Series Logo
Volume 00, 19xx
Quantum generalized cohomology
Jack Morava
Abstract.We construct a ring structure on complex cobordism tensored wi*
*th
Q, which is related to the usual ring structure as quantum cohomology is
related to ordinary cohomology. The resulting object defines a general*
*ized
two-dimensional topological field theory taking values in a category of*
* spectra.
Introduction
The conclusion of this paper is that the theory of two-dimensional topologi-
cal gravity has a remarkably straightforward homotopy-theoretic interpretation *
*in
terms of a generalized cohomology theory, completely analogous to the more fami*
*l-
iar interpretation of quantum ordinary cohomology as a topological field theory.
Two-dimensional gravity originated in attempts to integrate over the space of m*
*et-
rics on a Riemann surface; it was reformulated by Witten in terms of an algebra
of generalized Miller-Morita-Mumford characteristic classes for surface bundles*
*. In
the interpretation proposed here, this algebra is the coefficient ring of a gen*
*eralized
cohomology theory, and topological gravity becomes a topological-field-theory-l*
*ike
functor, which assigns invariants to families of algebraic curves just as a cla*
*ssical
topological field theory assigns invariants to individual curves; in this it re*
*sem-
bles algebraic K-theory, which assigns homotopy-theoretic invariants to familie*
*s of
modules over a ring.
Here is an outline of the argument. After a preliminary section which collec*
*ts
some background information, we define a generalized topological field theory i*
*n x2
in terms of (homotopy classes of) maps
__n ^n
ong: M g! M
from a compactified moduli space of curves marked with n points, to n-fold powe*
*rs
of a module-spectrum M. These maps preserve a monoidal structure (defined geo-
metrically in the domain by glueing curves together at marked points, but defin*
*ed
algebraically in the range); in the language of [14 x3, 17 x1.7], o**is a repre*
*senta-
tion of a certain cyclic operad. The existence of such a representation entails*
* the
existence of a (quantum) multiplication on the module-spectrum M (cf. x2.4); in
____________
1991 Mathematics Subject Classification. Primary 14H10, Secondary 55N35, 81*
*R10.
Author was supported in part by NSF Grant #9504234.
cO0000 American Mathematical Socie*
*ty
0000-0000/00 $1.00 + $.25 per pa*
*ge
1
2 JACK MORAVA
familiar cases this is the multiplicative structure defined by the WDVV equatio*
*n,
and when n = 0 we recover Witten's tau-function for the moduli space of curves.
That topological gravity and quantum cohomology are closely related is clear
from [37], but I suspect that the simplicity of the underlying geometry is not *
*widely
understood. The main technical lemma (x2.1) is a kind of splitting theorem (cf.*
* [21,
33]) for generalized Gromov-Witten classes; it is quite natural, in light of Ko*
*ntse-
vich's ideas about stacks of stable maps. These moduli objects have not yet been
shown to be smooth for curves of all genus for any variety more complicated tha*
*n a
point [21 x1], so our main example is still conjectural, but in fact smoothness*
* is much
more than we need; the constructions of this paper require only that the genera*
*lized
Gromov-Witten maps of x1.3 be local complete intersection morphisms. This is a
convenient working hypothesis, which can be weakened further by elaborating the
cohomological formalism; but this paper concerned with the consequences of this
assumption, not with proving it. For curves of genus zero the moduli stacks are
known to be smooth, if the target manifolds are convex in a suitable sense [4];*
* this
leads to a simple proof (x2.2) of the associativity of the quantum multiplicati*
*on, and
when the defining variety is a point, we can calculate the corresponding coupli*
*ng
constant (x2.3).
Two short appendices discusses some related issues. In particular, there is *
*rea-
son to think that the Virasoro algebra is a ring of `quantum generalized cohomo*
*logy
operations' for the main example. To state this more precisely requires a short*
* di-
gression about representions of the group of antiperiodic loops on the circle, *
*which
is included as the first appendix. A second appendix, added after the publicati*
*on of
this paper [in the Proceedings of the Hartford-Luminy Conference on the Operad
Renaissance, Contemporary Math. 202 (1997) 407-419], outlines a slight general-
ization of the main construction of this paper in terms of the physicists' `lar*
*ge phase
space' of deformations of quantum cohomology.
It is a pleasure to thank Ralph Cohen, Yuri Manin, and Alexander Voronov
for helpful conversations about the content of this paper; that the paper exist*
*s at
all, however, is the consequence of helpful conversations with Graeme Segal and
Edward Witten.
1. Notation and conventions
1.1. Let V be a simply-connected projective smooth complex algebraic variety
of real dimension 2d, with first Chern class c1(V ), and let H denote its second
integral homology group H2(V; Z). We will use a rational version
= Q[H x Z]
of the Novikov ring [25 x1.8] of V : its elements are Laurent polynomials
X
cff;kff vk
k2Z;ff2H
with coefficients cff;k2 Q. This ring has a useful grading, in which v has (coh*
*omo-
logical) degree two, and ff has degree 2. We will also use the nota*
*tion
k
v(k)= v_k!
