THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS



                   NITU KITCHLOO AND JACK MORAVA


       Abstract.This is very much an account of work in progress.  We sketch
       the construction of an Atiyah dual (in the category of T-spaces) for the*
 * free
       loopspace of a manifold; the main technical tool is a kind of Tits build*
 *ing for
       loop groups, discussed in detail in an appendix. Together with a new loc*
 *al-
       ization theorem for T-equivariant K-theory, this yields a construction o*
 *f the
       elliptic genus in the string topology framework of Chas-Sullivan, Cohen-*
 *Jones,
       Dwyer, Klein, and others. We also show how the Tits building can be used*
 * to
       construct the dualizing spectrum of the loop group. Using a tentative no*
 *tion
       of equivariant K-theory for loop groups, we relate the equivariant K-the*
 *ory of
       the dualizing spectrum to recent work of Freed, Hopkins and Teleman.
                            Introduction

If P ! M is a principal bundle with structure group G then LP ! LM is a
principal bundle with structure group

                          LG = Maps(S1, G),

and if the tangent bundle of M is defined by a representation V  of G then the
tangent bundle of LM is defined by the representation LV of LG. The circle group
T acts on all these spaces.


This is a report on the beginnings of a theory of differential topology for such
objects. Note that if we want the structure group LG to be connected, we need G
to be 1-connected; thus SU(n) is preferable to U(n). This helps explain why Cal*
 *abi-
Yau manifolds are so central in string theory, and this note is written assumin*
 *g this
simplifying hypothesis.


Alternately, we could work over the universal cover of LM; then ß2(M) would act
on everything by decktranslations, and our topological invariants become modules
over the Novikov ring Z[H2(M)]. From the point of view we're developing, these
translations seem to be what really underlies modularity, but this issue, like *
 *several
others, will be backgrounded here.


The circle action on the free loopspace defines a structure much closer to clas*
 *si-
cal differential geometry than one finds on more general (eg) Hilbert manifolds;
this action defines something like a Fourier filtration on the tangent space of*
 * this
infinite-dimensional manifold, which is in some sense locally finite. This lead*
 *s to
a host of new kinds of geometric invariants, such as the Witten genus; but this
filtration is unfamiliar, and has been difficult to work with [17]. The main co*
 *ncep-
tual result of this note [which was motivated by ideas of Cohen, Godin, and Seg*
 *al]
is the definition of a canonical equivariant `thickening' of a free loopspace, *
 *where
____________
   Date: 1 May 2004.
   1991 Mathematics Subject Classification. 22E6, 55N34, 55P35.
                                  1




2                   NITU KITCHLOO AND JACK MORAVA


the pulled-back tangent bundle admits a canonical filtration by finite-dimensio*
 *nal
equivariant bundles.  This thickening involves a contractible LG-space called t*
 *he
affine Tits building A (LG). This space occurs under various guises in nature: *
 *it
is a homotopy colimit of homogeneous spaces with respect to a finite collection*
 * of
compact Lie subgroups of LG. It is also the affine space of principal G-connect*
 *ions
on the trivial bundle over S1. We explore its structure in the appendix.


In the first section below, we recall why the Spanier-Whitehead dual of a finit*
 *e CW-
space is a ring-spectrum, and sketch the construction (due to Milnor and Spanie*
 *r,
and Atiyah) of a model for that dual, when the space is a smooth compact manifo*
 *ld.
Our goal is to produce an analog of this construction for a free loopspace, whi*
 *ch
captures as much as possible of its string-topological algebraic structure.  In*
 * the
second section, we introduce the technology used in our construction: pro-spect*
 *ra
associated to filtered infinite-dimensional vector bundles, and the topological*
 * Tits
building which leads to the construction of such a filtration for the tangent b*
 *undle.
In x3 we observe that recent work of Freed, Hopkins, and Teleman on the Verlinde
algebra can be reformulated as a conjectural duality between LG-equivariant K-
theory of a certain dualizing spectrum for LG constructed from its Tits buildin*
 *g,
and positive-energy representations of LG. In x4 we use a new strong localizati*
 *on
theorem to study the equivariant K-theory of our construction, and we show how
this recovers the Witten genus from a string-topological point of view.


We plan to discuss actions of various string-topological operads [15] on our co*
 *n-
struction in a later paper; that work is in progress.


We would like to thank R. Cohen, V. Godin, and A. Stacey for many helpful
conversations, and we would also like to acknowledge work of G. Segal and S.
Mitchell as motivation for many of the ideas in this paper.



                  1.The Atiyah dual of a manifold

If X is a finite complex, then the function spectrum F (X, S0) is a ring-spectr*
 *um
(because S0 is). If X is a manifold M, Spanier-Whitehead duality says that

                         F (M+ , S0) ~ M-TM  .

If E ! X is a vector bundle over a compact space, we can define its Thom space
to be the one-point compactification

                              XE := E+ .

There is always a vector bundle E? over X such that

                             E   E? ~=1N

is trivial, and following Atiyah, we write

                           X-E := S-N XE? .

With this notation, the Thom collapse map for an embedding M   RN is a map

                     SN = RN+! M  = SN M-TM ,




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS           3


and the midpoint construction

                         M+ ^ M  ! RN+= SN

defines the equivalence with the functional dual.  More generally, a smooth map
f : M ! N of compact closed orientable manifolds has a Pontrjagin-Thom dual
map
                         fPT : N-TN  ! M-TM

of spectra; in particular, the map S0 ! M-TM  dual to the projection to a point
defines a kind of fundamental class, and the dual to the diagonal of M makes
M-TM  into a ring-spectrum [13].

