Vetor Supersymmetry of Chern-Simons
Theory at Finite Temperature
D.G.G. Sasaki,
Centro BrasileirodePesquisas Fsias,
Rua XavierSigaud150,22290-180,Riode Janeiro, RJ,Brazil
S.P. Sorella,and V.E.R.Lemes
Departamentode FsiaTeoria
Institutode Fsia,UERJ
Rua S~aoFranisoXavier, 524
20550-013Maraan~a,RiodeJaneiro,Brazil
Reeived1Deember,1999
Theexisteneofthevetorsupersymmetryisanalysedwithintheontextofthenitetemperature
Chern-Simonstheory.
I Introdution
Sine many years the topologial three-dimensional
Chern-Simons [1, 2℄ theory is the soure of
ontinu-ousand renewedinterests,withmanyappliations
go-ingfrompureeldtheorytoondensedmatterphysis.
The Chern-Simonsgaugemodel hasbeenthe rst
ex-ampleofatopologialeldtheoryoftheShwarztype,
allowingfor theomputation ofseveral topologial
in-variants in knots theory [1℄. It is a remarkable fat
that these omputations an be performed within the
standardperturbationtheory[3℄. Moreover,the
Chern-Simonsprovidesanexampleofafullyultravioletnite
eldtheory,withvanishing-funtionandeld
anoma-lousdimensions[4℄. Thisfeaturereliesontheexistene
ofanadditionalglobalinvarianeoftheChern-Simons
ationwhihshowsuponlyaftertheintrodutionofthe
gauge xing and of the orresponding Faddeev-Popov
ghost term. Thisfurthersymmetryis knownasvetor
supersymmetry[5,2℄sineitsgeneratorsarryaLorentz
indexand,togetherwiththeBRSTsymmetry,giverise
toasupersymmetrialgebraoftheWess-Zuminotype.
Itworthmentioningthatthenonzerotemperature
ver-sionoftheChern-Simonsationisalsoavailable[6℄and
turnsouttoplayanimportantrole intheappliations
of three-dimensionalgauge theoriesto nite
tempera-tureeets. Therefore, it seemsnaturally to ask
our-selves if the vetor supersymmetry is still present in
theaseof anonzerotemperature. This is theaim of
the present letter. In partiular, we shall be able to
showthatthisquestionanbeansweredinthe
aÆrma-tive. In this sense, the fully quantized Chern-Simons
ationanbeonsidered asanexampleofa
superym-metri eld theory at nite temperature. The paper
isorganizedasfollows. In Set.2wepresentthe nite
temperature Chern-Simonsation and weanalyse the
existeneoftheaforementionedsupersymmetry. Set.3
willbedevotedto thestudyofsomeonsequenesand
totheonlusion.
II Finite temperature
Chern-Simons ation
InordertoanalysethepropertiesoftheChern-Simons
ation at nite temperature let us rst reall the
su-persymmetri struture of the zero temperature ase.
Adopting the Landau gauge, for the fully quantized
Chern-Simonsationwehave
S= Z
3 d
3
x
1
4 "
(A a
F
a
1
3 f
ab A
a
A
b
A
)+b
a
A a
a
(D
) a
Expression(II.1)isleftinvariantbythefollowing
nilpo-tentBRSTtransformations
sA a =(D ) a ; s a = 1 2 f ab b s a =b a ; sb a
=0:
(II:2)
Inaddition,theation(II.1)isknown[5, 2℄to possess
afurther rigid invariane whose generatorsÆ
arry a
vetorindex,i.e.
Æ =A ; Æ =0 Æ b= ; Æ A = " ; (II:3)
and,togetherwiththeBRSTtransformations,obeythe
followingrelations
s 2
=0; fÆ
;Æ
g=0 ;
fÆ
;sg=
+ eqs:of motion;
(II:4)
whih, losing on-shell onthe spae-time translations,
give rise to a supersymmetri algebra of the
Wess-Zuminotype.
Conerningnowthe nonzerotemperature ase,for
the quantized Chern-Simons ation in the imaginary
timeformalism[6℄,weobtain
S T = Z 0 d Z d 2 x 1 4 " (A a F a 1 3 f ab A a A b A )+b
A a (D ) a ; (II:5) d
where stands for the inverse of the temperature T.
