%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% ws-procs975x65.tex   :   27 January 2003
%% Text file to use with ws-procs975x65.cls written in Latex2E.
%% The content, structure, format and layout of this style file is the 
%% property of World Scientific Publishing Co. Pte. Ltd. 
%% Copyright 1995, 2002 by World Scientific Publishing Co. 
%% All rights are reserved.
%%
%% Main Text is 11/14pt                                   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[epsf,amstex]{ws-procs975x65}
\newcommand{\be}{\begin{eqnarray}}
\newcommand{\ee}{\end{eqnarray}}
\renewcommand{\vec}[1]{{\bf #1}}
\newcommand{\vecg}[1]{\mbox{\boldmath $#1$}}
%%%%%%%%%%%%%%%%%%%%%%%%%
%
% You can add your macros or additional packages here
%
%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Substitute the authors' names and the title (you can shorten it) in the  \markboth below.
% It is for running headers.
%
%%%%%%%%%%%%%%%%%%%%%%%%%

\markboth{\quad \small \sl A.V. Smilga \hfill}{\hfill\small \sl Low-dimensional sisters of 
Seiberg--Witten effective theory \quad }

\setcounter{page}{523}

\begin{document}

\title{ Low-dimensional sisters of Seiberg--Witten effective theory}


\author{A.~V. SMILGA\footnote{\,\uppercase{O}n leave of absence from \uppercase{ITEP, M}oscow, \uppercase{R}ussia.}}

\address{ SUBATECH, Universit\'e de
Nantes,\\  4 rue Alfred Kastler, BP 20722, Nantes  44307, 
France} %\footnote{\,\uppercase{O}n leave of absence from \uppercase{ITEP, M}oscow, \uppercase{R}ussia.} \\
% E-mail: smilga@math.univ-nantes.fr } 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

\abstracts{
We consider the theories obtained by dimensional reduction to $D=1,2,3$ of
$4D$ supersymmetric Yang--Mills theories and calculate there the effective 
low-energy lagrangians describing moduli space dynamics --- the low-dimensional analogs
of the Seiberg--Witten effective lagrangian. The effective theories thus obtained
are rather beautiful and interesting from a mathematical viewpoint. In addition, their
study allows one to understand better some essential features of $4D$ supersymmetric
theories, in particular the non-renormalization theorems. }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may put here the table of contents, just uncomment 3 lines below %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.5cm}
\tableofcontents
\newpage

\section{Introduction}
 Ian's scientific style had two attractive  features: {\it (i)} his works 
used, more often than not, rather non-trivial modern mathematical constructions; 
{\it (ii)} they were always based on a solid and clear physical idea.
This text also represents an exercise (a  review of exercises) in ``physical mathematics'',
 involving an interplay between purely mathematical geometric constructions and the simple
physical notion of effective lagrangians.

  Effective lagrangians/hamiltonians arise naturally in  theories involving two energy scales. 
Integrating out the ``fast'' variables (the degrees of freedom with large characteristic
excitation energy), one obtains the effective lagrangian involving only ``slow'' variables
which describes the low--energy dynamics. The classic example is the Born--Oppenheimer 
effective hamiltonian
describing the dynamics of nuclei in a molecule, obtained after integrating out the electronic
degrees of freedom. The Euler--Heisenberg effective lagrangian describing nonlinear soft photon
interactions, the effective chiral lagrangian for QCD, the Wilsonean renormalized effective lagrangian
(where modes with high frequency up to $\Lambda_{UV}$ are integrated out) all belong to this
class.

 Another lagrangian in this class is the famous Seiberg--Witten effective lagrangian \cite{SW}. 
Let us remind ourselves of its salient features. Consider the pure $4D$ ${\cal N} = 2$ supersymmetric Yang--Mills
theory. The lagrangian
written in terms of ${\cal N} = 1$ superfields 
is\,\footnote{\,Our convention is close to that of Ref. \cite{WB},
$\theta^2 = \theta^\alpha \theta_\alpha\, \bar\theta^2 = \theta_{\dot{\alpha}} \theta^{\dot{\alpha}} 
$ ,\  $\int d^2 \theta \, 
\theta^2 = \int d^2 \bar\theta \, 
\bar\theta^2 = 1$. In the following we will also use $(\sigma^\mu)_{\alpha \dot\beta} = 
\{1, \vecg{\tau}\}_{\alpha \dot\beta}$,
$(\bar\sigma^\mu)^{\dot\beta \alpha} = 
\{1, -\vecg{\tau}\}^{\dot\beta \alpha}$. 
Our Minkowski  metric
$\eta_{\mu\nu} = {\rm diag} (1,-1,-1,-1)$ differs in sign from Wess' and Bagger's conventions  
 and we include an extra
factor of $2$ in the definition of $V$. }
 \be
\label{N2SYM1}
 {\cal L} \ =\ \frac 1{g^2} {\rm Tr} \left \{ \int d^2 \theta \, W^\alpha W_\alpha  + 2 \int d^2\theta 
 d^2\bar\theta\,\Phi  e^{-V} \bar \Phi e^V   \right\}
 \ee
In the bosonic sector, it includes the gauge field $A_\mu$ and a complex scalar $\phi$ belonging
to the adjoint representation of the gauge group,
 \be  
 \label{N2SYMcomp}
 g^2{\cal L} \ =\ - \frac 12 {\rm Tr}\{ F_{\mu\nu}^2 \} + 2{\rm Tr} \{{\cal D}_\mu 
\bar \phi {\cal D}_\mu  \phi \}
-   {\rm Tr} \{ [\bar \phi, \phi]^2 \} \ + {\rm fermions}
 \ee
The lagrangian is most economically expressed as (see e.g. \cite{Lykken})
   \be
\label{N2SYM2}
{\cal L} \ =\  \frac 1 {g^2}  {\rm Tr} \int d^2 {\theta}  d^2\tilde{\theta} \, {\cal W}^2 \ ,
 \ee
 where ${\cal W}(x_L, \theta_\alpha, \tilde{\theta}_\alpha)$ is an
 ${\cal N} = 2$ chiral superfield 
  \be
 \label{Wcal}
{\cal W} \ =\ \Phi + i \sqrt{2} \tilde{\theta}^{\alpha} W_\alpha - \frac {\tilde{\theta}^2}4 
\bar D^2 \left( e^{-V} \bar \Phi e^V \right)\ . 
  \ee
The superfields $V, W_\alpha, \Phi, \bar\Phi $ live in ordinary superspace, $(x, \theta, \bar\theta)$.  
Besides the chirality conditions, $\bar D_{\dot{\alpha}}^a \, {\cal W} = 0$, the superfield (\ref{Wcal})
  satisfies the constraints
 \be
 \label{svjazW}
D^{a\alpha} D^b_\alpha \, {\cal W} \ =\  \bar D^a_{\dot{\alpha}} \bar D^{b\dot{\alpha}} \, \bar {\cal W} 
\ ,
 \ee
where $a,b=$ ({\sl no tilde, tilde}) are the global $SU(2)$ indices. The superfield ${\cal W}$ can be 
naturally expressed
in the framework of 
 the harmonic superspace approach (see the monograph \cite{kniga} and also \cite{sympl} ), 
but do not themselves depend on harmonics in the chosen basis.



 This theory has (infinitely) many different classical vacua. The supersymmetric vacuum has zero energy. At the
classical level, it has zero potential energy.
Note that the potential commutator term in (\ref{N2SYMcomp}) 
vanishes when $ [\bar\phi, \phi] = 0$, implying that
$\phi$ belongs to the Cartan subalgebra of the corresponding Lie algebra. Factorizing over gauge
transformations, this gives $r$ physical complex  parameters (where $r$ is the rank of the group) characterizing
the classical vacuum moduli space. When quantum corrections are taken into account, one could in principle 
expect the appearance of a non-trivial effective potential on the moduli space, such that the energy would 
generically 
be shifted from zero. For supersymmetric theories, quantum corrections {\it vanish}
at any order of perturbation theory; for the ${\cal N} =2$ theory, non-perturbative
corrections to the effective potential also vanish. However, corrections to the 
kinetic part of the lagrangian need not vanish
and they do not. The relevant slow variables are $r$ complex parameters $\phi^A$, mentioned above, and their 
${\cal N} =2$ superpartners involving fermions and also $r$ Abelian gauge fields. In the simplest
$SU(2)$ case, they can be combined into one ${\cal N} =2$ superfield ${\cal W} = \phi \, + \ldots\; .$\footnote{\,No spinor or matrix indices here!} The effective Seiberg--Witten lagrangian has the form
\be
\label{LSW}
  {\cal L} =  \int  d^4\theta  {F}({\cal W}) + \  {\rm c.c.}
 \ .
  \ee  
When expressed in components, this gives a non-trivial metric on moduli space. 
$F({\cal W})$ is a non-trivial, elliptic function taking account of the instanton contributions, etc.
Its asymptotic behavior at large ${\cal W}$ is simple, however: $F({\cal W}) = \frac {{\cal W}^2}{4\pi^2} 
 \ln {\cal W}$. This takes into
account only  the perturbative corrections, which appear only at the one-loop level.

This paper is devoted  to evaluation of the effective lagrangians in the theories obtained by the 
dimensional
reduction of (\ref{N2SYMcomp}) and also by the dimensional reduction of ${\cal N}=1$ SYM theories.
 
 Let us start by discussing the latter. In four dimensions, pure SYM theories do not possess a   
vacuum moduli space. The number of quantum vacua is finite, given by the dual Coxeter
 number (or, 
equivalently, the adjoint Casimir operator $c_V$) of
the gauge group \cite{Coxeter}. However, a moduli space does appear after dimensional reduction.
Consider first the theory reduced to $(0+1)$ dimensions. In such a theory, new gauge invariants
made of the spatial components of the gauge potential appear. The simplest such invariant is
${\rm Tr} \{ A_i^2 \}$. Indeed, the gauge transformation of $A_i^a$ is reduced now to multiplication
by a group matrix $O_{ab}$ and does not involve the derivative term.  
 The tree potential 
term $\propto {\rm Tr} \, [A_i, A_j]^2 $
vanishes when $ [A_i, A_j] = 0$, i.e. when $A_i$ belongs to the Cartan subalgebra. For $SU(2)$, this means
that $A_i$ can be gauge rotated to the form $c_i t^3$. The three variables $c_i$ characterize the vacuum moduli
space.  For an arbitrary gauge group, the moduli space is characterized by $3r$ parameters.
 
  Consider now the reduction to $(1+1)$ dimensions. Only two components of $A_i$ do not involve the derivative
term in their gauge transformation law and we have $2r$ physical moduli space parameters. When reducing to
$D=3$, only one component of the vector potential for each unit of rank is left, but there are also $r$  
Abelian gauge fields which are dual in three dimensions to  scalars, $\epsilon_{ijk}F_{jk} \leftrightarrow 
\partial_i  \Psi$. Thus, in three dimensions we have  $r + r = 2r$ parameters in the vacuum moduli
space.
 
 For ${\cal N} =2$ theories, the counting is basically the same, only we have to add $2r$ parameters
associated with the scalar fields. In other words, the corresponding effective lagrangians involve
 $5r$ bosonic degrees of freedom in the $1D$ case and $4r$ degrees of freedom in the $2D$ and $3D$ cases.

The paper is organized as follows. In the next section, we describe the supersymmetric quantum-mechanical
models representing effective lagrangians for the theories obtained after reduction to $(0+1)$ dimensions.
They represent non-standard (so called {\it symplectic}) supersymmetric sigma models. They are characterized
by a mismatch between the number of bosonic and fermionic degrees of freedom: for example, in the symplectic
$\sigma$ models of the first kind (obtained from ${\cal N} =1$ theories), we have 3 bosonic and 2 fermionic 
degrees of freedom for each unit of rank, while for  symplectic
$\sigma$ models of the second kind (obtained from ${\cal N} =2$ theories), we have $5r$ bosonic and $4r$
 fermionic 
degrees of freedom. We hasten to mention that the number of bosonic and fermionic 
{\it quantum states}
is still equal, as dictated by supersymmetry. We will  explain later why the existence of such an unusual 
${\cal N} =2$ sigma model\,\footnote{\,Our counting of ${\cal N}$ always refers to a number of minimal supercharge representations 
in a given
dimension. Thus, for $D=1$, ${\cal N}$ counts the number of {\it complex} supercharges, for $D=4$ it 
counts the number
of Weyl spinors, etc.\\[-2mm]~}
(it is not K\"ahler !) does not contradict the no-go  theorem proven in 
\cite{Freedman}.   

In Sect.\ 3, we discuss 2-dimensional effective theories. The theories obtained from
 ${\cal N} =1$ $4D$ SYM  represent
conventional K\"ahler sigma models. For extended SYM, the effective theories are more interesting
since they enjoy ${\cal N} = 4$ supersymmetry, but are not hyper-K\"ahler,
belonging to the class of so-called twisted sigma models \cite{GHR}.  

Sect.\ 4 is devoted to $3D$ effective theories. They are hyper--K\"ahler sigma models. In the simplest
$SU(2)$ case, the corresponding target space is the Atiyah--Hitchin manifold (the $(0+1)$ version
of this sigma
model describes also the dynamic of two BPS monopoles). In the $SU(N)$ case, the target space represents
a generalized Atiyah--Hitchin manifold associated with the dynamics of $N$ BPS monopoles.\,\footnote{\,To avoid confusion,
we note that they are the standard monopoles of the  $O(3)$ Georgi--Glashow model characterized by spatial position and a single
$U(1)$ phase.}
 For an arbitrary 
gauge group, the corresponding hyper--K\"ahler manifolds (not studied by mathematicians previously) 
are obtained after certain factorizations
(hyper--K\"ahler reductions) of  generalized AH manifolds.  

In Sect.\ 5 we discuss the relationship between effective lagrangians in different dimensions and
discuss in detail the {\it non-renormalization theorems} for $D=1,2,3$  and their relationship to the
conventional non-renormalization theorems in four dimensions.


