Papers
Topics
Authors
Recent
Search
2000 character limit reached

Moduli space of logarithmic states in critical massive gravities

Published 7 May 2019 in hep-th | (1905.02409v3)

Abstract: We take new algebraic and geometric perspectives on the combinatorial results recently obtained on the partition functions of critical massive gravities conjectured to be dual to Logarithmic CFTs throught the AdS$_3$/LCFT$_2$ correspondence. We show that the partition functions of logarithmic states can be expressed in terms of Schur polynomials. Subsequently, we show that the moduli space of the logarithmic states is the symmetric product $Sn \left( \mathbb{C}2 \right)$. As the quotient of an affine space by the symmetric group, this orbifold space is shown to be described by Hilbert series that have palindromic numerators. The palindromic properties of the Hilbert series indicate that the orbifolds are Calabi-Yau, and allow for a new interpretation of the logarithmic state spaces in critical massive gravities as Calabi-Yau singular spaces.

Authors (1)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (51)
  1. J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104 (1986) 207–226.
  2. M. Banados, C. Teitelboim, and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69 (1992) 1849–1851, arXiv:hep-th/9204099 [hep-th].
  3. S. Deser, R. Jackiw, and S. Templeton, “Three-Dimensional Massive Gauge Theories,” Phys. Rev. Lett. 48 (1982) 975–978.
  4. S. Deser, R. Jackiw, and S. Templeton, “Topologically Massive Gauge Theories,” Annals Phys. 140 (1982) 372–411. [Annals Phys.281,409(2000)].
  5. E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].
  6. A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three Dimensions,” JHEP 02 (2010) 029, arXiv:0712.0155 [hep-th].
  7. W. Li, W. Song, and A. Strominger, “Chiral Gravity in Three Dimensions,” JHEP 04 (2008) 082, arXiv:0801.4566 [hep-th].
  8. D. Grumiller and N. Johansson, “Instability in cosmological topologically massive gravity at the chiral point,” JHEP 07 (2008) 134, arXiv:0805.2610 [hep-th].
  9. V. Gurarie, “Logarithmic operators in conformal field theory,” Nucl. Phys. B410 (1993) 535–549, arXiv:hep-th/9303160 [hep-th].
  10. M. Flohr, “Bits and pieces in logarithmic conformal field theory,” Int. J. Mod. Phys. A18 (2003) 4497–4592, arXiv:hep-th/0111228 [hep-th].
  11. M. R. Gaberdiel, “An Algebraic approach to logarithmic conformal field theory,” Int. J. Mod. Phys. A18 (2003) 4593–4638, arXiv:hep-th/0111260 [hep-th].
  12. K. Skenderis, M. Taylor, and B. C. van Rees, “Topologically Massive Gravity and the AdS/CFT Correspondence,” JHEP 09 (2009) 045, arXiv:0906.4926 [hep-th].
  13. D. Grumiller and I. Sachs, “AdS (3) / LCFT (2) —> Correlators in Cosmological Topologically Massive Gravity,” JHEP 03 (2010) 012, arXiv:0910.5241 [hep-th].
  14. M. R. Gaberdiel, D. Grumiller, and D. Vassilevich, “Graviton 1-loop partition function for 3-dimensional massive gravity,” JHEP 11 (2010) 094, arXiv:1007.5189 [hep-th].
  15. D. Grumiller, W. Riedler, J. Rosseel, and T. Zojer, “Holographic applications of logarithmic conformal field theories,” J. Phys. A46 (2013) 494002, arXiv:1302.0280 [hep-th].
  16. Y. Mvondo-She and K. Zoubos, “On the combinatorics of partition functions in AdS33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT/LCFT22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT,” JHEP 05 (2019) 097, arXiv:1811.08144 [hep-th].
  17. J. Gray, A. Hanany, Y.-H. He, V. Jejjala, and N. Mekareeya, “SQCD: A Geometric Apercu,” JHEP 05 (2008) 099, arXiv:0803.4257 [hep-th].
  18. A. Hanany, N. Mekareeya, and G. Torri, “The Hilbert Series of Adjoint SQCD,” Nucl. Phys. B 825 (2010) 52–97, arXiv:0812.2315 [hep-th].
  19. D. Forcella, A. Hanany, Y.-H. He, and A. Zaffaroni, “The Master Space of N=1 Gauge Theories,” JHEP 08 (2008) 012, arXiv:0801.1585 [hep-th].
  20. D. Forcella, A. Hanany, Y.-H. He, and A. Zaffaroni, “Mastering the Master Space,” Lett. Math. Phys. 85 (2008) 163–171, arXiv:0801.3477 [hep-th].
  21. R. de Mello Koch, P. Rabambi, R. Rabe, and S. Ramgoolam, “Free quantum fields in 4D and Calabi-Yau spaces,” Phys. Rev. Lett. 119 no. 16, (2017) 161602, arXiv:1705.04039 [hep-th].
  22. R. de Mello Koch, P. Rabambi, R. Rabe, and S. Ramgoolam, “Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds,” JHEP 08 (2017) 077, arXiv:1705.06702 [hep-th].
