Contents

# Contents

## Idea

What is called light-cone gauge quantization is one approach to the quantization of sigma models whose target space has a lightlike Killing vector. The strategy of this approach is to gauge fix the metric and diffeomorphisms of the worldvolume such that also the worldvolume has a lightlike Killing vector field and such that this is mapped to the given one on target space.

This typically fixes most of the available gauge freedom, and the strategy is then to apply quantization to what remains. For more on this general idea see at quantization commutes with reduction.

Often this is considered for target space being Minkowski spacetime $\mathbb{R}^{d-1,1}$ and with $X^+ \coloneqq X^0 - X^1$ denoting one of its canonical lightlike coordinates. If $\tau$ denotes similarly a lightlike coordinate function on the worldvolume, then the condition of light-cone gauge reads

$X^+ = p^+ \tau$

for some proportionality constant $p^+$, the light cone momentum This is how light cone gauge appears in much of the physics literature.

If in addition, one assumes that the coordinate function $X^+$ on spacetime is periodic, hence that it actually runs along a circle fiber (which some authors take to be literally lightlike, while others consider the limit of boosts of a spacelike circle fiber), then the lightcone momentum $p^+$ becomes quantized in units of the inverse radius $R$ of this circle

(1)$p^+ \;=\; N/R \phantom{AAA} N \in \mathbb{Z} \,.$

The quantization of a sigma model in this situation is hence called the discrete light cone quantization or DLCQ, for short.

Often it turns out that negative values of $N$ in (1) can be neglected or integrated out, so that

$N \in \mathbb{N}$

becomes a natural number-parameter akin to that considered in the 't Hooft double line construction of gauge theories, and then the large N limit of the discrete light cone quantization becomes of interest.

## Applications

### Quantization of Green-Schwarz super $p$-brane sigma models

Light cone gauge quantization is the only method by which Green-Schwarz super p-brane sigma models have been quantized, to date.

### BFSS matrix model

Specifically, applying light-cone gauge quantization to the Green-Schwarz sigma model for the M2-brane on 11d Minkowski spacetime, combined with a certain regularization of the remaining l9ight-cone Hamiltonian yields the BFSS matrix model.

### BMN matrix model

Alternatively, applying the light cone gauge quantization of the Green-Schwarz sigma-model of the M2-brane not on Minkowski spacetime but, more generally, on 11d pp-wave spacetimes (which are Penrose limits of the near horizon geometry of the black M2-branes/M5-branes) yields the BMN matrix model.

## References

### General

For QCD:

Review

All the standard introductory texts on string theory have sections devoted to light-cone quantization. For instance

### Quantization of the M2-brane sigma-model to a matrix model

The Poisson bracket-formulation of the classical light-cone gauge Hamiltonian for the bosonic relativistic membrane and the corresponding matrix commutator regularization is due to:

On the regularized light-cone gauge quantization of the Green-Schwarz sigma model for the M2-brane on (super) Minkowski spacetime, yielding the BFSS matrix model:

Original articles:

Observation that the spectrum is continuous:

Review:

The generalization to pp-wave spacetimes (leading to the BMN matrix model):