Table Of Content

Orientability in Yang-Mills Theory over 0 Nonorientable Surfaces 1 0 2 Nan-Kuo Ho, Chiu-Chu Melissa Liu, and Daniel Ramras n a J 6 Abstract 1 The first two authors have constructed a gauge-equivariant Morse ] G stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface Σ. This space can be S identifiedwiththereallocusofthespaceofconnectionsonthepullback . h ofthisbundleovertheorientabledoublecoverofΣ. Inthiscontext,the t a normal bundles to the Morse strata are real vector bundles. We show m that these bundles, and their associated homotopy orbit bundles, are [ orientable for any n when Σ is not homeomorphic to the Klein bottle, and for n 3 when Σ is the Klein bottle. We also derive similar 4 ≤ orientability results when the structure group is SU(n). v 2 8 1 Introduction 8 4 . 0 Consider a finite stratification of a manifold S. If each stratum is µ µ 1 a locally closed submanifold of{SAw}ith codimension d , and the indexAset is µ 8 partially ordered so that for any λ, 0 : v ¯ i λ µ X A ⊂ A µ λ r [≥ a holds,then iscalledaMorsestratification. AMorsestratification gives µ {A } a Morse polynomial M (S;K) = tdµP ( ;K), where P ( ;K) denotes t t µ t A − the Poincaré polynomial with coefficients in the field K. TheMorse inequal- P ities state thatthereexists apolynomialR (t)with nonnegative coefficients K such that M (S;K) = P (S;K)+(1+t)R (t). t t K Under fairly general conditions, the Morse inequalities hold for K = Z . If, 2 moreover, thenormalbundleN toeach stratum isorientable, thenthese µ µ A Morse inequalities hold for any coefficient field K. 1 Atiyah and Bott studied the moduli space of flat G-connections over a Riemann surface via this Morse theoretical approach when the structure group G is compact and connected. One of their main results is the computation of the -equivariant Poincaré series P ( ;K) for the space of G tG Aflat Aflat flatconnections onaprincipalbundleover aRiemannsurface,where isthe G gauge group. They used the Yang-Mills functional, which is invariant under the action of the gauge group, as a Morse-type function and constructed a gauge equivariant Morse stratification on the space of all connec- µ {A } A tions on a principal bundle over a Riemann surface. The space of flat flat A connections sits inside of the unique open stratum and is a deformation ss A retract of via the Yang-Mills flow (Daskalopoulos [5], R˚ade [22]). Thus, ss A and are homotopy equivalent, and P ( ;K) = P ( ;K). Aflat Ass tG Aflat tG Ass With this Morse stratification, one can write down the -equivariant Morse G series of the space of all connections, A M ( ;K) = tdµP ( ;K), tG A tG Aµ µ I X∈ and the -equivariant Morse inequalities G M ( ;K) = P ( ;K)+(1+t)R (t), tG A tG A K where d is the codimension of the stratum , I is the index set of the µ µ A stratification, and