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].
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IASSNS-HEP-91/24
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: jack@math.jhu.edu
*