Series Logo
Volume 00, 19xx
Schur Q-functions and a Kontsevich-Witten genus
Jack Morava
Abstract.The Virasoro operations in Witten's theory of two-dimensional
topological gravity have a homotopy-theoretic interpretation as endomor*
*phisms
of an ordinary cohomology theory with coefficients in a localization of*
* I.
Schur's ring of Q-functions. The resulting theory has many of the feat*
*ures
of a vertex operator algebra.
Introduction
Smooth complex curves of genus g > 1 form a stack Mg with a compactification
defined by adjoining the divisor of stable singular curves (with double points,*
* but
only finitely many automorphisms). Ideas from string theory have led to great
progress in understanding the topology of these moduli objects in the large gen*
*us
limit. This note is concerned with an algebra introduced by Schur in connection
with the classification of projective representations of symmetric groups, and *
*its
relevance to these spaces. This algebra appears in disguise in classical work o*
*n the
Riemann moduli space and in more recent work of Witten and Kontsevich_on the
intersection theory of its Deligne-Mumford compactification M g, but its signif*
*icance
has become clear only in retrospect, as integral cohomology emerges from the fog
of Q.
The central construction of topological gravity is a partition function which
can be defined geometrically by a family
__
og : M g! MU ^Q[v]
of maps to the complex cobordism spectrum tensored with the rationals, and the
main result of this paper is the construction of a morphism
kw : MU ! H([q-11])
of ring-spectra, which (when composed with o) sends the fundamental class of
the moduli space of stable curves to a highest-weight vector for a naturally de*
*fined
Virasoro action on Q. This Kontsevich-Witten genus can be constructed in purely
algebraic terms from the theory of formal groups, but its natural context is the
topology of moduli spaces. The first section below is a summary of some backgro*
*und
from algebraic geometry, emphasizing homotopy theory, while the Hopf algebra of
____________
1991 Mathematics Subject Classification. Primary 14H10, Secondary 55N35, 81*
*R10.
The author was supported in part by the NSF.
cO0000 American Mathematical Socie*
*ty
0000-0000/00 $1.00 + $.25 per pa*
*ge
1
2 JACK MORAVA
Q-functions is the topic of the second. This whole subject is in many ways sti*
*ll
quite mysterious, and a final section argues that some features are most natural
in an equivariant context. An appendix summarizes the construction of a vertex
operator algebra following ideas of A. Baker.
I would like to thank M. Ando, A. Baker, F. Cohen, C. Y. Dong, C. Faber,
T. Jozefiak, M. Karoubi, E. Looijenga, G. Mason, S. Morita, and U. Tillmann for
conversations and correspondence about the material in this paper.
x1 Background from geometry
1.1. Teichm"uller theory describes the Riemann moduli space Mg as the quo-
tient of a complex 3(g - 1)-dimensional cell by an action of the mapping class
group
g = ss0Diff+(g)
of isotopy classes of orientation-preserving diffeomorphisms of a closed surfac*
*e g;
this action is properly discontinuous and almost free, in the sense that its is*
*otropy
groups are finite, so the resulting quotient is an orbifold. The components of *
*the
group of diffeomorphisms are contractible, and the map
BDiff+(g) = Bg ! Mg
from the homotopy-theoretic quotient of the Teichm"uller action to the geometric
quotient is a rational homology isomorphism. *
* __
There is no such essentially topological description of the compactification*
* M g,
but there are very interesting (proper) forgetful maps
__n __
ng: M g! M g
defined on the moduli stack of stable curves marked with n ordered smooth point*
*s.
[A marked curve is stable, as above, if it has only finitely many automorphisms;
the loss of the marking may render an irreducible component of the curve unstab*
*le,
and the forgetful map is understood to contract such a component to a point.] A*
*c-
cording to the conventions of Quillen, such a proper complex-oriented map betwe*
*en
homology manifolds defines an element
__
[ng] 2 MU-2nQ(M g)
of the ring (tensored with Q) of complex cobordism classes of maps to the moduli
space. These cobordism classes can be identified in terms of ordinary cohomology
by the ring isomorphism
MU*Q(X) ! H*(X; Q[tk|k 1])
which sends [ : V ! X] to the characteristic number polynomial
X
*mI(T *)tI ;
I
where T *is the formal cotangent bundle along and * isQthe Gysin or transfer
map in cohomology; I = i1; : :i:s a multi-index, tI = tikk, and mI denotes the
characteristic classPassociated to the monomial symmetric function indexed by t*
*he
partition of |I| = kik defined by I. [The tk correspond to the complete symme*
*tric
functions hk but we keep the standard notation.] This differs from the usual [29
x6.2] definition, which is expressed in terms of the normal bundle rather than *
*the
cotangent bundle, but the two constructions are related by a straightforward ch*
*ange
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 3
of basis for the characteristic classes, and the choice above leads most direct*
*ly to
the formulas of Witten.
