Curvature of the q-map

Multiplication of Eq. (7.86) by −e^−Ke^µ_a gives

e^µ_b =e^−K(X^µN^0J−N^µJ) ¯P_J^b. This equation shows that

−e^µ_b∂e¯ ^a_µ=e^−K(( ¯∂e^a_µ)N^µJ−X^µ( ¯∂e^a_µ)N^0J) ¯P_J^b =e^−K( ¯∂P_I^a)N^IJP¯_J^b. Using the above equation one then finds

ω^a_b ^(7.70)= −e^µ_b∂e¯ ^a_µ+ ¯e^µ_a∂e¯^b_µ =e^−K ( ¯∂P_I^a)N^IJP¯_J^b−P_I^aN^IJ(∂P¯_J^b) . Adding 0^(7.86)= δ^a_b∂K+e^−K∂(P_I^aN^IJP¯_J^b) to the above equation gives

ω^a_b =δ_b^a∂K+e^−K d(P_I^aN^IJ) ¯P_J^b−P_I^a( ¯∂N^IJ) ¯P_J^b

=δ_b^a∂K+e^−Kd(P_I^aN^IJ) ¯P_J^b +ie^−KP_I^aN^IKdFKL(X)N^LJP¯_J^b.

manifolds in the image of the supergravity r-map Kµ¯ν is real, we can choose e^a_µ to be real (see Eq. (7.62)). Recall how ¯M is realized as a C^∗-quotient of a conical affine special K¨ahler manifold M defined by a holomorphic prepoten-tial F : M → C (see Eq. (7.73) and below). This in particular defines the matrix-valued functions N = (N_IJ)I, J=0, ..., n, F = (F_IJ)I, J=0, ..., n, etc. on ¯M. Starting from the projective very special K¨ahler manifold ( ¯M, gM¯, JM¯), we now consider the one-loop deformed supergravity c-map. For c∈R, let

N⁰ :=N_(4n+4,0)⁰ ⊂M¯ ×R²ⁿ⁺⁴ ⊂R⁴ⁿ⁺⁴ (7.87) denote the domain where the one-loop deformed Ferrara-Sabharwal metric g_{F S}^c is positive definite (see Definition 5.5.1). As in Definition 5.5.1, we use standard real coordinates (ρ,φ,˜ ζ˜_I, ζ^I)I=0, ..., non the R²ⁿ⁺⁴ factor ofN⁰ and complex coor-dinates (X^µ)µ=1, ..., n on ¯M. Note that on N⁰, ρ > 0 and ρ+ 2c > 0. We define the following complex-valued one-forms on N⁰:

β⁰ :=ie^K/2

√ρ+ 2c ρ

n

X

I=0

X^IA_I, β^a :=

rρ+c ρ

n

X

I=0

P_I^adX^I =

rρ+c ρ σ^a, α⁰ :=− 1

2ρ

rρ+ 2c

ρ+c dρ−i ρ+c ρ+ 2c(dφ˜+

n

X

I=0

(ζ^Idζ˜_I−ζ˜_Idζ^I) +cd^cK)

! , α^a := i

√ρe^−K/2

n

X

I,J=0

P^a_IN^IJA_J (7.88)

(a = 1, . . . , n), where (P_I^a)_I=0,...,n = (P₀^a, P_µ^a)_µ=1,...,n = (−Pn

ν=1X^νe^a_ν, e^a_µ)_µ=1,...,n and A_I = dζ˜_I +Pn

J=0F_IJ(X)dζ^J, I = 0, . . . , n. Here, we trivially extend functions and one-forms from ¯M toN⁰, using the same notation (i.e. leaving out pullbacks). LetQ= span

R{J₁⁰, J₂⁰, J₃⁰}denote the (trivial) quaternionic structure onN⁰ obtained from the HK/QK correspondence (see Remarks 5.5.3 and 5.5.4).

Lemma 7.3.1 The coframe

(f^αΓ)α=1,2; Γ=1, ...,2n+2 = f^1A f^{1 ˜A} f^2A f^{2 ˜A}

!

