# String Theory notes by David Tong \[Links: [course site](https://www.damtp.cam.ac.uk/user/tong/string.html), [PDF](https://www.damtp.cam.ac.uk/user/tong/string/string.pdf)\] ## 3. Open strings and [[0156 D-brane|D-branes]] - boundary terms and BC - $\frac{1}{2 \pi \alpha^{\prime}}\left[\int_0^\pi d \sigma \dot{X} \cdot \delta X\right]_{\tau=\tau_i}^{\tau=\tau_f}-\frac{1}{2 \pi \alpha^{\prime}}\left[\int_{\tau_i}^{\tau_f} d \tau X^{\prime} \cdot \delta X\right]_{\sigma=0}^{\sigma=\pi}$ - first term zero by giving initial and final states - second term zero by choosing either Dirichlet or Neumann BC - Neumann: endpoints then move at the speed of light - Dirichlet: constant position in space - D-branes - $\partial_\sigma X^a=0 \quad$ for $a=0, \ldots, p$ - $X^I=c^I \quad$ for $I=p+1, \ldots, D-1$ - mode expansion - Neumann directions: $\alpha_n^a=\tilde{\alpha}_n^a$ - Dirichlet directions: $x^I=c^I \quad, \quad p^I=0 \quad, \quad \alpha_n^I=-\tilde{\alpha}_n^I$ ### 3.1 Quantisation - $x^a$ and $p^a$ only runs along the brane - mass formula: - $M^2=\frac{1}{\alpha^{\prime}}\left(\sum_{i=1}^{p-1} \sum_{n>0} \alpha_{-n}^i \alpha_n^i+\sum_{i=p+1}^{D-1} \sum_{n>0} \alpha_{-n}^i \alpha_n^i-a\right)$ - $a$ is a normalisation constant - open and closed - $S O(1, p) \times S O(D-p-1)$ requires $D=26$ and $a=1$, same as for closed strings: suggests that two theories are not different - ground state - $M^2=-\frac{1}{\alpha^{\prime}}$: again tachyonic, half of the closed string tachyon - this time confined to the brane - better understood compared to the closed string tachyon: the potential is known; the brane is unstable and dissolves into closed string modes - the global minimum of the potential is unknown - first excited states - longitudinal modes: $\alpha_{-1}^a|0 ; p\rangle \quad a=1, \ldots, p$ - identify with a photon (introduce a gauge field $A_a$ where $a=0,...,p$) - transverse modes: $\alpha_{-1}^I|0 ; p\rangle \quad I=p+1, \ldots, D-1$ - transform as scalars under the brane Lorentz group - interpreted as fluctuations of the brane - higher excited states - mass: $M^2=\frac{1}{\alpha'}(N-1)$ - maximal spin: $J_{\max }=N=\alpha^{\prime} M^2+1$ - "Regge trajectories" - superstring - removed also the open string tachyon - possible $D$-branes: Type IIA string theory has stable D$p$-branes with $p$ even; IIB with $p$ odd ### 3.2 Brane dynamics - Dirac action - $S_{D p}=-T_p \int d^{p+1} \xi \sqrt{-\operatorname{det} \gamma}$ - dofs and flunctuations - in static gauge, $X^a=\xi^a \quad a=0, \ldots, p$ - $X^I(\xi)=2 \pi \alpha^{\prime} \phi^I(\xi) \quad I=p+1, \ldots, D-1$ <- these are fluctuations ### 3.3 Multiple branes - mass for a string stretched between two branes - $M^2=\frac{|\vec{d}-\vec{c}|^2}{\left(2 \pi \alpha^{\prime}\right)^2}+$ oscillator modes - the first term can be thought of as the mass of a classical string stretched between the two branes - tachyonic if $|\vec{d}-\vec{c}|^2<4 \pi^2 \alpha^{\prime}$: i.e., when branes close to string scale - stacked branes - Chan-Paton factor to tell us which brane the string ends on - fields: $\left(\phi^I\right)_n^m$ and $\left(A_a\right)_n^m$ - as $N$ branes coincide, $U(1)^N$ gauge symmetry enhances to $U(N)$ ## 4. CFT - conformal transformation $g_{\alpha \beta}(\sigma) \rightarrow \Omega^{2}(\sigma) g_{\alpha \beta}(\sigma)$ - if metric dynamical, e.g. in the case of string theory in Polyakov formalism, this is a residual gauge symmetry - if metric is non-dynamical, this is physical symmetry - we will focus on this, and take