# Gas of Baby Universes in JT Gravity and Matrix Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generating Functional in Matrix Models

**Generating functional.**We consider ensemble of $N\times N$ Hermitian matrices [16,17,18,19] with potential $V\left(M\right)$. Let

**Particular case.**We consider a special case

**Double scaling limit.**This limit in matrix models has been introduced in [23], see [24,25,26] for review and refs therein. All the correlation functions ${\mathcal{Z}}_{n}^{matrix}({\beta}_{1},\cdots ,{\beta}_{n})$ in principle could be derived if the potential $V\left(x\right)$ or the spectral density/spectral curve $\rho \left(\mu \right)$ is known. To get a connection of the matrix model with JT gravity one has to go to the double scaling limit, see [1]. Consider a matrix model with a non-normalized spectral density

**Resolvents.**Similarly one has generating functional for correlation functions of resolvents

## 3. Generating Functional in JT Gravity

**Baby universes.**In cosmology [30,31,32,33,34,35,36,37,38], one usually deals with baby universes that branch off from, or join onto, the parent(s) Universe(s). In matrix theories one parent is a connected Riemann surface with arbitrary number of handles and at least one boundary. We assume that the lengths of boundaries of baby universities are small as compare with the boundary length of the parent, see Figure 1. Baby universes are attached to the parent by necks that have restricted lengths of geodesics at which the neck is attached to the parent, We assume that the lengths of the boundaries of baby universes are small compared to the length of the boundary of the parent, see Figure 1. Baby universes are attached to the parent with the help of thin necks. Thickness of the neck is defined as the geodesic length of the loop located at the thinness point of the neck, and this length is assumed to be essentially smaller than the length of theboundary of the parent, see Figure 1b,c. There are also restrictions on the area of the surface of baby universes, see [39,40] for more precise definitions. Cosmological baby universes in the parent-baby universe approximation interact only via coupling to the parent universes, that themselves interact via wormholes. In matrix models the baby universes always interact via their parents too and parents interact via wormholes, Figure 1b,c). One can expect that at large number of baby universes interaction between different parts of the system increases and this leads to phase transition (an analog of the the nucleation of a baby universe in [32]). We interpret the matrix partition function $\mathfrak{Z}$, defined by equation (22) as a partition function of the gas of baby universes.

## 4. Double Scaling Limit for the GUE

## 5. Deformation by an Exponential Potential

**Fine tuning.**It is obvious that fixing from the beginning the location of the eigenvalue one immediately gives restrictions on parameters of the potential of the matrix model. If we want to shift the location of the eigenvalues, ${S}_{[a,b]}\to {S}_{[0,b-a]}$ we have to make a shift in the potential, $V\left(x\right)\to V(x+a)$. For the quadratic potential this shift produces the linear term $\mathsf{\Delta}V\left(x\right)=jx$ and j can be determined from the location of the left point of the cut. For higher polynomial interaction the shift produces the linear term in the LHS of singular equation, as well as change of coupling constants. The shift in the exponential potential produces just a multiplication on positive constant.

