%Date: Thu, 27 Feb 1997 10:33:36 -0500
%From: Edward Witten
%have fun....
\input harvmac
Superhomework, Term 2
The purpose of the following is to describe facts that we will
need in order to study supersymmetric quantum field theories.
In doing so, we will mainly be interested in flat models (for the
supermanifolds on which the quantum field theory is formulated), and
in any event our supermanifolds will always be ``split'' (because
we will not include supergravity) which generally means that
one won't lose much that is essential by considering the flat model.
(Even without doing quantum field theory, one of the important things
one learns from these exercises is that the interesting relativistic
field theories
have supersymmetric generalizations, with some restriction
on the dimension of spacetime.)
We make one small change in notation. A manifold with $p$ even
and $q$ odd coordinates will be said to be of dimension $(p|q)$.
This frees up the comma for other purposes; for instance, if the
reduced space has a metric of signature $(n,m)$ (with $n+m=p$) we
might call this a manifold of dimension $(n,m|q)$.
I haven't tried to explain everything below. You will have to consult
Wess and Bagger, or other references, especially for the four-dimensional
gauge theory. However, if you understand these problems, you are probably
familiar with the classical facts that are most important background
for beginning to discuss supersymmetric quantum theories.
\def\R{{\bf R}}
(1) The first part is review from last fall. We consider a $(3|2)$
supermanifold $\R^{(3|2)}$
(or one can dimensionally reduce to $\R^{(n|2)}$ with $n<3$
by imposing invariances under some translations). We call
the fermionic coordinates $\theta^A$, $A=1,2$ and the bosonic
coordinates $y^{AB}$ (symmetric in $A$ and $B$).
The supersymmetry
generators are
\eqn\joggo{Q_A={\partial\over\partial \theta^A}+i\theta^B
{\partial\over \partial y^{AB}}}
and commute with
\eqn\joggo{D_A={\partial\over\partial \theta^A}-i\theta^B
{\partial\over \partial y^{AB}}}
which is used in writing Lagrangians.
The symbol $\epsilon^{AB}$ ($=-\epsilon^{BA}$) will denote a
translation-invariant volume form
on what Bernstein called the odd distribution -- the
odd subspace of the tangent bundle generated by the $D$'s.
The super-Poincar\'e group is generated by the $Q_A$, their anticommutators
$\partial/\partial y^{AB}$, and an $SU(2)\cong SO(3)$ for which the
$\theta^A$ and the $y^{AB}$ are in the two- and three- dimensional
representations, respectively.
A superfield is just a function $\Phi$ on $\R^{(3|2)}$. Let $X$
be a fixed Riemannian manifold with metric $g$. Let $\Phi$ be a map
from $\R^{(3|2)}$ to $X$. Picking local coordinates on $X$, we describe
$\Phi$ via functions $\Phi^I$ and write simply $g_{IJ}$ for
the pullback via $\Phi$.
(a) Then as we hopefully
remember from last fall
we can make the supersymmetric Lagrangian
\eqn\aba{L_0=\int d^3y\,d^2\theta \,\epsilon^{AB}g_{IJ}D_A\Phi^ID_B\Phi^J.}
Refresh your recollections; show that this is equivalent to an
ordinary sigma model with target space $X$ plus fermions and describe
the fermions.
Determine the conserved supercurrent.
(b) Now let $h$ be a smooth function on $X$ and let $L=L_0+L_1$ where
\eqn\baba{L_1=\int d^3y\,d^2\theta\,\Phi^*(h).}
Identify the reduced model. In particular, what is the potential
energy and what is the mass term for the fermions?
Suppose that $X$ is compact and that $h$ has only isolated critical
points. Can you
state a lower bound for the number of classical minima of the energy?
(After dimensional reduction to $\R^{(1|2)}$ this observation was of
course the starting point for my article on Morse theory in 1982. Even if $h$
does not have isolated critical points, there is a sense in which the
same lower bound on the number of vacua holds.)
(c)
If $X$ is noncompact it may be the case that there is no critical
point of $h$ and moreover that the set with $|dh|