REPRESENTING THE SUPER VIRASORO ALGEBRA BY MEROMORPHIC VECTORFIELDS ON THE GRADED RIEMANN SPHERE. Peter Bryant.

Transcription

1 REPRESENTING THE SUPER VIRASORO ALGEBRA BY MEROMORPHIC VECTORFIELDS ON THE GRADED RIEMANN SPHERE. Peter Bryant. Dept. Pure Maths fj Mathematical Statistics, 16 Mill Lane, Cambridge, U.K. CB21SB. Introduction. The Virasoro algebra V is the Z-graded algebra V = C <..., L_2, L_ 1, Lo, L 1, L2,... > with fundamental brackets [L m, L n ] = (m - n)l m + n. V arises naturally in string theories both because V is dense in 3(5 1 ), the algebra of smooth vectorfields on the circle, and because the Lie algebra of the conformal group in two dimensions consists of a pair of commuting copies of V. 3(5 1 ) is the Lie algebra of the group Dijj(5 1 ) of diffeomorphisrns of the circle. This group is associated with one of the fundamental invariances of the closed bosonic string theory, namely reparametrisation of the fundamental loop. The two commuting copies of V can be interpreted in terms of left and right-moving oscillations around the string loop. The string action is the harmonic map action IIdl)1I 2 (Hilbert-Schmidt norm) for smooth maps l) : 1: -+ RD so that classical (Euclidean) strings are minimal surfaces 1: worldsheet) isometrically embedded in RD. The quantum theory has as many symmetries as the classical theory only in D=26 so this is a necessary (and also sufficient) condition for a consistent quantum theory of closed bosonic strings. Since the Hilbert-Schmidt norm is invariant under conformal changes in the metric on the worldsheet, the energy of the minimal surface depends only upon the complex structure of 1:. 1: is a compact Riemann surface. In genus 0 (i.e 1: = CP1), the relationship between complex structure and the algebra V is well understood. In some remarkable recent work, Krichever & Novikov constructed analogues of the Virasoro algebra for Riemann surfaces of higher genus [4]. From the point of view of string theory it would be surprising if these new constructs were not just isomorphic to the Virasoro algebra as algebras. This is indeed the case [1], but the Krichever-Novikov algebras also contain some information (such as the genus) of a global character. For this reason BRST quantistion on higher genus Riemann surfaces using the Krichever-Novikov algebras has been considered [3]. In the current lecture I will review some recent work (in collaboration with Paolo Teofilatto & Marjorie Batchelor) on extending the Krichever-Novikov formalism to the graded setting. This should be relevent in superstring theories where one supersymmetrises the worldsheet. Ideally we would use super Riemann surfaces in place of ordinary Riemann

2 72 surfaces. For technical reasons, such as the lack of a Riemann-Roch theorem in the category of super Riemann surfaces, we can at the moment only work in the slightly more restrictive framework of graded Riemann surfaces [2],[7]. An alternative extension has been proposed independently by Bonora et al. [6]. The two pictures are in fact completely equivalent [7]. This research was funded by the Science & Engineering Research Council of Great Britain. 1. We consider the super Virasoro algebra SV, the Z-graded graded Lie algebra with even generators {L m ; m E Z} and odd generators {G r ; r E Z}. The brackets are [L m, L n ] (m - n) L m+n 1 [Lm,Gr] (2m - r)gm +r {Gr,G,} = 2L r+,. [We should really consider the centrally extended algebras; this is covered in {6], [7]]. Write R for the algebra of rational functions ( the meromorphic functions on CPI) and let Bdenote an odd generator (geometrically this is a local spinorfield). A representation of the super Virasoro algebra SV can be found amongst the graded derivations of the Z2-graded algebra AB as follows. By " :9" we mean the operator that acts on R@AB by taking Po + PI B to PI (po,pi rational). It is easily shown that :9 is an odd derivation of this graded algebra. The formulae displayed above have the following properties. (i) L m and G r are globally defined graded vectorfields on the graded Riemann sphere, ct» equipped with AE where E is the tautological bundle {2]. This bundle is distinguished by the fact that E E is the canonical bundle of the Riemann sphere hence E gives Cpl its unique spin structure. (ii) The graded conformal structure [7]is defined in terms of the odd derivation 0 = :9 +B:z which satisfies 0 2 = :z' 0 is interpreted as the supersymmetry generator (a commutator of supersymmetries is a translation). To define a graded (or super) Riemann surface one

