In this section, we develop a relation between the Gomis-Ooguri string theory and the discrete light cone quantization (DLCQ) of relativistic string theory. We will begin by reviewing the latter, following the review \cite{Bigatti:1997jy}. Roughly speaking, DLCQ is an approach to any quantum (field)theory where the theory is placed in a lightlike box. This discretizes the compact momentum in the lightlike direction—hence the \emph{D} in DLCQ. As a consequence, the spectrum of the theory is reorganized into Galilean blocks. The observation that DLCQ breaks manifest Lorentzian symmetries suggests a natural connection with the basic idea of this thesis—namely that of non-Lorentzian symmetries in string theory. In the DLCQ prescription it is conjectured that the Lorentzian symmetries are recovered in the limit where the compact discrete momentum quantum number becomes infinitely large. Sometimes, this is even taken as part of the definition of DLCQ itself. Here, we will not assume this for obvious reasons and always work with a fixed and finite compact momentum.\\
Historically, the main motivation to study DLCQ in the context of string theory comes from the BFSS formulation \cite{Banks:1996vh, Susskind:1997cw,Bigatti:1997jy, Banks:1999az} of M theory. Roughly speaking, the conjecture says that the fundamental degrees of freedom in M theory are captured by a matrix model. This is achieved by applying the DLCQ to the full eleven-dimensional theory which manifestly leads to the matrix model describing the quantum mechanics of $N$ $D0$ branes. Here, $N\in\mathds Z$ parametrizes the momentum in the compact lightlike direction. The conjecture states that the full eleven-dimensional information is recovered in the large $N\to\infty$ limit. This formulation is rather exotic from the point of view of General Relativity since a pre-existing spacetime manifold is not assumed from the start. This is in agreement with the general expectation that space and time should appear as emergent phenomena in a fundamental theory of quantum gravity. Note that what we have described here is not the original BFSS conjecture \cite{Banks:1996vh} in terms of an infinite momentum frame, but the refined one by Susskind \cite{Susskind:1997cw}.\\
As mentioned above, DLCQ is a form of light cone quantization\footnote{Lightcone quantization is reviewed in section \ref{sec:relstrquantization} on the quantization of relativistic string theory.} with an additional ingredient: one of the lightlike coordinates is taken to be periodic
\begin{align} <br>\big(x^-,x^+,x^{A'}\big)\quad\sim\quad\big(x^-,x^+ + 2\pi\,R,x^{A'}\big)\,. <br>\end{align}This can be understood intuitively as an identification of points along the lightcone as shown in figure \ref{fig:DLCQ}. The effect of this “lightlike compactification” is to quantize the canonically associated momentum \begin{align} p_+ = \frac{N}{R}\,, \end{align} for some non-negative number $N$. Similarly, we define a Hamiltonian canonically associated to the orthogonal lightlike coordinate $H=p_-$. From this we can see that [H,p_+]=0 and thus that $p_+$ is conserved. This, in turn, means that the Hilbert space splits into infinitely many subspaces, each characterized by a value of $N$: $\mathcal H = \bigoplus_N\mathcal H_N$. Let us now try to understand how the dynamics organizes itself in these. We note that we can define initial states by specifying $|x^-(i)\rangle$ on a slice $x^-=x^-(i)$ and evolving the state along $x^-$ to a final state $|x^-(f)\rangle$ as follows \begin{align} |x^-(f)\rangle = \rme^{\rmi H(x^-(f)-x^-(i))}|x^-(i)\rangle\,. \end{align} The fact that this prescription will not work for modes propagation along the $x^+$ direction will be ignored for the moment. We know that the original system is relativistic and satisfies the usual dispersion relation $p^2 + M^2 = 0$. Upon restricting to a Hilbert space with fixed $N=R\,p_+$ we can re-arrange the relativistic dispersion relation as \begin{align} H = \frac{p_{A’}^2}{2\,p_+} + E_M\,, \end{align} where $E_M = M^2/2p_+$ represents some constant, intrinsic energy that we will ignore here. This is the Hamiltonian of a free non-relativistic particle in nine dimensions and mass $p_+ = N/R$. This observation was, to our knowledge, first made by \cite{Pauli:1985ps}. One may now ask whether that is a coincidence and whether it survives when turning on interactions. The answer turns out to be that it is not a coincidence and follows purely from the study of the ten-dimensional Poincar\’e algebra in a lightcone basis.