## 6. Matrix Model for JT Gravity

#### 6.1. Potentials for Non-Normalized Density Distribution

#### 6.2. Effective Energy

#### 6.3. Phase Transition

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Saad, P.; Shenker, S.H.; Stanford, D. JT gravity as a matrix integral. arXiv
**2019**, arXiv:1903.11115. [Google Scholar] - Jackiw, R. Lower Dimensional Gravity. Nucl. Phys. B
**1985**, 252, 343–356. [Google Scholar] [CrossRef] - Teitelboim, C. Gravitation and Hamiltonian Structure in Two Space-Time Dimensions. Phys. Lett. B
**1983**, 126, 41–45. [Google Scholar] [CrossRef] - Mirzakhani, M. Growth of Weil-Petersson volumes and random hyperbolic surface of large genus. J. Differ. Geom.
**2013**, 94, 267–300. [Google Scholar] [CrossRef] - Eynard, B.; Orantin, N. Invariants of algebraic curves and topological expansion. arXiv
**2007**, arXiv:math-ph/0702045. [Google Scholar] [CrossRef] [Green Version] - Eynard, B.; Orantin, N. Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models. arXiv
**2007**, arXiv:0705.3600. [Google Scholar] - Witten, E. Two-dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom.
**1991**, 1, 243–310. [Google Scholar] - Kontsevich, M. Intersection theory on the moduli space of curves and the matrix airy function. Commun. Math. Phys.
**1992**, 147, 1–23. [Google Scholar] [CrossRef] - Manin, Y.I.; Zograf, P. Invertible Cohomological Field Theories and Weil-Petersson volumes. arXiv
**1999**, arXiv:math/9902051. [Google Scholar] [CrossRef] [Green Version] - Dijkgraaf, R.; Witten, E. Developments in Topological Gravity. Int. J. Mod. Phys. A
**2018**, 33, 1830029. [Google Scholar] [CrossRef] [Green Version] - Cotler, J.S.; Gur-Ari, G.; Hanada, M.; Polchinski, J.; Saad, P.; Shenker, S.H.; Stanford, D.; Streicher, A.; Tezuka, M. Black Holes and Random Matrices. J. High Energy Phys.
**2017**, 2017, 118. [Google Scholar] [CrossRef] - Saad, P.; Shenker, S.H.; Stanford, D. A semiclassical ramp in SYK and in gravity. arXiv
**2018**, arXiv:1806.06840. [Google Scholar] - Green, M.B.; Schwarz, J.H.; Witten, E. Superstring Theory; Cambridge University Press: Cambridge, UK, 1987; 469p. [Google Scholar]
- Witten, E. Noncommutative Geometry and String Field Theory. Nucl. Phys. B
**1986**, 268, 253–294. [Google Scholar] [CrossRef] - Aref’eva, I.Y.; Volovich, I.V. Two-dimensional gravity, string field theory and spin glasses. Phys. Lett.
**1991**, 255, 197–201. [Google Scholar] [CrossRef] - Wigner, E.P. On the statistical distribution of the widths and spacings of nuclear resonance levels. Proc. Cambr. Philos. Soc.
**1951**, 47, 790–798. [Google Scholar] [CrossRef] - Dyson, F.J. A Class of Matrix Ensembles. J. Math. Phys.
**1972**, 13, 90–97. [Google Scholar] [CrossRef] - Brezin, E.; Itzykson, C.; Parisi, G.; Zuber, J.-B. Planar diagrams. Commun. Math. Phys.
**1978**, 50, 35–51. [Google Scholar] [CrossRef] - Mehta, M.L. Random Matrices, 2nd ed.; Academic Press: New York, NY, USA, 1991. [Google Scholar]
- Migdal, A. Loop equations and 1/N expansion. Phys. Rep.
**1983**, 102, 199. [Google Scholar] [CrossRef] - Muskhelishvili, N.I. Singular Integral Equations; Noordhoff: Groningen, The Netherlands, 1953. [Google Scholar]
- Gakhov, F.D. Boundary Problems; Fizmatgiz: Moscow, Russia, 1977. (In Russian) [Google Scholar]
- Brézin, E.; Kazakov, V.A. Exactly Solvable Field Theories of Closed Strings. Phys. Lett. B