A=0, ..., n

:= β^A α^A

−¯α^A β¯^A

!

A=0, ..., n

(7.89)

in(T^∗N⁰)^Cdefines a unitary coframe for the one-loop deformed Ferrara-Sabharwal

metric, i.e. the metric reads g_{F S}^c =

n

X

A=0

(β^Aβ¯^A+α^Aα¯^A), (7.90)

and α^A, β^A areJ₁-holomorphic and fulfill

α^A=−J₂^∗β¯^A (A= 0, . . . , n). (7.91) Proof: Note that τ =−2iρq

ρ+2c

ρ+cα⁰, whereτ is given by Eq. (5.18). Further-more,

1

4dd^cK= i

2∂∂¯K= i

2σ^a∧σ¯^a = i 2

ρ

ρ+cβ^a∧β¯^a, β⁰∧β¯⁰ =e^Kρ+2c_ρ2 (X^IA_I)∧( ¯X^JA¯_J) and

α^a∧α¯^a = 1

ρe^−KN^IKP¯_K^aP_L^aN^LJA_I∧A¯_J ^(7.85)= −1

ρ(N^IJ −e^KX^IX¯^J)A_I∧A¯_J. Together with Eq. (5.16), this shows that the first fundamental two-form is given by

¯ ω1 = i

2

n

X

A=0

(β^A∧β¯^A+α^A∧α¯^A). (7.92) Note that

α^a∧β^a=ie^−K/2

√ρ+c

ρ P_I^aP¯_K^aN^KJA_J∧dX^I

(7.85)

= ie^K/2

√ρ+c

ρ (−A_I∧dX^I+e^KX¯^MN_{M I}X^JA_J∧dX^I)

(7.83)

= +ie^K/2

√ρ+c

ρ (dX^I∧A_I+∂K∧X^IA_I). (7.93) Together with Eq. (5.17) and the definitions of β⁰ and α⁰, this shows that

¯

ω2+i¯ω3 =

n

X

A=0

β^A∧α^A. (7.94)

The statements of the lemma follow immediately from Eqs. (7.92) and (7.94).

Before we proceed, we state a few more formulas that can be proven using the formulas in Remark 7.2.3 and the definitions of β^A, α^A. These formulas will be used in later proofs.

Remark 7.3.2 We have

N^IJAI∧A¯J =N^IJ(FIK −F¯IK)dζ^K∧dζ˜J =idζ^J ∧dζ˜J, (7.95) dX^{K (7.85)}=

(7.83)−X^K∂K−e^−K r ρ

ρ+cN^KJP¯_J^aβ^a, (7.96)

√i

ρe^−K/2A_J ^(7.85)=

r ρ

ρ+ 2cN_{J L}X¯^Lβ⁰−e^−KP_J^bα^b, (7.97)

√i

ρe^−K/2dζ^L= 2 i

√ρe^−K/2N^LMImA_M =−2iN^LMRe( i

√ρe^−K/2A_M)

(7.97)

= −2i r ρ

ρ+ 2cRe( ¯X^Lβ⁰) + 2ie^−KN^LMRe(P_M^b α^b), (7.98) d( ¯P_I^aN^IJ)N_{J L}X¯^L(7.82)= −P¯_I^aN^IJd(N_{J L}X¯^L)

(7.81)

= −

r ρ ρ+c

β¯^a+iP¯_I^aN^IJdF_{J L}(X) ¯X^L, (7.99) e^−KP¯_I^aN^IJP¯_M^b N^{M L}F_{J KL}(X)dX^{K (7.96)}=

(7.81)−e^−2K r ρ

ρ+cFe^abc(X)β^c, (7.100) where fora, b, c= 1, . . . , n,

F˜^abc(X) :=

n

X

I, J, K, L, M, N=0

P¯_L^aN^LIP¯_M^b N^{M J}P¯_N^cN^{N K}F_{IJ K}(X). (7.101)

