Riemann curvature under HK/QK

4.2. Inverting the QK/HK correspondence 89 This is the defining property of the quaternionic moment map written in terms of local frame(J₁,J2,J2). Therefore, we can now obtain

f_Q =´g(Z,Z)

kµ^Zk ^´νkµ^Zk= ^f˜1

Kf˜_H² g˜_H(Z, ˜^˜ Z) + ^Kν f˜₁

=_´ ¹

f˜₁f˜_H g˜(Z, ˜^˜ Z) + ¹ k˜f˜₁ = ¹

f˜_H, ι_Zg

kµ^Zk =_´ ¹

f˜_Hkµ^Zk^{twZ, ˜˜ fH, ˜ηH}(ιZ˜g˜_H) =_´¹ f˜₁ tw_{Z, ˜˜ f}

H, ˜η_H(λ₀), ι_Zω_i

kµ^Zk =´1 f˜₁tw_{Z, ˜˜ f}

H, ˜η_H(λ_i), η_Q =_´

ι_Zg

kµ^Zk+xJ₂,∇^gJ₃y (4.96)

= _´ ^ιZg

kµ^Zk^{´twZ, ˜˜ fH, ˜ηH} λ₀

f˜₁ ´ η˜_H f˜_H´η˜_H(Z^˜)

=tw_{Z, ˜˜ f}

H, ˜η_H

η˜_H f˜_H´η˜_H(Z^˜)

= ^η˜H f˜_H.

(4.100) Thus, the local twist data(U,Z,ω_Q,f_Q,η_Q)is indeed dual to(U, ˜^˜ Z, ˜ω_H, ˜f_H, ˜η_Q), lead-ing to the followlead-ing conclusion:

tw_Z,ωQ,fQ(Q) =tw_Z,ωQ,fQ˝tw_{Z, ˜˜ω}

H, ˜fH(H) = H, tw_Z,ωQ,fQ(g_Q) =tw_Z,ωQ,fQ

Kν

kµ^Zk^g|HQZK´ K f_Q

kµ^Zk^{2 g|HQZ}

= ^Kν

kµ^Zk^{g˜H|HHZ˜K´} K f_Q

kµ^Zk^{2 g˜H|HHZ˜}

= ^f˜1

K˜ g˜_H|_HHZ˜^K+ ^k˜ K˜

f˜₁²

f_Hg˜_H|_HHZ˜ = g.˜

(4.101)

Remark4.2.11. Now that we know that the HK/QK correspondence is a two-sided inverse of the QK/HK correspondence (for appropriate choices of the parameters involved), we shall refer to the tuple of HK/QK data(_{M, ˜}^˜ _{g,H, ˜Z,I1, ˜f1, ˜}ω_H, ˜fH, ˜gH) as being thehyperkähler dualof the tuple of QK/HK data(M,g,Q,Z,ω_Q,f_Q,g_Q), and the QK/HK data(M,g,Q,Z,ω_Q,f_Q,g_Q)as being thequaternionic dualof the HK/QK data(M, ˜^˜ g,H, ˜Z,I₁, ˜f₁, ˜ω_H, ˜f_H, ˜g_H)_.

Definition 4.2.13(Riemann product). The Riemann product:of two 2-formsα,βP Ω²Mon a manifoldMis given by

α:β=α?β+2αbβ+2βbα. (4.103) The point of introducing these products is that they can be used to construct tensor fields which have the same symmetries as the (lowered) Riemann curvature.

Definition 4.2.14(Abstract curvature tensor field). A(0, 4)-tensor fieldCPΓ(_T^0,4M on a manifoldM is said to be an abstract curvature tensor field if it satisfies for all vector fieldss,t,u,vonMthe following equations:

C(s,t,u,v) =´C(t,s,u,v) =´C(s,t,v,u),

C(s,t,u,v) +C(t,u,s,v) +C(u,s,t,v) =_0. ^(4.104) Lemma 4.2.15. For any two symmetric bilinear forms g,h and any two2-formsα,β, the tensor fields g?h andα:βare abstract curvature tensor fields.

Example 4.2.16. The Riemann curvature R_HPⁿ of the quaternionic Kähler metric g on the quaternionic projective spaceHPⁿ is given in terms of a local oriented or-thonormal frame(J₁,J₂,J₃)of its quaternionic bundleQby

g˝R_HPⁿ =´1

8 g?g+ ÿ3

i=1

(g˝J_i)_:(g˝J_i)

!

. (4.105)

The minus sign may seem strange when compared to other references which make use of the abstract index notation, but this is just a consequence of taking the (low-ered) Riemann curvature in the abstract index notation to be

R_abcd= g(R^g(e_c,e_d)e_b,e_a) =´g(R^g(e_a,e_b)e_c,e_d)_{. (4.106)} In terms of these products we can thus express the lowered Riemann curvature g˝R^g of a quaternionic manifold as the twist of the sum of the lowered Riemann curvature ˜g_H˝R^g˜H of its hyperkähler dual and certain tensor fields algebraically constructed out of ˜g_Hand ˜ω_Hthat are manifestly abstract curvature tensor fields.

Theorem 4.2.17. Let (M, ˜^˜ g,H, ˜Z,I₁, ˜f₁, ˜ω_H, ˜f_H, ˜g_H) be HK/QK data with quaternionic dual(M,g,Q,Z,ω_Q,f_Q,g_Q), and letT be the global twist map realising the HK/QK corre-˜ spondence. Then the Riemann curvatures R^gand R^g˜of the metrics g andg satisfy˜

g˝R^g= T^˜ K˜

f˜₁ g˜˝R^g˜+ ¹

8 ˜K g˜_H?g˜_H+ ÿ3

i=1

(g˜_H˝I_i)_:(g˜_H˝I_i)

!

´ K˜ 8˜k

1

f˜₁f˜_H ω˜_H:ω˜_H+ ÿ3

i=1

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

!!

,

(4.107)

where(I₁,I₂,I₃)is a local oriented orthonormal Kähler frame of H.

Proof. Since the Riemann curvatureR^hwith respect to any metrichis given by R^h(u,v)t = [∇^h_u,∇^h_v]t´∇^h_Luvt, (4.108)

4.2. Inverting the QK/HK correspondence 91

we have as a consequence of (4.78) and (3.34b) R^g(u¹,v¹)t¹ =T^˜

R^∇g˜+SHQ(u,v)t´ 1

f˜_Hω˜_H(u,v)∇^g˜_˜

Zt+S^HQ_Z˜ t

. (4.109) Hereu,v,t are ˜Z-invariant vector fields on ˜M, with global twistsu¹,v¹,t¹on M, and R^∇g˜+SHQ is the curvature of the connection∇^g˜+S^HQ, given by

R^∇g˜+SHQ(u,v)t=^h∇^g_u^˜+S^HQ_u ,∇^g_v^˜+S^HQ_v i

t´∇^g_L^˜

uvt´S^HQ_L

uvt

= R^g˜(u,v)t+^h∇^g_u^˜,S^HQ_v i t´

h

∇^g_v^˜,S^HQ_u i t +^hS^HQ_u ,S^HQ_v i

t´S^HQ

∇^g_u^˜vt+S^HQ

∇^g_v^˜ut

= R^g˜(u,v)t+∇^g_u^˜S^HQ

(v,t)_´∇_v^g˜S^HQ

(u,t) +^hS^HQ_u ,S^HQ_v i t, (4.110) where we have used the fact that∇^g˜is torsion-free. Moreover, sincetis ˜Z-invariant, we have

∇^g˜_˜

Zt=_LZ˜t+∇^g_t^˜Z^˜ =∇^g_t^˜Z.^˜ (4.111) Putting (4.110) and (4.111) together, we get

R^g(u¹,v¹)t¹ = T^˜

R^g˜(u,v)t+∇^g_u^˜S^HQ

(v,t)´

∇_v^g˜S^HQ (u,t) + ^hS^HQ_u ,S^HQ_v i

t´ 1

f˜_Hω˜_H(u,v)∇^g_t^˜Z^˜ +S^HQ_˜

Z t

. (4.112)

Substituting (4.79) into the above, making use of the identity

(∇_u^g˜I_H)v´(∇_v^g˜I_H)u=R^g˜(u,v)Z,^˜ (4.113) and carrying out simplifications (deferred to Section4.Ain the appendix) then yields

R^g(u¹,v¹)t¹

= T R^{˜ g˜}(u,v)t

´ 1 2 ˜f_H²

1 2

ÿ3

α,β=0

(ω˜_H(IαZ,˜ u)ω_H(I_βv,t)´ω_H(IαZ,˜ v)ω˜_H(I_βu,t))Iα˝I_βZ˜ +

ÿ3

α=0

˜

ω_H(u,v)ω˜_H(IαZ,˜ t)IαZ˜

!