**1990**, 236, 144. [Google Scholar] [CrossRef] - di Francesco, P.; Ginsparg, P.; Zinn-Justin, J. 2-D Gravity and random matrices. Phys. Rep.
**1995**, 254, 1–133. [Google Scholar] [CrossRef] [Green Version] - Marino, M. Les Houches lectures on matrix models and topological strings. arXiv
**2004**, arXiv:hep-th/0410165. [Google Scholar] - Eynard, B.; Kimura, T.; Ribault, S. Random matrices. arXiv
**2004**, arXiv:1510.04430. [Google Scholar] - Ambjorn, J.; Jurkiewicz, J.; Makeenko, Y.M. Multiloop correlators for two-dimensional quantum gravity. Phys. Lett. B
**1990**, 251, 517–524. [Google Scholar] [CrossRef] - Eynard, B. Topological expansion for the 1-Hermitian matrix model correlation functions. J. High Energy Phys.
**2004**, 2004, 31. [Google Scholar] [CrossRef] [Green Version] - Almheiri, A.; Polchinski, J. Models of AdS
_{2}backreaction and holography. J. High Energy Phys.**2015**, 2015, 14. [Google Scholar] [CrossRef] [Green Version] - Hawking, S.W.; Laflamme, R. Baby universes and the non-renormalizability of gravity. Phys. Lett.
**1988**, 209, 39–41. [Google Scholar] [CrossRef] - Lavrelashvili, G.V.; Rubakov, V.A.; Tinyakov, P.G. Disruption of Quantum Coherence upon a Change in Spatial Topology in Quantum Gravity. JETP Lett.
**1987**, 46, 167. [Google Scholar] - Giddings, S.B.; Strominger, A. Axion Induced Topology Change in Quantum Gravity and String Theory. Nucl. Phys. B
**1988**, 306, 890–907. [Google Scholar] [CrossRef] - Strominger, A. Baby Universes. In Quantum Cosmology and Baby Universes; World Scientific: Singapore, 1991; pp. 269–346. [Google Scholar]
- Hawking, S.W. Quantum coherence down the wormhole. Phys. Lett. B.
**1987**, 195, 337–343. [Google Scholar] [CrossRef] - Giddings, S.B.; Strominger, A. Baby Universes, Third Quantization and the Cosmological Constant. Nucl. Phys. B
**1989**, 321, 481–508. [Google Scholar] [CrossRef] - Coleman, S.R. Why There Is Nothing Rather than Something: A Theory of the Cosmological Constant. Nucl. Phys. B
**1988**, 310, 643–668. [Google Scholar] [CrossRef] - Volovich, I.V. Baby universes and the dimensionality of spacetime. Phys. Lett. B
**1989**, 219, 66–70. [Google Scholar] [CrossRef] - Hebecker, A.; Mikhail, T.; Soler, P. Euclidean wormholes, baby universes, and their impact on particle physics and cosmology. Front. Astron. Space Sci.
**2018**, 5, 35. [Google Scholar] [CrossRef] [Green Version] - Jain, S.; Mathur, S.D. World sheet geometry and baby universes in 2-D quantum gravity. Phys. Lett. B
**1992**, 286, 239–246. [Google Scholar] [CrossRef] [Green Version] - Ambjorn, J.; Barkley, J.; Budd, T.; Loll, R. Baby Universes Revisited. Phys. Lett. B
**2011**, 706, 86–89. [Google Scholar] [CrossRef] [Green Version] - Douglas, M.R.; Shenker, S.H. Strings in Less than One-Dimension. Nucl. Phys. B
**1990**, 335, 635–654. [Google Scholar] [CrossRef] - Gross, D.J.; Migdal, A.A. Nonperturbative Two-Dimensional Quantum Gravity. Phys. Rev. Lett.
**1990**, 64, 127. [Google Scholar] [CrossRef] - Brezin, E.; Zee, A. Universality of the correlations between eigenvalues of large random matrices. Nucl. Phys. B
**1993**, 402, 613–627. [Google Scholar] [CrossRef] - Bowick, M.; Brezin, E. Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B
**1991**, 268, 21. [Google Scholar] [CrossRef] - Bleher, P.M.; Its, A.R. Double scaling limit in the random matrix model: The Riemann-Hilbert approach. Commun. Pure Appl. Math.
**2003**, 56, 433–516. [Google Scholar] [CrossRef] [Green Version] - De Monvel, A.B.; Pastur, L.; Shcherbina, M. On the statistical mechanics approach in the random matrix theory: Integrated density of state. J. Stat. Phys.