\section{\boldmath{$D=1$}\,: Symplectic Sigma Models}
%\setcounter{equation}0

\subsection{${\cal N} =1$}

Consider the simplest example, namely massless ${\cal N} =1$ $4D$ SQED, 
with the lagrangian
   \be
\label{LSQED}
{\cal L} \ =\frac{1}{2e^{2}}  \int {\rm d}^2\theta\, W^2 \ +\  
   \int {\rm d}^4\theta \left[ \bar S \,e^{V} S+ 
\bar T \, e^{-V} T \right] \ ,
 \ee
($S,T$ are chiral multiplets carrying opposite electric charges)
reduced to $(0+1)$ dimensions. 
The effective lagrangian  (determined in \cite{SQED})
depends on the gauge potentials $A_i(t)$ and their superpartners: the photino fields $\psi_\alpha(t)$,
$\alpha = 1,2$. The charged scalar and spinor fields represent fast variables that should be 
integrated over. 
Now $A_i$, the auxiliary field $D$ and the spinor fields $\psi_\alpha$ can be combined  in a single
${\cal N} = 2$ $1D$ superfield
 \cite{my} (see also \cite{bp})\,\footnote{\,We follow the notations of Ref.\cite{sympl}, $$\bar \theta^\alpha = (\theta_\alpha)^\dagger,\ \ 
\bar \theta \theta = 
\bar \theta^\alpha \theta_\alpha, \  \bar \theta \sigma_k \theta = 
\bar \theta^\alpha (\sigma_k)_\alpha^{\ \beta} \theta_\beta$$
and the indices are raised and lowered with the help of
 the invariant tensors $\epsilon^{\alpha\beta} = - \epsilon_{\alpha\beta}$.}
  \be
\label{Vcomp}
\Gamma_k &=& A_k +  \bar \theta \sigma_k  \psi  + 
\bar \psi  \sigma_k \theta  
  + \epsilon_{kjp} \dot{A}_j \bar \theta
\sigma_p  \theta + D \bar \theta \sigma_k  \theta \nonumber \\
&& +\, i(\bar\theta \sigma_k \dot{ \psi} -
\dot{\bar \psi}
\sigma_k  \theta ) \bar\theta \theta + \frac {\ddot{A}_k}4
\theta^2 \bar\theta^2 \ .
 \ee  
The field (\ref{Vcomp}) satisfies the constraints
   \be
\label{albetgam}
D_{(\alpha} \Gamma_{\beta\gamma)} \ =\ 0\,, \quad \bar D_{(\alpha}\Gamma_{\beta\gamma)} = 0\ ,
  \ee
where $\Gamma_{\alpha\beta} = \Gamma_{\beta\alpha} \ =\ 
i(\sigma_k)_\alpha^{\ \gamma} \epsilon_{\beta\gamma}  \Gamma_k$
and 
\be
D_\alpha = \frac{\partial}{\partial \theta^\alpha} + i\bar\theta_\alpha
\frac{\partial}{\partial t}\,, \;\;
\bar D_\alpha = \frac{\partial}{\partial \bar\theta^\alpha} - i\theta_\alpha
 \frac{\partial}{\partial t}
\ee
are the covariant derivatives. Actually, $\Gamma_k$ are nothing but the spatial components of the
 former $4D$ superconnections
  \be
 \label{Gammu}
\Gamma_\mu =  \frac 14 (\bar \sigma_\mu)^{\dot \beta \alpha}
\bar D_{\dot \beta}  D_\alpha  V = \ A_\mu +
\ldots \ ,
  \ee
 the covariant
background derivatives having the form $\nabla_\mu  = \partial_\mu - i\Gamma_\mu$ \cite{Siegel}.  
In one-dimensional theory, $\Gamma_k$ is gauge invariant,
$$  \delta \Gamma_{\alpha\beta} \ \sim\ (D_\alpha \bar D_\beta + D_\beta \bar D_\alpha) 
(\Lambda - \bar \Lambda) 
= 0
$$ 
as follows from the (anti-)chirality of $\Lambda(\bar\Lambda)$ and the $1D$ relationship 
$\{D_\alpha, \bar D_\beta \}
\ =\ 2i\epsilon_{\alpha\beta} \partial_t$.

The effective supersymmetric and gauge--invariant action  is presented in the form 
  \be
\label{act1}
S \ =\ \int dt \int d^2 \theta d^2 \bar\theta\  F(\Gamma_k).
 \ee
By construction, it enjoys ${\cal N} = 2$ supersymmetry. The lagrangian is expressed in components as
follows
   \be
\label{Lcomp}
{\cal L} = \frac h {2} \dot A_k  \dot A_k + \frac {ih}{2}
\left({\bar \psi} \dot{\psi} - \dot{{\bar \psi}}  \psi 
\right) + \frac {\partial_k h}2 \epsilon_{kjp}  \dot A_j 
{\bar \psi} \sigma_p  \psi  \nonumber \\
+ \frac {hD^2}{2} - \frac {D \partial_k h}2 {\bar \psi} 
\sigma_k  \psi - \frac {\partial^2  h}8 
\bar{\psi}^2 \psi^2\ ,
 \ee
where 
 \be
\label{hF}
 h(\vec{A}) = - \frac 12 \,\partial^2 F(\vec{A})\ .
   \ee
This is a supersymmetric sigma model with conformally-flat $3D$ target space,
$ds^2 = h d\vec{A}^2$. However, it is not the conventional supersymmetric sigma
model associated with the de Rahm complex. The latter has only one pair of complex
supercharges $(Q, Q^\dagger) \equiv (d, d^\dagger)$. When the target space represents
a K\"ahler manifold, one can define an extra pair of supercharges (three such extra pairs 
for hyper--K\"ahler manifolds), but in our case the target space is 3--dimensional and definitely
not K\"ahler. 

 One also notices that the number of bosonic and fermionic degrees of freedom are not matched
in the usual way: in a conventional sigma model one has a complex fermion for each boson while
the lagrangian (\ref{Lcomp}) involves three bosonic dynamic variables and only two fermionic ones.
In  field theory, where each field is associated with an asymptotic quantum state, such a mismatch
would not be allowed by supersymmetry. But in supersymmetric quantum mechanics, there are no problems
with a mismatch of this kind: for each nonzero eigenvalue of the hamiltonian we still have 
two bosonic and two fermionic degenerate states
   $ |n\rangle  = \Phi_n(\vec{A}, \psi_\alpha)$.\footnote{\,As is well 
known, vacuum states with zero energy need not be paired. 
In Ref. \cite{weak}, we considered SQM models with non-standard ``weak'' supersymmetric
algebra. For such models, the exact pairing is absent also for the first excited state. But the algebra
of all models
considered in this paper is standard.}

 A reader might be somewhat confused at this point. The widely-known theorem \cite{Freedman}
seems to assert that ${\cal N} = 2$ sigma models can {\it only be defined} on K\"ahler
manifolds (and  ${\cal N} = 4$ models only on hyper--K\"ahler manifolds). However, this theorem
relies on two assumptions : {\it (i)} the theory considered should be a real field theory
with  at least 2 spacetime dimensions  and {\it (ii)} the kinetic term should have the standard form
$\propto g_{ab} \partial_\mu \phi^a \partial_\mu \phi^b$. For quantum mechanics, the first condition is not
satisfied and there are no restrictions. 

 In a standard sigma model, fermions are vectors in the tangent space. In our case, they belong to the spinor
representation of $SO(3) \equiv Sp(2)$. We will call this model a symplectic sigma model of the first kind
(see below for the second kind).   

  In our case, the function $F(\Gamma_k)$ has a particular form. At the tree level,
$F(\vecg{\Gamma}) = \vecg{\Gamma}^2/(6e^2)$  and $ h=1/e^2$. This gives the lagrangian of dimensionally-reduced photodynamics. 
Let us evaluate the one-loop correction to the metric. To this end, we need to calculate the loops
of charged superfields $S,T$ in a gauge background. It is convenient to do this in components. 
We choose the background $A_i = C_i +E_i t,\ \psi_\alpha = 0$ and calculate the charged scalar and fermion
loops. The corresponding contributions to the effective action have the form 
   \be
\label{detratio}
\Delta S_{\rm eff} =
-i \ln   \frac{\det^{\frac 12} (- D^2 I  + \frac i2
\sigma_{\mu\nu}  F_{\mu\nu}) }{\det^{\frac 12}\left(-D^2 I\right)} \ ,
   \ee
where $I$ is the $4 \times 4 $ unity matrix and the identity
 \be
\label{Dslash}
\left({\rm i}\,/\!\!\!\!{D}\right)^2 = - D^2 I  +
\frac{\rm i}{2} \sigma_{\mu\nu} F_{\mu\nu}\ ,
   \ee
$\sigma_{\mu\nu} = \frac 12 [\gamma_\mu, \gamma_\nu]$\ , has been used. We observe that a non-zero correction is
 due solely to magnetic interactions $\propto \sigma_{\mu\nu}  \, F_{\mu\nu}$. If the latter were absent, the 
fermion
and scalar contributions would exactly cancel. 
This feature is common to all
supersymmetric gauge theories, both non-Abelian and Abelian (see \cite{akhmedov} for more details). 
This fact is related to another known fact,
 namely that when the supersymmetric $\beta$
function is calculated in an {\it instanton}
background, only the contribution  of the zero modes survives \cite{shif}.

\begin{figure}
   \begin{center}
        \epsfxsize=170pt
       % \epsfysize=100pt
       % \vspace{-5mm}
        \parbox{\epsfxsize}{\epsffile{sestr1.eps}}
        \vspace{5mm}
    \end{center}
\caption{One-loop renormalization of the kinetic term in SQED. The internal lines
are Green's functions of the operator $(-D^2)$ with constant $A_i = C_i$. The vertices involve the magnetic
interaction $\propto \sigma_{0i}  \, E_i$.}
\label{loopSQED}
\end{figure}

  To lowest order in $F_{\mu\nu}$ (or $E_i$), the contribution (\ref{detratio}) can be represented by 
the graph 
in Fig.1. The constant background $\vec{C}$ gives a ``mass'' to the charged fields and the 
Euclidean propagator
has the form $1/(\omega^2 + \vec{C}^2)$.  The calculation gives
 \be
 \label{feract}
\Delta S_{\rm eff} \ =\  -i\cdot i_{\rm Wick} \cdot i^2 \cdot  \frac 12 \cdot \left( - \frac12 \right) \nonumber \\  
\cdot E_j E_k \, {\rm Tr} \{ \sigma_{0j}\sigma_{0k} \}
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \frac 1{(\omega^2 +
\vec{C}^2)^2}
\ = \  \frac {{\vec{E}}^2}{4|\vec{C}|^3}\ ,  
 \ee
where the  factor $1/2$ is the power of the determinant and the factor $-1/2$ comes from the 
expansion 
$$\ln \det \| I + \alpha \| = \ln \left[ 1 - \frac 12  {\rm Tr}\, \alpha^2 + 
\frac 13 {\rm Tr}\, \alpha^3 + \ldots \right]  \approx   - \frac 12  {\rm Tr}\, \alpha^2 $$
(Tr $\alpha = 0$ in our case).
 This immediately gives 
  \be
\label{hab}  
e^2h(\vec{C})  \ =\  1 + \frac {e^2} {2|\vec{C}|^3} + \ldots 
 \ee  

Let us discuss now non--Abelian theories. In the simplest case of the group $SU(2)$, the moduli space
involves the variables $c_k = A^3_k$ and their superpartners, which are combined in the
 superfield $\Gamma^3_k$.  The effective action again has the form (\ref{act1}), but the function
$F(\vecg{\Gamma}^3)$ is now different. Like in the Abelian case, it can be determined by calculating the loops of 
gauge and fermion fields in an Abelian background $A^{\rm cl}(t) = (C_i + E_i t)t^3$ (where $C$ stands
not only for ``constant'', but also for ``Cartan''). 

The graphs are conveniently calculated in the background gauge. We represent $A_\mu = A_\mu^{\rm cl} +
{\cal A}_\mu$, where ${\cal A}_\mu$ is the quantum fluctuation and   
  and add to the Lagrangian the gauge-fixing term
\be
\label{gaugfix}
- \frac 1{2g^2} (D_\mu^{\rm cl} {\cal A}_\mu)^2\ ,
\ee
where $D_\mu^{\rm cl} = \partial_\mu - {\rm i}
 \left[A_\mu^{\rm cl}, \, \cdot \right]$.  The coefficient chosen in Eq.\,(\ref{gaugfix}) defines
the ``Feynman background gauge'', which is more
convenient than others.
Adding (\ref{gaugfix}) to the lagrangian and
integrating by parts, we obtain for
 the gauge--field--dependent part of the Lagrangian
\begin{equation}
{\cal L}_{\cal A} =
-\frac{1}{2 g^2}\, {\rm Tr} \, \left(F_{\mu\nu}^2 \right) +
 \frac{1}{g^2} \, {\rm Tr}\,
\left\{ {\cal A}_\mu \left( D^2 g_{\mu\nu}{\cal A}_\nu  - 2i
\left[F_{\mu\nu}, {\cal A}_\nu  \right] \right)
 \right\} + \ldots \ ,
\end{equation}
where the dots stand for the terms of higher order in
${\cal A}_\mu$.
The ghost part of the Lagrangian is
\begin{equation}
{\cal L}_{ghost} = - 2\, {\rm Tr} \,\left( \bar{c} D^2 c \right)
\ + {\rm higher\ order\ terms}.
\end{equation}

Now we can integrate   over the quantum fields ${\cal A}_\mu$,
$c$ and  over the fermions using the relation
(\ref{Dslash}).   
We obtain  
   \be
\label{detrationab}
\delta S_{\rm eff} =
-i \ln  \left(\frac{\det^{\frac14}\left(- D^2 \, I + \frac i2
\sigma_{\mu\nu}  \left[F_{\mu\nu}, \, \cdot\right]\, \right)
\det^{1/4}\left(- D^2 I \right)}{\det^{\frac12}\left(-D^2\, g_{\mu\nu} +
2i\, \left[F_{\mu\nu}, \, \cdot\right]\right)}\right)\ .
   \ee
Again, the result would be zero in the absence of magnetic interactions. 
In this case, besides the fermion loop, the gauge field loop should also be
taken into account. The non-zero commutators, $\left[ F_{\mu\nu}, \, {\cal A}_\nu \right]$,
$ \left[F_{\mu\nu}, \, \lambda_\alpha \right]$,
imply that the quantum fields are charged with respect to the background, i.e.
that their color indices $a$ acquire the values $1,2$.

The fermion loop gives the same contribution to $S_{\rm eff}$ as in the Abelian theory:
the power of the determinant is now $1/4$ rather than $1/2$ (the theory involves a Weyl,
rather than Dirac, fermion), but this is compensated by the extra color factor 2. One 
can be convinced that the gauge boson loop contribution involves the factor -4 compared
to the fermion one (the factor $-2$, coming from the power of determinant $-1/2$ vs. $1/4$,
is explicitly seen in (\ref{detrationab})
and another factor $2$ comes from spin. This gives
 \be
\label{hnab}
g^2 h^{SU(2)}(\vec{C}) \ =\ 1 - \frac {3g^2}{2\vec{C}^2} + \ldots
 \ee
One notices at this point that exactly the { same} graphs determine the one--loop
renormalization of the effective charge in the corresponding $4D$ theories. The only difference
is that, in four dimensions, we have to substitute 
$$
     \int \frac{d\omega}{2\pi} \frac 1 {(\omega^2 +
\vec{C}^2)^2} \ \longrightarrow \ 
\int \frac{d^2p}{(2\pi)^4} \frac 1{(p^2 +
\vec{C}^2)^2}  \ \propto \ln \frac \Lambda {|\vec{C}|}\ .
  $$
In other words, the coefficients in (\ref{hab}), (\ref{hnab})  are rigidly related to the 
  one--loop $\beta$ function coefficients in the parent $4D$ theories. Indeed, the $\beta$ function
in non--Abelian SYM theory with $SU(2)$ gauge group involves the factor $-3$ compared to SQED.
We will return to the discussion of this point in the last section.

