THE TANGENT BUNDLE OF AN ALMOST-COMPLEX FREE
LOOPSPACE
JACK MORAVA
Abstract.The space LV of free loops on a manifold V inherits an action
of the circle group T. When V has an almost-complex structure, the tange*
*nt
bundle of the free loopspace, pulled back to a certain infinite cyclic c*
*over gLV,
has an equivariant decomposition as a completion of TV ( C(k)), where
TV is an equivariant bundle on the cover, nonequivariantly isomorphic to
the pullback of TV along evaluation at the basepoint (and C(k) denotes *
*an
algebra of Laurent polynomials). On a flat manifold, this analogue of Fo*
*urier
analysis is classical.
The purpose of this note is to show that the study of the equivariant tangent
bundle of a free smooth loopspace can be reduced to the study of a certain fini*
*te-
dimensional vector bundle over that loopspace - at least, provided the underlyi*
*ng
manifold has an almost-complex structure (e.g. it might be symplectic), and if *
*we
are willing to work over a certain interesting infinite-cyclic cover of the loo*
*pspace.
The first section below summarizes the basic facts we'll need from equivariant *
*dif-
ferential topology and geometry, and the second is a quick account of the unive*
*rsal
cover of a symmetric product of circles, which is used in the third section to *
*con-
struct the promised decomposition of the equivariant tangent bundle. It is inte*
*rest-
ing that the covering transformations and the circle act compatibly on the tang*
*ent
bundle of the covering, while their action on the splitting commutes only up to*
* a
projective factor.
1. The free loopspace and its universal cover
1.1 If V is a connected compact almost-complex manifold of real dimension 2n, t*
*he
space of smooth maps from the circle
S1 = {x 2 C | |z| = 1}
to V is an (infinite-dimensional) manifold LV , with local charts defined by t*
*he
vector bundle neighborhoods of [5 x13]; the tangent space at the loop oe is a v*
*ector
space
ToeLV = S1(oe*T V )
of sections of the pullback of the tangent bundle along oe. The circle group T *
*acts on
LV by rotating loops, and rotation through the angle ff lifts to the complex-li*
*near
transformation
ff* : ToeLV ! ToeOffLV
____________
Date: 1 May 2001.
1991 Mathematics Subject Classification. 58Dxx; 53C29, 55P91.
The author was supported in part by the NSF.
1
2 JACK MORAVA
which sends the section v of oe*T V to the section
ff*(v)(`) = v(` + ff)
of (oe O ff)*T V . The tangent bundle of LV is thus an (infinite-dimensional) c*
*omplex
T-equivariant vector bundle; in fact it is the free loopspace of T V . If V is *
*simply-
connected then the free loopspace will be connected; from now on I will assume
this.
Connections on vector bundles pull back, so a connection r on the tangent bundle
of V pulls back to a connection on oe*T V . Applied to the standard vector fiel*
*d d=d`
on the circle, the connection defines a derivation
Doe: ToeLV ! ToeLV
on the tangent space. It is easy to see that the diagram
ToeLV__Doe__//ToeLV
|ff*| ff*||
fflffl|DoeOfffflffl|
ToeOffLV____//ToeOffLV
commutes, or in other words that
ff*Doe= DoeOffff* .
If ijk(x) are the connection coefficients in local coordinates at x 2 V , then
Doevi(`) = `vi(`) + ijk(oe(`))o`ek(`)vj(`) = `vi(`) + Bij(`)vj(`)
and hence
DoeOffff*(v)i(`) = `vi(` + ff) + Bij(` + ff)vj(` + ff) ,
which is just ff*(Doev)i(`).
1.2 A Hermitian metric h on T V defines an inner product
Z
(v, w) = (v, w)h(`) d`
S1
on ToeLV . Given such a metric, let r be its associated connection; then [7 III*
* x2.1]
Z
(Doev, w) + (v, Doew) = d(v, w)h(`)) = 0 ,
S1
which is to say that Doeis a skew-adjoint differential operator acting on secti*
*ons
of a Hermitian vector bundle. The circle is compact and one-dimensional, so Doe*
*is
elliptic, with discrete (purely imaginary) spectrum.
If C is a closed subset of this spectrum, let PC be the projection onto the spa*
*n of
the eigenvectors with eigenvalues in C. When C = {h} is a single eigenvalue, Ph
will denote the projection onto the space of associated eigenvectors; thus
X
PC = Pc : ToeLV ! ToeLV
c2C
if C is finite.
If v is an eigenvector of -iDoewith eigenvalue h, then for any integer k,
Doe(v(`)eik`) = Doe(v(`))eik`+ ikv(`)eik`= i(h + k)v(`)eik`,
THE TANGENT BUNDLE OF AN ALMOST-COMPLEX FREE LOOPSPACE 3
so v(`)eik`will also be an eigenvector. The eigenvalues of -iDoetherefore fall *
*into
a (finite) set of equivalence classes in R=Z, and any v 2 ToeLV has a Fourier-l*
*ike
decomposition
X
v(`) ~ vk(`)eik`
k2Z
with coefficients in the (finite-dimensional) vector space ToeLX|[0,1)spanned by
eigenvectors of -iDoewith eigenvalues in the half-open interval [0, 1). Since
ff* O DoeO ff-1*= DoeOff,
eigenvectors of Doemap to eigenvectors of DoeOff, so the rotation operator ff* *
*pre-
serves this vector space. Up to completion there is thus a decomposition, natur*
*al
in T, of ToeLV as a sum
ToeLV |[0,1) ( k2ZC(k))
where C(k) is the complex one-dimensional representation of the circle in which
rotation by ff acts as multiplication by exp(ikff).
When the metric h is flat this Fourier decomposition is a familiar construction*
* (as
it is over the space of constant loops), but the interval [0, 1) is not closed,*
* so it is
not clear that this decomposition behaves well as the loop varies. That questio*
*n is
the topic of this note.
1.3 The construction sketched above is classical, but it is usually formulated *
*in
terms of holonomy [1]: a loop oe in V defines a periodic map
~oe: R ! R=2ßZ ! V
of the line to V , and if v is an eigenvalue of -iDoewith eigenvalue h, then
~v(t) = v(t)e(-ht=2ß)
(where e(t) abbreviates exp(2ßit)) is a section of the vector bundle ~oe*T V ov*
*er the
line, which satisfies the first-order differential equation D~oe~v= 0 . This se*
*ction is
consequently determined by its initial conditions: clearly ~v(0) = v(0), while
~v(2ß) = e(-h)v(0) ;
in other words, parallel transport (using the connection r) of the vector v(0) *
*around
the loop oe results in a twist by e(-h). The space ToeLV |[0,1)spanned by eigen*
*vectors
of -iDoewith eigenvalues in [0, 1) is thus isomorphic to the tangent space Toe(*
*0)V
of V at the basepoint oe(0), by the map which assigns to an initial vector, its
continuation by parallel transport around the loop. In this picture, exp(iDoe)*
* is
the holonomy operator, and the correspondence v 7! ~vsends eigenvectors of Doeto
eigenvectors of holonomy. This justifies the claim made above, that ToeLV |[0,1*
*)is
finite-dimensional: in fact it has complex dimension n.
1.4 A choice of connection on V thus assigns to a loop oe, a unitary automorphi*
*sm
of Toe(0)V . This defines the classical holonomy map
LV ! Un
which can also be interpreted as the composition
LV ! L(BUn) ~=Un x BUn ! Un
4 JACK MORAVA
obtained from the free loops on the classifying map for the complex vector bund*
*le
T V (think of BUn as a universal space for complex vector bundles with Hermitian
connection). Stably, the composition
LV ! Un U ! U=O
classifies the canonical polarization of T LV , that being an equivalence class*
* [2 x2]
of decompositions
T LV = T+ LV T- LV
of the tangent bundle, generalizing the classical decomposition of Fourier modes
into positive and negative frequencies.
