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Com mun . Math. Phys. 107, 483-513 (1986)

o m m u n i c a t i o n s in

athematical

Physics

O Springer-Verlag 1986

D eterm inants T orsion and Strings

D a n i e l S . F r e e d 1

Department o f Mathematics, M assachusetts Institute of Technology, Cam bridge, MA 02139,

USA

Abstract

W e a p p l y t h e r e s u lt s o f [- B F 1, B F 2 ] o n d e t e r m i n a n t s o f D i r a c

o p e r a t o r s t o S t ri n g T h e o r y . F o r t h e b o s o n i c s tr i ng w e r e c o v e r t h e h o l o -

m o r p h i c f a c t o r i z a t io n o f B e l a v i n a n d K n i z h i k . W i t t e n 's g l o b a l a n o m a l y

f o r m u l a i s u s e d t o g i v e s u ff ic i en t c o n d i t i o n s f o r a n o m a l y c a n c e l l a t io n i n t h e

h e t e r o t i c s t r i n g ( f o r a r b i t r a r y b a c k g r o u n d s p a c e t i m e s ) . T o p r o v e t h e l a t t e r

r e s u lt w e d e v e l o p c e r t a i n t o r s i o n i n v a r i a n t s r e l a t e d t o c h a r a c t e r i s ti c c l as s es o f

v e c t o r b u n d l e s a n d t o i n d e x t h e o r y .

S t ri n g T h e o r y h a s s p a w n e d a v i g o r o u s i n te r a c ti o n b e t w e e n m a t h e m a t i c s a n d

phys ics . Th i s in te rming l ing o f tw o qu i te s epara te in tu i t ions i s f ru i tfu l fo r bo t h

d i sc i pl in e s . O f p a r t i c u l a r v a l u e f o r m a t h e m a t i c i a n s a r e t h e c o n c r e t e e x a m p l e s

g e n e r a t e d a n d d i s c u s s e d b y p h y s i ci st s. T h e p u r p o s e o f t h is p a p e r i s t o c o n s i d e r t h e

r e l a t io n s h i p o f s o m e e x a m p l e s t o t h e c ir c le o f i d e a s s u r r o u n d i n g t h e A t i y a h - S i n g e r

I n d e x T h e o r e m .

A t i y a h a n d S i n g e r f ir s t d e m o n s t r a t e d t h e c o n n e c t i o n b e t w e e n d e t e r m i n a n t s (in

Q u a n t u m F i e ld T h e o r y ) a n d t h e f am i li es i n d ex t h e o r e m [ A S 1 ] . T h e i r c o n c e r n w a s

w i t h a n o m a l i e s, w h i c h t h e y i n t e r p r e t e d a s n o n t r i v ia t t o p o l o g y ( o v e r t h e r e a ls ) i n

t h e d e t e r m i n a n t l i n e b u n d l e ~ . W i t t e n ' s w o r k o n g l o b a l a n o m a l i e s [ W 1 ]

s u g g e s t e d a m o r e r ef in e d g e o m e t r i c p i c t u re : ~ h a s a n a t u r a l c o n n e c t i o n w h o s e

c u r v a t u r e a n d h o l o n o m y r e p r e s e n t t h e l o c a l a n d g l o b a l a n o m a l y , r e sp e c t iv e l y , z

T h e m a t h e m a t i c a l i d e a s u s e d t o c o n s t r u c t s u c h a c o n n e c t i o n a r e l a r g e l y d u e t o

Qui l l en [Q1] who , fo r en t i r e ly d i f f e ren t r easons , in t roduced a met r i c on 5e . The

c o n n e c t i o n o n L P w a s r i g o r o u s l y c o n s t r u c t e d i n [ B F 1 ] . T h e a n a l y t i c a l te c h n i q u e s

d e v e l o p e d b y B i s m u t i n [ B ] w e r e u s e d in [ B F 2 ] t o d e r i ve t h e f o r m u l a e f o r i ts

c u r v a t u r e a n d h o l o n o m y . W e d i s c u s s t h e s e d e v e l o p m e n t s i n S e c t. 1.

I n S e c t. 2 w e i n t e r p r e t s o m e k n o w n r e s u l t s a b o u t t h e b o s o n i c s tr in g i n t e r m s o f

t h e g e o m e t r y o f t h e d e t e r m i n a n t l in e b u n d le . W e e m p h a s iz e t h a t p u r e l y

t o p o l o g i c a l m e t h o d s d o n o t s u ffic e h er e. I n f a ct , o u r m a i n p u r p o s e i n t h i s p a p e r i s

1 Partially supported by an NSF Postdoctoral Research Fellowship

2 W itten explicitly stated that his wo rk could b e interpreted in term s of this connection


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484 D.S . F reed

to explain examples which dem and metho ds that go b eyon d the topological index

theorem. We consider f irs t the co nformal anomaly. In a recent paper Alvarez [A2]

explains a formal connect ion between the eonformal anomaly and the index

theorem . Althou gh we pro vide a g eometric setting for his observation, we feel that

our understanding here is incomplete . In the cr i t ical dimension d=26, the

conformal anomaly vanishes and the the ory can be formulated over the R iemann

mo duti space. The re is then a holo mo rphic factorization of the part i t ion

function, recently proved b y Betavin an d K nizh nik [B K] (cf. , [CCM R]). Th at is,

the par t i t ion funct ion is the nor m square of a ho lomo rphic funct ion on m odul i

space. We use the conn ection on ~ to recove r this result . A recent an no un cem ent

by Manin [M] gives an explicit formula for the part i t ion function.

Ou r other th em e is torsion. Witten's original examples of global gravitat ional

anom alies can be explained in terms o f torsion in the d eterm inan t line bundle. We

pointed this out to W itten, who applied this observation in man y contexts [W2,

WW]. Wit ten 's convict ion that tors ion phenomena are important in Str ing

Th eo ry inspired us to develop this ma terial further. Hence, in Sect. 3 we give an

exposit ion of the relevant torsion invarian ts in coho molo gy, and in Sect. 4 we show

how they apply to anomalies in the heterotic str ing. These anomalies were

discussed (in special cases) by W itten in [W 2], an d we follow his wor k closely. O ur

ma in the ore m in Sect. 4 gives sufficient con dition s for the canc ellation o f global

anomalies. This theorem applies to arbitrary spacetimes. Although torsion

invar iants enter to com pute the global anom aly, we emphasize that the global

ano ma ly is geometric, not topological. Tha t is, one ca nn ot rec over these results by

studying just the top ology o f cj ; for Y could be topologically tr ivial but have a

nontrivial connection. Because the typical elemen t of the map ping class group has

infinite order, and because the topolog y of spacetime is arbitrary, the ho lon om y

aro un d a typical loop is not com putab le from the Ch ern class. Rather, for certain

loops i t can be expressed in terms of torsion invariants on spacetime. The torsion

invariants relevant to the index theorem (hence to anomalies) l ie in K-theory,

thou gh we work arou nd the K-theo ry in this paper . They are related to ideas of

Dennis Sull ivan, and are more or less known to topologists. We defer their

consideration to IF1], where we discuss the connection with index theory.

Rather mo re algebraic topology than we would prefer enters o ur p roof of the

global an om aly cancellat ion. Mor e precisely, the relationship betwe en the torsion

invariants in K-theo ry and the torsion invariants in coho mo logy involves a

ordism invariant (C orollary 3.22). Recall tha t bord ism is the generalized

hom olog y theory def ined roughly by replacing s ingular chain with manifold in

the def ini t ion of ordinary s ingular homology. In [W3] Wit ten shows that the

global anomaly always vanishes in the 10 dimensional f ield theory l imit of the

E8 E8 hetero tic string, given the sam e topolog ical con dition we requ ire in Sect. 4.

The key ingredien t of his pro of is a very impressive bord ism calculation of Stong.

Such bordism com putatio ns seem an indispensible ingredient in an y discussion of

global anomalies.

The impetus to de velop these ideas came in part fro m an effort to explain some

formulae of Vafa IV]. 3 He der ived condi tions for mo dula r invar iance of the

3 These have been der ived by others as wel l [D HV W ]. My knowledg e of the S t r ing Theory and

li terature is far from com plete so my references to i t should n ot be presum ed defini t ive


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Determinants, Torsion, and Strings 485

heterotic string theory formulated on orbifolds. (His orbifolds are orbit spaces of

representations of finite groups.) Modular invariance is equivalent to the absence

of global anomalies, and so we were led to look for an interpretation of his

formulae in terms of the determinant line bundle. The material in Sect. 4 is a

prerequisite for that discussion, which will appear elsewhere [FV].

The examples discussed in this paper provided motivation and guidance for a

general theory of determinant line bundles [BFI, BF2, F2] and of torsion

invariants I-F1]. We hope that our expository efforts in Sects. 1 and 3 form a useful

introduction to these ideas.

1 Determin an ts an d Fam i li e s o f Dirac Op erators

We begin by reviewing determinants of operators in finite dimensions. Let V be a

finite dimensional complex vector space and A:V--* V an endomorphism of V.

