METRICS, CONNECTION S AN D GLUIN G THEOREM S

C. TAUBE S

1. INTRODUCTIO N

My goa l i n thes e lecture s i s t o provid e a n introductio n t o som e o f

the analyti c underpinning s fo r th e geometr y o f anti-sel f dualit y i n 4 -

dimensions. Anti-sel f dualit y i s rather specia l t o 4-dimension s an d th e

imposition of this condition on curvatures of connections on vector bun -

dles and als o on curvatures of Riemannian metric s has resulted i n som e

spectacular mathematics . I n th e ensuin g lectures , I wil l revie w som e

of the basi c geometry, bu t eve n so , I will assum e tha t th e reade r ha s a

generalist sor t o f background i n differential geometry , such as one might

obtain b y readin g a standar d tex t o n th e subjec t (Kobayash i an d No -

mizu's book s [32 ] come s t o mind) . Th e final lectur e consist s o f ope n

problems an d conjectures . (Actually , i t consist s of a series o f question s

that I would ask an alien spacefarer shoul d one land in my back yard an d

profess a profound understandin g of 4-dimensional geometry/topology. )

Before beginning , I shoul d poin t ou t som e o f th e fundamenta l ref -

erences fo r th e subject . Th e startin g referenc e i s th e manuscrip t b y

Atiyah, Hitchi n an d Singe r [5 ] where the basic geometry o f anti-self du -

ality is presented. M y presentation of the geometry borrows rather heav-

ily fro m [5] . Nex t come s th e boo k b y Free d an d Uhlenbec k [24 ] whic h

describes Simo n Donaldson' s remarkabl e first theore m whic h connect s

anti-self dualit y t o geometri c topology . Free d an d Uhlenbeck' s boo k

also describe s man y o f th e underlyin g analytica l issues . Th e nex t fun -

damental reference is the book by Donaldson and Kronheimer [20] . Thi s

marvelous manuscript details almost all you have to know to understan d

the anti-self dua l equations an d their application s a s of 1992. I also ad d

as a basi c reference, th e recen t pape r b y Kronheimer an d Mrowk a [31];

these contai n the lates t breakthrough s o n the subject . Finally , the wel l

equipped gaug e theor y librar y shoul d hav e a referenc e t o Floe r coho -

mology, s o I sugges t th e forthcomin g boo k b y Donaldson , Fukay a an d

Kotschick [18].

Some of the subject matte r i n these lectures will overlap with variou s

parts o f the aforementione d references . However , a s I remarke d above ,

my goal here i s to spotligh t th e analyti c issues , an d als o to presen t th e

1

http://dx.doi.org/10.1090/cbms/089/01