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the formation of colour - electric flux tubes provides an intuitive physical picture for permanent confinement of quarks within hadrons .
moreover , this picture seems to be realized within qcd , as has widely been observed in lattice simulations .
the mechanism leading to flux tube formation , however , is still subject to analytical and numerical investigations .
the shortcoming of the corresponding lattice calculations , however , is the fact that regarding sufficiently long flux tubes , i.  e.  large lattice sizes , becomes a numerically very difficult task . a simpler gauge theory which also exhibits confinement is @xmath0
; it can be used as a prototype for investigating the formation of flux tubes .
furthermore , it was realized many years ago @xcite that one can perform a duality transformation of the path integral in compact abelian gauge theories . in this way a new partition function is obtained which can be regarded as a limit of the dual non - compact abelian higgs model @xcite .
besides this fact , the dually transformed theory can be used as a very efficient tool for the calculation of expectation values in the presence of external charges @xcite .
the numerical advantages of such dual flux tube simulations also hold for the more general case of dual higgs models .    in this
talk we review the results obtained in dually transformed @xmath0 lattice gauge theory and present some first calculations in the dual abelian higgs model .
we will focus on one of the key questions within the dual superconductor picture , namely whether the confinement vacuum corresponds to a type - i or a type - ii superconductor , and we will investigate the role of quantum fluctuations of the dual degrees of freedom .
the expectation value of a physical observable like the electric field or the magnetic current in the presence of external sources ( represented by polyakov loops ) can be rewritten as averaging over the dual degrees of freedom , as has been shown in previous work @xcite .
the dual representation of the polyakov loops can be interpreted as a dual dirac sheet connecting the electric charges .
this dual formulation opens the possibility for very accurate numerical calculations for several reasons : contrary to standard lattice simulations , the confinement phase is the weakly coupled one in the dual theory , therefore there are less quantum fluctuations . even more important is the fact that it is not necessary to project the charge 
anticharge state out of the vacuum : on a lattice of same size charge pairs with arbitrary distance can be simulated with equal accuracy , and increasing the time extent of the loops ( i.  e.  of the lattice ) does not influence the quality of the result , either .
further advantages are the implementation of doubly charged flux tubes @xcite by double dirac strings which also yields results of equal accuracy , and the extension to periodically closed flux tubes .
these `` torelons '' are free of end effects and allow for a more reliable calculation of pure flux tube properties like the string tension .
an interesting physical task is the testing of the validity of a dual london equation for @xmath0 flux tubes @xcite . in the dual formulation
we were able to investigate rather long flux tube lengths up to 20 lattice spacings .
our results @xcite showed that the agreement between the @xmath0 data and the predictions of a classical model of maxwell and london equations are very good for small charge distances , but there is absolutely no agreement at large distances ( @xmath1 lattice spacings ) .
a possible interpretation within the dual superconductor picture is that the confining @xmath0 behaves as an effective type - i superconductor , rather than an extreme type - ii as an exact validity of the dual london equation would suggest .
in fact we got support for this conjecture also from a completely independent investigation : simulating doubly charged flux tubes .
we found @xcite that in four - dimensional @xmath0 there is an attractive interaction between flux tubes for the coupling @xmath2 approaching the phase transition .
thus one might look at @xmath0 lattice gauge theory from two points of view : from a `` microscopical '' it can be regarded as a double limit of a dual higgs model with both vanishing london penetration length and vanishing coherence length . from a `` macroscopical '' point of view it looks like a classical dual type - i superconductor .
this peculiarity is related to the fluctuations in the dual theory : the observed attraction between flux tubes was shown to be a purely `` quantum - mechanical '' effect of the dually transformed @xmath0 theory @xcite .
the action of the full abelian higgs model including a dual dirac string ( denoted with @xmath3 ) can be written in the form @xmath4 where @xmath5 is the dual gauge field and @xmath6 the dual higgs field , located on the sites of the dual lattice .
@xmath7 is the symmetry - breaking higgs potential with its minimum at @xmath8 . for the squared covariant derivative of the higgs
current the compact formulation has been chosen , because this allows for fluctuations of the fluxoid string @xcite . in the classical model @xmath9
is the squared london penetration depth , and @xmath10 the squared coherence length .
the double limit @xmath11 , @xmath12 exactly corresponds to the dually transformed @xmath0 theory discussed in the last section .
if we compare this model to the parameters of dual qcd @xcite after an abelian ansatz , we find the correspondence @xmath13 where @xmath14 is the ( chromo-)magnetic coupling , @xmath15 the location of the minimum and @xmath16 the strength of the corresponding continuum higgs potential , and @xmath17 is the lattice spacing .
these parameters can be chosen to reproduce very well the infrared behaviour of qcd .
the classical flux tube solutions of the higgs model on sufficiently fine lattices give very good agreement with corresponding continuum calculations @xcite .
it is now an interesting question how quantum and string fluctuations ( of the fluxoid string ) do effect the physical quantities , e.  g.  the string tension .
for the set of parameters which yields the desired classical solution , the system is not in the higgs phase any more on a reasonably fine lattice , if fluctuations are turned on .
for this reason we introduce a `` quantum scale '' @xmath18 , by substituting @xmath19 in the path integral by @xmath20 , which allows for a smooth interpolation between the classical limit ( @xmath21 ) and an ordinary simulation ( @xmath22 ) .
the changing of the string tension as a function of @xmath18 is depicted in fig .
[ sigma_n ] for two cases : considering only quantum fluctuations but keeping the physical fluxoid string fixed , and also considering string fluctuations , which leads to an increase of the energy per length ( until the system undergoes a phase transition to the coulomb phase ) .
( 0,0)(0,0 ) ( 40,160)(0,0)@xmath23    physical arguments suggest that the dual superconductor should be approximately at the borderline between type - i and type - ii .
this is fulfilled for the chosen classical solution .
investigating the influence of fluctuations on an effective landau - ginzburg parameter ( defined by the ratio of the individual contributions to the string tension ) showed us that the system is pushed towards type - i behaviour by fluctuations .
by dual simulations of flux tubes one can address a lot of interesting questions concerning the confinement mechanism .
here we focussed on the discussion whether the confinement vacuum is a dual type - i or type - ii superconductor , and on the meaning of fluctuations in dual theories .
similarly to the behaviour of @xmath0 in the vicinity of the phase transition , these fluctuations can produce an effective type - i superconductor also in the full abelian higgs model .    of course
, it is another interesting question , which set of ( bare ) parameters has to be used to describe the same physics as with the classical solution of the abelian higgs model .
for this task , however , it will be necessary to improve the control over string fluctuations in our simulations .
this work was supported by the fonds zur frderung der wissenschaftlichen forschung , project no .
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abelian gauge theories formulated on a space - time lattice can be used as a prototype for investigating the confinement mechanism . in @xmath0 lattice gauge theory 
 it is possible to perform a dual transformation of the path integral . simulating the obtained dual theory ( which corresponds to a certain limit of a dual higgs model ) including external sources 
 , we perform a very accurate analysis of flux tubes with respect to the dual superconductor picture . 
 dual flux tube simulations are also performed in the full abelian higgs model , in order to obtain non - perturbative control over quantum and string fluctuations , and for a comparison to the results of dual qcd .