QUANTUM GENERALIZED COHOMOLOGY 3
for the k-th divided power of v. If u : ! V is a map from a connected oriented
surface to V , then the degree of u is the class u*[] 2 H, where [] 2 H2(; Z) is
the fundamental class of the surface.
1.2. MU*(X) will denote the complex bordism of a CW -space X, and MU
is the spectrum representing the homology theory defined on base-pointed finite
complexes by
X 7! MU*(X) ^ = [S*; X ^ MU ];
the tensor product has been completed, so MU*(pt) ^ is the graded ring of formal
Laurent series in v, with coefficients from the graded group ring MU*(pt)[H]. It
will be convenient to write MU (V ) for the function spectrum F (V +; MU ).*
* [The
superscript + indicates the addition of a disjoint basepoint, but this refineme*
*nt will
often be omitted when the space is already encumbered with superscripts.] The
fiber product of spaces (or schemes) X and Y over Z will be denoted X xZ Y , and
the product of MU*(X) and MU*(Y ) over MU*(Z) will be denoted MU*(X) Z
MU*(Y ). The spectrum MU (V ) has an MU -algebra structure, and
MU (V n) = MU (V ) ^MU . .^.MU MU (V )
is its n-fold Robinson smash power [31] over MU . There is a map
TrV : MU (V ) ! MU
of MU -module spectra, which represents the transfer map
MU*(X ^ V +) ! MU*-2d(X)
defined by the (complex oriented) projection V ! pt, followed by multiplication
with vd (to shift dimensions).
According to Quillen, a proper complex-oriented map : P ! M between
smooth manifolds defines an element [] of MUk(M), where k is the codimension of
; more generally, a suitably oriented map between geometric cycles, with enough
of a normal bundle to possess rational Chern classes, will define an element of
MUkQ(M). Contravariant maps in cobordism are defined by fiber products, while
covariant map are defined by the obvious compositions. Finally, the bilinear fo*
*rm
bV : MU (V ) ^ MU (V ) ! MU (V ) ! MU
is the composition of the trace with the multiplication map of MU (V ).
The graded ring MU*(V ) is a technical replacement for MU*Q[v;v-1](V x H),
which is in some ways more natural; but the latter ring does not help with the
usual convergence problems, which (in the present framework) are consequences of
the failure of the map H x H ! ptto be proper.
1.3. _A_(marked)nalgebraic curve is stable if its group of automorphisms is
finite; M gwill denote the Deligne-Mumford-Knudsen space of such curves of arit*
*h-
metic genus g, marked with n ordered smooth points. These spaces are compact
orbifolds, of complex dimension 3(g - 1) + n 0; cases of low genus are thus so*
*me-
times exceptional. It is useful to understand n_tonbe a finite ordered_setn(or *
*ordinal
number), so that permutations of n can act on M g. More generally, M g(V; ff) w*
*ill
denote the stack [4 x3, 12] of stable maps of degree ff from a_curve_ofngenus_g*
*nmarked
with n ordered smooth points, to V ; there is a morphism from M g(V; ff) to M g
which assigns to a map (the stabilization of) its domain, and there is a morphi*
*sm to
4 JACK MORAVA
V nwhich evaluates a map at the marked points. The product of these is a perfect
(finite Tor-dimension [11 II x1.2]) proper morphism
__n __n n
nV;g;ff: M g(V; ff) ! M gx V
__n
of stacks. At a point_u : ! V of M_g(V; ff) defined by a smooth , the relative
tangent space to M ng(V; ff) over M ngdefines a K-theory class
[H0(; u*TV )] - [H1(; u*TV )]
of complex dimension d(1 - g) + , where d is the complex dimension *
*of V .
It seems likely that under reasonable hypotheses this map will be a local compl*
*ete
intersection morphism of stacks, and in particular that the K-theory class of i*
*ts
cotangent complex at a singular point (; u) will equal the holomorphic Euler cl*
*ass
of the pullback of u*TV to the normalization of . [In fact, Kontsevich [19 x1.4*
*] has
already sketched something very close to a local complete intersection structur*
*e for
this morphism.]. Similarly, let fflnV;g;ff(k) denote the proper complex-oriente*
*d map
__n+k __n
from M g (V; ff) to M g(V; ff) which forgets the final k marked points, and de*
*fine
__n+k __n n
nV;g;ff(k) = nV;g;ffO fflnV;g;ff(k) : M g (V; ff) ! M gx V :
There are also generalizations
__r+s __s+t __r+t
sV: M g (V; ff) xV sM h (V; fi) ! M g+h+s-1(V; ff + fi) ;
of Knudsen's glueing morphisms [17]. All these maps represent natural transform*
*a-
tions between moduli functors, so their normal bundles are reasonably accessibl*
*e.
In diagrams below, complicated subscripts and superscripts will be supressed wh*
*en
they are redundant in context.
