2.7 $6D$ superconformal field theories

^[1] and check section 2 of ^[2] for a different viewpoint based on pure spinors. Also, see this article ^[3] by Witten, where he explains how quantum field theory is nothing but a 1D quantum gravity on the worldline, and the Feynman diagrams are nothing but the 1D manifolds that need to be summed over.

Fundamental Physics theories can be viewed in 2 different formalisms. In physics, we consider entities like particles, strings or more general $p$-branes. How these branes feel the spacetime is called Worldvolume formalism. How the spacetime feels these branes is called Spacetime formalism¹ Some things are easier on one side, \& others are easier on the other. The difference is whether we are mapping to or from spacetime. We are essentially reducing the number of spacetime dimensions, but we now have to deal with more number of fields.

Newtonian mechanics is a map $x^i:\mathbb R\to \mathbb {R} ^{n}$, where $x^i$ are Klein-Gordon fields as explained in particleworldline. Here, $\mathbb {R} ^{n}$ is called the Euclidean target space, and $\mathbb R$ is called time. Special relativity (Lorentzian target space), the Heisenberg picture of Quantum Mechanics, the Heisenberg picture of Relativistic Quantum Mechanics, the worldline formalism of QFT, and the worldsheet formalism of string theory are the natural generalizations of Newtonian mechanics. All these theories also have a formulation in the target space (Koopman–von Neumann formulation, Schrödinger picture of QM \& RQM, QFT, String Field Theory).

Remark

A comparison between different theories.

\scalebox{0.7}{\parbox{\linewidth}{ \begin{table}[H] \begin{center} \begin{tabular}{|c|cc|} \hline Limits & \multicolumn{1}{l|}{Worldvolume formalisms} & \multicolumn{1}{l|}{Spacetime formalisms}
& \multicolumn{1}{l|}{$x^\mu$=operators} & \multicolumn{1}{l|}{$x^\mu$=label}
\hline \hline &Unknown M-theory formalisms & \multicolumn{1}{l|}{} \ \hline Compactified on $S^1$ & \multicolumn{1}{l|}{\color{Green}\checkmark} Worldsheet string theory $X^\mu:\mathbb{C}\to \mathbb {R}^{9,1}$} & \multicolumn{1}{l|}{SFT $\phi:\mathbb {R} ^{9,1}\to \text{Operators acting on string Fock space}$}\ \hline $l_s\to 0$: QFT & \multicolumn{1}{l|}{Worldline formalism $x^\mu:\mathbb R\to \mathbb {R}^{n-1,1}$} & \multicolumn{1}{l|}{QFT $\phi:\mathbb {R} ^{n-1,1}\to \text{Operators acting on Fock space}$ {\color{Green}\checkmark}} \ \hline Fock $\to$ Hilbert: RQM & \multicolumn{1}{l|}{Heisenberg picture $x^i:\mathbb R\to \mathbb {R} ^{n-1,1}$} & \multicolumn{1}{l|}{Schrödinger picture $\psi: \mathbb {R} ^{n-1,1} \to \mathbb C $ {\color{Green}\checkmark}} \ \hline $c\to \infty$: QM & \multicolumn{1}{l|}{Heisenberg picture $x^i:\mathbb R\to \mathbb {R} ^{n}$} & \multicolumn{1}{l|}{Schrödinger picture $\psi: \mathbb {R} ^{n} \to \mathbb C $ {\color{Green}\checkmark}} \ \hline $\hbar \to 0$: CM & \multicolumn{1}{l|}{\color{Green}\checkmark} Newtonian mechanics $x^i:\mathbb R\to \mathbb {R} ^{n}$} & \multicolumn{1}{l|}{Koopman–von Neumann mechanics $\psi: \mathbb {R} ^{n} \to \mathbb C $} \ \hline \end{tabular} \end{center} \caption{Summary of fundamental physics formalisms.} \end{table} }}

Note that the ``Fock $\to$ Hilbert’’ limit on the target space that gives Relativistic Quantum Mechanics can also be stated as the classical worldline gravity limit (matter on the worldline is still quantum). If we consider quantum gravity on the worldline, we need to sum over different worldlines, which gives rise to QFT (multiple particles) on the target space. Each worldline is nothing but a Feynman diagram. I didn’t include Classical Field Theory (Electromagnetism, GR, etc) in the table, but they, too, have Feynman diagrams; see ^[4]. It is often said that only tree-level diagrams contribute to classical field theory, but there are exceptions; see ^[5]. ^[6] studied $\hbar\to 0$ limit on both sides. If we only quantize worldline gravity, we get classical field theory; if we only quantize worldline matter, we get worldline quantum mechanics. Only when we quantize both do we get QFT. This worldline formalism for QFT is the main subject of this section. {\color{Green}\checkmark} tells which of the 2 formalisms is popular.

