Strings in AdS_3 and Spectral Flow

Studying string theory in $\text{AdS}_3$ is useful in understanding $\text{AdS/CFT}$ correspondence as it sheds the light on one half of the “million dollar” conjecture beyond gravity approximation. The recent progress with the exact $\text{AdS}_3\text{/CFT}_2$ duality makes it important to study the string theory and understand the isomorphism of the physical states on both the sides. In this blog-post, I will explain the bosonic version of the string theory with emphasis on something known as “Spectral Flow”.

The spectral flow was an answer to some earlier issues raised by the study of string theory on $\text{AdS}_3$ which involved existence of an upper bound in the internal energy of string (independent of the string coupling) and absence of the “long string” states in the spectrum. In 2000, Maldacena and Ooguri wrote seminal paper (actually it was a series of three papers 1, 2 and 3) which established the consistency of the theory and opened gates for further study in the field and the million dollar conjecture.

The worldsheet description of the theory takes the form of $SL(2,\mathbb{R})$ WZW model. Just like any story in physics, we first start with the classical version of the model. The action is $S = \frac{k}{8\pi\alpha^\prime}\int d^2\sigma\text{Tr}\left(g^{-1}\partial gg^{-1}\partial g\right)+k\Gamma_{\text{WZ}}$ . Here $g$ is the $SL(2,\mathbb{R})$ group element. Next we define left and right moving coordinates on the world sheet as $x^{\pm}= \tau\pm\sigma$ . Then the equation of motion is given by $\partial_{-}(\partial_{+}gg^{-1})=0$ such that the general solution is given by $g = g_{+}(x^{+})g_{-}(x^{-})$ . Furthermore the string is closed under $\sigma\to \sigma + 2\pi$ which leads to constraint $g_{+}(x^{+}+2\pi) = g_{+}(x^{+})M$ and $g_{-}(x^{-}-2\pi) = M^{-1}g_{-}(x^-)$ where the monodromy $M$ is defined only upto the conjugation by $SL(2,\mathbb{R})$ and classical solutions of WZW model are defined according to conjugacy class of $M$ . Of course we still need to impose the Virasoro constraint that the stress energy tensor vanishes on-shell (just like the bosonic strings theory on flat-space).

The classical stage is set. Now we study some examples. Consider the solution $g_{+}=Ue^{iv_{+}(x^{+})\sigma_{2}}$ and $g_{-}=e^{iu_{-}(x^{-})\sigma_{2}}V$ (it is easy to check that it satisfies equation of motion for $U = \mathbb{I}$ ) where $U$ and $V$ are constant $SL(2,\mathbb{R})$ matrices. The energy-momentum tensor of the solution is $T^{\text{AdS}}_{\pm\pm}=-k(\partial_{\pm}v_{+}/u_{-})^2$ . Now let there be excitations in the compact $\mathcal{M}$ of $\text{AdS}_3\times\mathcal{M}$ and let the $T^{\mathcal{M}}_{\pm\pm}=h$ then the Virasoro constraints imply $(\partial_{+}v_{+})^2=(\partial_{-}u_{-})^2=\frac{h}{k}$ . Solving for $v_{+}/u_{-}$ and substituting in the solution ansatz yields the result $g = U\begin{pmatrix}\cos\alpha\tau & \sin\alpha\tau\\-\sin\alpha\tau & \cos\alpha\tau\end{pmatrix}V$ where $\alpha = \pm\sqrt{\frac{4h}{k}}$ . $\sigma$ drops out and solution is only in terms of $\tau$ . This can be interpreted as “string collapsing to a point”. For $U=V=\mathbb{I}$ , it is essentially a particle sitting at the center of $\text{AdS}_3$ as shown in the figure.

PointString

Similarly, one can write spacelike solution in the form $g = U\begin{pmatrix}e^{\alpha\tau} & 0\\0 & e^{\alpha\tau}\end{pmatrix}V$ .

Now given a classical solution $g=\tilde{g}_{+}\tilde{g}_{-}$ , one can generate new solutions $g_{+}=e^{i\frac{w_R}{2}x^{+}\sigma_2}\tilde{g}_{+}$ and $g_{-}=\tilde{g}_{-}e^{i\frac{w_L}{2}x^{-}\sigma_2}$ which may be regarded as an action of the loop group $\hat{SL}(2,\mathbb{R})\times \hat{SL}(2,\mathbb{R})$ . This is known as the spectral flow in CFT literature.