2. Generalized topological field theories
2.1. We will be interested in generalized Gromov-Witten invariants defined
by the cobordism classes of these morphisms. I will assume that these maps are
local complete intersection morphisms; such maps between (possibly singular)
varieties have most of the topological transversality properties of maps between
smooth manifolds. In particular, they have well-behaved Gysin homomorphisms
and normal bundles [1 IV x4] and they thus define elements in complex cobordism
tensored with the rationals. [A variant approach is discussed below in x2.5.] O*
*ur
generalized Gromov-Witten invariants are the classes
X 2d(n+g-1)__n
OEnV;g(k) = [nV;g;ff(k)]ff v(k)2 MU (M g x V n) ;
ff2H
permutations of k define cobordant elements. Summing these classes over k defin*
*es
X __n
onV;g= v-d(n+g-1) OEnV;g(k) 2 MU0(M g x V n) ;
k0
the convergence problems mentioned in the preceding section do not appear when
the function ff 7! [nV;g;ff(k)] is supported in a proper cone in H. The tau-fun*
*ction
onV;gcan be interpreted geometrically as the cobordism class of the `grand cano*
*nical
QUANTUM GENERALIZED COHOMOLOGY 5
ensemble' of maps from a curve of genus g marked with n ordered smooth points, *
*to-
gether with an indeterminate number of further distinct smooth (unordered) poin*
*ts,
to V , but we will be more concerned with the homotopy class
__n n
onV;g: M g! MU (V )
it defines.
Proposition 2.1: The diagram
__r+s __s+t s __r+t
M g ^ M h ----! M g+h+s-1
?? ?
yo^o ?yo
s
MU (V r+s) ^ MU (V s+t)bV----!MU(V r+t)
commutes up to homotopy; alternately,
*s(or+tV;g+h+s-1) = vsdTrsV(or+sV;gV sos+tV;h) :
Sketch Proof, under the standing hypothesis above: The general case reduces by
induction to the case s = 1, which can be stated as a coproduct formula
X
*OEr+tV;g+h(k) = vd TrV (OEr+1V;g(i) V OE1+tV;h(j)):
i+j=k
This can be reformulated as the assertion that the two diagrams
F __i+r+1 __1+t+j V __r+t+k
i+j=kM g (V;?ff) xV M h (V; fi)----!Mg+h (V;?ff + fi)
?yFffl(i)xffl(j) ?yffl(k)
__r+1 __1+t __r+t
M g (V; ff) xV M h (V; fi) ----! M g+h(V; ff + fi)
and F __
r+1 __1+t V __r+t
ff+fi=flMg (V;?ff) xV M h (V;-fi)---!M g+h(V;?fl)
?yFffxfi ?yfl
__r+1 __1+t xTrV __r+t
M g x V r+1+tx M h ----! M g+hx V r+t
are fiber products; the claim follows by stacking the first diagram on top of t*
*he
second. The bottom diagram describes the stable maps of decomposable curves in
terms of the restrictions to their components. In the top diagram, the union is*
* to
be taken over partitions of the set with k elements into subsets of cardinality*
* i and
j; on the level of functors, this diagram asserts that the ways of sprinkling p*
*oints
on a curve decomposed into two components correspond to the ways of sprinkling
points on the components separately.
The last assertion is not entirely straightforward, because forgetting marked
points may destabilize a genus zero component of a stable marked curve; the mor-
phism V (k) will blow such components down to points. The union in the upper
left corner of the diagram is thus not necessarily disjoint: the fiber product *
*is ob-
tained from the disjoint union by identification along certain divisors [28 x3]*
*. The
point, however, is that the cobordism classes are defined by maps rather than by
subobjects; the fiber product class is equivalent to the sum of the classes def*
*ining
the disjoint union.
6 JACK MORAVA
2.2. Proposition 2.1 states that the triple (o*V;*MU (V ); bV ) defines a*
* topo-
logical field theory which takes values in the category of MU -module spectra,
where the usual monoidal structure defined by tensor product of modules over a
ring is replaced by the smash product of module-spectra over a ring-spectrum. T*
*he
domain of this generalized topological field theory is the monoidal category (S*
*table
Curves) with finite ordered sets as objects; morphisms are finite unions of mar*
*ked
curves. [This category, however, does not possess identity maps for its object*
*s.]
Both the domain and range of the generalized topological field theory are topo-
logical categories, and o*V *defines a homotopy class of maps from the space of
morphisms of the domain category to the space of morphisms of the range. These
homotopy classes preserve the composition of morphisms and thus define a functo*
*r.
In this generality, we need a version of Proposition 2.1 for Knudsen glueing of*
* two
points on a connected curve, but the changes required for this are minor.
The construction involves the bilinear map bV , but it has not otherwise used
the multiplicative structure on MU (V ). In fact the morphism
___3 3
o3V;0: S0 = M 0! MU (V )
defines a composition
3^id b2 ^id
*V : MU (V ) ^ S0 ^ MU-id^o0-----!MU(V 5)-V---! MU (V ) ;
and Proposition 2.1 has the following
Corollary 2.2 : The pair (MU (V ); *V ) is a homotopy commutative and homo-
topy associative ring-spectrum.
