Bases of representations of type A affine Lie algebras via q(15)

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),

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