AN ALGEBRAIC ANALOG OF THE VIRASORO GROUP JACK MORAVA Abstract.The group of diffeomorphisms of a circle is not an infinite-dim* *ensional algebraic group, though in many ways it behaves as if it were. Here we c* *on- struct an algebraic model for this object, and discuss some of its repre* *sen- tations, which appear in the Kontsevich-Witten theory of two-dimensional topological gravity through the homotopy theory of moduli spaces. [This * *is a version of a talk on 23 June 2001 at the Prague Conference on Quantum Groups and Integrable Systems.] 1. Some functors from commutative rings to groups 1.1 A formal diffeomorphism of the line, with coefficients in a commutative ring A, is an element g of the ring A[[x]] of formal power series with coefficients * *in A, such that g(0) = 0 and g0(0) is a unit. More precisely, the group of formal diffeomorphisms of the line, defined over A, is the set X G(A) = {g 2 A[[x]] | g(x) = gkxk+1 withg0 2 Ax } . k 0 Composition g0, g1 7! (g0O g1)(x) = g0(g1(x)) of formal power series makes this* * set into a monoid with e(x) = x as identity element, and it is an exercise in induc* *tion to show that such an invertible power series [ie with leading coefficient a uni* *t] possesses a composition inverse in G(A). Thus G defines a covariant functor from commutative rings to groups; in fact this functor is representable, in the sens* *e that G(A) is naturally isomorphic to the set of ring homomorphisms from the polynomi* *al algebra Z[gk | k 0][g-10] to A. Composition endows this representing algebra * *with the Hopf diagonal g(x) = (g 1)((1 g)(x)) , making G into a (pro-)algebraic group [9]. The kernel of the homomorphism ffl 7! 0 : G(A[ffl]=(ffl2)) ! G(A) can be given the structure of a Lie algebra, which is naturally isomorphic to t* *he Lie algebra over A spanned by the differentiation operators vk = xk+1_d_dx, k 0 . satisfying [vk, vl] = (l - k)vk+l. 1.2 There is a closely related functor ~Gfrom commutative rings to groups, which in some ways resembles the group of diffeomorphisms of the circle. This functor* * is ____________ Date: 15 June 2001. 1991 Mathematics Subject Classification. 81R10, 55S25. The author was supported in part by the NSF.. 1 2 JACK MORAVA quite representable - it is an ind-proalgebraic group - but it is close enough * *to being so to have some useful properties. In particular its Lie algebra, in the sense * *above, is spanned by operators vk with k 2 Z: thus k can be negative as well as positi* *ve. In the terminology of [5 x2.3] an element g of the Laurent series ring A((x)) := A[[x]][x-1]pis a nil-Laurent series if its coefficients gi for i < -1 arepnilpo* *tent. If A is the radical of A (ie the ideal of nilpotent elements) and Ared:= A= A th* *en the set of such nil-Laurent series is the inverse image of Ared[[x]] under the * *quotient homomorphism æ : A((x)) ! Ared((x)), and ~G(A) = {g 2 A((x)) | æ(g) 2 G(Ared)} P is the set of formal Laurent series g(x) = k -1 gkxk+1 2 A((x)) such that i) g0 is a unit, and ii) g-k is nilpotent, if k 1. This set is closed under composition of power series, and is in fact a group: W* *e can write g 2 ~Gas a sum g(x) = g+ (x) + g- (x-1) of an invertible formal power series g+ and a polynomial g- in x-1 with nilpote* *nt coefficients; this implies that the sum X g-1 = (-g- )kg-k-1+2 A((x)) k 0 is finite. If h = h+ + h- is another series of the same sort, we can thus make * *sense of the composition h- O g, so it suffices to show that h+ O (g+ + g- ) is well-* *defined; but g- , being a polynomial with nilpotent coefficients, is itself nilpotent, s* *o this composition can be written as a finite Maclaurin expansion X (Dkh+ )(g+ )gk-2 A((x)) , k 0 where kxn `n' Dkxn = 1_k!d__dxk= k xn-k ifn k , and is otherwise zero. To show the existence of (composition) inverses, we use * *the fact that h O g h O g+ mod I- ((x)) , where I- is the ideal generated by the coefficients of g- . [I would like to th* *ank M. Kapranov for suggesting this line of argument, which has substantially improved the result.] Because this ideal is generated by finitely many nilpotent element* *s, it is itself nilpotent, in the sense that (I- )n = 0 for n 0. It suffices to con* *struct an inverse for g under the assumption that g0 = 1, and that the rest of its coeffi* *cients lie in such a nil-ideal: for g+ has a composition inverse h+ , such that (g O h+ )(x) x mod I- ((x)) , and if u0 2 Ax is the coefficient of x in g O h+ then h(0)(x) = h+ (u-10x) is a* * formal series such that (g O h(0))(x) x mod I- ((x)). Under that hypothesis, then, l* *et h+(1)(x) = 2x - g(x) = x - ~g(x); then g(h+(1)(x)) = x - ~g(x) + ~g(x - ~g(x)) = x - ~g0(x)~g(x) + . .,. AN ALGEBRAIC ANALOG OF THE VIRASORO GROUP 3 the dots representing further Taylor's series-style corrections, so g O h+(1) x mod I2-((x)) . If