CONJUGATE SU(r)-CONNECTIONS AND HOLONOMY GROUPS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 18, Numbe 3, Pages S (99)05457-X Aticle electonically published on Septembe 9, 1999 CONJUGATE SU()-CONNECTIONS AND HOLONOMY GROUPS JIN-HONG KIM (Communicated by Chistophe Coke) Abstact. In this aticle we show that when the stuctue goup of the educible pincipal bundle P is SU() andq P is an SO()-subbundle of P, the ank of the holonomy goup of a connection which is gauge equivalent to its conjugate connection is less than o equal to [ ], and use the estimate to show that fo all odd pime, if the holonomy goup of the ieducible connection as above is simple and is not isomophic to E 8, F 4,oG,thenitis isomophic to SO(). 1. Intoduction In thei pape [7], S. Kobayashi and E. Shinozaki intoduced the concept of conjugate connection in a educible pincipal G-bundle P. By looking at pincipal bundles they had a bette undestanding of conjugate connections whose notion in affine diffeential geomety had been known fo a long time. Moeove, they showed that the goup consisting of automophisms of a Lie goup G fixing a Lie subgoup H, the automophism goup, induces a compatible action on the quotient space of connections modulo gauge goup on a pincipal G-bundle P that is educible to a pincipal H-subbundle Q. It was also obseved in [4] that the action does not depend on the eduction of a pincipal G-bundle to a subbundle. When we fix a Riemannian metic on the base manifold and G is compact, the automophism goup also acts on the moduli space of Yang-Mills connections and on the moduli space of anti-self-dual (ASD) connections, when the base manifold is 4-dimensional. Howeve, it tuns out that the inne automophism goup which consists of inne automophisms in the automophism goup tivially acts on the quotient space of connections. Thus, it is natual to conside the action on the quotient space of connections fo the oute automophism goup. It is well known that fo compact simple Lie goups only Lie goups of type A (>1), D ( 4), and E 6 have a nontivial oute automophism goup (see [10]). In case of G = SU(3), we showed a localization theoem of the moduli space of ieducible ASD connections in a educible pincipal SU(3)-bundle P along the moduli spaces of ieducible ASD connections in the SO(3)-subbundles Q of P (see [3]). One of the key ingedients of the poof was to know what is the holonomy goup of an ieducible connection which is gauge equivalent to its conjugate connection. This is elatively easy since the ank of the stuctue goup is just. But as the ank of the stuctue goup Received by the editos Apil, Mathematics Subject Classification. Pimay 53C c 1999 Ameican Mathematical Society

2 866 JIN-HONG KIM becomes highe, it does not seem to be a tivial poblem to know what ae the holonomy goups at all. In this pape, as a continuation of the pape [3] we give a shap ank estimate of the holonomy goup of an ieducible connection which is gauge equivalent to its conjugate connection, and as an easy consequence we pove that when the ank of the stuctue goup plus one is odd pime, if the holonomy goup is simple and not isomophic to E 8,F 4,oG, then it is isomophic to SO(). We oganize this pape as follows. In Section, we biefly ecall the definition of conjugate connection and an impotant theoem of Kobayashi and Shinozaki. We will show a shap ank estimate of the holonomy goup in Section 3.. Peliminaies In this section, we will set up some notations and ecall biefly the definition of conjugate connection in educible pincipal bundles (see [7] fo moe details) and a theoem fom [8], due to Kobayashi and Shinozaki, which motivates this pape. Let G be a Lie goup with Lie algeba g, and let H be a closed subgoup with Lie