Proposition 7.3.3

dβ⁰ = 1 2

1 + 2c ρ+ 2c

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)−id^cK

∧β⁰+

rρ+ 2c ρ+c

n

X

b=1

α^b ∧β^b,

dβ^a = c

2√

ρ+c√

ρ+ 2c(α⁰+ ¯α⁰)∧β^a−

n

X

b=1

ω^a_b∧β^b, dα⁰ = 1

√ρ+c√ ρ+ 2c

−(ρ+c) + 1 2

cρ ρ+ 2c

α⁰∧α¯⁰+ ρ ρ+ 2c

r ρ+c

ρ+ 2cβ⁰∧β¯⁰

−

r ρ+c ρ+ 2c

n

X

b=1

α^b∧α¯^b− c

√ρ+c√ ρ+ 2c

n

X

b=1

β^b∧β¯^b, dα^a = 1

2(

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)−id^cK)∧α^a+ ρ

√ρ+c√

ρ+ 2cβ⁰∧β¯^a

−

n

X

b=1

ω^a_b∧α^b−ie^K r ρ

ρ+c

n

X

b,c=1

˜h_abcα¯^b∧β^c,

where ˜h_abc =Pn

µ, ν, σ=1e^µ_ae^ν_be^σ_ch_µνσ for a, b, c = 1, . . . , n and (ω^a_b)a, b=1, ..., n is the

(pullback toN⁰ of the) connection one-form of the Levi-Civita connection on M¯ with respect to the given choice of coframe on M¯, i.e. (ω^a_b) is anti-Hermitian and fulfills dσ^a+Pn

b=1ω^a_b∧σ^b = 0, a= 1, . . . , n.

Proof: For β⁰ =ie^K/2

√ρ+2c

ρ X^IAI, we have dβ^{0 =4}

(7.81) (− 1

2ρ 1 + 2c ρ+ 2c

dρ+ 1

2dK)∧β⁰+ie^K/2

√ρ+ 2c

ρ dX^I∧A_I

= 1

2 1 + 2c ρ+ 2c

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)∧β⁰+ (∂K− i

2d^cK)∧β⁰ +ie^K/2

√ρ+ 2c

ρ dX^I∧A_I

= 1 2

1 + 2c ρ+ 2c

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)−id^cK

∧β⁰+

rρ+ 2c

ρ+c α^b∧β^b, since

α^a∧β^a(7.93)= +ie^K/2

√ρ+c

ρ dX^I ∧A_I+

r ρ+c

ρ+ 2c∂K∧β⁰. Forβ^a =q

ρ+c

ρ σ^a, we have dσ^a=−ω^a_b∧σ^b by the definition of (ω^a_b), so dβ^a=⁵− c

2ρ 1

ρ+cdρ∧β^a−ω^a_b∧β^b

= c

2√

ρ+c√

ρ+ 2c(α⁰+ ¯α⁰)∧β^a−ω^a_b∧β^b.

Forα⁰ =−_2ρ¹ q

ρ+2c ρ+c

dρ−i_ρ+2c^ρ+c(dφ˜+ζ^Idζ˜I−ζ˜Idζ^I+cd^cK)

, we find

dα⁰ =⁶i

−1 ρ +1

2

c

(ρ+c)(ρ+ 2c)

dρ∧Imα⁰+i 1 2ρ

r ρ+c

ρ+ 2c(2dζ^I∧dζ˜_I+cdd^cK)

= 2i

√ρ+c√ ρ+ 2c

ρ+c− 1 2

cρ ρ+ 2c

Reα⁰∧Imα⁰

| {z }

i 2α⁰∧¯α⁰

−1 ρ

r ρ+c ρ+ 2c

ρ α^b∧α¯^b− ρ²

ρ+ 2cβ⁰∧β¯⁰

− c

√ρ+c√

ρ+ 2cβ^b∧β¯^b,

4 ∂

∂ρ

^√_ρ+2c

ρ

=−_2ρ¹ 1 + _ρ+2c^{2c √}ρ+2c ρ 5 ∂

∂ρ

q_ρ+c

ρ =−_2ρ^c _ρ+c¹ q_ρ+c

ρ

since

α^b∧α¯^b = 1

ρe^−KP_K^bP¯_I^bN^IJN^KLAJ ∧A¯L (7.85)

= −1

ρN^{J L}A_J∧A¯_L+ 1

ρe^KX^JA_J ∧X¯^LA¯_L

(7.95)

= −i

ρdζ^L∧ζ˜L+ ρ

ρ+ 2cβ⁰∧β¯⁰ and dd^cK= 4ωM¯ = 2iσ^a∧σ¯^a= _ρ+c^2iρ β^a∧β¯^a.