u1 is the coefficient of x in g O h+(1), and we define h(1)(x) = h+(1)(u-11x* *), then it follows that g O h(1)is a series with all coefficients in I2-except that of x, * *which equals 1. Now we can iterate this process: induction defines a sequence h = h(0)O h(1)O . ... of compositions which will terminate in finitely many steps, defining the promi* *sed composition inverse for g. 1.3 If OE is a diffeomorphism of the circle with the property that OE(iz) = iOE(z) (where z 2 C with |z| = 1 and i is a primitive nth root of unity), then z 7! OE* *(z)n is an n-fold covering, which factors through a diffeomorphism of the circle sati* *sfying (zn) = OE(z)n . The group of such periodic diffeomorphisms thus defines an n-fold cover of Diff* * S1. The group-valued functor ~Ghas similar `covers': for simplicity let n = p be pr* *ime, e.g. two, and assume that A contains a nontrivial pth root i of unity: then G~1=p(A) = {g 2 ~G(A) | g(ix) = ig(x)} P is the subgroup of nil-Laurent series g(x) = k -1 gkpxkp+1 with g0 a unit, and when p is invertible in A (e.g. if A is a Q-algebra) the homomorphism g(x) 7! g(x1=p)p := g(p)(x) : ~G1=p! ~G induces an isomorphism of Lie algebras. This allows us to think of the group of invertible nil-Laurent series in A((x)) as a subgroup of the invertible nil-Lau* *rent series in A((x1=p)). 2. Some representations of these functors Certain standard representations of Diff S1 have analogs for ~G; because these * *are representations over the complexes, I will assume in this section that A is an * *algebra over a field of characteristic zero. 2.1 The A-bilinear form X g, h 7! := - k gk-1h-k-1 : A((x)) x A((x)) ! A k2Z is antisymmetric, and (aside from the subring of constants) is nondegenerate if* * A is a Q-algebra; it is an algebraic analog of the symplectic pairing g, h 7! resx=0gdh of [10 x1]. The set SpL (A) of `Laurent-symplectic' A-linear automorphisms of A((x)) which i) preserve the bilinear form <., .>, and ii) are continuous in the pro-discrete topology of A((x)) 4 JACK MORAVA defines a group-valued functor, analogous to the restricted symplectic group [1* *0 x5]. It is classical [12] that the residue of a differential over a local formal Lau* *rent series field is independent of the choice of uniformizer. This remains true over gene* *ral commutative rings A [13], which implies that the composition f 7! [h 7! h O f-1 ] : ~G(A) ! SpL(A) is a natural homomorphism between group-valued functors; thus the ~Ghas a natur* *al linear representation as automorphisms of the functor which sends A to the A- module A((x)). It is in any case elementary to see that the Lie algebra of ~G preserves the symplectic form: if x 7! x + fflxn+1 then xk 7! xk[1 + kfflxn] , dxl7! lxl-1[1 + (n + l)fflxn]dx so changes under such a transformation by l(n + k + l)ffl resx=0xn+k+l-1dx . The residue in this expression is zero unless n + k + l = 0, but in that case t* *he coefficient is zero; nothing in this argument requires that n be positive. 2.2 The residue pairing restricts to a bilinear form on the ring A[x, x-1] of L* *aurent polynomials, which has a canonical decomposition A[x, x-1] = A[x] + A[x-1] into Lagrangian subspaces. The symplectic form defines a Heisenberg algebra whi* *ch is essentially (when A is the field of real numbers) the identity component of * *the loop group of the circle. The Fock representation [10 x3, 11 x9.5] associated * *to this decomposition is an algebra of symmetric functions on the `positive-freque* *ncy' subspace A[x]. The restricted symplectic group acts as well on (a completion of* *) this representation, intertwining projectively with the action of the loops on the c* *ircle [10 x5, x7b; 11 x13.4]; this is simultaneously a (positive-energy) representati* *on of the Heisenberg algebra of the bilinear form, and (an extension of) the Lie algebra * *of ~G. The Segal-Sugawara construction expresses the action of the Virasoro generators as quadratic expressions in the Heisenberg group elements [14 x1.7]. 2.3 This Fock representation has an interpretation in terms of symmetric functi* *ons [4]; more generally, a certain class of twisted representations of the Virasoro* * algebra [2 x9.4], associated to Hall-Littlewood polynomials at roots of unity [7 III x8* *.12], fit naturally into this framework. For simplicity, let p be a fixed prime [e.g.