+ ¹ 2 ˜f_H

1 2

3

ÿ

α=0

˜

ω_H(Iαv,t)Iα˝I_Hu´ω˜_H(Iαu,t)Iα˝I_Hv+4 ˜g(Iα˝R^g˜(u,v)Z,^˜ t)IαZ˜

´ω˜_H(u,v)I_Ht

!

+ ¹ 4

ÿ3

α=0

˜

g_H(Iαu,t)Iαv´g˜_H(Iαv,t)Iαu+ g˜_H(Iαu,v)´g_H(Iαv,u)Iαt

! .

(4.114) Recall that any complex structure Kähler with respect to some metrich necessarily

commutes with the Riemann curvatureR^h(u,v)for any any vector fieldsuandvand thatR^h(u,v)is skew-self-adjoint with respect toh. In this case,I₁,I2,I3are all Kähler with respect to ˜g, and so commute withR^g˜, as does I₀ on account of just being the identity endomorphism field. Thus, we have

˜

g(I_α˝R^g˜(u,v)Z,^˜ t) =g˜(R^g˜(u,v)˝I_αZ,˜ t) =´g˜(I_αZ,˜ R^g˜(u,v)t). (4.115) As a result, we may now succinctly write (4.114) using the endomorphism field A introduced in (4.88) as

R^g(u¹,v¹)t¹

=T^˜

A˝R^g˜(u,v)t´ 1

2 ˜f_Hω˜_H(u,v)A˝I_Ht + ¹

4 ˜f_H ÿ3

α=0

˜

ω_H(I_αv,t)A˝I_H˝I_αu´ω˜_H(I_αu,t)A˝I_H˝I_αv + ¹

4 ÿ3

α=0

˜

g_H(Iαu,t)Iαv´g˜_H(Iαv,t)Iαu+ g˜_H(Iαu,v)´g˜_H(Iαv,u)Iαt

! .

(4.116)

Note that we have made use of the fact that A commutes with all the I_α, a conse-quence of the fact thatI₁,I2,I3 are Hermitian with respect to both ˜g and ˜g_H. Addi-tionally, because we have

˜

g_H(A¨,¨) = ^K˜

f˜₁g˜(¨,¨), (4.117) we may contract (4.116) withι_sg˜_Hunder the twist, wheresis some ˜Z-invariant vector field on ˜Mwith global twists¹onM, to obtain

g(R^g(u¹,v¹)t¹,s¹)

= ^K˜

f˜₁ g˜(R^g˜(u,v)t,s)_´ ¹ 2 ˜f_H

K˜

f˜₁ ω˜_H(u,v)g˜(I_Ht,s) + ¹

4 ˜f_H K˜ f˜₁

ÿ3

α=0

˜

ω_H(I_αv,t)g˜(I_H˝I_αu,s)´ω˜_H(I_αu,t)g˜(I_H˝I_αv,s) + ¹

4 ˜K ÿ3

α=0

˜

g_H(I_αu,t)g˜_H(I_αv,s)_´g˜_H(I_αv,t)g˜_H(I_αu,s) + g˜_H(Iαu,v)´g˜_H(Iαv,u)g˜_H(Iαt,s)

= ^K˜

f˜₁ g˜(R^g˜(u,v)t,s)´ K˜ 2˜k

1

f˜₁f˜_H ω˜_H(u,v)ω˜_H(t,s) + ^K˜

4˜k 1 f˜₁f˜_H

ÿ3

α=0

˜

ω_H(I_αv,t)ω˜_H(I_αu,s)´ω˜_H(I_αu,t)ω˜_H(I_αv,s) + ¹

4 ˜K ÿ3

α=0

˜

g_H(I_αu,t)g˜_H(I_αv,s)_´g˜_H(I_αv,t)g˜_H(I_αu,s) + g˜_H(Iαu,v)´g˜_H(Iαv,u)g˜_H(Iαt,s).

(4.118)

This may be rewritten in terms of the products?and:to yield (4.107).

4.2. Inverting the QK/HK correspondence 93 Remark4.2.18. The HK/QK Levi-Civita connection formula (4.78) and HK/QK cur-vature formula (4.107) are generalisations of the Levi-Civita and curcur-vature formulae for a special class of quaternionic Kähler manifolds known asq-map spacesderived by Cortés, Dyckmanns, Jüngling, and Lindemann in Sections 2.5 and 2.6 of [Cor+17].

Remark4.2.19. The HK/QK curvature formula (4.107) is also a refinement of Alek-seevsky’s decomposition of the curvature of quaternionic Kähler metrics, quoted in Theorem2.1.12, in the following sense. TheHPⁿpart arises from

T˜ 1

8 ˜K g˜_H?g˜_H+ ÿ3

i=1

(g˜_H˝I_i)_:(g˜_H˝I_i)

!!

= ¹

8 ˜K g?g+ ÿ3

i=1

(_g˝Ji)_:(_g˝Ji)

!

=νg˝R_HPⁿ.

(4.119)

Meanwhile, the quaternionic Weyl curvatureW_Q^g arises from T˜ K˜

f˜₁ g˜˝R^g˜´ K˜ 8˜k

1

f˜₁f˜_H ω˜_H:ω˜_H+ ÿ3

i=1

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

!!

. (4.120) The quaternionic Weyl curvature is by definition a tensor field with all the sym-metries of the Riemann or Weyl curvature which additionally commutes with the quaternionic bundleQ. In the lowered form, this amounts to requiringg˝W_Q^g to be an abstract curvature tensor field satisfying

g˝W_Q^g(s,t,J_ju,J_jv) = g˝W_Q^g(s,t,u,v), (4.121) where(J₁,J₂,J₃)is any local oriented orthonormal frame forQ. That it is an abstract curvature tensor field is manifest. Meanwhile, (4.121) follows from the proposition below.

Proposition 4.2.20. The tensor field W :=T^˜ K˜

f˜₁ g˜˝R^g˜´ K˜ 8˜k

1

f˜₁f˜_H ω˜_H:ω˜_H+ ÿ3

i=1

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

!!

(4.122) satisfies for any local oriented orthonormal frame(J₁,J2,J3)of Q

W(s,t,J_ju,J_jv) =W(s,t,u,v). (4.123) Proof. LetW = T^˜(W¹). Then it suffices to show thatW¹ satisfies

W¹(s,t,I_ju,I_jv) =W¹(s,t,u,v) (4.124) for local Kähler structuresI₁,I2,I3. But this is indeed the case as can be seen from the fact thatR^g˜ is the curvature of a locally hyperkähler metric and so commutes with

the local Kähler structures and the following short computation:

˜

ω_H:ω˜_H+ ÿ3

i=1

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

!

(s,t,I_ju,I_jv)

=4 ˜ω_H(s,t)ω˜_H(I_ju,I_jv) +2

3

ÿ

α=0

(ω˜_H(Iαs,I_ju)ω˜_H(Iαt,I_jv)´ω˜_H(Iαs,I_jv)ω˜_H(Iαt,I_ju))

=4 ˜ω_H(s,t)ω˜_H(u,v) +2 ÿ3

α=₀

(ω˜_H(I_αs,u)ω˜_H(I_αt,v)´ω˜_H(I_αs,v)ω˜_H(I_αt,u))

= ω˜_H:ω˜_H+ ÿ3

i=1

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

!

(s,t,u,v)_,

(4.125) where the penultimate step follows from making a replacement I_α ÞÑ I_j ˝I_α in the sum and then using the fact that I_H commutes with I_j, so that ˜ω_H(I_ju,I_jv) =

˜

ω_H(u,v).

The Riemann curvature and associated curvature invariants are in general much harder to compute in case of a quaternionic Kähler manifold than a locally hyperkäh-ler manifold. The above relation therefore simplifies such computation and makes computing, say, the curvature norm of the quadratic prepotential Ferrara–Sabharwal metrics tractable. As a result, we can establish the following fact.

Theorem 4.2.21. The 1-loop-deformed quadratic prepotential Ferrara–Sabharwal metrics have cohomogeneity1.

