Topological gravity in dimensions two and four
Jack Morava
Abstract.Recent work on gravity in two dimensions has a natural general-
ization to four dimensions.
1. Basic definitions
1.1 The (symmetric monoidal) two-category
(Gravity)d+1
has objects: compact oriented d-manifolds, with
o morphisms V0 ! V1 : (d + 1)-manifolds W with @W ~=V0opt V1, and
o diffeomorphisms "W ! W as two-morphisms.
The category Mor(V0; V1) with cobordisms from V0 to V1 as objects and diffeo-
morphisms (equal to the identity on the boundary) as morphisms, is a hom-object
in this two-category. Disjoint union defines the monoidal structure, and the
category has an orientation-reversing adjoint equivalence with its opposite.
1.2 The topological category
(Gravity)d+1
has compact Riemannian d-manifolds as objects, and the spaces
a
Mor(V0; V1) := (Metrics=Diff)(W )
V0optV1~=@W
as its hom-objects. Alternately: a morphism is a (d + 1)-dimensional cobordism,
together with (the equivalence class of) a Riemannian metric on it.
The group of diffeomorphisms which fix a frame at a point acts freely on the
space of Riemannian metrics on a complete manifold, so the morphism spaces of
(Gravity)d+1 are roughly just the classifying spaces [7] of the morphism catego*
*ries
of (Gravity)d+1.
____________
1991 Mathematics Subject Classification. 55P, 58D, 83C.
The author was supported in part by the NSF.
1
2 JACK MORAVA
1.3 When d = 1 the group of diffeomorphisms of a Riemann surface (of genus > 1)
has contractible components, and its mapping-class group = ss0(Diff) acts with
finite isotropy on Teichm"uller space, defining a rational homology isomorphism
BDiff~ B ~ Teichx E ! Teichx pt= M :
(Gravity)1+1 is thus very similar to the category Segal constructed to define c*
*on-
formal field theory.
1.4 A monoidal functor from the topological gravity category to some simpler
monoidal category, such as Hilbert spaces and trace-class maps, or modules over
a ring spectrum, defines a theory of topological gravity. Replacing the Hom-
objects in this category with their sets of components leads to TFT's in the se*
*nse
of Atiyah, and replacing the Hom-objects with their rational homotopy types lea*
*ds
to cohomological field theories.
2. Some examples
2.1 The `intersection homology' of a connected surface
7! ker[H1(; C) ! H1(@; C)]
is a simple example: if O 0is the composition of two surfaces along a boundary
component, then the induced map
o : BDiff() ! BU
fits in the commutative diagram
BDiff() x BDiff(0) ____//_BDiff( O 0)
oxo|| o||
fflffl| fflffl|
BU x BU _____________//_BU:
This defines a theory of topological gravity with values in a monoidal topologi*
*cal
category having one object, with the H-space BU of morphisms. [If we want to fu*
*se
along more than one boundary component, though, we need to be more careful.]
This functor has more structure: it takes values in symplectic lattices. The co*
*m-
position
BSp(Z) ! BSp(R) ~ BU ! B(U=Sp ) ~ SO=U
is a rational homology isomorphism, so o lifts to a map
oT : BDiff() ! SO=U
which sends a surface to its harmonic one-forms with the complex structure defi*
*ned
by the Hodge *-operator. This is a form of the Abel-Jacobi-Torelli map
7! SP1 : M ! A
which sends a surface to its Jacobian; note that the union of with its opposite
defines a quaternionic object, which maps to zero.
2.2 Kontsevich-Witten theory is a much deeper example, with the (rationalized)
complex cobordism ring-spectrum as target object. A toy version is easy to con-
struct:
TOPOLOGICAL GRAVITY IN DIMENSIONS TWO AND FOUR 3
Suppose @ has at most one component; then capping it off defines the closed
surface D := O D. The cobordism class of the bundle
[D ] : D xDiff()EDiff() ! BDiff()
is primitive, in the sense that it behaves additively under composition: the pu*
*ll-
back
*[ O 0] 2 MU-2 (BDiff() x BDiff(0))
under the composition
: Diff() x Diff(0) ! Diff( O 0)
is the sum
ffl*[D ] 1 + 1 ffl0*[0D] ;
where
ffl : Diff() ! Diff(D )
is the trivial extension [5].
This says we can pull O0apart as if it were made of taffy: the standard family *
*of
quadratic cones in R3 glued to (opt 0) x I defines a Diff-equivariant cobordism
from O 0to D t 0D.