Sketch proof: Smashing the morphism
__2+1 __1+2 4
A(2; 2) := (id ^ bV ^ id)(o2+10^ o1+20) : S0 = M 0 ^ M 0 ! MU (V )
with the identity map of MU (V 3) defines a map from MU (V 3) to MU (V 7);
arranging the seven copies of V into pairs and applying the trace map b three
times defines a collection of maps from MU (V 3) to MU (V ) indexed by the
possible groupings of the factors. By our conventions 2+1 and 1+2 are isomorphic
but not equal, so the notation for this associator class emphasizes that it dep*
*ends
on four points partitioned into two subsets, each containing two items. Ignori*
*ng
obvious involutions, there are three_different_partitions,_corresponding to the*
* maps
ss0, ss1, ss1 from the 0-manifold M 2+10x M1+20to M 40which send it to a degene*
*rate
curve of genus zero with two irreducible components, each carrying two marked
points (aside from the node); the cross-ratio identifies these configurations w*
*ith the
standard points 0,1, and 1 on the projective line. To verify associativity it s*
*uffices
to show that the homotopy class A(2; 2) is independent of the way the four poin*
*ts
are_partitioned into pairs; but by the proposition, A(2; 2) = o40O ssifactors t*
*hrough
M 40, where the three maps ssi become homotopic.
2.3. The morphism
X __k+3
o30= v-2d [M 0 (V ) ! V 3]v(k)
k0
defining this quantum multiplication_isnessentially the Gromov-Witten potential
[21]. Because the moduli spaces M gare not defined when 3(g - 1) + n is negativ*
*e,
QUANTUM GENERALIZED COHOMOLOGY 7
however, it is not clear that the resulting multiplicative structure on MU (V*
* )
possesses a unit. The class
qV := 1 *V 1 = v-2d(bV id)(o3V;0)
is the coupling constant for the topological field theory defined by MU*(V ); t*
*his
theory assigns to a connected surface of genus g with one boundary component, t*
*he
2gth power of 1 with respect to the product *V .
Corollary 2.3 : When V is a point, the coupling constant of the resulting topo-
logical field theory is
X __k+3
q = [M 0 ]v(k)2 MU0(pt);
k0
and the quantum product in MU*(pt) is x * y = qxy.
__k+3
In this formula [M 0 ] is the cobordism class of the manifold of configura-
tions of k + 3 points on a curve of genus zero; the sum is thus a cobordism
analogue of Manin's Hodge-theoretic invariant ' [26 x0.3.1]. With this structur*
*e,
(MU*(pt); *pt) is isomorphic to MU*(pt) with its usual multiplication, by a ho-
momorphism which sends 1*2gto q2g. More generally, the operation x 7! 1 *V x is
a module isomorphism, and it seems reasonable to hope that (1*V )-1(1) will be a
unit for *V .
Knudsen glueing defines a pair-of-pants product
__1 __1 __1 __3 __1 __1
+ : M g^ M h! M g^ M 0^ M h! M g+h
and it follows from x2.1 and the arguments above that the diagram
__1 __1 + __1
M g ^ M h ----! M g+h
?? ?
yo^o ?yo
MU ^ MU --*V--!MU
is homotopy-commutative; in other words,
__1
o1V *: M *! MU
is a kind of homomorphism of monoids. This is probably the most intuitive way to
think of the product in quantum cohomology, but from the present point of view *
*it
is a conclusion, not a definition.
2.4. A generalized topological field theory has an associated theory of top*
*o-
logical gravity, which assigns invariants to proper flat families of stable cur*
*ves._Suchn
a family Z, say of topological type (g; n), is defined by its classifying map t*
*o M g;
the pullback of onV;galong this morphism defines a class oV (Z) 2 [Z+ ; MU (V*
* n)].
If (for simplicity) we assume that V is a point, and write [Z] 2 H*(Z) for the
fundamental class of Z, then the image o*(Z) of [Z] in H*(MU ; ) under the map
induced on homology by o(Z) is a kind of absolute invariant of the family, obta*
*ined
by integrating o(Z) over [Z]. In particular, the vacuum morphism
0 ! 0
8 JACK MORAVA
is defined by the family of arbitraryFfinite_unions of unmarked stable curves; *
*the
infinite symmetric product SP1 ( g0 M 0g) is a rational model for its parameter
space. The resulting absolute invariant
X __0
o = exp( o*(M g)) 2 H*(MU ; Q[v; v-1])
g0
is Witten's tau-function for two-dimensional topological gravity [24]. The poin*
*t is
that the characteristic number homomorphism
MU*(M) ! H*(M; H*(MU ))
P
sends the class [ : P ! M] to a sum of the form I*cI()tI, where is the
stable normal bundle of , cI() is a certain polynomial (indexed by I = i1; : :):
in its Chern classes, tI = ti11: :i:s a product of elements in H*(MU ) = Z[ti|*
*i 1],
and * is the covariant (Gysin) homomorphism induced by . The stable normal
bundle of (k) is inverse to the tangent bundle along the fibers, which is the s*
*um
of the k line bundles defined by the tangent space to the universal curve at it*
*s k
marked points, so its Chern polynomials can be expressed as polynomials in the
Chern classes of these line bundles. Under the pushdown_(k)*,_these become
polynomials in the Mumford classes in the cohomology of M 0g.
2.5. Some aspects of Kontsevich-Witten theory suggest that Gromov-Witten
invariants can be defined more naturally in K-theory than in ordinary cohomolog*
*y.
