Holography and volume conjecture
imported>Pythagoras0님의 2014년 6월 10일 (화) 20:11 판
introduction
- asymptotic behavior of perturbative Chern-Simons invariants on knot complements $M$
- 3D-3D correspondence relates the CS invariants on M to supersymmetric quantities of the corresponding 3-dimensional quantum field theory which will be denoted by $T[M]$
- using holographic principle one can related the 3D theory $T[M]$ to M-theory on an anti de-Sitter space
M-theory
- still mysterious
- 11d physical theory
- fundamental objects are M2 (3d), M5(6d) branes
- let $M$ be a 3dimensional space obtained as knot complement
- in the study of dynamics of N M5-branes on $\mathbb{R}^{1,2}\times M$
3d-3d correspondence
- partition function $T_N[M]$ on $S_b^3$ = partition function $PGL(N)$ CS theory on $M$
$$ Z_{T_N[M]}[S_b^3](N,M;b)=Z^{\text{geom}}_{PGL(N)}[M;\hbar] $$ where $2\pi i b^2=\hbar$
holographic principle
- 3d $T_N[M]$ theory at large $N$ = M-theory on $\operatorname{AdS}_4\times M\times S^4$
- for large $N$
$$ Z_{T_N[M]}[S_b^3](N,M;b)=\exp(-S_0^{\text{gravity}}[\operatorname{AdS}_4\times M\times S^4])\frac{(b+b^{-1})^2N^3}{12\pi}\operatorname{Vol}(M)+\text{subleadings} $$
articles
- Gang, Dongmin, Nakwoo Kim, and Sangmin Lee. “Holography of Wrapped M5-Branes and Chern-Simons Theory.” arXiv:1401.3595 [hep-Th], January 15, 2014. http://arxiv.org/abs/1401.3595.
3d-3d correspondence
- Terashima, Yuji, and Masahito Yamazaki. “SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls.” Journal of High Energy Physics 2011, no. 8 (August 2011). doi:10.1007/JHEP08(2011)135.
- Dimofte, Tudor, Davide Gaiotto, and Sergei Gukov. “Gauge Theories Labelled by Three-Manifolds.” arXiv:1108.4389 [hep-Th], August 22, 2011. http://arxiv.org/abs/1108.4389.
holography
- Maldacena, Juan M. “The Large N Limit of Superconformal Field Theories and Supergravity.” arXiv:hep-th/9711200, November 27, 1997. http://arxiv.org/abs/hep-th/9711200.