Classification of Nahm pole solutions of the Kapustin-Witten equations on S 1 ×Σ× R +
aa r X i v : . [ m a t h . DG ] J a n CLASSIFICATION OF NAHM POLE SOLUTIONS OF THEKAPUSTIN-WITTEN EQUATIONS ON S × Σ × R + SIQI HE AND RAFE MAZZEO
Abstract.
In this note, we classify all solutions to the SU(n) Kapustin-Witten equationson S × Σ × R + , where Σ is a compact Riemann surface, with Nahm pole singularity at S × Σ × { } . We provide a similar classification of solutions with generalized Nahm polesingularities along a simple divisor (a “knot”) in S × Σ × { } . Introduction
An important conjecture by Witten [20] posits a relationship between the Jones polyno-mial of a knot and a count of solutions to the Kapustin-Witten equations. More specifically,let K be a knot in X = R or S , and fix an SU(n) bundle P over X × R + with associatedadjoint bundle g P . The Kapustin-Witten (KW) equations [11] are equations for a pair( A, Φ), where A is a connection on P and Φ is a g P -valued 1-form. We augment these withthe singular Nahm pole boundary conditions at y = 0 (where y is a linear variable on the R + factor), and with an additional singularity imposed along K × { } . The conjecture statesthat an appropriate count of solutions to the KW equations with these boundary conditionscomputes the Jones polynomial. One can define these equations when X is a more generalRiemannian 3-manifold, and in that case this gauge-theoretic enumeration may lead to new3-manifold invariants when K = ∅ , or to a generalization of the Jones polynomial for K lying in a general 3-manifold, see [21, 3].The core of all of this is to investigate the properties of the moduli space of solutions.Significant partial progress has been made, see [12, 13, 4, 15, 16], as well as Taubes’ recentadvance [18] regarding compactness properties.As usual in gauge theory, it is reasonable to seek to understand a dimensionally reducedversion of this problem. Thus suppose that X = S × Σ, where Σ is a compact Riemannsurface of genus g . Solutions which are invariant in the S direction are solutions of theso-called extended Bogomolny equations. General existence theorems for solutions of thesedimensionally reduced equations were proved in [6, 7]. In the present paper, we adaptarguments from [13] and prove that every solution to the KW equation on S × Σ × R + satisfying Nahm pole boundary conditions is necessarily invariant in the S direction. Thisleads to a complete classification of solutions in this special case. Theorem 1.1.
Consider the Kapustin-Witten equations on S × Σ × R + y for fields satisfyingthe Nahm pole boundary condition at y = 0 (with no knot singularity) and which convergeto a flat SL( n, C ) connection as y → ∞ . i) There are no solutions if g = 0 ; ii) There is a unique solution (up to unitary gauge equivalence) if g = 1 ; iii) If g > , there exists a solution if and only if the limiting flat connection as y → ∞ lies in the Hitchin section in the SL( n, C ) Hitchin moduli space, and in that case,this solution is unique up to unitary gauge.
Part ii) here largely comes from the uniqueness theorem in [12] for solutions on R × R + .The Hitchin section in part iii) is also known as the Hitchin component of the SL(n, R ) representation variety, cf. [10]. We recall that there are in fact n g equivalent Hitchincomponents, depending on the different choices of spin structure.Next suppose that the knot K ⊂ S × Σ is a union of ‘parallel’ copies of S , K = ⊔ i ( S × { p i } ). The Nahm boundary conditions at a knot require that we specify a weight k i , i.e., an ( n − k i , . . . , k in − ) ∈ N n − for each component K i . Theorem 1.2.
Consider the Kapustin-Witten equations on S × Σ × R + y for fields whichsatisfy the Nahm pole boundary condition with knot singularities with weights k i , as describedabove, along K × { } , where K = ⊔ i ( S × { p i } ) , and which converge to a flat SL( n, C ) connection, corresponding to a stable Higgs pair ( E , ϕ ) , as y → ∞ . i) There are no solutions when g = 0 ; ii) If g > and ρ is irreducible, there exists a solutions with these boundary conditionsat K if and only if there exists a holomorphic line subbundle L of E such that thedata set d ( E , ϕ, L ) = { ( p i , k i ) } , The definition of data sets d ( E , ϕ, L ) is recalled in Section 3.3. Remark.
We do not discuss the case g = 1 here. Indeed, it is not clear what the correctexistence theory for solutions with knot singularities should be in this case. Corollary 1.3.
There exists, up to unitary gauge, at most n g solutions to the KW equa-tions which converge to the given flat connection associated to ( E , ϕ ) and with Nahm singu-larity along K = ⊔ S × { p i } . The knot points p i and the weights k i determine the divisor D = P i p i P j k ij . Theorem 1.4. If deg D is not divisible by n , there exist no Nahm pole solutions to the KWequations with knot singularity along K . In particular, there are no solutions to the SU(2) extended Bogomolny equations with only a single knot singularity of weight . Acknowledgements.
The first author would like to thank Simon Donaldson for numer-ous helpful discussions. The second author was supported by the NSF grant DMS-1608223.2.
The Kapustin-Witten Equations and the Nahm Pole Boundary Conditions
We begin with some background materials on the Kapustin-Witten equations [11] andNahm pole boundary conditions:2.1.
The Kapustin-Witten Equations.