Then A induces an endomorphism

det A: det V-~det V

on the one dimensional line of totally antisymmetric tensors det V= m a x v ~ the

highest exterior power of V. For va, v2 ... , v, e V,

(det A) (v 1/x v2 A... A V,) = Av 1 A Av2 A. .. A

Av n

(1.1)

Since detV is a one dimensional vector space, the endomorphism detA is

multiplication by a complexnumber, which is the product of the eigenvalues of A.

We usually identify detD with this complex number.

Suppose now that W is another vector space and D: V-> W a linear map. If

dim V = dim W we can form the induced map

detD : det V-~det W

on the highest exterior powers. Without additional choices there is no natural way

to identify this with a complex number. Instead we must be content to regard

det O e (det V)* (det W)

as an element of a complex line. Note tha t detD vanishes if D has a kernel and is

nonzero otherwise. n fact there is an exact sequence

0 > kerD ~ V - ~ W - ~ cokerD > 0, (1.2)

and the multiplicative property of the determinant implies a natura l isomorphism

(det V) * @ (det W) ~- (det ker D)* (det coker D). (1.3)

If kerD = {0} then the right-hand side of (1.3) is canonically identified with C and

detD corresponds to I s C. If D is not invertible, then detD =0.

Introduce a parameter space Y and imagine that the operator Dy : Vy~ I/V

varies smoothly over y ~ Y The determinant construction at y s Y gives a point in

the complex line (det Vr)*N(det PITy).This fits together to yield a section detD of a

line bundle 5e--,Y. For operators in finite dimensions the line bundle 5 is

completely determined by the families of vector spaces {Vr} and {Wr} (which we


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486 D .S. Freed

a s s u m e f i t t o g e t h e r t o f o r m v e c t o r b u n d l e s o v e r E ) I n p a r t ic u l a r , i t i s i n d e p e n d e n t

o f t h e o p e r a t o r s

D r ) .

W e c a n p a s s t o o p e r a t o r s o n i n f in i te d i m e n s i o n a l s p a c e s b y r es t ri c ti n g t o

Fredholm operators, which by de f in i t ion have f in i t e d imens iona l ke rne l and

cokern e l ; then the f ibe r o f 5e is de f ined by the r igh t -ha nd s ide o f 1 .3 ). N o t ic e tha t

the l e f t-hand s ide no long er m ake s s ense as V and W are in f in it e d imens iona l . A lso ,

t o r ig o r o u s l y c o n s t r u c t t h e li n e b u n d l e ~ w e h a v e to a c c o u n t f o r t h e p o s s i b il i ty

t h a t t h e d i m e n s i o n o f k e r D y c a n j u m p a s y v ar ie s . S e e [ Q ] , f o r e x am p l e .) F o r

o p e r a t o r s i n i n fi ni te d i m e n s i o ns t h e h y p o t h e s i s d i m V = d i m W is r e p l a ce d b y

d i m

k e r D r =

d i m c o k e r D y , i.e ., b y t h e h y p o t h e s i s t h a t t h e F r e d h o l m o p e r a t o r s

D r

hav e index ze ro . Hen cefo r th , we wr i t e D fo r Dy .) Th e l ine bund le 5 ex i s ts fo r

a r b i t r a r y i n d e x , b u t t h e s e c t i o n d e t D r e q u i r e s t h e i n d e x z e r o h y p o t h e s i s . I f

i n d e x D + 0 , t h e n D is n e v e r i n ve r ti b le , s o i t i s n a t u r a l t o s e t d e t D - 0 .

W e a r e c o n c e r n e d w i th D i r a c o p e r a t o r s o n a c o m p a c t m a n i fo l ds . W e a l s o a ll o w

o p e r a t o r s o f D i r a c t y p e , w h i c h i n c l u d e s t h e s i g n a t u r e o p e r a t o r , R a r i t a -S c h w i n g e r

opera to r , s e l f -dua l opera to r , ~ -opera to r on a K / ih le r man i fo ld , e t c . Reca l l tha t

d e t e r m i n a n t s o f D i r a c o p e r a t o r s a r is e in fe r m i o n i c in t e g r a ti o n . T h e s e D i r a c

o p e r a t o r s d e p e n d o n B o s e fie ld s, w h i c h a r e p a r a m e t r i z e d b y a s p a c e Y, a n d t h e p a t h

i n t eg r a l q u a n t i z a t i o n d i c t at e s t h a t d e t D b e i n te g r a t e d o v e r Y. T h e p r e c e d i n g

d i s c u s si o n in d i c a te s t h a t d e t D c o m e s a s a section o f t h e d e t e r m i n a n t l in e b u n d l e

~e ~ Y. Bu t we can on ly in teg ra te funct ions o v e r Y. H e n c e w e n e e d t o f i n d a g l o b a l

b a s is f o r ~ , t h a t i s, a t ri v ia l iz a t io n o f ~ , s o a s t o e x p r e ss d e t D a s a f u n c t io n o n Y. A

g lob a l bas i s is a n onz ero s ec t ion s o f L~, and w i th r espec t to th i s g lob a l bas i s we can

w r i t e d e t D = f . s f o r s o m e f u n c t io n f . ) T h e o b s t r u c t i o n t o f in d i n g a t ri v i a li z a t io n i s

ca l l ed the anomaly.

T h i s c o n n e c t i o n b e t w e e n a n o m a l i e s a n d t h e d e t e rm i n a n t li ne b u n d l e w a s

p i o n e e r e d b y A t i y a h a n d S i n g er [ A S l ] . T h e r e is a to p o l o g i c a l o b s t r u c t i o n t o

t r iv ia l iz ing L~ , nam ely i t s in teg ra l f i r s t Ch ern c las s c~ ~) , an d they sho w ed tha t i t

c a n b e c o m p u t e d b y t h e A t i y a h - S i n g e r I n d e x T h e o r e m f o r F a m i l i e s [ A S 2 ] .

F u r t h e r m o r e , o v e r t h e r ea l s t h e f ir st C h e r n c h a r a c t e r chl Sg), w h i c h i s t h e i m a g e o f

c 1 ~ ) i n re a l c o h o m o l o g y , i s e x p r e s se d b y a n e x p l ic it c o h o m o l o g i c a l f o rm u l a . T h e

to r s io n in fo rm at ion in c l LP) i s on ly acces s ib le v ia K- th eo ry IF1] . ) T h i s

t o p o l o g i c a l o b s t r u c t i o n i s s t r o n g e n o u g h t o d e t e c t th e a n o m a l y i n m a n y s i t u a ti o n s .

N o t i c e , h o w e v e r , t h a t i f t h is t o p o l o g i c a l o b s t r u c t i o n c 1 ~ ) v a n i s h e s t h e re a r e

m a n y w a y s o f t ri v ia l iz i n g ~ , a n d t h e f u n c t i o n o b t a i n e d f r o m t h e s e c t i o n d e t D

d e p e n d s s t r o n g l y o n w h i c h t r iv i a li z a ti o n i s c h o se n . E x t r a g e o m e t r y is n e e d e d t o f ix

the t riv ia li za t ion , so as to f ix the de te rm inan t func t ion . Th a t ex t r a g eom et ry tu rns

o u t t o b e a c o n n e c t i o n V~se) on 5a . He nc e w e hav e th e k ey

Defini t ion 1.4. T h e geometr ic anomaly i s the obs t ruc t ion to t r iv ia l i z ing the

c o n n e c t i o n o n 5 e .

I n o t h e r w o r d s , t h e g e o m e t r i c a n o m a l y i s t h e o b s t r u c t i o n t o f in d i n g a g l o b a l f l a t

no nze ro s ec t ion s ee F ig . 1 ). I f one ex i s t s i t i s un iq ue up to a p has e o n ev ery

c o n n e c t e d c o m p o n e n t o f Y, a n d t h e r a t i o o f t h e s e c t io n d e t D t o t h e f l a t s e c ti o n is a

f u n c t i o n r e p r e s e n ti n g t h e d e t e r m i n a n t . T h e d e t e r m i n a n t b u n d l e a l s o h a s a m e t r ic

g~-~) and we t ak e the t r iv ia li z ing f l a t s ec t ion to have un i t no rm. ) Th i s c onn ec t io n

w a s c o n s t r u c t e d i n [ B F I ] , f o l lo w i n g c lo s e ly i d e a s o f Q u i U e n. W e r e m a r k


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Determinan ts Torsion and Strings 487

tha t the Green-Schwartz anomaly cancel la t ion [GS] involves a l - form on

Y, which enters geom etrically as a modification to the canonical conn ection V~e)

(cf. Sect. 4).

det

Fig. 1

Before describing the geom etric setup abstractly, let us consider a few physical

examples.

Example 1.5 Gravitational Anomalies

[AgW, A SZ ]) : Let X be a comp act even

dimensional sp in manifold , say of d imension n , and M et(X ) the space Riemannian

metrics on X . T he spin structure ~ on X determines a doub le cover G L(X) of the

frame bundle

GL X).

N o w l et

= {(g , f ) ~ M e t( X )x G- L(X): the frame f i s or tho norm al in the metr ic g}.