Prospectus: Chas and Sullivan [10] have constructed a very interesting product
on the homology of a free loopspace, suitably desuspended, motivated by string
theory. Cohen and Jones [15] saw that this product comes from a ring-spectrum
structure on                           *
                         LM-TM  := LM-e TM

where
                             e : LM ! M

is the evaluation map at 1 2 S1.  Unfortunately this evaluation map is not T-
equivariant, so the Chas-Sullivan Cohen-Jones spectrum is not in general a T-
spectrum.  The full Atiyah dual constructed below promises to capture some of
this equivariant structure. The Chas-Sullivan Cohen-Johes spectrum and the full
Atiyah dual live in rather different worlds: our prospectrum is an equivariant *
 *object,
whose multiplicative properties are not year clear, while the CSCJ spectrum has
good multiplicative properties, but it is not a T-spectrum.  In some vague sense
our object resembles a kind of center for the Chas-Sullivan-Cohen-Jones spectru*
 *m,
and we hope that a better understanding of the relation between open and closed
strings will make it possible to say something more explict about this.



                      2. Problems & Solutions

For our constructions, we need two pieces of technology:

Cohen, Jones, and Segal [16](appendix) associate to a filtration

                        E : . . .Ei  Ei+1  . . .

of an infinite-dimensional vector bundle over X, a pro-object

                   X-E : . .!.X-Ei+1 ! X-Ei ! . . .

in the category of spectra. [A rigid model for such an object can be constructed
by taking E to be a bundle of Hilbert spaces, which are trivializable by Kuiper*
 *'s
theorem. Choose a trivialization E ~=H x X and an exhaustive filtration {Hk} of
H by finite-dimensional vector spaces; then we can define
                                           ?
                       X-Ei = limS-Hk XHk\Ei  ,

with E?ithe orthogonal complement of Ei in the trivialized bundle E.] This pro-
object will, in general, depend on the choice of filtration. We will be interes*
 *ted in the
direct systems associated to such a pro-object by a cohomology theory; of course
in general the colimit of this system can be very different from the cohomology*
 * of
the limit of the system of pro-objects.




4                   NITU KITCHLOO AND JACK MORAVA


Example 2.1. If X = CP1 , j is the Hopf bundle, and E is

                 1j : . . .(k - 1)j   kj   (k + 1)j   . . .

then the induced maps of cohomology groups are multiplication by the Euler clas*
 *ses
of the bundles Ei+1=Ei, so

             H*(CP1-1j, Z) := colim{Z[t], t - mult} = Z[t, t-1] .

We would like to apply such a construction to the tangent bundle of a free loop*
 *space.
Unfortunately, these tangent bundles do not, in general, possess any such nice
filtration by finite-dimensional (T-equivariant) subbundles [17]!  However, suc*
 *h a
splitting does exist in a neighborhood of the constant loops:

                           M = LMT   LM

has normal bundle

                    (M   LM) = T M  C (C[q, q-1]=C)

(at least, up to completions; and assuming things complex for convenience). Here
small perturbations of a constant loop are identified with their Fourier expans*
 *ions
                               X
                                  anqn,
                               n2Z

with q = ei`. The related fact, that T LM is defined by the representation LV of
LG looped up from the finite-dimensional representation V of G, will be importa*
 *nt
below: for LV is not a positive-energy representation of LG.


The main step toward our resolution of this problem depends on the following re*
 *sult,
proved in x7 below.  Such constructions were first studied by Quillen, and were
explored further by S. Mitchell [26]. The first author has studied these buildi*
 *ngs
for a general Kac-Moody group [23]; most of the properties of the affine buildi*
 *ng
used below hold for this larger class.



Theorem 2.2. The topological affine Tits building

                       A(LG) := hocolimILG=HI

of LG is Tx~LG-equivariantly contractible.  In other words, given any compact
subgroup K   Tx~LG, the fixed point space A(LG)K is contractible.

[Here I runs over certain proper subsets of roots of G, and the HI are certain
compact `parabolic' subgroups of LG (see x7.2).]

Remark 2.3. The group LG admits a universal central extension LG. The natural
action of the rotation group T on LG lifts to LG, and the T-action preserves the
subgroups HI. Hence A(LG) admits an action of Tx~LG, with the center acting
trivially. We can therefore express A(LG) as

                       A(LG) = hocolimILG=HI

where HI is the induced central extension of HI.




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS           5


                    Other descriptions of A(LG)

This Tits building has other descriptions as well. For example:

1. A(LG) can be seen as the classifying space for proper actions with respect to
the class of compact Lie subgroups of Tx~LG.

2. It also admits a more differential-geometric description as the smooth infin*
 *ite
dimensional manifold of holonomies on S1 x G (see Appendix): Let S denote the
subset of the space of smooth maps from R to G given by

            S = {g(t) : R ! G, g(0) = 1, g(t + 1) = g(t) . g(1)} ;

then S is homeomorphic to A(LG). The action of h(t) 2 LG on g(t) is given by
hg(t) = h(t) . g(t) . h(0)-1, where we identify the circle with R=Z. The action*
 * of
x 2 R=Z = T is given by xg(t) = g(t + x) . g(x)-1.

3. The description given above shows that A(LG) is equivalent to the affine spa*
 *ce
A(S1 x G) of connections on the trivial G-bundle S1 x G.  This identification
associates to the function f(t) 2 S, the connection f0(t)f(t)-1.  Conversely, t*
 *he
connection rton S1xG defines the function f(t) given by transporting the element
(0, 1) 2 R x G to the point (t, f(t)) 2 R x G using the connection rt pulled ba*
 *ck
to the trivial bundle R x G.

Remark 2.4. These equivalent descriptions have various useful consequences. For
example, the model given by the space S of holonomies says that given a finite
cyclic group H   T, the fixed point space SH  is homeomorphic to S. Moreover,
this is a homeomorphism of LG-spaces, where we consider SH as an LG-space and
identify LG with LGH in the obvious way. Notice also that ST is G-homeomorphic
to the model of the adjoint representation of G defined by Hom (R, G).

Similarly, the map S ! G given by evaluation at t = 1 is a principal  G bundle,
and the action of G = LG= G on the base G is given by conjugation. This allows *
 *us
to relate our work to that of Freed, Hopkins and Teleman in the following secti*
 *on.