Let (
M, g ) be a Riemannian 4-manifold, and P an SU(n) bundle over M with the adjoint bundle g P . The Kapustin-Witten equations fora connection A and a g P -valued 1-form Φ are F A − Φ ∧ Φ + ⋆d A Φ = 0 , d ⋆A Φ = 0 . (1)When M is closed, all solutions to the KW equations are flat SL( n, C ) connections [11]. In-deed, in this setting, a Weitzenb¨ock formula shows that solutions must satisfy the decoupledequations(2) F A − Φ ∧ Φ = 0 , d A Φ = 0 , d A ⋆ Φ = 0 , or equivalently, F A = 0 where A := A + i Φ and d A ⋆ Φ = 0.Following [20, 21], the main case of interest here is when M = X × R + , where X isa closed 3-manifold and R + := (0 , ∞ ) with linear coordinate y . From now on, we fix aRiemannian metric on X with volume 1, and endow X × R + with the product metric. LASSIFICATION OF NAHM POLE SOLUTIONS 3
The Nahm Pole Boundary Condition.
Let G := SU(n), with Lie algebra g and choosea principal embedding ̺ : su (2) → g as well as a global orthonormal coframe { e ∗ a , a = 1 , , } of T ∗ X , which is possible since X is parallelizable. Next, choose a section e of T ∗ X ⊗ g P , e = P t a e ∗ a for some everywhere nonvanishing sections t a , a = 1 , ,
3, of the adjoint bundle g P which satisfy the commutation relations [ t a , t b ] = ǫ abc t c , and which lie in the conjugacyclass of the image of ̺ . This choice of e is called a dreibein form . Definition 2.1.
With all notation as above, the pair ( A, Φ) satisfies the Nahm poleboundary condition at y = 0 if, in some gauge, A = A + O ( y ǫ ) and Φ = ey + O ( y − ǫ ) for some ǫ > . The rationale for this name is that the dimensional reduction of the KW equations to R + are the Nahm equations, and in this case (0 , ey ) is a ‘standard’ solution of the Nahmequations with a so-called pole at y = 0. We remark also that as proved in [12], it issufficient to assume that A = O ( y − ǫ ), since the regularity theory for solutions shows thatthere is automatically a leading coefficient A .2.1.2. The Nahm Pole Boundary Condition with Knot Singularities.
A generalization of thisboundary condition incorporates certain ‘knot’ singularities at y = 0. Before describingthis, recall from [20] the model solution when G = SU(2) and X = R = R × C withcoordinate ( x , z = x + ix ). Introduce spherical coordinate ( R, s, θ ) in the ( z, y ) half-space: z = re iθ , R = p r + y , y = R sin s , r = | z | = R cos s . The model knot is the line( x , , ⊂ R × { } . Writing Φ = φ z dz + φ ¯ z d ¯ z + φ dx + φ y dy , the model solution of weight k takes the form A = − ( k + 1) cos s (1 + sin s ) k − (1 − sin s ) k (1 + sin s ) k +1 − (1 − sin s ) k +1 dθ (cid:18) i i (cid:19) ,φ z = 2( k + 1) e ikθ cos k sR (1 + sin s ) k +1 − R (1 − sin s ) k +1 (cid:18) (cid:19) ,φ = k + 1 R (1 + sin s ) k +1 + (1 − sin s ) k +1 (1 + sin s ) k +1 − (1 − sin s ) k +1 (cid:18) i i (cid:19) , φ y = 0 . (3)There is a less explicit model solution when G = SU(n), due to Mikhaylov [14]. Theweight in that case is an ( n − k = ( k , · · · , k n − ), and the corresponding solutionis denoted ( A mod k , Φ mod k ). As in the case n = 2, | A mod k | ∼ R − s and | Φ mod k | ∼ R − s − near z = 0 , y = 0.In general, given a knot K ⊂ X × { } , introduce local coordinates ( x , z = x + ix , y )near K , where K = { z = y = 0 } and t is a coordinate along K . We can use cylindricalcoordinates ( R, s, θ, x ) near K , where y = R sin s , z = R cos se iθ . Then, as in [20, 13], wemake the Definition 2.2.
With P and G as above, and K ⊂ X a knot, then ( A, Φ) satisfies Nahmpole boundary condition with knot K and weight k if in some gauge i) ( A, Φ) satisfies the Nahm pole boundary condition.away from knots K , ii) near K , A = A mod k + O ( R − ǫ s − ǫ ) , Φ = Φ mod k + O ( R − ǫ s − ǫ ) . The Boundary Condition at y = ∞ . We must also impose an asymptotic boundarycondition at the cylindrical end, as y → ∞ . We change to a temporal gauge, i.e., so that A y ≡
0. Then writing Φ = φ + φ y dy (so φ includes the φ part), the KW equations become SIQI HE AND RAFE MAZZEO flow equations ∂ y A = ⋆d A φ + [ φ y , φ ] ,∂ y φ = d A φ y + ⋆ ( F A − φ ∧ φ ) ,∂ y φ y = d ⋆A φ. (4)We shall assume that ( A, Φ) converges to a ”steady-state” ( y -independent) solution as y → ∞ , which is then necessarily a flat SL ( n, C ) connection. The y -independence, togetherwith the equations (2) yield that [ φ, φ y ] = d A φ y = 0; this shows that if φ y = 0, then A isreducible. Proposition 2.3. If ( A, Φ) satisfies the KW equations together with Nahm pole boundaryconditions (possibly with knots), and converges to an irreducible flat SL( n, C ) connection as y → ∞ , then φ y ≡ . Indeed, the hypothesis and the remark above shows that lim y → + ∞ φ y = 0. A well-knownvanishing theorem then implies that φ y ≡
0, see [17, Page 36] or [4, Corollary 4.7] for aproof. We assume henceforth, as in [18], that φ y ≡ M KWNP := { ( A, Φ) : KW( A, Φ) = 0 , ( A, Φ) converges to a flat SL( n, C ) connection as y → ∞ and satisfies the Nahm Pole boundary condition at y = 0 } / G , (5)and M KWNPK := { ( A, Φ) : KW( A, Φ) = 0 , ( A, φ, φ ) satisfies the Nahm poleboundary condition with knot K and converges to a flat SL ( n, C ) connection as y → ∞} / G , (6)where G is the space of gauge transformations preserving the boundary conditions.2.2. The Regularity theorems of Nahm pole Solutions.