algeba h. Let P be a pincipal G-bundle ove a manifold M with pojection π, andqa pincipal H-subbundle of P. In geneal, such a subbundle does not necessaily exist. We cove M by open sets U i with local sections ξ i : U i Q. Then, the tansition functions a ij : U i U j H ae defined by ξ j (x) =ξ i (x)a ij (x). A connection fom {A i } on P is defined as a family of g-valued 1-foms A i on U i which satisfies the following tansfomation ule fom A i to A j : (.1) A j = a 1 ij A ia ij + a 1 ij da ij on U i U j. Since A i is not defined on all of M, we define a g-valued 1-fom on P fom A i as follows: A = g 1 A i g + g 1 dg, g G, on π 1 (U i )=U i G. Then it is easy to see that A is a g-valued 1-fom on P,and it follows that ξi (A) =A i. Given σ Aut(G, H), we set A σ i = σ(a i ) and apply σ to (.1). Since we took local sections s i : U i Q and thus the tansition functions a ij s ae H-valued, we have A σ j = a 1 ij Aσ i a ij + a 1 ij da ij. Thus, {A σ i } defines a connection on P.Wecallittheσ-conjugate connection of A elative to Q, denoted A σ.notethata σ is not σ(a). In fact, σ(a) maynotbe a connection 1-fom. Now, let us set up some notations which will used late in this pape. Let Aut(G, H) be the goup of automophisms of G leaving all elements of H fixed, Inn(G, H) be the goup of all inne automophisms in Aut(G, H), and let Out(G, H) =Aut(G, H)/ Inn(G, H). Out(G, H) is called the oute automophism goup. Given σ Aut(G, H), the induced automophism of g is denoted also σ. Fo G = SU(), we fix an automophism σ given by a ā. Note that the automophism σ of SU() ( 3) is oute and a geneato of the oute automophism goup (see [10]).

3 CONJUGATE SU()-CONNECTIONS AND HOLONOMY GROUPS 867 We fix a point u 0 Q.WedenotebyH u0 (A) the holonomy goup of a connection A with efeence to u 0. We call a connection in P geneic if its holonomy goup coincides with G, and call a connection ieducible if its isotopy goup, consideed as a closed Lie subgoup of G, coincides with the cente of G (see [1] fo moe details). Finally, we state an impotant theoem of Kobayashi and Shinozaki [8] in this pape. Theoem.1. Let σ Aut (G, H) and A be a connection in P. Assume that A σ is gauge equivalent to A unde a gauge tansfomation ϕ. If we define an element a G by ϕ(u 0 )=u 0 a,then σ(g)=a 1 ga fo g H u0 (A). In paticula, if the holonomy goup is G, thenσis the inne automophism defined by a 1 above. As a consequence, Out (G, H) acts feely on the geneic pat of the quotient space of connections, and Aut (G, H) acts feely on the geneic pat of the famed quotient space of connections. 3. Main esults In this section we pove ou main esults in this pape. Fist, we begin with the following Theoem 3.1. Let P be a pincipal SU()-bundle ove a simply connected manifold M that is educible to an SO()-subbundle Q. LetAbe an ieducible connection in P. Assume that the σ-conjugate connection A σ of A is gauge equivalent to A unde a gauge tansfomation ϕ. Then, the holonomy goup H u0 (A) is a compact, connected, semisimple Lie subgoup of SU() of ank less than o equal to [. Remak 3.. The uppe bound in the theoem is shap because thee is an ieducible connection A which defines one in the pincipal SO()-subbundle Q and whose holonomy goup is SO(), and SO() has ank [. Notethatweneedthe ieducibility of the connection only to say that the holonomy goup is semisimple. To pove the theoem, we fist pove the following Lemma 3.3. Let P be a pincipal SU()-bundle whose base manifold is simply connected and paacompact. Then, evey holonomy goup of an ieducible connection is a compact, connected, semisimple Lie subgoup of SU(). Remak 3.4. Since it is well known that the holonomy goup is a connected Lie goup ([6], Theoem 4.), it emains to pove that the holonomy