Finally, the description of A(LG) as the affine space A(S1 x G) implies that the
fixed point space A(LG)K is contractible for any compact subgroup K   Tx~LG.



If E ! B is a principal bundle with structure group LG, then (motivated by ideas
of [14]) we construct a `thickening' of B:

Definition 2.5. The thickening of B associated to the bundle E is the LG-space

                B!(E) = E xLG A(LG) = hocolimIE=HI .

We will omit E from the notation, when the defining bundle is clear from contex*
 *t.

Remark 2.6. If P ! M is a principal G bundle, then LP ! LM is a principal
LG bundle. In this case, the description above gives L!M := LM!(LP ) a smooth
structure:
                L!M = {(fl, !) |  fl 2 LM,  ! 2 A(fl*(P ))}

where A(fl*(P )) is the space of connections on the pullback bundle fl*(P ).



Let Tx~LG be the extension of the central extension of LG by T acting as rota-
tions. On restriction to the subgroup Tx~HI, a unitary representation U of Tx~LG




6                   NITU KITCHLOO AND JACK MORAVA


decomposes into a sum of finite dimensional representations. We want to constru*
 *ct
a Thom Tx~LG-prospectrum A(LG)-U .


We consider the decomposition of the restriction of U to Tx~HI as a sum of irre-
ducibles:
                           U|T~xHI~=  UI(ff)

and let
                   UI(k) =   {UI(ff) | dim UI(ff)   k} .

Then we can define S-UI to be the Thom Tx~HI-prospectrum associated to the
filtered (equivariant) vector bundle

                    UI : . . .UI(k)   UI(k + 1)   . . .

over a point. If I   J then HI maps naturally to HJ, and there is a correspondi*
 *ng
morphism
                              UJ ! UI

of filtered vector bundles, given by inclusions UJ(k) ! UI(k).



Definition 2.7. We define A(LG)-U to be the Tx~LG-prospectrum

                  A(LG)-U = hocolimILG+ ^HI S-UI,

where LG+ denotes LG, with a disjoint basepoint.

Homotopy colimits in the category of prospectra can be defined in general, using
the model category structure of [11].



Remark 2.8. Given any principal LG-bundle E ! B, and a representation U
of LG, we define the Thom prospectrum of the virtual bundle associated to the
representation -U to be

            B-U!= E+ ^LG A(LG)-U = hocolimIE+ ^HI S-UI.

In particular, if P is the refinement of the frame bundle of M via a representa*
 *tion
V of G, then the tangent bundle of LM is defined by the representation LV of LG.

Definition 2.9. The Atiyah dual LM-TLM  of LM is the pro-spectrum L!M-LV .

We will explore this object further in x6.


                  3. The dualizing spectrum of LG

The dualizing spectrum of a topological group K is defined [24] as the K-homoto*
 *py
fixed point spectrum:
                      DK = KhK+= F (EK+ , K+ )K

where K+ is the suspension spectrum of the space K+ , endowed with a right K-
action. The dualizing spectrum DK admits a K-action given by the residual left
K-action on K+ . If K is a compact Lie group, then it is known that DK is the o*
 *ne
point compactification of the adjoint representation SAd(K). It is also known t*
 *hat
there is a K x K-equivariant homotopy equivalence

                          K+ ~=F (K+ , DK ) .




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS           7


It follows from the compactness of K+ that for any free K+ -spectrum E, we have
the K-equivariant homotopy equivalence


                        E ~=F (K+ , E ^K+ DK ) .


It is our plan to understand the dualizing spectrum for the (central extension *
 *of
the) loop group.


Theorem 3.1. There is an equivalence


                     DLG ~=holimILG+ ^HI SAd(HI)


of left LG-spectra.


Proof.We have the sequence of equivalences:


         DLG = F (ELG+ , LG+ )LG ~=F (ELG+ ^ A(LG)+ , LG+ )LG .


The final space may be written as


       holimIF (ELG+ ^HI LG+ , LG+ )LG = holimIF (ELG+ , LG+ )HI .


Now recall the equivalence of HI x HI-spectra:


(1)                  LG+ ~=F (HI +, LG+ ^HI DHI) .


Taking HI-homotopy fixed points implies a left HI-equivalence


            F (ELG+ , LG+ )HI = (LG+ )hHI ~=LG+ ^HI SAd(HI);


where we have used equation (1) at the end. Replacing this term into the homoto*
 *py
limit completes the proof.


Similarly, we have:


Theorem 3.2. There is an equivalence


                     DLG ~=holimILG+ ^HI SAd(HI)


of left LG-spectra.


Remark 3.3. The diagram underlying DLG or DLG can be constructed in the cat-
egory of spaces. Given an inclusion I   J, the orbit of a suitable element in A*
 *d(HJ)
gives an embedding HJ=HI   Ad(HJ), and the Pontrjagin-Thom construction for
this embedding defines an HJ-equivariant map


                      SAd(HJ)-! HJ+ ^HI SAd(HI)


which extends to the map


                  LG+ ^HJ SAd(HJ)-! LG+ ^HI SAd(HI)


required for the diagram. Moreover, composites of these maps can be made com-
patible up to homotopy.




8                   NITU KITCHLOO AND JACK MORAVA


  A conjectural relationship with the work of Freed, Hopkins and
                              Teleman

The discussion below assumes the existence of a hypothetical LG -equivariant K-
theory, whose value on a point in degree zero is the Grothendieck group of posi*
 *tive
energy representations (or equivalently, the group of characters of integrable *
 *repre-
sentations). The symmetric monoidal category whose objects are finite direct su*
 *ms
of irreducible positive energy representations, and whose morphism spaces consi*
 *st
of the (nonequivariant) isomorphisms of the vector spaces underlying the repres*
 *en-
tation (given the compactly generated topology) defines a candidate for a spect*
 *rum
representing such a functor: this is a topological category with an LG -action *
 *which
respects the symmetric monoidal structure.