We next recall the regularitytheory for this singular boundary condition at y = 0, as developed in [12, 13]. Still workingon X × R + , fix a smooth background connection ∇ , and write ∇ x , ∇ y for the covariantderivatives in the x ∈ X and y directions. Theorem 2.4. [12, 13]
Let ( A, Φ) satisfy the KW equations with Nahm pole boundarycondition, and write A = A + a , Φ = ey + b near y = 0 where a = O ( y ǫ ) , b = O ( y − ǫ ) .Then a and b are polyhomogeneous. Furthermore, the leading term A of A must correspond,under the intertwining provided by the dreibein e , with the Levi-Civita connection on X .If ( A, Φ) satisfies the Nahm pole boundary condition with a knot singularity along K of weight k at y = 0 , then writing A = A mod k + a, Φ = Φ mod k + b , where ( A mod k , Φ mod k ) is the model solution and a, b = O ( R − ǫ s − ǫ ) , then a, b are polyhomogeneous, i.e., haveexpansions in positive powers of R and s , and nonnegative integer powers of log R and log s ,with coefficients smooth in the tangential variables. These expansions are of product type atthe corner R = s = 0 . Remark.
We recall that a function (or section of some bundle) u is polyhomogeneous on X × R + at X × { } if, near any boundary point, u ∼ X j N j X ℓ =0 u jℓ ( x ) y γ j (log y ) ℓ as y → . Here x is a local coordinate on X and each coefficient u jℓ ( x ) is C ∞ , while γ j is a sequenceof complex numbers with real parts tending to infinity. In our setting, the γ j are explicitreal numbers calculated in [12] . LASSIFICATION OF NAHM POLE SOLUTIONS 5
The second polyhomogeneity statement, near K , may be phrased similarly once we intro-duce the blowup [ X × R + ; K × { } ] . This is a new manifold with corners of codimensiontwo obtained by replacing the knot K at y = 0 with its inward-pointing spherical normalbundle. The cylindrical coordinates ( x , R, s, θ ) are nonsingular on this space, and the twoboundaries are defined by { R = 0 } and { s = 0 } . A function or section u is polyhomoge-neous on this space if it admits a classical expansion as described above near each point inthe interior of the codimension one boundaries, while near the corner { R = s = 0 } it admitsa product type expansion u ∼ X j,k N j X ℓ =0 M j X m =0 u jkℓm ( x , θ ) s γ j R µ k (log s ) ℓ (log R ) m , where as before, each coefficient function is smooth in the variables t, θ along the corner. Inour setting the γ j and N j are the same numbers as in the previous expansion, while the µ k are real numbers calculated (somewhat less explicitly, i.e., only in terms of spectral data ofsome auxiliary operator) in [13] .The paper [5] considers various refined aspects of the higher terms in the expansion in y .We have described this precise regularity for the sake of completeness, but in fact, we donot use the full power of these expansions here, but only the estimates |∇ ℓx ∇ my a | C ≤ C ℓ,m y − m + ǫ , |∇ ℓx ∇ my b | C ≤ C l,m y − m + ǫ , |∇ ℓx ∇ mR ∇ ns a | C ≤ C ℓ,m,n R − ǫ − m s − ǫ − n , |∇ ℓx ∇ mR ∇ ns b | C ≤ C ℓ,m,n R − ǫ − m s − ǫ − n for any ǫ > and any ℓ, m, n ∈ N . The Extended Bogomolny Equations
We next recall the dimensional reduction of the Kapustin-Witten equations from S × Σ × R + to Σ × R + , obtained by considering fields invariant in the S direction. This waspreviously studied in [6, 7], and is closely related to the Atiyah-Floer approach to countingKapustin-Witten solutions [3].Assume on the one hand that the bundle P on S × Σ × R + is pulled back from Σ × R + .Changing notation slightly, given a solution ( b A, b Φ) of the KW equations on S × Σ × R + ,choose a gauge for which the S component of A vanishes and A y ≡ y = 0 and the asymptotic condition as y → ∞ ,Proposition 2.3 gives that φ y = 0, but we cannot gauge away the S component φ . Thusthe remaining fields are ( b A Σ , b Φ , b Φ Σ ). We regard b A Σ as a connection A on Σ, and write b Φ = φ , b Φ Σ = φ . These remaining fields satisfy the extended Bogomolny equations F A − φ ∧ φ − ⋆d A φ = 0 d A φ + ⋆ [ φ, φ ] = 0 ,d ⋆A φ = 0 . (7)On the other hand, given a solution ( A, φ, φ ) of the extended Bogomolny equations onΣ × R + , then denoting by π : S × Σ × R + → Σ × R + the natural projection, we define theconnection b A = π ∗ A and Higgs field b Φ = π ⋆ φ + π ⋆ φ dx . It is straightforward to check that( b A, b Φ) satisfies the KW equations.Let D = { ( p i , k i = ( k i , · · · , k in − )) } where for each i , k ij are non-negative integers withat least one of them nonzero. Definition 3.1.