Rotating the loop moves the basepoint, which changes the holonomy operator by
conjugation. It follows that the quotient map to the space ^Unof unitary conjug*
*acy
classes is invariant under T-translation. The space of conjugacy classes in a c*
*on-
nected Lie group is just the quotient of a maximal torus by the Weyl group, whi*
*ch
in this case is the space Tn= n of unordered n-tuples of points on the circle.
1.5 The theorem of Hurewicz implies that
ß1(LV ) ~=H2(V, Z) ,
so LV will not be simply-connected in general. The T-action on the loopspace li*
*fts
to an action on its universal cover, which commutes with the action of H2(V, Z)*
* by
covering transformations: consider the space of maps of a two-disk to V , modulo
the equivalence relation which identifies two maps if they agree on their bound*
*ary
circles, and if furthermore the map they then define, from a two-sphere to V , *
*is
nullhomotopic. The circle acts on this model by rotating the disks, while the s*
*econd
homotopy group of V acts by attaching a bubble at the center of a disk.
The fixed-point set LV Tconsists of constant loops, but the fixed point set of *
*the
circle action on the universal cover is a disjoint union of copies of V , index*
*ed
by H2(X, Z). The space of maps to a fibration is again a fibration, so the Hopf
2
construction implies that LS3 is a circle bundle over fLS; but the obvious circ*
*le
action on the domanin has S3 as its fixed-point set, so the circle action on the
universal cover can be more complicated than one might think.
On fundamental groups the holonomy map induces the homomorphism [2 x2]
b 7! 2c1(V ) . b : H2(V, Z) ! Z
defined by the Kronecker product with the first Chern class of T V ; if the Che*
*rn
class is rationally nontrivial, this defines an infinite cyclic cover gLVand a *
*map
gLV ! R x SU=SO ! R
which is equivariant with respect to covering transformations, such that a gene*
*rator
q of that group acts on R as translation by d = #(coker2c1(V )). I will call gLV
the holonomy cover of LV ; I'm indebted to Graeme Segal, for suggesting that
it is this cover, rather than the universal one, which is particularly interest*
*ing. It
will be convenient to extend this terminology to include LV itself as the holon*
*omy
cover, when the Chern class is (rationally) trivial.
THE TANGENT BUNDLE OF AN ALMOST-COMPLEX FREE LOOPSPACE 5
2. Morton's logarithm
2.0 The main result of this paragraph is the construction of a section
flog: fSPn(T) ! SPn(R)
of the quotient map from the space of unordered points on the line, to the univ*
*ersal
cover of the corresponding space of points on the circle. Soon after it was wri*
*tten, I
learned from Elmer Rees that it duplicates the argument of [3], published in 19*
*67.
As the construction is not very long, I have left it unchanged.
2.1 The nth symmetric power SPn(X) of a space X is the quotient Xn= n of the
n-fold cartesian product of copies of X by the symmetric group n. The action
of the symmetric group on Rn defined by permuting coordinates has an invariant
codimension one subspace Rn-10consisting of vectors whose coordinates sum to
zero, so we can decompose the symmetric power of the real line as a product
SPn(R) ~=Rn-10= n x R ,
where projection onto the last coordinate sends an n-tuple of real numbers to i*
*ts
average. Let
X
v 7! {vk} : Rn ! SPn(R)
denote the quotient map. The exponential
t 7! e(t) : R ! R=Z = T
from the line to the circle defines a map
e : SPn(R) ! SPn(T)
P P
which sends the configuration {vk} in R to the configuration [vk] in T. The
summation map
X X
[vk] 7! [ vk] : SPn(T) ! T
is a homotopy equivalence, for T is homotopy-equivalent to Cx , and the map
Y
{zi} 2 Cx 7! (z - zi) : SP n(Cx ) ! Cn-1 x Cx
[1 i n
which assigns to an unordered n-tuple of nonvanishing complex numbers, the monic
polynomial with those elements as its zeros, is a homeomorphism. It follows that
the universal cover
X X
fSPn(T) = {( [tk], t) 2 SPn(T) x R | t tk mod Z}
of SPn(T) splits, via the map
X X
( [tk], t) 7! ( [tk - t=n], t) : fSPn(T) ~=SP0n(T) x R
as a product.