1.2. The action of a diffeomorphism of the surface g on its homology defines
an integral symplectic representation of g, and thus a map
BDiff+(g) ! BSp(2g; Z) :
The group Sp(2g; Z) acts with finite isotropy on the contractible symmetric spa*
*ce
Sp(2g; R)=U(g), with quotient the space Ag of g-dimensional principally polariz*
*ed
abelian varieties, so the homomorphism induced by this map on rational homology
agrees with that defined by the construction which assigns to a Riemann surface
its Jacobian. By stability theorems of Harer and Ivanov, these maps are suitably
compatible for increasing g, and we can think of them as taking values in the
classifying space BSp(Z) of the infinite integral symplectic group. The composi*
*tion
of the obvious maps
BSp(Z) ! BSp(R) = BU ! B(U=O) = Sp(H)=U
with a final Bott isomorphism is a rational homology isomorphism which (away
from two [20 x3.15]) splits a copy of Sp(H)=U off the group completion BSp(Z)+ ,
cf. [5,30].
1.3.1. One of the main contentions of this paper is that the theory of topo-
logical gravity defines maps to the complex_cobordism spectrum which are natural
analogues, for the compactifications M g, of the classical Abel-Jacobi map desc*
*ribed
above. The physicists' definitions are motivated by ideas from statistical mech*
*anics:
using the language of cobordism, let
X vn __
og = [ng]___2 MU0Q[v](M g) ;
g0 n!
with v a bookkeeping variable of (cohomological) degree two. The element og is
the class of the space of configurations defined by an indefinite number of dis*
*tinct
but unordered smooth points on a stable curve of genus g; it is tempting to thi*
*nk
of this as the space of states of the `Mumford gas' of free particles on a Riem*
*ann
surface, with existence as their only attribute. The homotopy class representi*
*ng
this tau-function induces a homomorphism
__
og*: H*(M g; Q) ! H*(MU ; Q[v]) = Q[tk|k 1][[v]] ;
__
the image of the fundamental class [M g] of the moduli space under this homomor-
phism is essentially Witten's free energy Fg. [It is convenient to interpret ho*
*mology
with Q[v] coefficients to be v-adically completed; for a finite complex this im*
*plies
no change at all.]
1.3.2. To be more precise about this, we will need the slightly more sensiti*
*ve
characteristic number homomorphism which assigns to [] 2 MU-2n(X) the class
X n-l(I)
cl[] = *mI(T *)t0 tI ;
P
where l(I) = k>0ik is the length (or number of parts) of the partition I. The
ring Q[tk|k 0][t-10] of cohomology coefficients has now been enlarged to inclu*
*de
an invertible polynomial generator of degree zero; this corresponds [cf. x1.5 b*
*elow]
4 JACK MORAVA
to a natural extension of the Landweber-Novikov algebra of cobordism operations.
The monomial symmetric function mI is a sum
X
xd1oe(1):x:d:noe(n);
P
where di is a finite sequence of nonnegative numbers with di = |I| in which
k appears ik times; it is convenient to think of this sequence as having exactly
n terms, with zero appearing i0 = n - l(I) times. The stable cotangent bundle
T *along ngis the sum of the cotangent line bundles Li of the modular curve at
its marked points, and the characteristic class mI(T *) is obtained by substitu*
*ting
Euler classes e(Li) for the formal variables xi. Witten's characteristic number*
* [31
x2.4] Y __
= ( e(Li)di)[M ng]
1in
is invariant under permutations of the marked points, so
__n n! I
mI(T *)[M g] = __I! ;
Q
where I! = k0 ik!. This rational number is the image of
__
ng*mI(T *) 2 H*(M g; Q)
__
under the Gysin homomorphism defined by the map_from_M gto a point; the latter
is just evaluation on the fundamental class of M g, and the free energy [31 x2.*
*16]
can thus be recognized as the sum over g and n of terms of the form
__ X I tI
cl([ng]=n!)[M g] = __I!:
The classical Jacobian is similarly a configuration space of divisors, and the *
*sum
X
jg = [SP nCg]vn 2 MU0Q[v](Mg)
n0
of symmetric powers of the modular family of curves is the pullback of the prod*
*uct
of the universal torus bundle in MU-2g(BSp(2g; Z)) by the coupling constant
X
vg [CP (n)]vn :
n0
It is somewhat surprising that no expression seems to be known for the class of*
* this
universal bundle.