\\ \begin{figure}[t] \centering \includegraphics[width=.5\textwidth]{Figures/DLCQ.png} \caption{Schematic depiction of eq. \eqref{eq:DLCQiden}. One should see the points on the red dashed lines as identified. Note that the distance between these lines has no invariant meaning.} \label{fig:DLCQ} \end{figure} We consider the ten-dimensional Poincar\’e algebra—spanned by $(P_{\hat A}, J_{\hat A\hat B})$—and decompose the index $\hat A=(0,9,A’)=(+,-,A’)$. As before, $P_\pm = 2^{-1/2}(P_0\pm P_9)$, etc. Let us furthermore re-label some of the Lorentz generators as follows \begin{align} & G_{A’} = J_{A’+}\,, && S_{A’} = J_{A’-}\,,&& \Delta = J_{-+}\,, \end{align} and the other generators unchanged. It has a non-trivial sub-algebra larger than on spanned by $\{J_{A’B’},P_{A’}\}$, capturing the purely spatial symmetries. It is spanned by $(H=P_-,M=P_+,P_{A’},G_{A’},J_{A’B’})$ with the following non-zero commutation relations \begin{align} &[J_{A’B’},J_{C’D’}]=4\,\delta_{[A'[C’}J_{D’]B’]}\,,&&[P_{A’}, J_{B’C’}]=2\,\delta_{A'[B’}P_{c’]}\,,\notag \\ &[H,G_{A’}]=P_{A’}, && [G_{A’},P_{B’}] = \delta_{A’B’}\,M\,, \end{align} which is nothing but the Bargmann algebra in nine dimensions. Obviously the embedding presented here works for any dimension. We have already seen above that the lightcone-translations $P_+=M$ obtains the interpretation of a non-relativistic mass current. The Bargmann commutation relations also encode that fact as follows. Consider a finite boost with parameter $V^{A’}$ acting on the transversal momenta as \begin{align} \rme^{-\rmi\,V\cdot G}P_{A’}\rme^{+\rmi\,V\cdot G} = P_{A’} + M\,V_{A’}\,, \end{align} which has the form of a boost with velocity $V$ and only works when including the central extension $M$. Let us record the remaining commutators in the Poincar\’e algebra containing $(S_{A’},\Delta)$ \begin{align} &[S_{A’}, J_{B’C’}]=2\delta_{A'[B’}S_{C’]}\,, && [M,S_{A’}]=-P_{A’}\,,\notag\\ &[S_{A’},P_{B’}] = -\delta_{A’B’}H\,, && [S_{A’},G_{B’}] = -(J_{A’B’}+\Delta\delta_{A’B’}), \notag\\ &[\Delta, G_{A’}] = G_{A’}\,, && [\Delta, S_{A’}]=-S_{A’}\,,\notag\\ &[\Delta, M]=M\,, &&[\Delta, H]=-H, \end{align} These relations shows that the $\sigma^{A’}S_{A’}$ and the $\sigma\Delta$ transformations act on the value of $M$ nontrivially: $M\to M – \sigma^{A’}S_{A’}$ and $M\to (1+\sigma)M$, respectively. If we insist that the value of $M$ is fixed—as we do when quantizing the $X^+$ direction—we effectively break the Poincar\’e group to the Bargmann group. This is the explanation for the Galilean dispersion relation that we found above. More generally, any relativistic system that realizes $(S_{A’},\Delta)$ trivially will actually form a representation of the Bargmann algebra in one dimension lower.\\
Let us now consider a concrete realization of DLCQ: A scalar field $\Phi=\Phi(x^+,x^-,x^{A’})$ with arbitrary couplings. As above we consider the $x^+$ direction to be compact with radius $R$ and use the $x^-$ coordinate as a dynamical variable. It is then not hard to see that the canonical momentum reads $\pi = \partial\mathcal L/\partial\partial_-\Phi = \partial_+\Phi$, leading to the following equal time commutation relations \begin{align} \big[\Phi(x^{A’},x^+),\partial_+\Phi(y^{A’},y^+)\big] = \rmi\,\delta(x^{A’}-y^{A’})\,\delta(x^+-y^+)\,, \end{align} which does have a characteristically non-relativistic form. This can be made even more explicit by going to Fourier modes in the compact direction \begin{align} \Phi(x^{A’},x^+) = \Phi_0(x^{A’}) + \sum_{n=1}^\infty \Big(\Phi_n(x^{A’})\rme^{\rmi\,nx^+/R} + \Phi^\dagger_n(x^{A’})\rme^{-\rmi\,nx^+/R}\Big)\,. \end{align} We will refer to $\Phi_0$ as the zero mode which has no dynamics and can in principle be integrated out. For subtleties regarding the role of the zero mode see \cite{Hellerman:1997yu}. One can show that the generators $S_{A’}/\Delta$ are realized trivially on the given field content. Considering the modes in the Fourier expansion separately, the