Certification Paths 75 - Microsoft Home Page | Devices and Services
Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. ·...
Transcript of Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. ·...
![Page 1: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/1.jpg)
Title: Partition Functions of Three Dimensional Gravity
Date: Nov 12, 2007 03:00 PM
URL: http://pirsa.org/07110061
Abstract: We consider pure three dimensional quantum gravity with a negative cosmological constant. The torus partition function can be computedexactly as a sum over geometries, including all known quantum corrections. The answer provides important clues about the structure of quantumgravity; in particular, in order for the theory to be a proper quantum mechanical system some extra ingredients are needed beyond the usual realgeometries considered in general relativity. One possiblity is that complex geometries need to be included; this leads to holomorphically factorizedpartition functions. These partition functions provide a wealth of information about black hole microphysics. For example, the Hawking-page phasetransition can be studied exactly; it is a phase transition of the type described by Lee and Yang, which is associated with a condensation of zeros inthe complex temperature plane.
Pirsa: 07110061 Page 1/75
![Page 2: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/2.jpg)
Pirsa: 07110061 Page 2/75
![Page 3: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/3.jpg)
Pirsa: 07110061 Page 3/75
![Page 4: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/4.jpg)
Pirsa: 07110061 Page 4/75
![Page 5: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/5.jpg)
Pirsa: 07110061 Page 5/75
![Page 6: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/6.jpg)
Pirsa: 07110061 Page 6/75
![Page 7: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/7.jpg)
Pirsa: 07110061 Page 7/75
![Page 8: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/8.jpg)
Pirsa: 07110061 Page 8/75
![Page 9: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/9.jpg)
Pirsa: 07110061 Page 9/75
![Page 10: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/10.jpg)
Pirsa: 07110061 Page 10/75
![Page 11: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/11.jpg)
Pirsa: 07110061 Page 11/75
![Page 12: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/12.jpg)
Pirsa: 07110061 Page 12/75
![Page 13: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/13.jpg)
Pirsa: 07110061 Page 13/75
![Page 14: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/14.jpg)
Pirsa: 07110061 Page 14/75
![Page 15: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/15.jpg)
Pirsa: 07110061 Page 15/75
![Page 16: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/16.jpg)
Pirsa: 07110061 Page 16/75
![Page 17: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/17.jpg)
Pirsa: 07110061 Page 17/75
![Page 18: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/18.jpg)
Pirsa: 07110061 Page 18/75
![Page 19: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/19.jpg)
Pirsa: 07110061 Page 19/75
![Page 20: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/20.jpg)
Pirsa: 07110061 Page 20/75
![Page 21: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/21.jpg)
Pirsa: 07110061 Page 21/75
![Page 22: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/22.jpg)
Pirsa: 07110061 Page 22/75
![Page 23: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/23.jpg)
Pirsa: 07110061 Page 23/75
![Page 24: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/24.jpg)
Pirsa: 07110061 Page 24/75
![Page 25: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/25.jpg)
Pirsa: 07110061 Page 25/75
![Page 26: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/26.jpg)
Pirsa: 07110061 Page 26/75
![Page 27: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/27.jpg)
Pirsa: 07110061 Page 27/75
![Page 28: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/28.jpg)
Pirsa: 07110061 Page 28/75
![Page 29: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/29.jpg)
Pirsa: 07110061 Page 29/75
![Page 30: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/30.jpg)
Pirsa: 07110061 Page 30/75
![Page 31: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/31.jpg)
Pirsa: 07110061 Page 31/75
![Page 32: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/32.jpg)
Pirsa: 07110061 Page 32/75
![Page 33: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/33.jpg)
Pirsa: 07110061 Page 33/75
![Page 34: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/34.jpg)
Pirsa: 07110061 Page 34/75
![Page 35: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/35.jpg)
Pirsa: 07110061 Page 35/75
![Page 36: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/36.jpg)
Pirsa: 07110061 Page 36/75
![Page 37: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/37.jpg)
Pirsa: 07110061 Page 37/75
![Page 38: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/38.jpg)
Pirsa: 07110061 Page 38/75
![Page 39: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/39.jpg)
Pirsa: 07110061 Page 39/75
![Page 40: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/40.jpg)
Pirsa: 07110061 Page 40/75
![Page 41: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/41.jpg)
Pirsa: 07110061 Page 41/75
![Page 42: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/42.jpg)
Pirsa: 07110061 Page 42/75
![Page 43: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/43.jpg)
Pirsa: 07110061 Page 43/75
![Page 44: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/44.jpg)
Pirsa: 07110061 Page 44/75
![Page 45: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/45.jpg)
Pirsa: 07110061 Page 45/75
![Page 46: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/46.jpg)
Pirsa: 07110061 Page 46/75
![Page 47: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/47.jpg)
Pirsa: 07110061 Page 47/75
![Page 48: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/48.jpg)
Pirsa: 07110061 Page 48/75
![Page 49: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/49.jpg)
Pirsa: 07110061 Page 49/75
![Page 50: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/50.jpg)
Pirsa: 07110061 Page 50/75
![Page 51: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/51.jpg)
Pirsa: 07110061 Page 51/75
![Page 52: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/52.jpg)
Pirsa: 07110061 Page 52/75
![Page 53: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/53.jpg)
Pirsa: 07110061 Page 53/75
![Page 54: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/54.jpg)
Pirsa: 07110061 Page 54/75
![Page 55: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/55.jpg)
Pirsa: 07110061 Page 55/75
![Page 56: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/56.jpg)
Pirsa: 07110061 Page 56/75
![Page 57: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/57.jpg)
Pirsa: 07110061 Page 57/75
![Page 58: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/58.jpg)
Pirsa: 07110061 Page 58/75
![Page 59: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/59.jpg)
Pirsa: 07110061 Page 59/75
![Page 60: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/60.jpg)
Pirsa: 07110061 Page 60/75
![Page 61: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/61.jpg)
Pirsa: 07110061 Page 61/75
![Page 62: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/62.jpg)
Pirsa: 07110061 Page 62/75
![Page 63: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/63.jpg)
Pirsa: 07110061 Page 63/75
![Page 64: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/64.jpg)
Pirsa: 07110061 Page 64/75
![Page 65: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/65.jpg)
Pirsa: 07110061 Page 65/75
![Page 66: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/66.jpg)
Pirsa: 07110061 Page 66/75
![Page 67: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/67.jpg)
Pirsa: 07110061 Page 67/75
![Page 68: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/68.jpg)
Pirsa: 07110061 Page 68/75
![Page 69: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/69.jpg)
Pirsa: 07110061 Page 69/75
![Page 70: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/70.jpg)
Pirsa: 07110061 Page 70/75
![Page 71: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/71.jpg)
Pirsa: 07110061 Page 71/75
![Page 72: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/72.jpg)
Pirsa: 07110061 Page 72/75
![Page 73: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/73.jpg)
Pirsa: 07110061 Page 73/75
![Page 74: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/74.jpg)
Pirsa: 07110061 Page 74/75
![Page 75: Title: Partition Functions of Three Dimensional Gravity Date: Nov 12, 2007 … · 2019. 6. 19. · Pirsa: 07110061 Page 12/75. Pirsa: 07110061 Page 13/75. Pirsa: 07110061 Page 14/75.](https://reader036.fdocuments.net/reader036/viewer/2022071517/613a69870051793c8c01072a/html5/thumbnails/75.jpg)
Pirsa: 07110061 Page 75/75