Proof. We shall first show that the cohomogeneity of the 1-loop-deformed quadratic prepotential Ferrara–Sabharwal metricsg^1c_FS =: gis at least 1 by computing the cur-vature norm, given by

tr(_R²) = ¹ 4

ÿ

a,b,c,d

g(R^g(E¹_a,E_b¹)E_c¹,E¹_d)², (4.126) where the vector fieldsE¹_a constitute an orthonormal frame with respect to the Rie-mannian metricg. We have already seen in Example4.2.6that these metrics arise as images of the flat metric

˜

g=´(|dz₀|²+|dw₀|²) +

n´1ÿ

a=₁

(|dz_a|²+|dw_a|²), (4.127) on complex cotagent spaces of dimension real 4n, under the HK/QK correspon-dence. Therefore, by (4.107), the curvature norm is given by

tr(_R²) = ¹ 4

ÿ

a,b,c,d

C(E_a,E_b,E_c,E_d)², (4.128)

4.2. Inverting the QK/HK correspondence 95 whereCis a(0, 4)-tensor given by

C= ¹

8 ˜K g˜_H?g˜_H+ ÿ3

i=1

(g˜_H˝I_i)_:(g˜_H˝I_i)

!

´ K˜ 8˜k

1

f˜₁f˜_H ω˜_H:ω˜_H+

3

ÿ

i=₁

(ω˜_H˝I_i)_?(ω˜_H˝I_i)

! ,

(4.129)

and the vector fieldsEa constitute an orthonormal frame with respect to ˜g_H. Note that E_a do not need to be the twists of E¹_a, or even Z-invariant for that matter. A straightforward computation (deferred to Section4.Bin the appendix) now gives us

tr(_R²)

= ¹ K˜²



n(_5n+₁) +_{3 k}^{˜3 f˜}

13

f˜_H³ + (_n´1)_k^{˜ f˜1} f˜_H

!2

+₃ ^˜_k^{6 f˜}

16

f˜_H⁶ + (_n´1)_k^{˜2 f˜}

12

f˜_H²

!



=ν² n(5n+1) +3

ρ³

(ρ+2c)³ + (n´1)ρ (ρ+2c)

2

+3

ρ⁶

(ρ+2c)⁶ + (n´1)ρ² (ρ+2c)²

! . (4.130) We can write the mapρÞÑ tr(_R²)as a composition of two maps, bothRą0 ÑRą0, given by

ρÞÑ ρ

ρ+2c =:Y, YÞÑ ν²(n(5n+1) +3(Y³+ (n´1)Y)²+3(Y⁶+ (n´1)Y²). (4.131) The first map is clearly injective whencą0. As for the second map, suppose there existedY₁andY₂such that

ν²(n(5n+1) +3(Y₁³+ (n´1)Y₁)²+3(Y₁⁶+ (n´1)Y₁²)

= ν²(n(5n+1) +3(Y₂³+ (n´1)Y₂)²+3(Y₂⁶+ (n´1)Y₂²). (4.132) This can be rearranged into

(Y₁´Y₂)(Y₁+Y₂)((Y₁²+Y₂²+n´1)²+Y₁⁴+Y₂⁴+n´1) =0. (4.133) The only way the left-hand side can vanish forn ě 1 and Y1 andY2 both positive is if we hadY₁ = Y₂. So, the second map and hence the compositionρ ÞÑ tr(_R²) is injective. Since the curvature norm must be preserved by any isometry, it follows that any isometry ofgnecessarily sends any given constantρhypersurface to itself.

In other words,ghas cohomogeneity at least 1.

To prove that it has cohomogeneity at most and hence equal to 1, we make use of the involutionv Ø v˜in (3.42) to construct Killing fields ofgfrom ˜ω_H-Hamiltonian Killing fields of ˜g_H which Lie-commute with ˜Z. Killing fields of ˜g_H which Lie-commute with ˜Zmay be obtained by considering Killing fields of ˜gwhich preserve f˜₁,v₁,v₂,v₃ separately. (This is a sufficient condition, but not a necessary one, as the example of ˜Zshows.)

More concretely, we consider the following vector fields:

˜

u⁺_a =Re(zaBz₀ +z₀B_za´w₀Bwa´waB_wo), v˜⁺₀ =Re(Bw₀), v˜⁺_a =Re(Bwa),

˜

u^´_a =Im(zaBz₀ +z0Bza´w0Bwa´waBwo), v˜^´₀ =Im(Bw₀), v˜^´_a =Im(Bwa), (4.134)

witharanging from 1 ton´1. These Lie-commute with ˜Z, preservev₁,v₂,v₃, and are ˜ω_H-Hamiltonian. We make the following choice of Hamiltonian functions:

f˜_u˜⁺_a = ^˜k

2Re(´i(z_az₀´w_aw₀))´1, f˜_v˜⁺

0 = ^k˜

2Re(iw₀)´1, f˜_v˜⁺_a = ^k˜

2Re(´iw_a)´1, f˜_u˜^´

a = ^˜k

2Im(´i(z_az₀´w_aw₀))´1, f˜_v˜^´

0 = ^k˜

2Im(iw₀)´1, f˜_v˜^´

a = ^k˜

2Im(´iw_a)´1.

(4.135) With the choice of ˜η_Has in (4.61), the involutionvØv˜is given by

v=tw_{Z, ˜˜ f}

H, ˜η_H(v˜) + (f^˜_v˜+1)Z=v˜´

f˜_v˜´η˜_H(v˜) +1 f˜_H´η˜_H(Z^˜) ^Z˜

=v˜+^{8 ˜K}

k˜ (f^˜v˜´η˜_H(v˜) +1)Z.^˜

(4.136)

Therefore, we obtain u⁺_a =u˜⁺_a +

2 ˜Kc´1 2

Re

iz^a z₀

Z,˜

v⁺₀ =v˜⁺₀ +2 ˜KRe(iw0)Z,^˜ v⁺_a =v˜⁺_a ´2 ˜KRe(iwa)Z,^˜ u^´_a =u˜^´_a +

2 ˜Kc´1 2

Re

iz^a z0

Z,˜

v^´₀ =v˜^´₀ +2 ˜KRe(iw₀)Z,^˜ v^´_a =v˜^´_a ´2 ˜KRe(iw_a)Z.^˜

(4.137)

Carrying out the change of coordinates given in (4.65), we finally get u⁺_a =Re ´

n´1ÿ

b=1

X_aX_bBX_b+_BX

a´ζ₀Bζa´ζ_aB_ζ

0+2i ˜KcX^aBτ

! , v⁺₀ =^?2 Re(_Bζ0+i ˜Kζ₀Bτ), v⁺_a =^?2 Re(_Bζa´i ˜Kζ_aBτ), u^´_a =Im ´

n´1

ÿ

b=1

X_aX_bB_Xb+B_Xa´ζ₀B_ζa´ζ_aB_ζ

0 +2i ˜KcX^aBτ

! , v^´₀ =^?2 Im(B_ζ0+i ˜Kζ₀Bτ), v^´_a =^?2 Im(B_ζa´i ˜Kζ_aBτ).

(4.138)

These are Killing with respect tog. The diffeomorphisms generated by them and B_τ act transitively on the constant ρ hypersurfaces. Hence, the 1-loop-deformed quadratic prepotential Ferrara–Sabharwal metrics have cohomogeneity exactly 1.

Remark4.2.22. Note that (4.130) reduces to the curvature norm for the universal hy-permultiplet in (2.98) forn=1.

97

Appendix

This appendix includes details of the computation of the curvature formula in (4.107) and the curvature norm in (4.130) that were skipped over earlier in this chapter.

These computations have been carried out in collaboration with Danu Thung [CST20b;

CST20a].

4.A HK/QK curvature formula

In this section, we’ll show how to obtain (4.114) from (4.112). We’ll divide the com-putation into several lemmata describing the various pieces on the right-hand side of (4.112).

Lemma 4.A.1. The antisymmetrised covariant derivative of S^HQis given by ∇^g_u^˜S^HQ

(v,t)´

∇^g_v^˜S^HQ (u,t)

= ¹ 2

ÿ3

α=0

1

f˜_H² ω˜_H(Z,^˜ u)ω_µ(I_Hv,t)I_αZ˜ + ¹

f˜₁²λ₁(u) g˜(I_α˝I₁Z,v)I_αt+g˜(I_α˝I₁Z,˜ t)I_αv

´ 1

f˜_H² ω˜_H(Z,^˜ v)ω˜_H(I_αu,t)I_αZ˜ ´ 1

f˜₁² λ₁(v) g˜(I_α˝I₁Z,˜ u)I_αt+g˜(I_α˝I₁Z,˜ t)I_αu + ¹

2 ˜f_H ω˜_H(I_αv,t)I_α˝(I_H´I₁)u´ω˜_H(I_αu,t)I_α˝(I_H´I₁)v + ¹

2 ˜f₁

˜

ω_H(Iα˝I₁u,t) +g˜(Iαu,t)Iαv´ ω˜_H(Iα˝I₁v,t) +g˜(Iαv,t)Iαu + ¹

2 ˜f₁ g˜(I_αu,v)´g˜(I_αv,u)I_αt

!