**1995**, 79, 585–611. [Google Scholar] [CrossRef] - Pastur, L.; Shcherbina, M. Universality of the Local Eigenvalue Statistics for a Class of Unitary Invariant Random Matrix Ensembles. J. Stat. Phys.
**1997**, 86, 109–147. [Google Scholar] [CrossRef] [Green Version] - Tracy, C.; Widom, H. Level-Spacing Distributions and the Airy Kernel. Commun. Math. Phys.
**1994**, 159, 151–174. [Google Scholar] [CrossRef] [Green Version] - Tracy, C.; Widom, H. Fredholm Determinants, Differential Equations and Matrix Models. Commun. Math. Phys.
**1994**, 163, 33–72. [Google Scholar] [CrossRef] [Green Version] - Kuijlaars, A.B.J.; McLauglin, K.T.-R. Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external field. Commun. Pure Appl. Math.
**2000**, 53, 736–785. [Google Scholar] [CrossRef] - Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math.
**1999**, 52, 1335–1425. [Google Scholar] [CrossRef] - Brezin, E.; Hikami, S. Random Matrix Theory with an External Source; Springer: Singapore, 2016. [Google Scholar]
- Kazakov, V.A. A simple solvable model of quantum field theory of open strings. Phys. Lett. B
**1990**, 237, 212–216. [Google Scholar] [CrossRef] - Arefeva, I.Y.; Ilchev, A.C.; Mitruchkin, B.K. Phase structure of matrix NxN Goldstoune model in the large N limit. In Proceedings of the III International Symposium on Selected Topics in Statistical Mechanics, Dubna, Russia, 22–26 August 1984; pp. 20–26. [Google Scholar]
- Cicuta, G.M.; Molinari, L.; Montaldi, E. Large N phase transition in low dimensions. Mod. Phys. Lett. A
**1986**, 1, 125–129. [Google Scholar] [CrossRef] - Crnkovic, C.; Moore, G. Multicritical multi-cut matrix models. Phys. Lett. B
**1991**, 257, 322–328. [Google Scholar] [CrossRef] - Aref’eva, I.; Volovich, I. Gas of baby universes in JT gravity and matrix models. arXiv
**2019**, arXiv:1905.08207. [Google Scholar] - Bagrets, D.; Altland, A.; Kamenev, A. Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics. Nucl. Phys. B
**2016**, 911, 191–205. [Google Scholar] [CrossRef] [Green Version] - Stanford, D.; Witten, E. Fermionic Localization of the Schwarzian Theory. J. High Energy Phys.
**2017**, 2017, 8. [Google Scholar] [CrossRef] - Bethe, H.A. An Attempt to Calculate the Number Energy Levels of a Heavy Nucleus. Phys. Rev.
**1936**, 50, 332. [Google Scholar] [CrossRef] - Aref’eva, I.; Khramtsov, M.; Tikhanovskaya, M.; Volovich, I. Replica-nondiagonal solutions in the SYK model. J. High Energy Phys.
**2019**, 2019, 113. [Google Scholar] [CrossRef] [Green Version] - Garcia-Garcia, A.M.; Jia, Y.; Verbaarschot, J.J.M. Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N
^{2}. J. High Energy Phys.**2018**, 2018, 146. [Google Scholar] [CrossRef] - Garca-Garca, A.M.; Zacaras, S. Quantum Jackiw-Teitelboim gravity, Selberg trace formula, and random matrix theory. arXiv
**2019**, arXiv:1911.10493. [Google Scholar] - McGuigan, M. Dark Horse, Dark Matter: Revisiting the SO(16)x SO(16)’ Nonsupersymmetric Model in the LHC and Dark Energy Era. arXiv
**2019**, arXiv:1907.01944. [Google Scholar] - Jensen, K. Chaos in AdS
_{2}Holography. Phys. Rev. Lett.**2016**, 117, 111601. [Google Scholar] [CrossRef] [Green Version] - Engelsy, J.; Mertens, T.G.; Verlinde, H. An investigation of AdS
_{2}backreaction and holography. J. High Energy Phys.**2016**, 2016, 139. [Google Scholar] [CrossRef] - Maldacena, J.; Stanford, D.; Yang, Z. Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space. Prog. Theor. Exp. Phys.