2.2 A point of the standard simplex n-1 is an ordered n-tuple x = (x1, . .,.xn)
of real numbers between zero and one, subject to the constraint
X
xk = 1 ;
6 JACK MORAVA
if we define
X
m(x) = n-1 kxk ,
1 k n
then the sequence t1 = m(x), t2 = x1 + m(x), . .,. tn-1 = xn-2 + . .+.x1 +
m(x), tn = m(x) - xn of real numbers satisfies
X
ti= n - 1 ,
with nonnegative interpoint distances
t2 - t1 = x1, t3 - t2 = x2, . .,.t1 - tn = xn .
If oe(x) = (x2, . .,.xn, x1) then
m(oe(x)) = m(x) + x1 - 1=n ,
so the sequence
m(x) + x1 - 1=n, m(x) + x2 + x1 - 1=n, . .,.m(x) - 1=n
associated to oe(x) differs by a cyclic shift and a translation by 1=n from the*
* sequence
defined by x.
2.3 Let
X
x 7! ~(x) = {tk} : n-1 ! SPn(R) .
denote the result of forgetting the order on the points in this construction. *
*Its
composition
x 7! e(~(x)) : n-1 ! SP0n(T)
with the exponential takes values in the subspace of configurations on the circ*
*le
which sum to zero. The interior of the simplex maps onto the subspace of zero-
sum configurations with multiplicities at most one: indeed, any configuration of
n distinct points on the circle defines a sequence ø1, . .,.øn 2 T, ordered cou*
*nter-
clockwise from zero, with interpoint distances xk = øk+1 - øk defining a point *
*x in
the interior of the simplex. If we interpret these ø's to be real numbers in th*
*e unit
interval, then
X
m(x) = n-1 k(øk+1 - øk) + (ø1 - øn)
1 k n-1
can be rewritten as
X X X
ø1 + n-1[ (k - 1)øk - køk] = ø1 - n-1 øk ;
1 k n 1 k n 1 k n
thusPe O ~ applied to x recovers the original configuration, up to a translatio*
*n by
n-1 øk. The sum of the ø's is by assumption an integer m, so the cyclic shift
oem (x) recovers the original configuration.
A similar argument shows that e O ~ is one-to-one on the interior of the simple*
*x.
Its faces are defined by the vanishing of various barycentric coordinates; thes*
*e faces
map to configurations with points of multiplicity greater than one, so degenera*
*te
configurations which partition n into p parts correspond to faces of codimension
n - p. By induction the map is a bijection on these open faces, and is hence a
bijection on the whole of n-1.
THE TANGENT BUNDLE OF AN ALMOST-COMPLEX FREE LOOPSPACE 7
2.4 It follows that the (well-defined!) composition
~ O [e O ~]-1 : SP0n(T) ! SPn(R)
is a section of the map e restricted to the zero-sum configurations on the circ*
*le; it
defines an analog of the logarithm. If
flog0: SP0n(T) ! Rn-10= n
denotes its composition with projection onto the first factor of SPn(R), then t*
*he
equation
lfog0(X [tk]) = X {tk} .
makes a certain amount of sense, and flog0extends to define a section
flog(X [tk], t) = (X {tk - t=n}, t=n) : fSPn(T) ! SPn(R)
of the lift
X X X
{vk} 7! ~e( {vk}) = ( [vk], nv) : SPn(R) ! fSPn(T)
of e to the universal cover (v being the average of the vk).