Let ( A, φ, φ ) be a solution to the extended Bogomolny equations on Σ × R + . i) The fields ( A, φ, φ ) satisfy the Nahm pole boundary condition if the correspond-ing fields ( b A, b Φ) satisfy the Nahm pole boundary condition on S × Σ × R + . SIQI HE AND RAFE MAZZEO ii)
Similarly, ( A, φ, φ ) satisfies the Nahm pole boundary condition with knotdata D if the corresponding pull back fields ( b A, b Φ) satisfy the Nahm pole boundarycondition with knots at K i := S × { p i } with weight k i . The moduli space we shall consider are: M EBENP := { ( A, φ, φ ) : EBE( A, φ, φ ) = 0 , ( A, φ, φ ) converges to a flat SL( n, C )connection as y → ∞ and satisfies the Nahm Pole boundary condition at y = 0 } / G , (8)and M EBENPK := { ( A, φ, φ ) : EBE( A, φ, φ ) = 0 , ( A, φ, φ ) satisfies the Nahm poleboundary condition with knot and converges to a flatSL( n, C ) connection as y → ∞} / G , (9)where G is the gauge transformations that preserve the boundary condition.3.1. Hermitian-Yang-Mills Structure.
In [3, 20], it is observed that the extended Bo-gomolny equations have a Hermitian-Yang-Mills structure. By this we mean the following.Let E be complex vector bundle of rank n over Σ × R + with det E = 0. A choice of Her-mitian metric H on E induces an SU( n ) structure on this bundle, and we denote by g E theassociated adjoint bundle. Writing d A = ∇ dx + ∇ dx + ∇ y dy, and φ = φ dx + φ dx = 12 ( ϕ z dz + ϕ ¯ z d ¯ z ) , we define the operators D = ( ∇ + i ∇ ) d ¯ z = (2 ∂ ¯ z + A + iA ) d ¯ z, D = ad ϕ = [ ϕ, · ] = [( φ − iφ ) dz, · ] , D = ∇ y − iφ = ∂ y + A y − iφ . (10)Their adjoints with respect to H are denoted D † H i . The extended Bogomolny equations canthen be written in the elegant form[ D i , D j ] = 0 , i, j = 1 , , ,i (cid:16) [ D , D † H ] + [ D , D † H ] (cid:17) + [ D , D † H ] = 0 , (11)where Λ : Ω , → Ω is the inner product with the K¨ahler form (normalized as ( i/ dz ∧ d ¯ z when the metric on Σ is flat).The action D i → g − D i g of the gauge group G preserves the Hermitian metric; thecomplex gauge group is denoted G C . The smaller system [ D i , D j ] = 0 is invariant under G C ,while the full set of equations (11) is invariant only under G . The final equation is a realmoment map condition. Following Donaldson [2] and Uhlenbeck-Yau [19], geometric datafrom the G C -invariant equations play an important role in understanding the moment mapequation.3.2. Higgs Bundles and Flat Connections.
The appearance of Higgs bundles over Σin this story is motivated by the fact that the y -independent versions of the equations of(7), when in addition φ = 0, are simply the Hitchin equations.Recall that a Higgs bundle over Σ is a pair ( E , ϕ ) where E is a holomorphic bundle ofrank n with det E = 0 and ϕ ∈ H (End( E ) ⊗ K ). A Higgs pair (which is an alternatephrase for Higgs bundles) ( E , ϕ ) is called stable if for any holomorphic subbundle V with ϕ ( V ) ⊂ V ⊗ K , we have deg( V ) <
0, and polystable if it is a direct sum of stable Higgspairs.
LASSIFICATION OF NAHM POLE SOLUTIONS 7
Setting D = 0 in the extended Bogomolny equations (or alternately, considering onlythe equations for D and D on each slice Σ y := Σ × { y } ), we obtain the Hitchin equations:(12) F H + [ ϕ, ϕ ⋆ H ] = 0 , ¯ ∂ϕ = 0 . The initial term F H is the curvature of the Chern connection ∇ H associated to H and theholomorphic structure, and ϕ ⋆ H is the adjoint with respect to H . Irreducibility of the fields( A, ϕ + ϕ ⋆ H ) is defined in the obvious way. One may regard (12) as an equation for thefields ( A, ϕ ) or else for the Hermitian metric H ; we consider H as the variable here. Theorem 3.2. [8]
For any Higgs pair ( E , ϕ ) on Σ , there exists an irreducible solution H tothe Hitchin equations if and only if this pair is stable, and a reducible solution if and onlyif it is polystable. To any solution H of (12) we associate the flat SL( n, C ) connection D = ∇ H + ϕ + ϕ ⋆ H .This determines, in turn, a representation ρ : π (Σ) → SL( n, C ) which is well-defined up toconjugation. Irreducibility of the solution is the same as irreducibility of the representation,while complete reducibility corresponds to the fact that ρ is reductive. The map from flatconnections back to solutions of the Hitchin system is defined as follows: first find a harmonicmetric, cf. [1], which determines a decomposition D = D skew + D Herm into skew-Hermitianand Hermitian parts. After that, the further decomposition D Herm = ϕ + ϕ ⋆ H determines ϕ , and hence the Higgs bundle (( D skew ) , , ϕ ).Denoting by M Higgs := { ( E , ϕ ) } stable / G C the moduli space of stable SL( n, C ) Higgs bun-dle, we are then led to define(13) P NP ∞ : M EBENP → M
Higgs , P
NPK ∞ : M EBENPK → M
Higgs ;this is the map which assigns to a solution (
A, φ, φ ) of the extended Bogomolny equations itslimiting flat connection, and then, under Theorem 3.2, the corresponding Higgs bundle.The Hitchin fibration is the map π : M Higgs → ⊕ ni =2 H (Σ , K i ) π ( ϕ ) = ( p ( ϕ ) , · · · , p n ( ϕ )) , (14)where det( λ − ϕ ) = P λ n − j ( − j p j ( ϕ ). By [9], this is a proper map.We next introduce the Hitchin component (also called the Hitchin section). Choose aspin structure K and set B i = i ( n − i ). Now define the Higgs bundle ( E , ϕ ), where(15) E : = S n − ( K − ⊕ K ) = K − n − ⊕ K − n − +1 ⊕ · · · ⊕ K n − ϕ = √ B · · ·
00 0 √ B · · · √ B n − q n q n − · · · q . The constant √ B i in the ( i, i + 1) entry represents this multiple of the natural isomorphism K − n − + i → K − n − + i − ⊗ K , and similarly, H (Σ , K n − i ) ∋ q n − i : K − n − + i → K n − ⊗ K .The Hitchin component M Hit is the complex gauge orbit of this family of Higgs bundle,(16) M Hit := { ( E := S n − ( K − ⊕ K ) , ϕ as in (15)) } / G C . The following theorem explains its importance.