If we tensor with Q [and supress the grading] then the map okw representing
exp([]) 2 MU*Q(BDiff())
fits in the homotopy-commutative diagram
BDiff()+ ^ BDiff(0)+ _____//BDiff( O 0)+
|okw^okw| okw||
fflffl| fflffl|
MU Q ^Q MU Q____________//MU Q:
In both these examples, a surface is sent to some kind of configuration space of
points: such constructions take unions to products. Proper Kontsevich-Witten
theory involves much more complicated configuration spaces.
These constructions capture much of what's known about the stable cohomology
of moduli spaces. Mumford's conjecture, for example, is equivalent to the asser*
*tion
that okw defines an isomorphism on rational cohomology.
2.3 The Floer homology HF *(Y ) of a compact 3-manifold Y (e.g. a homology
sphere) is defined by the Chern-Simons functional on the space of connections C*
*(Y )
mod gauge equivalence on a (trivial) G-bundle over Y . It is periodically grade*
*d,
and has a kind of Poincare pairing. Following Cohen, Jones, and Segal, I assume
it is defined by an underlying spectrum HF (Y ). Because the space of connectio*
*ns
satisfies
C(Y0 t Y1) = C(Y0) x C(Y1) ;
it follows that
HF (Y0 t Y1) = HF (Y0) ^ HF (Y1) :
4 JACK MORAVA
Atiyah [1] saw that Floer homology is a topological field theory: when Y bounds
Z, the space A(Z) of Yang-Mills instantons on Z defines (by restriction to @Z) a
kind of Lagrangian cycle
[A(Z) ! C(Y )] 2 HF *(Y ) ;
and if
@Z = Y opt Y 0; @Z0= Y 0opt Y 00
then [A(Z)] ^ [A(Z0)] should map to [A(Z [Y Z0] under the pairing
HF (Y opt Y 0) ^ HF (Y 0opt Y 00) ! HF (Y opt Y 00) :
In fact Yang-Mills on Z presupposes a Riemannian metric, and there is a family
A(Z) xDiff(Z)EDiff(Z) ! C(@Z) x BDiff(Z)
of Lagrangian cycles; its hypothetical class
oA : BDiff(Z)+ ! HF (@Z)
should define a theory of topological gravity.
2.4 In honest Kontsevich-Witten theory the analogue okw of exp([]) is the class
X __n __
=n! 2 MU*Q(M g) ;
n0
__n
where is the cobordism class of a forgetful map to the Deligne-Mumford spa*
*ce
of stable algebraic curves of genus g, from a compactification of the space of *
*smooth
curves marked with n distinct points. Its characteristic number polynomial is
__
*mtot(-fake) 2 H*(M g; MUQ) ;
__n
where is the forgetful map from the Deligne-Mumford-Knudsen space M g; mtot
is the characteristic class defined by the total monomial symmetric function, a*
*nd
-fakeis the sum of the tangent line bundles to the modular curve at its marked
points. I'm indebted to Gorbunov, Manin, and Zograf for correcting my mis-
taken assertion [6] that fakeis the formal normal bundle of : above the divisor
__n-k+1 __1+k __n
M g x M0 on M gdefined by curves with two irreducible components, one of
genus zero, these two bundles differ stably by the sum of the pair of tangent l*
*ines
at the double point.
I claim that okw respects a monoidal structure defined by Knudsen's gluing map
: in the simplest case this means that
__1+ __1+ __+
M g ^ M h ______//_Mg+h
| |
| |
fflffl| fflffl|
MU Q ^Q MU Q_____//MUQ
commutes, or equivalently that
__n X __p:1 __q:1
*= x ;
p+q=n
TOPOLOGICAL GRAVITY IN DIMENSIONS TWO AND FOUR 5
__p:r __r
where 2 MU*Q(M g) is defined by the partially forgetful morphism r :
__p+r __p
M g ! M g: This follows because the diagram
` __p+1 __1+q_____//_n
p+q=nM g x M h M g+h
|"1x"1| ||
__ fflffl|_ __ fflffl|
M 1gx M 1h_________//_Mg+h
is a pullback in the category of smooth stacks; consequently
** = ("1 x "1)*"* :
But "*pulls back tangent lines at marked points to tangent lines at marked poin*
*ts,
so
*p;q((n)fake) = (p)fake 1 + 1 (q)fake;
and mtotis multiplicative, so
__n * (n)
*= *mtot(-fake)
is a sum of terms of the form
("1 x "1)*(mtot(-(p)fake) x mtot(-(q)fake)) ; QED:
2.5 Topological gravity coupled to the quantum cohomology of a (complex? sym-
plectic?) manifold V is a conjectural theory [8] defined by
X __n:k __k
=n! 2 MU*Q(M g x V k) :
n0
The configuration spaces are now suitable moduli spaces of stable maps from
curves to V , and the representing morphism
__k+ k
okw (V ) : M g ! [V ; MU Q ]
defines a functor to a monoidal category with objects N, morphisms
mor (j; k) = [V j+k; MU Q ];
and compositions defined by a Poincare trace
[V x V; MU Q ]! MU Q:
There is a natural monoidal functor n 7! [V n; MU Q ] to the usual category of *
*MU Q-
module spectra; this is the first two-dimensional case in which the target cate*
*gory
for a topological gravity theory has not been slightly degenerate.