Then . ~ -~ M et (X ) x X is a principal Spin(n) bundle w hich restr icts on {9} x X to

the bundle o f spin f rames on X for the metr ic 0. Not ice tha t the prod uct

M e t ( X ) x X carr ies a par t ia l metr ic a long the f ibers of the projec tion

M et (X) x X ~ M e t (X ) on to the f i rs t fac tor . The half -spinor representa t ions a + of

Spin(n) , appl ied to the bundle i f , y ie ld vector bundles over Met(X) x X, which

restrict on {g} x X to the bundles o f posit ive and negative spinors over X. F or each

fixed metric the chiral Dirac ope rato r can be defined as usual . Glo bally we obta in a

family of chira l Dirac o pera tors on X parameter ized b y Me t(X) .

The group of d i f feomorphisms Diff (X) ac ts on M et(X ) x X. The ac t ion on a

metric g is by pullb ack a nd th e diffeomorphism s act on X by definition. Restrict ing

to the gro up of diffeomorph isms Diff ,(X) w hich preserve the spin structure e, the

action l if ts to an action (possibly of a do ubl e covering o f Diff ,(X)) on o~. Tak ing

the quot ient we obta in a f ibra tion o f manifolds

Z = M e t ( X ) x X/Diff, X)

+x (1.6)

Y= Met X)/Diff , X).

There is a consistent spin structure a long the f ibers, so that spinors are defined .The

metric along the f ibers is preserved by the a ction o f Diff ,(X), so that the quotie nt Z

also carries a metric along the f ibers. Although Z is not globally a product in

general , the infinitesimal version of the prod uct structure M et (X) x X passes to the

quot ient ; there is a projec t ion P:

TZ~T~ertZ.

This should be tho ught of as a

con nec tion of the fibrati on of manifo lds (1.6) (see Fig. 2).

On e di f ficul ty we have no t ye t ment ioned is tha t in genera l Diff , (X) does not

act freely on M et(X) . The iso t ropy group a t 9 e M et(X ) is the isometry group of


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4 8 8 D S F r e e d

/ y

Fig 2 _ ~ _

t h e m e t r i c g, w h i c h is a c o m p a c t L i e g ro u p . I n t h e c a s e o f m a j o r i n t e re s t t o u s X =

i s a R i e m a n n s u r fa c e o f g e n u s > 2 a n d t h is i s o m e t r y g r o u p f o r m e t r i c s o f c u r v a t u r e

- 1 i s f in ite. T hu s th e q uo t ien t Y is an

orbifold

a n d c a n b e t r e a t e d b y s t a n d a r d

m e t h o d s ( cf . t h e d i s c u s s io n a b o u t g r o u p a c t i o n s a t t h e e n d o f t h is s e c ti on ) . A s w e

d iv ide ou t on ly by d i f f eom orph i sm s p rese rv ing a sp in s t ruc tu re , Y i s a f in i te cove r

o f t h e q u o t i e n t b y a l l d i f fe o m o r p h i sm s , t h e R i e m a n n m o d u l i s p a ce .

Example 1.7 a-Model

[ M N ] ) . A g a i n l et X b e a c o m p a c t e v e n d i m e n s i o na l s p in

m a n i f o l d , n o w w i t h a fi x e d m e t r ic , a n d s u p p o s e t h a t M i s a n a r b i t r a r y m a n i fo l d .

L e t E~M b e a H e r m i t i a n v e c t o r b u n d l e w i t h u n i t a r y c o n n e c t i o n V E). C o n s i d e r

t h e m a p p i n g s p a c e Y = M a p ( X , M ) . F o r e a c h (p e Y w e h a v e a p u l l b a c k b u n d l e

~o*E~X

with con nec t ion q~*V E). Th e D i rac op era to r on X coup les to th i s

con nec t ion (vec to r po ten t i a l ) to g ive a D i ra c o pe ra t o r on q~*E-valued sp in o r f ie lds.

T h i s f a m i ly is p a r a m e t r i z e d b y t h e m a p q ) ~ Y S e t Z = Y x X a n d c o n s i d e r t h e

e v a l u a t i o n m a p

e:Z~M.

W e o b t a i n t h e f o l l o w i n g d i a g r a m :

( E , V 0 , ( e , V

Z ~ M (1 .8)

Y

H e r e E ~ Z is t h e b u n d l e

e*E

with i t s pu l l ed ba ck con nec t ion V ~) = e* V~) . Over

eac h (0 e Y th is res t r ic ts to the bu nd le q~*E w ith co nn ec t ion q~*V E).

T h e P o l y a k o v f o r m u l a t i o n o f s tr in g t h e o r y c o m b i n e s th e s e t w o e x a m p l e s :

X = I ; is a R i e m a n n s ur fa c e a n d t h e b o s o n ic fie ld s a r e M e t ( ~ ) a n d M a p ( Z , M ) .

N e x t w e a b s t r a c t f r o m t h e s e e x a m p l e s t h e p r e c i s e g e o m e t r i c s e t u p w e n e e d .

Geometric Data 1.9.

(1 ) A s m o o t h f i b r a t i o n o f m a n i f o l d s ~ : Z x ~ y W e s u p p o s e

d i m X = n is e v en . T o d e fi ne a f am i l y o f D i r a c o p e r a t o r s w e n e e d t o a d d t h e

t o p o l o g i c a l h y p o t h e s i s t h a t t h e t a n g e n t b u n d l e a l o n g t h e f i b e r s T~ortZ~Z has a

f ixed sp in st ruc tu re . (Th i s i s s t rong er than as sum ing tha t X has a sp in s t ruc tu re ; the

s p i n s t r u c tu r e s o n t h e f i b e rs X y m u s t f it t o g e t h e r o v e r t h e p a r a m e t e r s p a c e Y I n

E x a m p l e 1 . 5 a b o v e t h i s h y p o t h e s i s i s g u a r a n t e e d b y d i v i d i n g o u t o n l y b y


7/31

Determ inants Tors ion an d St rings 489

d i f f e o m o r p h i s m s w h i c h p r e s e r v e t h e f ix e d sp i n s t r u c t u r e o n X . ) I f w e c o n s i d e r a

f ami ly o f J o per a to r s , s ay , then the sp in s t ru c tu re i s i r r e levan t .

(2) A m etr ic a lo ng the f ibers , tha t i s, a me tr ic g( rv rtz) on T~r tZ.

(3 ) A p ro jec t io n P : T Z ~ T w r t Z . T h e k e r n e l o f P i s a d i s t r i b u t i o n o f h o r i z o n t a l

c o m p l e m e n t s t o t h e v e r t i c a l ta n g e n t s p a c e s, a s i n d i c a t e d i n F i g . 2 .

(4) A co m plex (v i r tua l ) r ep resen ta t ion Q o f Sp in (n ) . Th i s i s u sed to spec i fy the

e x a c t c o m b i n a t i o n o f D i r a c - t y p e o p e r a t o r s i n t h e t h e o r y . It d e t e r m i n e s a b u n d l e

~ Z t o w h i c h t h e b a s i c D i r a c o p e r a t o r c o u p l e s . F o r e x a m p l e , if Q= 1 - (a + + a _) ,

t h e n w e h a v e t h e D i r a c o p e r a t o r m i n u s t h e s ig n a t u r e o p e r a t o r . ( a + + a _ is t h e t o t a l

sp in r ep resen ta t ion . )

(5 ) A c o m p l e x v e c t o r b u n d l e E ~ Z with a he rm i t i an m et r i c g(E) and com pa t ib le

con nec t ion F (~). Th i s desc r ibes ex t r in s ic d a ta (gauge f i e lds and pa r t i c le s ) a s in

Example 1 .7 .

Th e d at a (2) an d (3) in (1 .9) de term ine a co nn ec t ion F ~rvr~z) on the tan ge nt bu nd le

alo ng the f ibers of Z---}E I t is t h e p r o j e c t i o n o f t h e L e v i - C i v i t a c o n n e c t i o n o n Z

c o n s t r u c t e d b y c h o o s i n g a n a r b i t r a r y m e t r ic o n Y. A l l o f o u r c o n s t r u c t i o n s a r e

i n d e p e n d e n t o f a n y c h o i c e o f m e t r i c o n Y W e d e n o t e t h e c u r v a t u r e o f t h is

co nn ec tio n b y Y F v~'z~.

T h e d e t e r m i n a n t l in e b u n d l e ~ - -- } Y i s c o n s t r u c t e d f r o m t h e d a t a ( 1 .9 ) b y

p a t c h in g . B r ie fl y, o n e h a n d l e s t h e j u m p i n g k e r n e l s b y t h r o w i n g i n a f i n it e

d i m e n s i o n a l s p a c e o f l o w e i g e n m o d e s f o r t h e L a p l a c i a n s D * D a n d D D * . T h e n

(1 .2 ) a n d ( 1. 3) a r e u s e d t o p a t c h t o g e t h e r t h e d e t e r m i n a n t l in e s c o n s t r u c t e d f r o m

t h e s e l o w e i g e n m o d e b u n d l e s .

T h e o r e m 1 . 1 0 [ B F 1 ] . T h e d a t a (1.9) de termine func tor ia l l y a smoo th de terminan t

l ine bundle ~f~Y. I t carr ies the Qui l len metr ic

g S e )

( cons t ruc ted in [ Q ] ) a n d

comp at ib le con nect ion V (se~. I f the Dirac operators have index zero there is a sect ion

d e t D

of LP.

