We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a
AFFINELIEALGEBRAS,QUIVERVARIETIESANDSTATISTICALMECHANICS15HeisenbergalgebrabyBaranovsky[B]thatgeneralizestheGrojnowski/Nakajimaconstructiontohigherlevels.
Remark5.3.2.OnecanalsogiveageometricinterpretationofthefullFockspaceof[DJKMO2]withbasisindexedbyM[Λ]viathe“smooth”Ul-instantonmodulispace rM(r,l)whichhasthesamegeneratingfunctionforcohomology(seee.g.
[N4],Chapter5)asthefullFockspacewiththebasisM[Λ].ThetypesA(1)norA∞arere ectedintherespectiveactionofthegroupsZ/(n+1)ZorC onthemodulispace,andγ1,...,γlisthesetofone-dimensionalrepresentationsofthesegroupsthatdeterminethisaction.
6.AComparisonWithThePathSpaceRepresentation
Theauthorsof[DJKMO1]constructedthebasicrepresentationofAn(1)onthespaceofpaths.In[DJKMO2],thispathrealizationisgeneralizedtoarbitrarylevel.WenowcomparethegeometricpresentationL(v,w)withtheirs.Wewillslightlymodifythede nitionsof[DJKMO1]toagreewiththemoregeneralde nitionsof
[DJKMO2].
6.1.TheLevelOneCase.Abasicpathisasequencep=(λ0,λ1,...)ofintegersλi∈{0,1,...,n}.Thebasicpath
(
kfork∈Zsigni estheuniqueintegersuchthat
0≤k=kmodn+1.Let
Pb={p=(λ0,λ1,...)|λj=
i,
m(0),