metric to be flat - **dynamical v.s. non-dynamical** - any 2d gravity with both diffeo. and Weyl invariance will reduce to a conformally invariant theory when the background is fixed - any conformally invariant theory can be coupled to 2d gravity to give rise to a classical theory with both diffeo. and Weyl invariance - CFT only support **massless** excitations <- no length/mass scale - will not talk about particles and S-matrices - talk about correlations functions and how operators transform ### 4.1 Classical aspects - stress tensor: a cute trick - stress tensor is **traceless** <- scale invariance - $\delta g_{\alpha \beta}=\epsilon g_{\alpha \beta}$ - $0=\delta S=\int d^{2} \sigma \frac{\partial S}{\partial g_{\alpha \beta}} \delta g_{\alpha \beta}=-\frac{1}{4 \pi} \int d^{2} \sigma \sqrt{g} \epsilon T_{\alpha}^{\alpha}$ - in complex coordinates, $T_{z\bar z}=0$ - Noether current - $J^{z}=0 \quad$ and $\quad J^{\bar{z}}=T_{z z}(z) \epsilon(z) \equiv T(z) \epsilon(z)$ - $\bar{J}^{z}=\bar{T}(\bar{z}) \bar{\epsilon}(\bar{z}) \quad$ and $\quad \bar{J}^{\bar{z}}=0$ - free theory example - conformal only if free: any interaction term would break it - $T=-\frac{1}{\alpha^{\prime}} \partial X \partial X \quad$ and $\quad \bar{T}=-\frac{1}{\alpha^{\prime}} \bar{\partial} X \bar{\partial} X$ - EOM: - $\partial \bar{\partial} X=0$ - so $X(z, \bar{z})=X(z)+\bar{X}(\bar{z})$ ### 4.2 Quantum aspects - [[0030 Operator product expansion|OPE]] - OPE contain the same information as commutation relations as well as telling us how operators transform under symmetries - [[0106 Ward identity|Ward identity]] corresponding to [[0028 Conformal symmetry|conformal symmetry]] - $\frac{i}{2 \pi} \oint_{\partial \epsilon} d z J_{z}(z) \mathcal{O}_{1}\left(\sigma_{1}\right)=-\operatorname{Res}\left[J_{z} \mathcal{O}_{1}\right]$ - => $J_{z}(z) \mathcal{O}_{1}(w, \bar{w})=\ldots+\frac{\operatorname{Res}\left[J_{z} \mathcal{O}_{1}(w, \bar{w})\right]}{z-w}+\ldots$ - holomorphic part: $\delta \mathcal{O}_{1}\left(\sigma_{1}\right)=-\operatorname{Res}\left[J_{z}(z) \mathcal{O}_{1}\left(\sigma_{1}\right)\right]=-\operatorname{Res}\left[\epsilon(z) T(z) \mathcal{O}_{1}\left(\sigma_{1}\right)\right]$ - importance - knowing OPE between conformal current and an operator => knowing how a field transforms under conformal symmetry; - knowing how an operator transforms => know something about the OPE - [[0029 Primary operator|primary operator]] - under $\delta z=\epsilon z$ and $\delta \bar{z}=\bar{\epsilon} \bar{z}$, $\delta \mathcal{O}=-\epsilon(h \mathcal{O}+z \partial \mathcal{O})-\bar{\epsilon}(\tilde{h} \mathcal{O}+\bar{z} \bar{\partial} \mathcal{O})$ - => $T(z) \mathcal{O}(w, \bar{w})=\ldots+h \frac{\mathcal{O}(w, \bar{w})}{(z-w)^{2}}+\frac{\partial \mathcal{O}(w, \bar{w})}{z-w}+\ldots$ and $\bar{T}(\bar{z}) \mathcal{O}(w, \bar{w})=\ldots+\tilde{h} \frac{\mathcal{O}(w, \bar{w})}{(\bar{z}-\bar{w})^{2}}+\frac{\bar{\partial} \mathcal{O}(w, \bar{w})}{\bar{z}-\bar{w}}+\ldots$ - **primary operator** has no terms more singular than 2nd order - so can find out exactly how they transform under all conformal transformations - under finite conformal transformations, primary fields: - $\mathcal{O}(z, \bar{z}) \rightarrow \tilde{\mathcal{O}}(\tilde{z}, \overline{\tilde{z}})=\left(\frac{\partial \tilde{z}}{\partial z}\right)^{-h}\left(\frac{\partial \overline{\tilde{z}}}{\partial \bar{z}}\right)^{-\tilde{h}} \mathcal{O}(z, \bar{z})$ ### 4.3 Example: Free scalar field - propagator - $\left\langle\partial^2 