R (t) is a power series with nonnegative coefficients. In K their construction, the normal bundles N are complex vector bundles, thus µ orientable, so K can be any field. In order to compute the -equivariant G PoincaréseriesP ( ;K),oneneedsfouringredients: P ( ;K),d ,R (t), tG Ass tG A µ K and P ( ;K) for all = . Since the space of all connections over tG Aµ Aµ 6 Ass A a Riemann surface is an infinite dimensional complex affine space and thus contractible, P ( ;K) is just P (B ;K), the Poincaré series of the classi- tG A t G fying space of . As for P ( ), Atiyah and Bott found reduction formulas G tG Aµ [1, Proposition 7.12] that reduce the question to smaller groups. The Morse index d can be computed by Riemann-Roch [1, Equation (7.15)]. Most µ importantly, they showed that this stratification is -equivariantly perfect G [1, Theorem 7.14], i.e. R (t) = 0, and K P (B ;K) = P ( ;K) = tdµP ( ;K). t G tG A tG Aµ µ I X∈ In the end, this method produces a recursive formula for P ( ;K). tG Ass The first two authors defined a Yang-Mills functional on the space of connections over any nonorientable surface Σ in [12]. Using this Yang-Mills 2 functional,theyconstructeda -equivariantMorsestratificationonthespace G of connections over Σ. To be precise, consider the orientable double cover π :Σ˜ Σ,andletP˜ = π P over Σ˜ denotethepullbackofaprincipalbundle ∗ → P over Σ. The non-trivial deck transformation of Σ˜ induces an involution on the space õf connections of P˜ whose fixed point set is exactly the space A of connections of P. Ho and Liu define the Yang-Mills functional L on A A to be the restriction of the Yang-Mills functional L˜ on the fixed point set of ˜. The absolute minimum of L is zero, achieved by flat connections on P. A The gradient flow of L defines a -equivariant Morse stratification on µ G {A } . Indeed, the Morse stratification is just the intersection of with µ A {A } A the Morse stratification ˜ . This procedure also tells us that the normal µ {A } bundle N to each stratum in is the fixed locus of the normal bundle µ µ N˜ to each stratum ˜ in A˜, whAich is complex (we will discuss in detail µ µ A A the various involutions on vector bundles in Section 3). In other words, the normal bundle N to each Morse stratum is a -equivariant real vector µ µ A G bundle and hence is not automatically orientable. The -equivariant Morse series of this stratification is µ G {A } M ( ;K) = tdµP ( ;K). tG A tG Aµ µ I X∈ Since the Yang-Mills strata admit gauge-invariant tubular neighborhoods (see [23] for a construction), one can use the stratification to obtain the -equivariant Morse inequalities G M ( ;K) = P ( ;K)+(1+t)R (t). tG A tG A K A priori, we cannot assume orientability of the normal bundles N , so the µ Morse inequalities holds only for K = Z . To compute the Poincaré se- 2 ries P ( ;K), we again need four ingredients: P ( ;K), d , R (t), and tG Ass tG A µ K P ( ;K) for all = . Reduction formulas for P ( ;K) and a for- tG Aµ Aµ 6 Ass tG Aµ mula for d were given in [12, 13]. On the other hand, the computation of µ P ( ;K) is rather difficult when K = Z due to the existence of 2-torsion tG A 2 elements in integral cohomology (see [12, Section 5.3] [13, Section 2] for more details) and one is encouraged to consider rational coefficients. Hence we need to establish orientability of the normal bundles. Let X denote the homotopy orbit space EG X. Then (N ) is hG ×G µ h also a real vector bundle over ( ) . In this paper, we fix the structGure Aµ h group of the principal bundle P to beGthe unitary group U(n) or the special unitary group SU(n). Our main result is: 3 Theorem 1. Suppose that either (i) χ(Σ) = 0 (so that Σ is homeomorphic to the Klein bottle) and n 3, or (ii) χ(Σ) = 0 and n is any positive ≤ 6 integer. Then (N ) is an orientable vector bundle over ( ) for all µ. µ h Aµ h As a consequence, NG is an orientable vector bundle over forGall µ. µ µ A In [14], the first two authors discuss how far this stratification is from being perfect, i.e. what the power series R (t) looks like. They define K the notion of antiperfection, which leads to some conjectural formulas for P ( ;K). tG Ass Thomas Baird [4] has recently proven the formula conjectured in [14] for the -equivariant Poincaré series of the space of flat U(3)-connections over G a non-orientable surface. His argument relies on Yang-Mills theory, and in particular uses our orientability results. Thus Baird’s work may be viewed as a concrete application of the results in this paper. 2 Preliminaries Let Σ˜ be a Riemann surface. Let Pn,k denote the degree k principal U(n)- Σ˜ bundle on Σ˜. Let ρ : U(n) GL(n,C) be the fundamental representation, → and let E = Pn,k Cn be the associated complex vector bundle over Σ˜. Σ˜ ×ρ Then E is a rank n, degree k complex vector bundleequippedwith a Hermi- tian metric h, and the unitary framebundleU(E,h) of the Hermitian vector n,k bundle (E,h) is isomorphic to P as a C principal U(n)-bundle. Σ˜ ∞ 2.1 Hermitian, (0,1)-, and (1,0)-connections n,k n,k Let (P ) be the space of U(n)-connections on P , which can be identi- A Σ˜ Σ˜ fiedwith (E,h), thespaceof Hermitian connections on(E,h) (connections A on E which are compatible with the Hermitian structure h, cf. [26, pp.76]) . Itisacomplex affinespacewhosevector spaceof translations is Ω1(adPn,k), Σ˜ Σ˜ where the complex structure is given by the Hodge star (cf. [1]). Let (E) denote the space of (0,1)-connections ∂¯ : Ω0(E) Ω0,∗1(E), and let C(E) Σ˜ → Σ˜ C′ denote the space of (1,0)-connections ∂ : Ω0(E) Ω1,0(E). Recall that a Σ˜ → Σ˜ (0,1)-connection (resp. (1,0)-connection) defines a holomorphic (resp. anti- holomorphic) structure