From the Virasoro Algebra to Krichever--Novikov Type Algebras and Beyond

0.25 MB·English

by  Martin Schlichenmaier

FROM THE VIRASORO ALGEBRA TO
KRICHEVER–NOVIKOV TYPE ALGEBRAS
3 AND BEYOND
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2 MARTINSCHLICHENMAIER

n
a ABSTRACT. StartingfromtheVirasoroalgebraanditsrelativesthegeneraliza-
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tion to higher genus compact Riemann surfaces was initiated by Krichever and

1 Novikov. The elements of these algebras are meromorphic objects which are
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holomorphic outside a finiteset of points. A crucial and non-trivial point isto

establish an almost-grading replacing the honest grading in the Virasoro case.
]
G Such an almost-grading isgiven by splitting the set of points of possible poles
intotwonon-emptydisjointsubsets. KricheverandNovikovconsideredthetwo-
A
point case. Schlichenmaier studied the most general multi-point situation with
.
h arbitrarysplittings. HerewewillreviewthepathofdevelopmentsfromtheVira-
t soroalgebratoitshighergenusandmulti-pointanalogs.Thestartingpointwillbe
a
aPoissonalgebrastructureonthespaceofmeromorphicformsofallweights.As
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sub-structuresthevectorfieldalgebras,functionalgebras,Liesuperalgebrasand
[ therelatedcurrentalgebrasshowup.Allthesealgebraswillbealmost-graded.In

detailalmost-gradedcentralextensionsareclassified. Inparticular,forthevector
1
v fieldalgebraitisessentiallyunique. Thedefiningcocyclearegiveningeometric
5 terms.Someapplications,includingthesemi-infinitewedgeformrepresentations
2 arerecalled. Finally,someremarksonthebyKricheverandSheinmanrecently
7 introducedLaxoperatoralgebrasaremade.
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.
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1 1. INTRODUCTION
:
v Lie groups and Lie algebras are related to symmetries of systems. By the use
Xi of the symmetry the system can be better understood, maybe it is even possible
to solve it in a certain sense. Here we deal with systems which have an infinite
r
a numberofindependent degreesoffreedom. TheyappearforexampleinConformal
Field Theory (CFT),see e.g. [2], [67]. Butalso in the theory of partial differential
equationsandatmanyotherplacesin-andoutsideofmathematicstheyplayanim-
portant role. The appearing Lie groups and Lie algebras are infinite dimensional.
Someofthesimplest nontrivial infinitedimensional Liealgebras aretheWittalge-
bra and its central extension the Virasoro algebra. We will recall their definitions
in Section 2. In the sense explained (in particular in CFT)they are related to what
Partial support by the Internal Research Project GEOMQ11, University of Luxembourg, is
acknowledged.
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2 MARTINSCHLICHENMAIER
is called the genus zero situation. For CFT on arbitrary genus Riemann surfaces
the Krichever-Novikov (KN) type algebras, to be discussed here, will show up as
algebrasofglobalsymmetryoperators.
These algebras are defined via meromorphic objects on compact Riemann sur-
faces S of arbitrary genus with controlled polar behaviour. More precisely, poles
areonly allowed atafixed finiteset ofpoints denoted by A. The“classical” exam-
ples are the algebras defined by objects on the Riemann sphere (genus zero) with
possiblepolesonlyat{0,¥ }. Thisyieldse.g. theWittalgebra, theclassical current
algebras, including theircentralextensions theVirasoro,andtheaffineKac-Moody
algebras [21]. For higher genus, but still only for two points where poles are al-
lowed,theyweregeneralised byKricheverandNovikov[26],[27],[28]in1987. In
1990 the author [37], [38], [39], [40] extended the approach further to the general
multi-point case.