The algebraic K-theory of a reasonable stack, tensored with the rationals, agre*
*es
with the algebraic K-theory of its quotient space [15 x7], but (perhaps because*
* of
this) the K-theory of stacks seems to have received little attention otherwise.*
* The
following assumes that the standard direct image construction for perfect proper
maps of schemes [36_x3.16.4]_generalizesnto stacks.
Let ssg;n: Cng! M gbe the universal stable curve; because the range is smoot*
*h,
this is a perfect proper morphism. Let
__n n
Cng;ff(V ) := M g;ff(V ) x__MngCg ;
__
from now on I will supress the subscripts. The projection ss: C(V ) ! M (V ) to
the first factor, being the pullback of a perfect morphism, is again perfect. L*
*et U :
C(V ) ! V be the universal evaluation morphism; the vector bundle U*TV defines
an_element_of K(C(V )), and its hypothetical direct image ss*U*TV := (V ) 2
K(M (V )) is a reasonable candidate for the normal bundle of V .
Because V is itself proper and perfect, we can define generalized Gromov-
Witten classes
X __n
V *mI((V ))tI 2 K*(M g x V n) K K*MU ;
I
Q i
where I is a multiindex as above, tI = ktkkis a basis for K*MU , and mI deno*
*tes
the K-theory characteristic class associated to the monomial symmetric function
by the correspondence which assigns gamma operations [5 V x3] to the elementary
symmetric functions. By the Hattori-Stong_theorem,nsuch a sum can be identified
with a class in the localization MU*(M g x V n)[CP (1)-1] of complex cobordism.
If V is a local complete intersection morphism, this approach to defining
Gromov-Witten invariants agrees with the definition in x2.1. In any case some s*
*uch
hypothesis seems to be needed to make the arguments of Prop. 2.1 work.
QUANTUM GENERALIZED COHOMOLOGY 9
3.Some questions
3.1. Kontsevich and Witten [20, 24, 37] show that the tau-function for the
vacuum state of two-dimensional topological gravity is a lowest weight vector f*
*or
a certain representation of the Virasoro algebra. This Lie algebra bears a stri*
*king
resemblance to the Lie algebra defining the Landweber-Novikov algebra of opera-
tions in complex cobordism, but the relation between these two structures is not
well-understood. I have included as an appendix a construction for the Kontsevi*
*ch-
Witten representation, starting from a representation of a certain loop group of
antiperiodic functions on the circle, following [8]. One point of the appendix *
*is that
the representation theory of this loop group is essentially trivial.
On the other hand, the usual complex cobordism functor takes values in the
monoidal category of Z=2Z-graded G-equivariant sheaves over the moduli scheme
Spec MU*(pt) of formal group laws, with the Landweber-Novikov group G of for-
mal coordinate transformations acting by change of coordinate; but this category
is equivalent, after tensoring with Q[v; v-1], to the category of Z=2Z-graded v*
*ec-
tor spaces. It therefore seems not completely unreasonable to conjecture that t*
*he
group of antiperiodic loops is a kind of motivic group for the generalized quan*
*tum
cohomology theory defined by MU .
As for products in V , the functor o*V *seems to behave very naturally (cf. *
*[19]);
in particular, it is reasonable to expect that
onV0xV1;g= onV0;gMU* onV1;g:
3.2. The work in this paper was originally motivated by a desire to under-
stand topological gravity and quantum cohomology from the point of view of Floer
homotopy theory [6,7], but such questions have been supressed here. It may be
helpful, however, to observe that the circle group T acts on the universal cover
gLVof the free loopspace of V , with V x H as fixed point set, so we can think *
*of
MU*(V ) as its tTMU*Q-cohomology [16 x15]. The Tate cohomology tTMU*Q(gLV)
is a rough approximation to the Floer MU -homotopy type of gLV, and we might
hope to understand the relation between these invariants as a localization theo*
*rem
for Tate cohomology.
More specifically, given a compact pointed Riemann surface (; x), let
(D; 0) ! (; x)
be a holomorphically embedded closed disk; the boundary @D separates the sur-
face into components 0 and 1 , with x the point at infinity, as in [30 x8.11]*
*. Let
Hol(; D; V ) denote the space of continuous maps from to V which are holomor-
phic on 0 and 1 . This is a manifold, with tangent space
H0(0 ; u*0TV ) H0(1 ; u*1TV )
at u 2 Hol(; D; V ); here the sections of the pullback bundles are to be holomo*
*rphic
on the interior and smooth on the boundary. Restriction to the boundary defines
a map u 7! @u to the free loopspace of V , but the homotopy class of u1 define*
*s a
canonical contraction of @u, so this restriction map factors naturally through *
*a lift
to the universal cover gLVof LV .
This map is Fredholm, with index equal to the holomorphic Euler characterist*
*ic
of u*TV . [More precisely: since u will usually not be holomorphic, u*TV can't *
*be
expected to be holomorphic either; but u*TV restricts to a holomorphically triv*
*ial
10 JACK MORAVA
bundle on an annulus containing @D, so u*0TV extends to a holomorphic bundle
"u*TV on . Then O("u*TV ) is the index at u.] Moreover, away from maps which
collapse @D to a point, this map appears to have a good chance to be proper.
