# ${\mathrm{w}}_{1+\infty}$ symmetry Recently, it has been understood that [[0009 Soft theorems|soft theorems]] are closely related to [[0060 Asymptotic symmetry|asymptotic symmetries]]. At tree-level and restricting to positive helicity, the symmetries of General Relativity in flat space can be assembled into the loop algebra of the wedge algebra of $\rm{w}_{1+\infty}$. Notation-wise, writing a positive-helicity graviton with conformal weight as $G^+_{\Delta}$, one can define the [[0390 Conformally soft theorems|conformally soft]] gravitons $H^k$ by$H^k=\lim _{\varepsilon \rightarrow 0} \varepsilon G_{k+\varepsilon}^{+}, \quad k=2,1,0,-1, \ldots.$We then expand it first in the antiholomorphic coordinate $\bar{z}$,$H^k(z, \bar{z})=\sum_{n=\frac{k-2}{2}}^{\frac{2-k}{2}} \frac{H_n^k(z)}{\bar{z}^{n+\frac{k-2}{2}}},$followed by an expansion in $z$:$H_n^k(z)=\sum_{a \in \mathbb{Z}-\frac{k+2}{2}} \frac{H_{a, n}^k}{z^{a+\frac{k+2}{2}}}.$Some leading modes can be identified: - $\Delta=k=2$: an extra one sitting at the top - $\Delta=k=1$: leading soft graviton theorem - $S L(2, \mathbb{R})_R$-doublet: $\bar h =\pm1/2$ - generates supertranslations - in particular, $H_{\pm \frac{1}{2}, \pm \frac{1}{2}}^1$ are global translations - $\Delta=k=0$: subleading soft graviton theorem - $S L(2, \mathbb{R})_R$-triplet: $\bar h =0,\pm1$ - $H^0_{0,0}=2\bar{L}_0$, $H^0_{0,\pm1}\propto \bar{L}_{\pm1}$ - form a closed subalgebra - $\Delta=k=-1$: subsubleading soft graviton theorem - $S L(2, \mathbb{R})_R$-quadruplet: $\bar h =\pm1/2,\pm3/2$ One can then proceed to derive the algebra. The derivation starts from the [[0114 Celestial OPE|celestial OPE]], which comes from the [[0078 Collinear limit|collinear]] limit of scattering amplitudes. To turn the result into a form that is identified with $\mathrm{w}_{1+\infty}$, we also need the following redefiniton:$w_{a, m}^p=\frac{1}{\kappa}(p-m-1)!(p+m-1)!H_{a, m}^{-2 p+4}.$The algebra is then found to be$\left[w_{a, m}^p, w_{b, n}^q\right]=[m(q-1)-n(p-1)] w_{a+b, m+n}^{p+q-2}.$ In AdS, this algebra is deformed. The deformed algebra was first found by [[2023#Taylor, Zhu (Dec)|Taylor and Zhu]] by deforming the flat space amplitude with a perturbative cosmological constant. Then [[2024#Bittleston, Bogna, Heuveline, Kmec, Mason, Skinner|Bittleston et al.]] derived it using twistor techniques. It reads$\left[w_{a, m}^p, w_{b, n}^q\right]=[m(q-1)-n(p-1)] w_{a+b, m+n}^{p+q-2}-\Lambda[a(q-2)-b(p-2)] w_{a+b, m+n}^{p+q-1}.$ ## Refs - from celestial OPE - talk at [Cornell](https://www.cornell.edu/video/andrew-strominger-the-holographic-principle-in-flat-space) - main [[2021#Guevara, Himwich, Pate, Strominger]] and [[2021#Strominger]] - currents that generate them constructed in Section 5 of [[2021#Himwich, Pate, Singh]] - from WS string theory - [[2021#Jiang (Oct)]] - [[AdamoMasonSharma2021]] - from twistor space - [[AdamoMasonSharma2021]] - with [[0359 Supersymmetry|SUSY]] - [[2021#Jiang (Aug)]] - [[Ahn2021]] - with higher