This extension was not a straight-forward generalization. The crucial point is
to introduce a replacement of the graded algebra structure present in the “classi-
cal” case. Krichever and Novikov found that an almost-grading, see Definition 4.1
below, will be enough to do the usual constructions in representation theory, like
triangular decompositions, highest weight modules, Verma modules which are de-
mandedbytheapplications. In[39],[40]itwasrealizedthatasplittingofAintotwo
disjoint non-empty subsets A=I∪O is crucial for introducing an almost-grading
and the corresponding almost-grading was given. In the two-point situation there
is only one such splitting (up to inversion) hence there is only one almost-grading,
which in the classical case is a honest grading. Similar to the classical situation a
Krichever-Novikovalgebra,shouldalwaysbeconsideredasanalgebraofmeromor-
phicobjectswithanalmost-grading comingfromsuchafixedsplitting.
I like to point out that already in the genus zero case (i.e. the Riemann sphere
case) with more than two points where poles are allowed the algebras will only be
almost-graded. Infact, quite anumberofinteresting newphenomena willshowup
alreadythere,see[41],[15],[16],[8].
In this review no proofs are supplied. For them I have to refer to the original
articlesand/ortotheforthcomingbook[53]. Forsomeapplicationsjointlyobtained
withOleg Sheinman, see also [66]. Formore on the Witt and Virasoro algebra see
forexamplethebook[18].
After recalling the definition of the Witt and Virasoro algebra in Section 2 we
startwithdescribingthegeometricset-upofKrichever-Novikov(KN)typealgebras
inSection3. WeintroduceaPoissonalgebrastructureonthespaceofmeromorphic
forms(holomorphic outsideofthefixedsetAofpointswherepolesareallowed)of
all weights (integer and half-integer). Special substructures will yield the function
algebra, the vector field algebra and more generally the differential operator alge-
bra. Moreover, wediscuss alsotheLiesuperalgebras ofKNtypedefinedviaforms
KRICHEVER-NOVIKOVTYPEALGEBRAS 3
of weight -1/2. An important example role also is played by the current algebra
(arbitrary genus-multi-point) associated toafinite-dimensional Liealgebra.
In Section 4 we introduce the almost-grading induced by the splitting of A into
“incoming” and“outgoing” points, A=I∪O.
In Section 5 we discuss central extensions for our algebras. Central extensions
appear naturally in the context of quantization and regularization of actions. We
give for all our algebras geometrically defined central extensions. The defining
cocycle for the Virasoro algebra obviously does not make any sense in the higher
genusand/or multi-point case. Forthegeometric description weuseprojective and
affine connections. In contrast to the classical case there are a many inequivalent
cocycles and central extensions. If we restrict our attention to the cases where we
canextendthealmost-gradingtothecentralextensionstheauthorobtainedcomplete
classification anduniqueness results. Theyaredescribed inSection5.3.
In Section 6 we present further results. In particular, we discuss how from
the representation of the vector field algebra (or more general of the differential
operator algebra) on the forms of weight l one obtains semi-infinite wedge rep-
resentations (fermionic Fock space representations) of the centrally extended al-
gebras. These representations have ground states (vacua), creation and annihila-
tion operators. We add some words about b−c systems, Sugawara construction,
Wess-Zumino-Novikov-Witten (WZNW) models, Knizhnik-Zamolodchikov (KZ)
connections, anddeformations oftheVirasoroalgebra.
Recently, a new class of current type algebras the Lax operator algebras, were
introducedbyKricheverandSheinman[25],[29]. IwillreportontheminSection7.
In the closing Section 8 some historical remarks (also on related works) on
Krichever-Novikov type algebras and some references are given. More references
canbefoundin[53].
2. THE WITT AND VIRASORO ALGEBRA
2.1. The Witt Algebra. The Witt algebra W, also sometimes called Virasoro al-
gebra without central term1, is the complex Lie algebra generated as vector space
bytheelements {e |n∈Z}withLiestructure
n
[e ,e ]=(m−n)e , n,m∈Z. (1)
n m n+m
Oneofitsrealization isascomplexification oftheLiealgebraofpolynomialvector
fieldsVect (S1)onthecircleS1,whichisasubalgebraofVect(S1),dieLiealgebra
pol
¥
ofallC vectorfieldsonthecircle. Inthisrealization
d
e :=−iexpinj , n∈Z. (2)
n dj
1Inthebook[18]argumentsaregivenwhyitismoreappropriatejusttouseVirasoroalgebra,as
Wittintroduced“his”algebrainacharacteristic pcontext.Nevertheless,Idecidedtostickheretothe
mostcommonconvention.