Finally, α^a = ^√i_ρe^−K/2P¯_I^aN^IJA_J fulfills dα^a = (− 1

2ρdρ−1

2dK)∧α^a+ i

√ρe^−K/2d( ¯P_I^aN^IJ)∧A_J + i

√ρe^−K/2P¯_I^aN^IJdFJ L(X)∧dζ^L

(7.97)

= 1 2

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)∧α^a−(i

2d^cK+ ¯∂K)∧α^a +

r ρ

ρ+ 2cd( ¯P_I^aN^IJ)NJ LX¯^L∧β⁰−e^−Kd( ¯P_I^aN^IJ)P_J^b∧α^b + i

√ρe^−K/2P¯_I^aN^IJdF_{J L}(X)∧dζ^L

(7.99)

=

(7.98)

1 2

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)∧α^a−(i

2d^cK+ ¯∂K)∧α^a

(((((((((((((((((((

+i r ρ

ρ+ 2c

P¯_I^aN^IJdF_{J L}(X) ¯X^L∧β⁰− ρ

√ρ+c√ ρ+ 2c

β¯^a∧β⁰

−e^−Kd( ¯P_I^aN^IJ)P_J^b ∧α^b

((((((((((((((((((((((

−2i r ρ

ρ+ 2c

P¯_I^aN^IJdF_{J L}(X)∧Re( ¯X^Lβ⁰) + 2ie^−KP¯_I^aN^IJdF_{J L}(X)N^LM ∧Re(P_M^b α^b)

(7.81),(7.79)

=

(7.100)

1 2(

r ρ+c

ρ+ 2c(α⁰+ ¯α⁰)−id^cK)∧α^a+ ρ

√ρ+c√

ρ+ 2cβ⁰∧β¯^a

−ω^a_b∧α^b+ie^−2K r ρ

ρ+cFe^abc(X) ¯α^b∧β^c. Finally, note that

F˜^abc(X) = −e^3K˜h_abc(ImX). (7.102)

6 ∂

∂ρ

−i_2|ρ|^σ1 q _ρ+c

|ρ+2c|

=−i_2|ρ|^σ1 q _ρ+c

|ρ+2c|

−_ρ¹+¹₂(ρ+c)(ρ+2c)^c

Recall that ¯θ₁ = −¹₂q

ρ+2c

ρ+c Imα⁰− ¹₄d^cK and ¯θ₂+iθ¯₃ =q

ρ+c

ρ+2cβ⁰ (see Remark 5.5.3). The H-part of the Levi-Civita connection is given by (see Eq. (7.48))

p= p¹₁ p¹₂ p²₁ p²₂

!

=









−¹₄(−id^cK+q

ρ+2c

ρ+c ( ¯α⁰−α⁰)) −q

ρ+c ρ+2cβ⁰ qρ+c

ρ+2cβ¯^{0 1}₄(−id^cK+q

ρ+2c

ρ+c ( ¯α⁰−α⁰))







 .

Corollary 7.3.4 The E-part of the Levi-Civita connection with respect to the frame (E_Γ) for the one-loop deformed q-map is given by (Θ^Γ_∆) = q^A_B t^A_˜

B

−t¯^A˜_B q¯^A˜_˜

B

!

with (q^A_B) =q,

q=









i

4d^cK+¹₄^√_ρ+c^1√_ρ+2c

3ρ+_ρ+2c^4c2

( ¯α⁰−α⁰) −q

ρ+c ρ+2cα^b qρ+c

ρ+2cα¯^a ω^a_b+¹₄(−id^cK+ ^√_ρ+c^ρ√_ρ+2c( ¯α⁰−α⁰))δ^a_b









and

t= (t^A_B˜) =









2c ρ+2c

qρ+c

ρ+2cβ^{0 √}_ρ+c^c√_ρ+2cβ^b

√ c ρ+c√

ρ+2cβ^a ie^Kq _ρ

ρ+c˜habcα^c







 .