O v e r t h e p o i n t s i n Y w h e r e D is in v e r t ib l e t h e s e c t io n d e t D g i v e s a t r i v ia l i z at i o n o f

~a . In t e rm s o f tha t t r iv ia l i za t ion the Qu i l l en met r i c i s g iven by

Ildet Dll[se)=det D * O . (1.11)

T h e L a p l a c i a n A = D * D i s a s e l f - ad jo in t opera to r , bu t on a in f in i t e d imens iona l

s p a c e , s o t h a t ( 1 .1 ) c a n n o t b e u s e d t o d e f i n e d e t D * D . R a t h e r , w e u s e a p r o c e d u r e

d u e t o R a y a n d S i n g e r [ R S ] - z e t a - f u n c t i o n r e g u t a r i z a t i o n . L e t { 2 } b e t h e

e i g e n v a lu e s o f D D , l i s t ed accord ing to the i r mu l t ip l i c i t i e s , and s e t

s)= ~ 1 = T r ( ( D , D ) _ ~ ) 1.12)

Th en ~( s ) i s f in it e and ho lom orp h ic fo r R es su f f i c ien t ly l a rge , and has a

m e r o m o r p h i c c o n t i n u a t i o n w h i c h i s r e g u l a r a t s = 0 . D e f i n e

d e t D * D = e - c ( ) . (1.13)

Th e con nec t ion V ze) on the s ec t ion de tD i s a l so exp l i c i t where D is inver tib le :

V~'~)(detD) = Tr ( t~D D -~ ) (de t D ) . (1 .14 )


8/31

490 D.S . Freed

He re I~ is the c on ne cti on 17~Tvrtz) op era tin g po intwise, b ut wi th a co rre ctio n te rm

due to the chang ing volum e forms on th e fibers of Z ~ Y. The resulting conne ction

ly is u nitary on the bundles of L 2 spinor fields. N ote that formally

Tr(lY~v)D - 1)= 6 In detD, which is the usual expression in the physics l i terature.

Again we use a zeta function to define the right-h and side of (1.14). Set

og(s)= Tr ((DD*)- SffD D - i), (1.15)

wh ich is fin ite for R es >>0. The mero mo rphic extension is more co mplicated than

for ~(s); in particu lar,

co(s)

has a pole at s= 0 in general. W e use the regular par t to

define

Tr(f fD D - 1) = (s~o(s)) (0). (1.16)

These constructions extend to all of L by patching.

In the preceding paragra ph we used the canonical sect ion detD to t rivial ize ~ ,

where D is invertible. Suppose instead th at w e work over a region in Y where kerD

and cokerD have constant dimension. We no longer require that D have index

zero. Ch oos e s m oo thl y varyin g bases {(0~}for ke rD an d {~,} for kerD *. T he ~0~and

~p, are harm on ic spinors. Passing to d eterm inants we obtain a n onz ero section s of

[cf. (1.3)]. In term s of this section th e Quillen m etric is

llsli~ao - de t(q~, tpp) de t D *D . (1.17)

det (~bl,~bj)

Here 0P,, ~Pa) and (~bl,~bj) are the matrices of L 2 inner prod ucts of the har mo nic

spinors. T he prime in det D*D deno tes th e omission o f zero m odes. This

de ter m ina nt is defined using a ~-function, as in (1.12) an d (1.13), wh ere 2 no w runs

only ov er non zero eigenvalues. Similarly, the con nectio n form is given by

Tr is defined as in (t .15) and (1.i6), b ut restrict ing to the o rthog ona l com plem ent

of kerD*.

The ma in theorems in [B F2] determine the curvature and ho lono my of Le. Of

course, the curv ature and holo nom y are the obstruct ions to f inding a global f lat

section of 5e. The curvature formula is re lated to Bismut s heat equat ion appro ach

to the index theo rem for families [B] a nd represents the local oeometric anomaly,

while the ho lono my formula is Wit ten s

global 9eometric anomaly

[WI ] .

Theorem 1.19 [BF 2]. Th e curvature of the determ inant line bundle ~ ~ Y is the 2-

fo r m

f2 -~ = [2zri ~x A(f2 T . .. r)) ch(Of2,rvo,,z))ch(f2(E))],2).

d and ch are the usual polynomials

, / - / f2/47z ~

A(f2) = V d e t ~sinh-O ;4;) ch(f2) = Tr

e a/2= .

(1.20)

The integ rand is a differential form of mixed d egree on Z which is integra ted over

the fibers in the fibration of manifolds Z--* Y. Th e 2-form co mp on ent of the


9/31

Determinan ts Torsion and Strings 491

result ing differential form on Y is the curva ture of ~ . Fo r families of ~ ope rato rs

replace the .~ poly nom ial by the To dd polynom ial. O n the level of real

cohomology (1.19) is the Atiyah-Singer Index Theorem for families [AS2].

The h olo no m y formula is mo re com plicated to state. Let ? : $1 ~ y b e a loop. By

pul lback w e obtain a geometr ic family of Dirac operators parametr ized by SL

Th us there is an (n + 1)-manifold P fibered ove r S ~, a me tric and spin struc ture

along the fibers, etc. Introduce an arbitrary metric 9 tsl) and end ow S a with i ts

trivial spin structure. Then P acquires a metric and spin structure and so a self-

adjoint Dirac operator A (coupled to V~E). Since our constructions are

independent of the metr ic on the base , we must scale away the choice of metr ic on

the circle. Replace g~Sl) in the preceding by g~Sl)/e2 for a parameter e, and let A~

denote the Dirac operator for the scaled metric. Set

q, =~/-invar iant of A, , h ,= dim ke rA ~, ~ ,=(~/ ,+h~).

The q-invariant is a spectral invariant defined by analytic contin uatio n as the value

o f

sgn2

a t s= 0 lAPS] . An easy a rgument shows ~ (mo dl ) i s con t inuous in e.

heorem 1.21 [WI, BF2]. The holonom y of V ~e) around ? is

hol(?) = ( - 1) dxD l i m e -2~I~ .

~

Here index D is the numerical index of the D irac oper ator on X (for any f ixed value

of the parameter on the circle).

There will be ma ny applications of these form ulae in succeeding sections. We

simply remark at this stage that q-invariants are difficult to compute directly

unless the metr ic has some special symmetry. We wil l be able to compute the

holonomy in examples by topological methods involving tors ion.

An y geo me try adde d to the basic geometric da ta (1.9) is reflected by extra

structure in the determinant l ine bundle. We will have occasion to use group

act ions and holomorphic s t ructures .

Proposit ion 1.22 IF2].

Le t G be a group acting on Z which preserves all of the data

( fib rat ion ~z, spin structure, g T~tZ), pro jec tio n P , g(e), V (E) in (I .9). Furthermore, we

assume tha t a lift of the action to spinors is given. The n G acts on the determinant line

bund le ~q preserv ing its Quillen metr ic 9 ~~) and conn ectio n V ~e).

I f G acts f reely we can equal ly wo rk on the qu ot ient family of operators . In general,

though, w e resor t to equivar iant con struct ions - equivar iant bundles with

connection, equivariant cohomology, equivariant K-theory, etc. This arises twice

in String Theory. The m odu li space of Riem ann surfaces has an orbifotd structure,

so we can t reat i t us ing the equivar iant geome try of the mapping class group on the

Teichmiil ler space. On the other hand, the spacetime M is sometimes taken to be

an orbifold [DHVW, V] which a lso requires equivar iant techniques [FV].

Consider now the ~-operator on a Riemann surface. Because we are in

dimension two it is elliptic. (For a higher dimensional complex manifold it is the


10/31

492 D.S. Freed

op erat or 0 + 0* which is ell iptic.) Quil len [Q ] considers the family of Jo pe ra to rs

obta ined by varying the holom orphic s t ructure on som e extrinsic vector bundle E.

The dete rm inant l ine bund le L~e then h as a holo m orph ic structure, and one mu st

verify that the co nnec tion V ~), the co nstruc tion of which is manifestly not

holom orphic [note the adjoint in (1 .15)], i s the unique holo mo rphic connect ion

com pat ible w ith the me tric g~-~). This w as don e in [ B FI , T he ore m 1.21]. In string

theory the holomorphic structure on the Riemann surface also varies. The

following extension of [BF1, The orem 1.21] guarantees that the co nnection on A

is holomorphic in this case.

Proposit ion 1.23 [F2].

Suppose in

(1.9)

that X is a Riema nn surface and the

representation Q is chosen so tha t D is the ~ operator, possibly coupled to a

holomorphic bundle. Let Z ~ Y be a holomorphic fib ra tion o f Riem ann surfaces.

Assume that the projection P is complex. Finally, if there is an extrinsic Hermitian

bundle E we take it to have a holomorphic structure and unique compatible

holomorphic connection. Th en the determinant line bundle Lf has a holomorphic

structure, and V(~) is the holomorphic connection compatible with the QuiIIen metric.

Furthermore, if i nd e xD = 0 then the natural section is holomorphic.