above describes a set of Schrödinger fields $\Phi_n$ with masses $n/R$ and commutation relations \begin{align} &[\Phi_m,\Phi_n]=0\,, && [\Phi_m(x^{A’}),\Phi_n^\dagger(y^{A’})] = -\delta_{mn}\delta(x^{A’}-y^{A’})\,, \end{align} as expected. For a given $N = MR$ we can only have a finite number of allowed processes. $N=1$, for example, only allows for one $\Phi_1$ particle in the spectrum which cannot split further into constituents. For $N=2$ we have two different quanta $\Phi_1$ and $\Phi_2$ for which we can consider one of four processes: $\Phi_2\to\Phi_2$, $\Phi_2\to\Phi_1\Phi_1$, $\Phi_1\Phi_1\to\Phi_2$, and $\Phi_1\Phi_1\to\Phi_1\Phi_1$. It is not hard to see that the number of allowed processes—and interesting physics—grows exponentially with $N$. Obviously, it is easier to do calculations for low values of $N$. Hence it is natural to ask how large $N$ has to be for the DLCQ to capture the essential features of a given physical situation. In the review \cite{Bigatti:1997jy} the authors give an estimate for a system of size $\rho$ and find that $N\gtrsim \rho\,M$.\\
In the above example, we have considered the DLCQ of a quantum field theory. Let us now consider the analogous situation for the DLCQ of string theory \cite{Susskind:1997cw}. Most of the features carry over from field theory or have already been discussed in section \eqref{sec:relstrquantization}. The conservation of mass turns into the conservation of string length since the range of $\sigma$ must be $N/R$. This implies that, e.g., a string of length $2/R$ can split into two strings of length $1/R$. As before, the number of allowed processes grows exponentially with $N$. Secondly, and even more interestingly, we are forced to consider string modes with non-zero winding around the lightlike direction $X^+(\tau,\sigma + 2\pi\ell_s)=X^+(\tau,\sigma) + 2\pi R w$. In other words, there are string modes with a non-zero global winding charge \begin{align} W = \int_0^{2\pi \ell_s}\,d\sigma\,\frac{\partial X^+(\sigma)}{\partial\sigma} = 2\pi R\cdot w\,, \end{align} where $w$ is referred to as the winding number as usual. This sounds very much analogous to the winding strings of ordinary string theory on a compact spatial direction. However, there is one crucial difference, namely that $X^+$ is not an independent coordinate in light cone quantization. This implies that the mass shell condition is altered with the consequence that winding strings have energy of order $R/\alpha’$, whereas unwound strings have energies of order $R/\alpha’N$, see \cite{Susskind:1997cw}. This implies that the unwound string becomes lighter and lighter in the large $N$ limit. In other words—the relative energy of wound strings with respect to unwound strings diverges in the limit $N\to\infty$. However, for large but finite $N$ the wound strings can still be produced as virtual states even if the entire process has $W=0$. This should remind us of the analogous (but reversed) situation in Gomis-Ooguri string theory discussed above. This simple observation will lead us to a T-duality relation between the DLCQ of string theory and Gomis-Ooguri string theory.\\
To make the duality precise, let us first give another viewpoint of DLCQ following Seiberg \cite{Seiberg:1997ad,Hellerman:1997yu}, relating the IMF and the DLCQ point of view more concretely. Let us regularize the light-cone frame by introducing a parameter $\varepsilon$ as follows \begin{align}\label{eq:regularizedNullFlat} d\tilde s^2 = -2\,d\tilde x^+ d\tilde x^- + \varepsilon^2\,\big(d\tilde x^+\big)^2 + \big(d\tilde x^{A’}\big)^2\,, \end{align} so that the light cone frame is realized as $\varepsilon\to 0$. For finite $\varepsilon$, however, the coordinate $\tilde x^+$ is spacelike and we can use some of the established results and intuition from the previous sections. Assuming that the compact direction is periodic with length $x^+\sim x^+ + 2\pi R$ is misleading, since the actual invariant radius is \begin{align} \tilde R_s = R\cdot \varepsilon\,, \end{align} which vanishes as $\varepsilon\to 0$. Hence we see that the DLCQ can be seen as the limit of a spacelike reduction where the size of the compact direction shrinks to zero. At the same time, however, we observe that the spacelike momentum \begin{align} P_+ = \frac{N}{\tilde R_s} = \frac{1}{\varepsilon}\,\frac{N}{R} \end{align} diverges as $\varepsilon\to 0$ since we keep $N/R$ fixed. On the other hand, however, we also know that $P_+$ is not a boost invariant property of the theory. Phrased differently, one can use longitudinal boosts with parameter $\beta\propto \tilde R_s/R = \varepsilon$ to regularize the value of $P_+$. Let us instead look directly at the invariant mass formula \eqref{eq:quantumlevelmatching} and use the expression for the metric \eqref{eq:regularizedNullFlat} to express $P^2 = -2\,P_+P_- – \varepsilon^2P_- + P_{A’}^2$ and thus find \begin{align} H = \frac{1}{\varepsilon}\,P_- = \frac{R}{2\,N}\Big(P_{A’}^2 + \frac{2}{\alpha’}(N+\tilde N -2)\Big) + \mathcal O(\varepsilon^2)\,, \end{align} which is formally very similar to the Gomis-Ooguri dispersion relation \eqref{eq:GODispersion} in the $\varepsilon\to 0$ limit. We will show now that this is not an accident. As derived in \cite{Susskind:1968} the level matching condition actually turns into $wN = N -\tilde N$. All of this suggests a T-duality like relation between the DLCQ of string theory and Gomis-Ooguri string theory as follows \begin{align}\label{eq:longRto1oR} N\quad\leftrightarrow\quad w\,, &&\mathrm{and} && R\quad\leftrightarrow\quad \ell_s^2/R\,. \end{align} Both theories decouple part of the spectrum: DLCQ does not have winding modes, whereas Gomis-Ooguri does not have momentum modes. This statement needs a qualifier because it only applies to the spectrum of asymptotic states. We have seen in both cases that there are wound string modes in DLCQ and unwound modes in Gomis-Ooguri string theory. In both cases, they have vanishing transverse momentum $P_{A’}=0$ and $N=1=\tilde N$ (for bosonic string theory where we ignore tachyons). Both states are not allowed as asymptotic states but appear as virtual states in scattering processes. In the case of Gomis-Ooguri they were identified as Newtonian gravitons that effectively transmit the Newtonian instantaneous gravitational interaction.\\
Let us make this relation even more precise by considering the T-duality relation of string theory on the background \eqref{eq:regularizedNullFlat}. Here, we essentially follow \cite{Gomis:2000bd}, though in a reverse order. To do so we first notice that the regularized lightcone background \eqref{eq:regularizedNullFlat} is not Minkowski. Thus we need to apply the Buscher rules \cite{Buscher:1987qj} mapping non-trivial background geometries to one another.\footnote{We will have a lot more to say about T-duality rules in the later chapters, in particular chapter \ref{sec:SUGRA}.} The non-zero components read \begin{align} &G_{++} = \tilde G_{++}^{-1} = \varepsilon^{-2}\,, && G_{–} = – \tilde G_{++}^{-1}\tilde G_{-+}^2 = -\varepsilon^{-2}\,, \notag \\ &B_{+-} = \tilde G_{++}^{-1}\tilde G_{+-} = -\varepsilon^{-2}\,, &&\rme^{-2\Phi} = \tilde G_{++}\rme^{-2\tilde\Phi} = \varepsilon^2\,\tilde g_s^{-2}\,. \end{align} If we furthermore note that the string sigma model has an overall factor of $\alpha’^{-1}=\ell_s^{-2}$ we see that the above is equivalent to the NCOS limit \eqref{eq:GOlimit} if we identify $\varepsilon^2 = \omega^{-1}$. The limit $\varepsilon\to 0$ is well-defined due to the critical $B$ field. What used to be the compact null direction $x^-$ has turned into a compact spatial direction. This time, however, the invariant radius turns out to be \begin{align} R_s = \frac{R}{\varepsilon}\,, \end{align} which goes to infinity as $\varepsilon\to 0$—and follows from the compositions of the T-duality rule $\tilde R_s = \alpha’/ R_s$ together with \eqref{eq:longRto1oR}. This is in agreement with the general intuition that the decompactification limit decouples part of the spectrum. As we showed around eq. \eqref{eq:masswithB} this decoupling is modified in a subtle way by tuning the background $B$ field to a critical value. The fact that we obtained the relation between the two sigma-models from a singular limit of ordinary T-duality motivates us—and motivated the authors of \cite{Danielsson:2000gi,Danielsson:2000mu,Gomis:2000bd}—to refer to the relation as longitudinal T-duality. For a targetspace point of view on this duality, see chapter \ref{sec:Dualities}.