´ 1

2 ˜f₁ ω˜_H(u,v)I₁t+ ^2˜k

f˜_H g˜(I_α˝R^g˜(u,v)Z,^˜ t)I_αZ.˜ (4.139) Proof. We begin by computing the covariant derivative

∇_u^g˜S^HQ (v,t)

= ¹ 2

ÿ3

α=0

´d ˜f_H(u)

f˜_H² ω˜_H(Iαv,t)IαZ˜ + ^2˜k

f˜_Hg˜(Iα˝(∇^g˜)²_u,vZ^˜),t)IαZ˜ + ¹

f˜_Hω˜_H(I_αv,t)I_α˝∇^g_u^˜Z^˜ +^{d ˜f1}(u)

f˜₁² (λ_α(v)I_α˝I₁t+λ_α(t)I_α˝I₁v)

´ 1

f˜₁(g˜(I_α˝∇^g_u^˜Z,^˜ v)I_α˝I₁t+g˜(I_α˝∇^g_u^˜Z,^˜ t)I_α˝I₁v)

.

(4.140)

Using the definitions of ˜f₁, ˜f_H,I_Hand replacingI_α byI_α˝I₁in some of the terms, we can rewrite this as

∇_u^g˜S^HQ (v,t)

= ¹ 2

3

ÿ

α=0

1

f˜_H² ω˜_H(Z,^˜ u)ω˜_H(Iαv,t)IαZ˜ + ^2˜k

f˜_H g˜(Iα˝(∇^g˜)²_u,vZ^˜),t)IαZ˜ + ¹

2 ˜f_Hω˜_H(I_αv,t)I_α˝(I_H´I₁)u+ ¹

f˜₁²λ₁(v)(g˜(I_α˝I₁Z,˜ v)I_αt+g˜(I_α˝I₁Z,˜ t)I_αv) + ¹

2 ˜f₁(g˜(Iα˝I₁˝(I_H´I₁)u,v)Iαt+g˜(Iα˝I₁˝(I_H´I₁)u,t)Iαv)

= ¹ 2

3

ÿ

α=0

1

f˜_H² ω˜_H(Z,^˜ u)ω˜_H(Iαv,t)IαZ˜ + ^2˜k

f˜_H g˜(Iα˝(∇^g˜)²_u,vZ^˜),t)IαZ˜ + ¹

2 ˜f_Hω˜_H(I_αv,t)I_α˝(I_H´I₁)u+ ¹

f˜₁²λ₁(v)(g˜(I_α˝I₁Z,˜ v)I_αt+g˜(I_α˝I₁Z,˜ t)I_αv) + ¹

2 ˜f₁((ω˜_H(I_α˝I₁u,v) +g˜(I_αu,v))I_αt+ (ω˜_H(I_α˝I₁u,t) +g˜(I_αu,t))I_αv)

. (4.141) Now, we can use the fact that ˜ω_H˝I_α˝I₁is antisymmetric whenα=1 and symmet-ric otherwise to conclude that antisymmetrisinguandvin (4.141) gives (4.139).

Lemma 4.A.2. The commutator of S^HQ with itself is given by h

S_u^HQ,S^HQ_v i t

= ¹ 4

ÿ3

α,β=0

1

f˜_H² ω˜_H(I_µu, ˜Z)ω˜_H(I_βv,t)_´ω˜_H(I_αv, ˜Z)ω˜_H(I_βu,t)I_β˝I_αZ˜ + ¹

f˜₁²

˜

g(I_α˝I₁Z,˜ u)g˜(I_β˝I₁Z,v)_´g(I_α˝I₁Z,˜ v)g˜(I_β˝I₁Z,˜ u)I_α˝I_βt +g˜(I_α˝I₁Z,˜ v)g(I_β˝I₁Z,˜ t)I_β˝I_αu´g˜(I_α˝I₁Z,˜ u)g˜(I_β˝I₁Z,˜ t)I_β˝I_αv

+ ¹

4 ˜f₁f˜_H ÿ3

α=0

2 ˜ω_H(u,v)g˜(Iα˝I₁Z,˜ t)IαZ˜

´ 1

k˜

f˜H´ f˜1

ω_H(_Iαv,t)_Iα˝I1u´ω_H(_Iαu,t)_Iα˝I1v.

(4.142)

4.A. HK/QK curvature formula 99 Proof. First we compute the composition ofS^HQ with itself to be

S^HQ_u ˝S^HQ_v t

= ¹ 2

ÿ3

α=0

1

f˜_Hω˜_H(Iαu,S^HQ_v t)IαZ˜ + ¹

f˜₁ g˜(Iα˝I₁Z,˜ u)Iα˝S^HQ_v t+g˜(Iα˝I₁Z,˜ S^HQ_v t)Iαu

= ¹ 4

ÿ

α,β=0

1

f˜_H² ω˜_H(I_αu,I_βZ˜)ω˜(I_βv,t)I_αZ˜ + ¹

f˜₁f˜_H

˜

ω_H(Iαu,I_βt)g˜(I_β˝I₁Z,˜ v) +ω˜_H(Iαu,I_βv)g(I_β˝I₁Z,˜ t)IαZ˜ +g˜(Iα˝I₁Z,˜ u)ω˜_H(Iαv,t)Iα˝I_βZ˜ +g˜(Iα˝I₁Z,˜ I_βZ˜)ω˜_H(I_βv,t)Iαu + ¹

f˜₁²

˜

g(I_α˝I₁Z,˜ u) g˜(I_β˝I₁Z,˜ v)I_α˝I_βt+g˜(I_β˝I₁Z,˜ t)I_α˝I_βv + g˜(I_α˝I₁Z,˜ I_βt)g˜(I_β˝I₁Z,˜ v) +g˜(I_α˝I₁Z,˜ I_βv)g˜(I_β˝I₁Z,˜ t)I_αu

. (4.143) Making the replacementI_αÞÑ I_β˝I_αin some of the terms and swapping the labelsα andβin others, we obtain

S^HQ_u ˝S^HQ_v t

= ¹ 4

ÿ

α,β=0

1

f˜_H² ω˜_H(Iαu, ˜Z)ω˜(I_βv,t)I_β˝IαZ˜ + ¹

f˜₁f˜_H

˜

ω_H(Iαu,t)g˜(I_β˝I₁Z,˜ v) +ω˜_H(Iαu,v)g(I_β˝I₁Z,˜ t)I_β˝IαZ˜ +g˜(Iα˝I₁Z,˜ u)ω˜_H(Iαv,t)Iα˝I_βZ˜ +g˜(Iα˝I₁Z,˜ I_βZ˜)ω˜_H(I_βv,t)Iαu + ¹

f˜₁²

˜

g(I_β˝I₁Z,˜ u) g˜(I_α˝I₁Z,˜ v)I_β˝I_αt+g˜(I_α˝I₁Z,˜ t)I_β˝I_αv

+ g˜(I_α˝I₁Z,˜ t)g˜(I_β˝I₁Z,˜ v) +g˜(I_α˝I₁Z,˜ v)g˜(I_β˝I₁Z,˜ t)I_β˝I_αu . (4.144) Note that ˜g(I_α˝I₁Z,˜ I_βZ˜)vanishes unlessα=1. Moreover, certain pairs of terms are symmetric inuandv, as isω_H(Iαu,v)wheneverα‰0. These terms drop out under

antisymmetrisation, leaving us with h

S_u^HQ,S^HQ_v i t

= ¹ 4

ÿ3

α,β=0

1

f˜_H² ω˜_H(Iµu, ˜Z)ω˜_H(I_βv,t)´ω˜_H(Iαv, ˜Z)ω˜_H(I_βu,t)I_β˝IαZ˜ + ¹

f˜₁²

g˜(Iα˝I₁Z,˜ u)g˜(Iβ˝I₁Z,v)´g(Iα˝I₁Z,˜ v)g˜(Iβ˝I₁Z,˜ u)Iα˝Iβt +g˜(I_α˝I₁Z,˜ v)g(I_β˝I₁Z,˜ t)I_β˝I_αu´g˜(I_α˝I₁Z,˜ u)g˜(I_β˝I₁Z,˜ t)I_β˝I_αv

+ ¹

4 ˜f₁f˜_H ÿ3

α=₀

2 ˜ω_H(u,v)g˜(I_α˝I₁Z,˜ t)I_αZ˜

´g˜(Z, ˜^˜ Z) ω˜_H(Iαv,t)Iα˝I₁u´ω˜_H(Iαu,t)Iα˝I₁v

! .

(4.145) Equation (4.142) now follows by rewriting ˜g(Z, ˜^˜ Z) =k^˜´1f˜_H´f˜₁.