**2016**, 2016, 12C104. [Google Scholar] [CrossRef] - Harlow, D.; Jafferis, D. The Factorization Problem in Jackiw-Teitelboim Gravity. J. High Energy Phys.
**2020**, 2020, 1–32. [Google Scholar] [CrossRef] [Green Version] - Aref’eva, I.; Volovich, I. Notes on the SYK model in real time. Theor. Math. Phys.
**2018**, 197, 1650–1662. [Google Scholar] [CrossRef] [Green Version] - Aref’eva, I.; Volovich, I. Spontaneous symmetry breaking in fermionic random matrix model. J. High Energy Phys.
**2019**, 2019, 114. [Google Scholar] [CrossRef] [Green Version] - Okuyama, K. Replica symmetry breaking in random matrix model: A toy model of wormhole networks. Phys. Lett. B
**2020**, 803, 135280. [Google Scholar] [CrossRef] - Wang, H.; Bagrets, D.; Chudnovskiy, A.L.; Kamenev, A. On the replica structure of Sachdev-Ye-Kitaev model. J. High Energy Phys.
**2019**, 2019, 57. [Google Scholar] [CrossRef] [Green Version] - Aref’eva, I.; Khramtsov, M.; Volovich, I. Revealing nonperturbative effects in the SYK model. Theor. Math. Phys.
**2019**, 201, 1585–1605. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J.; Qi, X.-L. Eternal traversable wormhole. arXiv
**2018**, arXiv:1804.00491. [Google Scholar] - Kim, J.; Klebanov, I.R.; Tarnopolsky, G.; Zhao, W. Symmetry Breaking in Coupled SYK or Tensor Models. Phys. Rev. X
**2019**, 9, 021043. [Google Scholar] [CrossRef] [Green Version] - Garcia-Garcia, A.M.; Nosaka, T.; Rosa, D.; Verbaarschot, J.J.M. Quantum chaos transition in a two-site SYK model dual to an eternal traversable wormhole. Phys. Rev. D
**2019**, 100, 026002. [Google Scholar] [CrossRef] [Green Version] - Aref’eva, I.Y.; Volovich, I.V. Cosmological daemon. J. High Energy Phys.
**2011**, 2011, 102. [Google Scholar] [CrossRef] [Green Version] - Coleman, S.R.; Lee, K. Escape From the Menace of the Giant Wormholes. Phys. Lett. B
**1989**, 221, 242–249. [Google Scholar] [CrossRef] - Maldacena, J.; Turiaci, G.J.; Yang, Z. Two dimensional Nearly de Sitter gravity. arXiv
**2019**, arXiv:1904.01911. [Google Scholar] - Lin, H.W.; Maldacena, J.; Zhao, Y. Symmetries Near the Horizon. J. High Energy Phys.
**2019**, 2019, 49. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Zhang, P. Entanglement Entropy of Two Coupled SYK Models and Eternal Traversable Wormhole. J. High Energy Phys.
**2019**, 2019, 33. [Google Scholar] [CrossRef] [Green Version] - Freivogel, B.; Godet, V.; Morvan, E.; Pedraza, J.F.; Rotundo, A. Lessons on Eternal Traversable Wormholes in AdS. J. High Energy Phys.
**2019**, 2019, 122. [Google Scholar] [CrossRef] [Green Version] - Betzios, P.; Kiritsis, E.; Papadoulaki, O. Euclidean Wormholes and Holography. J. High Energy Phys.
**2019**, 2019, 42. [Google Scholar] [CrossRef] [Green Version] - Cotler, J.; Jensen, K.; Maloney, A. Low-dimensional de Sitter quantum gravity. arXiv
**2019**, arXiv:1905.03780. [Google Scholar] - Dijkgraaf, R.; Gopakumar, R.; Ooguri, H.; Vafa, C. Baby universes in string theory. Phys. Rev. D
**2006**, 73, 066002. [Google Scholar] [CrossRef] [Green Version] - Blommaert, A.; Mertens, T.G.; Verschelde, H. Fine Structure of Jackiw-Teitelboim Quantum Gravity. J. High Energy Phys.
**2019**, 2019, 66. [Google Scholar] [CrossRef] [Green Version] - Blommaert, A.; Mertens, T.G.; Verschelde, H. Clocks and Rods in Jackiw-Teitelboim Quantum Gravity. J. High Energy Phys.
**2019**, 2019, 60. [Google Scholar] [CrossRef] [Green Version] - Iliesiu, L.V.; Pufu, S.S.; Verlinde, H.; Wang, Y. An exact quantization of Jackiw-Teitelboim gravity. J. High Energy Phys.
**2019**, 2019, 91. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Gas of baby universes. (