In principle, everything in my \href{http://ksr.onl/FP}{FP notes} can also be again discussed in this section, but I will only briefly discuss things. In this section, we consider (0+1)-D theories that have no space.

Sigma model: The historical terminology is related to the sigma meson. But the current definition is that if a worldvolume theory has some manifold as target space, then it is a sigma model.

Non-linear sigma model: If the target space is not a linear manifold (affine space), then the theory is called a non-linear sigma model. Example: In Newtonian mechanics, if a particle is confined to a sphere, we can consider that sphere as target space. If we consider the entire Euclidean space as the target space, then we also need to include normal forces, which are basically Lagrange multipliers enforcing the constraint to be on the sphere.

Note that (classical or quantum) gravity theories with extended objects, such as the 10D SUGRA that has 1-branes ^[7] and 11D SUGRA that has 2-branes cannot be described in worldline formalism.

\etocsettocdepth.toc{subsection} \subsection{Classical \& quantum mechanics} In this section, we discuss nonrelativistic and relativistic particles. Almost all string theory textbooks start with particles before considering the string. But I have never seen in any book/notes a proper interpretation of the particle case, such as what it means to consider the particle as a sigma model. I will discuss it here, but since subsection particleworldline is my own interpretation, beware of mistakes \& email me for any corrections.

\subsubsection{Newtonian mechanics} Historically, the first sigma model was discovered by Newton. Newton directly gave the equations of motion (EoM) on the worldline \& interpreted the meaning of the worldline EoM directly on target space. Later, Lagrange gave the worldline Lagrangian, but he also directly interpreted its meaning in the target space.

Galilean spacetime

See the below image and ^[8] and chapter 5 of ^[9]. Spacetime with Galilean invariance is a fibre bundle with the worldline as the base space. Nonrelativistic metric: In the limit $c\to\infty$, $ {\displaystyle ds^{2}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}}$ gives 2 metrics $\dfrac{ds^{2}}{c^2}=-dt^{2}=g_{tt}dt^{2}\Rightarrow g_{tt}=-1$ and the spatial metric $ {\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}}$. The spatial metric is not invariant under Galilean boosts, which is why Galilean spacetime is not a trivial fibre bundle (Cartesian product). But for the sake of intution, I will write as $x^i:\mathbb R\to \mathbb {R} ^{n}$ even though, technically speaking, the target space is a fibre bundle, and I should write ${\displaystyle \pi:{ST}\to \mathbb{T},}$ where $\mathbb{ST}$ is the spacetime and $\mathbb{T}$ is the worldline.

Wick rotation of the Worldline $\mathbb{T}$ gives an Euclidean line $\mathbb{R}$.

\begin{equation} {\displaystyle S= \int dt \left( {\frac {1}{2}}m\sum_i (\dot {x}^i)^{2}-V({x}^i)\right)} \end{equation} The above is the worldline action, and the below is my interpretation of what the quantities mean on the worldline

$x^i:\mathbb R\to \mathbb {R} ^{n}$ is a group of classical scalar fields living on the worldline. Each field outputs a single real number. They are not particles² living on the worldline because particles don’t have a value at each point on the worldline.
Target spacetime symmetry: In the target space, we have Galilean coordinate invariance. This on the worldline becomes field redefinitions, it is not a real flavour symmetry, and it’s a fake redundancy that doesn’t give any conservation laws. Note that under field redefinitions, the target space metric $g_{ij}$ becomes non-trivial. If the target space has real symmetries, which need \href{https://en.wikipedia.org/wiki/Killing_vector_field}{Killing vector fields}, then that gives real flavour symmetry. If we neglect the electromagnetic interaction between protons and neutrons and since their masses are almost the same, we have an approximate $SU(2)$ flavour symmetry, which gives the conservation of Isospin ($I_3$) from the Noether’s theorem. Similarly, energy and momentum are also charges of the flavour symmetry.
$m$: Mass on the target space has the interpretation as the measure of inertia. More precisely, the mass on the target space is the central charge of the Bargmann group (it is a central extension of the Galilei group). On the worldline, it is the central charge of a flavour symmetry instead of the spacetime symmetry. If we add another mass $m’$ then its coordinates again have another flavour symmetry whose central charge is $m’$.
Worldline mass: Target spacetime mass becomes the central charge of a flavour symmetry on the worldline. But if we have a potential $V({x}^i)=\frac{1}{2}k\sum_i ( {x}^i)^{2}$. Then, the Lagrangian will separate, and each scalar field will not interact with others. \begin{equation} {\displaystyle S= \int dt \sum_im\left( {\frac {1}{2}} (\dot {x}^i)^{2}-\frac{1}{2}\omega^2 ( {x}^i)^{2}\right)} \end{equation} This is literally the Klein–Gordon Lagrangian for each scalar field separately with mass $\omega=\sqrt{\frac{k}{m}}$. The scalar field solutions in higher dimensions will be of the form $e^{-ip\cdot x}$ where $p^2=-m^2$. In 1D, it just becomes $p^\mu p_\mu=g_{tt}E^2=-\omega^2\Rightarrow E=\pm\omega$ and the solutions are $e^{\pm i\omega t}$. We get many solutions in higher dimensions, so we need to sum them up using the Fourier transformation. Here, we just sum over the 2 to get sinusoidal solutions. If we have $V({x}^i)=\frac{1}{2}\sum_i k_i( {x}^i)^{2}$, then the target spacetime symmetry is broken, and similarly the flavour symmetry on the worldline is broken, and each field has different mass $\omega_i=\sqrt{\frac{k_i}{m}}$.
More complicated interactions: Free particles on the target space gave us massless free scalar fields. If there was a Harmonic potential interaction on the target space, then each free scalar field became massive with mass given by the harmonic mode frequencies $\omega_i$. But still, on the worldline, there was no interaction so far between scalar fields. All these scalar fields get coupled to each other for more complicated potentials like the target space radial inverse square law. We might simplify the problem by using a field redefinition corresponding to the target space spherical coordinate transformation. But Cartesian coordinates had a nice Klein–Gordon interpretation on the worldline. But the fields $\theta,\phi$ don’t have such nice interpretations as they are coupled to the field $r$. Note that if we Taylor expand the $\operatorname{s i n}^{2} \theta$ we get infinite interactions between the 3 worldline fields. \begin{equation}\label{worldlineinteractingmetric} L=\frac{1} {2} m [ \dot{r}^{2}+r^{2} ( \dot{\theta}^{2}+\operatorname{s i n}^{2} \theta\dot{\phi}^{2} ) ]-V ( r ). \end{equation} Similarly, in generic coordinates, the metric term in the Lagrangian $\frac{1}{2} m g_{ij} (\vec{x}) \dot{x}^i \dot{x}^j - V(\vec{x})$ can be Taylor expanded to get infinite interactions between the worldline classical scalar fields.

