3.3 Global aspects of the twist construction
3.3.2 Global twists
The idea of gluing together local twists in order to get a consistent map of invariant tensor fields on two different manifolds is made precise by the following definition.
Definition 3.3.4(Global twist map). Given manifoldsMand ˜Mequipped with twist data(Z,ω,f)and(Z, ˜˜ ω, ˜f), a global twist mapTis anR-linear map
T :Γ(T‚,˛M)Z ÑΓ(T‚,˛M˜)Z˜ (3.85) sendingZ-invariant tensor fields onMto ˜Z-invariant tensor fields on ˜M, in
particu-lar
´1 f Z,1
f ω,1 f
ÞÑ(Z, ˜˜ ω, ˜f), (3.86) such that there exist
• open coverstUΛu of M andtU˜Λuof ˜M and surjective local diffeomorphisms tψΛ:UΛ ÑU˜Λu, all indexed by the same settΛu,
• local twist data(UΛ,Z,ω, f,ηΛ)satisfying for anyZ-invariant tensor fieldS ψ˚ΛT(S)|U˜Λ =twZ,f,ηΛ(S|UΛ). (3.87) Furthermore, if such a mapTexists,(M, ˜˜ Z, ˜ω, ˜f)is said to be atwistof(M,Z,ω,f). Remark 3.3.5. Any Z-invariant tensor subbundle is generated by (compactly sup-ported)Z-invariant tensor fields. So, if we have a global twist map T sending Z-invariant tensor fields on M to ˜Z-invariant tensor fields on ˜M, we automatically have a map, identified withTitself, sendingZ-invariant tensor subbundles onMto Z-invariant tensor subbundles on ˜˜ M.
Example 3.3.6. We specialise Example3.3.3to the case where theZ-action is aU(1) -action and ω is integral. Let UΛB be a contractible open cover of Band let hΛΣ be functions on the (contractible) overlapsUΛB XUΣB such that
ηΛB ´ηΣB =dhΛΣ. (3.88)
Then forαaZ-invariant form onM, we have onUΛXUΣ
twZ,f,ηΛ(α) =φ˚ΛΣtwZ,f,ηΣ(α), (3.89) where φΛΣ sends (τ,pB) to (τ+hΛΣ(pB),pB). Since Z = Bτ is Hamiltonian, the integrality ofω is equivalent to the integrality ofωB. This in turn is equivalent to saying that we can choose the setsUΛB and the functions hΣΛ so that on the triple overlapsUΛBXUΣBXUΠB, we have the cocycle condition
hΣΠ´hΛΠ+hΛΣ”0 (mod 1). (3.90) Phrased differently, this means that if we regardMand ˜MasU(1)-bundles onB, then their first Chern classes are related byc1(M˜) = c1(M) + [ωB], where [ωB]denotes the (integral) cohomology class ofωB.
The definition of twists ˜Mof a manifoldMand of global twist mapsTbetween tensor fields onMand ˜Mdoes not make any guarantees about existence or unique-ness. It’s not too hard to see that global twist maps, if they exist at all, are generally not unique. This is because the composition of any global twist map between tensor fields onMand ˜Mwith the linear action induced by either a twist automorphism of Mor of ˜Mis again going to be a global twist map. As for existence, the construction in Examples3.1.5,3.3.3, 3.3.6can always be carried out whenever the twist vector field Zinduces aU(1)-action and the twist formω is integral. So in these cases, a twist certainly exists.
A necessary criterion of existence is given by Theorem 3.3.9below, which also establishes the relationship of our formulation of the twist with that of Swann. But first, we need the following lemma and an easy corollary thereof.
Lemma 3.3.7. Let(M, ˜˜ Z, ˜ω, ˜f) be a twist of (M,Z,ω,f)realised by a global twist map T and let p P M and p˜ P M be points such that for all Z-invariant functions h on M,˜ we have h(p) = T(h)(p˜). Then for any Z-invariant function h0 on M that is identically zero in some open neighbourhood V of p, the function T(h0)is identically zero in some open neighbourhoodV of˜ p that is independent of h˜ 0.
