# Factorisation problem By virtue of [[0001 AdS-CFT|AdS/CFT]], the partition function of a quantum gravitational theory in AdS, $Z_{AdS}$, should equal to that of the CFT, $Z_{CFT}$. If the CFT lives on more than one connected components, say two, then $Z_{CFT}=Z_{CFT1}Z_{CFT2}$ due to locality, i.e., it should factorise. The gravitational partition function, however, does not obviously factorise because there are [[0278 Euclidean wormholes|Euclidean wormholes]] connecting the two boundaries. These solutions contribute to the partition function via saddle point approximation, $Z_{AdS}=\sum e^{-I}$, where $I$ is the Euclidean action and the sum is over saddles. This is a very interesting observation but also a big puzzle. Resolving this important puzzle would very likely lead to new insight into holography and quantum gravity. Many resolutions have been proposed, some of which listed below. ## Refs - [[0154 Ensemble averaging]] - originals - [[WittenYau1999]] - [[2004#Maldacena, Maoz]] - classical solutions - [[2021#Marolf, Santos]] ## Making issues manifest - [[MarolfSantos2021]] - Euclidean wormholes in various models - [[2021#Cotler, Jensen]] - [[2021#Mahajan, Marolf, Santos]] - [[2021#Saad, Shenker, Stanford, Yao]] - half wormholes disappear after averaging ## Proposed solutions - Possibility 1: even in higher dimensions the bulk theory is dual to an [[0154 Ensemble averaging|ensemble]] of boundary theories - [[2020#Marolf, Maxfield (a)]] - Possibility 2: semi-classical gravity only know about some coarse-grained information about the boundary theory (e.g. the random nature of OPE coefficients), which explains lack of factorisation - [[2020#Pollack, Rozali, Sully, Wakeham]] - [[2020#Belin, de Boer]] - [[2020#Altland, Sonner]] - Possibility 3: wormhole contributions are cancelled out - Possibility 4: wormholes not allowed - Possibility 5: wormholes gauge-equivalent to non-wormholes - Gesteau's talk at MURI - [[2021#Eberhardt]] - string partition function factorises on a Euclidean wormhole - [[Verlinde2021]] - factorisation v.s. entanglement - [[2021#Saad, Shenker, Stanford, Yao]] - include half wormholes - [[2021#Mukhametzhanov]] - quoted by Shenker (see [[Rsc0031 TASI 2021]] Shenker lecture) - Eberhardt: start with thermal AdS -> perturbation around it resums to something equal to BH; you could start with BH and perturbations around it would give thermal AdS - half-wormhole: looking at SYK model, perturb around half-WH, resum the perturbations (finite number of terms due to fermionic property) and find that the last term in the series is a full WH; vice versa - [[2021#Saad, Shenker, Yao]] - works with a specific member of the ensemble -> will factorise - works with alpha-states - exclusion rules for half-wormholes - [[Garcia-GarciaGodet2021]][](https://arxiv.org/abs/2107.07720) - adding D-branes - half-wormholes restore factorisation - [[GotoKusukiTamaokaUgajin2021]][](https://arxiv.org/pdf/2108.08308.pdf) - averaging over product states gives entangled states - some CFT interpretation of the [[0308 Half-wormhole]] - gauging a bulk solution - [[BeniniCopettiDiPietro2022]][](https://arxiv.org/pdf/2203.09537.pdf) - factorised models - [[2021#Blommaert, Iliesiu, Kruthoff]] - [[2022#Blommaert, Iliesiu, Kruthoff]] - [[BanerjeeDorbandErdmengerMeyerWeigel2022]][](https://arxiv.org/pdf/2202.11717.pdf) - using [[0423 Berry phase]] - [[Banks2022]][](https://arxiv.org/abs/2203.08855) - non-factorisation comes from time averaging that is necessary to apply the hydrodynamic approximation - factorisation in [[0089 Chern-Simons theory|Chern-Simons theory]] - [[2022#Benini, Copetti, Di Pietro]]: gauging a 1-form symmetry (should be equivalent to summing over topology) - [[2023#Melnikov]]: [[0607 Topological QFT|topological QFT]] description of CS theory - [[2023#Chua, Jiang]]: factorisation of [[0162 No-boundary wavefunction|HH state]] in [[0002 3D gravity|3d gravity]] - [[2024#Gesteau, Marcolli, McNamara]] - renormalisation using counter-wormholes ## Stability - [[2021#Mahajan, Marolf, Santos]] - double cone is stable to brane nucleation ## Related - [[0514 Lorentzian factorisation problem]]