SIQI HE AND RAFE MAZZEO
Theorem 3.3. [10]
Every element in M Hit is a stable Higgs pair. Furthermore, the mapassigning to each element of ⊕ ni =2 H (Σ , K i ) the unique solution of the Hitchin equationscorresponding to the associated Higgs pair is a diffeomorphism to one of the n g choices forthe Hitchin component; thus its inverse, the restriction of the Hitchin fibration π | M Hit , isalso a diffeomorphism.
Note that the image of this map is only one component of the space of all irreducible flatSL( n, R ) connections, which explains the name ‘Hitchin component.’3.3. The Kobayashi-Hitchin Correspondence.
We now recall the Kobayashi-Hitchincorrespondence for the extended Bogomolny equations moduli space [3, 6, 7].As noted earlier, from the Hermitian structure in (11) and the commutation relationship[ D , D ] = 0, we obtain a Higgs bundle ( E y , ϕ y ) on each slice Σ × { y } . The commutationrelationship [ D , D ] = [ D , D ] = 0 means that parallel transport by D identifies theseHiggs bundles for different values of y .Suppose first that the solution of the extended Bogomolny equations satisfies the Nahmpole boundary condition without knots. As explained in more detail in [7, Section 4],there is a holomorphic line subbundle L ⊂ E determined by the property that the paralleltransports (under D parallel transport) of its sections vanish at the fastest possible rateas y →
0, measured with respect to the Hermitian metric H . In other words, a solution ofthe extended Bogomolny equations satisfying these boundary conditions determines a triple( E , ϕ, L ), consisting of a Higgs bundle and a line subbundle.More generally, consider any triple ( E , ϕ, L ) where L is any holomorphic line subbundleof E . Define holomorphic maps f i := 1 ∧ ϕ · · · ∧ ϕ i − ∈ H (Σ; L − i ⊗ ∧ i E ⊗ K i ( i − ) , ≤ i ≤ n. Note that Z ( f j ) − Z ( f j − ) = P i k ij p i for some k ij ∈ N . Setting k i := ( k i , · · · , k in − ) ∈ N n − ,then we define the knot data set to be d ( E , ϕ, L ) := { ( p i , k i ) } . Note the important specialcase (which holds by noting that f n = 0 everywhere): Proposition 3.4. [7, Section 4] If d ( E , ϕ, L ) = ∅ , then ( E , ϕ ) ∈ M Hit and L ∼ = K n − . We then state the main equivalences between the extended Bogomolny equations modulispaces and the spaces of triples ( E , ϕ, L ), first for data in the Hitchin component and thenfor general data. Theorem 3.5. [6, 7]
There is a diffeomorphism of moduli spaces M EBENP ∼ = M Hit . More specifically, recall the map P NP ∞ from (13) . i) For any ( E , ϕ ) ∈ M Hit , there exists a unique Nahm pole solution ( A, φ, φ ) ∈ M EBENP such that P NP ∞ ( A, φ, φ ) = ( E , ϕ ) ; ii) Given any Higgs bundle ( E , ϕ ) / ∈ M Hit , there is no solution to the extended Bo-gomolny equations which converges to the flat connection determined by ( E , ϕ ) . Inother word, ( P NP ∞ ) − ( E , ϕ ) = ∅ . Theorem 3.6. [6, 7]
Fix a data set D = { ( p i , k i = ( k i , · · · , k in − )) } . If ( E , ϕ ) is any stableHiggs bundle over Σ with genus g (Σ) > , there exists a solution to the extended Bogomolnyequations satisfying the general Nahm pole boundary condition with knot singularities at p i with weight k i if and only if there exists a line bundle L ⊂ E such that d ( E , ϕ, L ) = D . Inother words, there is a bijection M EBENPK ∼ = { ( E , ϕ, L ) } / G C , where the pairs ( E , ϕ ) on the right are stable Higgs bundles and L ⊂ E is a line subbundle. LASSIFICATION OF NAHM POLE SOLUTIONS 9
Remark.
Notice that in the second result, when knot singularities are allowed, we do notclaim that this bijection of moduli spaces is a diffeomorphism. Indeed, while the space oftriples ( E , ϕ, L ) maps onto the space of all stable Higgs pairs, i.e., onto the entire Hitchinmoduli space, it is not clear that this space of triples is even a manifold.As a second remark, if ( E , ϕ ) is polystable, it seems likely that there are no solutions tothe extended Bogomolny equations which satisfy Nahm pole boundary conditions with knotsingularities which converge to ( E , ϕ ) . However, we do not prove this. A Weitzenb¨ock Identity for the Kapustin-Witten Equations
In this section, we establish a Weitzenb¨ock identity analogous to the one in [13], and usethis to show that all solutions to the KW equations over M := S × Σ × R + are invariantin the S direction, hence determine solutions to the extended Bogomolny equations. In allthe following, we use coordinates x ∈ S , z ∈ Σ and y ∈ R + .4.1. Weitzenb¨ock Identity.