* * p = 2] and let C((x)) := V0 C((x1=p)) := V be the extension of the field of formal complex Laurent series defined by adjoi* *ning a pth root of x. The Galois group of the field extension V=V0 is cyclic of orde* *r p, generated by the automorphism x1=p7! ix1=p, and the bilinear form satisfies = , AN ALGEBRAIC ANALOG OF THE VIRASORO GROUP 5 so the invariant subspace V0 is a bilinear submodule. More generally, V = a2Z=pZVa splits into orthogonal bilinear submodules Va spanned by series of the form X g = gsxs s -1 in which s = (k + a=p) with k and a nonnegative integers, 0 a p - 1. The r* *e- striction of the Fock representation of the group of nil-Laurent series over C(* *(x1=p)) to ~G1=pcan be interpreted (using the isomorphism of x1.3) as a representation * *of ~Gon the (completed) tensor product of the rings S(Va) of symmetric functions. These rings have Hopf algebra structures, which are usually described in terms * *of exponentials: the Witt functor assigns to a commutative ring A the multiplicati* *ve group W(A) = (1 + xA[[x]])x P of formal series w(x) = wkxk with constant coefficient w0 = 1. This is natura* *lly isomorphic to the set of ring homomorphisms from a polynomial algebra on gener- ators {wk, k > 0} to A; the group structure endows this representing ring with * *the structure of a (commutative and cocommutative) Hopf algebra. The involution w(T ) 7! w*(x) = w(-x) respects the product, and so defines a Z=2Z-action on W. The Hopf algebra of Schur Q-functions represents the kernel of the norm homomorphism w 7! w . w* : W ! W ; in other words it represents the functor which sends a ring to the group of pow* *er series q(x) with q(0) = 1 over that ring, which satisfy the relation q(x)q(-x) * *= 1. This ring is torsion-free, and we can reformulate the relation above in the uni* *versal example as the assertion that the formal logarithm logq(x) is an odd power seri* *es in T . More generally, the group of pth roots of unity acts on W by w(x) 7! w(i* *x), and Y w(x) 7! w(iapx) = N(w)(x) : W ! W a2Z=pZ is the Frobenius homomorphism of Witt theory. The Hopf algebra representing its kernel can be described as an algebra of Hall-Littlewood symmetric functions evaluated at a pth root of unity. For our purposes it can most conveniently be understood in terms of power series w(x) with w(0) = 1 such that the projection of logw(x1=p) to V0 is zero. The polynomial algebra underlying the Fock represe* *n- tation thus splits (over Q) as a product of Hopf algebras, indexed by a 2 Z=pZ;* * its ath component is the Fock representation of the Heisenberg group defined by Va. 2.4 The primitives in these Hopf algebras acquire natural normalizations from t* *he Heisenberg algebra: the fractional divided powers s fls = ___x___ (s + 1) 6 JACK MORAVA satisfy (n+m)=p-1dx = resx____________=(1-+ßn=p)-(m=p)1ffin+m,0 sin(nß=p) , 2 so the elements 1_ fl(a) k := | sin(aß=p)|-2 fl (k+a=p), k 2 Z+ define a normalized symplectic basis for Va when a is not congruent to zero mod p. When p = 2, this defines the Virasoro representation with c = 1 and h = 1=16 studied in the Kontsevich-Witten theory of two-dimensional topological gravity * *[1]; that theory has a conjectural generalization [3,6] in which the more general re* *pre- sentations defined above (with c = 1 and h = (p2-1)=48) play a similar role. Fr* *om a geometric point of view, these representations are mysterious: they are someh* *ow homotopy-theoretic, and do not arise in any natural way from the Lie algebra of vector fields on the circle; instead, they seem to be related to automorphisms * *of the cohomology of infinite-dimensional complex projective space, along the lines la* *id out in this paper, t hrough work of Madsen and Tillmann [8]. References 1. P. Di Francesco, C. Itzykson, J.-B. Zuber, Polynomial averages in the Kontse* *vich model, CMP 151 (1993) 193-219 2. I. Frenkel, J. Lepowsky, A. Meurman, Vertex operators and the Monster, Acade* *mic Press Pure and Applied Mathematics no. 134 (1988) 3. T. Jarvis, T. Kimura, A. Vaintrob, Moduli spaces of higher spin curves and i* *ntegrable hierar- chies, available at math.AG/9905034 4. Jing Naihuan, Vertex operators, symmetric functions, and the spin group n, * *J. Algebra 138 (1991) 340-398 5. M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space, avai* *lable at math.AG/0107143 6. E. Looijenga, Intersection theory on Deligne-Mumford compactifications (afte* *r Witten and Kontsevich), Seminaire Bourbaki 768, Asterisque 216 (1993) 7. I. McDonald, Symmetric functions and Hall polynomials, (2nd ed), OUP (1995) 8. I. Madsen, U. Tillmann, The stable mapping-class group and Q(CP1+), Aarhus p* *reprint 14 (1999); and further work, in progress 9. J. Morava, Noetherian localisations of categories of cobordism comodules. An* *n. of Math. (1985) 1-39. 10.G. Segal, Unitary representations of some infinite-dimensional groups, Comm.* * Math. Phys. 80 (1981) 301-342. 11._-, A. Pressley, Loop groups, OUP (1986) 12.J.P. Serre, Corps Locaux, Hermann (1968) 13.J. Tate, Residues of differentials on curves, Ann. Sci. Ecole Norm. Sup. 1 (* *1968) 149-159 14.E. Witten, On the Kontsevich model, and other models of two-dimensional grav* *ity, preprint, IASSNS-HEP-91/24 Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: jack@math.jhu.edu