1.4. The construction o can be extended to stable curves which are not neces-
sarily connected. From the point of view of homotopy theory the space of unorde*
*red
configurations of points in X is
a
Q(X) = En xn Xn
n0
but over the rationals this has the homotopy type of the infinite symmetric pro*
*duct
SP1 (X). We will therefore take
__ a __
Q(M ) = Q( M g)
g0
as a model for the space of stable curves,_connected_or_not;_its rational homol*
*ogy is
the symmetric tensor algebra on g0 H*(M g; Q), with M 0and M 1interpreted as
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 5
one-point spaces._It is natural to think of the fundamental_homology_class of t*
*he
orbifold SPnM g as the quotient in this tensor algebra of [M g]n by n!, so
__ X __ 3(g-1)
[Q(M )] = exp( [M g]v )
g0
__
defines a kind of fundamental class for Q(M ) with coefficients in Q[v]. Becaus*
*e no
marked curve is stable when 3(g -1)+n < 0, surfaces of small genus play_a_sligh*
*tly
anomalous role in these formulas; for example, it is useful to define [M 1] = -*
*_1_12.
The hard Lefschetz theorem (or the theory of mixed Hodge structures) defines
an action of sl2 on the rational homology of a projective orbifold,_with its fu*
*nda-
mental class as a highest weight vector,_so the homology of Q(M ) inherits an a*
*ction
of sl2[v; v-1] in which the class [Q(M )] has conformal weight zero [18 x2.6]. *
*The
construction of og extends multiplicatively to define an element
__
o 2 MU0Q[v](Q(M ))
which sends this fundamental class to the partition function
__ X
oW = o*[Q(M )] = exp( Fg) 2 Q[tk|k 1][[t0; v]] :
g0
These results from physics suggest that the (immensely complicated) mod-
uli space of curves has quite interesting homotopy-theoretic approximations, but
(unlike the somewhat similar situation in algebraic K-theory) we do not yet und*
*er-
stand these stabilizations in terms of universal mapping properties. That o tak*
*es
the fundamental class of the moduli space to a highest-weight vector for the na*
*tural
endomorphisms of Q is the central result of Kontsevich-Witten theory, but it is
not yet a characterization.
1.5. The parameters tk have an intrinsic interpretation as polynomial genera-
tors for the extended Landweber-Novikov Hopf algebra
S = Z[tk| k 0][t-10]
of cooperations in complex cobordism. The universal stable cohomology operation
is a ring homomorphism
MU*(X) ! MU* S
and the characteristic number homomorphism of x1.1 can be defined as the com-
position of this map with the Thom map from cobordism to ordinary cohomology.
The universal operation sends the Euler class e of a complex line bundle to
X
t(e) = tkek+1 ;
k0
it follows that the algebra S represents the group of formal origin-preserving *
*diffeo-
morphisms of the line, and that the characteristic number homomorphism induces
the classifying map for the universal formal group law
X +S Y = t(t-1(X) + t-1(Y ))
of additive type. The Lie algebra of formal vector fields on the line is close*
*ly
related to the Virasoro algebra, but the cobordism ring is more closely related*
* to
the untwisted charge one basic representation [21] than to the representation [*
*31
x2.59] defined by topological gravity.
6 JACK MORAVA
x2 Background from algebra
2.1. The simplest definition of the algebra of Q-functions is as the quotie*
*nt
of the polynomial algebra Z[qk|k 1] by the ideal generated by the coefficients*
* of
the relation
q(T 1_2)q(-T 1_2) = 1 ;
where X
q(T 1_2) = qkT 1_2k
k0
is a generating function with q0 = 1 [14 x7]. This algebra has a natural gradin*
*g,
which does not fit very comfortably1with the conventions of algebraic topology;*
* we
will assign the formal variable T _2cohomological degree one, but we will not a*
*ssume
that elements of odd degree anticommute. The diagonal homomorphism
X
qi7! qj qk
i=j+k
makes into a bicommutative Hopf algebra over Z, and the relation
X
(-1)iq2i= 2q2i+ 2 (-1)k-1qkq2i-k
i-1k1
implies that the square of a generator can be expressed as a sum of monomials in
which no qk appears with exponent greater than one. It follows that is a free
module over the integers, with a basis of square-free monomials; similarly, [1_*
*2] is
a polynomial algebra generated by the elements q2k+1, while the reduction of
modulo two is an exterior algebra. Being torsion-free, embeds in Q = Q,
and its defining relations imply that the power series
X 2xk 1_
logq(T 1_2) = ______T k+2
i0 2k + 1
contains only odd powers of T 1_2. Newton's identity
(2k + 1)q2k+1= 2(x0q2k+ x1q2k-2+ . .+.xk)
shows that the classes 2xk are integral, e.g. 2x0 = q1 and 2x1 = 3q3 - q1q2.