on E if and only if ∂¯2 = 0 (resp. ∂2 = 0) (cf. [7, Section 2.2.2]); now Ω0,2 = 0 (resp. Ω2,0 = 0) since dimCΣ˜ = 1, so the Σ˜ Σ˜ integrability condition ∂¯2 = 0 (resp. ∂2 = 0) holds automatically. The local holomorphic (resp. anti-holomorphic) sections are solutions to ∂¯s = 0 (resp. 4 ∂s = 0). (E) and (E) are complex affine spaces whose vector spaces of ′ C 0,1C 1,0 translations are Ω (End(E)) and Ω (End(E)), respectively (cf. [1]). Σ˜ Σ˜ Given a Hermitian connection : Ω0(E) Ω1(E), let : Ω0(E) ∇ Σ˜ → Σ˜ ∇′ Σ˜ → Ω1,0(E) and : Ω0(E) Ω0,1(E) be the (1,0) and (0,1) parts of . Σ˜ ∇′′ Σ˜ → Σ˜ ∇ n,k Then and define isomorphisms j : (P ) (E) and ∇ 7→ ∇′′ ∇ 7→ ∇′ A Σ˜ → C n,k j : (P ) (E) of real affine spaces. Their differentials ′ A Σ˜ → C′ j :Ω1(adPn,k) Ω0,1(EndE), j : Ω1(adPn,k) Ω1,0(EndE), ∗ Σ˜ Σ˜ → Σ˜ ∗′ Σ˜ Σ˜ → Σ˜ are complex linear and conjugate linear, respectively. More explicitly, j and j are C (Σ˜,R)-linear, so they are induced by real vector bundle map∗s ′ ∞ ˜j : T ∗ adPn,k (T )0,1 EndE and ˜j : T adPn,k (T )1,0 EndE, Σ˜∗ ⊗ Σ˜ → Σ˜∗ ⊗ ′ Σ˜∗ ⊗ Σ˜ → Σ˜∗ ⊗ respectively. Given any point x Σ˜, let dz = dx+idy be a local basis of (T )1,0 and let X,Y u(n). The∈n X +iY gl(n,C), and Σ˜∗ p ∈ ∈ 1 1 ˜j(Xdx+Ydy) = (X +iY)dz¯, ˜j (Xdx+Ydy) = (X iY)dz. ′ 2 2 − n,k Thecomplexstructureon(T adP ) isgivenbytheHodgestar: (Xdx+ Σ˜∗⊗ Σ˜ x ∗ Ydy) = Ydx+Xdy. It is straightforward to check that ˜j is complex linear − and ˜j is conjugate linear. ′ Given a (0,1)-connection ∂¯ on a Hermitian vector bundle (E,h) over Σ˜, there is a unique connection on E which is compatible with h and such that = ∂¯(see e.g. [26, pp∇.78]). We denote this canonical Hermitian ′′ connection∇by . Themapj j 1 : (E) (E)isgivenby∂¯ ( ), ∇h,∂¯ ′◦ − C → C′ 7→ ∇h,∂¯ ′ where ( ) is the (1,0)-part of . ∇h,∂¯ ′ ∇h,∂¯ LetE bethecomplexdualofE (seee.g. [20,pp.168-169]). ThenE isa ∨ ∨ rank n, degree k complex vector bundle equipped with a Hermitian metric − h inducedbyh. Moreexplicitly, if e ,...,e is alocal orthonormalframe ∨ 1 n { } oftheHermitianvector bundle(E,h), thenitsdualcoframe e ,...,e isa { ∨1 ∨n} local orthonormal frame of the Hermitian vector bundle (E ,h ). The map ∨ ∨ v h(,v) defines a conjugate linear bundlemap E E which induces an ∨ 7→ · → isomorphism I : (E,h) = (E ,h ) of Hermitian vector bundles. We have h ∼ ∨ ∨ n,k U(E ,h ) = U(E,h) = P . ∨ ∨ ∼ ∼ Σ˜ A (0,1)-connection ∂¯ on E induces a (0,1)-connection ∂¯ on E and a ∨ ∨ (1,0)-connection ∂ on E . This gives a map j : (E) (E ). The map ∨ ∨ 1 ′ ∨ C → C v h(,v) defines an isomorphism E = E of C complex vector bundles, 7→ · ∼ ∨ ∞ which induces an isomorphism j : (E) (E ) of complex affine spaces. 