The map e is equivariant with respect to the translation actions of R and T on
SPn(R) and SPn(T), so its lift ~eis R-equivariant as well. The composition e O ~
(or, more correctly, its inverse) is essentially just an identification of the *
*quotient
of SP n(T) by the circle action, so it is unreasonable to hope that the sections
constructed here might be R-equivariant.
Nevertheless, the map flogis equivariant with respect to translation by the sub*
*group
Z of R.
3. A decomposition of the tangent bundle
3.1 Suppose, then, that V is a connected and simply-connected almost complex
manifold of real dimension 2n. The holonomy of the connection associated to a
Hermitian metric on V defines a T-invariant map
^H: LV ! ^Un= SPn(T) ,
and thus a lifting
~H: gLV! fSPn(T)
to a map of covers. The composition
j = flogO ~H: gLV! SPn(R)
continuously assigns to an element oe0of the cover, a choice of n real eigenval*
*ues of
-iDoe, whose eigenvectors (up to suitable twists by C(k)) span ToeLV |[0,1). Let
oe07! oe0:= h2j(oe0)Ph : Toe0gLV! Toe0gLV
be the function which assigns to oe0, the projection onto the span of the eigen*
*vectors
supported in j(oe0). This is a continuous T-equivariant family of projections, *
*and
its image
TV = Image T gLV
8 JACK MORAVA
is a T-equivariant complex n-plane bundle over gLV. The equivariant bundle mono*
*mor-
phism
TV ( k2ZC(k)) ! T gLV
is a global analog of the local Fourier expansion in x1.2.
3.2 Covering translations will not preserve this decomposition, but x2.4 implie*
*s the
existence of an equivariant isomorphism
q*TV ~=TV C(d) ,
such that the extension [6 x6.5]
1 ! Cx ! H ! Z x T ! 0
defined by the bilinear form
(k0, c0), (k1, c1) 7! ck10c-k01
on Z x T acts on the polarization
T gLV= TV ( k , 0C(k)) ,
by a lift of the action of Z x T on gLV.
Acknowledgements This note began in conversations with Ralph Cohen during
the August 2000 conference on equivariant homotopy theory at Stanford. I also
owe a great deal to conversations with Matthew Ando, and to Tom Goodwillie and
Graeme Segal (for trying to straighten out my thinking about holonomy). I owe
Jim Martino ten million dollars in thanks for help with MAPLE calculations at a
crucial stage in this project. I would also like to thank Clarence Wilkerson an*
*d Bill
Dwyer for encouraging correspondence, and Elmer Rees for telling me about [3,4].
That this note exists at all is due to Tom Mrowka, who saw that I had mistakenly
assumed [e.g. in the case V = S2] that the holonomy map was inessential; many
thanks to him for taking the matter seriously.
References
1. M. Berger, Sur les groupes d'holonomie homog'ene des vari'et'es a connexion *
*affine et des vari'et'es
riemanniennes, Bull. Soc. Math. France 83 (1955) 279-330
2. R.L. Cohen, J.D.S. Jones, G.B. Segal, Floer's infinite dimensional Morse the*
*ory and homotopy
theory, in The Floer Memorial Volume, Birkhäuser, Progress in Mathematics 13*
*3 (1995)
297-326
3. H.R. Morton, Symmetric products of the circle, Proc. Cambridge Phil. Soc. 63*
* (1967) 349-352
4. J. Mostovoy, Geometry of truncated symmetric products and real roots of real*
* polynomials,
Bull. London Math. Soc. 30 (1998) 159-165
5. R. Palais, Foundations of global nonlinear analysis, Benjamin (1968)
6. C. Voisin, Mirror Symmetry, SMF/AMS Texts (1999)
7. R.O. Wells, Differential analysis on complex manifolds, Springer GTM 65 (198*
*0)
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: jack@math.jhu.edu