%Date: Wed, 19 Feb 1997 17:46:42 -0500
%From: Edward Witten
\input harvmac
Term 2
Problem Set 5
(1) Consider in two dimensions an $SO(n)$ gauge theory with
scalar fields $\phi$ consisting of $n-1$ copies of the $n$ dimensional
representation of $SO(n)$ and a potential energy
\eqn\guggo{V=\sum_{i=1}^{n-1}\lambda_i\left[(\phi_i,\phi_i)-a_i^2\right]^2
+\sum_{i1$) exchanges the two copies of Poincar\'e.
Deduce that for $n=4$ this theory is ``reducible'' as the product
of two decoupled theories, in one of which one copy of Poincar\'e acts
and in the other the second.
(c) Use the bose-fermi correspondence to verify this and identify
the factors. You should be able to draw an inference about the
symmetries of the quantum sine-Gordon model at a special value
of the coupling. (You drew the same inference last week in a different
way.)
(d) Now consider the case $n=3$. Show that $R$ and $R'$ are
supersymmetries, so that the model is supersymmetric for $n=3$.
Can you verify this using the bose-fermi correspondence?
A reference for this problem is my paper ``Some Properties Of
The $(\bar\psi\psi)^2$ Model in Two Dimensions'' which will be available
from Paula Bozzay in C building.
(3) The following problem is actually one that was assigned in the fall but
not discussed (see problem set 3 from term one, which
you can also consult for some additional background).
It is the beginning of a series that will hopefully
prepare us to do some things with supersymmetric gauge theories
in two and four dimensions.
We work locally for the present; it takes some more work to make
a global version.
\def\R{{\bf R}}\def\Z{{\bf Z}}
We consider a (4,4) supermanifold $W$ with
the structure described in Bernstein's lectures -- two complex
conjugate integrable distributions $A_+$ and $A_-$ of dimension $(0,2)$
such that $\{A_+,A_-\}$ generates $TM/(A_+\oplus A_-)$, $A_+$ and $A_-$
are endowed with volume forms.
A chiral superfield is a field $\phi$ that is a function on
$W/A_-$. If $M$ is a complex manifold, a chiral map $\Phi:W\to M$ is
one such that, for $f$ a local holomorphic function on $M$,
$\Phi^*(f)$ is a chiral function on $W$. In that case, $\Phi^*(\bar f)$ is
antichiral, that is, it is a function on $W/A_+$.
Let $K$ be a function on $W$ and consider the Lagrangian
\eqn\uggu{L=\int_Wd^4yd^4\theta\, \Phi^*(K).}
$d^4y\,d^4\theta$ is the section of the Berezinian coming from the
volume forms on $A_\pm$.
(a) Show that $L$ is invariant under $K\to K+f+\bar f$, where $f$ is
a local holomorphic function on $W$.
(b) Recall that any $K$ determines the two-form
$\omega=-i\bar\partial\partial K$
which for a suitable class of $K$'s is the Kahler class of a Kahler
metric $g$ on $M$. Of course, $\omega$ is invariant under the
transformation considered in (a).
Show that $L$ can be written just in terms of $\omega$ (or equivalently
$g$) and that the ``reduced'' part of $L$ is the harmonic map action,
specialized to maps to a Kahler manifold:
\eqn\huggu{L=\int_{W_{red}} d^4y g_{i\bar j}(d X^i,dX^{\bar j})}
where $ X^i$ are local holomorphic coordinates on $M$.
(c) Specialize now to the flat model with $W=\R^{4,4}$.
$A_+$ is generated by
\eqn\ppp{D_A={\partial\over\partial\theta^A}-
i\sigma^{m}_{A\dot A}\bar\theta^{\dot A}
{\partial\over\partial y^m}}
where $\theta^A$ are odd coordinates transforming as ``positive''
spinors of $Spin(1,3)=SL(2,{\bf C})$, $\bar \theta^{\dot A}$ are
their complex conjugates, and $y^m$ are even coordinates.
$\sigma$ is the isomorphism $S_+\otimes S_-= V$.
$A_-$ is generated by
\eqn\ppp{D_{\dot A}={\partial\over\partial\theta^{\dot A}}-
i\sigma^{m}_{A\dot A}\bar\theta^{ A}
{\partial\over\partial y^m}.}
Impose invariance under a space-like translation of $W$, say
a translation of $y^3$ (the metric being $(dy^0)^2-\sum_{i=1}^3 (dy^i)^2$
to get a model on $\R^{3,4}$. Show that this is equivalent to
the $\R^{3,2}$
$\sigma$ model described in problem 1(e) of term 1, problem
set 3 (but specialized to the
case that $M$ is Kahler). Here is an attempt to explain the strategy:
under $Spin(1,2)$, the two spinor representations of $Spin(1,3)$
become isomorphic. So after the reduction, $A_+$ and $A_-$ are
naturally isomorphic. One can in an $SO(1,2)$-invariant way
take the ``real'' combination $\chi^A=\theta^A+\bar\theta^A$ of
the $\theta$'s and $\bar\theta$'s. The $\R^{3,2} $ we want
has even coordinates $y^i$, $i=0,1,2$, and odd coordinates
$\chi^A$. By ``integrating over the fibers'' of a map from $\R^{3,4}$
to $\R^{3,2}$, one reduces \huggu\ (specialized to $y^3$-independent maps)
to the $\R^{3,2}$ Lagrangian of problem 1(e).
\end