X(\sigma) X\left(\sigma^{\prime}\right)\right\rangle=-2 \pi \alpha^{\prime} \delta\left(\sigma-\sigma^{\prime}\right)$ - can obtain either from path integral or canonical quantisation - solving for the propagator - using $\partial^2 \ln \left(\sigma-\sigma^{\prime}\right)^2=4 \pi \delta\left(\sigma-\sigma^{\prime}\right)$ - $\implies$ $\left\langle X(\sigma) X\left(\sigma^{\prime}\right)\right\rangle=-\frac{\alpha^{\prime}}{2} \ln \left(\sigma-\sigma^{\prime}\right)^2$ - fundamental field has $X(z) X(w)=-\frac{\alpha^{\prime}}{2} \ln (z-w)+\ldots$ - so not nice - whereas $\partial X(z) \partial X(w)=-\frac{\alpha^{\prime}}{2} \frac{1}{(z-w)^{2}}+$ non-singular - stress tensor and normal ordering - $T=-\frac{1}{\alpha^{\prime}}: \partial X \partial X: \equiv-\frac{1}{\alpha^{\prime}} \operatorname{limit}_{z \rightarrow w}(\partial X(z) \partial X(w)-\langle\partial X(z) \partial X(w)\rangle)$ - then $\langle T\rangle=0$ - from OPE, we can get the following primaries - $\partial X$ has $h=1$ and $\bar h=0$ - $:e^{ikX}:$ has weight $h=\bar h=\alpha^\prime k^2/4$ - also, $T$ itself is not a primary, and has weights $(h,\tilde{h})=(2,0)$ ### 4.4 The central charge - see [[0033 Central charge]] - for a free scalar, $c=\tilde{c}=1$; with $D$ non-interacting free scalars, $c=\tilde{c}=D$ - suggest that they measure the number of dof - transformation of energy - $\tilde{T}(\tilde{z})=\left(\frac{\partial \tilde{z}}{\partial z}\right)^{-2}\left[T(z)-\frac{c}{12} S(\tilde{z}, z)\right]$ - $S(\tilde{z},z)$: Schwarzian - [[0306 Weyl anomaly|Weyl anomaly]] - $\left\langle T_\alpha^\alpha\right\rangle=-\frac{c}{12} R$ - holds for all states - reason: it comes from UV divergences, and at short distances, all finite energy states look the same - since we also have $\left\langle T_\alpha^\alpha\right\rangle=-\frac{c}{12} R$, when $R\ne0$, we must have $c=\tilde{c}$ - derivation - start with - $\partial_z T_{z \bar{z}}(z, \bar{z}) \partial_w T_{w \bar{w}}(w, \bar{w})=\bar{\partial}_{\bar{z}} T_{z z}(z, \bar{z}) \bar{\partial}_{\bar{w}} T_{w w}(w, \bar{w})=\bar{\partial}_{\bar{z}} \bar{\partial}_{\bar{w}}\left[\frac{c / 2}{(z-w)^4}+\ldots\right]$ - using - $\bar{\partial}_{\bar{z}} \bar{\partial}_{\bar{w}} \frac{1}{(z-w)^4}=\frac{1}{6} \bar{\partial}_{\bar{z}} \bar{\partial}_{\bar{w}}\left(\partial_z^2 \partial_w \frac{1}{z-w}\right)=\frac{\pi}{3} \partial_z^2 \partial_w \bar{\partial}_{\bar{w}} \delta(z-w, \bar{z}-\bar{w})$ - n.b. $1/z$ secretly depends on $\bar{z}$ - get $T_{z \bar{z}}(z, \bar{z}) T_{w \bar{w}}(w, \bar{w})=\frac{c \pi}{6} \partial_z \bar{\partial}_{\bar{w}} \delta(z-w, \bar{z}-\bar{w})$ - under Weyl transformation - $\delta\left\langle T_\alpha^\alpha(\sigma)\right\rangle=-\frac{1}{2 \pi} \int \mathcal{D} \phi e^{-S}\left(T_\alpha^\alpha(\sigma) \int d^2 \sigma^{\prime} \omega\left(\sigma^{\prime}\right) T_\beta^\beta\left(\sigma^{\prime}\right)\right)$ - using $TT$ OPE and integrating by parts - $\delta\left\langle T_\alpha^\alpha\right\rangle=\frac{c}{6} \partial^2 \omega$ - then $\left\langle T_\alpha^\alpha\right\rangle=-\frac{c}{12} R$, at least infinitesimally - dimensional argument and reparameterisation invariance ensure its validity for finite transformations - [[0406 Cardy formula|Cardy formula]] - $S(E) \sim \sqrt{c E}$ - so $c$ counts the density of high energy states ### 4.5 The [[0032 Virasoro algebra|Virasoro algebra]] - commutators from OPE - $\left[L_m, L_n\right]=\left(\oint \frac{d z}{2 \pi i} \oint \frac{d w}{2 \pi i}-\oint \frac{d w}{2 \pi i} \oint \frac{d z}{2 \pi