This hypothesis provides us with a convenient language. We expect to return to
the underlying technical issues in a later paper.

The center of LG acts trivially on DLG, defining a second grading on K*LG(DLG);
we will use a formal variable z to keep track of the grading, so
                                 M   *,n
                     K*LG(DLG) =    KLG (DLG)zn.
                                  n
The spectral sequence for the cohomology of a cosimplicial space, in the case of
K*LG(DLG), has
                      Ei,j2= colimiIKjHI(SAd(HI)) .

This spectral sequence respects the second grading given by powers of z. In a s*
 *equel
to this paper, we will show that this spectral sequence collapses to give

          K*LG(DLG) = colimIK*HI(SAd(HI)) ~=colimIK*-r-1HI(pt) ,

where r is the rank of G. Therefore, this group admits a natural Thom class giv*
 *en
the system {S(Ad(HI))} of spinor bundles for the adjoint representations of the
parabolics HI. In section 11 of [19], the authors construct an explicit map bet*
 *ween
the Verlinde algebra and this colimit, as follows:

To a positive energy representation corresponding to a dominant character ~, we
associate the LG-equivariant bundle given by L-~-æ   S(N), where L-~-æ is the
canonical line bundle over the coadjoint orbit of the regular element ~ + æ, and
S(N) is the spinor bundle of the normal bundle to the coadjoint orbit.  Such an
orbit is of the form LG=H for some parabolic subgroup H, and its normal bundle
is Ad(H), so this element defines a class in KLG(DLG). The same can be done for
antidominant weights. This suggests the following:

Conjecture 3.4. For following map is an isomorphism in homogeneous degree zn,
for n 6= 0:
 M         M                                           M   *,n
    Vkzk+h    Vkz-(k+h)~=colimIKHI(pt) ! Kr+1LG(DLG) =    KLG (DLG)zn
 k 0       k 0                                          n

where Vk is the Verlinde algebra of level k, h is the dual Coxeter number of G,*
 * and
r is its rank.

Example 3.5. To illustrate this in an example, consider the case G = SU(2). In
this case r = 1, h(G) = 2. Here the groups HI are given by

      H0 = SU(2) x S1,  H1 = S1 x SU(2),  H0 \ H1 = T = S1 x S1 .




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS           9


The respective representation rings may be identified by restriction with subal*
 *gebras
of KT(pt) = Z[u 1, z 1] :

         KH0(pt) = Z[u + u-1, (z=u) 1],  KH1(pt) = Z[z 1, u + u-1] .

Now consider the two pushforward maps involved in the colimit:

              '0 : KT(pt) ! KH0(pt),  '1 : KT(pt) ! KH1(pt)

A quick calculation shows that for k > 0, we have
                        (     k   k      -1

               'j(zk) =  (z=u) Sym (u + u  ),  j = 0
                         zk,                    j = 1
                         (     k   k-1      -1

             'j(zku-1) =  (z=u) Sym   (u + u  ),  j = 0
                          0,                      j = 1 ,

where Symk(V ) denotes the k-th symmetric power of the representation V , e.g.
Symk(u + u-1) = uk + . .+.u-k.


We also have a similar formula for negative exponents:
                       (      k   k-2      -1

             'j(z-k) =  -(u=z) Sym   (u + u  ),  j = 0
                        z-k,                      j = 1
                        (      k   k-1      -1

           'j(z-ku-1) =  -(u=z) Sym   (u + u  ),  j = 0
                         0,                        j = 1 .

The colimit is the cokernel of

                 '1   '0 : KT(pt) -! KH1(pt)   KH0(pt)

Now consider the decomposition

            Z[u 1, z 1] = Z[u + u-1, z 1]   u-1Z[u + u-1, z 1] .

It is easy to check from this that the cokernel for nontrivial powers of z is i*
 *somorphic
to the cokernel of '0 restricted to u-1Z[u + u-1, z 1] and hence is

       M      Z[u + u-1]                M       Z[u + u-1]
          _________________k+1-1(z=u)k+2    _________________k+1-1(u=z)k+2

       k 0<Sym   (u + u  )>              k 0<Sym   (u + u  )>

which agrees with the classical result [18].

Remark 3.6. We can calculate the equivariant K-homology KLG *(A(LG)) using
the same spectral sequence. This establishes an isomorphism between K*LG(DLG)
and KLG *(A(LG)).  Results of [19] suggest that the latter group calculates the
Verlinde algebra, which is yet another motivation for the conjecture.  Recall a*
 *lso
that A(LG) is the classifying space for proper actions (i.e. with compact isotr*
 *opy)
so our conjecture is a topological analog of the Baum-Connes conjecture for fin*
 *ite
groups [7]

Question. Given a manifold LM, with frame bundle LP , we can construct a
spectrum
                    DLM  := holimILP+ ^HI SAd(HI)

It would be very interesting to understand something about KT(DLM ).




10                  NITU KITCHLOO AND JACK MORAVA


                      4.Localization Theorems

If E is a T-equivariant complex-oriented multiplicative cohomology theory, and *
 *X is
a T-space, we have contravariant (j*) and covariant (j!) homomorphisms associat*
 *ed
to the fixedpoint inclusion
                             j : XT   X ,

satisfying
                          j*j!(x) = x . eT( ) ;

if the Euler class of the normal bundle   is invertible, this leads to a close *
 *relation
between the cohomology of X and XT.


More generally, if f : M ! N is an equivariant map, then its Pontrjagin-Thom
transfer is related to the analogous transfer defined by its restriction

                            fT : MT ! NT

to the fixedpoint spaces, by a `clean intersection' formula:

                  j*NO f!(-) = fT!(j*M(-) . eT( (f)|MT )) .

Definition 4.1. The fixed-point orientation defined by the Thom class

                   Thy( (fT)) = Th( (fT)) . eT( (f)|MT

for the normal bundle of the inclusion of fixed-point spaces is the product of *
 *the usual
Thom class with the equivariant Euler class of the full normal bundle restricte*
 *d to
the fixed-point space.