As before, let P be an SU( n ) bundle over M := S × Σ × R + ,and fix a connection b A and a g P -valued 1-form b Φ on M ; assume that b A = b A y = b Φ y ≡ d b A = d A + dx ∧ ∇ and b Φ = φ + φ dx . We also fix a product metric on S × Σ × R + with orientation dx ∧ dA Σ ∧ dy .Now write F b A = F A + B A ∧ dx ; the Bianchi identity d b A F b A = 0 is equivalent to(17) ∇ F A + d A B A = 0In the following, we write ⋆ and ⋆ for the Hodge star operators on M and S × Σ, respec-tively.We first compute F b A − b Φ ∧ b Φ + ⋆ d b A b Φ= ( F A − φ ∧ φ + ⋆ ( ∇ φ − d A φ )) + ( B A − [ φ, φ ] − ⋆d A φ ) ∧ dx ,d ⋆ b A b Φ = d ⋆A φ − ∇ φ . (18)Next, for any ǫ ∈ (0 , M ǫ := S × Σ × [ ǫ, ǫ − ]. Then ˆ M ǫ | KW | = ˆ M ǫ | F A − φ ∧ φ + ⋆ ( ∇ φ − d A φ ) | + | B A − [ φ, φ ] − ⋆d A φ | + | d ⋆A φ − ∇ φ | = ˆ M ǫ | F A − φ ∧ φ − ⋆d A φ | + |∇ φ | + | B A | + | [ φ, φ ] + ⋆d A φ | + | d ⋆A φ | + |∇ φ | + ˆ M ǫ χ, (19)where(20) χ := 2 h F A − φ ∧ φ − ⋆d A φ , ⋆ ∇ φ i − h B A , ⋆d A φ + [ φ, φ ] i − h d ⋆A φ, ∇ φ i . The inner product here is h A, B i := − Tr( A ∧ ⋆ B ). Lemma 4.1.
We have the following identities: i) h F A , ⋆ ∇ φ i − h B A , ⋆d A φ i = ∇ Tr( F A ∧ φ ) ∧ dx + d Tr( B A ∧ φ ) ∧ dx , ii) h ⋆d A φ , ⋆ ∇ φ i + h B A , [ φ, φ ] i + h ⋆d A φ , ⋆ ∇ φ i = ∇ Tr( φ ∧ ⋆d A φ ) ∧ dx − ∇ Tr( φ ∧ d A ⋆ φ ∧ dx )iii) h φ ∧ φ, ⋆ ∇ φ i = ∇ Tr( φ ∧ φ ∧ φ ∧ dx ) Proof.
For (i), we compute h F A , ⋆ ∇ φ i − h B A , ⋆ d A φ i = Tr( F A ∧ ∇ φ ) ∧ dx − Tr( B A ∧ d A φ ) ∧ dx = ∇ Tr( F A ∧ φ ) ∧ dx − Tr( ∇ F A ∧ φ ) ∧ dx + d Tr( B A ∧ φ ) ∧ dx − Tr( d A B A ∧ φ ) ∧ dx = ∇ Tr( F A ∧ φ ) ∧ dx + d Tr( B A ∧ φ ) ∧ dx , where the last step uses (17).Next, for (ii), h ⋆d A φ , ⋆ ∇ φ i = ∇ Tr( φ ∧ ⋆d A φ ) ∧ dx − Tr( φ ∧ ∇ ( ⋆d A φ )) ∧ dx = ∇ Tr( φ ∧ ⋆d A φ ) ∧ dx − Tr( φ ∧ ⋆B A ∧ φ ) ∧ dx , h d ⋆A φ, ∇ φ i = − Tr( ∇ φ ∧ d A ⋆ φ ) ∧ dx = −∇ Tr( φ ∧ d A ⋆ φ ∧ dx ) + Tr( φ ∧ B A ∧ ⋆φ ) ∧ dx , and h B A , [ φ, φ ] i = − Tr( B A ∧ ⋆φφ − B A ∧ φ ∧ ⋆φ ) ∧ dx . Adding these three equalities yields (ii). The proof of (iii) is straightforward. (cid:3)
Corollary 4.2.
We have ˆ M ǫ χ = ˆ M ǫ ∇ Tr(2 F A ∧ φ − φ − φ ∧ ⋆d A φ + φ ∧ d A ⋆ φ ) ∧ dx + 2 ˆ M ǫ d Tr( B A ∧ φ ) ∧ dx . Lemma 4.3.
Let A ρ + φ ρ be a flat SL ( n, C ) connection over S × Σ , and write A ρ = A ρ + A ρ Σ , φ ρ = φ ρ + φ ρ Σ . i) If we write F A ρ = B A ρ ∧ dx + E A ρ , then B A ρ = 0 ; ii) Up to a unitary gauge transformation, we can assume A ρ and φ ρ are invariant inthe Σ directions and A ρ Σ , φ ρ Σ are invariant in the S directions. iii) Up to a unitary gauge transformation, φ ρ = 0 .Proof. Items i) and ii) follow from the fact that π (Σ × S ) = π (Σ) × π ( S ).For iii), observe that A ρ and φ ρ come from the contribution of π ( S ) → SL ( n, C ). Since π ( S ) is abelian, and A ρ + iφ ρ is an unitary connection, we obtain that φ ρ = 0. (cid:3) We now prove vanishing of the second part of the boundary contribution:
Lemma 4.4.