The metrics (\ref{hab}), (\ref{hnab}) and the relation (\ref{hF}) allow one to restore the 
corresponding prepotentials: 
  \be
e^2 F^{\rm SQED}(\vecg{\Gamma})  &=& - \frac {\vecg{\Gamma}^2}{3} + 
\frac {e^2 \ln |\vecg{\Gamma}|}{|\vecg{\Gamma}|} + ...  \nonumber \\
g^2 F^{SU(2)}(\vecg{\Gamma})  &=& - \frac {\vecg{\Gamma}^2}{3} -  
\frac {3g^2\ln |\vecg{\Gamma}|}{|\vecg{\Gamma}|} + ...
 \ee
(we replaced $\vecg{\Gamma}^3 \to \vecg{\Gamma}$ in the non--Abelian case).
Consider now an arbitrary simple, compact Lie group. The classical potential energy vanishes
 when $[A_j, A_k] = 0$,
which implies  that $A_j$ lies in the Cartan subalgebra (and is effectively Abelian). 
This gives $3r$ bosonic variables in the effective lagrangian.
They are supplemented by $2r$ Abelian gluino variables. These variables are organized in $r$ 
superfields $\vecg{\Gamma}^{A = 1,\ldots, r}$ defined as in Eqs.\,(\ref{Vcomp}), (\ref{albetgam}). 
$\vecg{\Gamma}^A$ represent dimensionally-reduced  Abelian superconnections. Thus, the effective
lagrangian has the form\,\footnote{\,It is interesting that 
such a lagrangian describes also the dynamics of $r$ extremal 
Reissner--Nordstr\"om
black holes (representing classical solutions in ${\cal N} =2$ $4D$ supergravity) \cite{Strom}.\\[-2mm]~} $\int d^4\theta F(\vecg{\Gamma}^A)$ 
and the only question is what the function
$F(\vecg{\Gamma}^A)$ is. Again, we choose an Abelian gauge field background and perform the calculation
over quantum fields. The latter must have non-zero commutators with the background. They 
are classified according to the {\it roots} of the corresponding Lie algebra. Actually, we have to add the 
contributions of the loops
corresponding to each such (positive) root. The result is (see \cite{BO} for more details) 
  \be
\label{Fsumj}
g^2 F(\vecg{\Gamma}^A) \ =\   - \sum_j \left[  \frac 2{3c_V}
\left(\vecg{\Gamma}^{(j)}\right)^2 +  
\frac {3g^2}{|\vecg{\Gamma}^{(j)}|} \ln |\vecg{\Gamma}^{(j)}|\right]
\ ,
 \ee 
where $\vecg{\Gamma}^{j} = \alpha_j(\vecg{\Gamma}^A)$ and $\alpha_j$ are the roots. For example, for $SU(3)$
we have the sum of three terms with 
 \be
\label{formSU3}
\hspace{-5mm}
\alpha_1(\vecg{\Gamma}^A) = \vecg{\Gamma}^3\ , \ \ \ \  \alpha_2(\vecg{\Gamma}^A) = 
\frac{-\vecg{\Gamma}^3 + \sqrt{3} 
\vecg{\Gamma}^8}2\ , \ \ \ \  
\alpha_3(\vecg{\Gamma}^A) = \frac{\vecg{\Gamma}^3 + \sqrt{3} \vecg{\Gamma}^8}2\ . 
   \ee

\subsection{${\cal N} = 2$}
The same program can be carried out for SQM models obtained by dimensional reduction from ${\cal N} =2$ 
$4D$ theories. Consider first Abelian theory. ${\cal N} =2$ SQED has the same charged matter 
content as ${\cal N} =1$ theory, but involves an extra neutral chiral multiplet $\Phi$.
The lagrangian acquires two new terms 
 \be
\label{delLN2}
 \Delta L \ =\  \int d^4\theta \, \bar \Phi \Phi +\ 
\left[\sqrt{2}e \int d^2\theta \, \Phi ST + {\rm c.c.} \right] \ .
   \ee
The lowest component of $\Phi$ 
 gives two extra degrees of freedom in the vacuum moduli space, which thereby becomes 
5--dimensional.\footnote{\,The moduli can be represented as spatial 
components of the gauge potential in $6D$ SQED, from 
which the
${\cal N} =2$  $4D$ theory is obtained by dimensional reduction.} The vector superfield $V$ and the
 chiral superfield
$\Phi$ can be unified in a single ${\cal N}\!=\!4$ (in SQM sense) harmonic gauge superfield and the effective
 lagrangian
can be formulated in the terms of the latter \cite{sympl}. We use here a more conventional approach,
using ${\cal N} =2$ superfields. The effective action depends on $\Gamma_J = (\vecg{\Gamma}, \sqrt{2} 
\, {\rm Re} \{\Phi\}, \sqrt{2} \, {\rm Im} \{ \Phi \})$
(interpreted as superconnection in the ``grandmother'' $6D$ theory) and
must have the form
 \be
 \label{act5}
S = \int dt \int d^2 \theta d^2 \bar\theta \ {\cal K} (\vecg{\Gamma}, \bar\Phi, \Phi)\, .
 \ee
Now ${\cal N} =2$ symmetry is manifest here. The action (\ref{act5})  is invariant under the
{\it additional} ${\cal N}=2$ supersymmetry transformations
 \be
\label{trans}
\delta \bar\Phi &=& \frac {2i}3 \epsilon^\alpha (\sigma_k)_\alpha^{\ \beta}
 D_\beta \Gamma_k\,, \nonumber \\
\delta \Phi &=&  \frac {2i}3 \bar \epsilon_\alpha (\sigma_k)_\beta^{\ \alpha}\bar D^\beta \Gamma_k\,,
\nonumber \\
\delta \Gamma_k &=& -i\epsilon^\alpha (\sigma_k)_\alpha^{\ \beta} D_\beta \Phi -
i \bar \epsilon_\alpha (\sigma_k)_\beta^{\ \alpha} \bar D^\beta \bar \Phi\,,
 \ee
provided that 
  \be
\label{harm5}
  \frac {\partial^2{\cal K}}{\partial \Gamma_k^2} +
2 \frac {\partial^2{\cal K}}{\partial \bar\Phi \partial \Phi} \equiv \frac  {\partial^2{\cal K}}
{\partial \Gamma_J^2}
 \ =\ 0\ ,
  \ee
i.e. ${\cal K}$ is a 5--dimensional harmonic function \cite{DE}. Unifying $\vec{A}$ and $\phi, \bar\phi$
in a single 5-dimensional vector $A_J$ and two spinors from the multiplets $\Gamma_k$ and $\Phi$
in a single 4--component complex spinor $\eta_\alpha$ lying in the fundamental (spinor) representation 
of $SO(5) \equiv Sp(4)$, we can write the following component expression for the lagrangian \cite{N2}
   \be
\label{LO5}
 {\cal L} \ = \ h \left[ \frac 12 \dot{A}^2_J + \frac i2 (\bar\eta
\dot{\eta} - \dot{\bar\eta} \eta ) \right] + \frac i2
\partial_J h \dot{A}_K \ \bar\eta \sigma_{JK} \eta + \nonumber \\
\frac 1{24} \left( 2 \partial_J \partial_K h - \frac 3h \partial_J h
\partial_K h \right) \left( \bar\eta \gamma_J \eta \ 
\bar\eta \gamma_K \eta - \eta C \gamma_J \eta \ 
\bar \eta  \gamma_K C \bar \eta \right)\ ,
   \ee
where $\gamma_K$ are 5--dim Dirac matrices, $\sigma_{JK} = (1/2) (\gamma_J \gamma_K - \gamma_K
\gamma_J )$ and $C$ is the antisymmetric matrix of charge conjugation, $C\gamma_J^T = -\gamma_J C$.
The metric $h$ is related to  ${\cal K}$ by $h = -(1/2) \partial^2{\cal K}/\partial \vec{A}^2$.
 
The lagrangian (\ref{LO5}) describes a sigma model defined on a conformally-flat  5--dimensional target space.
We will call it a symplectic sigma model of the second kind. A generalized symplectic model of the second kind
depends in this approach on  $r$ sets of  ${\cal N} = 2$ superfields 
$\Gamma_J \equiv (\vecg{\Gamma}^A, \Phi^A, \bar\Phi^A)$. The action
    \be
 \label{act5r}
S = \int dt \int d^2 \theta d^2 \bar\theta \ {\cal K} (\Gamma^A_J)\, .
    \ee
enjoys extended  ${\cal N} = 4 $ supersymmetry,  provided the following generalized harmonicity 
conditions are satisfied \cite{sympl}
   \be
\label{harm5r}
\frac {\partial^2 {\cal K}}{\partial \Gamma_I^{A} \partial \Gamma_I^{B}}
\ =\ 0\ , \ \ \ \ \ \ \ \ \ \
 \frac {\partial^2 {\cal K}}{\partial \Gamma_I^{[A} \partial \Gamma_J^{B]}}
\ =\ 0\,.
    \ee


In the Abelian case,  the effective action has the form (\ref{LO5}) with the {\it same} metric $h$ as
in the ${\cal N} = 1$ $4D$ SQED case discussed above. Indeed, we can choose the background with zero $\phi$,
in which case the effective action is given by the  graph drawn in Fig.\ref{loopSQED}. Now, $O(5)$ invariance
dictates that the metric also has the  form (\ref{hab}) in a generic background $C_J$ with $\vec{C}^2$
being substituted by $C_J^2$. The prepotential can be chosen as 
  \be
\label{prepot}
e^2{\cal K} \ =\ - \frac {R^2}{3} + \frac {\rho^2}2 \ +\ 
\frac {e^2}{R} \ln \left(R + \sqrt{R^2 + \rho^2} \right)
\ ,
   \ee
where $R^2 = \vecg{\Gamma}^2$ and $\rho^2 = 2\bar\Phi \Phi$. Note that
${\cal K}$ need not be (and is not) $O(5)$ invariant.  

In non-Abelian ${\cal N} = 2$ SYM theory with $SU(2)$ gauge group, the calculations are readily done
in the same way as before. The only modification is that there are now two Weyl fermions and 
an additional adjoint scalar. The ghost determinant is canceled
by the adjoint scalar determinant and we obtain
   \be
\label{detratioN2}
\delta S_{\rm eff} =
-i \ln  \left(\frac{\det^{\frac 12}\left(- D^2 \, I + \frac i2
\sigma_{\mu\nu}  \left[F_{\mu\nu}, \, \cdot\right]\, \right)}
{\det^{\frac 12}\left(-D^2\, g_{\mu\nu} +
2i\, \left[F_{\mu\nu}, \, \cdot\right]\right)}\right)\ .
   \ee
This gives the expression 
    \be
\label{hnabN2}
g^2 h^{SU(2)}_{{\cal N}=2}(C_J) \ =\ 1 - \frac {g^2}{|C_J|^3} 
 \ee
 for the metric. The respective coefficients in the correction in the Abelian and non-Abelian
cases conform with the respective coefficients in the corresponding $4D$ beta functions.

The structure of the expressions (\ref{hnabN2}) and    (\ref{hnab}) is similar, but there is one essential
difference. Eq.\,(\ref{hnabN2}) contains no ellipsis! The expression for the metric is {\it exact}:
higher-loop corrections vanish. The proof of this {\it non-renormalization theorem} is simple. Dimensional
counting tells us that an  n--loop correction to the metric
should be proportional to $(A_J A_J)^{-3n/2}$. But this is not  harmonic  for $n \geq 2$ and is excluded
by supersymmetry requirements. We will discuss the relationship of this non-renormalization theorem to
the $4D$ non-renormalization theorem (in ${\cal N}=2$ theories two and higher loop contributions to the 
beta function vanish) in  Sect. 5.

We want to emphasize that the absence of the corrections to the metric does not mean the absence of
the corrections to the effective lagrangian. The latter  involves  higher derivative corrections, which 
do not vanish
either at the one--loop or at the two- and higher-loop level \cite{sestry}.
Thus, the singularity of the metric at $A_J^2 = 0$ has no great physical meaning: the effective lagrangian
involves uncontrollable higher--derivative corrections there anyway. 

The effective lagrangian can also be found  for an arbitrary gauge group. Again we have to sum over all
positive roots. The prepotential is
 \be
 \label{prepotr}
g^2 {\cal K}  \ =\  - \sum_j \left\{ \frac 2{3c_V} \left[ 
\left(R^{(j)}\right)^2 -
\frac 32 \left(\rho^{(j)}\right)^2 \right] + \right. \nonumber
\\ \left. 
\frac {2g^2}{R^{(j)}} \ln \left[ R^{(j)} + 
\sqrt{\left(R^{(j)}\right)^2 + \left(\rho^{(j)} \right)^2}
 \right] \right\}\ ,
   \ee
where $\left(R^{(j)}\right)^2 = \left(\vecg{\Gamma}^{(j)}\right)^2$,
 $\left(\rho^{(j)}\right)^2 = 2 \bar \Phi^{(j)}
\Phi^{(j)}$ and $\vecg{\Gamma}^{(j)} = \alpha_j(\vecg{\Gamma}^A)$, 
 $\Phi^{(j)} = \alpha_j(\Phi^A)$. 
 



\section{\boldmath{$D=2$}\,: K\"ahler and Twisted Models}
%\setcounter{equation}0

\subsection{${\cal N}=1:$ Unfolding  the ring}
Consider first Abelian theory. As was noted previously, in two dimensions we have two, rather than three,
moduli, representing the components of the gauge potential in the reduced dimensions. 
The bosonic part of the effective lagrangian can be evaluated in the same way as in the $1D$ case
by calculating the loop diagram in Fig.\ref{loopSQED}. The only difference is that the loop integral is
now two--dimensional. We obtain 
  \be
\label{LbosD2}
e^2 {\cal L}_{\rm eff}^{\rm bos} \ =\ \frac {(\partial_\alpha A_j)^2}2  \left[ 1 + 
\frac {e^2}{2\pi A_j^2} + \ldots 
\right]\ ,
 \ee
$\alpha = 1,2$ and $j = 1,2$. 
This describes a sigma model on a 2--dimensional target space. One can, of course, introduce 
the complex 
coordinate $\sigma = (A_1 + iA_2)/\sqrt{2}$. 

