# Off-shell strings The off-shell string formalism is an approach to string theory that allows strings to propagate on backgrounds that do not satisfy the Einstein equation, i.e., off-shell backgrounds. Some advantages of this formulation are: - one can derive target space effective action directly from string worldsheet, without having to obtain EOM first; - one can compute $n$-point correlators without having to take the insertions to infinity; - allows e.g. a conical singularity which is an off-shell configuration so that the entropy derivation can be performed. ## Notations - $I_g$: string action at genus $g$ correction ($I_g^{\text{eff}}=-Z_g$) - $I_0$: classical string (EFT) action - $I_{g\ge1}$: quantum effective actions - $K_{g,n}$: CFT correlator for inserting $n$ vertex operators onto the genus $g$ worldsheet - $K_0=\sum\limits_{n}K_{0,n}$ - string amplitude - $g\ge1$: $A_{\mathrm{g}, n}=\int[\mathrm{d} \tau] \frac{K_{\mathrm{g}, n}}{\operatorname{Vol}(\mathrm{KG})}$ - $g=0$: $A_{0, n}=\frac{\partial}{\partial \log \epsilon} K_{0, n}$ (T1) - sum over $n$: $Z_0=\sum_n A_{0, n}=\frac{\partial}{\partial \log \epsilon} K_0$ - $\epsilon$: UV cutoff on the worldsheet - $n$: number of operator insertions - $\mathcal{M}_{0,n}$: moduli space - $\mathcal{P M}_{0, n}$: pre-moduli space - $\mathcal{M}_{0,n}=\mathcal{P M}_{0, n} / \mathrm{SL}(2, \mathbb{C})$ - $\mathcal{P M}_{0, n}^{(\epsilon)}$: regulated pre-moduli space - $\mathcal{M}_{0, n}^{(\epsilon)}:=\mathcal{P M}_{0, n}^{(\epsilon)} / \mathrm{SL}(2, \mathbb{C})$: regulated moduli space ## Two serious problems and resolutions - theory depends on an arbitrary choice of the Weyl frame, $\omega$ - not a problem: changing $\omega$ can be absorbed into field redefinition of target space fields - corresponds to renormalisation of worldsheet QFT - non-compact conformal Killing group for the sphere diagram - Tseytlin does not deal with it by fixing three points, which is not generalisable to off-shell case - the effect of Tseytlin's sphere prescription is to mod out by the gauge orbits (*not* dividing by the volume of the gauge group) ## III. Defining the partition function off-shell - choice of Weyl frame - $g_{a b}=e^{2 \omega} \gamma_{a b}$ - off-shell gauge fixing - define a function $\tilde{\mathcal{L}}[X, g]$ that is the unique Weyl-invariant function which agrees with $\mathcal{L}_{\mathrm{QFT}}[X, \gamma, \omega]$ when $g_{a b}=e^{2 \omega} \gamma_{a b}$ - then can repeat Faddeev-Popov gauge fixing for this $\tilde {\mathcal{L}}$ - field redefinition - under a small Weyl rescaling - $\frac{\delta}{\delta \omega(z)} Z[\omega]=\langle\langle T(z)\rangle\rangle_\omega=\sum_a \beta^a\left\langle\left\langle\mathcal{O}_a\right\rangle\right\rangle_\omega$ $\propto$ EOM - perturbatively two nearby Weyl frames give equivalent results up to a field redefinition - conformal perturbation theory - any QFT can be regarded as a CFT with a gas of vertex operator insertions - $\mathcal{L}=\mathcal{L}_{\mathrm{CFT}}+\sum_a \epsilon^{2\left(h_a-1\right)} \phi^a \mathcal{O}_a$ - QFT amplitude: - $K_{\mathrm{g}}(\tau)=\sum_{n=0}^{\infty} K_{\mathrm{g}, n}(\tau)$ - at order $\phi^n$: $K_{\mathrm{g}, n}=\sum_{a_1 \ldots a_n}\left(\prod_n \epsilon^{2\left(h_a-1\right)} \phi^a\right) \frac{1}{n !}\left\langle\left\langle V_{a_1} \ldots V_{a_n}\right\rangle\right\rangle_{\mathrm{CFT}}$ - momentum basis: $\mathcal{O}_a \sim \exp \left(i P_\mu X^\mu\right) \mathcal{O}_{\aleph}$ - if $\mathcal{O}_a$ is an $(1,1)$ primary $\to$ physical state with $P^2+M^2=0$ - string amplitude - $g>0$: - $A_{\mathrm{g}, n}=\int[\mathrm{d} \tau] \frac{K_{\mathrm{g}, n}}{\operatorname{Vol}(\mathrm{KG})}$ - n.b. we need to *choose* $\hat g$ s.t. CKG acts as a normal KG (doable only when $g>0$) - $g=0$: - T1: $A_{0, n}=\frac{\partial}{\partial \log \epsilon} K_{0, n}$ - so $Z_0=\frac{\partial}{\partial \log \epsilon} K_0$ - wrongly implies that there is a tree-level tadpole associated with the tachyon - $I_0^{\mathbf{T} \mathbf{1}}=-\frac{\partial K_0}{\partial \log \epsilon}=\sum_a \beta^a \frac{\partial K_0}{\partial \phi^a}$ - T2: $Z_0=\left(\frac{\partial}{\partial \log \epsilon}+\frac{1}{2} \frac{\partial^2}{(\partial \log \epsilon)^2}\right) K_0$ - $I_0^{\mathbf{T} 2}=I_0^{\mathbf{T} 1}+\frac{1}{2}\left(\sum_{a, b} \beta^a \beta^b \frac{\partial^2 K_0}{\partial \phi^a \partial \phi^b}+\beta^b \frac{\partial \beta^a}{\partial \phi^b} \frac{\partial K_0}{\partial \phi^a}\right)$ - n.b. taking derivative in $\log \epsilon$ is related to the $\beta$ functions - n.b. the action vanishes on-shell: $\beta=0$ - on-shell amplitude - $S_{\mathrm{g}, n}=A_{\mathrm{g}, n} \prod_n \delta\left(P_a^2+M_a^2\right)$, i.e., forcing external legs on-shell - two-point amplitude - ... - string tadpoles - $A_{0,1}\ne0$ ## IV. Renormalisation and propagating strings - divergences (separating degenerations): a single string propagator becomes long - momentum-dependent ones: when the propagator satisfies mass-shell - momentum-independent ones: when vertex operators approach each other - regulated propagator - ... - locality and cutoff - $\epsilon\to0$: WS theory exactly local; target space gives S-matrix - $\epsilon\sim1$: WS non-local; target space confusing (discretised worldsheet?) - perturbatively in $\epsilon$ (WS perturbatively local) - $\log \epsilon^{-1} \sim 1$: $I^\text{eff}_0$ local perturbatively in $l_s$ (local over scales $\Delta X\gg l_s$) - $\log \epsilon^{-1} \gg 1$: $I^\text{eff}_0$ non-local even at scales $\Delta X\gg l_s$ ## V. Obtaining the tree-level S-matrix - takeaway: T1 or T2 is equivalent to quotienting by the gauge orbits of $\mathrm{SL}(2,\mathbb{C})$ - the gauge orbits of $\mathrm{SL}(2,\mathbb{C})$ - refer to each element of $\mathcal{M}_{0, n}^{(\epsilon)}:=\mathcal{P M}_{0, n}^{(\epsilon)} / \mathrm{SL}(2, \mathbb{C})$ as $\Omega$ - i.e., each $\Omega$ is a gauge orbit of $\mathrm{SL}(2,\mathbb{C})$ - the nontrivial contribution to $\Omega$ comes from $\frac{\mathrm{SL}(2, \mathbb{C})}{\mathrm{SU}(2)}=H_3$, where $\mathrm{SU}(2)$ is the compact rotation group - regulated volume in $H_3$ per solid angle: - $\int_0^{\log (a / \epsilon)+O\left(\epsilon^2\right)} \mathrm{d} \lambda \sinh ^2(\lambda)=\frac{a^2}{8} \epsilon^{-2}+\frac{1}{2} \log (\epsilon)+b+O\left(\epsilon^2\right)$ - n.b. for any ball of radius $r$ in $H^n$: - $\operatorname{Vol}(B(r))=\operatorname{Vol}\left(S^{n-1}\right) \int_0^r \sinh ^{n-1}(t) d t$ - integrate up to $\log \epsilon$: these are orbits for which each pair of points are at least $\epsilon$ apart on the sphere - n.b. for $n=2$, $\operatorname{Vol}(\Omega)=\infty$: will not consider in this section; for $n\ge3$, the (regulated) volume is finite: - ![[AhmadainWall2022a_fig5.png]] - then T2 gives $\lim _{\epsilon \rightarrow 0}\left(\frac{\partial}{\partial \log \epsilon}+\frac{1}{2} \frac{\partial^2}{(\partial \log \epsilon)^2}\right) \operatorname{Vol}(\Omega) \propto 1$ - generic momenta: fixing 3 points - when $n=3$ (6 real parameters), only one orbit $\mathrm{SL}(2,\mathbb{C})$ orbit - $K_{0, n}=\operatorname{Vol}\left(\Omega_3\right) F_n$ - for all $n\ge3$ - $F_n$ is the tree level amplitude defined by fixing 3 points - to show this, start with $n$ insertions and act with $\mathrm{SL}(2,\mathbb{C})$ on *all* $n$ insertions - $A_{0,n}=F_n$ - by applying T2 to both sides above - this shows that Tseytlin's prescription is the same as fixing 3 points ## VI. Obtaining the classical EOM - goal: show that the tree-level string action $I_0$ obtains the correct EOM to all orders in perturbation theory in $n$ - plan: show at orders $n=1,2$; then show it for marginal primaries when $n\ge0$, which is sufficient, as $n=1,2$ dominate in non-marginal cases ## Refs - originals - [[1985#Fradkin, Tseytlin (PRB158)]] - [[1985#Fradkin, Tseytlin (NPB261)]] - [[1985#Fradkin, Tseytlin (PRB160)]] - Tseytlin's proposal - [[1988#Tseytlin]] - [[1989#Tseytlin]] - careful study [[2022#Ahmadain, Wall (a)]] - entropy calculation [[2022#Ahmadain, Wall (b)]] - [[1988#Tseytlin]]: some useful results needed for [[1994#Susskind, Uglum]]