2 ′ ′ ∨ C → C It is straightforward to check that j j 1 = j 1 j : (E) (E). ′ ◦ − 2− ◦ 1 C → C′ 5 2.2 Gauge groups LetGL(E)betheframebundleofthecomplexvectorbundleE. LetU(E,h) be the unitary frame bundle of the Hermitian vector bundle (E,h) as in the previous subsection. Then GL(E) is a principal GL(n,C)-bundle over Σ˜, and U(E,h) is a principal U(n)-bundle over Σ˜. Let Aut(E) be the (infinite dimensional) group of complex vector bundle isomorphisms E E, → and let Aut(E,h) be the (infinite dimensional) group of Hermitian bundle isomorphisms (E,h) (E,h). (See [1, Section 2] for details.) Then → Aut(E) = AutGL(E)andAut(E,h) =AutU(E,h); Aut(E,h)is asubgroup ∼ ∼ of Aut(E). Aut(E) acts on (E) by u ∂¯= u ∂¯ u 1 and Aut(E,h) acts on (E,h) − C · ◦ ◦ A by u = u u 1. More explicitly, relative to a local orthonormal frame, − ·∇ ◦∇◦ a (0,1)-connection on E is of the form ∂¯= ∂¯ +B, 0 where ∂¯ is the usual Cauchy-Riemann operator and B is a gl(n,C)-valued 0 (0,1)-form; a unitary connection is of the form = d+A, ∇ where d is the usual exterior derivative and A is a u(n)-valued 1-form. An element u in the gauge group Aut(E) is locally a GL(n,C)-valued function, and acts on the form B by B uBu 1 (∂¯ u)u 1; (1) − 0 − 7→ − an element uin the gauge group Aut(E,h) is locally a U(n)-valued function, and acts on the form A by A uAu 1 (du)u 1. (2) − − 7→ − In particular, if u GL(n,C) (resp. U(n)) is a constant gauge transfor- ∈ mation, then it acts on B (resp. A) by B uBu 1 (resp. A uAu 1 − − 7→ 7→ ). Given u Aut(E) and x Σ˜, u : E E is a complex linear iso- x x x ∈ ∈ → morphism for all x Σ˜. The dual of u is a complex linear isomorphism x ∈ (u ) : (E ) (E ) = (E ) . It induces a complex linear isomorphism x ∨ x ∨ x ∨ ∨ x → (u ) :(E ) (E ) = (E ) . x ∨ x ∨ → x ∨ ∼ ∨ x Define u Aut(E ) by (u ) = (u ) . Then u u defines a group ∨ ∨ ∨ x x ∨ ∨ ∈ −→ homomorphism Aut(E) Aut(E ). The isomorphism I : E = E allows ∨ h ∼ ∨ → 6 us to identify Aut(E) with Aut(E ). We let I˜ : Aut(E) Aut(E ) be ∨ h ∨ → this h-dependent identification, and let φ : Aut(E) Aut(E) be defined h → by u I˜ (u ). Then φ can be described explicitly as follows. Let u h ∨ h 7→ ∈ Aut(E), and let A GL(n,C) be the matrix of u : E E with respect x x x ∈ → to an orthonormal basis of (E ,h ). Then φ (u) = (At) 1. Note that x x h x − φ : Aut(E) Aut(E) is an involution, and the fixed locus Aut(E)φh = h → Aut(E,h). 3 Involution Let Σ be a closed nonorientable surface, and let π : Σ˜ Σ be its orientable → double cover. Then Σ˜ is a Riemann surface, and the non-trivial deck transformation is an anti-holomorphic, anti-symplectic involution τ :Σ˜ Σ˜ such → that π τ =π. ◦ 3.1 The action of τ on holomorphic structures Thereisananti-holomorphic,anti-symplecticmapτ : (E,h) (τ E,τ h) ∗ ∗ n,k n, k A A → A n, k given by ∇ 7→ τ∗∇. Note that τ∗PΣ˜ ∼= PΣ˜ − , so A(τ∗E,τ∗h) ∼= A(PΣ˜ − ). We have (τ ) = τ ( ), (τ ) = τ ( ), ∗ ′ ∗ ′′ ∗ ′′ ∗ ′ ∇ ∇ ∇ ∇ so there are maps τ : (E) (τ E), (τ E) (E), ∂¯ τ ∂¯ ∗ ′ ∗ ∗ ′ ∗ C → C C → C 7→ τ : (E) (τ E), (τ E) (E), ∂ τ ∂, ∗ ′ ∗ ′ ∗ ∗ C → C C → C 7→ such that τ τ is the identity map. ∗ ∗ ◦ Define τ := j τ j 1 = τ j j 1 : (E) (τ E). Then τ is − ∗ ′ − ∗ given by ∂¯ 7→C τ∗(∇◦h,∂¯A)′◦. In the res◦t of◦this suCbsectio→n,Cwe study the effCect of τ on the Harder-Narasimhan filtration. C Let denote E equipped with a (0,1)-connection (holomorphic struc- E ture), so that can be viewed as a point in (E). Let E C 0 = = 0 1 r E ⊂ E ⊂ ··· ⊂ E E be the Harder-Narasimhan filtration, so / is semi-stable. Set j j 1 E E − j = j/ j 1, nj = rankC j, kj = deg j. D E E − D D 7 The Atiyah-Bott type of is E k k k k k k 1 1 r r 1 r µ( ) = ,..., ,..., ,..., , where > > . (3) E n n n n n ··· n 1 1 r r 1 r (cid:16) (cid:17) n1 nr Recall that r|ankC{zE =}n and|deg{Ez =}k. Let r r k k k k k k 1 1 r r 1 r I = ,..., ,..., ,..., > > , n = n, k = k , n,k j j n n n n n ··· n 1 1 r r 1 r n(cid:16) (cid:17)(cid:12) Xj=1 Xj=1 o n1 nr (cid:12)(cid:12) | {z } | {z } and for µ I let = (E) µ( ) = µ . The Harder-Narasimhan n,k µ ∈ C {E ∈ C E } strata of (E) are µ I (cf.(cid:12)[1, Section 7]). µ n,k C {C | ∈ } (cid:12) Using the isomorphism I : E =(cid:12) E defined in Section 2.1, we may h ∼ ∨ identify (τ E) with (τ E ). Then τ : (E) (τ E ) is given by ∗ ∗ ∨ ∗ ∨ τ C, where , C,τ are holomoCrphiCc vecto→r bCundles over Σ˜, while ∗ ∨ ∨ ∗ ∨ E 7→ E E E E is an anti-holomorphic vector bundle over Σ˜. ∨ E For j = 0,...,r, define a holomorphic subbundle ( ) of by ∨ j ∨ E − E ( ) = α α(v) = 0 v ( ) . E∨ −j x { ∈ Ex∨ | ∀ ∈ Ej x} Then ( ) =(cid:0)( / ) (cid:1). The Harder-Narasimhan filtration of (E ) ∨ j j ∨ ∨ ∨ E − E E E ∈ C is given by 0 = ( ) ( ) ( ) ( ) = ∨ r ∨ (r 1) ∨ 1 ∨ 0 ∨ E − ⊂ E − − ⊂ ··· ⊂ E − ⊂ E E Notice that ( ∨) i/( ∨) (i+1) ∼= ( i+1/ i)∨ = ( i+1)∨. E − E − E E D Set = ( ) /( ) . Then j ∨ (r j) ∨ (r j+1) H E − − E − − Hj ∼= (Dr+1−j)∨, rankCHj = nr+1−j, degHj = −kr+1−j. Hence the Atiyah-Bott type of is ∨ E k k k k k k r r 1 1 r 1 µ = ,..., ,..., ,..., , where > > . −n −n −n −n − n ··· −n r r 1 1 r 1 (cid:16) (cid:17) nr n1 Forj|= 0,.{.z.,r,de}finea|holom{ozrphics}ubbundleτ ( ) ofτ ( ) = τ j ∗ ∨ C E − C E E by τ ( ) = τ ( ) . The Harder-Narasimhan filtration of τ ( ) is given j ∗ ∨ j C E − E − C E by 0 = τ ( ) τ ( ) τ ( ) = τ ( ) r (r 1) 0 C E − ⊂ C E − − ⊂ ··· ⊂ C E C E 8 Let = τ ( ) /τ ( ) . Then j (r j) (r j+1) K C E − − C E − − Kj ∼= τ∗Hj ∼= τ∗(Dr+1−j)∨ = τC(Dr+1−j), and rankC j =nr+1 j, deg j = kr+1 j. K − K − − The Atiyah-Bott type of τ ( ) is C E k k k k r r 1 1 µ = ,..., ,..., ,..., . −n −n −n −n r r 1 1 (cid:16) (cid:17) nr n1 From the above dis|cussion{,zwe co}nclud|e: {z } Lemma 2. Let τ : (E) (τ E) = (τ E ) be defined as above, and C C → C ∗ ∼ C ∗ ∨ define τ : Aut(E) Aut(τ E) by u τ φ (u), where φ is defined as in ∗ ∗ h h C → 7→ Section 2.2. Define τ :I I by 0 n,k n, k → − k k k k k k k k 1 1 r r r r 1 1 ,..., ,..., ,..., ,..., ,..., ,..., , n n n n 7→ −n −n −n −n 1 1 r r r r 1 1 (cid:16) (cid:17) (cid:16) (cid:17) n1 nr nr n1 Then| {z } | {z } | {z } | {z } 1. τ : (E) (τ E) = (τ E ) maps bijectively to . C C → C ∗ ∼ C ∗ ∨ Cµ Cτ0(µ) 2. τ isequivariantwithrespecttotheAut(E)-actionon (E)andAut(τ E)- ∗ C C action on (τ E), i.e., ∗ C τ (u ∂¯)= τ (u) τ (∂¯), u Aut(E), ∂¯ (E). C · C · C ∈ ∈ C 3.2 The degree zero case Let P Σ be a principal U(n)-bundle, and let P˜ = π P be the pull back ∗ → principal U(n)-bundle on Σ˜. We first review some facts about P˜ (see [12, Section 3.2] for details). The pull back bundle P˜ = Pn,0 = Σ˜ U(n) is ∼ Σ˜ ∼ × topologically trivial. We wish to describe an involution τ˜ : P˜ P˜ which is → U(n)-equivariant, covers the involution τ : Σ˜ Σ˜, and satisfies P = P˜/τ˜ . s → Fixing a trivialization P˜ = Σ˜ U(n), any such involution must be given ∼ × by τ˜ : Σ˜ U(n) Σ˜ U(n), (x,h) (τ(x),s(x)h), for some C map s ∞ × → × 7→ s : Σ˜ U(n) satisfying s(τ(x)) = s(x) 1. − → The topological type of a principal U(n)-bundle P Σ is classified by c1(P) ∈ H2(Σ;Z) ∼= Z/2Z. Let PΣn,+ and PΣn,− denote→the principal U(n)- bundles on Σ with c = 0 and c = 1 in Z/2Z, respectively. Let τ be the 1 1 ǫ 9 involution on Pn,0 = Σ˜ U(n) defined by a constant map s(x) = ǫ U(n). Σ˜ × ∈ We must have ǫ2 = In, so detǫ = ±1. Then PΣñ,0/τǫ ∼= PΣn,± if detǫ = ±1. We choose ǫ to be the diagonal matrix diag( 1,1,...,1), and define τ = ± ±n,0 n, ± τǫ . Then PΣ˜ /τ± ∼=PΣ ±. ±Let E = Pn,0 Cn = Σ˜ Cn, where ρ : U(n) GL(n,C) is the Σ˜ ×ρ ∼ × → fundamental representation. Then τ induces an involution τ : E = Σ˜ ± ± ∼ × Cn E = Σ˜ Cn given by (x,v) (τ(x),ǫ v). Thetwo involutions τ+,τ → ∼ × 7→ ± − give two isomorphisms τ E = E, which induce isomorphisms ∗ ∼ A(E,h) ∼= A(τ∗E,τ∗h), C(E) ∼= C(τ∗E), Aut(E) ∼= Aut(τ∗E). Therefore, we have involutions τ : (E,h) (E,h), τ : (E) (E), τ : Aut(E) Aut(E), ± ± ± A → A C → C → A C C and τ : (E) (E) is Aut(E)-equivariant with respect to the Aut(E)- ± C → C actionCon (E). We have C A(PΣn,±) = A(E,h)τA± ∼= C(E)τC±, Aut(PΣn,±) ∼= Aut(E,h)τC±, where Aut(E,h) Aut(E) is the group of unitary gauge transformations of ⊂ the Hermitian vector bundle (E,h). The following two equivariant pairs are isomorphic: A(PΣn,±),Aut(PΣn,±) ∼= C(E)τC±,Aut(E,h)τC± . (cid:16) (cid:17) (cid:16) (cid:17) 3.3 SU(n)-connections Let Qn Σ be a principal SU(n)-bundle. Then Qn is topologically trivial. Σ → Σ Wefixatrivialization Qn = Σ SU(n),whichallows ustoidentify thespace Σ ∼ × (Qn) of connections on Qn with the vector space of su(n)-valued 1-forms A Σ n,+ Σ on Σ. Let P = Σ U(n) be the trivial U(n)-bundle on Σ, as before. The Σ ∼ × short exact sequence of vector spaces Tr 0 su(n) u(n) u(1) 1 → → → → induces a short exact sequence of infinite dimensional vector spaces 0 (Qn) (Pn,+) Tr (P1,+) 0. → A Σ → A Σ → A Σ → 10