2.2. The generators qi of can be interpreted as specializations at t = -1 of
Hall-Littlewood symmetric functions qk(; t) of the eigenvalues of a positive-de*
*finite
self-adjoint matrix , defined by
X 1_ 1 - -1T 1_2t
q;t(T 1_2) = qk(; t)T 2k= det ___________1_;
1 - -1T 2
the primitive element xk thus becomes the power sum tr -2k-1. The canonical
symmetric bilinear Z[t]-valued form on the algebra of Hall-Littlewood functions
defines a positive-definite inner product on Q regarded as a quotient of that a*
*lgebra
[23 III x8.12]. The product and coproduct of are dual with respect to this bil*
*inear
form; in fact [1_2] is a positive self-adjoint Hopf algebra in the sense of Zel*
*evinsky
[23 I x5 ex 25]. The classical Q-functions are the orthogonalization of the bas*
*is of
square-free monomials with respect to this inner product.
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 7
2.3. The inner product on Q defines a skew bilinear form on the complexi-
fication of its space of primitives, which allows us to interpret Q as a standa*
*rd
representation of a Heisenberg algebra, or alternately [26, appendix I] of the *
*group
of antiperiodic loops on the circle. The latter group has two connected compone*
*nts,
and in some ways [13, appendix 8] it is natural to think of as Z x Z=2Z-graded
algebra, with qk in homological degree (k - 1; 1); butpin_other contexts the Z=*
*2Z-
grading appears as an action of the Galois group of Q( 2) over Q.
If, using the notation of [32 x1.7], we write
21_2ff-k-1_2= -xk
when k > 0 then the operators
X 1 1 X
L0 = ff-kffk + __ ; Ln = _ ffkffn-k ; n 6= 0
k>0 16 2
define a twisted [9 x9.4] charge one representation of the Virasoro algebra on *
*Q.
There appear to be deep general connections between self-dual Hopf algebras and
vertex operator algebras [1,13].
2.4. It is striking that these Q-functions have been important in algebraic
topology for more than a generation: in Cartan's 1960 Seminar [4 x17] the integ*
*ral
homology of Sp(H)=U = 5O appears as the quotient + of a polynomial algebra
on generators q+kof degree 2k, modulo the ideal generated by the coefficients o*
*f the
relation
q+ (T )q+ (-T ) = 1 ;
where X
q+ (T ) = qkT k;
k0
the algebras and + thus differ only by a doubling of the grading. The cohomol-
ogy of the corresponding finite-dimensional homogeneous spaces has been studied
more recently along similar lines [16].
In light of the results of Karoubi cited in x1.2, an algebra of Q-functions *
*forms a
substantial part of the stable integral homology of the space of Abelian variet*
*ies; the
primitives x+kare dual to Mumford's classes 2k+1. The stable rational cohomology
of the Riemann moduli space contains a polynomial algebra on the Mumford classes
k of degree 2k, and the dual homology Hopf algebra contains the polynomial
algebra [24] on dual primitives ^k. The map of x1.2 kills the even classes ^2k,*
* while
2k-17! (-1)kBk_2k2k-1;
where Bk is the kth Bernoulli number [29 x6.2]; the even Mumford classes are
detected by the constructions of [28]. Stabilization implies that
^k 2 H2k(Mg; Q)
for sufficiently large g, and (since these classes vanish on decomposables) it *
*follows
from the characteristic number formula of x1.1 that
og*(^k) = vtk+1 ;
the maps og are however not ring homomorphisms with respect to the usual multi-
plication [26] on complex cobordism.
8 JACK MORAVA
2.5. The importance of the theory of symmetric functions in the work of Kont-
sevich and Witten was discovered by Di Francesco, Itzykson, and Zuber [7 x3.2],
but it was Jozefiak [17] who saw the connection with Q-functions. In our formal*
*ism,
their map
tk 7! -(2k - 1)!! tr -2k-1 = -(2k - 1)!!xk ;
[where the `odd' factorial (2k + 1)!! is the product of the odd integers less t*
*han or
equal to 2k + 1, with (-1)!! = 1 by convention] defines a homomorphism from the
extended Landweber-Novikov algebra to [1_2]. The image of the partition function
oW under this map satisfies a large family of differential equations [22] whic*
*h can
be summarized impressionistically by the assertion that
"oW(xi) = oW (xi+ ffii;1)
is an sl2-invariant highest weight vector for the natural Virasoro action on (a*
* com-
pletion of) Q. The formal series oW is divergent at xi = ffii;1, but this clai*
*m can
be reformulated precisely as
"LnoW = 0 ; n -1 ;
where "Lnis the linear shift of Ln defined by the map which sends x1 to x1 - 1,*
* cf.