Suppose that ( A, φ, φ ) is a solution to the extended Bogomolny equations. i) If ( A, φ ) satisfies the Nahm pole boundary conditions at y = 0 , with or without knotsingularities, then lim ǫ → ˆ S × Σ ×{ ǫ } Tr( B A ∧ φ ) = 0;ii) If ( A, φ ) converges to a flat SL( n, C ) connection as y → ∞ , then lim ǫ → ˆ S × Σ × (1 /ǫ ) Tr( B A ∧ φ ) = 0 . LASSIFICATION OF NAHM POLE SOLUTIONS 11
Proof.
First consider i). Away from knots, Theorem 2.4 gives that A ∼ A LC + O ( y − ǫ ) forany ǫ >
0, which implies that B A = B A LC + O ( y − ǫ ) + dy ∧ ( B A ) y . (The ‘LC’ subscriptdenotes Levi-Civita.) The dy component vanishes in the integration so we may disregardit. In addition, since we are using the product metric, B A LC = 0. Finally, since φ ∼ ey , weconclude that B A ∧ φ ∼ O ( y − ǫ ), so there are no boundary contributions in this region.Near a knot K , we use spherical coordinates ( R, s, x ) as before, and consider the bound-ary term as R →
0. By Theorem 2.4, B A ∼ B A mod + O (1) ∼ O (1) because B A mod . Inaddition, φ ∼ R − , so B A ∧ φ ∼ R − . Since the volume form is R dRdsdx , this boundarycontribution vanishes too.Part ii) follows directly from the previous lemma. (cid:3) The other terms in χ are derivatives with respect to x , and hence vanish once we integrateover S . Corollary 4.5.
Under the previous assumptions, ´ M χ = 0 . S -invariance. In summary, we may now conclude the
Theorem 4.6.
Any solution to the KW equations over S × Σ × R + satisfying Nahm poleboundary condition at y = 0 (possibly with knot singularities at K = S × D × { } ), andwhich converges to a flat SL( n, C ) connection as y → ∞ , is S invariant and reduces to asolution of the extended Bogomolny equations. In addition, A ≡ .Proof. By Corollary 4.5, any solution to the KW equations with these boundary and as-ymptotic conditions must satisfyEBE(
A, φ, φ ) = 0 , ∇ φ = 0 , B A = 0 , ∇ φ = 0 , where EBE is the extended Bogomolny equation operator.By Lemma 4.3, up to gauge we can assume that ( A, φ, φ ) converges to ( A ρ , φ ρ ,
0) as y →∞ , where A ρ , φ ρ is S invariant. Since ( A, φ, φ ) is a solution to the extended Bogomolnyequations , Theorem 3.5 and Theorem 3.6 imply that ( A, φ, φ ) is S invariant. From B A = ∇ φ = 0, we obtain d A A = 0 and [ φ, A ] = 0. Irreducibility of solutions tothe extended Bogomolny equations with Nahm pole boundary conditions give finally that A = 0. (cid:3) The projection map π : S × Σ × R + → Σ × R + naturally induces morphisms π ⋆ : M EBENP → M
KWNP ,π ⋆ : M EBENPK → M
KWNPK
We obtain from this the
Corollary 4.7. π ⋆ : M EBENP → M
KWNP and π ⋆ : M EBENPK → M
KWNPK are bijections. Classification
We are now able to complete our main theorem.5.1.
Case 1:
Σ = S .Proposition 5.1. There is no Nahm pole solution to the KW equations on S × S × R + .Proof. By Theorem 4.6, all such solutions must be S -invariant and reduce to solutions ofthe Extended Bogomolny equations. Hence any such solution would lead to a stable Higgsbundle over S with nonvanishing Higgs field. However, these do not exist [8]. (cid:3) Case 2:
Σ = T . We next classify Nahm pole solutions over T × R + .Let M = T × R + with flat metric g . If A is a connection, then d A = ∇ ⊥ A + ∇ y , where ∇ ⊥ A is the covariant derivative on T .We quote the following identity for solutions of the KW equations from [15, 12]: ˆ M ǫ | KW ( A, Φ) | = ˆ M ǫ ( | F A | + |∇ ⊥ A | + |∇ y φ + ⋆φ ∧ φ | + h Ric( φ ) , φ i ) + 2 ˆ ∂M ǫ φ ∧ F A , (21)where M ǫ := T × ( ǫ, ǫ ) and ⋆ is the Hodge star operator on T . Proposition 5.2. If ( A, Φ) is a solution to the KW equations over T × R + satisfying theNahm pole boundary conditions, then F A = 0 , ; ∇ ⊥ A Φ = 0 , ∇ y φ + ⋆φ ∧ φ = 0 . (22) Proof.
From [5], F A ∼ F A LC + O ( y ) = O ( y ), where A LC is the Levi-Civita connection on T , but since the metric on T is flat, F A LC = 0. This shows that lim ǫ → ´ T ×{ ǫ } φ ∧ F A = 0.Furthermore, since ( A, Φ) converges to a flat connection on T , Lemma 4.3 implies thatlim ǫ → ´ T ×{ ǫ } φ ∧ F A = 0. (cid:3) Proposition 5.3.