goup is closed and semisimple. In geneal, the statement in the lemma may not be tue. Thus, we would like to point out that the statement by S.K. Donaldson and P.B. Konheime, line 4 fom the top on page 13 in [1], that it can be shown that the holonomy goup is a closed Lie subgoup of G is false, in geneal. In fact, S. Kobayashi [5] gave a vey simple counteexample to thei claim: Let H be an open Lie subgoup of a Lie goup G, and let Q be a pincipal H-bundle. Let P be the pincipal G-bundle extending Q. Since H G,we have Q P.That is,qis a pincipal H-subbundle of P. Now, taking a connection in P which defines in the pincipal H-subbundle Q and whose holonomy is H completes the counteexample. So, the point of this

4 868 JIN-HONG KIM lemma is that if the stuctue goup is SU(), then the holonomy goup is always a closed Lie subgoup of SU(). Poof. To pove the lemma, we fist note that evey non-semisimple subalgeba of type A is educible by E. Catan and E. B. Dynkin (e.g., see [), when we use the odinay (o standad) epesentation. So, if the Lie algeba of the holonomy goup H u0 (A), in shot holonomy algeba, is not semisimple, then its centalize would be lage than the cente of SU(), Z. Thus, the holonomy goup is semisimple. Now, the poof follows immediately fom the esult of K. Yosida [11] (see also [6]) that evey connected semisimple Lie subgoup of GL(; C) is closed and the fact that SU() is closed in GL(; C). Poof (Theoem 3.1). We denote by [x 1,...,x ] the matices of the fom 1x1... (x i eal), 1x fo the sake of simplicity. We next define polynomial functions g 1,g,... on the Lie algeba u() by λ λ k t(exp( X)) = t( 1 k!( 1) X k ) k = = k=0 k=0 k=0 λ k k!( 1) k t(x k ) λ k k! g k(x). Then, it can be shown (see Theoem XII..5 in [6]) that g 1,...,g ae algebaically independent and geneate the algeba of polynomial functions on u() invaiant by ad(u()). Let T be the subgoup of U() consisting of diagonal elements. Then, its Lie algeba t consists of the matices of the fom [x 1,...,x ] (x i eal). Note that when esticted to t, the polynomial functions have the following special fom: g k ([x 1,...,x ]) = fo all k. By Lemma 3.3, the holonomy goup H u0 (A) is a compact, connected, semisimple Lie subgoup of SU(). Thus, it emains to pove that the ank of the holonomy goup H u0 (A) islessthanoequalto [. To do this, let l be the Lie algeba of the holonomy goup H u0 (A). Then, if we define an element a in SU()byϕ(u 0 )=u 0 a as in Theoem.1, we have (3.1) j=1 x k j σ(x) =a 1 Xa, X l.

5 CONJUGATE SU()-CONNECTIONS AND HOLONOMY GROUPS 869 Thus, when k is an odd intege (1 k ) it follows fom (3.1) g k (X) =g k (a 1 Xa)=g k (σ(x)) = g k ( X) =g k ( X T )=g k ( X) =( 1) k g k (X) = g k (X), X l. Hence, we obtain (3.) g k (X) =0 fo all X l and all odd integes k (1 k ). Set Z k = {[x 1,...,x ] g k ([x 1,...,x ]) = 0} fo all integes k (1 k ). We will fist estimate the dimension of the complete intesection Z = k=odd 1 k of the algebaic vaieties Z k. To do this, assuming that + 1 is even, it suffices to compute geneically the maximal ank of the following ( ) +1 -matix g 1 g x x g 3 g x x = 3x x g g x 1... x x 1... x It is easy to see fom the Vandemonde fomula that the deteminant of the ( ) +1 ( +1 ) -submatix (3.4) 3x x x 1... x +1 of the matix (3.3) is 3 5 ( 1) ( 1)(+1) 8 (x i x j ), Z k i<j 1 i,j +1 which is not zeo geneically, i.e., the matix (3.4) has ank ( ) +1 geneically. Thus, the dimension of Z is less than o equal to ( ) +1 = 1 [ =. Thecasethatis odd can be shown similaly. On the othe hand, since Z always contains a vecto space of the fom {[x 1, x 1,...,x [, x [ ] x i eal} ( even), o {[x 1, x 1,...,x [, x [,0] x i eal} ( odd), whose dimension is [, we see that the dimension of Z is actually equal to [.