4 MARTINSCHLICHENMAIER
TheLieproductistheusualLiebracketofvectorfields.
Ifweextendthesegenerators tothewholepunctured complex planeweobtain
d
e =zn+1 , n∈Z. (3)
n
dz
Thisgives another realization ofthe Witt algebra asthe algebra of those meromor-
phicvectorfieldsontheRiemannsphereP1(C)whichareholomorphicoutside {0}
and{¥ }.
Letzbethe(quasi)globalcoordinate z(quasi,becauseitisnotdefinedat¥ ). Let
w=1/zbethelocalcoordinateat¥ . AglobalmeromorphicvectorfieldvonP1(C)
willbegivenonthecorresponding subsetswhere zresp. waredefinedas
d d
v= v (z) , v (w) , v (w)=−v (z(w))w2. (4)
1 2 2 1
(cid:18) dz dw(cid:19)
Thefunction v willdetermine the vectorfield v. Hence, wewillusually justwrite
1
v andinfactidentifythevectorfieldvwithitslocalrepresentingfunctionv ,which
1 1
wewilldenotebythesameletter.
Forthebracketwecalculate
d d d
[v,u]= v u−u v . (5)
(cid:18) dz dz (cid:19)dz
The space of all meromorphic vector fields constitute a Lie algebra. The subspace
of those meromorphic vector fields which are holomorphic outside of {0,¥ } is a
Liesubalgebra. Itselementscanbegivenas
d
v(z)= f(z) (6)
dz
where f isameromorphicfunctiononP1(C),whichisholomorphicoutside{0,¥ }.
ThoseareexactlytheLaurentpolynomials C[z,z−1]. Consequently, thissubalgebra
has the set {e ,n∈Z} as basis elements. The Lie product is the same and it can
n
beidentifiedwiththeWittalgebra W .
Thesubalgebraofglobalholomorphicvectorfieldsishe ,e ,e i . Itisisomor-
−1 0 1 C
phictotheLiealgebra sl(2,C).
The algebra W is more than just a Lie algebra. It is a graded Lie algebra. If
we set for the degree deg(e ):=n then deg([e ,e ])=deg(e )+deg(e ) and we
n n m n m
obtainthedegreedecomposition
W = W , W =he i . (7)
n n n C
Mn∈Z
Notethat[e ,e ]=ne ,whichsaysthatthedegreedecompositionistheeigen-space
0 n n
decomposition withrespecttotheadjoint actionofe onW.
0
Algebraically W canalsobegivenasLiealgebraofderivationsofthealgebra of
Laurentpolynomials C[z,z−1].
KRICHEVER-NOVIKOVTYPEALGEBRAS 5
2.2. The Virasoro Algebra. In the process of quantizing or regularization one is
often forced to modify an action of a Lie algebra. A typical example is given by
the product of infinite sums of operators. Quite often they are only well-defined if
acertain“normalordering” isintroduced. Inthiswaythemodifiedactionwillonly
be a projective action. This can be made to an honest Lie action by passing to a
suitablecentralextension oftheLiealgebra.
Forthe Wittalgebra the universal one-dimensional central extension istheVira-
soro algebra V. As vector space it is the direct sum V =C⊕W. If we set for
x∈W, xˆ:=(0,x), and t :=(1,0) then its basis elements are eˆ , n∈Z and t with
n
theLieproduct
1
[eˆ ,eˆ ]=(m−n)eˆ − (n3−n)d −mt, [eˆ ,t]=[t,t]=0, (8)
n m n+m 12 n n
for2alln,m∈Z. Ifwesetdeg(eˆ ):=deg(e )=nanddeg(t):=0thenV becomes
n n
a graded algebra. The algebra W will only be a subspace, not a subalgebra of V.
Itwillbeaquotient. Insomeabuse ofnotation weidentify theelementxˆ∈V with
x ∈ W. Up to equivalence and rescaling the central element t, this is beside the
trivial(splitting) centralextensiontheonlycentralextension.