[19]. This defines a charge one vertex operator algebra embedded in a completion
of Q, cf. [8].
There is reason [22 x7, 23 III x7 ex 7] to expect that these results general*
*ize to
Hall-Littlewood functions at other roots of unity.
x3 The Kontsevich-Witten genus
3.1. Quillen's theorem establishes a bijection between one-dimensional formal
group laws over a commutative ring A, and homomorphisms from the complex
cobordism ring to A, so we can define a formal group law
X +Q Y = kw-1(kw (X) + kw(Y ))
over the localization [q-11], and hence a Q-function valued genus of complex ma*
*n-
ifolds, by specifying its exponential series to be
X
kw-1(T ) = (2k - 1)!!x-10xkT k+1:
k0
The homomorphism
kw : MU ! S ! [q-11]
classifying this group law factors through the map classifying the universal ad*
*ditive
law of x1.5 and therefore defines a group law of additive type.
3.2. This Kontsevich-Witten genus is defined by a nonstandard orientation on
an ordinary cohomology theory represented by the generalized Eilenberg-Mac Lane
space associated to [q-11]; its vertex algebra structure is the source of Witte*
*n's
Virasoro operations.
Some properties of this genus are unfamiliar to the point of pathology: not
only is its exponential series integral, for example, but modulo an odd prime i*
*t is
polynomial as well. The point of this section is that the orientation defining *
*this
genus can be interpreted as the formal completion of a coordinate, in the sense*
* of
[10 x1.8], on a T-equivariant cohomology theory taking values in sheaves of mod*
*ules
over a certain abelian group object in the category of ringed spaces. This glob*
*al
object is semi-classical, and some of the strangeness of the Kontsevich-Witten *
*genus
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 9
is a property not of the group object itself, but of a rather inconvenient coor*
*dinate
on it.
It seems simplest to present this construction in two parts: the first step *
*will
define a local version over a basic group object Q. The genus itself will then*
* be
defined by a family of group objects parametrized by a Grassmannian of positive-
definite self-adjoint matrices.
3.3.1. The basic idea comes from the classical theory of functions of one co*
*m-
plex variable: if g is an entire function, of exponential type in every right h*
*alf-plane,
then under certain circumstances there is an asymptotic relation
X
g(n)zn ~ 0 ;
n2Z
such equations may look more familiar written in the form
X X
g(n)zn ~ - g(-n)z-n :
n0 n1
If G(z) is the left-hand sum, G(z-1) will denote the sum on the right. When g is
rational [e.g. g = 1] such relations are familiar, but a less trivial example i*
*s defined
by
g(w) = (1 + ffw)-1
with 0 < ff < 2. The resulting (Mittag-Leffler) function
X zn
expff(z) = _________
n0 (1 + ffn)
thus has the asymptotic expansion
X z-n
expff(z) ~ - _________
n1 (1 - ffn)
for z outside a sector of angular width ffss centered on the positive real axis*
*, cf.
[12 x11.3.23]. This function is especially interesting when ff is a rational nu*
*mber
between zero and one, but we will be concerned only with exp1_2(z) which, up to
normalization, is the Laplace transform of Gaussian measure. Using the duplicat*
*ion
formula for the gamma function, its asymptotic expansion takes the form
X
exp1_2(z) ~ -ss-1_2z-1 (2n - 1)!!(-2z2)-n
n0
for z outside a sector of width 1_2ss centered on the positive real axis; in pa*
*rticular
the expansion is valid along the entire imaginary axis. The odd function
sin1_2(z) = -_i2[exp1_2(iz) - exp1_2(-iz)]
therefore satisfies
X
sin1_2(x) ~ ss-1_2x-1 (2n - 1)!!(2x2)-n
n0
as x approaches infinity in either direction along the real axis.
10 JACK MORAVA
3.3.2. Now consider the behavior on the real line of the entire function
ffl(z) = (_ss_2z)1_2sin1_2((z_2)1_2)
defined by the power series
ss_1_2X(-1_2z)n_= X __(-z)n__:
2 n0 (n + 3_2) n0 (2n + 1)!!
The argument above implies that
X
ffl(x) ~ ffl(x-1) = (2n - 1)!!x-n-1
n0
for x large and positive; it follows that ffl is monotone decreasing for positi*
*ve x.