In Sect. 4 we shall need to kno w ho w the ~-invariant varies with the differential

geometric parameters (metrics and connections). The appropriate formula is

conta ined in the w ork o f At iyah-Patodi-Singer , though the precise form that w e

use can be found in [BF2, Theorem 2.10] . Consider a geometr ic family of

odd

dimensiona l spin manifolds X , which is defined exactly as in (1.9). Attac hed to each

man ifold Xy is a self-adjoint Dirac opera tor, and so an ~-invariant r This is a

funct ion on the pa rame ter space wi th values in R / Z . Its d ifferential is given b y the

usual index formula.

Proposition 1.24. The variation of ~r is the 1-form

d~Y~- ~XA(~ ~(T ~rtZ))ch(O~(Tve~tZ')ch(~-~(E')](1, modl).

2 The Bosonic String

The determ inants which arise in the bos onic str ing i l lustrate the ideas of Sect. 1. W e

consider the Po lyak ov formulat ion of the bosonic s tr ing in a f iat Eucl idean

ba ck gr ou nd p e. As usual, all metrics are p ositive definite. O ur first goal is to give a

geom etric description o f the conform al anom aly. In the cri t ical dimension d = 26

there is the holo mo rphic factofizafion of Belavin and Kn izhnik - the part i t ion

funct ion is the norm square o fa h olom orphic funct ion on mod ul i space . W e der ive

i t using the connect ion on the determinant l ine bundle . As we have nothing to ad d

to other aspects of the theo ry, w e will be extremely b rief in ou r exposit ion.

Le t Z b e a R iem ann surface of genus >= 2. The cases of genus 0 and 1 can be

treated similarly, and for simplici ty we omit the mod ifications needed to treat these

cases. The partition function (for fixed genus) is

Z = ~ [ d g ] 5

[dq~]exp ( 1 )

g~U~tm ~o~Map


11/31

Determinan ts, Torsion, and Strings 493

Th e act ion 1 ~ (dq~,dq~)ohas ma ny symm etry groups: the t rans la tion group of R a,

/-. Z

the diffeom orphism grou p of S, an d the gro up COg(X) of posit ive functions on S,

acting as con form al rescalings o f the m etric. T he m easures [dg], [d~0] in this

expression are the natural

formal

mea sures divided b y the volum e (of the orbits) of

the sym me try groups. N ow the integral ov er ~o is a stan dard Gau ssian, a nd so

Z = ~ [dg]fdet'A ~ -a/2

Met Z) \ 1 ~ o ] . 2.2)

He re det 'A 0 is the regularized p rod uct of the non zero eigenvalues of the L aplacian

A oon functions. T he factor (1, 1)0, which is the total volum e of the m etric g, o ccurs

because of the t rans la t ion symmetry . By a s tandard change of var iables which we

will not duplicate here (see [DP] for a nice exposit ion), the part i t ion function is

reexpressed as

Z = S [dg]' (det'A~ - /2 f det'O*~L ~ (2.3)

M et X) \ ~

] \de t ( , , i )J

N o w [dg]'is a new formal m easure on Met(S) , which reflec ts the prod uct s t ructure

Met (S) = Conf (S)

COg(S).

Con f(S) denote the space of conformal s t ructures on

S; i t i s the quot ient M et(S) /C~ (S) . The second determinant i s the Jacob ian f rom

the change o f variables. The o per ator JL (sometimes deno ted P1 in the physics

l i te ra ture) i s the J opera tor coupled to the holomorphic tangent bundle

L ~ S .

He re we use the fact that a metric on S determ ines a com plex structure. O f course,

the seco nd dete rmin ant in (2.3) also depend s on the metric. T he {i} are a basis of

holomorphic quadratic differentials (which represent deformations on the

Teichmiiller space). W e choo se the i to be invariant unde r

COg(X).

Note tha t by

Serre duali ty kerff* is dual to the space of holomorphic quadratic differentials .

Also, the mea sure

[d9]'

depen ds o n this choice of basis . Ou r purp ose is to focus on

the d eterm inants in (2.3), not on the derivation of the m easure, so we defer to

[DP] for details .

W e ask w hether the p roduc t of de terminants in (2.3) passes to the quot ient

Conf(S)= Met(S)/COg(S), i .e., whether it is invariant under conformal rescalings

of the m etric. A ny v ariat ion is called a conformal anomaly, and the precise form ula

has b een c om pu ted directly [-Fr, A1 ]. Indeed, this is the calculati on wh ich fixes the

dimen sion d = 26, as this is the only dim ension in w hich the co nform al an om aly

vanishes. In [A 2] Alvarez observe d that form ally the families index the orem yields

the sam e result. As the gr oup COg(S) is contractible, o ne ca nno t at tach purely

topological significance to his computation. Rather, we interpret his observation

in terms o f the geo me try of determ inant l ine bundles. H is calculation is then the

curv ature of a connection, n ot a Ch ern class.

In the lang uage of (1.9) take Y= Me t(S) and Z = M et(S) S w i th the natura l

vert ical metric and projection. No spin structure is necessary as we consider ~-

operators. The repres entation ~ is chosen so that VQ= - d/2 + L, where - d/2 is the

negat ive o f the tr iv ia l d /2 -dimensiona l bund le and L ~ M e t ( S ) S is the ho lo -

morphic tangent bundle to S . (The spacet ime M is assumed to have even dimension.)

This data describes the family of operators

J-d/Z+L

on the Riemann surface,


12/31

494 D.S. Freed

param etrize d by the metric. Let ~a ~ Me t (2;) be the determin ant l ine bundle for this

family. Although these operators have nonzero index, the kernel and cokernel

have co nstan t dimens ion as the metric varies. In fact, i t is an easy consequ ence of

the Kodaira vanishing theorem (or Weitzenb6ck formula) tha t kerJL=0. Fur-

thermore , ker J i s the space of holom orphic funct ions on S , which a lways consis ts

only of the const ant functions. By Serre duality we can identify kerJ* with the dua l

of the space of hol om orp hic 1-forms. Let {co,} be a basis of holo mo rph ic l-form s

locally on Met(S). This da ta determines a section s of &o whos e nor m square is

given by (1.17):

\ (1, det'J*~- ~-a/2 ( det 'J*gL

(2.4)

1)0 de t (co~, co p) ,] \det(~bi, ~bj)

The co~ and ~i occ ur in the deno mi nat or because the Serre duality m entio ned

abov e. Since Ag = O*O we see tha t llslI 2 differs from the pr od uc t of deter mi nan ts in

(2.3) only by t he fa ctor [det (c% coa)]-d/Z. Bu t a s we ca n c hoo se co= to be inv arian t

und er COg(S), since the ff ope rato r only depen ds on the underlyi ng confo rmal

structure, and since the L 2 inner p rod uct on l-form s is conform ally invariant, this

d iscrepancy is i r relevant to the comput a t ion of the conformal anomaly . In o ther

words, the conformal anomaly vanishes precisely when the function Ilsll2 on

Met(Z) is invar iant under the ac t ion of Cg(S).

There is a natural l if t of the acti on of

C~(S)

to M et(S ) x Z - the con forma l

rescalings act tr ivially on Z - which induces an action on the determinant l ine

bund le ~ . We claim that s is invariant unde r this action. Fo r the section s is

essentially the ~ operato r, and the comp lex structu re o f 2 is unc hang ed by

rescaling the m etric. T he oth er ing redients in s - the co~, q~i,and cons tan t func t ions -

are explicit ly chosen to be conformally invariant. I t follows that the parti t ion

funct ion is confor mal ly invar iant if 9 (se) is preser ved by C~(S) . In Sect. 1 we

intro duced a unitar y conn ection V se) on ~ . Since a metric con nectio n deter mines

the metric thr ou gh its ho lon om y, it suffices to pro ve th at V -~) is invariant. Fo r this

we use the following

Lem ma 2.5.

Let ~-~ Y be a line bundle w ith connection and (# a connected Li e 9roup

acting free ly on ~ and Y Denote the connection 1-fo rm (o n the correspondin9

principal C* bundle) by co~'~) and its curvature by f2 ~e~. Suppose that fo r each X in

the Lie algebra of (~,

(i) ~xco s~= O, (ii) tx2~'~)= O.

Then the connection passes to the quotient bundle ~ / f ~ Y/f.

The lem ma is a simple application of the Car tan form ula for the Lie derivative.

Alvarez's inde x calcu latio n [A2] is the verification of con dit ion (ii) for the

bosonic str ing.

roposition 2.6.

Th e curvature 0 (se) o f the determinant line bundle fo r the

~ - d [ 2 L

family vanishes/ f d = 26.

Proof We apply Theorem 1.19. Let

i

x = ~ 0 (L) ~ 02 (M et( S) x S)


13/31

Determin ants Torsion and Strings 495

be the curvature of the holom orphic tangent bundle L. The index densi ty for the

opera tor is the Todd genus

x x 2

Todd(Y2 (L)) = 1 + ~ + ]~ + .. .. (2.7)

The C hern character of the auxi lia ry bundle - d/2 + L is

X 2

ch ~2 -a/z +L)) = (1 -- d/2) + x + - ~ + . . . . (2.8)

Then (2(~e) is 2zri time s the integ ral ov er X of the pr od uc t of (2.7) and (2.8). Since

1 + ~- + ~ (1 - d/2) + x + ~ 4) 24

we obta in

d -

2 6 )

~2 s~) = i \ 48n ] (f2(L))2' (2.9)

whi ch vanis hes identically if d = 26.