The full effective lagrangian involves, besides $A_j$, their supersymmetric partners, which are 
 two-component photino fields. We see that, in this case, there is perfect matching between the 
number
of bosonic and fermionic degrees of freedom. Actually, the Alvarez--Gaume--Freedman theorem 
\cite{Freedman}
dictates that the only two--dimensional ${\cal N} = 2$ supersymmetric theory with standard 
sigma--model
kinetic term like in (\ref{LbosD2}) is the supersymmetric K\"ahler sigma model. The K\"ahler
 potential 
${\cal K}$ (${\cal L} = \int d^4\theta {\cal K}$ ) can be 
recovered from the metric. In the case under consideration, it can be chosen as
 \be
\label{Kab}
e^2 {\cal K}(\bar\Phi, \Phi) \ =\ \bar\Phi \Phi + \frac {e^2}{4\pi} \ln \Phi 
\ln \bar\Phi \ .
 \ee 
Now, $\Phi$ is a chiral superfield that is related to the gauge--invariant  
superconnections $\Gamma_j$ in reduced
dimensions in the  following way. Consider the superfield 
$\Sigma  = (\Gamma_1 + i\Gamma_2)/\sqrt{2}$. From the definition (\ref{Gammu}) and
the $2D$ anticommutation relations between $D_\alpha$ and $\bar D_{\dot{\alpha}}$,
we deduce that $\Sigma$ satisfies the constraints
   \be
  \label{contwist}   
 \bar {\cal D}_1 \Sigma \ =\ 
{\cal D}_2 \Sigma  = 0\ 
   \ee
and represents a so-called {\it twisted} chiral multiplet. It 
  differs from the standard one by a pure convention: 
$\Sigma$ is obtained from $\Phi$ by interchanging
$\theta_2$ and $\bar\theta^2$. This means that the change $\Phi \to \Sigma$, in any
 standard action 
involving $\Phi$, would not change anything except the sign due to the change of sign 
of $d^4\theta$. For example, the tree Lagrangian is expressed as
  \be
\label{illustr}
e^2 {\cal L}^{2d}_{\rm tree} =  - \frac 1{2} 
\int d^4\theta (\Gamma_1^2 + \Gamma_2^2) =\ -\int  d^4\theta \bar\Sigma \Sigma 
\ \equiv \ \int  d^4\theta \bar\Phi \Phi.
 \ee



It is very instructive to derive the effective $2D$ lagrangian {\it directly}, elucidating its 
relationship with the SQM effective lagrangian (\ref{Lcomp}) discussed in the previous section.
To do this, consider the original theory, not on $R^2$ and not  on $R^1$, but
rather on $R^1 \times S^1$. By adjusting the
length $L$  of the circle, one can interpolate between the $1D$ and $2D$ pictures \cite{akhmedov}. 

The Lagrangian (\ref{Lcomp}) was obtained after integrating out the charged fields in $1D$ theory. 
Thinking in $1D$ terms, we now have an infinite number of charged fields, representing the coefficients 
in the Fourier series
  \be
  \label{Fourier}
  f(z,t) \ =\ \sum_{n = -\infty}^\infty  f_n(t) e^{i nz/L} \ .
  \ee
 The relevant variables in the effective Lagrangian are still
 the zero Fourier modes of the vector potential $\vec{A} \equiv (A_{j = 1,2}, A_3)$ and its superpartners.
The expression (\ref{Lcomp}) is replaced by the infinite 
sum\,\footnote{\,The notation $e_1$ indicates that we are dealing with the coupling constant in $1D$ theory,
\mbox{$[e_1] \sim m^{3/2}$}.}
 \be
 \label{LR1S1}
e_1^2{\cal L}  \ =\
 \left[ 1 +  \sum_{n = -\infty}^\infty \delta h \left(A_j, A_3 + \frac {2\pi n}{ L} \right)\right] (\dot A_j^2
+ \dot A_3^2) 
+\ {\rm other\ terms}.
\ee
In the limit $L \to 0$, only one term in the sum survives and we reproduce the
 previous $1D$ result (with $\delta h = e_1^2/(2|\vec{A}|^3)$). But for large $L \gg e_1^{-2/3}$, all terms
 are essential.
In the limit
 $L \to \infty$, we can replace the sum by an integral, $\sum_n \longrightarrow \frac L{2\pi}
\int dA_3$. This integral depends on $A_j$, but not on $A_3$: the expression in square brackets in
Eq.\,(\ref{LR1S1}) gives 
  \be
  \label{htilde}
  \tilde{h} \ =\ 1 + \frac {e_2^2}{2\pi A_j^2}\ ,
  \ee
with $e_2^2 = e_1^2L$. This agrees, of course, with (\ref{LbosD2}). 
Actually, in the  limit $L \to \infty $, the effective lagrangian cannot
depend on $A_3$. For large $L$,
 the range where $A_3$ changes is very small, $0 \leq A_3 \leq 2\pi/L$, and the eigenmodes of the
hamiltonian  $\Psi_n(A_3) \sim \exp\{inA_3\}$ with $n \neq 0$ acquire large energy and decouple;
only the mode $n=0$ survives.  To take this
limit carefully, we cannot just set $A_3 = 0$ in Eq.\,(\ref{LR1S1}), however, but 
should perform
the functional integral of $e^{iS}$ ($S$ is obtained from Eq.\,(\ref{Lcomp}) by substituting $\tilde{h}$ 
for $h$)
over  $\prod_t dA_3(t)$ first. Doing this and integrating out also the auxiliary field $D$, we
arrive at the result
   \be
e_2^2{\cal L}_{2D} = \frac 12 g_{jk} \dot A^j  \dot A^k + \frac {i\tilde{h}}{2}
\left({\bar \psi} \dot{ \psi} - \dot{{\bar \psi}}  \psi 
\right) + i \tilde{h} \omega_j^{ab} \dot A^j \bar\psi \sigma^{ab} \psi + \nonumber \\
  \frac{1}{8 \tilde{h}} \left[ (\partial_j \tilde{h})^2 -  \tilde{h}
 (\partial^2 \tilde{h�}) \right]
 \left(\bar{\psi}\right)^2 \left(\psi\right)^2\ ,
\label{LKahl}
    \ee
where we have raised the index of the vector
$A^j$ indicating its contravariant
nature, $g_{jk} = \tilde{h} \delta_{jk}$, 
$\sigma^{ab} = \frac i2 \epsilon^{abc} \sigma^c = \frac i2 
\epsilon^{ab} \sigma^3$ ($a,b = 1,2$ ) is the
generator of rotations in the tangent space, and
  \be
\label{omega2}
\omega_i^{ab} = \frac 12 \left[ \delta^a_i \, \partial^b \log{(\tilde{h})} -
\delta^b_i \, \partial^a \log{(\tilde{h})} \right]
   \ee
is the spin connection on a conformally flat
manifold with the natural choice of the  zweibein,
$e_j^a = \sqrt{\tilde{h}} \delta_j^a$.

 In deriving (\ref{LKahl}), we went
 over from the lagrangian ${\cal L}_{1D}$ to the
$2D$ lagrangian density ${\cal L}_{2D} = {\cal L}_{1D}/L$ . (Normally, 
${\cal L}_{1D}$ is a spatial integral of ${\cal L}_{2D}$, but 
we have dealt up to now 
only with the terms depending on zero spatial Fourier modes, in which case the spatial
integral is reduced to multiplication by $L$.)  
  

  
The lagrangian (\ref{LKahl}) coincides with the standard lagrangian of the
K\"ahler supersymmetric sigma model \cite{sigmod} 
in the QM limit. In particular, the coefficient of the 4--fermion term represents 
a $2D$ scalar curvature.\footnote{\,Incidentally, although the bifermion 
term in (\ref{Lcomp}) can be interpreted
in terms of a $3D$ spin connection, the 4--fermion term (before or after  integrating out
 $D$) is {\it not} expressed in terms of $3D$ curvature.}
The full $(1+1)$ effective lagrangian could be obtained
by taking into account the higher Fourier harmonics $\propto \exp\{inz/L\}$
of $A_j(z,t)$ and $\psi_\alpha(z,t)$ in the background.  

 The result (\ref{Kab}) can be readily generalized for an arbitrary non--Abelian 
gauge group. 
The K\"ahler potential depends on $r$ complex chiral superfields $\Phi^A$ and has the same 
sum--over--the--roots structure
as the $1D$ prepotential in Eq.\,(\ref{Fsumj}), 
   \be
\label{Ksumj}
g^2 {\cal K}(\Phi^A) \ =\   \sum_j \left[  \frac 2{c_V} \bar\Phi^{(j)} 
\Phi^{(j)} - 
\frac {3g^2}{4\pi} \ln   \bar\Phi^{(j)}  \ln   \Phi^{(j)}
\right]
\, ,
 \ee 
where $\Phi^{(j)} = \alpha_j(\Phi^A)$.



\subsection{${\cal N}=2:$ Twisted Sigma Model}

We start again by analyzing the Abelian theory. The effective lagrangian now involves 
two complex bosonic variables 
 \be
  \label{fisi}
\sigma = (A_1 + iA_2)/\sqrt{2},\ \ \ \ \ \ \ \
\phi = (A_4 + iA_5)/\sqrt{2} 
\ .
  \ee
The one--loop calculation brings about a non-trivial metric in the target space  ($\sigma, \bar\sigma, \phi,  \bar\phi $).
This metric can be related
to the SQM 5--dimensional metric by integrating the latter
over $A_3$ in the same way as the K\"ahler metric (\ref{LbosD2})  
was obtained from the metric 
 of the SQM model in the ${\cal N} =1$ case:
    \be
\label{metrfisi}
e^2 \left. ds^2_{1+1}\right|_{{\cal N} =2} \ =\ 
\left(1 +  
\int_{-\infty}^\infty \frac {dA_3}{2\pi}
\delta h_{0+1} \right)  \ = \nonumber \\
\left[1  + \frac {e^2}{4\pi (\bar\phi \phi + \bar\sigma \sigma) } \right]  
 (2 d\bar\sigma d\sigma
+ 2 d\bar\phi d\phi)\ .  
    \ee
We expect the effective action to have a
$\sigma$ model form. One might worry at this point, because
the metric (\ref{metrfisi}) is not
hyper-K\"ahler (the Ricci tensor and the scalar
curvature do not vanish), while the hyper-K\"ahler property of
the metric was shown to be  necessary in order for
the standard (1+1) $\sigma$ model to enjoy ${\cal N}= 4$
supersymmetry \cite{Freedman}. In our case, 
${\cal N}= 4$ supersymmetry is present but the metric is not
hyper-K\"ahlerian, and this seems to present a paradox.
The resolution is  that the  $\sigma$ model to hand  
is  not standard \cite{N2}. 
   Indeed, the bosonic part of the Lagrangian involves, besides the standard kinetic term 
$h\left( \partial_\alpha \bar \sigma 
\partial_\alpha \sigma   + \partial_\alpha \bar \phi
\partial_\alpha \phi \right) $, the "twisted" term 
$\propto \epsilon_{\alpha\beta} \partial_\alpha  \sigma
\partial_\beta \phi $ and $\propto \epsilon_{\alpha\beta}
 \partial_\alpha  \bar\sigma
\partial_\beta \bar\phi$. To understand where the twisted term comes from, consider a 
charged-fermion loop in the background
   \be
   \label{backfisi}
   \sigma = \sigma_0 + \sigma_\tau \tau + \sigma_z z,\ \ \ \ 
\phi = \phi_0 + \phi_\tau \tau + \phi_z z
  \ee
  ($\tau$ is the Euclidean time). The contribution to the 
effective action is $\propto \ln \det \| {\mathfrak D}\|$, 
where ${\mathfrak D}$ is the 6--dimensional Euclidean Dirac 
operator, which can be written in the form
   \be
   \label{Dir6} 
   {\mathfrak D}  \ =\ i \frac{\partial}{\partial\tau} + \gamma_3
   \frac{\partial}{\partial z} -i (\gamma_1 A_1 + \gamma_2 A_2 +
   \gamma_4 A_4 + \gamma_5 A_5 )\ .
     \ee     
 Now, if $A_4$ and $A_5$ were absent, we could write 
 ${\mathfrak D} = \gamma_4 (i\tilde{\gamma}_\mu {\cal D}_\mu )$ , with
 $$\mu = 1,2,3,4;\  {\cal D}_4 = \frac{\partial}{\partial\tau} ,\ 
 {\cal D}_3 = \frac{\partial}{\partial z},\ {\cal D}_{1,2}=  - i A_{1,2}; \  
\tilde{\gamma_4} = \gamma_4, \tilde{\gamma}_{1,2,3} = -i\gamma_4 \gamma_{1,2,3} $$
and then use the squaring trick
  \be
  \label{square}
  \det \| {\mathfrak D} \| \ =\ \det \| i \tilde{\gamma}_\mu {\cal D}_\mu \| \ =\ 
\det\,\!^{1/2} \left\| - {\cal D}^2 + \frac i2 
\tilde{\sigma}_{\mu\nu} F_{\mu\nu} \right\|\ ,
   \ee
with $F_{14} = -\partial A_1 /\partial \tau$, etc. The effective action
would be proportional to 
    \be
       \label{intFF}
{\rm Tr} \{\sigma_{\mu\nu} \sigma_{\alpha\beta} \}
F_{\mu\nu} F_{\alpha\beta} \int \frac {d^2p}{4\pi^2} 
\frac 1{(p^2 + 2\bar\sigma \sigma )^2}\ \propto F_{\mu\nu}^2\ ,
  \ee
  which gives the renormalization of the kinetic term, while the twisted term does not appear.
 The squaring trick also works in the case where $A_{4,5}$ are non-zero, but do not depend on $\tau,z$. 
Then $2\bar\phi \phi$ is just added to $-{\cal D}^2$ in Eq.\,(\ref{square}) and to $2\bar\sigma \sigma$
 in Eq.\,(\ref{intFF}), leading to Eq.\,(\ref{metrfisi}).
 But in the generic case, the fermion determinant cannot 
be reduced to $\det^{1/2} \| - {\cal D}^2 + \frac i2 \sigma_{\mu\nu} F_{\mu\nu} \|$. The basic reason for 
this impasse is that one cannot adequately ``serve'' six components of the gradient with only five $\gamma$
 matrices.\footnote{\,By the same reasoning, the squaring trick does not work for Weyl 2--component fermions in 4 dimensions:
 the three Pauli matrices that are available in that case are not enough to do the job.}
  As a result, the extra twisted term in the determinant appears.
  
  We need not perform an explicit calculation here, as the twisted and all other terms in the Lagrangian are
 fixed by supersymmetry. The twisted ${\cal N} = 4$ supersymmetric $\sigma$ model was constructed almost 
20 years
 ago \cite{GHR}. At that time it did not attract much attention. Recently, there has been a revival of interest 
in the
 GHR model: it arose in some  string--related problems \cite{brany,DiaSei}. It also arises as 
the 
effective $(1+1)$ Lagrangian in the case under study.

It was shown that, for the ${\cal N} = 4$ supersymmetric generalization to be possible, 
the conformal factor in the metric $h(\bar\sigma, \sigma, \bar\phi, \phi)$ should satisfy the
 harmonicity 
condition
  \be
  \label{harm4}
 \frac {\partial^2 h}{\partial \bar\sigma \partial \sigma } +
 \frac {\partial^2 h}{\partial \bar\phi \partial \phi} \ =\ 0\ .
  \ee
  Obviously, (\ref{metrfisi}) satisfies the condition everywhere except at the origin. The relationship 
between (\ref{harm4}) and the 5--dimensional harmonicity condition for the metric in the effective 
SQM model (\ref{LO5}) is also obvious. Indeed, integrating a $D$--dimensional harmonic function 
over one of the coordinates, like in (\ref{metrfisi}), we always arrive at a $(D-1)$--dimensional
 harmonic 
function.