On the other hand it is clear from its power series expansion that ffl is monot*
*one
decreasing on the negative real axis, so
ffl : (R; 0) ! (R+ ; 1)
is a bijection. The open interval
Q = (0; 1)+ - {1}
can now be made an abelian group by interpreting ffl to be the exponential of a
group law defined in a neighborhood of infinity on the projective line: if x an*
*d y
are large (i. e. nonzero) real numbers, then
x +1 y = _xy__x + y
will again be a large real number. It is easy to see that the resulting composi*
*tion
on P 1(R) - {0} is associative, with 1 as identity element; the inverse of x is*
* just
-x. We can therefore construct a formal group law +1fflover R by requiring that
ffl(x) +1fflffl(y) ~ ffl(x +1 y)
for x and y large and positive. This group law as the completion at the identit*
*y of
the analytic composition
X +fflY = ffl(ffl-1(X) +1 ffl-1(Y ))
with translation-invariant one-form dffl-1(T )-1; its exponential is the series*
* ffl. The
involution
[-1]ffl(T ) = ffl(-ffl-1(T ))
interchanges (0; 1) and (1; 1), identifying Q with the group completion of the *
*semi-
group defined on (0; 1) by +ffl. This group object is smooth, but not analytic;*
* its
exponential map has trivial domain of convergence.
3.3.3. The usual sheaf of smooth real-valued functions defines the structure*
* of
a ringed space on Q, but its law of addition is also compatible with the sheaf *
*of
real-analytic functions formally completed at the origin. The composition
x 7! (x) = exp(-ffl-1(x)-1) : Q ! Rx+
is a homomorphism to the real multiplicative group; pulling the sheaf over Gm (*
*R)
defined by T-equivariant K-theory back along this map yields a T-equivariant co-
homology theory taking values in the category of sheaves of modules over Q. Thi*
*s is
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 11
a kind of ordinary equivariant cohomology theory with coefficients in Q; its Ch*
*ern-
Dold character sends the standard one-dimensional representation of the circle *
*to
, regarded as a section of the structure sheaf. The Hirzebruch genus
__x___= ss-1_2__(2x)1_2__
ffl(x-1) sin1_2((2x)-1_2)
of the associated complex orientation is the reciprocal of a power series in x-*
*1 with
trivial constant term. It defines a function on P 1(R) analytic aside from a ju*
*mp
discontinuity at 0.
3.4. By replacing the function ffl with
ffl (x) = (_ss_2x)1_2tr sin1_2((x_2)1_2)
we obtain a T-equivariant theory taking values in sheaves over a family of abel*
*ian
group objects Q parametrized by equivalence classes of self-adjoint positive-d*
*efinite
matrices . It is remarkable that if is an np x np matrix and t is a pth root of
unity, then it follows from the definition of x2.2 that
q;t(T 1_2) = q-1;t-1(T -1_2) ;
in particular,1if is an endomorphism of an even-dimensional vector space, and
q(T _2) denotes the generating function for the Hall-Littlewood functions at -1
associated to the matrix -1, then
q(T -1_2) = q(T 1_2) :
It seems reasonable to expect that this T-equivariant theory is a real version *
*of a
theory over the quotient of some completion of by the coefficients of the
relation 1 1
q(T -_2)q(-T _2) = 1 :
Appendix: vertex operator algebras and self-dual Hopf algebras
This appendix has been added in July 1998 to the published version of this
paper, which has appeared in `Homotopy theory via algebraic geometry and repre-
sentation theory', ed. S. Priddy and M. Mahowald, in Contemporary Mathematics.
It is a kind of commentary on Andy Baker's construction of vertex operator alge*
*bras
associated to even unimodular lattices using ideas from the theory of (bicommut*
*a-
tive) Hopf algebras. I suspect that these methods will have further use, and I *
*have
tried here to summarize Baker's results in the language of group-valued functor*
*s,
and to include some references to related work [e. g. on -rings [3]] which might
otherwise be overlooked.