The vanishin g of O (s) in 26 dim ensi ons is univers al - it holds for any family o f

Riemann surfaces.

Proposition 2.6 is condition (i i) in Lemma 2.5. To prove the vanishing of the

conform al ano ma ly in d = 26, i t remains to verify condi tion (i) . Unfo rtunat ely, this

is where our understanding falls short . Direct verification would involve

comp uting the connect ion form in (1 .I 8), contrac ted a long the orbi ts of C~(Z) . But

this is the original, direct calc ulati on of the vari atio n of [[s II 2, whi ch we are trying to

avo id. Ro ug hl y sp eaki ng, the co nn ec tio n fro m co z:) is the first der iva tiv e o f ]1 1]2

and the curvature (2 (-~) is the second derivative. Ou r goal is to use some geom etric

principle to show that in this particular situation it suffices to com put e second

derivatives. O ne po ssible geom etric principle is holo mor phic ity: If s is a

holom orphic sec t ion of a f lat Herm it ian holom orphi c l ine bundle , then the

vanishin g of the cu rva tur e O81og [[s [[ 2 im pl ies th at lo g [[s ][2 is harm onic . Indeed , it is

possible to embe d Met(X) in a comp lex space and 5 in a hol om orp hic l ine bundle,

with the rescal ing ac t ion extending appropr ia te ly . However , we are unable to

pro ve that the ha rm on ic func tion log I[s1[2 is constant . For now we must content

ourselves with a geometric

interpretation

of the conformal anom aly ca lcula t ion ,

though we hoped for a geometr ic derivation.

Restr ic t to the cr it ica l d imension d = 26. Because the conformal anom aly

vanishes , we now have a l ine bundle ~ C o n f ( X ) together with a metric ,

connection , an d non vanish ing section s. The next step is to divide out by the action

of the connec ted com ponen t Di f f0(Z) of the diffeom orphism group. W e pau se first

to review the basic geometry of the quot ient Te ich Z )=Conf Z ) /Di f fo X) , the

Teichmi;tller sp ace .

A conformal structure on ~ determines a *-operator on l-forms (since the *-

opera t or of a Riemannian metr ic is conformally invar iant in the middle

dimension), and .2 = _ 1. Thu s * defines an alm ost com plex structu re on Z, and for

trivial dimensional reasons this structure is integrable. In fact, Conf(Z) is exactly


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496 D .S. Freed

t h e s p a c e o f c o m p l e x s t r u c t u r e s o n Z . N o w C o n f ( Z ) i ts e lf i s a c o m p l e x m a n i f o l d .

F o r t h e t a n g e n t s p a ce a t * ~ C o n f ( Z ) c o n si s ts o f r e al e n d o m o r p h i s m s A o f th e r e al

t a n g e n t b u n d l e t o Z w i t h * A + A * = 0 . A n a l m o s t c o m p l e x s t r u c t u r e is d e f i n e d b y

s e n d i n g A t o * A . O n e c a n v e ri fy t h a t t h e t o r s i o n t e n s o r f o r t h is a l m o s t c o m p l e x

s t r u c t u r e v a n i s h e s , s o t h a t C o n f ( Z ) is a c o m p l e x m a n i f o l d . ( W e w i ll n o t d e a l w i t h

t h e t e c h n i c a l i t ie s o f i n f in i t e d i m e n s i o n a l m a n i f o l d s h e r e ; s ee [ F T 1 ] f o r d e ta i ls .) T h e

p r o d u c t s p a c e C o n f ( Z ) x Z c a rr i es a n a t u r a l c o m p l e x s t r u c t u r e . O n t h e fi rs t f a c t o r

i t i s t h e s t r u c t u r e j u s t d i s c u s s e d , a n d o n { * } x Z i t r es t r ic t s t o t h e c o m p l e x s t r u c t u r e

o n Z . F u r t h e r m o r e , t h e p r o j e c t i o n C o n f ( Z ) x X ~ C o n f ( Z ) is c l ea r ly h o l o m o r p h i c .

N o w t h e o b v i o u s a c t i o n o f D i f f0 ( Z o n t h e s e s p a c es is f re e a n d p r e s e r v e s t h e

c o m p l e x s t r u c t u r e s . H e n c e t h e q u o t i e n t

Z = Co nf ( Z ) x Z /Di f f o ( Z )

$~ (2.10)

Y = C o n f ( Z )/ D i f fo ( 2 ) = T e ich( Z )

i s a h o l o m o r p h i c f i b r a t i o n o f R i e m a n n s u r f ac e s . Z is c a l le d t h e universa l

Teichmii l ler) curve.

A s i n (1 .6 ) t h e p r o j e c t i o n T ( C o n f ( Z )

Z ) - - - , T Z

passe s t o a

p r o j e ct io n P :

TZ --* T, .or tZ

o n t h e q u o t i e n t . I t i s c l e ar t h a t P is a c o m p l e x m a p p i n g .

T h e u n i f o r m i z a t i o n t h e o r e m p r o v i d e s a R i e m a n n i a n v i e w o f t h e s e sp a ce s.

R e c a l l t h a t u n i f o r m i z a t i o n is a s e c ti o n o f M e t ( Z ) ~ C o n f ( ~ ) w h i c h a ss ig n s to e a c h

~ C o n f ( Z ) a m e t r i c o f c o n s t a n t c u r v a t u r e - I . T h e a c t i o n o f D i f f o ( ) p r e s e rv e s t h e

s p a c e M e t _ l ( Z ) o f a ll s u c h m e t r ic s , a n d s o ( 2.1 0) c a n b e i d e n t i f ie d w i t h

Z = M e t _ 1 (2 ) x

Z / D i f f o Z )

2 . 1 1 )

Y = M e t _ l ( Z ) / D i f fo (Z) = Te ich (Z)

As in ( 1 .6 ) t he un ive r sa l cu r v e Z in he r i t s a pa r t i a l m e t r i c g TvmZ)a n d p r o j e c t i o n

P : T Z ~ T , m Z .

C l e a r l y t h i s p r o j e c t i o n a g r e e s w i t h t h e o n e o b t a i n e d i n ( 2 .1 0). A l so ,

t h e m e t r ic g r ,~ r t Z ) i s K / i h l e r o n e a c h f i b e r N y ( si nc e a n y R i e m a n n i a n m e t r i c o n a

R i e m a n n s u r f a c e i s K / i h l e r ) . H e n c e t h e h o l o m o r p h i c p i c t u r e ( 2 . 1 0 ) a n d R i e m a n -

n i a n p i c t u r e ( 2.1 1) a r e c o m p a t i b l e , i n t h e s e n s e o f P r o p o s i t i o n 1 .2 1.

A s o u r c o n s t r u c t io n s a r e i n d e p e n d e n t o f a n y m e t ri c o n t h e b a s e Y = T e i c h ( Z ) ,

w e h a v e n o t i n t r o d u c e d o n e . W e r e m a r k i n p a ss i ng , th o u g h , t h a t t h e n a t u r a l m e t r i c

o n M e t _ l ( Z ) d e s c en d s t o t h e T e i ch m f i ll e r sp a c e, a n d t h e l if t t o M e t _ l ( Z ) x Z

d e s c e n d s t o t h e u n i v e r sa l cu r v e. T h e s e t u r n o u t t o b e c o m p a t i b l e w i t h t h e c o m p l e x

p i c t u r e - t h e y a r e K / i h l e r m e t r i c s [ F T 2 ] . O n T e i c h ( Z ) t h i s i s c a l l e d t h e Weil-

P e t e r s s o n m e t r i c .

W e c a n c a r r y a l o n g t h e h o l o m o r p h i c t a n g e n t b u n d l e t o Z i n t h e s e c o n s t r u c -

t io n s , a n d e n d u p w i t h a h o l o m o r p h i c H e r m i t i a n l in e b u n d l e

L ~ Z .

L e t u s r e s u m e o u r d i s c u s s i o n o f t h e b o s o n i c s t r i n g . A t t h i s s t a g e w e h a v e a l in e

bu nd le 5 ~ Co nf ( Z ) w i th m e t r i c g (z~), co nn ec t i on V a '), an d no nv an i sh ing s ec t ion s .