To construct the full action, consider, along with the standard chiral multiplet $\Phi$ satisfying 
the conditions ${\cal D}_\alpha \Phi= 0$, a {\it twisted} chiral multiplet $\Sigma$, which 
satisfies 
the constraints (\ref{contwist}). As we have seen, the action (depending on only $\bar\Sigma$
and $\Sigma$) can be expressed in terms of standard chiral multiplets.
 However, one can write non-trivial Lagrangians involving 
{\it both} $\Phi$ and $\Sigma$. The twisted $\sigma$ model is determined by the expression
  \be
  \label{LGHR}
  {\cal L} \ =\ \int d^2 \theta d^2 \bar\theta 
  \ {\cal K} (\bar\Phi, \Phi; \bar\Sigma, \Sigma)\ ,
     \ee
     where the prepotential ${\cal K}$ satisfies the harmonicity condition,
  \be
  \label{harm4K}
 \frac {\partial^2 {\cal K}}{\partial \bar\Sigma \partial \Sigma } +
 \frac {\partial^2 {\cal K}}{\partial \bar\Phi \partial \Phi} \ =\ 0\ .
  \ee
The condition (\ref{harm4K}) is required if we want the theory to be ${\cal N} = 4$ supersymmetric. 
This is best seen by expressing the lagrangian in components \cite{N2} and observing that the lagrangian
is symmetric under interchange of fermionic variables entering the twisted and untwisted multiplets
only for harmonic ${\cal K}$. The composition of this discrete symmetry and ${\cal N} =2$ supersymmetry,
which is manifest in (\ref{LGHR}),  brings about two extra supersymmetries, mixing $\phi$ and $\sigma$ with
the fermion components of ``alien'' ${\cal N} =2$ multiplets. 

One of the possible choices for ${\cal K}$ 
(two functions ${\cal K}$ and 
$${\cal K}' \ =\ {\cal K} + f(\bar\sigma, \phi) + \bar f(\sigma, \bar\phi) + g(\sigma, \phi) + 
\bar g(\bar\sigma, \bar\phi)$$
result, up to a total derivative, in one and the same lagrangian.)
leading to the metric (\ref{metrfisi}) is \cite{Hitchin,DiaSei}
   \be
   \label{Kres}
   e^2 {\cal K} \ =\ {\bar\Sigma \Sigma - \bar\Phi \Phi}\  +
   \frac {e^2}{4\pi} \left[ F\left( \frac {\bar\Sigma \Sigma }{\bar\Phi \Phi} \right)  - 
\ln \Phi \ln \bar\Phi \right]\ ,
 \ee
 where
   \be
   \label{Spence}
   F(\eta) \ =\ \int_1^\eta \frac {\ln(1+\xi)}\xi \ d\xi
    \ee
 is the Spence function. This gives, besides (\ref{metrfisi}), a twisted term
   \be
 \label{twist} 
 {\cal L}^{\rm twisted} = \  - \frac {e^2}{4\pi(\bar\sigma \sigma + \bar\phi \phi)} 
 \left[ \frac
 {\sigma}{\bar\phi} \epsilon_{\alpha\beta} (\partial_\alpha \bar\sigma)(\partial_\beta \bar\phi) 
 +\frac  {\bar\sigma} \phi 
 \epsilon_{\alpha\beta}(\partial_\alpha \sigma)(\partial_\beta \phi) 
 \right]  
    \ee
in the lagrangian. 
The twisted term is a  2--form $F$. Its external derivative $dF$
can be associated with the torsion. The above--mentioned freedom  
to choose
${\cal K}$ corresponds to adding to $F$ the external derivative
of the 1-form generated from the functions 
$f,\bar f, g, \bar g$. The torsion is invariant under such a 
change.

Consider now a generic non--Abelian case. For a simple Lie group of 
rank $r$, the effective lagrangian is
\be
  \label{Lr}
  {\cal L} \ =\ \int d^2 \theta d^2 \bar\theta 
  \ {\cal K} (\bar\Phi^A, \Phi^A; \bar\Sigma^A, \Sigma^A)\ ,
     \ee
where $A = 1,\ldots,r$ and the expression for ${\cal K}$ is derived exactly in the same
way as in previous cases. We have
  \be
   \label{Kr}
   g^2 {\cal K} \ =\ \sum_j \left\{ \frac 2{c_V} 
\left[\bar\Sigma^{(j)} \Sigma^{(j)} - \bar\Phi^{(j)} \Phi^{(j)} \right] \right.
\nonumber \\ 
\left. -   \frac {g^2}{2\pi} \left[ F \left( \frac {\bar\Sigma^{(j)} \Sigma^{(j)} }
{\bar\Phi^{(j)} \Phi^{(j)}} \right)  - \ln \Phi^{(j)} \ln \bar\Phi^{(j)}
 \right]
\right\}
\ ,
 \ee
where $\Sigma^{(j)} = \alpha_j(\Sigma^A)$, etc. The prepotential (\ref{Kr}) 
satisfies a generalized harmonicity condition
   \be
  \label{harm4Kr}
 \frac {\partial^2 {\cal K}}{\partial \bar\Sigma^A \partial \Sigma^B } +
 \frac {\partial^2 {\cal K}}{\partial \bar\Phi^A \partial \Phi^B} \ =\ 0
  \ee
for all $A,B$.


 \section{\boldmath{$D=3$}\,: K\"ahler and Hyper--K\"ahler Models}
%\setcounter{equation}0 

\subsection{${\cal N} =1:$ Dual photon}

The effective lagrangian for $3D$,  ${\cal N} = 2$ (in $3d$ sense) SQED 
depends on only one gauge-invariant superconnection  in the reduced 
dimension $\Gamma_3$. Its component expansion (in Wess--Bagger notation) is 
 \be
\label{Gam3}
\Gamma_3 \ = \ A_3 - \frac 12 \epsilon_{\mu\rho\alpha} F_{\mu\rho}\,
 \theta \sigma_\alpha \bar\theta - D \theta \sigma_3 \bar\theta +
\frac 14 (\partial^2 A_3) \theta^2 \bar\theta^2
+ {\rm fermion\ terms}\ ,
 \ee 
where $F_{\mu\rho}$ is the $3D$ electromagnetic field ($\mu, \rho = 0,1,2$). The bosonic terms in the
effective lagrangian are
 \be
\label{act3}
{\cal L} \ =\ \int {\cal F} (\Gamma_3) d^4\theta \ =\ h(A_3) \left[ \frac 12 (\partial_\mu A_3)^2 - 
\frac 14   F_{\mu\rho}  F_{\mu\rho} + \frac {D^2}2\right] \ ,
  \ee
where $h = -{\cal F}''/2$.


It is convenient at this stage to perform a {\it duality transformation}. To this end, write the functional
integral corresponding to the lagrangian (\ref{act3}) in the form
 \be
 \label{dualint}
\int \, \prod dF \, d\Psi \, \exp\left\{i \int d^3x \left( {\cal L} + \frac 12 \epsilon_{\mu\rho\alpha} 
 F_{\mu\rho}
\partial_\alpha \Psi \right) \right\}\,.
  \ee
Integrating this over $\prod d\Psi$ brings about the Bianchi constraints $\epsilon_{\mu\rho\alpha}
 \partial_\alpha F_{\mu\rho} = 0$, which are solved by the standard relation
 $ F_{\mu\rho} = \partial_{[\mu} A_{\rho]}$. But let us instead do  the integral in 
Eq.\,(\ref{dualint})
over  $\prod dF$ first.  We are left with
   \be
  \label{dualZ}
   \prod \, d\Psi  \exp \left\{i \int d^3x \left[ \frac h2 (\partial_\mu A_3)^2 + 
  \frac 1{2h}  (\partial_\mu \Psi)^2 \right] \right\} \, .
   \ee
The integrand in the exponent is the dual lagrangian (the bosonic part thereof). 
The scalar field $\Psi$ is the dual photon.

Let us now introduce  the field 
 \be
\label{Bdef}
B \ =\ -\frac {{\cal F}'(A_3)}2,
 \ee
 so that $\partial_\mu B = h \, \partial_\mu A_3$. Introducing a
complex variable $\phi = (B + i\Psi)/\sqrt{2}$, we can write ${\cal L}_{\rm dual}$ in the K\"ahler
form $\int d^4\theta \, {\cal K} (\bar\Phi, \Phi)$. The relation between the K\"ahler potential ${\cal K}$ and
the function ${\cal F}$ can be inferred from Eq.\,(\ref{Bdef}). 

For the effective lagrangian of $3D$ SQED, the particular form of the metric and prepotentials 
${\cal F}$, ${\cal K}$ 
can be found via the ``unfolding the ring'' procedure. 
We have to integrate the one--loop correction to the $2D$ metric in (\ref{htilde}) over one
of the components $A_j$, in the same way as we earlier integrated the correction to the $1D$
metric to derive (\ref{htilde}). We obtain
  \be
\label{h3}
e^2 h_{3D} \ =\ 1 + \frac {e^2}{4\pi |A_3|} + \ldots
 \ee
(with 3--dimensional charge $e$). The metric is singular at $A_3 = 0$. This point
 separates two completely independent sectors in the moduli space (and in the theory !) 
with positive and negative $A_3$.  We will assume for definiteness that $A_3$ is positive. 
The prepotential entering (\ref{act3}) can be recovered from
the metric as 
 \be
\label{F3K3}
-e^2{\cal F}(\Gamma_3) \ =\ \Gamma_3^2  + \frac {e^2}{2\pi} \Gamma_3 \ln \Gamma_3 + \ldots \ .
  \ee
The K\"ahler potential depends only on $\Delta =  (\bar\Phi + \Phi)/\sqrt{2}$ and is given at the 
one--loop level
by a similar formula, namely
  \be
  \label{K3}
{\cal K} \ =\ e^2 \left[ \Delta^2 + \frac 1{2\pi} \Delta \ln \Delta \right]\, .
    \ee
The generalization to the non--Abelian case is straightforward. The generalized K\"ahler 
potential is
  \be
 \label{K3r}
  {\cal K}(\Delta^A) \ =\ 
g^2 \sum_j \left\{ \frac 2{c_V} 
(\Delta^{(j)})^2   
 -   \frac {3}{2\pi} \Delta^{(j)} \ln \Delta^{(j)}   \right\} + \ldots
\ ,
 \ee
where $\Delta^A =  (\bar\Phi^A + \Phi^A)/\sqrt{2}$ and $\Delta^{(j)} = \alpha_j(\Delta^A)$.


This is not  yet the end of the story, however.  Considerations
of supersymmetry alone do not exclude the presence of a {\it superpotential}
$\sim {\rm Re} \, \int d^2\theta \, F(\Delta^A) $ on top of the K\"ahler 
potential in the effective lagrangian. Indeed, such a superpotential is generated  
in non-Abelian $3D$ theories  by a 
{\it non-perturbative} mechanism \cite{3Dinst}. The mechanism is roughly the same
as  the known instanton mechanism for generating a superpotential in $4D$ ${\cal N} =1$ SYM
theory with matter \cite{ADS}. In three dimensions, instantons 
are t' Hooft--Polyakov monopoles.
They have two fermion ``legs''  (zero modes) which lead to generation of gluino condensate.
The superpotential can be recovered from the condensate. In the simplest $SU(2)$ case,
it has the form 
  \be
 \label{suppot}
  F(\Delta) \ \sim\ g^4 \exp \left\{ -  {2\sqrt{2} \pi \Phi }  \right\} .
  \ee
The superpotential (\ref{suppot}) lifts the degeneracy of the valley. Actually, the
scalar potential 
$U \sim  \exp \left\{ - {2\sqrt{2}\pi (\phi + \bar\phi) }  \right\}$
corresponding to the superpotential (\ref{suppot}) 
(the exponent $2\sqrt{2}\pi (\phi + \bar\phi) = 4\pi A_3/g^2$ is nothing but the 
$3D$ instanton action)
does {\it not} 
have a minimum at a finite value of $\phi$ (as it does not in the massless   ${\cal N} =1$ 
supersymmetric QCD --- this is a typical ``runaway vacuum'' phenomenon). In 
4-dimensional SQCD, this
can be cured by giving a mass to the matter fields: the supersymmetric
vacuum would then occur at a finite value of $\phi$. But in the framework given here, the form of the lagrangian of the descendants 
is dictated by the original $4D$
theory and we have to conclude that the $3D$ ${\cal N} =1$ sister simply does not exist 
as a consistent theory.   


In principle, this could also happen in the $2D$ ${\cal N} =1$ theory, 
where the appearance
of the superpotential is also not excluded by the symmetry considerations. Moreover, 
non-perturbative instanton solutions also exist in the $2D$ case \cite{Wit2D,2Dinst}. 
They appear
in any $2D$ gauge theory involving only adjoint fields, due to the non-triviality of 
$\pi_1$({\small gauge\ group}), by the same token as ordinary BPST instantons 
in four dimensions
appear,
due to non-triviality of $\pi_3(G)$. Now, $\pi_1[SU(N)] = 0$, but  if only adjoint fields are present,
the gauge group is globally $SU(N)/Z_N$ and involves $N-1$ topologically distinct
non-contractible loops and, correspondingly, $N-1$ different types of instantons.

 However, these
instantons do not generate a superpotential in this case for two reasons.
\begin{itemize}
\item 
 The minima of the classical action are realized on delocalized constant gauge field strength configurations (like
in the Schwinger model). The non--Abelian $2D$ instantons do not know about the scalar fields (unlike the monopoles) 
and cannot lift the degeneracy on the moduli space. 
\item As was shown in \cite{2Dinst}, the instantons involve 
$N-1$ {\it pairs} of fermion zero modes for each  fermion flavor. 
In our case, there are two flavors and an instanton involves  altogether $4(N-1)$ instanton legs, which 
 is too many to generate the fermion condensate
and superpotential.  
 \end{itemize}


\subsection{${\cal N}=2:$ Taub--NUT, Atiyah--Hitchin  and their Relatives}

The effective lagrangian here involves $4r$ moduli: three components of the vector-potential in reduced
dimensions and a dual photon for each unit of the rank. One way to derive the effective
lagrangian is to first determine the  effective lagrangian for the theory defined on
$R^2 \times S^1$ (it represents a twisted sigma model involving an infinite sequence of the Fourier
modes associated with the circle) and unfold the circle as was explained in detail in Sect. 3.
 When the length $L$ of the circle becomes large, we can replace the sum over the modes by the integral.
In the Abelian case, we obtain \cite{seliv}
  \be
 \label{Ld3dodual}
e_3^2{\cal L} \ =\ \left(\frac 12 + \frac {e^2}{8\pi |\vec{A}|} \right)
\left[ (\partial_\mu \vec{A} )^2 + (\partial_\mu \tau )^2 \right] \nonumber \\
- \frac {ie^2}{4\pi} \vecg{\omega}(\vec{A}) \epsilon_{\mu\nu} \partial_\mu \tau \partial_\nu \vec{A}\ ,
 \ee
where $g_3^2 = g_2^2 L$ is the 3--dimensional gauge coupling constant, $\mu = 1,2$
(we have not yet added excited  Fourier modes of the slow variables),   
 and  $\vecg{\omega} (\vec{A}) \equiv   \omega(\vec{A}) = cos \theta \, d\phi$ is a 
1-form which coincides with 
the Abelian connection describing a Dirac monopole in the space of $\vec{A}$.
 The variables $\vec{A}$ live on $R^3$, whereas the variable $\tau$ lives on the dual circle,
$0 \leq \tau \leq 2\pi/L$. The second term in Eq.\,(\ref{Ld3dodual}) comes from the twisted term  
 (\ref{twist}).