Lemma 4.A.3. The endomorphism field S^HQsatisfies the equation

∇_t^g˜Z^˜ +S^HQ_Z˜ t

= ¹ 2

I_H´1

k˜ f˜_H

f˜₁I₁

t+¹ 2

ÿ3

α=0

1

f˜_Hω˜_H(I_αZ,˜ t) + ¹

f˜₁g˜(I_α˝I₁Z,˜ t)

I_αZ.˜ (4.146) Proof. This follows from a short computation:

∇^g_t^˜Z^˜ +S^HQ_˜

Z t

= ¹

2(I_H´I₁)t + ¹

2 ÿ3

α=0

1

f˜_Hω˜_H(IαZ,˜ t)IαZ˜ + ¹

f˜₁ g˜(Iα˝I₁Z, ˜˜ Z)Iαt+g˜(Iα˝I₁Z,˜ t)IαZ˜

= ¹

2(I_H´I₁)t´ g˜(Z, ˜^˜ Z) 2 ˜f₁ I₁t + ¹

2

3

ÿ

α=0

1

f_Hω˜_H(IαZ,˜ t) + ¹

f˜₁ g˜(Iα˝I₁Z,˜ t)

IαZ˜

= ¹ 2

I_H´1

k˜ f˜_H

f˜₁I₁

t+¹ 2

ÿ3

α=0

1

f˜_H ω˜_H(I_αZ,˜ t) + ¹

f˜₁g˜(I_α˝I₁Z,˜ t)

I_αZ,˜

(4.147)

where in the final step, we have used the definition of ˜f_Hi.e. ˜f_H= k^˜(f^˜₁+g˜(Z, ˜^˜ Z). Finally, we put together all of the above results to get the main lemma of this section.

4.A. HK/QK curvature formula 101 Lemma 4.A.4. The endomorphism field S^HQsatisfies the equation

∇^g_u^˜S^HQ

(v,t)´

∇^g_v^˜S^HQ (u,t) +^hS^HQ_u ,S^HQ_v i

t´ 1

f˜_H ω˜_H(u,v)∇^g_t^˜Z^˜ +S^HQ_Z˜ t

=´ 1 2 ˜f_H²

1 2

3

ÿ

α,β=0

(ω˜_H(IαZ,˜ u)ω_H(I_βv,t)´ω_H(IαZ,˜ v)ω˜_H(I_βu,t))Iα˝I_βZ˜ +

3

ÿ

α=0

˜

ω_H(u,v)ω˜_H(IαZ,˜ t)IαZ˜

!

+ ¹ 2 ˜f_H

1 2

ÿ3

α=0

˜

ω_H(Iαv,t)Iα˝I_Hu´ω˜_H(Iαu,t)Iα˝I_Hv+4˜kg˜(Iα˝R^g˜(u,v)Z,^˜ t)IαZ˜

´ω˜_H(u,v)I_Ht

!

+ ¹ 4

ÿ3

α=0

˜

g_H(I_αu,t)I_αv´g˜_H(I_αv,t)I_αu+ g˜_H(I_αu,v)´g_H(I_αv,u)I_αt .

(4.148) Proof. Substituting (4.139), (4.142), and (4.142) into the left-hand side and cancelling terms yields

∇^g_u^˜S^HQ

(v,t)_´∇^g_v^˜S^HQ (u,t) +^hS^HQ_u ,S^HQ_v i

t´ 1

f˜_Hω˜_H(u,v)∇^g_t^˜Z^˜ +S^HQ_˜

Z t

= ¹

f˜₁ D_1,1(u,v,t) + ¹

f˜₁² D_1,2(u,v,t) + ¹

f˜_H D_H,1(u,v,t) + ¹

f˜_H² D_H,2(u,v,t),

(4.149)

where the tensor fieldsD_1,1,D_1,2are given by D_1,1(u,v,t)

= ¹ 4

ÿ3

α=0

g˜(Iαu,t)Iαv´g˜(Iαv,t)Iαu+ g˜(Iαu,v)´g˜(Iαv,u)Iαt , D_1,2(u,v,t)

= ¹ 2

ÿ3

α=₀

λ₁(u) g˜(I_α˝I₁Z,˜ v)I_αt+g˜(I_α˝I₁Z,˜ t)I_αv

´λ₁(v) g˜(Iα˝I₁Z,˜ u)Iαt+g˜(Iα˝I₁Z,˜ t)Iαu +¹

4 ÿ

α,β

˜

g(I_α˝I₁Z,˜ u)g˜(I_β˝I₁Z,˜ v)_´g˜(I_α˝I₁Z,˜ v)g˜(I_β˝I₁Z,˜ u)I_α˝I_βt

+g˜(I_β˝I₁Z,˜ v)g˜(I_α˝I₁Z,˜ t)I_α˝I_βu´g˜(I_β˝I₁Z,˜ u)g˜(I_α˝I₁Z,˜ t)I_α˝I_βv , (4.150)

while the tensor fieldsD_H,1,D_H,2are given by D_H,1(u,v,t)

= ¹ 4

ÿ3

α=0

ω˜_H(Iαv,t)Iα˝I_Hu´ω˜_H(Iαu,t)Iα˝I_Hv+4˜kg˜(Iα˝R^g˜(u,v)Z,^˜ t)IαZ˜

´1

2ω˜_H(u,v)I_Ht, D_H,2(u,v,t)

= ¹ 2

ÿ3

α=0

˜

ω_H(Z,^˜ u)ω˜_H(Iαv,t)´ω˜_H(Z,^˜ v)ω˜_H(Iαu,t)´ω˜_H(u,v)ω˜_H(IαZ,˜ t)IαZ˜ +¹

4 ÿ3

α,β=0

˜

ω_H(I_βu, ˜Z)ω˜_H(I_αv,t)_´ω˜_H(I_βv, ˜Z)ω˜_H(I_αu,t)I_α˝I_βZ.˜

(4.151) The expressions forD_1,2andD_H,2can be simplified further by absorbing some of the terms in the single summations into the double summation by rewriting

λ₁(¨)Iα = ¹ 2

ÿ3

β=0

(g˜(I_β˝I₁Z,˜ ¨) +g˜(I₁Z,˜ I_β¨))Iα˝I_β,

˜

ω_H(Z,^˜ ¨)I_α = ¹ 2

ÿ3

β=0

(ω˜_H(I_βZ,˜ ¨)_´ω˜_H(I_β¨, ˜Z))I_α˝I_β.

(4.152)

This gives us the following expressions forD_1,2andD_H,2: D_1,2(u,v,t)

= ¹ 4

3

ÿ

α,β=0

˜

g(Iα˝I₁Z,˜ v)g˜(I₁Z,˜ I_βu)´g˜(Iα˝I₁Z,˜ u)g˜(I₁Z,˜ I_βv)Iα˝I_βt +g˜(I₁Z,˜ Iβu)g˜(Iα˝I₁Z,˜ t)Iα˝Iβv´g˜(I₁Z,˜ Iβv)g˜(Iα˝I₁Z,˜ t)Iα˝Iβu

= ¹ 4

ÿ3

α=₀

˜

g_λ(I_αu,v)_´g˜_λ(I_αv,u)I_αt+g˜_λ(I_αu,t)I_αv´g˜_λ(I_αv,t)I_αu , D_H,2(u,v,t)

=´1 2

ÿ3

α=0

˜

ω_H(u,v)ω˜_H(IαZ,˜ t)IαZ˜ +¹

4 ÿ3

α,β=0

˜

ω_H(I_βZ,˜ u)ω˜_H(I_αv,t)´ω˜_H(I_βZ,˜ v)ω˜_H(I_αu,t)I_α˝I_βZ,˜

(4.153)

where we have made use of the shorthand ˜g_λ introduced in (4.85). Plugging these simplified expressions into (4.149) and noting that the standard hyperkähler elemen-tary deformation is

˜ g_H= ^K˜

f˜₁ g˜+ ^K˜

f˜₁² g˜_λ (4.154)

gives us the required expression (4.148).

4.B. Ferrara–Sabharwal curvature norm 103

4.B Ferrara–Sabharwal curvature norm

In this section, we’ll compute the curvature norm (4.128) of the quadratic prepoten-tial Ferrara–Sabharwal metric. As preparation, we first introduce some notation and prove a general lemma about traces.