Proof. Assume without loss of generality thatVis Z-invariant. Then there exists a Z-invariant function h1 such thath1(p) ‰ 0 and whose support is contained in V.
Thush0h1is identically zero onM, and soT(h0)T(h1) =T(h0h1)is identically zero on ˜M.
However,T(h1)(p˜) = h1(p)‰0 by hypothesis. SinceT(h1)is continuous, there exists a neighbourhood ˜V of ˜p on which T(h1) is nowhere vanishing. But since T(h0)T(h1)is identically zero on ˜V, so must beT(h0). Note that ˜Vis independent of h0.
Corollary 3.3.8. Let(M, ˜˜ Z, ˜ω, ˜f)be a twist of(M,Z,ω,f)realised by a global twist map T and let p P M and p˜ P M be points such that for all Z-invariant functions h on M, we˜ have h(p) = T(h)(p˜). Then if two Z-invariant functions h,h1on M coincide in some open neighbourhood V of p, the functions T(h),T(h1)coincide in some open neighbourhoodV of˜
˜
p that is independent of h,h1.
Theorem 3.3.9. Let M be a manifold of dimension n equipped with twist data (Z,ω,f) such that every point pP M is contained in a hypersurface transversal to Z which intersects any Z-flowline in at most one point. Let(M, ˜˜ Z, ˜ω, ˜f)be a twist of(M,Z,ω,f)realised by
3.3. Global aspects of the twist construction 65 a global twist map T. Then there exists a double surjection
P
M M˜
π π˜ (3.91)
where P is a manifold of dimension n+1equipped with a1-formηˆ such that
d ˆη=π˚(ω), (3.92)
and for any Z-invariant vector field u on M withη-horiziontal liftˆ u, we haveˆ
˜
π(uˆ) =T(u), (3.93)
where T(u)is understood as a section of the pullback bundle π˜´1TM constant along the˜ mapπ.˜
Proof. Define Pto be the set of pairs of points(p, ˜p) P MˆM˜ such that for all Z-invariant functionshon M, we have
h(p) =T(h)(p˜). (3.94) We shall show thatPhas all the required properties.
Given a point(p, ˜p)PP, we can by hypothesis find a hypersurfaceN1containing p and transversal to Z such that any Z-flowline intersects it at most once. Let N be an open set ofN1 containing pand letha|N ben´1 coordinate functions on N.
These can be extended ton´1 functions ha|N1 with compact support on N1. Then we define the value of the functionha at any other point p1 P Mto be equal to the value of the functionha|N1 at p2 if the Z-flowline through p1 intersects N1 in some point p2 and equal to zero if the Z-flowline through p1 does not intersect N1. This extends the coordinate functionsha|N onNton´1Z-invariant functionshaon M.
Without loss of generality, we may assumeha yields a diffeomorphismφ: N Ñ Rn´1. Given any Z-invariant function h on M, this yields a map Fh : Rn´1 Ñ R given by
Fh =h|N˝φ´1. (3.95)
Construct a new Z-invariant function h1 = Fh(ha) := Fh(h1,h2, . . .) on M that re-stricts tohon N. If we letVN be the (Z-invariant) open neighbourhood of p in M given by the union of all theZ-translates of N, handh1 agree on VN. By Corollary 3.3.8, the functions T(h),T(h1) agree on some open neighbourhood ˜VN of ˜p that is independent ofh,h1.
Restrictions of the global twist mapTare local twists. This may be used to locally verify that
T(Fh(ha)) =Fh(T(ha)). (3.96) So if a point (p1, ˜p1) P VNˆV˜N satisfies ha(p1) = T(ha)(p˜1) for all the ha, then it satisfies h1(p1) = T(h1)(p˜1). Thus for any (p1, ˜p1) P VN ˆV˜N, we have ha(p1) = T(ha)(p˜1)if and only ifh(p1) =T(h)(p˜1)for allh(i.e. if and only if(p1, ˜p1)PP).