When $L$ is very large, the size of the dual circle is very small
which would normally imply that the excitations related to nonzero Fourier modes of $\tau$
 become heavy and decouple. This is exactly what happened  when we reconstructed  the $2D$ effective Lagrangian  from
the $1D$ one in Sect. 3 according to this method.  But in the case under consideration,
it would not be correct just to cross out the terms
involving $\partial_\mu \tau$: The presence of the twisted term 
$\propto \epsilon_{\mu\nu} $ prevents us from
doing it.

To understand this, consider a trivial toy model,
 \be
 \label{toy}
 {\cal L} = \frac 12 (\dot{x}^2 + \dot{y}^2) + B x \dot{y} \ \ \Longrightarrow  \ 
\ H = \frac 12 \left[ p_x^2  + (p_y - Bx)^2 \right]\ ,
 \ee
where $x \in R^1$, while $y$ is restricted to lie on a small circle,
$0 \leq y \leq \alpha$. The lagrangian (\ref{toy}) describes a particle living on a   cylinder and 
moving in a  constant magnetic field . Now, if the magnetic field $B$ were absent, the higher
 Fourier modes of the variable $y$ would be heavy and the low--energy spectrum would be continuous,
corresponding to free motion along $x$ direction.
When $B \neq 0$, for {\it each} Fourier mode of the variable $y$, we obtain
 the {\it same} oscillatory spectrum. Only the position of the center of the orbit, and not the energy,
depends on $p_y^{(n)} = 2\pi n/\alpha$.

Thus, we cannot  suppress the variable $y$ in the Lagrangian (\ref{toy}). Likewise,  we cannot  suppress 
the variable $\tau$ in Eq.\,(\ref{Ld3dodual}). What we can do, however, is to trade it with another variable
 using the  duality trick. Performing the same transformations as in the ${\cal N} =1$ case, we arrive
at the lagrangian of a sigma model 
living on a target space with the metric 
   \be
\label{TAUBNUT}
ds^2 \ =\  \left( 1 + \frac {e^2}{4\pi |\vec{A}|}
\right) d\vec{A}^2 + \frac {
\left(d\Psi - \frac {e^2}{4\pi} \omega  \right)^2}
{\left( 1 + \frac {e^2}{4\pi |\vec{A}|}
\right)} \ .
 \ee
The dual variable $\Psi$ describes the dual photon.

We want to emphasize that the effective lagrangian thus obtained represents a 
{\it conventional} sigma model ---  the twisted term 
disappears after the duality transformation. A conventional ${\cal N} =4$ sigma model must be 
hyper--K\"ahler. Indeed,
the metric (\ref{TAUBNUT}) describes a well-known hyper--K\"ahler Taub--NUT manifold
\cite{Eguchi}. 

Consider now SYM theory and let us start with the case of $SU(2)$. The result is
immediately written down by substituting  $-2g^2$ for   $e^2$  in all above formulae.  
The metric thus obtained (Taub--NUT with negative mass term) is also hyper--K\"ahler.
However, in contrast to the regular Taub--NUT metric, it is singular at $|\vec{A}|
= 2\pi/g^2$ and represents an {\it orbifold}. Note that the singularity occurs at small
values of $|\vec{A}|$, where the Born--Oppenheimer approximation is not valid.
As was explained earlier, we cannot neglect higher--derivative terms
in the effective lagrangian  in this region. Moreover, the very notion of the effective
lagrangian makes no sense any more. Still, the presence of a singularity in the double--derivative
part of the lagrangian is somewhat irritating.

Remarkably, the singularity actually disappears  when one takes into account
{\it non-perturbative} effects associated with instantons (coinciding with 
 't Hooft--Polyakov monopoles). Instantons bring about
 corrections to the metric of the form
$\sim \exp\{-4\pi n|\vec{A}|/g^2 \}$ ($n$ is the topological charge). 
They are irrelevant asymptotically, but
are very  important  for small values of $|\vec{A}|$: their re-summation gives a  {\it smooth}
hyper--K\"ahler Atiyah--Hitchin metric.\,\footnote{\,This was the conjecture of Ref. \cite{SW1}, confirmed by direct evaluation of the one-instanton contribution
to the metric in Ref. \cite{Tong}. (Multi-instanton contributions need not be calculated. The Hyper-K\"ahler nature
of the metric fixes them once the one--instanton contribution is known.)}
There exists an explicit expression for the AH metric. It involves elliptic functions and is not so
simple. An interested reader may look it up in  \cite{AH} where it is shown that its asymptotics
coincide with Taub--NUT and that the corrections to these asymptotics are exponential.


Another remarkable fact is that the same AH metric describes the low--energy dynamics of two BPS
monopoles \cite{GM}. In this case, the vector $\vec{A}$ acquires the meaning of the monopole
separation $\vec{r}$ and $\Psi$ of their relative phase. The (smoothened) singularity occurs 
when the distance between the monopoles is of the same order as the size of the monopole cores.
The classical trajectories of the monopoles represent geodesics on the AH manifold.

       
In the same way as before, we can write the effective lagrangian for an arbitrary
gauge group,
  \be
  \label{L3d}
  g^2{\cal L}\ =\ \frac 12 (\partial_\mu \vec{A}^A )  (\partial_\mu \vec{A}^B ) Q_{AB} + 
\frac 12  J_\mu^A Q^{-1}_{AB} J_\mu^B \ ,
  \ee
  where
  \be
  \label{QiJ}
 Q_{AB} &=& \delta_{AB} - \frac {g^2}{2\pi} \sum_j \frac
 {\alpha_j^A \alpha_j^B}{|\vec{A}^{(j)}|}\ , \nonumber \\
 J_\mu^A &=&  \partial_\mu \Psi^A +
 \frac {g^2}{2\pi} \sum_j \vecg{\omega}(\vec{A}^{(j)})
 \partial_\mu \vec{A}^{(j)} \alpha_j^A \ ,
  \ee
$\vec{A}^{(j)} = \alpha_j(\vec{A}^A) \equiv  \alpha_j^A \vec{A}^A $.

These are asymptotic expressions for the metric. They involve 
singularities and their structure is complicated. However, re-summation of instanton corrections
should patch up these singularities. The result of such a re-summation gives a smooth hyper--K\"ahler
manifold. Let me give arguments in favor of this conclusion.

{\bf (i)} Consider first the unitary groups \cite{CH}. The Cartan subalgebra of $SU(N)$ consists of 
traceless $N \times N$
diagonal matrices. As far as the effective lagrangian (\ref{L3d}) is concerned, we have four 
such matrices:
diag $(\vec{A}_1, \ldots, \vec{A}_N)$ and diag $(\Psi_1, \ldots, \Psi_N)$, 
$\sum_m \vec{A}_m = \sum_m \Psi_m = 0$.
 There are $N(N-1)/2$
positive roots, $\alpha_{ml}(\vec{A}) = \vec{A}_m - \vec{A}_l,
m < l = 1,\ldots,N$.
Substituting
 it in Eq.\,(\ref{L3d}), we obtain the metric 
   \begin{eqnarray}
  \label{CHmetr}
   ds^2\ = \ A_{ml} d\vec{A}_m d\vec{A}_l + A^{-1}_{ml} \Lambda_m \Lambda_l\ ,
  \end{eqnarray}
where  $A$ is the following $N\times N$ matrix:
  \be
  \label{Aml}
 A_{mm} &=& 1 - \frac {g^2}{4\pi} \sum_{l\neq m} \frac 1{|\vec{A}_m - \vec{A}_l|}\ \ \ \ 
\ \ \ \ \ \ \ \ \ \ ({\rm no\ summation\ over } \ m), \nonumber \\
A_{ml}  &=& \frac {g^2}{4\pi |\vec{A}_m - \vec{A}_l|},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (m \neq l)\ ,
 \ee
 and
 $$ \Lambda_m = d\Psi_m +  \frac {g^2}{4\pi} \sum_{l \neq m}
\omega(\vec{A}_m - \vec{A}_l) \ .$$


This metric happens to describe the dynamics of $N$ well--separated BPS monopoles \cite{GM}. 
$\vec{A}_m \equiv \vec{r}_m$ and
$\Psi_i$ are interpreted as the positions and phases of individual monopoles. The condition 
$\sum_m \vec{r}_m = 0$
(and similarly for phases) means that the trivial center of mass motion is separated out. 
The classical dynamics is described by the  following 
equations of motion\,\footnote{\,Here $g$ is interpreted as the monopole magnetic charge. The equations (\ref{eqmotmon})
are  classical as far as the variables $\vec{r}_l$ are concerned, but the quantization of the dynamic 
variables $\Psi_l$ has already been carried out. 
The  spectral parameters $q_l$ are quantized to (integer)/$g$ and are interpreted as the 
electric charges of the corresponding dyons.}
 \be
 \label{eqmotmon}
 \ddot{\vec{r}}_l - \frac {g^2}{4\pi} \sum_{m\neq l}^N \frac {\ddot{\vec{r}}_{lm} }{\vec{r}_{ml}} + 
\frac {g^2}{8\pi} \sum_{l \neq m = 1}^N \frac{2\left[\dot{\vec{r}}_{ml} \times \vec{r}_{ml}
\right] \cdot \dot{\vec{r}}_{ml} - \vec{r}_{ml} (\dot{\vec{r}}_{ml}^2) }{r_{ml}^3} \nonumber \\
- \frac {g}{4\pi} \sum_{m \neq l  } (q_{ml}) \dot{\vec{r}}_{ml} \times \frac {\vec{r}_{ml}}{r_{ml}^3} + 
\frac 1{8\pi} \sum_{m \neq l} \frac { q_{ml}^2 \vec{r}_{ml}}{r_{ml}^3} \ = 0\ , \nonumber \\
q_l \ =\ gA^{-1}_{lm} \left[ \dot{\Psi}_m + \frac {g^2}{4\pi}
\sum_{n \neq m} \vecg{\omega}( \vec{r}_{nm} ) \dot{\vec{r}}_{nm} \right] \ =\ {\rm const}\ ,
 \ee
where $\vec{r}_{ml} = \vec{r}_m - \vec{r}_l,\ q_{ml} = q_m - q_l$. The equations of motion for the 
effective lagrangian
(\ref{L3d}) have a similar form, with time derivatives being replaced by $\partial_\mu$ (and $\vec{r}$
by $\vec{A}$). 

The metric (\ref{CHmetr}) is singular
for certain small values of the distances between the monopoles $|\vec{r}_{ml}|$.  
These singularities can be patched, however, and with all probability {\it are} patched by the instanton
corrections.
A conjecture of existence and uniqueness can now be formulated: 
there is only one smooth
hyper-K\"ahler manifold of dimension $4(N-1)$ ( a {\it generalized} Atiyah--Hitchin manifold) 
 with the asymptotics (\ref{CHmetr}). (I bet there is, though, as far as I know, this has not yet 
been proven  
mathematically in an  absolutely rigorous way.)  An explicit expression for the generalized AH 
metric is not known.

{\bf (ii)} 
 $Sp(2r)$. There are $r$ long positive
roots $\alpha_m (\vec{r}) =  \vec{r}_m $
and $r(r-1)$
short positive roots   $\alpha_{ml}(\vec{r}) =
(\vec{r}_m \pm \vec{r}_l)/2$ ($m < l = 1,\ldots, r$ ; $\vec{r}_m$ are mutually orthogonal and
 linearly independent). The
metric reads
\begin{eqnarray}
\label{C}
ds^2  &=& \sum_m (d{\bf r}_m)^2 - \frac{g^2}{4\pi} \sum_{\pm}
\sum_{m<l}
\frac{(d {\bf r}_l \pm d{\bf r}_m)^2}
 {\vert{\bf r}_l \pm {\bf r}_m \vert}
-   \frac{g^2}{2\pi}  \sum_m \frac{(d{\bf r}_m)^2}{r_m} +\ {\rm phase\ part} \nonumber \\[1mm]
 &\equiv&
Q_{ml} d{\bf r}_m d{\bf r}_l +\ {\rm phase\ part}\ . 
\end{eqnarray}
The full metric is restored from Eqs.\,(\ref{L3d}, \ref{QiJ}).

An important observation is that the
corresponding effective Lagrangian (the QM version thereof)
is obtained from  the effective Lagrangian describing the dynamics
 of $2r+1$ BPS monopoles
numbered by the integers $j=-r,\ldots, r$
by imposing the constraints
\be
\label{A2C}
\vec{r}_{-r} + \vec{r}_{r} \ = \dots = \vec{r}_{-1} + \vec{r}_{1} \ =\ 2\vec{r}_0 \ =\ 0 \ , \nonumber \\[1mm]
\Psi_{-r} + \Psi_{r} \ = \dots = \Psi_{-1} + \Psi_{1} \ =\ 2\Psi_0 \ =\ 0\ .
 \ee
 We {\it are} allowed to impose these constraints because they are compatible with the equations
 of motion (\ref{eqmotmon}).
The corresponding  metric is hyper-K\"ahler. It has to be, due to ${\cal N} = 4$ supersymmetry, absence
of the twisted term and the theorem \cite{Freedman}. 
One can also prove it more
directly, reproducing the result (\ref{C})
by the  hyper-K\"ahler reduction procedure worked out in \cite{GR}. 

{\bf (iii)} $SO(2r+1)$. The system of roots is the same as for $Sp(2r)$, only the long and short roots are 
interchanged: there are now $r(r-1)$ long roots
$(\vec{r}_m \pm \vec{r}_l)/\sqrt{2}$ and $r$ short roots $\vec{r}_m/\sqrt{2}$.
 The metric reads
\begin{eqnarray}
\label{B}
ds^2\ =\
\sum_m (d{\bf r}_m)^2 - \frac{g^2}{2\pi \sqrt{2}} \left[ \sum_{\pm}
\sum_{m<l}
\frac{(d {\bf r}_l \pm d{\bf r}_m)^2}
 {\vert{\bf r}_l \pm {\bf r}_m \vert} + \sum_m \frac{(d{\bf r}_m)^2}{r_m}
\right] \nonumber \\
+\ {\rm phase\ part}\ .
\end{eqnarray}
This metric is obtained from the Gibbons-Manton type  metric for $2r$ BPS monopoles
numbered by the integers $j=-r, \ldots, r,\; j\neq 0$ by
imposing the constraints
  \be
\label{A2B}
\vec{r}_{-r} + \vec{r}_{r} \ = \dots = \vec{r}_{-1} + \vec{r}_{1} \ =\ 0 \ , \nonumber \\[1mm]
\Psi_{-r} + \Psi_{r} \ = \dots = \Psi_{-1} + \Psi_{1} \  =\ 0\
 \ee
 and rescaling $ds^2$ and $g^2$.
The constraints (\ref{A2B}) are compatible with the equations of motion.

Note that we obtained the effective Lagrangian for $Sp(2r)$ out of that for $SU(2r+1)$ and not 
$SU(2r)$, as one could naively expect in view of the embedding
$Sp(2r) \subset SU(2r)$. Likewise, the moduli space for $SO(2r+1)$ is
obtained out of  $SU(2r)$  and not  $SU(2r+1)$. This is  due to
the fact that
magnetic charges are coupled to co-roots rather than roots.

The effective lagrangian for $SO(2r)$ can also be readily written. It can also be interpreted in
monopole terms and the corresponding manifold is related to a generalized AH manifold for the
system of $2r$ monopoles by hyper--K\"ahler reduction accompanied by a certain mass {\it deformation}
(suppressing interactions between certain monopoles) \cite{seliv}.