Since fT!(-) = fT*PT(- . Th( (fT))), in this new notation the clean intersection
formula becomes
                           j*NO f! = fTy O j*M

with a new Pontrjagin-Thom transfer

                     fTy(-) = fT*PT(- . Thy( (fT))) .

In the case of most interest to us (free loopspaces), we identified the normal *
 *bundle
above, in x2; using that description, we have
                                 Y
                eT( (M   LM)) =       (e(Li) +E [k](q)),
                                06=k2Z;i

where the Li are the line bundles in a formal decomposition of T M, q is the Eu*
 *ler
class of the standard one-dimensional complex representation of T, and +E is the
sum with respect to the formal group law defined by the orientation of E. It may
not be immediately obvious, but it turns out that such a formula implies that t*
 *he
fixed-point orientation defined above will have good multiplicative properties.


Such Weierstrass products sometimes behave better when `renormalized', by divid-
ing by their values on constant bundles [2].  If E is KT with the usual complex
(Todd) orientation, we have

                       e(L) +K [k](q) = 1 - qkL ;

but for our purposes things turn out better with the Atiyah-Bott-Shapiro spin
orientation; in that case the corresponding Euler class is

                         (qkL)1=2- (qkL)-1=2.




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS          11


The square roots make sense under the simple-connectivity assumptions on G men-
tioned in the introduction:


To be precise, let V be a representation of LG with an intertwining action of T*
 *. We
restrict ourselves to representations V which (for lack of a better name [?]) w*
 *e call
symmetric, i.e. such that V is equivalent to the representation of LG obtained *
 *by
composing V with the involution of LG which reverses the orientation of the loo*
 *ps.
The restriction of the representation V to the constant loops T x G   Tx~LG has
a decomposition                    X

                          V = V T     Vk qk
                                   k6=0
where Vk are representations of G, and q denotes the fundamental representation
of T. Let V (m) be the finite dimensional subrepresentation
                                    X
                      V (m) = V T        Vk qk ;
                                   0<|k| m

the symmetry assumption implies that Vk = V-k as representations of G, so this
can be rewritten                 X
                   V (m) = V T       Vk (qk   q-k) .
                                0<k m
At this point we need the following



Proposition 4.2. If G is a compact Lie group, and W is an m-dimensional complex
spin representation of G, then the representation W~ = W   (qk   q-k) of T x G
admits a canonical spin structure.

Proof.The representation qk   q-k of T admits a unique spin structure. Since W
is also endowed with a spin structure, the representation W   (qk   q-k) admits*
 * a
canonical spin structure defined by their tensor product.

Remark 4.3. If G is simply connected, then any representation W of G admits a
unique spin structure.

This justifies the square roots of the formal line bundles appearing in the res*
 *triction
of T LM to M. The resulting renormalized Euler class

                        Y  (qkL)1=2- (qkL)-1=2
                           __________________k=2-k=2

                        k6=0   q   - q

is a product of terms of the form

                (1 - qkL)(-qkL)-1=2(q-kL)1=2(1 - qkL-1)

divided by terms of the form

                    (1 - qk)(-qk)-1=2(q-k)1=2(1 - qk) ,

(where now all k's are positive) yielding a unit

                              Y  (1 - qkL)(1 - qkL-1)
                    fflT(L) =    ___________________k2
                              k 1      (1 - q )

in the ring Z[L  ][[q]].




12                  NITU KITCHLOO AND JACK MORAVA


Following 4.1, we can reformulate the localization theorem in terms of a new or*
 *ien-
tation, obtained by multiplying the ABS Thom class by this unit, to get precise*
 *ly
the Mazur-Tate normalization

                                      Y  (1 - qkL)(1 - qkL-1)
             oe(L, q) = (L1=2- L-1=2)    ___________________k2
                                      k 1      (1 - q )

for the Weierstrass sigma-function as Thom class for a line bundle L. This exte*
 *nds
by the splitting principle to define the orientation giving the Witten genus [3*
 *2].

                     5. One moral of the story

Since the early 80's physicists have been trying to interpret

                            M 7! KT(LM)

as a kind of elliptic cohomology theory; but of course we know better, because *
 *we
know that mapping-space constructions (such as free loop spaces) don't preserve
cofibrations.


Now it is an easy exercise in commutative algebra to prove that

                          Z((q)) := Z[[q]][q-1]

is flat over

                             KT = Z[q  ] ,

for the completion of a Noetherian ring, eg Z[q], at an ideal, eg (q), defines *
 *a flat
[6](x10.14) Z[q]-algebra Z[[q]].  Flat modules pull back to flat modules [8] (C*
 *h I
x2.7), so it follows that Z((q)) is flat over the localization

                      Z[q  ] := Z[q][q-1] = Z[q, q-1] .

The Weierstrass product above is a genuine formal power series in q, so for que*
 *stions
involving the Witten genus it is formally easier to work with the functor defin*
 *ed on
finite T-CW spaces by

                  X 7! K*T(X)  Z[q ]Z((q)) := K*^T(X) .

This takes cofibrations to long exact sequence of Z((q))-modules. Its real virt*
 *ue,
however, is that it satisfies a strong localization theorem:



Theorem 5.1. If X is a finite T-CW space, then restriction to the fixedpoints
defines an isomorphism

                         j* : K^T(X) ~=K^T(XT) .

Proof, by skeletal induction; based on the

Lemma 5.2. If C   T is a proper closed subgroup, then

                            K*^T(T=C) = 0 .




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS          13


Proof.If C is cyclic of order n 6= 1, then

        K^T(T=C) = KT(T=C)  Z[q ]Z((q)) = Z[q]=(qn - 1)  Z[q]Z((q))

is zero, since                         X
                       -1 = (qn - 1) .    qnk = 0 .
                                       k 0
On the other hand, if C = {0}, then

                        K^T(T ) = Z  Z[q]Z((q)) ,

with Z a Z[q]-module via the specialization q ! 1; but by a similar argument, t*
 *he
resulting tensor product again vanishes.