T h e a c t i o n o f D i ff o (Z ) o n C o n f ( Z ) l if ts t o a n a c t i o n o n C o n f ( Z ) x Z , a s i n E x a m p l e

1 .5 . A l l o f o u r g e o m e t r i c d a t a i s i n v a r i a n t u n d e r D i ff o (Z ), s o b y P r o p o s i t i o n 1 .2 2 a n

a c t i o n i s i n d u c e d o n ~ . T h e a c t i o n p r e s e rv e s g (se) a n d V ~ ). F u r t h e r m o r e , i t i s a

s t a n d a r d f a c t i n R i e m a n n s u r f a c e t h e o r y t h a t D i ff o ( ~) a c t s f re e l y o n C o n f ( Z ) , si n ce

w e a s s u m e t h a t t h e g e n u s > 2 . ( I n t h e p h y s i c s l i t e r a t u r e o n e s t a te s , t h e r e a r e n o

c o n f o r m a l K i ll i n g v e c t o rs o n N . ) S o t h e re is a n i n d u c e d l i ne b u n d l e 5 ~ T e i c h o n


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De terminan ts Tors ion and St rings 497

the qu o t ie n t w i th m et r i c g (so ) and con nec t ion V se). A l te rna t ive ly , w e ca n s ta r t

d i r ec t ly w i th the geo m et r ic da ta impl ic i t in (2 .10 ), (2.11 ) and the a s soc ia ted 8_ 1a + L

f a m i l y . T h e r e s u l t i n g d e t e r m i n a n t l i n e b u n d l e , Q u i l l e n m e t r i c , a n d c o n n e c t i o n

a g r e e w i th t h e p r e v i o u s o n e s o b t a i n e d o n t h e q u o t i e n t . T h i s i s t h e c o n t e n t o f t h e

func t io r ia l i ty s t a tem en t in Th eor em 1 .10 , and i t i s c l ea r f rom the con s t ruc t ions . The

s e c ti o n s o f ~ ~ C o n f ( Z ) p a s s e s t o a s e c ti o n o f t h e q u o t i e n t ~ ~ T e i c h ( 2 : ) i f w e t a k e

care to c ho ose the loca l base s {~bi} and {o ) ,} to be D i f fo (Z) - invarian t . W e ca n go

f u rt h e r. T h e s p a c e o f h o l o m o r p h i c 1 - fo r m s v a r ie s h o l o m o r p h i c a l l y w i t h t h e

c o m p l e x s t r u c t u r e , s o f o r m s a h o l o m o r p h i c v e c t o r b u n d l e o v e r T e i c h ( Z ) . W e

r e q u i r e t h a t c0~ b e l o c a l h o l o m o r p h i c s e c ti o n s. S i m i l a r r e m a r k s a p p l y t o t h e

h o l o m o r p h i c q u a d r a t i c d i ff er en ti al s. N o w P r o p o s i t i o n 1 . 2 3 i m m e d i a t e l y im p l i e s

P ropos i t ion 2 .12 .

T h e b u n d le

~ T e i c h ( 2 ; )

i s holomorphic and V se) i s the unique

ho lomorph ic connec t i on com pa t ib le w i th t he me t r i c g s e) . Fur therm ore , s i s a

holomorphic sect ion .

T h e b o s o n i c s t r in g h a s n o a n o m a l i e s i n t h e s e n s e d e s c r i b e d i n S e ct. 1 ; t h e r e i s n o

o b s t r u c t i o n t o d e f in i n g t h e d e t e r m i n a n t s w h i c h a r i se . In d e e d , w e h a v e r e a l i z ed t h e

pa rti t io n fu nc tio n a s ][s I[2 fo r a s ec t ion s o f ~ - - .T e ic h (Z ) . N o w we ask w he th e r [1 [[ 2

i s a l s o th e n o r m s q u a r e o f a h o l o m o r p h i c f unc t i on .

P r o p o s i t i o n

2 . 1 3 [ B K , C C M R ] . In t he cr i t i ca l d imens ion d = 2 6 t he par t i t i on

func t i on f o r t he boson i c s t r i ng i s t he norm square o f a ho lomorph ic f unc t i on F on

Teich (~) .

Proof . I t su ffic es t o p r o d u c e a h o l o m o r p h i c s e c ti o n o f ~ T e i c h ( Z ) w i t h u n i t

n o r m . F o r t h i s w e o b s e r v e t h a t t h e c u r v a t u r e o f V -~) v a n i s h e s b y t h e p r e v i o u s

ca lcu la t ion (2 .6 ) . Then s ince Te ich (Z) i s s imply connec ted the re i s a g loba l f l a t

s ec t ion s o o f un i t no rm , un iqu e up to an overa l l phase . Th i s s ec t ion i s a l so

h o l o m o r p h i c , a s V -~) i s a h o l o m o r p h i c c o n n e c t i o n . T h e d e s i r e d h o l o m o r p h i c

func t ion i s F = S/So.

P r o p o s i t i o n 2.1 3 i s t h e h o l o m o r p h i c f a c t o r i z a t io n r e fe r re d t o i n t h e

i n t r o d u c t i o n .

3 T o r s i o n

P h y s i c i s t s a r e m o s t f a m i l i a r w i t h c o h o m o l o g y v i a d i f f e r e n t i a l f o r m s a n d t h e d e

R h a m t h e o ry . A s m o o t h m a n i fo l d M h a s a n e x t e r io r d e r iv a t iv e o p e r a t o r d w h i c h

f it s i n t o t h e d e R h a m c o m p l e x . T h e d e v i a t i o n o f t h is c o m p l e x f r o m e x a c t n e s s

d e fi ne s t h e d e R h a m c o h o m o l o g y o f M . I n e a c h d i m e n s i o n t his c o h o m o l o g y i s a

r e a l v e c t o r sp a c e . T o p o l o g i s t s, o n t h e o t h e r h a n d , u s e si n g u l ar c h a i n s a n d c o c h a i n s

t o d e fi ne h o m o l o g y a n d c o h o m o l o g y , w h i c h t h e n a p p e a r a s a b e li a n g r o u p s . T h e

f u n d a m e n t a l t h e o r e m o f d e R h a m s ta t es t h a t t h e t e n s o r p r o d u c t o f t h e se

c o h o m o l o g y g r o u p s w i t h th e r ea l n u m b e r s g iv e s t h e d e R h a m c o h o m o l o g y g r o u p s.

H o w e v e r , tors ion in fo rm at ion is lo s t in th i s p roces s. Reca l l tha t an e lem en t 9 o f an

a b e l i a n g r o u p G is s a i d t o b e t o r s i o n i f s o m e m u l t ip l e k - 9 ( k a n i n te g e r) v a n i s h e s.

T h e t o r s i o n e l e m e n t s in G f o r m a s u b g r o u p T o r G , a n d t h e r e is a n e x a c t s e q u e n c e

O ~ T o r G ~ G ~ F r e e G ~ O . (3.1)


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498 D .S. Freed

N o t e t h a t G p r o j ec ts on t o i ts fr ee pa r t , bu t t he r e is no G o d- g i ven p r o j ec t i on f r om G

t o T o r G .

R e c e n t ly to r s io n p h e n o m e n a h a v e e n t e r e d s t ri ng t h e o ry , m a i n l y t h r o u g h t h e

w o r k o f E d W i tt e n [ W l , W 2 , W W ] . O u r g o a l i n th i s s ec t io n is t o d e v e l o p t h e

t o r s io n i n v a r i a n t s in c o h o m o l o g y r e le v a n t to W i tt e n 's w o r k . T h e a n a l o g o u s

i nva r i an t s i n K - t h eo r y a l so p l ay a r o l e , bu t w e s t op s ho r t o f exp l a in i ng t hem he r e .

(We wil l deve lop th em in [F1] . ) S til l, t hose invar ian t s en te r the d i scuss ion th rou gh

an alyt ic exp ress ions ( involving ~- invar iants ), and pla y a role in Sect. 4 .

T h e au t ho r f aces he r e t he unenv i ab l e t a s k o f exp la i n i ng t hes e ideas, w h i ch

p e r h a p s a r e n o t c o m p l e t e l y s t a n d a r d i n m a t h e m a t ic s , t o a n a u d i e n c e p a r t l y

compos ed o f ( b r ave ) phys i c i s t s , w ho a r e p r e s umab l y un f ami l i a r w i t h i n t eg r a l

coho m ol ogy . Fo r t una t e l y , W i t ten has k i nd l y p r ov i ded expos it ions o f som e basi cs

[ W 3, W W ] , an d w e enc ou r ag e t he r eade r t o cons u l t t he s e re f e rences first. T he

beau t if u l book [ B T ] s hou l d a l s o s e r ve a s a u s e f u l gu ide . I n ou r p r e s en t expos i t o r y

acc ou nt we om i t p roofs , which a re in a ny case n o t to o d i ff icult .

L e t M b e a r e a s o n a b l e sp a ce . T h e n f o r t h e c o h o m o l o g y g r o u p s o f M w e h a v e

by (3 .1) an exa c t s eque nce

0 ~ T o r

H t( M ) -- * H I ( M ) ~

Fr ee H I(M) ~ 0. (3.2)

[ S t a nd a r d n o t a t i on i den t if ie s

H t ( M ) = H i (M ; Z )

as t he i n teg r a l cohom ol ogy . ] T he

un iver sa l co e f f i c i en t t heor em i den t if ie s t he s e g r oups i n t e r m s o f ho m ol o gy g r oups :

T o r

H t ( M ) ~_

T o r H , _ 1 M ) , F r e e

H i ( M ) ~-

F r e e

H t ( M ) .

(3.3)

Th e i som orph isms in (3.3) a re non can onic a l , tha t i s, involve choices . Th e c ano nica l

ve rsio n of (3.3) is

T o t H i (M ) ~- H o m ( T o r H i _ I (M ) , ~ / ~ ' ) ,

(3.4)

F r ee H t ( M ) -~ H om ( Fr e eH l ( M ) , ~E).