3. THE KRICHEVER-NOVIKOV TYPE ALGEBRAS
3.1. The Geometric Set-Up. For the whole article let S be a compact Riemann
surfacewithoutanyrestrictionforthegenus g=g(S ). Furthermore,letAbeafinite
subsetofS . Laterwewillneedasplitting ofAintotwonon-empty disjoint subsets
I and O, i.e. A=I∪O. Set N :=#A, K :=#I, M :=#O, with N =K+M. More
precisely, let
I=(P ,...,P ), and O=(Q ,...,Q ) (9)
1 K 1 M
be disjoint ordered tuples of distinct points (“marked points”, “punctures”) on the
Riemannsurface. Inparticular, weassume P 6=Q foreverypair (i,j). Thepoints
i j
inI arecalled the in-points, thepoints inOtheout-points. Sometimesweconsider
I andOsimplyassets.
Inthearticlewesometimesrefertotheclassicalsituation. Bythisweunderstand
S =P1(C)=S2, I={z=0}, O={z=¥ } (10)
ThefollowingFigures1,2,3exemplifythedifferent situations:
Our objects, algebras, structures, ... will be meromorphic objects defined on
S which are holomorphic outside of the points in A. To introduce the objects let
K =KS bethecanonicallinebundleofS ,resp. thelocallyfreecanonically sheaf.
The local sections of the bundle are the local holomorphic differentials. If P∈S
is a point and z a local holomorphic coordinate at P then a local holomorphic dif-
ferential can be written as f(z)dz with a local holomorphic function f defined in a
2Hered l istheKroneckerdeltawhichisequalto1ifk=l,otherwisezero.
k
6 MARTINSCHLICHENMAIER
FIGURE 1. Riemannsurfaceofgenuszerowithoneincomingand
oneoutgoing point.
FIGURE 2. Riemann surface ofgenus twowith one incoming and
oneoutgoing point.
P1 Q1
P2
FIGURE 3. Riemann surface of genus two with two incoming
pointsandoneoutgoing point.
neighbourhood of P. A global holomorphic section can be described locally with
respect to acovering by coordinate charts (U,z) by asystem of local holomor-
i i i∈J
phic functions (f) , which are related by the transformation rule induced by the
i i∈J
coordinate changemapz =z (z)andthecondition fdz = f dz yielding
j j i i i j j
dz −1
j
f = f · . (11)
j i
(cid:18)dz (cid:19)
i
Moreover, ameromorphic section of K is given as acollection of local meromor-
phicfunctions (h) forwhichthetransformation law(11)stillistrue.
i i∈J
KRICHEVER-NOVIKOVTYPEALGEBRAS 7
Inthefollowingl iseitheranintegerorahalf-integer. If l isanintegerthen
(1)K l =K ⊗l forl >0,
(2)K 0=O,thetriviallinebundle, and
(3)K l =(K ∗)⊗(−l ) forl <0.
Here as usual K ∗ denotes the dual line bundle to the canonical line bundle. The
dual line bundle is the holomorphic tangent line bundle, whose local sections are
the holomorphic tangent vector fields f(z)(d/dz). If l is a half-integer, then we
first have to fix a “square root” of the canonical line bundle, sometimes called a
theta-characteristics. Thismeanswefixalinebundle LforwhichL⊗2=K .
After such a choice of L is done we set K l =K l =L⊗2l . In most cases we
L
will drop the mentioning of L, but we have to keep the choice in mind. Also the
fine-structure ofthealgebras weareabouttodefinewilldepend onthechoice. But
themainproperties willremainthesame.
Remark3.1. ARiemannsurfaceofgenusghasexactly22g non-isomorphic square
roots of K . For g= 0 we have K = O(−2), and L =O(−1), the tautological
bundle, is the unique square root. Already for g = 1 we have 4 non-isomorphic
ones. As in this case K = O one solution is L = O. But we have also other
0
bundlesL,i=1,2,3. NotethatL hasanon-vanishingglobalholomorphicsection,
i 0
whereas this is not the case for L ,L , L . In general, depending on the parity of
1 2 3
dimH(S ,L), one distinguishes even and odd theta characteristics L. For g=1 the
bundleO isanodd,theothersareeventhetacharacteristics.