3 73 has only to demand that 0 transforms homogeneously when one moves from one coordinate patch to the next. By direct computation one verifies that [O,L m ] [O,G r ] 1 (-2(m+ l)zm)o «r + Hence the given graded vectorfields are graded conformal in the sense that they preserve the graded line bundle generated by the graded conformal structure O. (iii) L m is graded holomorphic (i.e. has no poles anywhere on Cpl) only when m = -1,0, +1 and G; is graded holomorphic only when r =-t, +t. These five elements generate a graded subalgebra 8pl(2, C) C SV of dimension (3,2) which is the graded Lie algebra of the group of superconformal automorphisms of the super Riemann sphere. The even part of this algebra (generated by L_ 1, La, L+ 1 ) is of course isomorphic to 81(2, C). 2. The idea of Krichever & Novikov was to find vectorfields on a general compact Riemann surface X analoguous to the L m on ct» and then compute their brackets. One uses the Riemann-Roch theorem to give existence and uniqueness of appropriate vectorfields on X. In our case we must in addition treat spinorfields (= - t-differentials). Proposition 2.1. Suppose X is a compact Riemann surface ofgenus g 2: O. For t =1, t, and E a generic line bundle on X with Chern number C(E) = 2t(g - 1) + 9 we have DimHO(X;,,,)/ E) = 1. Selecting points P+, P: on X (to play the roles of the north & south poles on Cpl), consider a general compact graded Riemann surface structure AK,lj2 on X (i.e. a choice of spin structure). When genu8(x) 9 and go = 3/2.g, consider the divisors on X : (m = (m +1- go)p+ + (1 - go 1 1 1]r = (r + 2- g)p+ + (2-9 (iv) We can relate the generators (hence the whole) of SV to the algebra of supersmooth graded vectorfields on the 'supercircle' by simply restricting them to the equator on Cpl. The resulting injective map thus relates our representation of SV with the standard superstring. m)pr)p_. [the r' 8 are always half-integer, whilst for the m'8, we must take integers or half-integers to ensure that the coefficients of P:I: in (m lie in Z]. Now the line bundles (;", ifr associated

4 74 with the divisors (m, TJr respectively have the following Chern numbers: C( (m) = 3g 2, C(fJr) = 2g - 1. Hence 2.1 implies the existence of e'm E HO(X j IeX f'r E H O(Xjle;//2 iir). Furthermore we can always construct graded conformal vectorfields em, fr on (X, AIe1j2) out of these objects. This is done by sending an object written in local graded coordinates (T' = (Po + PI O):Z to (T = (Po + PIO) :z + DO - PI) :9' The construction of em, fr makes sense because" ()t/' is a local generator of the spinorfields guaranteed that we obtain a globally defined graded conformal vectorfield given the local formulae above and the transformation rules for vectorfields/spinors on X [7]. It is convenient to fix these graded conformal vectorfields by choice of a local graded coordinate system (z, 0) near P+ together with the following normalisation condition at P em = -zm- go(1+0(z))zoz +2'(m+1-go)(1+0(z))OoO) fr = and It is Given the local graded coordinates (z, 0) near P_ we can find (determined) constants b m, Cr such that em, fr near p: have the local forms near P_ : em = z-go-mbm(l + O(z))z:z + - go - m)(l + O(z))O:0) ir = zl/2- g-r cr(1+ O(z))( 0_ - se 0:_). oz Computing the graded Lie brackets of these generators gives the Z-graded vectorspace EB Cem EEl EB C]; a super Lie algebra structure of the following type. go [em, en] = E K;"n em+n-,,::=-go go [em,frl = E K;"rfm+r-,,::=-go g/2 {fr, f.} = E K:. er+.-, 1::=-g/2 [The only other known coefficients are those given as in = etc.] K!:n =(m - n) 1 1 Kgo = -m - r + -g mr 2 4 K g/ 2 =2 r.. and K;.g/2. These are obtained from This algebra is called the graded Krichever-Novikov algebra and written SV(X). Notice that the brackets do not preserve the Z-grading (as the ordinary Virasoro brackets do) - we have a go-graded super Lie algebra in the terminology of [4].