A.1. The functor which assigns to a commutative ring A, the abelian group
(under multiplication) of formal series
X
h(T ) = 1 + hiT i2 (1 + T A[[T ]])x := W0(A) ;
i1
is represented by the polynomial Hopf algebra S := Z[hi|i 1] with comultiplica-
tion X
(hi) = hj hk :
i=j+k
12 JACK MORAVA
W0(A) is essentially the classical Witt ring of A [6 V x2, 2], and its represen*
*t-
ing algebra can be identified with the usual ring S of symmetric functions. More
precisely, there is a natural isomorphism of the set W0(A) with the set of ring
homomorphisms from S to A, such that the map induced by agrees with mul-
tiplication of power series. The functor W(A) which assigns to A the set of all
invertible power series over A is represented by the tensor product S[h0; h-10]*
* of the
usual ring of symmetric functions with a Hopf algebra representing the multipli*
*ca-
tive groupscheme. It will be useful to know that W0(A) has a natural commutative
ring-structure * characterized by the identity
(1 + aT ) * (1 + bT ) = (1 + abT ) :
If L is a free abelian group of finite rank l [e. g. a lattice, with dual L**
* =
Hom ab(L; Z)], then it is easy to see [for example by choosing a basis] that the
functor
A 7! Hom ab(L; W0(A)) = L* Z W0(A)
is also represented by a Hopf algebra,*which can be taken to be an l-fold tensor
product of copies of S. I will write L S for this representing object, which Ba*
*ker
calls S(L). The point of the definition at the beginning of [1 x3] is to prese*
*nt a
coordinate-free definition of this object: a homomorphism from L to W0(A) defin*
*es
a family 7! hi() of maps from L to A, such that
X
hi( + 0) = hj()hk(0) ;
i=j+k
the ring L*S thus represents the tensor product functor in a natural way, witho*
*ut
specifying a basis. The generating function
X
h (T ) = hi()T i
i0
is a convenient substitute for such a choice.
It is tempting to think of L* as a constant groupscheme, and to interpret
L* Z W0 as a tensor product in a category of group-valued functors. A natural
internal product on a suitable category of commutative and cocommutative Hopf
algebras has*been studied by Goerss [11] and by Hunton and Turner [14].
In fact L S is only a part of the vertex operator algebra associated to a la*
*ttice;*
the full construction is usually described as the graded tensor product Z[L](L *
*S)
of the symmetric functions with the group ring of L. In this context it is natu*
*ral
to think of this group ring as graded, with the element in degree <; >, and to
give S its usual grading. In view of the discussion above we can define VOA (L)*
* to
be the Hopf algebra representing the functor L* Z W.
A.2. If H is a commutative and cocommutative Hopf algebra, projective of
finite rank over a base ring k, then the module
H* = Hom k(H; k)
is again a bicommutative Hopf algebra. If
A 7! Hom k-alg(H; A) := H(A)
is the group-valued functor H represents, then its dual Hopf algebra H* represe*
*nts
the Cartier dual functor
A 7! Hom gp-valued functors(H; Gm )(A) ;
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 13
cf. [6 II x1 no. 2.10]. A self-duality on such a (commutative and cocommutative,
projective and finite) Hopf algebra can thus be defined either as an isomorphism
H ! H*
of Hopf algebras, or as an isomorphism
H ! H*
of abelian group-valued functors; alternately, such a structure can be defined *
*either
as a nondegenerate pairing
H H ! k
of algebras [with suitable properties [23 I x5 ex. 25, 13]] or as a nondegenera*
*te
pairing
H x H ! Gm
of abelian group-valued functors.
Similar dualities exist more generally, in particular in the context of loca*
*lly
finite graded Hopf algebras. I will write S* for the ring of symmetric functio*
*ns
given its usual grading, and S* for its graded dual; the Hall inner product dis*
*cussed
by MacDonald then defines an isomorphism
S* ! S*
of graded algebras, with the convention that H* is the graded algebra H-*, i. e.
with its grading negated. A self-duality on a locally finite graded Hopf algebr*
*a H
can thus be defined by a nondegenerate graded pairing
H H ! k
with suitable properties. If we think of the grading on H as an action of the
multiplicative group on H, then H will have the inverse action.
In terms of group-valued functors, the Hall duality defines a homomorphism
W0 x W0 ! Gm
which is familiar classically as part of the theory of the Artin-Hasse exponent*
*ial [6
V x4 no. 4.3]. It seems to be simpler to work not with graded rings but with the
formal completion of W0 at the origin, i. e. to consider the subfunctor ^W0defi*
*ned
on complete local rings A by series h(T ) with coefficients hiin the maximal id*
*eal of
A such that hi! 0 as i ! 1. Since W0 is a commutative ring-valued functor, the
duality can be constructed in terms familiar from the theory of Frobenius algeb*
*ras.
