Matrix model approach to minimal Liouville gravity revisited, J. Phys. A 48 (2015) 18FT01 [arXiv:1502.05575] [INSPIRE].
B. Dubrovin, Integrable systems in topological field theory, Nucl. Phys. B 379 (1992) 627 [INSPIRE].
G. Tarnopolsky, Five-point Correlation Numbers in One-Matrix Model, J. Phys. A 44 (2011) 325401 [arXiv:0912.4971] [INSPIRE].
A. Belavin, M. Bershtein and G. Tarnopolsky, A remark on the three approaches to 2D Quantum gravity, JETP Lett. 93 (2011) 47 [arXiv:1010.2222] [INSPIRE].
V. Belavin, Torus Amplitudes in Minimal Liouville Gravity and Matrix Models, Phys. Lett. B 698 (2011) 86 [arXiv:1010.5508] [INSPIRE].
L. Spodyneiko, Minimal Liouville gravity on the torus via the Douglas string equation, J. Phys. A 48 (2015) 065401 [INSPIRE].
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
J.L. Cardy, Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
I. Runkel, Boundary structure constants for the A-series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].
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].
B. Ponsot and J. Teschner, Boundary Liouville field theory: Boundary three point function, Nucl. Phys. B 622 (2002) 309 [hep-th/0110244] [INSPIRE].
I.K. Kostov, B. Ponsot and D. Serban, Boundary Liouville theory and 2D quantum gravity, Nucl. Phys. B 683 (2004) 309 [hep-th/0307189] [INSPIRE].
J.-E. Bourgine, K. Hosomichi and I.K. Kostov, Boundary transitions of the O(n) model on a dynamical lattice, Nucl. Phys. B 832 (2010) 462 [arXiv:0910.1581] [INSPIRE].
J.-E. Bourgine and K. Hosomichi, Boundary operators in the O(n) and RSOS matrix models, JHEP 01 (2009) 009 [arXiv:0811.3252] [INSPIRE].
I.K. Kostov, Boundary correlators in 2D quantum gravity: Liouville versus discrete approach, Nucl. Phys. B 658 (2003) 397 [hep-th/0212194] [INSPIRE].
J.L. Jacobsen and H. Saleur, Conformal boundary loop models, Nucl. Phys. B 788 (2008) 137 [math-ph/0611078] [INSPIRE].
G. Ishiki and C. Rim, Boundary correlation numbers in one matrix model, Phys. Lett. B 694 (2011) 272 [arXiv:1006.3906] [INSPIRE].
E.J. Martinec, G.W. Moore and N. Seiberg, Boundary operators in 2D gravity, Phys. Lett. B 263 (1991) 190 [INSPIRE].
K. Hosomichi, Minimal Open Strings, JHEP 06 (2008) 029 [arXiv:0804.4721] [INSPIRE].
A. Belavin and C. Rim, Bulk one-point function on disk in one-matrix model, Phys. Lett. B 687 (2010) 264 [arXiv:1001.4356] [INSPIRE].
A.A. Belavin and A.B. Zamolodchikov, On Correlation Numbers in 2D Minimal Gravity and Matrix Models, J. Phys. A 42 (2009) 304004 [arXiv:0811.0450] [INSPIRE].
J.-E. Bourgine, G. Ishiki and C. Rim, Bulk-boundary correlators in the hermitian matrix model and minimal Liouville gravity, Nucl. Phys. B 854 (2012) 853 [arXiv:1107.4186] [INSPIRE].
P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, in proceedings of the Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, Boulder, Colorado, U.S.A., 3-28 June 1992, pp. 277-469 [hep-th/9304011] [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
P.H. Ginsparg, M. Goulian, M.R. Plesser and J. Zinn-Justin, (p, q) string actions, Nucl. Phys. B 342 (1990) 539 [INSPIRE].
N. Seiberg and D. Shih, Branes, rings and matrix models in minimal (super)string theory, JHEP 02 (2004) 021 [hep-th/0312170] [INSPIRE].
P. Di Francesco and D. Kutasov, Integrable Models of Two Dimensional Quantum Gravity, Springer, Boston Massachusetts U.S.A. (1991), pp. 35-51.
R. Pandharipande, J.P. Solomon and R.J. Tessler, Intersection theory on moduli of disks, open KdV and Virasoro, arXiv:1409.2191 [INSPIRE].
A. Buryak, Equivalence of the open KdV and the open Virasoro equations for the moduli space of Riemann surfaces with boundary, Lett. Math. Phys. 105 (2015) 1427 [arXiv:1409.3888] [INSPIRE].
A. Buryak, Open intersection numbers and the wave function of the KdV hierarchy, Moscow Math. J. 16 (2016) 27 [arXiv:1409.7957] [INSPIRE].
M. Bertola and D. Yang, The partition function of the extended r-reduced Kadomtsev-Petviashvili hierarchy, J. Phys. A 48 (2015) 195205 [arXiv:1411.5717] [INSPIRE].
B. Balthazar, V.A. Rodriguez and X. Yin, The c = 1 String Theory S-matrix Revisited, arXiv:1705.07151 [INSPIRE].
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ArXiv ePrint: 1708.06380
Weston Visiting Professorship at Weizmann Institute. On leave from Lebedev Physical Institute, Moscow. (Vladimir Belavin)
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Aleshkin, K., Belavin, V. & Rim, C. Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk. J. High Energ. Phys. 2017, 169 (2017). https://doi.org/10.1007/JHEP11(2017)169
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DOI: https://doi.org/10.1007/JHEP11(2017)169