E x a m p l e s . (1 ) C on s i de r t he r ea l p r o jec t ive p l ane R P 2= 2 / ( Z / 2 ) . Since the 2-

s phe r e is s i mp l y connec t ed , t he f un dam en t a l g r ou p ~ I ( R P 2) = Z / 2 . I t f o ll ow s t ha t

t he f ir st ho m ol o gy g r oup , w h i ch i s the ab e l i an i za ti on o f t he f un dam en t a l g r oup , is

H I ( R P z) = Z / 2 . F r o m (3 .3 ) w e ob t a i n H 2( I R P 2) = Z / 2 . T h e gen e r a t o r is r ep r e s en -

ted b y a non t r iv ia l f la t line bundle , the q uo t ien t o f the t r iv ia l line bund le o ver S z by

t he non t r i v ia l Z / 2 ac t i on cove r i ng t he an t i poda l map . T he h o l o no m y o f t he fl a t

conn ec t i on a r ou nd t he non t r i v i a l loop i n R I P 2 is mu l t i p li ca t ion by - I .

(2) The lens

s p c e

Ln , k= S2n-1 / (TZ / k ) , w her e w e i den t i f y S 2n 1 s t he un i t

v e c to r s i n ~ a n d t h e g e n e r a t o r o f Z / k acts as m ul t ip l icat io n by e 2~i/~. T h en as in (1)

w e d e d u c e H 2 ( L , , k ) = Z / k . In fact ,

[ Z / k ,

/ e v e n ,

H Z ( L , , k ) = ] O , l o d d , / < 2 n - l , ( 3 . 5 )

[ Z , / = 2 n - l ,

T h i s exampl e occu r s i n [ W 2 , W W ] .

(3) S imp ly conn ec te d spaces can a l so exhib i t to r s io n in th e i r in tegra l

c o h o m o l o g y . F o r t h e L i e g r o u p E 8 w e h a v e H6 Es)=Z/2.


17/31

D e t e r m i n a n t s T o r s i o n a n d S t r in g s 4 9 9

(4) Se t Y = Met(Sl)/Diff(S~). T h e n n~(Y) = no( D if f (S l ) ) = 7Z/992. (W e gloss

ov er the f ac t tha t D i f f (S 1) doe s no t ac t f r ee ly on M et (S1 ) ; s t r i c t ly speak ing , we

s h o u l d w o r k w i t h t h e e q u i v a r i a n t q u o t i e n t. ) I t fo l l o w s t h a t H 2 ( y ) = ~g /9 92 . T h e

d e t e r m i n a n t s c o n s i d e r e d i n [ W 1 ] a r e s e c t io n s o f a l in e b u n d l e ~ Y , w h i c h i s

de te rm ined topo log ic a l ly b y c~(~ ) ~ 2~ /992 . W i t t en s g lob a l a no m aly de tec t s th i s

t o p o l o g i c a l i n v a r ia n t o f t h e d e t e r m i n a n t l in e b u n d l e . I n g e n e r a l, h o w e v e r , t h e

g l o b a l a n o m a l y i s not d e t er m i n e d b y t he t o p o l o g y o f ~ , b u t r a t h e r d e p e n d s o n t h e

con nec t ion V ze), a s exp la ined in Sec t. 1 . O n ly fo r to r s io n loo ps i s the g loba l

a n o m a l y p u r e l y t o p o l o g i c a l .

F i x a n i n te g r al c o h o m o l o g y c l as s c e HZ(M ). A cco rdin g to (3 .2) ther e i s a w el l -

de f ined f r ee pa r t

cfr e FreeH Z(M )~-Hom (Hz(M ), 7~),

w h i c h c a n b e d e t e c t e d b y

eva lua t ing on l -d imens iona l cyc les . In t e rm s o f d i f fe ren t ia l fo rms , c i s r ep rese n ted

b y a c l o s e d / - f o r m y ~ O Z (M ) , a n d i f F :

Q - ~ M

i s a m a p o f a c l o s e d / - m a n i f o l d ( o r

d i f f e ren t i ab le cyc le ) Q in to M , then Q ca rr i e s a ho m olo gy c las s [Q] ~ Hz(M), a n d

= ~ F* (y ) . (3 .6 )

t2

T h e a n g l e b r a c k e t s d e n o t e t h e p a i r i n g Ht(M)H~(M)~2~. So (3 .6 ) desc r ibes an

a n a l y t i c m e t h o d o f d e t e c t i n g F r e e H Z ( M ) , a s is w e l l - k n o w n t o p h y s i c is ts .

T h e t o r s i o n i s m o r e d i ff ic u lt to d e t e c t . N o t e f r o m (3 .2 ) t h a t t h e t o r s i o n p a r t o f c

i s we l l -de f ined on ly wh en c fr~ = 0 . Th e k ey to de tec t ing the in teg ra l in fo rm at ion in

c b e y o n d t h e r a t i o n a l i n f o r m a t i o n i s t o s t a r t w i t h t o r s i o n i n f o r m a t i o n i n

h o m o l o g y . T h u s f ix p mT o r H z _ I( M ) . W e w i ll u s e p t o d e n o t e b o t h t h is h o m o l o g y

c las s and a cyc le r ep resen t ing the h om olo gy c lass . S ince p is a to r s ion c lass , k - p = 0

i n H z - a ( M ) f o r s o m e p o s i t iv e i n te g e r k . C h o o s e a n / - c h a i n q ~ Cz(M) with ~q = k . p

(as chains ) . T hu s 0 q = p is an in tegra l chain , so van ishes ( rood 1). L et ~ q

d e n o t e t h is c h a i n w i t h c o e ff ic i en t s r e d u c e d ( r o o d 1 ); t h e n ~ q i s c l o se d . H e n c e ~ q

v

r e p r e s e n t s a n e l e m e n t o f Hz(M; Z/k) . [ N o t e Z / k C I I~ /Z C N /Z ; the cyc l ic g ro up ~g/k

i s k . . . . . . W e o f ten wr i t e Hz(M;(I~/7Z) o r Hz(M; N /Z) e v e n t h o u g h ~ q

l iv e s i n t h e m o r e p r e c is e g r o u p Hz(M; Z/k) . F i n a ll y , w e c a n e v a l u a t e t h e i n te g r a l

1

c las s c on ~ q to ob ta in

( ~ q]---,c) ~ Z / k . (3.7)

T h i s i s t h e b a s i c t o r s i o n i n v a r i a n t i n c o h o m o t o g y . N o t i c e t h a t i n g e n e r a l i t d e p e n d s

o n q ( n o t j u s t o n p ) . H o w e v e r , i f c is a t o r s i o n c l as s t h e n w e h a v e t h e f o l l o w i n g

Pro po s i t ion 3 .8 . Suppo se that c r*e = O. Th en c is a torsion class, wh ich by (3.4) can be

identified w ith an elemen t c r ~ H o m ( T o r H z _ I ( M ) ; O _ JZ ). Furthermore,

under this identification. In pa rticular, this Z/ k invariant depends only on p, not on q.


18/31

500 D S Freed

D e n n i s S u l l iv a n u s e s t h e se i n v a r i a n t s t o s t u d y g e n e r a l p r o p e r t i e s o f m a n i f o l d s f f

H is po in t o f v iew is tha t an in teg ra l c oh om olo gy c las s can be spec i fi ed by i t s ~E

pe r io ds (3.6) an d i t s O~/;E per iod s (3 .7). (This i s [M S, Sect . 2] . ) O u r pre sen t in te res t i s

n o t i n t h e s e t o p o l o g i c a l d i s c u s si o n s, b u t r a t h e r i n a n a l y t i c e x p r e s s i o n s fo r t h e s e

invar ian t s w hen c i s a cha rac te r i s t i c c l a s s o f a vec to r bund le .

Cons ide r f i r s t a l ine bund le

L ~ M .

E n d o w L w i th a H e r m i t ia n s t r u c t u re a n d

i o~L); h e n

n i t a r y c o n n e c t i o n , a n d d e n o t e t h e c u r v a t u r e 2 - f o r m b y

0 ~L).

Set 7 =

i s a c l o s e d f o r m r e p r e s e n ti n g t h e r e a l c o h o m o l o g y c l a ss c l L) R~ HZ M; ~) . N o w

s u pp o se f :

P ~ M

i s a l o o p ( P = S 1 ) w h i c h r e p r e s e n t s a t o r s i o n e l e m e n t i n H i ( M ) .

T h e n w e c a n f i nd a 2 - m a n i f o ld Q w h o s e b o u n d a r y d Q c o n s i st s o f k d i s j o in t c o p i e s

o f P , a n d a m a p F : Q ~ M w h i c h re s tr ic ts o n e a c h b o u n d a r y c o m p o n e n t aQ) to

th e m a p f :

P ~ M

( see F ig . 3 ). Cons ide r the expres s ion

i l n h o l ( P ) + 1 S F *(? ,) ( m o d l ) . (3 .9 )

2 -7 kQ

( T h e s ig n s a r e e x p la i n e d b y t h e fa c t th a t t h e l o g h o l o n o m y i s

minus

the in teg ra l o f

t h e c o n n e c t i o n f o r m a r o u n d t h e lo o p . ) B y S t o k e s t h e o r e m k t im e s t h is e x p r e ss i o n

van i shes . There fo re , i t i s a topo log ica l invar ian t ( t ak ing va lues in

Z/k) ,

a n d i t

s h o u l d c o m e a s n o s u r pr is e t h a t

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