It will suffice to define a suitable `trace' morphism from ^W0to Gm ; the pairi*
*ng will
then be the result of this trace applied to the Witt ring product. But if h 2 ^*
*W0(A)
then h(1) 2 Ax , and
h; g 7! (h * g)(1)
will do the trick : : :
A.3. The principal observation in this appendix is that some of Baker's form*
*u-
lae can be simplified considerably by use of the self-dual nature of the underl*
*ying
Hopf algebra. His formula 2.4 can be summarized as follows: let
Y(h(w)) = h(z + w) h(-(z + w)-1) 2 (S S)[[z; z-1]][[w]]
14 JACK MORAVA
P
be defined by expanding the coefficient in the right-hand term as i0(-w)iz-i-*
*1;
here h(T ) 2 S[[T ]] is the analogue of the generating function h(T ). This for*
*mula
extends multiplicatively to define a homomorphism
Y : S ! (S S)[[z; z-1]]
of (graded) ringoids, the point being that multiplication is not always defined*
* in
the object on the right. The assertion, more precisely, is that for any b and b*
*0 in
S, the product Y(b)Y(b0) is in fact defined, and equal to Y(bb0). This is not q*
*uite
Baker's definition of the vertex operator, but the two agree under the identifi*
*cation
of S S with End(S) given by Hall duality: the commutative product in the former
algebra is a resource not available in the latter, and it is just what is neede*
*d to
make sense of the assertion that Baker's formula is multiplicative.
The definition of Cartier duality in terms of an internal Hom functor on the
category of commutative groupschemes makes it natural to identify L Z W0 with
Hom (L* Z W0; Gm ); the isomorphism
L* ! L
defined by the pairing*on a self-dual lattice thus extends to a (grade-negating)
isomorphism between L S and LS . The analogue of the generating function
h (T ) is a generating function h (T ) 2 (LS )[[T ]], and the formula above for*
* Y
extends immediately to this more general context [1 x3.3]. To define Y on the w*
*hole
of VOA (L), it remains to construct Y on elements of the group ring of L : Bake*
*r's
formula is
Y() = h (z) h (-z-1)z ;
where z 2 End(Z[L])[z; z-1] is the operation 7! z<;> : : :
Much of the preceding construction appears to generalize in a relatively str*
*aight-
forward way from the classical algebra of symmetric functions to the Hopf Z[t]-
algebra HL of Hall-Littlewood symmetric functions [23 III x5 ex. 8]: this can be
defined as the polynomial algebra on generators qi related to the power sum sym-
metric functions pn by the formula
X X T n
q(T ) = qiT i= exp( (1 - tn)pn ___)
i0 n1 n
[23 III x2.10]. As t ! 0; q(T ) ! h(T ), which suggests that Baker's formula f*
*or
Y might have an interesting analogue in HL. Taking formal logarithms, we can
rewrite that formula in terms of `reduced' power sums "pn= pn=n as
X 1 - t|m|m
Y("pn) = _______n p"mzm-n ;
m2Z-{0} 1 - t n
where n is positive and p-m = (-1)m+1 pm for m negative.
The theory of Q-functions is one of the original motivations for this accoun*
*t:
these are elements of the self-dual Hopf algebra which represents the functor
A 7! {h 2 W0(A) | h(-T ) = h(T )-1} :
An analogous construction exists for any prime p: if ! is a primitive pth root *
*of
unity, the subfunctor
Y
{h 2 W0 | h(!kT ) = 1}
0kp-1
SCHUR Q-FUNCTIONS AND A KONTSEVICH-WITTEN GENUS 15
is also represented by a self-dual Hopf algebra. Expressed in terms of power su*
*ms,
the relation defining this quotient becomes pn = 0 when p divides n. It is also
possible to think of these algebras as quotients of HL defined by specializing t
to a root of unity [23 III x7 ex. 7]; perhaps requiring that its order be prime*
* is
superfluous. Considerable effort [13,15] has been devoted to constructing aspec*
*ts
of a vertex algebra structure on HL and on the rings of Q-functions, and it may*
* be
worth noting that when t is a primitive pth root of unity, the formula above fo*
*r Y
continues to make sense on the quotient of HL defined by sending pn to zero when
p divides n; but I am not sure if this fits with the thinking in [9 x9.2.51], a*
*nd at
the moment I don't know how to fit a lattice into the picture.
As Baker notes, some constructions of VOA theory are strikingly close to con-
structions familiar in algebraic topology. As a closing footnote I'd like to su*
*ggest
the possible relevance of the classifying space BUT for T-equivariant K-theory *
*in
this context: the vertex operator Y can be interpreted as a homotopy-class of m*
*aps
BUT x BUT ! BUT[[z; z-1]] ;
or in terms of representation theory as a construction which relates [or `fuses*
*'] two
representations of T specified at the points 0; 1 on the projective line, to de*
*fine a
representation at 1. This seems very close to the point of view of Huang-Lepows*
*ky
[and Segal] : : :
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IASSNS-HEP-91/24
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212*
*18
E-mail address: jack@math.jhu.edu