Let e be a dreibein which is parallel along T . Then (0 , ey ) is the onlysolution to (22) .Proof. Use the temporal gauge in the y -direction, so ∇ y = ∂ y . Then ∇ y φ + ⋆φ ∧ φ = 0 is justthe Nahm equations. Uniqueness of solutions to the Nahm equations with these boundaryconditions implies that φ ≡ ey for some dreibein e . Up to a unitary gauge transformation,we can write e = P i =1 dx i t i where de = 0, dx i is an orthogonal basis of T ⋆ T and thetriplet t i ∈ g P satisfies [ t i , t j ] = ǫ ijk t k . Finally, ∇ ⊥ A Φ = 0 together with de = 0 implies that A ⊥ = 0. (cid:3) Case 3: g (Σ) > .Proposition 5.4. Let ( A, Φ) be a solution to the KW equations on S × Σ × R + y satisfyingNahm pole boundary conditions and which converges to a flat SL( n, C ) connection ( A ρ , φ ρ ) as y → ∞ . If g (Σ) > , then there exists a unique solution if and only if ρ is S independentand lies in the Hitchin component.Proof. By Theorem 4.6, all Nahm pole solutions are S invariant and thus satisfy theextended Bogomolny equations and the statement then follows from Theorem 3.5. (cid:3) Case 4: Knots.
Suppose now that the Nahm pole boundary condition has an addi-tional singularity along the knot K = ∪ i K i where K i = S × { p i } ⊂ S × Σ with weight k i = ( k i , · · · , k in − ). Theorem 5.5.
There is no solution ( A, Φ) to the KW equations over S × S × R + y satisfyingthe Nahm pole boundary conditions with knots K i and weight k i , and which converges to aflat SL( n, C ) connection as y → ∞ .On the other hand, solutions to these equations with these boundary and asymptoticconditions on S × Σ × R + exist when g (Σ) > if and only if there exists a line subbundle L ⊂ E , where ( E , ϕ ) is the Higgs data corresponding to the flat bundle at infinity, such that d ( E , ϕ, L ) = { ( p i , k i ) } . LASSIFICATION OF NAHM POLE SOLUTIONS 13
Proof.
By Proposition 4.6, solutions in either case are necessarily S -invariant. As thereare no Higgs bundles with non-vanishing Higgs field over S , there is no solution over S × S × R + . The rest of the statement is just Theorem 3.6. (cid:3) Corollary 5.6.
Let ρ be an irreducible flat SL( n, C ) connection. Then there exists at most n g solutions to the KW equations satisfying Nahm pole boundary condition with a knotsingularity along K at y = 0 and which converges to ρ in the cylindrical end.Proof. Denote by ( E , ϕ ) the Higgs bundle corresponding to ρ . By Theorem 5.5, existenceof a solution is equivalent to the existence of a line bundle L ⊂ E for which d ( E , ϕ, L ) = { p i , k i = ( k i , · · · , k in − ) } . The knot data determines the divisor D = P i p i ( P n − j =1 k ij ), andwe have Z ( f n ) = D where f n := 1 ∧ ϕ ∧ · · · ∧ ϕ n − . If L D is the line bundle associated to D ,then L n = L − D ⊗ K n ( n − . However, this determines L only up an n th root of unity: if N isany line bundle with N n = O , then ( L ⊗ N ) n = L n . There are n g choice of N , hence n g possible solutions. However, it is not necessarily the case that each ( L ⊗ N ) j is a subbundleof E , so there may not be n g actual solutions. (cid:3) Theorem 5.7.
Let D = P i p i ( P n − j =1 k ij ) be the divisor determined by the given knot data.If deg D is not divisible by n , then there exists no solution.Proof. Let L D be the line bundle associated to D and ( E , ϕ ) the Higgs bundle determinedby ρ . Suppose there exists a solution; then there exists a subbundle L ⊂ E such that L n = L − D ⊗ K n ( n − . Therefore, deg D = − n deg( L ) + n ( n − g − n dividesdeg D . (cid:3) Corollary 5.8.
Let K = S × { p } ⊂ S × Σ with weight and suppose g (Σ) > . Thenthere is no SU(2) solution to the KW equations with Nahm pole singularity and knot K . We now focus on the special case where ρ lies in one of the “non-Hitchin” components ofSL(2 , R ) Higgs bundles. These components are described as follows. Let ℓ be a line bundlewith 0 < deg ℓ < g − E = ℓ − ⊕ ℓ, ϕ = (cid:18) αβ (cid:19) , where α ∈ H ( ℓ − ⊗ K ) and β ∈ H ( ℓ ⊗ K ) are nontrivial sections. Then the zeroes of f := 1 ∧ ϕ : ℓ → K coincide with those of α , and the number of zeroes counted withmultiplicity equals 2 g − − ℓ . Proposition 5.9.
With all notation as above, fix the knot data D = P i p i k i . (i) If deg D = 2 g − − ℓ , then there exists a unique Nahm pole solution if andonly if D = α and no solution otherwise; (ii) if g − > deg D > g − − ℓ , there is no solution.Proof. With L D the line bundle for D , by Theorem 5.5 the necessary condition for existenceof a Nahm pole solution is that there exists L ⊂ E such that L = L − D ⊗ K . For (i), ifdeg( D ) = 2 g − − ℓ , then deg L = deg ℓ . However, since E has rank 2 and deg E = 0,there is a unique subbundle of positive degree, so L = ℓ . By the form of the Higgs bundle,we conclude that D = α . For (ii), if 2 g − > deg D > g − − ℓ , if there is solutionwith line bundle L , then 0 < deg L < deg ℓ , which is impossible. (cid:3) References [1] Kevin Corlette. Flat G -bundles with canonical metrics. J. Differential Geom. , 28(3):361–382, 1988.[2] Simon K. Donaldson. Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stablevector bundles.
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Simons Center for Geometry and Physics, StonyBrook University, Stonybrook, NY 11794
E-mail address : [email protected] Department of Mathematics, Stanford University, Stanford,CA 94305
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