% \begin{equation} % \eta = 2dX^-dX^+ + \delta_{ab}dX^adX^b. % \end{equation} % If we make the additional assumption that not just $\delta\eta=0$ but separately also $\delta dX^-=0$ we are forced to set $\mathbf{S}_a=0=\Delta$, thus yielding the Bargmann algebra. It is then natural to interpret $X^-$ as nonrelativistic time. So it seems we can embed nonrelativistic symmetry into relativistic relativistic symmetry of a lightcone-direction. Note the similarity to ambient formulations of conformal field theories .
\newpage \section{Summary}\label{sec:SummString} \noindent In this chapter, we have given a lightning introduction to relativistic string theory. We have tried to focus on the features that are essential to this thesis. We have motivated the spectrum of superstring theories, and thereof especially the massless NSNS states $(G,B,\Phi)$ which also parametrize the background given in the non-linear sigma model \eqref{eq:nonlinsigma}. In subsection \ref{sec:curvedTS}, we have shown that the Polyakov model has a Weyl anomaly for generic backgrounds. Cancellation of the quantum inconsistency leads to differential constraints on the background, which we identify as the targetspace equations of motion. Finally, we have introduced and discussed string theory on a background with a compact direction $\mathsf{R}^9\times\mathsf{S}^1$, the winding modes, and T-duality.\\
In section \ref{sec:GOstringtheory}, we have introduced Gomis-Ooguri string theory as a two-dimensional CFT with a targetspace interpretation. The worldsheet symmetries have shown to be the same as for relativistic string theory, leading to the same BRST system and thus the same critical dimension. The rigid targetspace symmetries are different, making this a non-Lorentzian string theory. It has a non-trivial spectrum of winding modes $w>0$ with a characteristically Galilean dispersion relation. We have also mentioned that the scattering of such states involves an instantaneous gravitational interaction.\\
The Gomis-Ooguri string shares many features with relativistic string theory. As shown in section \ref{sec:alphaptozero} this is not a coincidence since it can be obtained as a critical $\alpha’\to 0$ limit. This also gives a physical interpretation to the smaller Hilbert space of the Gomis-Ooguri theory. The limit $\alpha’\to 0$ limit is an infinite tension limit that makes all the states infinitely heavy, leaving an empty theory. However, in the prescription by \cite{Gomis:2000bd} we also tune the Kalb-Ramond field to a critical value which “salvages” part of the spectrum: the winding modes with $w>0$.\\
In section \ref{sec:DLCQTduality}, we review the discrete lightcone quantization (DLCQ) of string theory. This is an exotic quantization procedure where string theory is placed in a lightlike box with a fixed momentum in the lightlike direction. This breaks Lorentzian symmetries to Bargmann symmetries and organizes the relativistic spectrum into non-relativistic blocks. The conjecture states that one recovers Lorentzian physics in the limit of infinite compact momentum. Here, we notice a duality relation between the DLCQ of string theory at finite compact momentum and Gomis-Ooguri string theory. This duality relation is known as longitudinal T-duality.