Definition 4.B.1(h-Kulkarni–Nomizu product). Given a metrich, and two endomor-phism fieldsB₁,B₂self-adjoint with respect to it, theirh-Kulkarni–Nomizu product B₁?hB₂PΓ(End(_Λ²TM))is defined by

h((B₁?hB₂)(s^t),u^v) = (h˝B₁)_?(h˝B₂)(s,t,u,v). (4.155) Definition 4.B.2(h-Riemann product). Given a metric h, and two endomorphism fieldsB¹₁,B₂¹ skew-self-adjoint with respect to it, theirh-Riemann productB¹₁:hB₂¹ P Γ(_End(_Λ²TM))is defined by

h((B₁¹ :hB₂¹)(s^t),u^v) = (h˝B₁¹)_:(h˝B¹₂)(s,t,u,v). (4.156) Lemma 4.B.3. Given a metric h, self-adjoint endomorphism fields B₁,B₂, and skew-self-adjoint endomorphism fields B¹₁,B¹₂, we have

tr((B₁?hB₁)_˝(B₂?hB₂)) =2(tr(B₁˝B₂)²_´tr((B₁˝B₂)²)), tr((B₁¹ :hB¹₁)_˝(B₂¹ :hB¹₂)) =6(tr(B₁¹ ˝B¹₂)²+tr((B¹₁˝B₂¹)²)) tr((B₁?hB₁)˝(B₂¹ :hB¹₂)) =tr((B₂¹ :hB₂¹)˝(B₁?hB₁))

=2(tr(B₁˝B¹₂)²´3 tr((B₁˝B₂¹)²)).

(4.157)

Proof. Lette_aube an orthonormal basis forhand letε_abe the sign ofh(e_a,e_b). Now we compute:

tr((B₁?hB₁)˝(B₂?hB₂))

= ¹ 4

ÿ

a,b,c,d

εaε_bεcε_dh((B₁?hB₁)ea^e_b,ec^e_d)h((B2?hB2)ec^e_d,ea^e_b)

= ¹ 4

ÿ

a,b,c,d

ε_aε_bε_cε_d(h˝B₁)_?(h˝B₁)(e_a,e_b,e_c,e_d)(h˝B₂)_?(h˝B₂)(e_c,e_d,e_a,e_b)

= ^ÿ

a,b,c,d

εaε_bεcε_d(h(B₁ea,ec)h(B₁e_b,e_d)´h(B₁ea,e_d)h(B₁e_b,ec)) (h(B₂e_c,e_a)h(B₂e_d,e_b)´h(B₂e_c,e_b)h(B₂e_d,e_a))

= ^ÿ

a,b,c,d

ε_aε_bε_cε_d(h(B₁e_a,e_c)h(B₁e_b,e_d)h(B₂e_c,e_a)h(B₂e_d,e_b)

´h(B₁ea,e_d)h(B₁e_b,ec)h(B2ec,ea)h(B2e_d,e_b)

´h(B₁e_a,e_c)h(B₁e_b,e_d)h(B₂e_c,e_b)h(B₂e_d,e_a) +h(B₁e_a,e_d)h(B₁e_b,e_c)h(B₂e_c,e_b)h(B₂e_d,e_a))

=2(tr(B₁˝B₂)²_´tr((B₁˝B₂)²)).

(4.158)

The next computation proceeds similarly, so we omit steps:

tr((B¹₁:hB¹₁)˝(B¹₂:hB¹₂))

= ^ÿ

a,b,c,d

ε_aε_bε_cε_d(h(B₁¹e_a,e_c)h(B¹₁e_b,e_d)_´h(B₁¹e_a,e_d)h(B¹₁e_b,e_c) +2h(B¹₁e_a,e_b)h(B¹₁e_c,e_d)) (h(B¹₂ec,ea)h(B₂¹e_d,e_b)´h(B¹₂ec,e_b)h(B₂¹e_d,ea) +2h(B¹₂ec,e_d)h(B¹₂ea,e_b))

=₂(_tr(B¹₁˝B₂¹)²_´tr((B¹₁˝B¹₂)²+_{2 tr}(B₁¹ ˝B¹₂˝B₁^1:h˝B^1:h₂ )

´tr(B₁¹ ˝B¹₂˝B₁^1:h˝B¹₂)´tr(B¹₁˝B₂¹ ˝B¹₁˝B^1:h₂ ) +2 tr(B₁¹ ˝B^1:h₂ )²)

=6(tr(B¹₁˝B₂¹)²+tr((B¹₁˝B¹₂)²)),

(4.159) where in the last step, we have used thatB¹₁andB¹₂are skew-self-adjoint. Likewise, we have

tr((B₁?hB₁)_˝(B¹₂:hB¹₂))

= ^ÿ

a,b,c,d

ε_aε_bε_cε_d(h(B₁e_a,e_c)h(B₁e_b,e_d)´h(B₁e_a,e_d)h(B₁e_b,e_c))

(h(B¹₂e_c,e_a)h(B₂¹e_d,e_b)´h(B¹₂e_c,e_b)h(B₂¹e_d,e_a) +2h(B₂¹e_c,e_d)h(B₂¹e_a,e_b))

=2(tr(B₁˝B₂¹)²´tr((B₁˝B¹₂)²+tr(B₁˝B¹₂˝B^:h₁ ˝B^1:h₂ )´tr(B₁˝B¹₂˝B^:h₁ ˝B¹₂)

=2(tr(B₁˝B₂¹)²´3 tr((B₁˝B¹₂)²)),

(4.160) where we have used thatB₁is self-adjoint andB₂¹ is skew-self-adjoint.

Now we specialise to the case of the quadratic prepotential Ferrara–Sabharwal metrics. To apply the above lemma, we need to first rewrite the curvature norm in terms of traces of the above form.

Lemma 4.B.4. The curvature norm in(4.128)may be written as tr(_R²) =_tr

1

8 ˜K idTM?g˜HidTM+ ÿ3

i=1

I_i:g˜H I_i

!

´ k˜ 8 ˜K

f˜₁

f˜_H (A˝I_H)_:g˜H(A˝I_H) +

ÿ3

i=1

((A˝I_H˝I_i)_?g˜H (A˝I_H˝I_i)

!!2

,

(4.161) where A is the endomorphism field introduced in(4.88), namely

A= ^K˜

f˜₁ g˜^´1_H ˝g˜ =id_TM|_HHZ˜^K+^˜k f˜₁

f˜_Hid_TM|_HHZ˜. (4.162) Proof. Using the symmetries of the Riemann tensor, we may rewrite (4.128) as

tr(_R²) = ¹ 4

ÿ

a,b,c,d

C(E_a,E_b,E_c,E_d)C(E_c,E_d,E_a,E_b)_{. (4.163)}

4.B. Ferrara–Sabharwal curvature norm 105 Equation (4.161) then follows from the definitions of the ˜g_H-Kulkarni–Nomizu and g˜_H-Riemann products, and the following observations:

˜

g_H =g˜_H˝id_TM, ω˜_H =k^˜g˜˝I_H= ^k˜

K˜ f˜₁g˜_H˝A˝I_H. (4.164)

Next we collect the various trace computations we will need into a lemma.

Lemma 4.B.5. The endomorphism fieldsid_TM,I_i,A,I_H on the complex cotangent bundle T^˚Cⁿsatisfy the following trace formulae:

tr(id_TM) =4n, tr(I_i) =0, tr(I_i˝I_j) =´4nδ_ij, tr((I_i˝I_j)²) =´(´1)^δij4n, tr(A˝I_H) =0, tr((A˝I_H)²) =´tr(A²) =´4 n´1+k^˜2 f˜₁²

f˜_H²

! , tr((A˝I_H)⁴) =tr(A⁴) =4 n´1+k^˜4 f˜₁⁴

f˜_H⁴

!

, tr(A˝I_H˝I_i) =0, tr((A˝I_H˝I_i)²) =tr(A²), tr(A˝I_H˝A˝I_H˝I_i) =´tr(A²˝I_i) =0,

tr((A˝I_H˝A˝I_H˝I_i)²) =´tr(A⁴), tr((A˝I_H˝I_i˝I_j)²) =_´(´1)^δijtr(A²), tr(A˝I_H˝I_i˝I_j) =´δ_ijtr(A˝I_H) +

ÿ3

k=1

e_ijktr(A˝I_H˝I_k) =0, tr(A˝I_H˝I_i˝A˝I_H˝I_j) =δ_ijtr(A²)´

3

ÿ

k=1

e_ijktr(A²˝I_k) =δ_ijtr(A²), tr((A˝I_H˝I_i˝A˝I_H˝I_j)²) = (´1)^δijtr(A⁴),

(4.165) whereδ_ijis the Kronecker delta ande_ijkis the Levi-Civita symbol.

Proof. The trace of the identity endomorphism field on any vector bundle E is the rank ofE. Thus, since we can writeAas

A= ^K˜

f˜₁ g˜_H^´1˝g˜=idTM|_HHZ˜^K+k^˜ f˜1

f˜_HidTM|_HHZ˜, (4.166) it immediately follows that for all integersm, we have

tr(A^m) =4 n´1+k^˜m f˜₁^m f˜_H^m

!