**a**) One parent, (

**b**) two parents connected by the wormhole, (

**c**) three parents by the wormhole. Here $|{\beta}_{1}|>>|{\beta}_{i}|$ for $i\ge 2$ and ${b}_{i}\le {b}_{c}$ for $i\ge 2$.

**Figure 2.**Four different choices of perturbations of the gaussian ensemble by the exponential potential ${V}_{1}=J{e}^{\omega x}$: (

**a**) $J>0,\phantom{\rule{0.166667em}{0ex}}\omega >0$; (

**b**) $J>0,\phantom{\rule{0.166667em}{0ex}}\omega <0$; (

**c**) $J<0,\phantom{\rule{0.166667em}{0ex}}\omega >0$; (

**d**) $J<0,\phantom{\rule{0.166667em}{0ex}}\omega <0$.

**Figure 3.**Non–normalized density plot $\rho =\rho \left(\lambda \right)$ for the exponential potential with negative $\omega =-1$ (

**a**) and positive $\omega =1$ (

**b**) and different regularization parameter $\mathsf{\Lambda}$.

**Figure 4.**The plot of non–normalized density for the quadratic potential deformed by the exponential potential for $\omega =1$ and different values of the regularization parameter $\mathsf{\Lambda}$: $\mathsf{\Lambda}=5$ for (

**a**) and $\mathsf{\Lambda}=6$ for (

**b**).

**Figure 5.**(

**a**) Relations between ${m}^{2}$ and J for fixed $\mathsf{\Lambda}$ and $\omega $. (

**b**) ${J}_{cr}$ vs ${m}^{2}$ at $\omega =1$. The legend is the same as at (

**a**).

**Figure 6.**The potential supported the density ${\rho}_{norm,0}\left(E\right)$ for different parameter $\mathsf{\Lambda}$.

**Figure 9.**(

**a**) The points show the values of ${J}_{cr}$ for $\rho \left(\lambda \right)$ given by (66) for different $\mathsf{\Lambda}$. (

**b**) ${J}_{cr}$ vs ${m}^{2}$ for the same set of $\mathsf{\Lambda}$ as at (

**a**). $\mathsf{\Lambda}=5,6,7.5,10,12.5,15$ are shown by red, brown, magenta, blue, cyan and green colors. Here $\omega =-1$.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aref’eva, I.; Volovich, I.
Gas of Baby Universes in JT Gravity and Matrix Models. *Symmetry* **2020**, *12*, 975.
https://doi.org/10.3390/sym12060975

**AMA Style**

Aref’eva I, Volovich I.
Gas of Baby Universes in JT Gravity and Matrix Models. *Symmetry*. 2020; 12(6):975.
https://doi.org/10.3390/sym12060975

**Chicago/Turabian Style**

Aref’eva, Irina, and Igor Volovich.
2020. "Gas of Baby Universes in JT Gravity and Matrix Models" *Symmetry* 12, no. 6: 975.
https://doi.org/10.3390/sym12060975