The simplest case of the monoidal axiom asserts the commutativity of
__1+ __1+ __+
M g ^ M h ____________//Mg+h
| |
| |
fflffl| fflffl|
[V; MU Q ]^MU Q [V; MU]Q____//MUQ ;
__3
the map from M 0to [V 3; MU Q ]then defines a new (quantum) product. A relative
version of this construction uses Gromov-Witten classes
__k n n
2 [M g(V ) x V ; BT ^ MU Q ]*
6 JACK MORAVA
which record the tangent lines at the marked points; summing over n-fold cap
products with a cycle z 2 H*(V; H*(BT)) defines a theory based on configuration
spaces with marked points restricted to lie on z. This recovers the Gromov-Witt*
*en
potential and the WDVV family of quantum multiplications.
3. General nonsense
Monoidal functors between monoidal categories form a monoid, much as homomor-
phisms between abelian groups form an abelian group. Manin and Zograf [4] sugge*
*st
that we think of these families of theories as parametrized by the Picard group*
* of
invertible objects. These objects can be identified with the points of Spec MUQ,
which are one-dimensional formal group laws; but there is no natural way to com-
pose them. Kontsevich-Witten theory [2] suggests the natural parametrizing obje*
*ct
is the Hopf algebra Q of Schur Q-functions: these are Hall-Littlewood symmetric
functions of the eigenvalues of a positive-definite matrix , evaluated at t = -*
*1.
The Kontsevich-Witten genus MU ! Q defines a formal group law with Q-function
coefficients; its exponential (aside from normalization) is the asymptotic expa*
*nsion
as ! +1 of the Mittag-Leffler exponential
X Trn
_________1:
_n)
n0 (1 + 2
There is a natural twisted charge one action of the Virasoro algebra on the Q-
functions, and the image under okw of the fundamental class
X __ a __
exp( [M g]) 2 H*(SP 1( M g); Q)
of the space of not-necessarily-connected curves is an sl2-invariant highest-we*
*ight
vector, or vacuum state.
Recently Eguchi et al, Dubrovin, Getzler [3], and others have begun to extend t*
*his
fundamental result of Kontsevich-Witten theory to topological gravity coupled to
quantum cohomology. The relevant Virasoro representation appears to be defined
on a group of loops on the torus H*(V; R=Z), twisted by the endomorphism 1_2H+t*
*X,
where H; X; Y generate the standard sl2 action on the Hodge cohomology of V , a*
*nd
t = c1(V )=! depends on the K"ahler class.
This twisted torus is mysterious even when V is a point. The adjoint operation *
*on
(Gravity)1+1 defines an involution on the Picard group of invertible theories, *
*and
the analogy with Abel-Jacobi theory suggests that Q represents its skew-adjoint
part.
TOPOLOGICAL GRAVITY IN DIMENSIONS TWO AND FOUR 7
References
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* in Pure Math.
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*vich model, CMP
151 (1993) 193 - 219
3. E. Getzler, The Virasoro conjecture for Gromov-Witten invariants, math.AG/98*
*12026
4. Yu.L. Manin, P. Zograf, Invertible cohomological field theories and Weil-Pet*
*ersson volumes,
math.AG/9902051
5. E.Y. Miller, The homology of the mapping-class group, J. Diff. Geometry 28 (*
*1986) 1 - 14
6. J. Morava, Quantum generalized cohomology, in Contemporary Math. 202; expand*
*ed ed. at
math.QA/9807058
7. U. Tillmann, On the homotopy of the stable mapping-class group, Inventiones *
*Math. 130
(1997) 257 - 276
8. E. Witten, Two-dimensional gravity and intersection theory on moduli space, *
*Surveys in Diff.
Geo. 1 (1991) 243 - 310
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212*
*18
E-mail address: jack@math.jhu.edu