As iswellknown, allelds =(A;;;b) arerequired
to obey periodi boundary onditions along the
om-patieddiretion [7℄,namely
(x;)= X 1 n= 1 n (x)e i!n ; n (x)= 1 Z 0
d(x;)e i!n
;
(II:6)
where the !
n
are the so-alledMatsubara frequenies
[7℄ ! n = 2n : (II:7)
Weemphasizehere that theghostelds ;,although
beingantiommuting variables,haveto be periodi in
. Asweshallseeinthefollowing,thispropertywillbe
ruialfortheexisteneofasupersymmetristruture
at nonzero temperature. In order to write down the
nitetemperatureChern-Simonsationintermsofthe
Matsubaramodes n
,weidentify the-diretion with
the x 3
variable and we introdue the followinguseful
two-dimensionalnotation A n =(A n ; n
); ;;=1;2;
" 3 =" ; " " =Æ : (II:8)
Thus,fortheationweobtain
mations(II.2)read sA an = an +f ab P l A bl (n l) ; s an = i! n an +f ab P l bl (n l) s a = 1 2 f ab P l bl (n l) ; s an =b an ; sb an
=0:
(II:11)
Moreover,itanbehekedthatthenonzero
tempera-tureation(II.9)isleftinvariantbythefurther
follow-ingrigidtransformationsÆ
;Æ,namely
Æ A an =i! n " an ; Æ an =" an ; Æ an =A an ; Æ b an = an ; Æ an
=0;
(II:12) and ÆA an = " an ; Æ an = an ; Æ an
=0;
Æb an = i! n an ; Æ an
=0:
(II:13)
ThegeneratorsÆ
;Ægiverise,togetherwiththeBRST
operators,tothefollowingalgebrairelations
fÆ;sg n = i! n n
+ eqs:of motion;
fÆ ;sg n = n
+ eqs:of motion;
fÆ
;Æ
g=0;
Æ 2
=0:
(II:14)
We see therefore that the supersymmetri struture
(II.4) of the zero temperature Chern-Simons persists
alsointheaseofanonvanishingtemperature. In
par-tiular, it is easily reognized that the operator Æ of
eqs.(II.13)orrespondstothegeneratorÆ
ofeqs.(II.3)
alongtheompatieddiretion =x 3
. Itisalsoworth
underlining herethat the existeneof a nonzero
tem-peraturesupersymmetrialgebrareliesontheperiodi
boundary onditions required for the Faddeev-Popov
ghosts;. Asiswellknown,thispropertyfollowsfrom
thegaugeinvarianeofthenonzerotemperatureation
S
T
. Moreover,thesupersymmetryturnsouttobe
ru-ialinordertoensurethatnophysialexitationsshow
upin thenonzero temperaturease, as it will be
dis-ussedinthenextsetion. Inotherwords,thenonzero
temperature Chern-Simonsation remains a
topologi-altheory,withnoloalphysialdegreesoffreedom.
III Conlusion
Ithasbeenalreadyunderlinedthatinthezero
temper-atureasetheexisteneofthevetorsupersymmetryis
deeplyrelated to thetopologial nature ofthe
Chern-Simonsterm. Wereallinfatthatthesupersymmetry
showsuponlyaftertheintrodutionoftheghostelds.
Asaonsequene,itfollowsthattheontributions
om-ingfromthepropagatingomponentsofthegaugeeld
areexatlyompensatedbythoseorrespondingtothe
ghosts,resultinginthewellknownultravioletniteness
ofthetheory. Thismeansthatthearenoloalphysial
degreesof freedom,i.e. that thetheoryis topologial.
Theexisteneofasupersymmetristrutureinthease
of nonzero temperature suggests a similar behaviour
forthenitetemperatureversionoftheChern-Simons.
This fat an be easily onrmed in the abelian ase
byshowingthat thepartitionfuntion turnsouttobe
independent from the temperature, implying the
van-ishingofallrelevantthermodynamiquantities. Letus
omputein fat the partition funtion for the abelian
Maxwell-Chern-Simonsation
Z =e F
= Z
DA DDbDe SMCS ; (III:15) with S MCS = Z 0 d Z d 2 x g 4 F F +i 1 2 " A A +b A : (III:16)
WehaveintroduedaonstantginordertotakeintoaounttheMaxwellterm. Ofourse,thepureChern-Simons
ontributionisreoveredinthelimitg!0. ForthefreeenergyF weobtainthefollowingresult
where L 2
stands for the two-dimensionalarea.
Obvi-ously,expression(III.17)doesnotdependfrominthe
limitg !0. Again,there isaomplete ompensation
between theghost and the gaugesetors, as expeted
fromtheexisteneofthesupersymmetry. Theanalysis
oftheultravioletnitenessofthenonabeliannite
tem-peratureaseaswellastheomputationofthevauum
expetation valueof Polyakovloops areunder
investi-gation.
Aknowledgements
The Conselho Naional de Desenvolvimento
Ciento e Tenologio CNPq-Brazil, the Funda~ao
de Amparo aa Pesquisa do Estado do Rio de Janeiro
(Faperj) and theSR2-UERJare aknowledged forthe
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