6 870 JIN-HONG KIM Now, let t be a maximal tous of the Lie algeba l. Since the maximal tous t of l can be assumed to be contained in a maximal tous of su() of the fom {[x 1,...,x ] x 1 +x +...+x =0}, the above agument togethe with (3.) shows immediately that t is contained as a vecto space in the complete intesection Z, which implies that the ank of the Lie algeba l is less than o equal to [. This completes the poof. Using Theoem 3.1 and the classification of compact, connected, simple Lie goups of E. Catan, we get the following Theoem 3.5. Let P be a pincipal SU()-bundle ove a simply connected manifold M that is educible to an SO()-subbundle Q. LetA be an ieducible connection in P. Assume that the σ-conjugate connection A σ of A is gauge equivalent to A and that is a positive odd pime intege. If the holonomy goup H u0 (A) is simple and is not isomophic to E 8, F 4,oG,thenH u0 (A)is isomophic to SO(). Remak 3.6. The theoem extends the esult fo = 3 in [3] to all positive odd pime integes. It is not clea to the autho whethe o not thee is an ieducible connection which is gauge equivalent to its conjugate connection and whose holonomy goup is isomophic to E 8, F 4,oG. Poof. Since is odd pime, the cente of the holonomy goup is isomophic eithe to the cente of SU(), Z o to {0}. Fom the classification of compact, simple Lie goups by E. Catan, we have only Lie goups of type A l,b l,c l,d l,e 6,E 7,E 8,F 4 and G. It is also well known (see Theoem II.4.10 and V.6.38 in [9]) that the cente of Lie goups A l,b l,c l,d l,e 6,E 7,E 8,F 4,andG is isomophic to Z l, 0, Z, Z, Z 3, Z, 0, 0, and 0, espectively. Since thee is no compact simple Lie goup whose cente is Z and whose ank is less than o equal to 1, the holonomy goup must be isomophic to B k fo k (1 k ), unless H u0 (A) is isomophic to E 8,F 4,o G. Howeve, if H u0 (A) is isomophic to B k fo 1 k 1, it is easy to see that the centalize of H u0 (A) is stictly lage than Z. Thus, the holonomy goup H u0 (A) mustbe isomophic to B [, unless H u 0 (A) is isomophic to E 8,F 4,oG. This completes the poof. Now, a few concluding emaks ae in ode. Ou ultimate goal of this pogam is to use the above esults to deduce infomation about the geomety of 4-manifolds o the set of stable holomophic stuctues, and so on, as we did in [3]. To do this we fist need to extend above esults to the semisimple case, and we hope that we can deal with these poblems in the futue. Refeences [1] S.K. Donaldson and P.B. Konheime, The Geomety of fou-manifolds, Oxfod Univesity Pess (1994). MR 9a:57036 [ E. B. Dynkin, Semisimple subalgebas of semisimple Lie algebas, Math. Sbonik N. S. 30 (195), ; Ame. Math. Soc. Tans., Se., 6 (1957), MR 13:904c [3] J.H. Kim, Conjugate SU(3)-connections and a mod vanishing theoem, pepint (1998). [4], Conjugate Non-abelian monopoles and Localization of moduli spaces of non-abelian monopoles, pepint (1997). [5] S. Kobayashi, pivate communication.

7 CONJUGATE SU()-CONNECTIONS AND HOLONOMY GROUPS 871 [6] S.Kobayashi and K.Nomizu, Foundations of Diffeential Geomety, Vol I and II, Wiley, New Yok, MR 97c:53001a [7] S. Kobayashi and E. Shinozaki, Conjugate Connections in Pincipal Bundles, Geomety and Topology of Submanifolds VII, Wold Scientific Publ.(1995), MR 98c:53035 [8], Conjugate Connections and Moduli Spaces of Connections, Tokyo J. Math. 0 (1997), MR 98c:53036 [9] M. Mimua and H. Toda, Topology of Lie Goups, I and II, Tans. of Math. Mono. Vol. 91, AMS, MR 9h:55001 [10] J. A. Wolf, Spaces of constant cuvatue, Publish o Peish Inc., [11] K. Yosida, A theoem concening the semi-simple Lie goups, Tohoku Math. J., 43 (1937), Depatment of Mathematics, Univesity of Califonia, Bekeley, Califonia addess: jinkim@math.bekeley.edu Cuent addess: Depatment of Mathematics, 51 Mathematical Sciences, Oklahoma State Univesity, Stillwate, Oklahoma addess: jinkim@math.okstate.edu