i}\right) z^{m+1} w^{n+1} T(z) T(w)$ - $\left[L_m, L_n\right]=(m-n) L_{m+n}+\frac{c}{12} m\left(m^2-1\right) \delta_{m+n, 0}$ - central term missing if we just do coordinate transformations - because the conformal transformation is a diff followed by a Weyl rescaling - the Weyl rescaling gives the central extension - Verma module - built from raising operators $L_{-n}$, $n>0$ - they are irreps of the Virasoro algebra - vacuum: $h=0$ - it has the max number of symmetries: annihilated by $L_0$ too (not $L_{-n}$ though: would be inconsistent with the central term) - unitarity - $\left.\left|L_{-1}\right| \psi\right\rangle\left.\right|^ 2=\left\langle\psi\left|L_{+1} L_{-1}\right| \psi\right\rangle=\left\langle\psi\left|\left[L_{+1}, L_{-1}\right]\right| \psi\right\rangle=2 h\langle\psi \mid \psi\rangle \geq 0$ - $\left.\left|L_{-n}\right| 0\right\rangle\left.\right|^ 2=\left\langle 0\left|\left[L_n, L_{-n}\right]\right| 0\right\rangle=\frac{c}{12} n\left(n^2-1\right) \geq 0$ - so $h\ge0$ and $c\ge0$ ### 4.6 The [[0025 Operator-state correspondence|state-operator map]] - now relates primary operators to primary states, both independently defined earlier - free scalar example - inverting the expansion of the primary field $\partial X$ - $\alpha_n=i \sqrt{\frac{2}{\alpha^{\prime}}} \oint \frac{d z}{2 \pi i} z^n \partial X(z)$ - using $\partial X\partial X$ OPE, get - $[\alpha_m,\alpha_{n}]=m\delta_{m+n,0}$ - same as canonical quantisation! - vacuum: - $\Psi_0\left[X_f\right]=\int^{X_f(r)} \mathcal{D} X e^{-S[X]}$ - indeed annihilated by $\alpha_m$, $m\ge0$ - excited states - $\alpha_{-m}|0\rangle=\left|\partial^m X\right\rangle$ - open string, or CFT with boundaries - infinite strip mapped to the upper half plane - $T_{\alpha \beta} t^\beta$ still a conserved current, $t^\alpha$ being tangent to the boundary - BC: $T_{\alpha \beta} n^\alpha t^\beta=0 \quad$ at $\operatorname{Im} z=0$ - i.e., $T_{z z}=T_{\bar{z} \bar{z}}$ - extension to the whole plane: - $T_{z z}(z)=T_{\bar{z} \bar{z}}(\bar{z})$ - for closed string, we have both $T$ and $\bar{T}$ in the whole plane; now we just have $T$ in the whole plane, which is the same as having both of them in the upper-half plane - consequently, only one set of Virasoro generators ## 6. String interactions ### 6.1 What to compute - S-matrix - via operator-state map, they becomes sphere worldsheets with vertex operator insertions - Weyl invariance requires the vertex operators be on-shell: this is related to why we can only compute on-shell correlation functions in string theory - sum over topology - $\chi=2-2 h=2(1-g)$ - $h$: number of handles; $g$: genus - string coupling is played by $e^{\lambda}\equiv g_s$ - because of the $e^{-2 \lambda(1-g)}$ in from of the action ### 6.2 Closed string amplitudes at tree level - on a sphere, can use diff $\times$ Weyl to transform the metric to a flat metric - remnant gauge symmetry: $SL(2,\mathbb{C})/\mathbb{Z}_2=PSL(2,\mathbb{C})$ - consider transformations that are smooth everywhere on the sphere - $z \rightarrow \frac{a z+b}{c z+d}$, $ad-bc=1$ - this is the conformal Killing group - tachyon amplitudes - take $m$ tachyons - $\mathcal{A}^{(m)}\left(p_1, \ldots, p_m\right)=\frac{g_s^{m-2}}{\operatorname{Vol}(S L(2 ; \mathbf{C}))} \int \prod_{i=1}^m d^2 z_i\left\langle\hat{V}\left(z_1, p_1\right) \ldots \hat{V}\left(z_m, p_m\right)\right\rangle$ - where $g_s \int d^2 z e^{i p_i \cdot X} \equiv g_s \int d^2 z \hat{V}\left(z, p_i\right)$ - result - after computing the Gaussian integrals - $\mathcal{A}^{(m)} \sim \frac{g_s^{m-2}}{\operatorname{Vol}(S L(2 ; \mathbf{C}))} \delta^{26}\left(\sum_i p_i\right) \int \prod_{i=1}^m d^2 z_i \prod_{j<l}\left|z_j-z_l\right|^{\alpha^{\prime} p_j \cdot p_l}$ - 4-point amplitude - can use $SL(2,\mathbb{C})$ to fix three points - one integral to perform - $\mathcal{A}^{(4)} \sim g_s^2 \delta^{26}\left(\sum_i p_i\right) \frac{\Gamma\left(-1-\alpha^{\prime} s / 4\right) \Gamma\left(-1-\alpha^{\prime} t / 4\right) \Gamma\left(-1-\alpha^{\prime} u / 4\right)}{\Gamma\left(2+\alpha^{\prime} s / 4\right) \Gamma\left(2+\alpha^{\prime} t / 4\right) \Gamma\left(2+\alpha^{\prime} u / 4\right)}$ - "Virasoro-Shapiro amplitude" - poles in the amplitude - first pole: $s=-\frac{4}{\alpha^{\prime}}$ - mass of tachyon - other poles: $s=4(n-1) / \alpha^{\prime}$ - mass of higher states of closed string - i.e. the amplitude sums up an infinite number of tree-level diagrams exchanging all the different intermediate states - by looking at powers of momentum - $\mathcal{A}^{(4)} \sim \sum_{n=0}^{\infty} \frac{t^{2 n}}{s-M_n^2}$ - conclude: the highest spin particle at level $n$ has spin $J=2n$ (matches with the results from canonical quantisation) - comment about duality - we sum over both $s$ and $t$ channels in usual field theory, but not in string theory - in string theory, can sum over either channel: "duality" - high energy - at each higher $n$, the exchange of higher spin particles gives a worse divergence; but the infinite sum is softer than than finite order result - graviton scattering - computing graviton scattering (with matter) shows that the amplitude is the same as [[0554 Einstein gravity|Einstein-Hilbert action]] with $\kappa^2 \approx g_s^2\left(\alpha^{\prime}\right)^{12}$ ### 6.3 Open string scattering - coupling - Euler: $\chi=2-2 h-b$ - $g_{\text {open }}^2=g_s$ - conformal Killing group - now has BC: $\operatorname{Im} z=0$ $\implies$ $a,b,c,d\in \mathbb{R}$ - so we now have $S L(2 ; \mathbb{R}) / \mathbb{Z}_2$. - Veneziano amplitude - fixed three points using residual gauge symmetry; remaining point is free but restricted to $x\in[0,1]$ due to operator ordering (since they are on the boundary) - $\mathcal{A}^{(4)} \sim g_s\left[B\left(-\alpha^{\prime} s-1,-\alpha^{\prime} t-1\right)+B\left(-\alpha^{\prime} s-1,-\alpha^{\prime} u-1\right)+B\left(-\alpha^{\prime} t-1,-\alpha^{\prime} u-1\right)\right]$ - poles at $s=\frac{n-1}{\alpha^{\prime}}$: spectrum of the open string - D-brane tension - tension tells us the coupling between brane and gravity - in our new language, it's the strength of interaction between a closed string state and an open string - see this diagram: - ![[Rsc0006_fig39.png]] - $T_p \sim \frac{1}{l_s^{p+1}} \frac{1}{g_s}$ ### 6.4 One-loop amplitude - let us look at the torus diagram (genus-1) - to construct a torus - identify $z \equiv z+2 \pi \quad$ and $\quad z \equiv z+2 \pi \tau$ - $\tau$ is some complex number - moduli space - the space of conformally inequivalent tori, parameterised by $\tau$ - equivalent ones (related by modular transformations) - $T$ transformation: $\tau \rightarrow \tau+1$ and $S$ transformation $\tau \rightarrow-1 / \tau$ - a general one constructed using their combinations: - $\tau \rightarrow \frac{a \tau+b}{c \tau+d}$ - $ad-bc=1$, and they are *integers* - fundamental domain - $|\tau| \geq 1 \quad$ and $\quad \operatorname{Re} \tau \in\left[-\frac{1}{2},+\frac{1}{2}\right]$ - we need to integrate over it because we sum over metrics - $SL(2,\mathbb{Z})$ invariant measure: - $\int \frac{d^2 \tau}{(\operatorname{Im} \tau)^2}$ - residual gauge symmetry - conformal Killing group for torus: $U(1)\times U(1)$ - higher genus - moduli space has dimension $3g-2$ - no conformal Killing vectors when $g>1$ - one loop partition function - $Z[\tau]=\operatorname{Tr} e^{-2 \pi(\operatorname{Im} \tau)\left(L_0+\tilde{L}_0\right)} e^{-2 \pi i(\operatorname{Re} \tau)\left(L_0-\tilde{L}_0\right)} e^{2 \pi(\operatorname{Im} \tau)(c+\tilde{c}) / 24}$ - i.e. vacuum amplitude on the torus is an sum over all states in the theory with the above weighting - define $q=e^{2 \pi i \tau} \quad, \quad \bar{q}=e^{-2 \pi i \bar{\tau}}$ - then $Z[\tau]=\operatorname{Tr} q^{L_0-c / 24} \bar{q}^{\tilde{L}_0-\tilde{c} / 24}$ - for a single free scalar field - $Z_{\text {scalar }}[\tau] \sim \frac{1}{\sqrt{\alpha^{\prime} \operatorname{Im} \tau}} \frac{1}{(q \bar{q})^{1 / 24}} \prod_{n=1}^{\infty} \frac{1}{1-q^n} \prod_{n=1}^{\infty} \frac{1}{1-\bar{q}^n}$ - for the string partition function - $Z_{\text {string }}=\int d^2 \tau \frac{1}{(\operatorname{Im} \tau)} \frac{1}{\left(\alpha^{\prime} \operatorname{Im} \tau\right)^{13}} \frac{1}{q \bar{q}}\left(\prod_{n=1}^{\infty} \frac{1}{1-q^n}\right)^{24}\left(\prod_{n=1}^{\infty} \frac{1}{1-\bar{q}^n}\right)^{24}$ - $Z_{\text {string }}=\int \frac{d^2 \tau}{(\operatorname{Im} \tau)^2}\left(\frac{1}{\sqrt{\operatorname{Im} \tau}} \frac{1}{\eta(q)} \frac{1}{\bar{\eta}(\bar{q})}\right)^{24}$ (using the Dedekind eta function) ## 7. Low energy effective actions - first, just write down the naive generalisation: - $S=\frac{1}{4 \pi \alpha^{\prime}} \int d^2 \sigma \sqrt{g} g^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X^\nu G_{\mu \nu}(X)$ - to see that it actually comes from the closed string graviton - expand $G_{\mu \nu}(X)=\delta_{\mu \nu}+h_{\mu \nu}(X)$ - then $Z=\int \mathcal{D} X \mathcal{D} g e^{-S_{\text {Poly }}-V}=\int \mathcal{D} X \mathcal{D} g e^{-S_{\text {Poly }}}\left(1-V+\frac{1}{2} V^2+\ldots\right)$ - where $V=\frac{1}{4 \pi \alpha^{\prime}} \int d^2 \sigma \sqrt{g} g^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X^\nu h_{\mu \nu}(X)$ - this is just the graviton vertex operator - in other words, the curved metric $G_{\mu \nu}$ is indeed built from the quantised gravitons ### 7.1 Einstein's equation - $\alpha^\prime$ expansion: ok if radius of curvature is much bigger that string length scale $\sqrt{\alpha'}$ - beta functions - $\beta_{\mu \nu}(G) \sim \mu \frac{\partial G_{\mu \nu}(X ; \mu)}{\partial \mu}$ - need it to vanish to have conformal invariance at quantum level - compute it by a nice choice of coordinates - $G_{\mu \nu}(X)=\delta_{\mu \nu}-\frac{\alpha^{\prime}}{3} \mathcal{R}_{\mu \lambda \nu \kappa}(\bar{x}) Y^\lambda Y^\kappa+\mathcal{O}\left(Y^3\right)$ - then the action is $S=\frac{1}{4 \pi} \int d^2 \sigma \partial Y^\mu \partial Y^\nu \delta_{\mu \nu}-\frac{\alpha^{\prime}}{3} \mathcal{R}_{\mu \lambda \nu \kappa} Y^\lambda Y^\kappa \partial Y^\mu \partial Y^\nu$ - can compute the beta function by treating it as a 2d QFT - get $\beta_{\mu \nu}(G)=\alpha^{\prime} \mathcal{R}_{\mu \nu}=0$: vacuum Einstein's equation - beta functions and Weyl invariance - break down of conformal invariance on flat worldsheet is the same as breakdown of Weyl invariance on a curved worldsheet - in dim-reg, considering $g_{\alpha \beta}=e^{2 \phi} \delta_{\alpha \beta}$ - $S=\frac{1}{4 \pi \alpha^{\prime}} \int d^{2+\epsilon} \sigma e^{\phi \epsilon} \partial_\alpha X^\mu \partial^\alpha X^\nu G_{\mu \nu}(X)$\approx \frac{1}{4 \pi \alpha^{\prime}} \int