%From: "Eric D'Hoker"
%Date: Wed, 26 Mar 1997 16:08:48 -0500 (EST)
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%%%% STRING THEORY : Problem Set 9, March 27, 1997
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\def\det{{\rm det}}
\def\Det{{\rm Det}}
\def\tr{{\rm tr}}
\def\Tr{{\rm Tr}}
\def\12{{1 \over 2}}
\def\ker{{\rm Ker}}
\def\O{{\cal O}}
\def\G{{\cal G}}
\centerline{{\bf STRING THEORY}}
\centerline{ Problem Set \# 9}
\centerline{March 27, 1997}
\bigskip
\bigskip
In this problem set, we investigate the so-called lightcone gauge
quantization method. The key ingredient of this method is to eliminate
at the classical level the constraints resulting from Diff($\Sigma$)
invariance (and also other symmetries for superstring theories),
by solving for the constraints explicitly. Quantization is then
carried out on the remaining ``transverse degrees of freedom" only,
and automatically results in a Hilbert space, with positive definite norm.
Solving the constraints is achieved by choosing a null vector $\xi$ in
Minkowski space-time (i.e. a vector on the lightcone, whence the
name of the method). Poincare symmetry will be realized in the
quantum theory provided Poincare generators can be constructed
that obey the Poincare algebra, and this is possible again only in
the critical dimension.
Let $M={\bf R}^D$ be flat Minkowski space-time of dimension $D$, with
flat metric $\eta= {\rm diag} (-,+,\cdots,+,+)$. We choose the null vector
$\xi = (1,0,\cdots,0,1)$, introduce associated lightcone coordinates
$$
x^\pm = {1 \over \sqrt 2} (x^0 \pm x^{D-1})
$$
and denote the remaining ``transverse coordinates" by $x^i$, $i=1,\cdots,
D-2$. The scalar product in $M$ reads
$$
v\cdot w = v^\mu w^\nu \eta _{\mu \nu} = -v^+ w^- - v^- w^+ + v^i w^i.
$$
The string worldsheet $\Sigma$ is taken to be an annulus with global
complex coordinates $z, \bar z$ and worldsheet metric $g=2|dz|^2$.
\bigskip
\noindent
{\bf Closed Bosonic String}
\medskip
We specialize here to the closed oriented bosonic string; the open
and unoriented strings may be constructed by truncation.
The classical field equations are $\partial _z \partial _{\bar z} x^\mu
(z,\bar z)=0$, and are solved by
$$
x^\mu (z, \bar z) = q^\mu + i p ^\mu \ln |z|^2 +
i \sum _{n\not=0} {1 \over n} ( x^\mu _n z^{-n} + \tilde x ^\mu _n
\bar z ^{-n} )
\eqno (1)
$$
The classical constraints are
$$
\partial _z x \cdot \partial _z x =
\partial _{\bar z} x \cdot \partial _{\bar z} x = 0
\eqno (2)
$$
The generators of the Poincare algebra are given by $p^\mu$ and
by the Lorentz generators
$$
J^{\mu \nu} = q^\mu p^\nu - q^\nu p ^\mu + S^{\mu \nu}
+ \tilde S^{\mu \nu}
\eqno (3)
$$
where the spin operators of the massive states are defined by
$$
S^{\mu \nu} = -i\sum _{n=1} ^\infty
\{ x^\mu _{-n} x^\nu _n - x^\nu _{-n} x^\mu _n \}
\qquad
\tilde S^{\mu \nu} = -i\sum _{n=1} ^\infty
\{ \tilde x^\mu _{-n} \tilde x^\nu _n
- \tilde x^\nu _{-n} \tilde x^\mu _n \}
\eqno (4)
$$
a) Show that it is consistent with the field equations for $x^\mu$
to make a choice of coordinates $z, \bar z$ such that
$x^+ _n = \tilde x^+_n=0$ for all
$n\in {\bf Z}, ~ n\not=0$.
b) In these coordinates, solve for the classical constraint equations (2),
by solving explicitly for $x^-(z,\bar z)$ in terms of
$x^i(z,\bar z)$, for $x^-_n$ in terms of $x^i_n$ and for $\tilde x_n^-$
in terms of $\tilde x_n ^-$.
c) Quantize the independent degrees of freedom $x^i _n$ and $\tilde x^i _n$
by requiring for $m,n \in {\bf Z}$ and $i,j=1,\cdots,D-2$ that
$$
[x^i _m, \tilde x ^j _n]=0
\qquad \qquad
[x^i _m, x ^j _n]=[\tilde x^i _m, \tilde x ^j _n]
= \delta ^{ij} \delta _{m+n,0}.
\eqno (4)
$$
It is convenient to work on a sector of Fock space with definite
eigenvalue of the operator $p^+$, so that $p^+$ may effectively be
regarded as a c-number.
Define a suitable renormalization for the composite field $x^-$.
This renormalization will be unique up to a constant, which we
denote by $a$ for later reference.
Show that the oscillators $ p^+ x^-_n$ (and similarly $p^+ \tilde x_n ^-$)
obey a Virasoro algebra with central charge $D-2$.
d) Show that the algebra of the transverse Lorentz group, SO(24),
generated by $J^{ij}$ closes for any $D$ and any value of $a$.
Show that the commutators $[J^{ij}, J^{k-}]$ obey the
Lorentz algebra of SO(1,D-1), for all values of $a$ and $D$.
e) From the general structure of the Lorentz generators, show
that the commutators $[J^{i-}, J^{j-}]$ must be of the form
$$
[J^{i-}, J^{j-}]= {1 \over (p^+)^2} \sum _{n=1} ^\infty
\Delta _n (x_{-n} ^i x_n ^j - x_{-n} ^j x_n ^i ) + \{ x \to \tilde x \}.
\eqno (5)
$$
Determine $\Delta _n$, and find the conditions under which $\Delta _n$
vanishes for all $n\in {\bf Z}, ~n\not=0$.
(This is a necessary condition for the
closure of the Lorentz and Poincare algebras.)
f) Show that the remaining Poincare structure relations are obeyed.
g)
We define the Del Giudice Di Vecchia Fubini (DDF) operators (for
simplicity only in the left moving sector) by
$$
V^{i} (k) = \oint dz \partial _z x^i _+ e^{ik\cdot x_+ }
$$
where
$$
x^i (z, \bar z) = x^i _+ (z) + x^i _- (\bar z).
$$
Show that, for a suitable choice of momenta $k$, the operators
map positive norm states into positive norm states, and physical
states into physical states.
\bigskip
\bigskip
\noindent
{\bf Ramond-Neveu-Schwarz strings}
\medskip
Generalize the set-up and the analysis carried out above to the case
of the Ramond-Neveu-Schwarz string.
\end
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