The functor K*^Textends to an equivariant cohomology theory on the category of
T-CW spaces, which sends a general (large) object X to the pro-Z((q))-module

                   {K*^T(Xi) | Xi2 finiteT - CW    X} ,

[5] (appendix). We can thus extend the claim above:


Corollary 5.3. For a general T-CW-space X, the restriction-to-fixedpoints map

                           K*^T(X) ! K*^T(XT)

is an isomorphism of pro-objects. Moreover, if the fixedpoint space XT is a fin*
 *ite
CW-space, then the pro-object on the left is isomorphic to the constant pro-obj*
 *ect
on the right.

The free loopspace LX of a CW-space X is weakly T-homotopy equivalent to a
T-CW-space, by a map which preserves the fixed-point structure [25] (x1.1).


Theorem 5.4.
                       M 7! K^T(LM) := KTate(M)

is a cohomology theory, after all!

Remark 5.5. This seems to be what the physicists have been trying to tell us
all along: they probably thought (as the senior author did [27]) that the formal
completion was a minor technical matter, not worth making any particular fuss
about.  Of course our construction is a completion of a much smaller (elliptic)
cohomology theory, whose coefficients are modular forms, with the completion map
corresponding to the q-expansion. The geometry underlying modularity is still [*
 *9]
quite mysterious.

Remark 5.6. A cohomology theory defined on finite spectra extends to a coho-
mology theory on all spectra [1]; moreover, any two extensions are equivalent, *
 *and
the equivalence is unique up to phantom maps.  For the case at hand, it is clear
that this cohomology theory is equivalent to the formal extension K((q)), where*
 * K
is complex K-theory, with q a parameter in degree zero. Hovey and Strickland [2*
 *0]
have shown that an evenly graded spectrum does not support phantom maps, so
our cohomology theory is uniquely equivalent, as a homotopy functor, to K((q)).


However, there is more to our construction than a simple homotopy functor:  it
comes with a natural (fixed-point) orientation, which defines a systematic theo*
 *ry
of Thom isomorphisms. In the terminology of [4], it is represented by an ellipt*
 *ic




14                  NITU KITCHLOO AND JACK MORAVA


spectrum, associated to the Tate curve over Z((q)); its natural orientation is *
 *defined
by the oe-function of x4.



Remark 5.7. Let HT denote T-equivariant singular Borel cohomology with ratio-
nal coefficients; then H*T(pt) = Q[t], where t has degree two. We have a locali*
 *zation
theorem
               H*T(X)[t-1] = H*T(XT)[t-1] = H*(XT)   Q[t  ]

for finite complexes, and can therefore play the same game as before, and obser*
 *ve
that there is a unique extension H^T(-) of HT(-)[t-1] to all T-spectra, with a
localization theorem valid for an arbitrary T-spectrum X:

                          H^*T(X) = ^H*T(XT) .

Jones and Petrack [21] have constructed such a theory over the real numbers, to-
gether with the analogous fixedpoint orientation - which in their case is (a ra*
 *tional
version of) the ^A-genus.

Results of this sort (which relate oriented equivariant cohomology theories on *
 *free
loopspaces to cohomology theories on the fixedpoints, with related (but distinc*
 *t)
formal groups, is part of an emerging understanding of what homotopy theorists
call `chromatic redshift' phenomena, cf. [2, 3, 31].

Remark 5.8. Our completion of equivariant K-theory is the natural repository for
characters of positive-energy representations of loop groups; it is not preserv*
 *ed
by the orientation-reversing involution ~ 7! ~-1 of T. It is in some sense a ch*
 *iral
completion.

Remark 5.9. The completion theorem above is a specialization of Segal's original
localization theorem [29], which says that KT(X), considered as a sheaf over the
multiplicative groupscheme Spec KT = Gm (cf. [28]), has for its stalks over gen*
 *eric
(ie nontorsion) points, the K-theory of the fixed point space. The Tate point

                        SpecZ((q)) ! SpecZ[q  ]

is an example of such a generic point, perhaps too close to zero (or infinity) *
 *to have
received the attention it seems to deserve.

In fact, we can play a similar game for any oriented theory ET. Let E denote the
union of the one-point compactifications of all representations of T which do n*
 *ot
contain the trivial representation (cf.  [25]).  Then the theory ET ^ E satisfi*
 *es a
strong localization theorem.

                6. Toward Pontrjagin-Thom duality

One might hope for a construction which associates to a map f : M ! N of
manifolds (with suitable properties), a morphism

                      LfPT : LN-TLN  ! LM-TLM

of prospectra. This seems out of reach at the moment, but some of the construct*
 *ions
sketched above can be interpreted as partial results in this direction.

In particular, there is at least a cohomological candidate for a PT dual

                       jPT : LM-TLM  ! M-TM




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS          15


to the fixedpoint inclusion j : M   LM.  To describe it, we should first observe
that there is a morphism
                         M-TLM  ! L!M-TLM

of prospectra, constructed by pulling back the tangent bundle of LM along the
fixedpoint inclusion. To be more precise we need to note that the T-fixedpoints*
 * of
the thickened loopspace is a bundle L!MT ! M with contractible fiber; the choice
of a section defines a composition

               ~j: M-TLM  ~ ((L!M)T)-TLM  ! L!M-TLM  ;

of course M-TLM  = (M-TM )- , where   = TM   (C[q  ]=C) is the normal bundle
described in x2.


Now according to the localization theorem above, the map induced on K^Tby ~jis
an isomorphism, so it makes sense to define

     j!:= (~j*)-1 O OE-1 : K^T(M-TM ) ! K^T(M-TM-   ) ! K^T(LM-TLM  ) ,

where OE  is the Thom pro-isomorphism associated to the filtered vector bundle *
 * .