This allows us to realise the intersection ofPwith the open neighbourhoodVNˆ V˜N of an arbitrary point(p, ˜p)PPas a level set of a map fromVNˆV˜N ĎMˆM˜ to Rn´1given by
(p1, ˜p1)ÞÑ ha(p1)´T(ha)(p˜1). (3.97)
Since the functionsharestrict to a coordinate chart containingponN, it follows that the differential of the above map is surjective. Hence, we conclude using the implicit function theorem thatPis a submanifold ofMˆM˜ of codimensionn´1. From the local realisation of the twist, we know that ˜M has the same dimension as M i.e.n.
So,Pis a manifold of dimension
2n´(n´1) =n+1 (3.98)
which inherits from the canonical projections prM and prM˜ on MˆM, the maps˜ π and ˜πto Mand ˜Mrespectively.
In particular, if two pointsp,p1are related by the flow alongZ, then(p, ˜p)lies in Pif and only if (p1, ˜p)does so as well. Likewise, since functions on ˜M of the form T(f)are ˜Z-invariant, if two points ˜p, ˜p1 are related by the flow along ˜Z, then (p, ˜p) lies inPif and only if(p, ˜p1)does so as well.
All this tells us so far is that the mapsπand ˜πare submersions. To see that they are indeed surjections, we make use of the fact that the local diffeomorphisms ψΛ are surjective. Note that the graphs of the mapsψΛ are contained in P. So, given a point p P UΛ, we have (p,ψΛ(p))in the preimageπ´1(p)of p. Likewise, given
˜
pPU˜Λ, there exists somepsuch thatψΛ(p) =p˜ and we have(p, ˜p)in the preimage π˜´1(p˜)of ˜p.
The space of tangent vector fields onPregarded as aC8(P)-module is spanned by vector fields of the form
u‘u1 =u‘(T(u) +aZ˜), (3.99) whereu is a Z-invariant vector field on M, a is a constant, and we think of TPas being a subbundle of the pullback of the bundle
T(MˆM˜)–pr´1MTM‘pr´1M˜ TM (3.100) along the inclusion map from P into MˆM. Note that we have slightly abused˜ notation to letuandu1denote sections of the pullback bundles pr´1MTMand pr´1M˜ TM constant along prM and prM˜ respectively. Since such vector fields span the space of all tangent vector fields onP, in order to define the 1-form ˆη, it’s enough to specify how it acts on sections of the form (3.99):
ˆ
η(u‘(T(u) +aZ˜)) =a. (3.101) In particular the ˆη-horizontal lift ˆuof anyZ-invariant vector fielduis given by
ˆ
u=u‘T(u), (3.102)
from which (3.93) immediately follows.
Meanwhile, to deduce (3.92), we make use of the following consequence of the naturality of Lie-commutators. Ifu,v are vector fields onM andu1,v1 vector fields on ˜M, then we have
Lu‘u1(v‘v1) =Luv‘Lu1v1. (3.103) In conjunction with (3.50), this implies that ifu,v,wareZ-invariant vector fields on Msatisfying
Luv=w, (3.104)
3.3. Global aspects of the twist construction 67 then their ˆη-horizontal lifts ˆu, ˆv, ˆwsatisfy
Luˆvˆ =Luv‘LT(u)˝T(v) =w‘(T(w)´ω(u,v)Z˜)
=wˆ ´0‘ω(u,v)Z,˜ L0‘Z˜vˆ =LZ˜ ˝T(v) =0.
(3.105)
This in turn implies the following:
d ˆη(u, ˆˆ v) =uˆ(ηˆ(vˆ))´vˆ(ηˆ(uˆ))´ηˆ(Luˆvˆ) =ω(u,v),
d ˆη(0‘Z, ˆ˜ v) = (0‘Z˜)(ηˆ(vˆ))´vˆ(ηˆ(0‘Z˜))´ηˆ(L0‘Z˜vˆ) =0. (3.106) It follows that d ˆη = π˚(ω)since vector fields of the form ˆuand 0‘Z˜ spanΓ(TP). We have essentially retrieved Swann’s construction of the twist map in [Swa10].
From the above proof, we see that ifZand ˜ZinducedU(1)-actions onMand ˜M, then Pwould be aU(1)-principal bundle over both Mand ˜M, with ˆηbeing a connection 1-form forπ: PÑ Mwith curvatureω.
We can go the other way as well and show that the existence of a double surjec-tion is also a sufficient criterion for the existence of global twists.