. (4.167)

The explicit expression for Aalso makes it clear that Acommutes with I_i. We also know thatI_Hcommutes withI_i. Our next goal shall be to show thatI_HandA com-mute as well.

As a matter of general fact, we have already noted that I_H is skew-self-adjoint with respect to ˜gby virtue of the Killing equation for ˜Z. In the specific case of the complex cotangent bundleM =T^˚Cⁿ, the explicit expression forI_Hin (4.18), i.e.

I_H =i ´dz₀^g˜B_z0+dw₀^g˜B_w0+

n´1

ÿ

a=1

(´dz_a^g˜B_za+dw_a^g˜B_wa)

!

, (4.168)

additionally tells us thatI_H² =´id_TM, implying I_His ˜g-orthogonal, and that

I_HZ˜ =´I₁Z,˜ I_H˝I_iZ˜ = I_i˝I_HZ˜ =´I_i˝I₁Z,˜ (4.169) implyingHHZ˜ andHHZ˜^Kare invariant subbundles of I_H. Thus,I_Hcommutes with Aas well. This allows us to reduce the trace of any string consisting of A,I_H,I_i in some order to one of the following traces:

tr(A^m), tr(A^m˝I_i), tr(A^m˝I_H), tr(A^m˝I_H˝I_i). (4.170) The first trace we have already dealt with. The next two traces are traces of endo-morphism fields that are skew-self-adjoint with respect to the metric ˜g˝A^´m, and so vanish. The last one also vanishes by virtue of the following chain of equalities:

tr(A^m˝I_H˝I_i) = ¹ 2

ÿ3

j,k=1

e_ijktr(A^m˝I_H˝I_j˝I_k) = ¹ 2

ÿ3

j,k=1

e_ijktr(I_j˝A^m˝I_H˝I_k)

= ¹ 2

ÿ3

j,k=1

e_ijktr(A^m˝I_H˝I_k˝I_j) =´tr(A^m˝I_H˝I_i),

(4.171) where in the first line we have used the fact thatI_jcommutes withAandI_H, and in the second line, we have used the cyclic invariance of traces.

Now, we have all the ingredients we require to prove the main lemma of this section.

Lemma 4.B.6. The curvature norm in(4.128)is given by tr(_R²)

= ¹ K˜²



n(5n+1) +3 k˜³ f˜₁³

f˜_H³ + (n´1)k^˜ f˜₁ f˜_H

!2

+3 k˜⁶ f˜₁⁶

f˜_H⁶ + (n´1)k^˜2 f˜₁² f˜_H²

!

. (4.172)

4.B. Ferrara–Sabharwal curvature norm 107 Proof. We begin by expanding the square inside the trace in (4.161) and using the cyclic invariance of the trace to write

64 ˜K²tr(_R²)

=tr



(idTM?g˜_HidTM)²+ ÿ3

i,j=1

(I_i:g˜_H I_i)˝(I_j:g˜_H I_j) +k^˜2 f˜₁²

f˜_H² ((A˝I_H)_:g˜H(A˝I_H))² +k^˜2 f˜₁²

f˜_H² ÿ3

i,j=1

((A˝I_H˝I_i)_?g˜H(A˝I_H˝I_i))_˝((A˝I_H˝I_j)_?g˜H(A˝I_H˝I_j)) +2

ÿ3

i=1

(id_TM?g˜Hid_TM)_˝(I_i:g˜H I_i)_´2˜k f˜₁

f˜_H(id_TM?g˜Hid_TM)_˝((A˝I_H)_:g˜H (A˝I_H))

´2˜k f˜₁ f˜_H

3

ÿ

i=1

(id_TM?g˜_Hid_TM)˝((A˝I_H˝I_i)_?g˜H (A˝I_H˝I_i))

´2˜k f˜₁ f˜_H

ÿ3

i=1

(I_i:g˜H I_i)_˝((A˝I_H)_:g˜H(A˝I_H))

´2˜k f˜₁ f˜_H

ÿ3

i,j=1

(I_i:g˜H I_i)˝((A˝I_H˝I_j)_?g˜H(A˝I_H˝I_j)) + 2˜k² f˜₁²

f˜_H²

3

ÿ

i=1

((A˝I_H)_:g˜H(A˝I_H))˝((A˝I_H˝I_i)_?g˜H(A˝I_H˝I_i))

! .

(4.173) Next, we use (4.157) to obtain

64 ˜K²tr(_R²)

=2(tr(id_TM)²_´tr(id_TM)) +6 ÿ3

i,j=1

(tr(I_i˝I_j)²+tr((I_i˝I_j)²)) +6˜k² f˜₁²

f˜_H² (tr((A˝I_H)²)²+tr((A˝I_H)⁴)) +2˜k² f˜₁²

f˜_H² ÿ3

i,j=1

(tr(A˝I_H˝I_i˝A˝I_H˝I_j)²´tr((A˝I_H˝I_i˝A˝I_H˝I_j)²)) +4

ÿ3

i=1

(tr(I_i)²+3 tr(id_TM))_´4˜k f˜₁

f˜_H (tr(A˝I_H)²_´3 tr((A˝I_H)²)

´4˜k f˜₁ f˜_H

ÿ3

i=1

(tr(A˝I_H˝I_i)²´tr((A˝I_H˝I_i)²) +3(tr(A˝I_H˝I_i)²+tr((A˝I_H˝I_i)²)))

´4˜k f˜₁ f˜_H

ÿ3

i,j=1

(tr(A˝I_H˝I_i˝I_j)²´3 tr((A˝I_H˝I_i˝I_j)²) +4˜k² f˜₁²

f˜_H² ÿ3

i=1

(tr(A˝I_H˝A˝I_H˝I_i)²_´3 tr((A˝I_H˝A˝I_H˝I_i)²)).

(4.174)

Finally, substituting the trace formulae (4.165) into the above gives us 64 ˜K²tr(_R²)

=2((4n)²_´4n) +6 ÿ3

i,j=1

((4n)²δ_ij+ (´1)^δij4n) +6˜k² f˜₁²

f˜_H² (tr(A²)²+tr(A⁴)) +2˜k² f˜₁²

f˜_H² ÿ3

i,j=1

(δ_ijtr(A²)²´(´1)^δijtr(A⁴)) +4 ÿ3

i=1

12n´4˜k f˜₁

f˜_H(3 tr(A²))

´4˜k f˜₁ f˜_H

3

ÿ

i=₁

(´tr(A²) +3 tr(A²))´4˜k f˜₁ f˜_H

3

ÿ

i,j=₁

(´1)^δij3 tr(A²) +4˜k² f˜₁² f˜_H²

3

ÿ

i=₁

(3 tr(A⁴))

=64



n(5n+1) +3 k˜³ f˜₁³

f˜_H³ + (n´1)k^˜ f˜₁ f˜_H

!2

+3 k˜⁶ f˜₁⁶

f˜_H⁶ + (n´1)k^˜2 f˜₁² f˜_H²

!

. (4.175)

109

Chapter 5

Deformations of quaternionic Kähler structures

In this chapter, we revisit the local twist and prove two useful lemmata about it.

Lemma5.1.1is a generalisation of Proposition3.1.13that lets us write the composi-tion of two twists as a single twist, while Lemma5.1.8introduces a notion of differ-entiating twists with respect to twist data.

We then use Lemma 5.1.1to write the one-loop deformation of a quaternionic Kähler manifold as a twist of an elementary deformation of the quaternionic Kähler manifold (Theorem 5.2.1 and Definition 5.2.2). Note that this differs from Macia and Swann [MS14] in that we are taking the elementary deformation and twist of the quaternionic Kähler manifold directly rather than its QK/HK dual. In fact, as we show in Proposition5.2.11, the QK/HK correspondence may be thought of as a certain limit of the one-loop deformation.

Meanwhile, Lemma 5.1.8 is used to derive three sets of geometric evolution equations on the space of quaternionic Kähler metrics on a contractible open set U, namely the naïve, reparametrised, and rescaled one-loop flow equations (Propo-sitions5.3.2,5.3.5, and5.3.7respectively).

Finally, in Subsection5.3.4, we conclude with an outline of speculative directions to be explored in the future.

5.1 Local twists revisited

In Chapter4, we saw that applying the HK/QK correspondence to a hyperkähler metric with a rotating Killing field followed by the QK/HK correspondence gives us back the same hyperkähler metric up to an overall scaling. However, applying the QK/HK correspondence to a quaternionic Kähler metric with a Killing field fol-lowed by the HK/QK correspondence gives us back the same quaternionic Kähler manifold up to an overall scaling forcertainchoices of ˜f_H, ˜K, ˜kas described in (4.72).

