Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

3

tticecoupling.

3.Chiralsymmetrybreakingin nitevolume

Sincephysicsdoesnotdependontheboxsizein0c-phase,oneshouldshowthatchiralsymmetryisspontaneouslybrokenin nitevolumeinordertoproperlyreproducephysicsinthisphase7.Theorderoflimitsareimpor-tantandonehastotakethelargeNlimitbeforetakingthequarkmasstozeroat nitephsyicalvolume.ThelowlyingspectrumofthemasslessDiracoperatorshowsevidenceforspontaneouschiralbreakingsincetheeigenval-ues,iλ,scaleslikez=λΣNl4withzobeyingauniversaldistribution.Thechiralcondesate,Σ,isindependentofl.ResultsofacalculationofthechiralcondensateonthelatticeisshowninFig.2.Resultsfromchiralrandommatrixtheory8wereusedtoextractthechiralcondensateata xedN,Landlatticecoupling.Twolowestnon-zeroeigenvaluesintheQ=0andQ=1topologicalsectorswereusedtoshowconsistency.TheplotshowsthatthereisalimitasN→∞andthislimitisindependentofL.Results

obtainedatdi erentlatticecouplingsyieldl3

c