d^{2+\epsilon} \sigma(1+\phi \epsilon) \partial_\alpha X^\mu \partial^\alpha X^\nu G_{\mu \nu}(X)$ - this remains even if $\epsilon\to0$: $S=\frac{1}{4 \pi \alpha^{\prime}} \int d^2 \sigma \partial_\alpha X^\mu \partial^\alpha X^\nu\left[G_{\mu \nu}(X)+\alpha^{\prime} \phi \mathcal{R}_{\mu \nu}(X)\right]$ - from stress tensor: $T_{\alpha \beta}=+\frac{4 \pi}{\sqrt{g}} \frac{\partial S}{\partial g^{\alpha \beta}}=-2 \pi \frac{\partial S}{\partial \phi} \delta_{\alpha \beta}$ - $\Rightarrow \quad T_\alpha^\alpha=-\frac{1}{2} \mathcal{R}_{\mu \nu} \partial X^\mu \partial X^\nu$ - we can define beta function via $T_\alpha^\alpha=-\frac{1}{2 \alpha^{\prime}} \beta_{\mu \nu} \partial X^\mu \partial X^\nu$; this gives the same beta function as before ### 7.2 Other couplings - 2-form field - $B_{\mu\nu}$ is like $A_\mu$ but for a 2d object (worldsheet) - gauge symmetry $B\to B + dC$ - field strength $H=d B$ - dilaton - $S=\frac{1}{4 \pi \alpha^{\prime}} \int d^2 \sigma \sqrt{g}\alpha'\Phi(X)R^{(2)}$ - it breaks Weyl invariance - but it's ok: this term has $\alpha'$ in front, and its breaking will be canceled by loop corrections from other terms - when it is a constant, $S_{\text {dilaton }}=\lambda \chi$ - define $\Phi_0=\operatorname{limit}_{X \rightarrow \infty} \Phi(X)$ - then $g_s=e^{\Phi_0}$ - beta function for all fields - $\left\langle T_\alpha^\alpha\right\rangle=-\frac{1}{2 \alpha^{\prime}} \beta_{\mu \nu}(G) g^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X^\nu-\frac{i}{2 \alpha^{\prime}} \beta_{\mu \nu}(B) \epsilon^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X^\nu-\frac{1}{2} \beta(\Phi) R^{(2)}$ - $\beta_{\mu \nu}(G)=\alpha^{\prime} \mathcal{R}_{\mu \nu}+2 \alpha^{\prime} \nabla_\mu \nabla_\nu \Phi-\frac{\alpha^{\prime}}{4} H_{\mu \lambda \kappa} H_\nu^{\lambda \kappa}$ - $\beta_{\mu \nu}(B)=-\frac{\alpha^{\prime}}{2} \nabla^\lambda H_{\lambda \mu \nu}+\alpha^{\prime} \nabla^\lambda \Phi H_{\lambda \mu \nu}$ - $\beta(\Phi)=-\frac{\alpha^{\prime}}{2} \nabla^2 \Phi+\alpha^{\prime} \nabla_\mu \Phi \nabla^\mu \Phi-\frac{\alpha^{\prime}}{24} H_{\mu \nu \lambda} H^{\mu \nu \lambda}$ ### 7.3 Low energy EFT - look for 26-dim action that reproduces $\beta(G)=\beta(B)=\beta(\phi)=0$ - $S=\frac{1}{2 \kappa_0^2} \int d^{26} X \sqrt{-G} e^{-2 \Phi}\left(\mathcal{R}-\frac{1}{12} H_{\mu \nu \lambda} H^{\mu \nu \lambda}+4 \partial_\mu \Phi \partial^\mu \Phi\right)$ - Einstein frame - $S=\frac{1}{2 \kappa^2} \int d^{26} X \sqrt{-\tilde{G}}\left(\tilde{\mathcal{R}}-\frac{1}{12} e^{-\tilde{\Phi} / 3} H_{\mu \nu \lambda} H^{\mu \nu \lambda}-\frac{1}{6} \partial_\mu \tilde{\Phi} \partial^\mu \tilde{\Phi}\right)$ - corrections to Einstein's equations - $\alpha'$ correction and $g_s$ corrections - $g_s$ corrections are topological, but the beta functions arise from UV regulations - moduli space have points where the Riemann surface degenerates (eg a circle pinching off), so the UV divergences do care about global topology ### 7.4 Some simple solutions - compatifications - $\mathbf{R}^{1,3} \times \mathbf{X}$ - where $\mathbf{X}$ is a Ricci-flat compact manifold - brane tensions - fundamental string does not depend $g_s$ - D-brane: $1/g_s$ - magnetic brane: $1/g_s^2$ - coupling to gravity is $T\kappa^{2}\sim T/g_s^2$ - so gravitational backreaction of string and D-brane can be neglected - but magnetic brane is is order one - non-critical dimensions - extra term in EFT: $S=\frac{1}{2 \kappa_0^2} \int d^D X \sqrt{-G} e^{-2 \Phi}\left(\mathcal{R}-\frac{1}{12} H_{\mu \nu \lambda} H^{\mu \nu \lambda}+4 \partial_\mu \Phi \partial^\mu \Phi-\frac{2(D-26)}{3 \alpha^{\prime}}\right)$ - linear dilaton CFT - solve $\partial_\mu \Phi \partial^\mu \Phi=\frac{26-D}{6 \alpha^{\prime}}$ - choose $\Phi=\sqrt{\frac{26-D}{6 \alpha^{\prime}}} X^1 \quad D<26$; $\Phi=\sqrt{\frac{D-26}{6 \alpha^{\prime}}} X^0 \quad D>26$ - an exact CFT underlying it; can solve to all orders in $\alpha$ ### 7.5 D-branes revisited: background gauge fields - vertex operator associated to photon leads to - $S_{\text {end-point }}=\int_{\partial \mathcal{M}} d \tau A_a(X) \frac{d X^a}{d \tau}$ - i.e. the end points of the open string is charged under the background gauge field - the beta function - action splits up: $S=S_{\text {Neumann }}+S_{\text {Dirichlet }}$ - $S_{\text{Dirichlet}}$ describes free fields, and does not affect beta function - vanishing of beta function gives $\partial_b F_{a c}\left[\frac{1}{1-4 \pi^2 \alpha^{\prime 2} F^2}\right]^{a b}=0$ - Born-Infeld action - a corresponding action that reproduces the EOM above is given by - $S=-T_p \int d^{p+1} \xi \sqrt{-\operatorname{det}\left(\eta_{a b}+2 \pi \alpha^{\prime} F_{a b}\right)}$ - to leading order in $\alpha$, get Maxwell: - $S=-T_p \int d^{p+1} \xi\left(1+\frac{\left(2 \pi \alpha^{\prime}\right)^2}{4} F_{a b} F^{a b}+\ldots\right)$ ### 7.6 The [[0299 Dirac-Born-Infeld action|DBI action]] - consider both the gauge field and the fluctuation of the brane - DBI action: - $S_{D B I}=-T_p \int d^{p+1} \xi \sqrt{-\operatorname{det}\left(\gamma_{a b}+2 \pi \alpha^{\prime} F_{a b}\right)}$ - where $\gamma_{a b}=\frac{\partial X^\mu}{\partial \xi^a} \frac{\partial X^\nu}{\partial \xi^b} \eta_{\mu \nu}$ - dof - $D$ fields in total - but we only want $D-p-1$ transverse physical dof - extra ones removed by using reparameterisation invariance - leading order - $S=-\left(2 \pi \alpha^{\prime}\right)^2 T_p \int d^{p+1} \xi\left(\frac{1}{4} F_{a b} F^{a b}+\frac{1}{2} \partial_a \phi^I \partial^a \phi^I+\ldots\right)$ - free Maxwell coupled to free massless scalars ($\phi^I=X^I / 2 \pi \alpha^{\prime}$) - coupling to closed string fields - $S_{D B I}=-T_p \int d^{p+1} \xi e^{-\tilde{\Phi}} \sqrt{-\operatorname{det}\left(\gamma_{a b}+2 \pi \alpha^{\prime} F_{a b}+B_{a b}\right)}$ - $\gamma_{a b}=\frac{\partial X^\mu}{\partial \xi^a} \frac{\partial X^\nu}{\partial \xi^b} G_{\mu \nu}$ - $\Phi=\Phi_0+\tilde{\Phi}$ - gauge invariance - actually $F$ alone is not gauge invariant - the combination $B_{a b}+2 \pi \alpha^{\prime} F_{a b}$ is - open string deposits $B$ charge on the brane, where it is converted to $A$ charge ### 7.7 The [[0071 Yang-Mills|YM]] action - consider $N$ coincident D-branes - there is no non-abelian generalisation of DBI, but the low energy action can be written down - $S=-\left(2 \pi \alpha^{\prime}\right)^2 T_p \int d^{p+1} \xi \operatorname{Tr}\left(\frac{1}{4} F_{a b} F^{a b}+\frac{1}{2} \mathcal{D}_a \phi^I \mathcal{D}^a \phi^I-\frac{1}{4} \sum_{I \neq J}\left[\phi^I, \phi^J\right]^2\right)$ - when we separate two D-branes, i.e. giving vev to scalar fields - then the mass of W-boson can be computed, i.e. Higgs mechanism - $M_W^2=\left(\phi_2-\phi_1\right)^2=T^2\left|X_2-X_1\right|^2$ - it's the mass of the string stretched between two branes - when the branes are well-separated, they are heavy, and the positions are described by diagonal elements $\phi^I_1,...,\phi^I_N$; when they are close, they become light and responsible for dynamics of the brane: the full $N\times N$ matrix needed to describe the positions