In general, such a procedure will give an honest deformation of the original quaternionic Kähler metric, and since our goal is to construct interesting examples of quaternionic Kähler manifolds, we are obviously interested in describing these deformations. But in order to do so, we first need to establish two additional results regarding the local twist, one regarding the compositions of the local twist, and one regarding derivatives of the local twist map with respect to twist data.

Lemma 5.1.1. If(U,Z,ω,f,η)and(U,aZ, ˜˜ ω¹,af˜¹, ˜η¹)are local twist data on some mani-fold M with a being a nonzero constant and

Z˜ =_´¹

f tw_Z,f,η(Z) =_´ ¹

f´η(Z)^{Z, (5.1)}

then so is

(U,Z,ω², f²,η²):= (U,Z,a¹d(f^˜1η´η˜¹),a¹ff˜¹,a¹(f^˜1η´η˜¹)), (5.2) for any nonzero constant a¹. Moreover, the local twist maps with respect to the above tuples of local twist data satisfy

tw_{aZ, ˜˜ f}1,aη˜¹˝tw_Z,f,η =tw_Z,f²_,η². (5.3) Proof. Since(U,Z,ω,f,η)and(U,aZ, ˜˜ ω¹,af˜¹, ˜η¹)constitute local twist data, we must have

ι_Zω =ι_Zdη=´df, ι_aZ˜ω˜¹ =´ aι_Zd ˜η¹

f´η(Z) =´ad ˜f¹. (5.4) It then follows that

ι_Zω² =a¹ι_Zd(f^˜1η´η˜¹) =a¹ι_Z(d ˜f¹^η+ f^˜1dη´d ˜η¹)

=a¹(_´η(Z)d ˜f¹´ f˜¹df´(f´η(Z)d ˜f¹)

=´a¹(f^˜1df+ fd ˜f¹) =´d(a¹ff˜¹) =´df².

(5.5)

In addition, since f´η(Z)andaf˜¹´η˜¹(aZ˜), and so ˜f¹´η˜¹(Z), are nowhere vanish-ing,

f²´η²(Z) =a¹ff˜¹´a¹(f^˜1η(Z)_´η˜¹(Z))

= a¹(ff˜¹´ f˜¹η(Z)´(f´η(Z))η˜¹(Z^˜))

= a¹(f´η(Z))(f^˜1´η˜¹(Z))

(5.6)

is nowhere vanishing as well. So, (U,Z,ω²,f²,η²) indeed constitutes local twist data.

To prove (5.3), it’s enough to show that it holds for functions and 1-forms since local twists preserve contractions and tensor products by definition. That it holds for functions is clear, since the local twist is just the identity in that case. For a 1-formα, we introduce the shorthandα¹ =tw_Z,f,η(α)and compute

tw_{aZ, ˜˜ f}1,aη˜¹˝tw_Z,f,η(α) =tw_{aZ, ˜˜ f}1,aη˜¹(α¹) =α¹´α¹(aZ˜) af˜¹ η˜¹

=α¹+^α(Z)

ff˜¹ η˜¹ =α´α(Z)

f η+^α(Z) ff˜¹ η˜¹

=α´α(Z)

a¹ff˜¹ a¹(f^˜1η´η˜¹) =tw_Z,f²_,ω²(α).

(5.7)

Corollary 5.1.2. If(U,Z,ω,f,η)is local twist data on some manifold M and a₁,a₂,a₃are nonzero constants such that

a₂

f +a₃ (5.8)

is nowhere vanishing, and if(U, ˜Z, ˜ω, ˜f, ˜η)are the local twist data dual to(U,Z,ω,f,η), then we have the following relation of local twist maps:

tw_a

1Z,a˜ ₁(a2f˜+a3),a2η˜˝tw_Z,f,η =tw_Z,f+^a2

a3,η. (5.9)

5.1. Local twists revisited 111 Proof. This is just a specialisation of Lemma5.1.1to the case

˜

η¹ = a₂η,˜ f˜¹ =a₂f˜+a₃, a= a₁, a¹ = ¹ a3

. (5.10)

Remark5.1.3. The limita₃ Ñ0 is well-defined and reproduces (3.30), i.e. local twists with respect to tuples of twist data dual to each other are the inverses of each other.

In order to state the other lemma, we need two new definitions.

Definition 5.1.4(Local log-derivative twist). The local log-derivative twist dltw_Z,f,η with respect to local twist data(U,Z,ω,f,η)is a gradedC⁸(U)-linear map

dltw_Z,f,η :Γ(_T^‚,˛U)_{Ñ Γ}(_T^‚,˛U) (5.11) of tensor fields, satisfying the Leibniz rule over tensor products and contractions, whose action on an arbitrary 1-formαis given by

dltw_Z,f,η(α) = ^α(Z)

f(f´η(Z))^{η. (5.12)} Remark5.1.5. The fact that dltw_Z,f,η isC⁸(U)-linear and satisfies the Leibniz prop-erty with respect to tensor products forces it to vanish on 0-forms.

Definition 5.1.6 (Local derivative twist). The local derivative twist map dtw_Z,f,η with respect to local twist data(U,Z,ω,f,η)is the composition

dtw_Z,f,η =_{twZ,f,η˝dltwZ,f,η. (5.13)} Example 5.1.7. The local log-derivative twists of a vector fieldu, a symmetric bilin-ear formg, and an endomorphism field Amay be computed to be

dltw_Z,f,η(u) =´ η(u)

f(f´η(Z))^{Z, dltwZ,f,η}(g) = ²

f(f´η(Z))^{η ιZg,} dltw_Z,f,η(A) = ¹

f(f ´η(Z))[A,ηbZ].

(5.14)

Thus, their local derivative twists are given by dtw_Z,f,η(u) =´ η(u)

(f´η(Z))^{2 Z, dtwZ,f,η}(g) = ²

f²ηtw_Z,f,η(ι_Zg), dtw_Z,f,η(A) = ¹

f(f´η(Z))[tw_Z,f,η(A),ηbZ].

(5.15)

Lemma 5.1.8. Given local twist data(U,Z,ω,f,η)on some manifold M, we have for all tensor fields S on M, the following identity:

d

da(tw_Z,f+a,η(S)) ˇ ˇ ˇ ˇa=0

=dtw_Z,f,η(S). (5.16)

Proof. Let(U, ˜Z, ˜ω, ˜f, ˜η)denote the local twist data dual to(U,Z,ω,f,η), as usual.

We begin by showing that theC⁸(U)-linear map tw_{Z, ˜˜ f, ˜η}

d

da(tw_Z,f+a,η(¨)) ˇ ˇ ˇ ˇa=0

(5.17) satisfies the Leibniz property over tensor products. Letαbe an arbitrary 1-form and βan arbitraryk-form. Then we have

twZ, ˜˜ f, ˜η

d

da(tw_Z,f+a,η(αbβ)) ˇ ˇ ˇ ˇa=0

= twZ, ˜˜ f, ˜η

d

da(tw_Z,f+a,η(α)btw_Z,f+a,η(β)) ˇ ˇ ˇ ˇa=0

= twZ, ˜˜ f, ˜η

d

da(tw_Z,f+a,η(α)) ˇ ˇ ˇ ˇa=0

btw_Z,f,η(β) + tw_Z,f,η(α)b d

da(tw_Z,f+a,η(β)) ˇ ˇ ˇ ˇa=0

= twZ, ˜˜ f, ˜η

d

da(tw_Z,f+a,η(α)) ˇ ˇ ˇ ˇa=0

bβ+αbtwZ, ˜˜ f, ˜η

d

da(tw_Z,f+a,η(β)) ˇ ˇ ˇ ˇa=0

.

(5.18) A similar chain of equalities shows that the map (5.17) satisfies the Leibniz property over contractions as well. All that now remains to be shown in order to prove the lemma is that (5.17) coincides with the local log-derivative twist for functions and 1-forms. Lettingαbe an arbitrary 1-form, we have

tw_{Z, ˜˜ f, ˜η} d

da(tw_Z,f+a,η(α)) ˇ ˇ ˇ ˇa=0

= tw_{Z, ˜˜ f, ˜η} d

da

α´ α(Z) f +aη

ˇ ˇ ˇ ˇa=0

=twZ, ˜˜ f, ˜η

α(Z) f² η

= ^α(Z) f²

η´η(Z^˜) f˜ η˜

= ^α(Z) f²

1+ ^η(Z) f´η(Z)^η˜

η= ^α(Z) f(f´η(Z))^η.

(5.19)