# Irreversibility theorems Irreversibility theorems are statements about a quantum field theories under RG flow. In 2, 3, 4 dimensions, they are referred to as $c$, $F$, $a$-theorems respectively. In a more general setting, consider a $D$-dimensional CFT coupled to a $d$-dimensional planar static defect ($d<D$). When $D=2$ and $d=1$, it is called a $g$-theorem. When $D=d$, it reduces to the study of QFT with no defects. Important to the discussion is the so-called RG charge. They decrease under RG flow and characterise the fixed points. ## Identifying the RG charge - in general, it should be the universal coefficient of the free energy induced by the defect in a $D$-sphere - constant term for odd $d$ and log term for even $d$ - special case of $D=d$ - in even $D=d$, it's the Euler term in [[0306 Weyl anomaly|conformal anomaly]], or the logarithmic coefficient of the free energy of the CFT in a Euclidean $d$-dimensional sphere, or the logarithmic universal coefficient of the [[0301 Entanglement entropy|entanglement entropy]] of a sphere - in odd $D=d$, it's the constant term of entanglement entropy of a sphere, or equivalently the constant term of the free energy of the CFT in a Euclidean $d$-dimensional sphere ## Refs - reviews - [[2023#Casini, Landea, Torroba]] summarises and reviews many previous works - c-theorem ($D=d=2$) - original proof by [[1986#Zamolodchikov]], using reflection positivity of stress tensor correlators - [[2023#Hartman, Mathys (Oct)]]: proof using [[0417 Averaged null energy condition|ANEC]] - a-theorem ($D=d=4$) - [[1988#Cardy]]: conjectured the theorem - [[2011#Komargodski, Schwimmer]]: derives it by matching the anomaly in dilaton scattering - [[2023#Hartman, Mathys (Sep)]]: proof using [[0417 Averaged null energy condition|ANEC]] - [[2024#Hartman, Mathys]]: obtains a sum rule for $c$ which does not obey an irreversibility theorem - $F$-theorem ($D=d=3$) - [[2012#Casini, Huerta]]: using [[0218 Strong subadditivity|strong subadditivity]] - $g$-theorem ($D=2,d=1$) - [[1991#Affleck, Ludwig]] - [[2003#Friedan, Konechny]] - general $D$ and $d$ - [[2023#Casini, Landea, Torroba]] ## 2d c-theorem - [[1986#Zamolodchikov]] - complementary proof using Lorentz symmetry and [[0218 Strong subadditivity|SSA]] - [[CasiniHuerta2004]][](https://arxiv.org/abs/hep-th/0405111) - [[CasiniHuerta2006]][](https://arxiv.org/abs/cond-mat/0610375) - new proof of c-theorem: [[2023#Hartman, Mathys (Oct)]] ## Cardy's conjecture - $\left\langle T_{a}^{a}\right\rangle=\sum B_{i} I_{i}-2(-)^{d / 2} A E_{d}+B^{\prime} \nabla_{a} J^{a}$ - statement: $(A)_{\mathrm{UV}} \geq(A)_{\mathrm{IR}}$ ## See also - [[0350 Holographic c-theorem]] - [[0306 Weyl anomaly]]