  23. R. P. Stanley, “Hilbert functions of graded algebras,” Advances in Mathematics 28 no. 1, (1978) 57–83.
  24. A. Bagchi, S. Lal, A. Saha, and B. Sahoo, “Topologically Massive Higher Spin Gravity,” JHEP 10 (2011) 150, arXiv:1107.0915 [hep-th].
  25. A. Bagchi, S. Lal, A. Saha, and B. Sahoo, “One loop partition function for Topologically Massive Higher Spin Gravity,” JHEP 12 (2011) 068, arXiv:1107.2063 [hep-th].
  26. C.-C. Zhou and W.-S. Dai, “Canonical partition functions: ideal quantum gases, interacting classical gases, and interacting quantum gases,” Journal of Statistical Mechanics: Theory and Experiment 2018 no. 2, (2018) 023105, arXiv:1710.02819.
  27. A. Fronczak, “Microscopic meaning of grand potential resulting from combinatorial approach to a general system of particles,” Physical Review E 86 no. 4, (2012) 041139, arXiv:1205.4986.
  28. G. Siudem, “Partition function of the model of perfect gas of clusters for interacting fluids,” Reports on Mathematical Physics 72 no. 1, (2013) 85–92.
  29. A. Fronczak, “Cluster properties of the one-dimensional lattice gas: The microscopic meaning of grand potential,” Physical Review E 87 no. 2, (2013) 022131.
  30. G. Siudem, A. Fronczak, and P. Fronczak, “Exact low-temperature series expansion for the partition function of the zero-field Ising model on the infinite square lattice,” Scientific reports 6 (2016) 33523.
  31. J. Lucietti and M. Rangamani, “Asymptotic counting of BPS operators in superconformal field theories,” J. Math. Phys. 49 (2008) 082301, arXiv:0802.3015 [hep-th].
  32. Cambridge University Press, 1994.
  33. P. Pouliot, “Molien function for duality,” JHEP 01 (1999) 021, arXiv:hep-th/9812015 [hep-th].
  34. S. Benvenuti, B. Feng, A. Hanany, and Y.-H. He, “Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics,” JHEP 11 (2007) 050, arXiv:hep-th/0608050 [hep-th].
  35. B. Feng, A. Hanany, and Y.-H. He, “Counting gauge invariants: The Plethystic program,” JHEP 03 (2007) 090, arXiv:hep-th/0701063 [hep-th].
  36. H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces. University lecture series. American Mathematical Soc. https://books.google.co.za/books?id=pjjgNhDVgcgC.
  37. H. Nakajima, “Quiver varieties and McKay correspondence–lectures at Hokkaido University, 2001 Dec.”.
  38. M. Forger, “Invariant polynomials and Molien functions,” Journal of Mathematical Physics 39 no. 2, (1998) 1107–1141.
  39. W. N. Traves, “Differential operators on orbifolds,” Journal of Symbolic Computation 41 no. 12, (2006) 1295–1308.
  40. H. Nakajima and K. Yoshioka, “Instanton counting on blowup. 1.,” Invent. Math. 162 (2005) 313–355, arXiv:math/0306198 [math.AG].
  41. H. Nakajima, “Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces,” Annals of Mathematics (1997) 379–388, arXiv:9507012.
  42. W. P. Johnson, “The curious history of Faà di Bruno’s formula,” The American mathematical monthly 109 no. 3, (2002) 217–234.
  43. M. Hochster and J. L. Roberts, “Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay,” Advances in mathematics 13 no. 2, (1974) 115–175.
  44. D. Ridout, “Non-Chiral Logarithmic Couplings for the Virasoro Algebra,” J. Phys. A45 (2012) 255203, arXiv:1203.3247 [hep-th].
  45. J. Raeymaekers, “Quantization of conical spaces in 3D gravity,” JHEP 03 (2015) 060, arXiv:1412.0278 [hep-th].
  46. Y. Mvondo-She, Combinatorial and Analytic properties of partition functions in AdS/LCFT. PhD thesis, 2019.
  47. A. A. Bytsenko, M. Chaichian, and R. Luna, “Vector Generation Functions, q-Spectral Functions of Hyperbolic Geometry and Vertex Operators for Quantum Affine Algebras,” J. Math. Phys. 58 no. 12, (2017) 121701, arXiv:1707.01553 [math-ph].
  48. A. A. Bytsenko and M. Chaichian, “Multipartite Generating Functions and Infinite Products for Quantum Invariants,” arXiv:1702.02208 [math-ph].
  49. L. Bonora, A. A. Bytsenko, and A. E. Goncalves, “Chern–Simons invariants on hyperbolic manifolds and topological quantum field theories,” Eur. Phys. J. C76 no. 11, (2016) 625, arXiv:1606.02554 [hep-th].
  50. A. A. Bytsenko and M. Chaichian, “S𝑆Sitalic_S-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants,” Nucl. Phys. B907 (2016) 258–285, arXiv:1602.06704 [hep-th].
  51. B. Sturmfels, Algorithms in invariant theory. 2nd ed. Wien: Springer, 2 ed., 2008. http://link.springer.com/book/10.1007%2F978-3-211-77417-5.
Citations (5)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free