Proof: q is anti-Hermitian and t is symmetric. A straightforward calculation shows that the equations given in Proposition 7.3.3 agree with equations (7.50) and (7.51), whenq and t are given as above.

From now on, we restrict ourselves to the undeformed q-map, i.e. we set c= 0.

Proposition 7.3.5 The E-part of the curvature two-form with respect to the frame (E_Γ) is given for any quaternionic K¨ahler manifold in the image of the

q-map by (R_E^Γ_∆) = r^A_B s^A_˜

B

−¯s^A˜_B r¯^A˜_˜

B

!

with (r^A_B) = r,

r =

































 1

2 α⁰∧α¯⁰−β⁰∧β¯⁰ +

n

X

C=0

α^C ∧α¯^C−β^C ∧β¯^C α^b∧α¯⁰+ ¯β^b∧β⁰+ie^K˜h_bcdα¯^c∧β^d

α⁰ ∧α¯^a+ ¯β⁰∧β^a +ie^K˜hacdα^c∧β¯^d

1 2δ^a_b

n

X

C=0

(α^C ∧α¯^C −β^C∧β¯^C)

−(β^a∧β¯^b+ ¯α^a∧α^b)

−e^2Kh˜_adc˜h_ceb(α^d∧α¯^e+ ¯β^d∧β^e)



































and (s^A_˜

B) = s,

s = 0 0

0 ie^K˜h_abc(β⁰∧β¯^c+ ¯α⁰∧α^c) +e^2K˜h_abf˜h_{f de}α¯^d∧β^e−2S_abcdα^c∧β¯^d

! ,

where Sabcd :=−1

2e^2K

(˜hbcf˜hf ad−4˜hbc˜had) + (˜hacf˜hf bd−4˜hac˜hbd) + (˜habf˜hf cd−4˜hab˜hcd) + 4˜h_ah˜_bcd+ 4˜h_b˜h_cda+ 4˜h_c˜h_dab+ 4˜h_d˜h_abc

.

Proof: First, we calculate dq:

dq⁰₀ = i

4dd^cK+3

4(d¯α⁰−dα⁰)^(d¯α

0=−dα⁰)

= −1

2∂∂¯K− 3 2dα⁰

=−1

2β^c∧β¯^c+3

2(α⁰∧α¯⁰−β⁰∧β¯⁰+α^c∧α¯^c),

dq⁰_b =−dq^b₀ =−dα^b

=−1

2(α⁰+ ¯α⁰ −id^cK)∧α^b−β⁰∧β¯^b+ ¯ω^b_c∧α^c+ie^K˜h_abcα¯^a∧β^c,

dq^a_b =dω^a_b +1

2δ^a_b(∂∂¯K−dα⁰)

=dω^a_b +1

2δ^a_b(β^c∧β¯^c+α⁰∧α¯⁰−β⁰∧β¯⁰+α^c∧α¯^c).

Then with

(q^A_C∧q^C_B) =

−α^c∧α¯^c −¹₂(id^cK+ ¯α⁰ −α⁰)∧α^b−ω¯^b_c∧α^c

−¹₂(id^cK+ ¯α⁰−α⁰)∧α¯^a+ω^a_c∧α¯^c ω^a_c∧ω^c_b−α¯^a∧α^b

!

and

(t^A_C ∧¯t^C_B) = 0 0

0 e^2K˜h_adc˜h_cebα^d∧α¯^e

! , we obtain

r^(7.56)= dq+q∧q−t∧¯t

=





















 1

2(α^c∧α¯^c−β^c∧β¯^c) +3

2(α⁰∧α¯⁰−β⁰∧β¯⁰)

α^b∧α¯⁰+ ¯β^b∧β⁰+ie^K˜h_bcdα¯^c∧β^d

α⁰∧α¯^a+ ¯β⁰∧β^a +ie^K˜h_acdα^c∧β¯^d

dω^a_b+ω^a_c∧ω^c_b −α¯^a∧α^b−e^2K˜h_adch˜_cebα^d∧α¯^e +1

2δ^a_b(β^c∧β¯^c+α^c∧α¯^c+α⁰∧α¯⁰−β⁰∧β¯⁰)





















 .