A good general theory of PT duals would provide us with a commutative diagram
                                  PT
                      LN-TLN   -Lf---!LM-TLM
                          ??              ?
                          yjPTN           ?yjPTM
                                  PT
                        N-TN   --f--!  M-TM  ,

so it follows from the constructions above that

                         Lf! := j!NO f!O (j!M)-1

defines a formally consistent theory of PT duals for K^T.


For example, (the evaluation at 1 of) the composition

 KTate(M) = K^T(LM) ~=K^T(LM-TLM  ) ! K^T(M-TM ) ! KT(S0) ~=KTate(pt)

is (the q-expansion of) the Witten genus; more generally, our ad hoc constructi*
 *on
Lf! for K^Tagrees with the covariant construction fy defined for the underlying
manifold by the fixedpoint (or oe) orientation.


The uncompleted K-theory KT(LM-TLM  ) is also accessible, through the spectral
sequence of a colimit, but our understanding of it is at an early stage. It is *
 *of course
not a cohomological functor of M, but it does not seem unreasonable to hope that
some of its aspects may be within reach through similar PT-like constructions.


These fragmentary constructions suffice to show that K^T(LM-TLM  ) has enough
of a Frobenius (or ambialgebra) structure to define a two-dimensional topologic*
 *al
field theory, which assigns to a closed surface of genus g, the class

                         ßy fflT(T M)g 2 Z((q)) ,

with fflT the characteristic class defined in x4, and ßy the pushforward of M t*
 *o a
point defined by the fixed-point (oe) orientation.  When g = 0 this is the Witt*
 *en
genus of M, and when g = 1 it is the Euler characteristic.




16                  NITU KITCHLOO AND JACK MORAVA


Finally: our construction is, from its beginnings onward, formulated in terms of
closed strings.  Stolz and Teichner [30] have produced a deeper approach to a
theory of elliptic objects, which promises to incorporate interesting aspects o*
 *f open
strings as well. However, their theory is in some ways quite complicated; and o*
 *ur
hope is that their global theory, combined with the quite striking computational
simplicity of the very local theory sketched here, will lead to something really
interesting.


           7.Appendix: The Tits Building of a loop group



Let G be a simply-connected compact Lie group of rank n.  Let LG denote the
loop group of G.  For the sake of convenience, we will work with a smaller (but
equivalent) model for LG, which we now describe:


Let GC be the complexification of the group G. Since GC has the structure of a
complex affine variety, we may define the group LalgGC to be the group of poly-
nomial maps from C* to GC. Let LalgG be the subgroup of LalgGC consisting of
maps taking the unit circle into G   GC. The inclusion LalgG   LG is a homotopy
equivalence in the category of T-spaces. We begin by making our constructions w*
 *ith
LalgG. We then use these to draw conclusions about the (smooth) Tits building
A(LG)


Fix a maximal torus T of G, and let ffi, 1   i   n be a set of simple roots. We
let ff0 denote the highest root.  Each root ffi, 0   i   n determines a compact
subgroup Gi of G. More explicitly, Gi is the semisimple factor in the centraliz*
 *er of
the codimension one subtorus given by the kernel of ffi. Each Gimay be canonica*
 *lly
identified with SU(2) via an injective map

                           'i: SU(2) -! G .

We use these groups Gito define corresponding compact subgroups Giof LalgG as
follows:             `     '          `     '

         Gi= {z 7! 'i  ac bd}      if   ac bd 2 SU(2),  i > 0

                       `        '        `    '

          G0 = {z 7! '0  caz-1 bzd}  if   ac bd 2 SU(2),


Remark 7.1. Note that each Gi is a compact subgroup of LalgG isomorphic to
SU(2). Moreover, Gi belongs to the subgroup G of constant loops if i   1. The
circle group T preserves each Gi, acting trivially on Gi for i   1, and nontriv*
 *ially
on G0.

Definition 7.2. For any proper subset I   {0, 1, . .,.n}, define the parabolic *
 *sub-
group HI   LalgG to be the group generated by T and the groups Gi, i 2 I. For
the empty set, we define HI to be T . It follows from 7.1 that each HI is prese*
 *rved
under the action of T.

Remark 7.3. The groups HI are compact Lie [26]. Moreover, HI is isomorphic
to its image in G, under the evaluation map ev(1) : LalgG ! G. Notice that for
I = {1, . .,.n}, HI = G. Notice also that T acts nontrivially on HI if and only*
 * if
0 2 I.




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS          17


We are now ready to define the Tits building A(LalgG).

Definition 7.4. Let A(LalgG) be the homotopy colimit:

                    A(LalgG) = hocolimI2CLalgG=HI

where C denotes the poset category of proper subsets of {0, 1, . .,.n}.

We now come to the main theorem:

Theorem 7.5. The space A(LalgG) is Tx~LalgG-equivariantly contractible.  In
other words, given a compact subgroup K   Tx~LalgG, then the fixed point space
A(LalgG)K is contractible.

Proof.A proof of the contractibility of A(LalgG) was given in [26]. We use some
of the ideas from that paper, but our proof is different in flavour.


Mitchell expresses the space A(LalgG) as the following:

                     A(LalgG) = (LalgG=T x  )= ~
                                                                       O
where   is the n-simplex, and (aT, x) ~ (bT, y) if and only if x = y 2  I and
aHI = bHI. Here we have indexed the walls of   by the category C, and denoted
                      O
the interior of  I by  I.


Let LalgG   Rd be the Lie algebra of the extended loop group Tx~LalgG. Consider
the affine subspace
                      A = LalgG + d   LalgG   Rd .

The adjoint action of LalgG on A is given by

             Adf(z)(~(z) + d) = Adf(z)(~(z)) + zf0(z)f(z)-1 + d

This action extends to an affine action of Tx~LalgG. The identification of A wi*
 *th
A(LalgG) is given as follows. Let   be identified with the affine alcove:

           = {(h + d) 2 Lie(T ) + d |  ffi(h)   0, i > 0, ff0(h)   1} .