Weset
l l l
F :=F (A):={f isaglobalmeromorphic sectionofK |
suchthat f isholomorphic overS \A}. (12)
l
We will drop the set A in the notation. Obviously, F is an infinite dimensional
C-vectorspace. Recallthat inthecase ofhalf-integer l everything depends onthe
thetacharacteristic L.
The elements of the space Fl we call meromorphic forms of weight l (with
respect to the theta characteristic L). In local coordinates z we can write such a
i
l
formas fdz ,with f alocalholomorphic, resp. meromorphicform.
i i i
Specialimportantcasesoftheweightsarethefunctions(l =0),thespaceisalso
denotedbyA,thevectorfields(l =−1),denoted byL,thedifferentials (l =1),
andthequadratic differentials (l =2).
Nextweintroduce algebraic operations onthespaceofallweights
l
F := F . (13)
lM∈1Z
2
Theseoperations willallowustointroduce thealgebras weareheading for.
8 MARTINSCHLICHENMAIER
3.2. Associative Structure. The natural map of the locally free sheaves of rang
one
K l ×K n →K l ⊗K n ∼=K l +n , (s,t)7→s⊗t, (14)
definesabilinearmap
·:Fl ×Fn →Fl +n . (15)
Withrespecttolocaltrivialisationsthiscorrespondstothemultiplicationofthelocal
representing meromorphicfunctions
(sdzl ,tdzn )7→sdzl ·tdzn =s·t dzl +n . (16)
Ifthere isno danger ofconfusion then wewillmostly usethe samesymbol forthe
sectionandforthelocalrepresenting function.
Thefollowingisobvious
Proposition 3.2. The vector space F is an associative and commutative graded
(over 1Z)algebras. Moreover,A =F0 isasubalgebra.
2
Definition3.3. Theassociative algebra A istheKrichever-Novikov function alge-
bra(associated to(S ,A)).
Ofcourse,itisthealgebraofmeromorphicfunctionsonS whichareholomorphic
l
outside of A. The spaces F are modules over A. In the classical situation A =
C[z,z−1],thealgebraofLaurentpolynomials.
3.3. Lie Algebra Structure. Next we define a Lie algebra structure on the space
F. Thestructure isinducedbythemap
Fl ×Fn →Fl +n +1, (s,t)7→[s,t], (17)
whichisdefinedinlocalrepresentatives ofthesectionsby
dt ds
(sdzl ,tdzn )7→[sdzl ,tdzn ]:= (−l )s +n t dzl +n +1, (18)
(cid:18) dz dz(cid:19)
andbilinearly extendedtoF.
Proposition3.4. [43],[53]
(a)Thebilinear map[.,.]definesaLiealgebra structureonF.
(b)ThespaceF withrespectto·and[.,.]isaPoissonalgebra.
Nextweconsidercertainimportantsubstructures.
3.4. The Vector Field Algebra and the Lie Derivative. For l =n =−1in (17)
weendupinF−1 again. Hence,
Proposition 3.5. The subspace L =F−1 is a Lie subalgebra, and the Fl ’s are
LiemodulesoverL.
KRICHEVER-NOVIKOVTYPEALGEBRAS 9
As forms of weight −1 are vector fields, L could also be defined as the Lie
algebra of those meromorphic vector fields on the Riemann surface S which are
holomorphic outside of A. The product (18) gives the usual Lie bracket of vector
fieldsandtheLiederivative fortheiractions onforms. Duetoitsimportance letus
specialize this. Weobtain (naming thelocal functions withthe samesymbol asthe
section)
d d df de d
[e,f](z)=[e(z) ,f(z) ]= e(z) (z)− f(z) (z) , (19)
| dz dz (cid:18) dz dz (cid:19)dz
dg de
(cid:209) (g)(z)=L (g) =e.g = e(z) (z)+l g(z) (z) (dz)l . (20)
e | e | | (cid:18) dz dz (cid:19)
Definition3.6. ThealgebraL iscalledKrichever-Novikovtypevectorfieldalgebra
(associated to(S ,A).