5 75 We can find a vectorspace decomposition where SV(X)+ and SV(X)_ denote the graded subalgebras generated by the em, Ir that are graded holomorphic at P+ (respectively P_) and SV(X)o is the complementary subspace spanned by the remaining generators. SV(X)o is not a subalgebra for 9 1. By contrast, for genus 0, we denoted by SVo the intersection of SV + (generated by indices m -1, r - t) and SV_ (generated by indices m +1, r +t). To be specific, SV(X)o is the graded vectorspace SV(X)o = C < e_go+2,., ego+2 > $C < l-g- 1/2, ' ",/g+1/2 >. Notice that Dim(SV(X)o = (3g - 3, 2g - 2». In fact the elements of SV(X)o represent infinitesimal superteichmuller deformations of the super Riemann surface corresponding to (X, AK,;{\ This gives an interesting description of the tangent space to super Teichmuller space [cf. [4]]. 3. The authors of [1] show that near P+ and P_ we can linearly transform the basis of even vectorfields so that it agrees locally with the standard Virasoro algebra. In this sense the generalised grading is an artifact dependent on choice of basis. From the construction of these even generators, however, we should expect the Krichever-Novikov algebras to possess some information of a global character. Indeed the genus is clearly encoded in the explicit bracket relations worked out above. These remarks extend to the graded setting and show that, whilst the quantisation of higher genus (graded) Riemann surfaces using Krichever-Novikov algebras is consistent with the flat-space quantisation, it nonetheless has a distinctive character of its own.

6 76 We finish with the following open question. Is it possible to construct K richever-nooiko» algebras for non-split super Riemann surfaces? One primary ingredient would appear to be a super Riemann-Roch theorem. To summarise the contents of this lecture, we have identified a reasonable analogue SV(X) of the Krichever-Novikov algebras over an arbitrary compact graded Riemann surface. It has been suggested that the subspace SV(X)o represents super Teichmuller deformations of the associated split super Riemann surface. A super "Krichever-Novikov algebra" should probably consist of superconformal (not merely superanalytic) vectorfields - the existence of such objects in the larger category of all super Riemann surfaces cannot be guaranteed at present. References. [1] J.Alberty, A.Taormina& P. van Baal Relating Kac-Moody, Virasoro and Kricheoer- Noviko» algebras [Comm.Math.Phys. 120 (1989) 249 ] [2] M.Batchelor & P.Bryant Graded Riemann surfaces [Comm.Math.Phys. 114 (1988) 243] [3] L.Bonora, M.Bregola, P. Cotta-Ramusino & M.Martellini Virasoro-type algebras and ERST operators on Riemann surfaces [Phys.Lett. 205(1)B (1988) 53] [4] I.Krichever & S.Novikov Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons [Funet.Anal. & Appl. 21(2) (1988) 46] [5J I.Krichever & S.Novikov Virasoro-type algebras, Riemann surfaces and strings in Minkowski space [Funet.Anal. & Appl. 21(4) (1987) 47J [6J P.Bryant Graded Riemann surfaces and Kricbeuer-Nooikov algebras, submitted to Lett.Math.Phys. [7J L.Bonora, M.Martellini, M.Rinaldi & J.Russo Neveu-Schwarz and Ramond-type superalqebras, Trieste preprint, April 1988.