We can elaborate this construction, by considering the space Hol(^x; V ) of
holomorphic disks in centered at x, together with a map to V, continuous and
holomorphic away from @D as above; since we're enlarging things, we may as
well include trivial disks too. This thickening has the same homotopy type as t*
*he
preceding space, but now T acts by rotating loops. More generally, we can allow
the moduli of to_vary_as well, thus defining a space of maps over a thickening*
*_of
the moduli space M . Restriction maps this space to a similar thickening of M x*
*gLV,
defining a_candidate_for a proper T-equivariant Fredholm map, and thus an eleme*
*nt
of MU-2OT(M x gLV), which restricts to the classical Gromov-Witten invariant at
the fixed point set of T.
Appendix I : MU*(pt) as a Virasoro-Landweber-Novikov bimodule
The Virasoro algebra is the Lie algebra of a central extension of the group *
*D of
diffeomorphisms of a circle; it is true generally [30 x13.4] that D acts projec*
*tively
on a positive energy representation of a loop group, and in this appendix I will
sketch the construction of an action of the double cover D(2) of D on the basic
representation of the twisted loop group
LTwist= {f 2 LT|f = (f)}
of functions from the circle R=Z to T = {z 2 C||z| = 1} which are invariant und*
*er
the involution (f)(x) = f(x + 1_2)-1.
Because the loop functor preserves fibrations, the exact sequence of the exp*
*o-
nential function e(x) = e2ssixyields an exact sequence
0 ! Z ! LR ! LT0 ! 0
of abelian groups with involution, the group on the right being the identity co*
*m-
ponent of the group of untwisted loops. The associated exact sequence
0 ! LRZ=2Z! LTwist! H1(Z=2Z; Z) ! 0
of cohomology groups presents the antiperiodic loops as a canonically split ext*
*ension
of the group Z=2Z of constant loops with value plus or minus one, by a vector s*
*pace
of antiperiodic functions.
Now LT0 contains a subgroup T of constant loops, and D contains the subgroup
R of rotations, so the lift of a positive-energy projective unitary representat*
*ion
of LT to an honest unitary representation of an extension fLTof LT by a circle
group C restricts to a representation of a semidirect product R : E, where E is*
* an
extension of Z x T by C which splits over the identity component. The irreducib*
*le
positive-energy projective representations of LT are classified by their restri*
*ction to
representations of RxC xT [30 x9.3]; an irreducible representation of T is clas*
*sified
by its weight, and the corresponding integer defined by C is the level. Howeve*
*r,
the identity component of fLTwist has a trivial subgroup of constant loops: its
representation theory is effectively weightless.
There is, however, an interesting basic representation of fLTwist; one const*
*ruc-
tion, modelled on [34 x2], begins with the skew bilinear form defined on LRZ=2Z
QUANTUM GENERALIZED COHOMOLOGY 11
by Z
1
B(f0; f1) = 2_ss f0(x + 1_2)f1(x)dx :
0
The group D(2) of smooth orientation-preserving maps g of R to itself satisfying
g(x + 1_2) = g(x) + 1_2acts on this symplectic space, by
1_
g; f 7! g02f O g:
The complexified space of antiperiodic functions admits the decomposition
LRZ=2Z C = A+ A- ;
A+ being the subspace of functions on the circle which extend inside the unit d*
*isk.
There is a standard [34 x9.5] unitary representation of fLTwist on the symmetric
algebra S(A+ ) associated to this polarization; the basis
1_ n+ 1_
an = -ssi-2 (-1_2) 2(n + 1_2)-1e((n + 1_2)x)
for the complexification satisfies
B(an; am ) = (2m + 1)ffin+m+1;0 :
The polarization is defined by a nonstandard complex struction in which conju-
gation acts by an = -ia-n-1, making iB(a; a) a positive-definite Hermitian form
on A+ . [This complex structure differs from the standard one by a transformati*
*on
which is diagonal in the basis an; this operator is real but unbounded.] The ac*
*tion
of D(2) on LRZ=2Zmakes it reasonable to interpret antiperiodic functions as sec-
tions of a bundle of half-densities on the circle; the complexification of this*
* bundle
admits the nonvanishing flat section
1_ 1_ 1_
(2ss)2 e(x + 1_8)(dx)2 = (dZ)2 ;
where Z = e(x). It follows from Euler's duplication formula that
1_ 1_
an(dx)2 = (2n + 1)!!Z-n-1 (dZ)2
when n is nonnegative.
The Lie algebra of D(2) now acts on S(A+ ) with generators (cf. [20 x1.2, 37
x2]) P
Lk = 1_4n2Z ak-n-1an if k 6= 0 ;
P 1
= 1_2n0 a-n-1an + __16if k = 0 :
Convenient polynomial generators tn for the algebra S(A+ ), regarded as a ring *
*of
holomorphic functions on A- , can be defined by expanding an element f of A- as
X 1_
tn(f)Z-n-1 (dZ)2 ;
n0
similar generators Tn, constructed by writing this element as
X
Tn(f)a-n-1 ;
n0
satisfy the equation
tn = (2n + 1)!!Tn :
12 JACK MORAVA
The Virasoro generators (which are not derivations) act on these elements so th*
*at
LkTn = (n - k + 1_2)Tn-k if n k
= 0 otherwise .