General facts about Loop groups [22, 26] show that the surjective map

                 LalgG x   -! A,  (f(z), y) 7! Adf(z)(y)

has isotropy HI on the subspace  I. Hence it factors through a Tx~LalgG-equivar*
 *iant
homeomorphism between A(LalgG) and the affine space A. Notice that any com-
pact subgroup K   Tx~LalgG admits a fixed point on A(LalgG). Hence, the space
A(LalgG)K is also affine. This completes the proof.

We now define the smooth Tits building

Definition 7.6. Let A(LG) be the homotopy colimit:

           A(LG) = hocolimI2CLG=HI = LG xLalgGA(LalgG) .

It is clear from the proof of the above theorem, that A(LG) is Tx~LG-equivarian*
 *tly
homeomorphic to the affine space LGxLalgGA which is homeomorphic to the affine
space A(S1 x G) of connections on the trivial bundle S1 x G. This shows that:

Theorem 7.7. The smooth Tits building A(LG) is Tx~LG-equivariantly contractible.




18                  NITU KITCHLOO AND JACK MORAVA


Recall [26] that A(LalgG) is homeomorphic to the space

Salg= {g(t) : [0, 1] ! G  | g(t) = f(e2ßit) . exp(tX); f(z) 2  algG, X 2 Lie(G)}

We have a corresponding smooth version

S = {g(t) : [0, 1] ! G  | g(t) = f(e2ßit) . exp(tX); f(z) 2  G} = LG xLalgGSalg

which is clearly homeomorphic to A(LG). It remains to identify S with the space

            S = {g(t) : R ! G, g(0) = 1, g(t + 1) = g(t) . g(1)} ;

This is straightforward, and is left to the reader.




           THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS          19


                             References

1. F. Adams, A variant of E.H. Brown's representability theorem, Topology, Vol.*
 * 10 (1971)
   185-198.
2. M. Ando, J. Morava, A Renormalized Riemann-Roch formula and the Thom isomorp*
 *hism for
   the free loopspace, in the Milgram Festschrift, Contemp. Math. 279 (2001)
3. __, __-, H. Sadofsky, Completions of Z=(p)-Tate cohomology of periodic spect*
 *ra, Geom-
   etry & Topology 2 (1998) 145 - 174
4. _-, M. Hopkins, N. Strickland, Elliptic spectra, the Witten genus and the th*
 *eorem of the
   cube, Inv. Math. 146 (2001) 595 - 687
5. M. Artin, B. Mazur Etale homotopy, Springer LNM 100 (1969)
6. M. Atiyah, I. MacDonald, Commutative Algebra
7. P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and *
 *K-theory of
   group C*-algebras, p. 240 - 291 in C*-algebras: 1943-1993, Contemporary Math*
 * 167, AMS
   (1994)
8. N. Bourbaki, Algebre Commutatif
9. JL Brylinski, Representations of loop groups, Dirac operators on loop space,*
 * and modular
   forms, Topology 29 (1990) 461 - 480
10.M. Chas, D. Sullivan, String topology, available at math.AT/9911159
11.J.D. Christensen, D.C. Isaksen, Duality and prospectra (in progress)
13.R. Cohen, Multiplicative properties of Atiyah duality, available at math.AT/*
 *0403486
14.___, V. Godin, A polarized view of string topology, available at math.AT/030*
 *3003
15.__-, J.D.S Jones, A homotopy-theoretic realization of string topology, avail*
 *able at
   math.GT/0107187
16.__-, __-, G. Segal, Floer's infinite-dimensional Morse theory and homotopy t*
 *heory, in
   The Floer memorial volume, Progr. Math 133, Birkhäuser (1995)
17.__-, A. Stacey, Fourier decomposition of loop bundles, available at math.AT/*
 *0210351
18.D.  Freed,  The  Verlinde  Algebra  is  Twisted  Equivariant  K-Theory,  ava*
 *ilable  at
   math.RT/0101038
19.D. Freed, M. Hopkins, C. Teleman, Twisted K-theory and loop group representa*
 *tions I,
   available at math.AT/0312155
20.M. Hovey, N.P. Strickland, Morava K-theories and localization, Mem. Amer. Ma*
 *th. Soc. 139
   (1999), No.666.
21.J.D.S Jones, S.B. Petrack, The fixed point theorem in equivariant cohomology*
 *, Transactions
   of the Amer. Math. Soc., Vol. 322, No. 1 (1990) 35-49.
22.V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press, 1985
23.N. Kitchloo, Topology of Kac-Moody groups, Thesis, M.I.T., 1998.
24.J. Klein, The dualizing spectrum of a topological group, Math. Ann 319 (2001*
 *) 421 - 456
25.G. Lewis, J.P. May, M. Steinberger, Equivariant stable homotopy theory, Spri*
 *nger LNM
   1213 (1986)
26.S.A. Mitchell, Quillen's theorem on buildings and the loops on a symmetric s*
 *pace, Enseign.
   Math. 34 (1988) 123-166
27.J. Morava, Forms of K-theory, Math Zeits. 201 (1989)
28.I. Rosu, Equivariant K-theory and equivariant cohomology, Math. Zeits. 243 (*
 *2003) 423-448.
29.G. Segal, Equivariant K-theory, Inst. Hautes 'Etudes Sci. Publ. Math. No. 34*
 * (1968) 129-151.
30.S. Stolz, P. Teichner, What is an elliptic object? available at math.ucsd.edu
31.T. Torii, On degeneration of one-dimensional formal group laws and stable ho*
 *motopy theory,
   AJM 125 (2003) 1037-1077
32.D. Zagier, Note on the Landweber-Stong elliptic genus, in Elliptic curves an*
 *d modular
   forms in algebraic topology, ed. P. Landweber, Springer 1326 (1988)

  Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
  E-mail address: nitu@math.jhu.edu, jack@math.jhu.edu