The constraints (\ref{A2C}), (\ref{A2B}) have the form of ``mirrors'' in the monopole 
configuration space. In the $Sp(2r)$ case, the mirror passes through one of the monopoles while in the
$SO(2r+1)$ case it does not. Such ``mirrors'' appeared earlier in string--related problems and were 
christened
{\it orientifolds} \cite{H}. For me, they just represent graphical pictures describing the embedding
of the symplectic and orthogonal groups into unitary ones.

{\bf (iv)} $G_2$. This is the simplest exceptional group. 
 There are three long ( $\vec{r}_1 - \vec{r}_2, \vec{r}_1 - \vec{r}_3, \vec{r}_2 - \vec{r}_3$)
 and three short ($\vec{r}_{1,2,3}$)
positive roots (the constraint $\vec{r}_1 + \vec{r}_2 + \vec{r}_3 =
 0 $  being imposed). The metric reads
 \be
 \label{G}
 ds^2 \ =\ \sum_{m=1}^3 d\vec{r}_m^2 - \frac {g^2}{2\pi}
 \left( \sum_{m>l=1}^3 \frac {(d\vec{r}_m  - d\vec{r}_l)^2}{|\vec{r}_m  - \vec{r}_l|}  + 3 \sum_{m=1}^3 
\frac {d\vec{r}_m^2}{|\vec{r}_m |} \right) + \ldots\ .
  \ee
It can be obtained out of the metric for $Sp(6)$
  \be
 \label{Sp6}
 ds^2 \ =\ \sum_{m=1}^3 d\vec{r}_m^2 -
 \frac {g^2 }{4\pi}
 \left( \sum_\pm \sum_{m>l=1}^3 \frac {(d\vec{r}_m  \pm d\vec{r}_l)^2}{|\vec{r}_m  \pm \vec{r}_l |}  +
 2 \sum_{m=1}^3 \frac {d\vec{r}_m^2}{|\vec{r}_m |} \right) + \ldots
  \ee
  by rescaling and  imposing the [compatible with $Sp(6)$ equations of motion] constraints
\be
  \label{C3-2-G2}
\vec{r}_1 + \vec{r}_2 + \vec{r}_3 \ =\ 0\ , \ \ \ \ \ \ \
\Psi_1 + \Psi_2 + \Psi_3 \ =\ 0\ .
  \ee
This ``3--fold mirror'' is (or could be called) an orientifold of new type.
  Again, we obtained ${\cal L}_{\rm eff}$ for  $G_2$ out of   ${\cal L}_{\rm eff}$ for  $Sp(6)$, 
though    
$G_2$ is embedded not into
$Sp(6)$, but into the dual algebra $SO(7)$.

The effective Lagrangian for $F_4$  can be related to the moduli space of
26 monopoles. (26 is the lowest dimension of a unitary group in which
$F_4$ can be embedded. This follows from the fact that the representation 
{\bf 26} of $F_4$ has the lowest dimension.) $E_6$ can be embedded into
$SU(27)$ and hence the corresponding effective Lagrangian is
 related to the moduli space of
27 monopoles. Now, the shortest representation in $E_7$ has the
dimension 56 and we need at least 56 monopoles in this
case. Finally, $E_8 \subset SU(248)$ and for this we need 248 monopoles. The moduli
space of 248 monopoles can also be used as a universal starting point
to describe the dynamics
of $F_4, E_6$ and $E_7$, if one
follows the chain of embeddings $F_4 \subset E_6 \subset E_7 \subset E_8
\subset SU(248)$.

The  explicit formulae we have written refer to the asymptotic region
 where non-perturbative effects are suppressed. The corresponding metrics
involve singularities at small $|\vec{r}^{(j)}|$.
As for the $SU(N)$ case, a reasonable conjecture is that  these
singularities are sewn up by instantons  for any simple Lie group, giving a unique
 smooth hyper-K\"ahler metric with the asymptotics
    \be
   ds^2 \ =\ d\vec{r}^A Q_{AB} d\vec{r}^B + \ldots \ .
   \ee
It is  natural to conjecture that this metric is obtained
from the multimonopole Atiyah-Hitchin
metrics by the same hyper-K\"ahler reduction procedure as above. To the best of my knowledge,
 hyper--K\"ahler manifolds thus obtained have
not been studied before by mathematicians.  

\section{Non--renormalization Theorems}
%\setcounter{equation}0
 There are several proofs of the well--known fact that two-- and higher--order corrections to
the $\beta$ function in the $4D$ ${\cal N} = 2$ supersymmetric Yang--Mills theory vanish.
 We will discuss here two such proofs: {\it (i)} diagrammatic (historically, this was the first) and 
{\it (ii)} the one following from holomorphy.

{\it Supergraphs. } 
The diagrammatic proof is based on the technique of supergraphs. The simplest non-renormalization 
theorem states that all loop corrections to the {\it superpotential} (the term $\int \, d^2\theta F(\Phi_i)$ 
in the lagrangian, $\Phi_i$ are chiral superfields) vanish. We refer the reader to the textbooks
\cite{WB,Siegel,West} for its proof,  recalling
in more detail how it is done for gauge couplings (following Refs.\cite{ShiVa}), the subject of our
interest here. 

Consider for simplicity SQED.\footnote{\,The generalization to the
non--Abelian case is relatively straightforward, but it 
involves some subtleties associated with infra-red singularities of the theory \cite{ShiVa,SmiVa}
 which we do not discuss here.}
We explained above [see Eq.\,(\ref{detratio})] how the one--loop correction 
to the effective action in any dimension can be evaluated. It does not vanish here. 
The effective charge in ${\cal N} =1$ theory is given by
 \be
 \label{eren4}
  \frac 1 {e^2_{\rm phys}} \ =\ \frac 1{e_0^2} + \frac 1{4\pi^2} \ln \frac {\Lambda_{UV}} {m_0} + \ldots\ ,
  \ee
where $m_0$ is the {\it bare} charged field's mass.
\begin{figure}
   \begin{center}
        \epsfxsize=170pt
        %\epsfysize=100pt
        %\vspace{-5mm}
        \parbox{\epsfxsize}{\epsffile{sestr2.eps}}
        \vspace{5mm}
    \end{center}
\caption{Two-loop contribution to the effective action. 
Solid lines are chiral field superpropagators $\langle \Phi \bar \Phi \rangle$ evaluated in the classical
gauge background and the dashed
line is the propagator of the (quantum part of the) vector superfield $V$. The bar on the solid line marks
the $\bar \Phi$ end.} 
\label{2loop}
\end{figure}

The relevant 
two--loop graph is drawn in Fig. \ref{2loop} (there are actually two such supergraphs giving the same
contribution with the superfields $S$ or $T$ in the loop).
 According to the supergraph Feynman rules \cite{Siegel,West},  each vertex
involves the integral $\int d^8z = \int d^4x  \, d^2\theta d^2\bar\theta $ and the whole contribution
of the graph in Fig. \ref{2loop} is
 $ \int d^4x_1 \, d^2\theta_1  d^2\bar\theta_1  K$, where
$$ {\cal K} = \frac i2 \int d^8z_2 \langle \Phi_1 \bar \Phi_2 \rangle  \langle  \Phi_2 \bar \Phi_1  \rangle
\langle v_1 v_2 \rangle \ .$$   
 Here $\Phi$ stands for charged chiral superfields $S,T$, $v$ is quantum vector superfield
 and $\langle \Phi_1 \bar \Phi_2 \rangle, \ \langle v_1 v_2 \rangle$  are quantum superpropagators 
 evaluated in the external background $V_{\rm cl}$.  Now, $ \langle v_1 v_2 \rangle$ does not depend
on the external field or its gauge. The charged field propagators are gauge--dependent:
    \be
&&\langle S_1 \bar S_2 \rangle \to e^{i\Lambda_1} \langle S_1 \bar S_2 
\rangle e^{-i\bar\Lambda_2}\ ,\\[1mm] \nonumber
&&\langle T_1 \bar T_2 \rangle \to e^{-i\Lambda_1} \langle T_1 \bar T_2 \rangle e^{i\bar\Lambda_2}\ .
  \ee
The point is, however, that the {\it integrand} $K$ is gauge--independent and should thereby be
{\it locally}\,\footnote{\,Locality follows from the presence of 
an infra-red cut-off (non-zero mass) in the theory.\\[-2mm]~}
expressed via the gauge--invariant superfield $W_\alpha$. 
But $W_\alpha$ is a chiral superfield and
the integral over $d^2\bar\theta$ of any function of $W$ vanishes.  Therefore 
$\int d^2\theta d^2\bar\theta \, {\cal K} =0$,
Q.E.D. The same reasoning applies also to an arbitrary multi-loop 
graph.\footnote{\,To be precise, the integrand could depend 
on {\it both} $W$ and $\bar W$,
in which case the integral  $\int d^2\theta d^2\bar\theta \, {\cal K}$ need not vanish. One can be
convinced, however, that such contribution is a supersymmetric generalization of higher derivative
$\sim F^4$  terms in the Euler--Heisenberg effective lagrangian. {\it Such} corrections to the
effective lagrangian are indeed present, but this does not affect the renormalization of the gauge
coupling.}

We hasten to comment that this does {\it not} mean that multi-loop contributions to $\beta$ 
function in ${\cal N} =1$ supersymmetric QED vanish. Higher loops appear when expressing
$m_0$, entering Eq.\,(\ref{eren4}), into $m_{\rm phys}$. As was already mentioned above, the physical
mass {\it is} renormalized in spite of the fact that the mass term in the Lagrangian is not. 
Indeed,
the physical mass is defined as the pole of the fermion propagator $\propto 1/(Z/\!\!\!p - m_0)$, 
where $Z$
describes the renormalization of the {\it kinetic} term 
$$\propto \int d^2\theta d^2\bar\theta \left( \bar S e^V S + \bar T e^{-V} T \right) \ . $$
We have $m_0 = Z  m_{\rm phys}$ which leads to an exact relation expressing the charge
renormalization via the matter $Z$ factor,
 \be
\frac 1 {e^2_{\rm phys}} \ =\ \frac 1{e^2_0} + \frac 1{4\pi^2} \ln 
\frac {\Lambda }{m_{\rm phys} } - \frac  1{4\pi^2} \ln Z \ . 
  \ee
In particular, using knowledge of $Z$ at the one--loop level 
 \be
  \label{mrenorm}
 Z\ = \ 1 -  \frac {e_0^2}{4\pi^2}  \ln \frac \Lambda {m_{\rm phys}}\ ,
  \ee
we obtain the two--loop renormalization of the charge
  \be  
\label{ephys}
\frac 1{e_{\rm phys}^2} \ =\  \frac 1{e_0^2} + 
\frac 1{4\pi^2} \ln \frac {\Lambda}{m_{\rm phys}} 
+ \frac {e_0^2}{16\pi^4} \ln \frac {\Lambda}{m_{\rm phys}} + \ldots \ .
   \ee

Now, ${\cal N} = 2$ supersymmetric electrodynamics involves an extra neutral chiral superfield
$\Upsilon$. The Lagrangian involves its kinetic term and the extra superpotential term
$\propto \int d^2\theta \, \Upsilon ST $.  The latter is not renormalized: this is the standard
$F$ term non-renormalization theorem. The point is that this superpotential term is related
by extended supersymmetry to the charged-field kinetic term. Hence, non-renormalization of the
superpotential {\it implies} in  ${\cal N} = 2$ theory non-renormalization of the kinetic term,
which implies the absence of the mass renormalization. In other words, in  ${\cal N} = 2$ theory,
$m_{\rm phys} = m_0$ and hence only the first term in the $\beta$ function survives. All 
 ${\cal N} = 2$ theories with vanishing 1--loop contribution to the  $\beta$ function are 
finite.  The ${\cal N} = 4$ SYM theory belongs to this class.

The proof just given uses the formalism of ${\cal N} =1$ supergraphs.  ${\cal N} = 2$ symmetry is used
indirectly via the requirement of equality of renormalization factors for the standard kinetic and superpotential
terms. One can also define and calculate supergraphs in  ${\cal N} = 2$ (harmonic) superspace. In this case,
the absence of the corrections is manifest \cite{kniga}.     

\vspace{.5mm} 

{\it Holomorphy}. An alternative elegant proof comes from the analysis of the Seiberg--Witten 
effective lagrangian.
As was mentioned in the beginning, ${\cal N} =2$ supersymmetry dictates the form (\ref{LSW}) 
for the effective
lagrangian, where $F({\cal W})$ is an {\it holomorphic} function of the  ${\cal N} =2$ superfield
(\ref{Wcal}) (the Abelian version thereof). 
Consider this function for large ${\cal W}$. Going around the large circle
( multiplying ${\cal W}$ by $e^{2i\pi}$ ), we should obtain the 
{\it same} theory.\footnote{\,Actually, it is sufficient to 
multiply  ${\cal W}$ by $e^{i\pi}$. It is ${\cal W}^2$
rather than    ${\cal W}$ which has direct physical meaning, 
the lowest component of ${\cal W}^2$
coinciding with the true moduli $u = {\rm Tr}\, \phi^2$. }
Knowing that,  asymptotically,
   \be
  {\cal L}_{\rm eff} \sim  {\rm Re}  \,    
\int d^4\theta  \,  \frac 1 {2g^2 ({\cal W})} \, {\cal W}^2 
  \ee
(This is written for non--Abelian $SU(2)$ theory. The same expression with 
$e^2({\cal W})$ substituted
for $g^2({\cal W})$ holds  in ${\cal N} =2$ SQED),  
only two possibilities are allowed:  {\it (i)} $g^2 ({\cal W})$ is a constant (this possibility is
realized for ${\cal N} =4$ non--Abelian gauge theories) or {\it (ii)} . $g^{-2} ({\cal W})$ involves
a term $\sim \ln {\cal W}$, which corresponds to one--loop renormalization. When multiplying
${\cal W}$ by $e^{2i\pi}$, the logarithm is shifted by an imaginary constant. This gives a change
$\sim {\rm Im}\, \int d^4 \theta \, {\cal W}^2$ in the lagrangian, which is a { total derivative}
($\theta$ term). In Abelian theory, the $\theta$ term is never relevant. In non--Abelian theory it {\it might}
have been relevant, but it is not in this case: one can be convinced that multiplying
 ${\cal W}^2$ by $e^{2i\pi}$ amounts to the shift $\theta \to \theta + 4\pi$. 

 The higher--order coefficients $\beta_2 , \beta_3$, etc should vanish. Were, e.g., $\beta_2$ nonzero, the 
coefficient $g^{-2} ({\cal W})$ in  ${\cal L}_{\rm eff}$ would involve the contribution
$\sim \ln \left|1 + c  \ln (\Lambda_{\rm UV}/ {\cal W} ) \right| $, which is not holomorphic and 
 not allowed. On the other hand,
nothing prevents  the function $f({\cal W})$ from  having 
contributions $\sim {\cal W}^{-n}$, which vanish   asymptotically. 
 Indeed,   $f({\cal W})$ {\it does} involve such contributions brought about by  instantons \cite{SW}.