On the other hand the group G of invertible formal power series (under composi-
tion) in Z-1 acts on A- , interpreted as a free module over the ring of power s*
*eries
in Z-1; the Lie algebra of this group is spanned by vector fields
vk = Z-k+1d=dZ with k 0;
which act on S(A+ ) (as derivations) such that
vktn = (n - k + 1)tn-k when n k
= 0 otherwise:
We can thus identify MU*C(pt) with S(A+ ) as a comodule over the Landweber-
Novikov algebra, in a way which makes it a Virasoro representation as well. [The
complex coefficients are only a technical convenience.] The resulting bimodule
defines a kind of Morita equivalence of the category of cobordism comodules to *
*the
category of representations of fLTwist, given the (weak) monoidal structure def*
*ined
by the fusion product [35 x7] of representations.
Appendix II : the large phase space of deformations
Some recent advances [9] in quantum cohomology seem to fit very naturally in
the homotopy-theoretic framework sketched above. This appendix has been added
in July 1998 to the body of the published paper; I have used the opportunity to
add references to recent work on the construction of Gromov-Witten invariants
discussed in x2.1.
II.1. If H is a commutative Hopf algebra over a ring k and B is a finitely-
generated Z-module, then the functor defined on the category of commutative k-
algebras by
A 7! B Hom k-alg(H; A)
is represented by a Hopf k-algebra which might be denoted B H : if B is free of
rank b then a choice of basis defines an isomorphism of B H with the b-fold ten*
*sor
product of copies of H. A variant of this construction occurs in the theory of *
*vertex
operator algebras; in that context H is the algebra of symmetric functions and B
is a positive even lattice.
This appendix suggests a conjectural interpretation for a family of deforma-
tions of the quantum cohomology of a smooth algebraic variety V in terms of a
similar construction, in which the role of B is played by the cohomology of V ;
in the basic example, H is an algebra of Schur Q-functions [28]. The parameter
space for this family is the `large phase space' of Witten [37 x3c], defined wh*
*en
topological gravity is coupled to quantum cohomology; the more usual space of
deformations of quantum cohomology proper is then called the `small phase space*
*'.
In impressionistic terms this large phase space is essentially a tubular neighb*
*orhood
of the moduli space of holomorphic maps from a Riemann surface to V , inside the
space of all smooth maps. A two-dimensional quantum field theory is a kind of
measure on such a space of smooth (or perhaps continuous) maps, but it seems to
QUANTUM GENERALIZED COHOMOLOGY 13
be reasonable to think of these measures as supported near the (finite-dimensio*
*nal)
subspace of holomorphic maps. The resulting hybrid structure can thus be inter-
preted as a homotopy-theoretic family of deformations of a reasonably familiar *
*kind
of algebro-geometric object.
This is a summary of work in progress; one way to paraphrase the basic idea *
*is
that (even though quantum cohomology is not in any very natural sense a functor*
*),
we might interpret it as a cohomology theory taking values in the abelian categ*
*ory
of bicommutative Hopf algebras over Q. The interesting examples have further
structure, and the point of this note is that some, at least, of these extra st*
*ructures
seem to have natural interpretations in a Hopf-algebraic context.
I am indebted to Andy Baker and to Ezra Getzler for very helpful discussions
of this and related material.
__n __n
II.2. I will simplify notation here, writing M (V ) for M g;ff(V ) when ind*
*exing
components is unnecessary. Gromov-Witten invariants in the context of algebraic
geometry have now been defined rigorously by Behrend and Fantechi [2,3]; related
results in the symplectic context have been announced by Li and Tian [23, cf. a*
*lso
32]. We will be especially interested in the forgetful evaluation maps
__n+k __n k
ffln(k) : M (V ) ! M (V ) x V ;
the spaces of_nonzero_tangentnvectors at marked points are principal Cx -orbifo*
*ld
bundles over M (pt) which can be pulled back over the domain of ffl, and
__n k k
v-k(d-1)fflnT(k) 2 [M (V ) x V+ ; BT+ ^ MU ]
will denote its cobordism class enriched by the memory of the tangent bundles at
the forgotten points.