derivative coupling - [[2021#Mago, Ren, Srikant, Volovich]]: Jacobi identity - [[2021#Jiang (Aug)]]: no EFT correction to chiral symmetry algebra - from classical phase space - [[2021#Freidel, Pranzetti, Raclariu (Dec)]] - [[2024#Geiller]]: [[0456 Newman-Penrose charges|Newman-Penrose]], higher Bondi aspects, etc - quantum correction in [[0234 Self-dual gravity|SDGR]] - [[2021#Ball, Narayanan, Salzer, Strominger]] - [[2022#Bittleston]] - [[2023#Bittleston, Heuveline, Skinner]] - ${\rm w}_{1+\infty}$ gravity - [[PopeRomansShen1990]][](http://dx.doi.org/10.1016/0550-3213(90)90539-P) - from [[0419 Carrollian CFT|Carrollian CFT]] - [[2023#Saha]] - with a cosmological constant - [[2023#Taylor, Zhu (Dec)]] - massive particles - [[2023#Himwich, Pate]] - from twistor space - [[2024#Kmec, Mason, Ruzziconi, Srikant]] - deformed algebra by a cosmological constant - [[2023#Taylor, Zhu (Dec)]]: treating $\Lambda$ perturbatively and then fix the algebra by hand - [[2024#Bittleston, Bogna, Heuveline, Kmec, Mason, Skinner]]: rigorous derivation using twistor space - a two-parameter deformation from curved twistor space - [[2024#Bogna, Heuveline]] ## [[0071 Yang-Mills|YM]] - definitions - $R^{k, a}(z):=\lim _{\varepsilon \rightarrow 0} \varepsilon O_{k+\varepsilon}^{a,+}(z), \quad k=1,0,-1,-2, \ldots$ - $R^{k, a}(z, \bar{z})=\sum_{n=\frac{k-1}{2}}^{\frac{1-k}{2}} \frac{R_n^{k, a}(z)}{\bar{z}^{n+\frac{k-1}{2}}}$ - examples - $\Delta=k=1$: leading soft current (KM) - singlet, $h=1,\bar h=0$ - $\Delta=k=0$: subleading soft currents - doublet, $h=1/2,\bar h =\pm1/2$ - $\Delta=k=-1$: generated by the first two levels - triplet, $h=0, \bar{h}=0,\pm1$ --- \[*In the remainder, we turn to an alternative set of notations that make the algebra look neater.*\] ## Generators - $\mathrm{w}^{q}(z, \bar{z}) \equiv \frac{1}{\kappa}(-1)^{2 q} \Gamma(2 q) \lim _{\varepsilon \rightarrow 0} \mathbf{L}\left[\mathcal{O}_{3-q, 1-q+\varepsilon}\right](z, \bar{z})$\quad q=1, \frac{3}{2}, 2, \frac{5}{2}, \cdots$ - define mode expansion: $\mathrm{w}^{q}(z, \bar{z})=\sum_{m, n} \frac{\mathrm{w}_{m, n}^{q}}{z^{3-q+m} \bar{z}^{q+n}}$ - modes that do not mix $SL(2,R)$ primaries and descendent are: - $\widehat{\mathrm{w}}_{n}^{q} \equiv \mathrm{w}_{q-2, n}^{q}, \quad 1-q \leq n \leq q-1$ - examples: - $\widehat{\mathrm{w}}_{m}^{2}=\bar{L}_{m}$ - $\widehat{\mathrm{w}}_{m, n}^{\frac{3}{2}}=P_{m, n}$ n.b. $\Delta =3$ ## The wedge subalgebra - $\left[w_m^p, w_n^q\right]=[m(q-1)-n(p-1)] w_{m+n}^{p+q-2}$ - where $p=1,3/2,2,5/2,...$ and $1-p \leq m \leq p-1$ - aka $GL(\infty,\mathbb{R})$ - n.b. the closed $p=2$ subalgebra is the $c=0$ [[0032 Virasoro algebra|Virasoro algebra]] ## The loop algebra of ${\mathrm{w}}_{1+\infty}$ - from $w[k, l]\left(z_1\right) w[m, n]\left(z_2\right) \sim \frac{k n-l m}{z_{12}} w[k+m-1, l+n-1]\left(z_2\right)$ - define $w[k, l](z)=\sum_{r \in \mathbb{Z}} \frac{w[k, l]_r}{z^{r+1}}$ - then $\left[w[k, l]_r, w[m, n]_s\right]=(k n-l m) w[k+m-1, l+n-1]_{r+s}$