Theorem 3.3.10. Let M be a manifold of dimension n equipped with twist data(Z,ω,f) and let
P
M M˜
π π˜ (3.107)
be a double surjection, with P being a manifold of dimension n+1equipped with a1-form ˆ
ηsuch that
d ˆη=π˚(ω). (3.108)
Let XP, ˆZ,ZP be vector fields on P satisfying
π˚(XP) =0, π˜˚(ZP) =0, π˚(Zˆ) =Z, ˆ
η(XP) =1, ηˆ(Zˆ) =0, ZP =Zˆ + f XP. (3.109) Let T be the map sending Z-invariant tensor fields on M to tensor fields onM induced by˜ pulling back functions on M alongπand taking theη-horizontal lift of vector fields on Mˆ and pushing them down ontoM along˜ π. Then˜
T
´1 f Z
,T
1 f ω
,T
1 f
(3.110) is twist data onM and T is a global twist map.˜
Proof. Let h be a Z-invariant function and α be a Z-invariant 1-form on M. Then the pullbacks π˚(h) and ˜π˚(T(h)) on P are equal while the pullbacks π˚(α) and π˜˚(T(α))onPagree on ˆη-horizontal vector fields. So if ˆuis the ˆη-horizontal lift of a vector fielduonM, we have
π˜˚(T(α))(uˆ) =π˚(α)(uˆ) =α(u),
˜
π˚(T(α))(XP) = 1
f π˜˚(T(α))(ZP)´1
f π˜˚(T(α))(Zˆ) =´1
f α(Z). (3.111)
Note that the choice of auxiliary local twist data (U,η) on M gives a 1-form θ := ηˆ´π˚(η)onP|U satisfying
dθ =0, θ(XP) =ηˆ(XP) =1, θ(ZP) =ηˆ(ZP)´η(Z) = f´η(Z). (3.112) The closed 1-form θ defines integral hypersurfaces in P and the nonvanishing of θ(XP) =1 andθ(ZP) = f´η(Z)corresponds to the fact that the integral hypersur-faces are transversal to bothπ and ˜π. We may identify one such hypersurfaceUP withU=π(UP). This gives us a surjective local diffeomorphismψ:UÑπ˜(UP) =: U.˜
The tangent vectors toUPare elements of the kernel ofθ. We may check
θ(uˆ+η(u)XP) =η(u)ηˆ(XP)´π˚(η)(uˆ) =η(u)´η(u) =0. (3.113) The vector fields ˆu+η(u)XPare thus tangent toUPand may be identified withuon U. To obtainψ˚T(h)andψ˚T(α), we simply pull back ˜π˚(T(h))and ˜π˚(T(α))along the mapι:U–UP ãÑ Pgiven by the inclusion under the above identification:
ψ˚T(h) =ι˚˝π˜˚(T(h)) =ι˚˝π˚(h) =h=twZ,f,η(h), ψ˚T(α)(u) =ι˚˝π˜˚(T(α))(u) =π˜˚(T(α))(uˆ+η(u)XP)
(3.111)
= α(u)´1
f α(Z)η(u) = (twZ,f,η(α))(u).
(3.114)
Compatibility with tensor products and contractions then tells us thatTrestricts to local twist maps for all tensor fields. Proposition3.1.13may then be used to locally verify that
(Z, ˜˜ ω, ˜f):=
T
´1 f Z
,T
1 f ω
,T
1 f
(3.115) constitutes twist data.
The only thing remaining to check in order to conclude thatTis indeed a global twist map is to show that there exist tuples of local twist data(UΛ,ηΛ) such that tUΛu andtU˜Λ := π˜(UΛ,P)u constitute open covers of M and ˜M respectively. For this, we note that for any arbitrary point(p, ˜p)P P, we can find a sufficiently small hypersurfaceUP containing (p, ˜p) which is transversal to both π and ˜π such that π|UP is a diffeomorphism onto its image U in M. There is then a unique (closed) XP-invariant 1-form which vanishes onUP such that θ(XP) = 1. Then ˆη´θ is an XP-invariant horizontal 1-form and so it is the pullbackπ˚(η)of some 1-formηon Usuch that dη= ω|Uand f´η(Z) =θ(ZP)is nowhere vanishing.