Now let X
z = tk;ickzi2 H*(BT; H*(V; Q)) ;
where {zi} is a basis for H*(V; Q), be a homogeneous class of degree zero: we
thus interpret the coefficient tk;ito be an indeterminate of degree |zi| - 2k. *
*In the
language of physics, the elements zi are the `primary fields' of a topological *
*field
theory defined by the quantum cohomology of V , while ckzi is the kth `topologi*
*cal
descendant' of zi. Using this terminology we can generalize the constructions o*
*f x2
above, replacing the class OEng(k) defined there with
X __n
OEng(k; z) = ng;ff*(fflnT;g;ff(k) \ kz) ffv(k)2 H*(M g x V n; ) ;
ff2H2(V;Z)
the cap product being (the Q-linear extension of) the Kronecker pairing
H*(BT; H*(V; Z)) H*(V; H*(BT; Z)) ! Z :
The product with z leaves the coproduct formula 2.1 essentially unchanged, and
the sum X
ong(z) = v-d(n+g-1) OEng(k; z)
k0
still defines a generalized topological field theory, and thus a family of mult*
*ipli-
cations, which specializes when z is the fundamental class of V to our previous
construction. If, for_example,_V is a single point, then o0(z) is a formal func*
*tion
from H*(BT; Q) to H*(M ; Q), which can alternately be interpreted as an element
14 JACK MORAVA
of the tensor product of the symmetric algebra on H*(BT; Q) and the cohomology
of the moduli space of curves. On the other hand, the composition
BT ! BU ! MU
of the Thom map for cobordism with the map induced by the inclusion of the circ*
*le
in the unitary group defines a canonical isomorphism
S(H*(BT; Q)) ! H*(MU ; Q) ;
the element o0(z) thus defines a homomorphism from the homology of the moduli
space to H*(MU ; Q) = MU*Q. In this case there is a unique primary field z0 = *
*1,
and the tk's defined by its topological descendents are the standard generators*
* of
the Landweber-Novikov algebra.
II.3. At the opposite extreme we might suppose that V is nontrivial and set
ti;k= 0 if i; k > 0; in this model there are then nontrivial primary fields, bu*
*t only
the field z0 = 1 has topological descendants. If we identify the symmetric alge*
*bra
on H*(BT; Q) with MU* Q as above, then on (z) becomes the class in MU*(V n)
defined by the space of stable maps from a curve marked with n smooth points,
together with an indeterminate number of further distinct but unordered smooth
points which map to the subvariety z of V . This yields a deformation of the qu*
*antum
multiplication on MU*(V ); its parameter space is the classical `small phase sp*
*ace'
of deformations, enlarged slightly (since we permit nontrivial descendents of z*
*0) to
yield a theory interpretable in terms of cobordism rather than ordinary cohomol*
*ogy.
When V is a point, for example, the deformation z 2 MU* Q[v; v-1] replaces the
coupling constant q = q(v) of x2.3 with q(zv).
In the general case (with no hypothesis that the topological descendants of *
*any
primary fields vanish) we can interpret the descendants of z0 = 1 as lying in t*
*he
cobordism ring and rewrite on as an element of
__n n
S(H*(V; H*(BT; Q))) H*(M x V ; ) ;
which can be expressed in terms of reduced cohomology as
__n n
S(H*(BT; Q)) S(H"*(V; H*(BT; Q))) H*(M x V ; ) :
This is in turn isomorphic to the symmetric MU*-algebra
__n n
SMUQ (MU*Q(V )) MUQ MU*(M x V ) ;
and we can think of on (z) as a formal family of deformations, parametrized by *
*(the
space underlying) H*(V; H*(BT; Q)), of a generalized topological field theory
__n n
on (z) : M ! F(V+ ; MU ) :
II.4. Recently Eguchi [9, cf. also 13] and his coworkers have studied an act*
*ion
of the Virasoro algebra on a large phase space model for the quantum cohomol-
ogy of CP (n), which generalizes the Virasoro action on what can be interpreted*
* as
the (large) quantum cohomology of a point. There is reason to believe [29] that
the latter Virasoro action can be understood most naturally not in terms of com-
plex cobordism but instead in terms of cohomology with coefficients in a VOA-li*
*ke
structure defined by the ring of Schur Q-functions; the resulting theory has g*
*ood
integrality properties, and its Virasoro structure is a consequence of Hopf-alg*
*ebraic
properties of . There is a natural (Kontsevich-Witten) genus
kw : MU* ! [q-11]
QUANTUM GENERALIZED COHOMOLOGY 15
relating these constructions, which is essentially an isomorphism over the rati*
*onals,
and it is natural to ask if this interpretation of topological gravity can be e*
*xtended
to ecompass the quantum cohomology of algebraic varieties.
This is a subject for further research, but it is at least reasonable in ter*
*ms of
the Hopf algebra structures. From that point of view the map defined at the end
of x3 identifies the rational homology of BT with the primitives P* of Q, which
provides an interpretation of the large phase space of deformations for the qua*
*ntum
cohomology of V as the spectrum of H(V;Z)Q. Connected, graded Hopf algebras
over the rationals are primitively generated, and there is an internal tensor p*
*roduct
which sends the Hopf algebra H0 (resp. H1 with primitives P0 (resp. P1) to the *
*Hopf
algebra with primitives P0 P1; the Hopf algebra of functions on the large phase
space is just this tensor product construction, applied to Q and S(H*(V )). An
internal tensor product on the category of bicommutative Hopf algebras has beco*
*me
important recently in other parts of algebraic topology, and an integral versio*
*n of
the construction sketched above would seem to be within reach. Questions related
to Cartier duality and self-adjointness need to be explored, but it seems likel*
*y that
these ideas will lead to a VOA-like structure on this large quantum cohomology,*
* at
least in relatively simple cases like CP (n).
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Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212*
*18
E-mail address: jack@math.jhu.edu