IntheclassicalcasethisgivestheWittalgebra.
3.5. The Algebra of Differential Operators. In F, considered as Lie algebra,
A =F0 isanabelianLiesubalgebra andthevectorspacesum F0⊕F−1=A ⊕
L isalsoaLiesubalgebra ofF. Inanequivalentwayitcanalsobeconstructed as
semi-direct sumofA considered asabelian Liealgebra and L operating onA by
takingthederivative.
Definition 3.7. This Lie algebra is called the Lie algebra of differential operators
ofdegree≤1ofKNtype(associated to(S ,A))andisdenotedbyD1.
Inmoredirectterms D1=A ⊕L asvectorspacedirectsumandendowedwith
theLieproduct
[(g,e),(h,f)]=(e.h− f.g,[e,f]). (21)
ThespacesFl willbeLie-modulesoverD1.
Itsuniversalenvelopingalgebrawillbethealgebraofalldifferentialoperatorsof
arbitrarydegree[40],[42],[46].
3.6. TheSuperalgebraofHalfForms. Nextweconsidertheassociative product
·F−1/2×F−1/2→F−1=L. (22)
Weintroduce thevectorspaceandtheproduct
S :=L ⊕F−1/2, [(e,j ),(f,y )]:=([e,f]+j ·y ,e.j − f.y ). (23)
Usually we will denote the elements of L by e,f,..., and the elements of F−1/2
byj ,y ,....
Thedefinition(23)canbereformulatedasanextensionof[.,.]onL toa“super-
bracket”(denoted bythesamesymbol)onS bysetting
dj 1 de
[e,j ]:=−[j ,e]:=e.j =(e − j )(dz)−1/2 (24)
dz 2 dz
10 MARTINSCHLICHENMAIER
and
[j ,y ]:=j ·y . (25)
We call the elements of L elements of even parity, and the elements of F−1/2
elementsofoddparity. Forsuchelements xwedenotebyx¯∈{0¯,1¯}theirparity.
Thesum(23)canalsobedescribed as S =S0¯⊕S1¯, where Si¯isthesubspace
ofelementsofparity i¯.
Proposition 3.8. [52] The space S with the above introduced parity and product
isaLiesuperalgebra.
Definition3.9. ThealgebraS istheKrichever-NovikovtypeLiesuperalgebra(as-
sociatedto(S ,A)).
Classically this Lie superalgebra corresponds to the Neveu-Schwarz superalge-
bra. Seeinthiscontextalso[10],[3],[5].
3.7. Jordan Superalgebra. Leidwanger and Morier-Genoux introduced in [30] a
Jordansuperalgebra intheKrichever-Novikov setting, i.e.
J :=F0⊕F−1/2=J0¯⊕J1¯. (26)
Recall that A = F0 is the associative algebra of meromorphic functions. They
l
definethe(Jordan) product ◦isviathealgebrastructures forthespaces F by
f ◦g:= f ·g ∈F0,
f ◦j := f ·j ∈F−1/2 (27)
j ◦y :=[j ,y ] ∈F0.
Byrescaling theseconddefinitionwiththefactor1/2oneobtains aLieantialgebra.
See[30]formoredetailsandadditional results onrepresentations.
3.8. Current Algebras. Westart with ga complex finite-dimensional Liealgebra
andendowthetensorproduct g=g⊗CA withtheLiebracket
[x⊗ f,y⊗g]=[x,y]⊗ f ·g, x,y∈g, f,g∈A. (28)
The algebra g is the higher genus current algebra. It is an infinite dimensional
Lie algebra and might be considered as the Lie algebra of g-valued meromorphic
functions ontheRiemannsurface withpoles onlyoutside of A. Notethat weallow
alsothecaseofganabelian Liealgebra.
Definition3.10. ThealgebragiscalledcurrentalgebraofKricheverNovikovtype
(associated to(S ,A)).
Sometimesalsothenameloopalgebraisused.
In the classical case the current algebra g is the standard current algebra g =
g⊗C[z−1,z]withLiebracket
[x⊗zn,y⊗zm]=[x,y]⊗zn+m x,y∈g, n,m∈Z. (29)