Vertex Operators in CM

Rutherford scattering

\subsubsection{Relativistic mechanics}

\subsubsection{Quantum mechanics} ^[10]

Vertex Operators in QM

\begin{equation} V_{\mathbf k}(t)=e^{i\mathbf k\cdot \hat{x}} \end{equation} These are nothing but the momentum translation operators in Quantum Mechanics. These Vertex Operators are extremely useful in scattering amplitudes.

Scattering

Hydrogen atom as 1d quantum gravity

Section 5 of ^[11].

\subsubsection{Relativistic quantum mechanics}

Old Covariant Quantization of relativistic particle

^[12]

Light cone quantization of relativistic particle

5.1 in ^[13] and chapter 11 of ^[14].\

BV quantization of relativistic particle

^[15]

Pure spinor

^[16]

\etocsettocdepth.toc{paragraph} \subsection{Perturbative QFT (sum over worldlines)}

Main reviews are ^[17].The full nonlinear worldline structure of Yang-Mills theory was recently understood ^[18]. For a massive spin 1 theory, see ^[19]. Target space SUSY is discussed in ^[20]. For perturbative 1D gravity, see ^[21]. Also see ^[22] for decretized 1D quantum gravity.

``Given this state of affairs one might hope that the world line could also play a role in describing the coupling of RR-background fields once Ramond states can be successfully implemented in it. This is the purpose of this note. Since bosonization is not available on the world line and neither is the operator product expansion it is not, a priory, clear how to construct the analog of spin fields on the world line. What does carry over from the world sheet to the world line, are equal time commutators, via contour integrals, which contains much less information, making it harder to guess what the “resolution” of the world line fermion should be.” From ^[23].

Orbifolds

^[24]

\subsection{Worldline nonperturbative QFT} ^[25]

Remark

Even if we assume that worldline formalism can only perturbative QFT, it still doesn’t mean that it is less fundamental than target space QFT because we can uniquely derive the target space Lagrangian from the worldline theory~^[26]. So, in principle, it contains the same amount of information as QFT.

\subsection{Gravity} ^[27] and also see ^[28] in the context of AdS/CFT. Many papers that cite ^[29] are also relevant.

\subsection{Non-geometric target space} ^[30]
\href{https://ncatlab.org/nlab/show/Connes-Lott-Chamseddine-Barrett+model}{Connes-Lott-Chamseddine-Barrett model}:

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Normally, people call these as first quantized vs second quantized formalisms, but that’s wrong, as the so-called ``first quantized” worldvolume versions of QFT and String Theory are multi-particle, unlike the historical first quantized single particle RQM. But as Weinberg says in his volumes, this terminology is already wrong in the context of QFT because QFT is not obtained by quantizing the inconsistent RQM (an already quantum theory) but by quantizing classical fields. ↩
Particles on the worldline are trivial. Their position must always be the same \& their velocity is $0$. ↩