\vspace{1mm}

 We have seen that low-dimensional sisters of $4D$ ${\cal N} =2$ have similar properties: the perturbative
corrections to the effective lagrangians vanish beyond one loop. In sect. 2.2 we explained why: extended
supersymmetry dictates a special form for the prepotentials. In $1D$, resp. $2D$ theories, the
prepotentials in (\ref{act5}), resp. (\ref{LGHR}) living in $R\!\!\!\!R^5$, 
resp.  $R\!\!\!\!R^4$  must be harmonic functions of their arguments .
For  $3D$ theories, extended supersymmetry requires the metric to be hyper--K\"ahler. In the asymptotics,
the metric (\ref{TAUBNUT}) involves a   harmonic function $1 + e^2/(4\pi |\vec{A}|)$ living
 in $R\!\!\!\!R^3$.
A more detailed analysis shows that the harmonicity follows from the hyper--K\"ahler nature of the metric
(i.e. extended supersymmetry) and from its $U(1)$ isometry corresponding to shifting the phase 
$\Psi$ (this isometry shows up  in the 
asymptotics). Actually, in the cases when such an isometry is present, the K\"ahler potential 
of a hyper--K\"ahler metric can be obtained from a certain 3-dim harmonic prepotential by a Legendre
(physically -- by duality) transformation \cite{Hitchin,seliv}.\,\footnote{\,The full Atiyah--Hitchin metric, which involves, besides one loop, also non-perturbative
instanton corrections does not
have this isometry and cannot be expressed via a 3-dim harmonic function. It can be expressed, however,
via certain more complicated generalized harmonic functions \cite{AHharm}. 
Their physical meaning is yet to
be revealed.}     

Now, in $4D$ theories the moduli space is   $R\!\!\!\!R^2 \equiv C\!\!\!\!C^1$ 
and harmonicity there is the same as 
analyticity\,! In other words, the  proof of the $4D$ non-renormalization theorem based on 
holomorphy has  direct low--dimensional counterparts. Nonrenormalizability is a family property
of all sisters.

What about the diagrammatic proof ?

The $4D$ diagrammatic proof quoted above involved two parts: {\it (i)} the ${\cal N} =1$
non-renormalization theorem and {\it (ii)} the 
${\cal N} = 2$ relationship between the kinetic and superpotential
terms. This relationship holds also in low dimensions, but there is no 
${\cal N} =1$ non-renormalization theorem anymore. Indeed, the theorem was based on the the fact that
the $4D$ effective lagrangian had the form $\int d^2\theta \, W^2$, while the two- and higher-loop supergraphs
suggested the form $\int d^2\theta  d^2\bar\theta \, X(W, \bar W)$, which could be only reconciled if $X=0$. But
in low dimensions, the effective lagrangian does not have a chiral form, but instead represents an integral 
 $\int d^2\theta  d^2\bar\theta$ of a local density depending not on $W$, but rather on
superconnections $\Gamma_k$ in reduced dimensions [see e.g. Eq.\,(\ref{act1})]. This can well be reconciled
with what follows from the diagram in Fig. 2. 

Indeed, direct component calculations of the two--loop corrections to the effective action in the 
$D=1, \ {\cal N} =1$ Abelian theory showed that they do not vanish \cite{Abel}. 
One obtains instead of Eq.\,(\ref{hab})
  \be
\label{hab2}  
e^2h(\vec{C})  \ =\  1 + \frac {e^2} {2|\vec{C}|^3}  - \frac {3e^4}{4|\vec{C}|^6 } + \ldots 
 \ee  
 for the metric. In addition, the two--loop contribution is not related to any $Z$-factor, unlike in four dimensions:  the latter just cannot be defined in quantum mechanics.

In Ref.\cite{Abel}, the result (\ref{hab2}) was obtained after rather cumbersome calculations 
where the contribution of several graphs was added. Using the ${\cal N} = 1$ supergraph technique, 
only one graph in Fig. 2 should be evaluated and the calculation is rather simple \cite{SmiVa}. 
In ${\cal N} = 2$ theory, a similar graph with the exchange of $\Upsilon$ field should be added.
It has exactly the same structure and gives exactly the same contribution, but with the opposite
sign. The cancellation is manifest. Unfortunately, this simple cancellation pattern does not hold
at the 3--loop level and higher. Again, one obtains zero after adding up several different
supergraphs. In other words, it is hardly possible to prove non-renormalization of ${\cal N} = 2$
theories using the formalism of ${\cal N} = 1$ supergraphs. 

On the other hand, it is very reasonable to suppose that the diagrammatic proof of
non--renormalizability based on the technique of harmonic supergraphs \cite{kniga} can be extended
to low dimensions. This question is currently under study.
%\vspace{-0.6cm}

\section{Conclusions}
To make distinction with the Introduction where main results concerning the nature and character
of different sisters were outlined in words and the main body of the paper where the relevant formulae 
were written, we give here the same information in the 
 form of a table.  
%\vskip -0.4cm 
\begin{table}[ph]
\tbl{Pure SYM: the family of effective theories.\vspace*{3pt} }
{
\begin{tabular}{|c|p{5.5cm}|p{5cm}|}
\hline
\ & \ & \ \\[-2mm]
\  &\qquad \qquad \qquad ${\cal N} =1$ &  \qquad \qquad\qquad  ${\cal N} =2$ \\[1mm]
\hline 
\ & \ & \ \\[-2mm]
$D =1$ & Symplectic $\sigma$ model of the first kind & Symplectic $\sigma$ model of the second kind \\[1mm]
\hline
\ & \ & \ \\[-2mm]
$D =2$ & K\"ahler $\sigma$ model & Twisted $\sigma$ model \ (GHR)\\[1mm]
\hline
\ & \ & \ \\[-2mm]
$D =3$ & K\"ahler $\sigma$ model with superpotential.  Run-away vacuum & Hyper--K\"ahler $\sigma$ model \\[1mm]
\hline
\ & \ & \ \\[-2mm]
$D =4$ & No moduli space. Discrete vacua & SW effective theory \\[1mm]
\hline
 \end{tabular}}
 

\end{table}

%\vskip -0.5cm
\noindent
{\bf Acknowledgments}\\
%\section*{Acknowledgments}
I thank  E. Akhmedov, E. Ivanov, and A. Vainshtein for collaboration
and many  illuminating discussions. Special thanks are due to A. Vainshtein who read the
draft and made many very useful comments. 

{\it  My thanks are also due to my friend and collaborator Konstantin Selivanov. Alas, 
 Kostya 
is not with us anymore. He died young, as Ian did. This paper is a tribute to the memory
of Ian and to the memory of Kostya.}

 
\newpage
\begin{thebibliography}{40}
\addcontentsline{toc}{section}{References}


\bibitem{SW}  N.~Seiberg and E.~Witten, Nucl. Phys. B {\bf 426}, 
 19 (1994).

\bibitem{WB}  J.~Wess and J.~Bagger, {\it Supersymmetry and supergravity} 
(Princeton Univ. Press, Princeton,  1983).

\bibitem{Lykken} J.~D.~Lykken, 
{\em Introduction to supersymmetry},
hep-th/9612114.
%%CITATION = HEP-TH 9612114;%%

\bibitem{kniga} A.S.~Galperin,
E.A.~Ivanov, V.I.~Ogievetsky, and E.S.~Sokatchev,
{\it Harmonic Superspace} (Cambridge Univ. Press, 2001).

\bibitem{sympl}  
E.A.~Ivanov and A.V.~Smilga,
{\em Symplectic sigma models in superspace}, Nucl. Phys. B, in Press, 
hep-th/0402041.
%%CITATION = HEP-TH 0402041;%%

\bibitem{Coxeter}  E.~Witten, Nucl.\ Phys.\  B {\bf 202},  253  (1982); 
JHEP {\bf 9802}, 006  (1998);\\
A.~Keurentjes, A.~Rosly, and A.V.~Smilga, Phys. Rev. D {\bf 58}, 
081701  (1998);\\ V.G.~Kac and A.V.~Smilga, in
{\it The Many Faces of the Superworld}, ed.
M.A.~Shifman (World Scientific, 2000) [hep-th/9902029]; A.~Keurentjes, JHEP  {\bf 9905}, 001  (1999).

\bibitem{Freedman} L.~Alvarez--Gaum\'e and D.Z.~Freedman,
Commun. Math. Phys. {\bf 80}, 443  (1981).

 \bibitem{GHR}
S.J.~Gates, Jr., C.M.~Hull, and M.~Ro\^cek, Nucl.
Phys. B {\bf 248},  157 (1984). 


\bibitem{SQED} A.V.~Smilga, Nucl. Phys. B {\bf 291}, 241 (1987).

\bibitem{my} E.A.~Ivanov and A.V.~Smilga, Phys. Lett. B
{\bf 257},  79 (1991).

\bibitem{bp} V.P.~Berezovoj and A.I.~Pashnev, Class. Quant. Grav. {\bf 8},  2141 (1991).

\bibitem{Siegel} See, e.g., 
S.J.~Gates, M.T.~Grisaru, M.~Rocek, and W.~Siegel,
{\it Superspace, Or One Thousand And One Lessons In Supersymmetry},
Front.\ Phys.\ {\bf 58}, 1 (1983) \\{} 
[hep-th/0108200].
%%CITATION = HEP-TH 0108200;%%

\bibitem{weak} A.V.~Smilga, Phys. Lett. B {\bf 585} (2004) 173. 

\bibitem{akhmedov} E.A.~Akhmedov and A.V.~Smilga, Yad. Fiz. {\bf 66},  2290 (2003).

\bibitem{shif} V.A.~Novikov, 
M.A.~Shifman, A.I.~Vainshtein, and V.I.~Zakharov, Nucl. Phys. B {\bf 229},  381 (1983), 
Phys. Lett. B {\bf 166}, 329  (1986); for a modern review see M.A.~Shifman and
A.I.~Vainshtein,  in:
M.A.~Shifman, {\it ITEP Lectures on Particle Physics and Field
Theory}, (World Scientific, 1999) v.\,2, p.\,485 [hep-th/9902018].

\bibitem{Strom} A.~Maloney, M.~Spradlin, and A.~Strominger, JHEP {\bf 0204}, 003  (2002).

\bibitem{BO} A.V.~Smilga, JHEP {\bf 0204}, 054  (2002).

\bibitem{DE} D.E.~Diaconescu and R.~Entin, Phys. Rev. D
{\bf 56},  8045 (1997).

\bibitem{N2} A.V.~Smilga, Nucl. Phys. B {\bf 652},  93 (2003).

\bibitem{sestry} K.~Becker and M.~Becker, Nucl. Phys. B {\bf 506},
 48 (1997).

\bibitem{sigmod} D.Z.~Freedman and P.K.~Townsend, Nucl. Phys. B
{\bf 250},  689 (1985).

\bibitem{brany} E.~Witten, Nucl. Phys. B {\bf 403},  159 (1993).
 
\bibitem{DiaSei} D.E.~Diaconescu and N.~Seiberg, JHEP {\bf 9707}, 001  (1997).

\bibitem{Hitchin} N.J.~Hitchin,  A.~Karlhede, U.~Lindstrom, and M.~Ro\^cek,
Commun. Math. Phys. {\bf 108}, 535 (1987). 

\bibitem{3Dinst} I.~Affleck, J.~Harvey, and E.~Witten,  Nucl. Phys. B {\bf 206},
 413 (1982); N.M.~Davies, T.J.~Hollowood, V.V.~Khoze, and M.P.~Mattis, 
Nucl. Phys. B {\bf 559},  123 (1999).

\bibitem{ADS} I.~Affleck, M.~Dine, and N.~Seiberg, Nucl. Phys. B {\bf 241}, 
493  (1984); {\bf 256}, 557  (1985). 

\bibitem{Wit2D} E.~Witten,  Nuovo Cim. A {\bf 51},  325 (1979).

\bibitem{2Dinst} 
A.V.~Smilga, Phys. Rev. D {\bf 49},  6836 (1994);  {\bf 54},  7757 (1996).

\bibitem{seliv} K.G.~Selivanov and A.V.~Smilga, JHEP {\bf 0312}, 027 (2003).

\bibitem{Eguchi} See, e.g., T. Eguchi, P.~B.~Gilkey, and A.~J.~Hanson, 
Phys. Rept. {\bf 66}, 213  (1980).

\bibitem{SW1}
N.~Seiberg and E.~Witten,
{\em Gauge dynamics and compactification to three dimensions},
hep-th/9607163.
%%CITATION = HEP-TH 9607163;%%

\bibitem{Tong} 
N.~Dorey, V.~V.~Khoze, M.~P.~Mattis, D.~Tong and S.~Vandoren,
%``Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin
%manifold,''
Nucl.\ Phys.\ B {\bf 502}, 59 (1997)
[hep-th/9703228].
%%CITATION = HEP-TH 9703228;%%

\newpage
\bibitem{AH}
M.~Atiyah and N.~Hitchin, {\it
The Geometry and Dynamics of Magnetic Monopoles} (Princeton Univ. Press, 1988). Useful formulae can also be found in 
A. Hanany and B. Pioline, JHEP {\bf 0007}, 001 (2000) [hep-th/0005160].

\bibitem{GM}
G.W.~Gibbons and N.S.~Manton,
Phys.\ Lett.\ B {\bf 356}, 32 (1995).


\bibitem{CH}
G.~Chalmers and A.~Hanany,
Nucl.\ Phys.\ B {\bf 489}, 223 (1997).


\bibitem{GR}
G.W.~Gibbons and P.~Rychenkova,
%{\em Hyper-K\"ahler quotient construction of BPS monopole moduli spaces}, 
Commun. Math. Phys. {\bf 186}, 581 (1997)\\{} [hep-th/9608085].
%%CITATION = HEP-TH 9608085;%%


\bibitem{H}
A.~Hanany, B.~Kol and A.~Rajaraman,
JHEP {\bf 9910}, 027 (1999) [hep-th/9909028];\\
A.~Hanany and B.~Kol,
JHEP {\bf 0006}, 013 (2000) [hep-th/0003025];\\
A.~Hanany and J.~Troost,
JHEP {\bf 0108}, 021 (2001) [hep-th/0107153].

\bibitem{West} P.C.~West, {\it Introduction to supersymmetry and supergravity}  (World Scientific, 1990).

\bibitem{ShiVa}
M.A.~Shifman, A.I.~Vainshtein, and V.I.~Zakharov, Phys. Lett. B {\bf 166}, 334 (1986);
M.A.~Shifman and A.I.~Vainshtein, Nucl. Phys. B {\bf 277},  456 (1986).

\bibitem{SmiVa} A.V.~Smilga and  A.I.~Vainshtein, {\it Background field calculations and nonrenormalisation 
theorems in 4d supersymmetric gauge theories and their low--dimensional descendants}, hep-th/0405142. 

\bibitem{AHharm} I.T.~Ivanov and M.~Ro\^cek, Commun. Math. Phys. {\bf 182}, 
 291 (1996) [hep-th/9512075];\\ I.~Bakas, Fortsch. Phys. {\bf 48}, 9 (2000) hep-th/9903256].

\bibitem{Abel} A.V.~Smilga, Nucl. Phys. B {\bf 659}, 424 (2003) [hep-th/0205044].

\end{thebibliography}



\end{document}