Thus, our construction of the twist and that of Swann are essentially equivalent.
That said, there is a slight increase of generality in that the requirement of integrality of the twist formω in Swann’s construction may be relaxed to rationality. This is illustrated in the following example.
Example 3.3.11. Let M = Rą0ˆRˆT4be coordinatised by(r,τ,x1,y1,x2,y2) sub-ject to the identifications
(x1,y1,x2,y2)„(x1+1,y1,x2,y2)„(x1,y1+1,x2,y2)
„(x1,y1,x2+1,y2)„(x1,y1,x2,y2+1). (3.116)
3.3. Global aspects of the twist construction 69 Then we have twist data(Z,ω, f)on it given by
Z=Bτ, ω =dr^dτ+dx1^dy1+λdx2^dy2, f =r. (3.117) whereλis some real parameter that shall be used to control whether ωis rational or not. Working with an atlas is pretty cumbersome, so we will instead go to the universal coverMof M(which happens to be contractible in this case), while keep-ing track of the identifications (3.116). On M, we introduce for every pair(a,b)of integers, the auxiliary 1-form
ηa,b= (r+1)dτ+ (x1+a)dy1+λ(x2+b)dy2. (3.118) Note that these all satisfy
f´ηa,b(Z) =´1, (3.119)
which is nowhere vanishing. As a result Z˜ =´ Z
f´ηa,b(Z) = Z=Bτ. (3.120) The variousηa,bdiffer fromη0,0by the exterior derivative of the function
ha,b= ay1+λby2. (3.121)
From the proof of Proposition3.3.1, we know that the local twists with respect toηa,b differ from that with respect toη0,0by a diffeomorphismφ1given by the differential equation
φ0=idM, dφs
ds ˇ ˇ ˇ ˇs=t
(φ´1t (¨)) =ha,b(¨)Bτ. (3.122) The solution to this is
φ1(r,τ,x1,y1,x2,y2) = (r,τ+ay1+λby2,x1,y1,x2,y2). (3.123) Note that making the identifications (3.116) forces us to make the following identifi-cations in addition:
τ„τ+a+λb, (3.124)
where(a,b)is an arbitrary pair of integers. Ifλis rational with standard form p/q, then the set of integer linear combinations of 1 andλisq´1Z. Otherwise, the set is dense inR. Thus, it is only whenλ(and henceω) is rational that the local twist map onMdescends to a well-defined global twist map sendingBτ-invariant tensor fields onMtoBτ-invariant tensor fields on a manifold ˜Mobtained fromMby making the identifications
(τ,x1,y1,x2,y2)„(τ´y1,x1+1,y1,x2,y2)„(τ,x1,y1+1,x2,y2)
„(τ´λy2,x1,y1,x2+1,y2)„(τ,x1,y1,x2,y2+1). (3.125)
71
Chapter 4
To locally hyperkähler manifolds and back again
In this chapter, we describe Haydys’ QK/HK correspondence in terms of the twist (Theorem4.1.11). Note that though the correspondence itself is not new, the origi-nal formulation in [Hay08] was different and made use of Swann bundles and the hyperkähler quotient. We then identify in Propositions4.2.7and4.2.10the precise conditions for which this is inverse to the twist description of the opposite HK/QK correspondence due to Macia and Swann [MS14]. Finally, as a generalisation of the results in [Cor+17], we use the various twist formulae developed in Chapter 3to derive identities relating the Levi-Civita connection and Riemann curvature of a quaternionic Kähler manifold to that of the locally hyperkähler manifold associ-ated to it via the QK/HK correspondence (Proposition4.2.8 and Theorem 4.2.17).
We then use this to show that the 1-loop-deformed Ferrara–Sabharwal metrics with quadratic prepotential have cohomogeneity exactly 1 (Theorem4.2.21).
All the results in this chapter with the exception of Proposition4.2.5due to Macia and Swann [MS14] are original. Proposition4.2.8and and Theorems4.2.17and4.2.21 have been proved in collaboration with Danu Thung [CST20b;CST20a].