This can be brought into the form stated above using (see Eq. (7.72)) dω^a_b+ω^a_c∧ω^c_b =−δ^a_bβ^c∧β¯^c−β^a∧β¯^b+e^2K˜h_adch˜_cebβ^e∧β¯^d.

Since e^a_µ is real, we have e^a_µde^µ_b = ¯e^a_µd¯e^µ_b ^(7.70)= ω¯^a_b + ¯m^a_b, where m^a_b :=e^a_µ∂e^µ_b+ ¯e^b_µ∂¯e^µ_a. Using d(˜habc) = ˜hdbce^d_µde^µ_a+ ˜hadce^d_µde^µ_b+ ˜habde^d_µde^µ_c, we calculate

dt^a_d=ie^K˜h_abc(x) dK∧α^c+1

2(α⁰+ ¯α⁰−id^cK)∧α^c+β⁰∧β¯^c

−ω¯^c_d∧α^d−ie^K˜hcde(x) ¯α^d∧β^e +ie^K ˜h_dbc(¯ω^d_a+ ¯m^d_a) + ˜h_adc(¯ω^d_b+ ¯m^d_b) + ˜h_abd(¯ω^d_c+ ¯m^d_c)

∧α^c.

With

q^A_C ∧t^C_B ^ω

a c=−¯ω^c_a

= 0 0

0 −ie^K˜hcbeω¯^c_a∧α^e+₄ⁱe^Kh˜abe(−id^cK+ ¯α⁰−α⁰)∧α^e

!

and

t^A_C∧q¯^C_B = 0 0

0 ie^K˜h_aceα^e∧ω¯^c_b+₄ⁱe^Kh˜_abe(−id^cK+ ¯α⁰−α⁰)∧α^e

! , we obtain

s^(7.57)= dt+q∧t+t∧q¯= 0 0 0 s^a_b

! , where

s^a_b =ie^K˜h_abc

(dK+ ¯α⁰−id^cK)∧α^c+β⁰∧β¯^c−ie^K˜h_cdeα¯^d∧β^e +ie^K(˜hdbcm¯^d_a+ ˜hadcm¯^d_b+ ˜habdm¯^d_c)∧α^c

= 8e^2Kh˜_abc˜h_dβ¯^d∧α^c+ie^Kh˜_abc(β⁰∧β¯^c+ ¯α⁰∧α^c) +e^2K˜h_abch˜_cdeα¯^d∧β^e +ie^K(˜hdbcm¯^d_a+ ˜hadcm¯^d_b+ ˜habdm¯^d_c)∧α^c.

In the last equality, we used

dK−id^cK= 2 ¯∂K=− i

h(x)h_µ(x)e^µ_dβ¯^d=−8ie^K˜h_dβ¯^d. Now, using

m^a_b =−e^a_ρe^µ_be^σ_cΓ^ρ_µσβ^c and (see (7.64) and (6.4))

Γ^ρ_σµ =− i 2h

hh^ρκhκµσ −hσδ_µ^ρ−hµδ^ρ_σ+ 1 2x^ρhµσ

, e^ν_ae^λ_a =K^¯νλ =−4h(x)h^νλ(x) + 2x^νx^λ, we find

˜hdbcm¯^d_a =ie^K(˜hbcf˜hf da−4˜hbc˜hda+ 4˜hah˜bcd+ 4˜hd˜habc) ¯β^d. Hence,

s^a_b =ie^Kh˜abc(β⁰∧β¯^c+ ¯α⁰∧α^c) +e^2K˜habfh˜f deα¯^d∧β^e

−e^2K

(˜h_bcf˜h_{f ad}−4˜h_bc˜h_ad) + (˜h_acf˜h_{f bd}−4˜h_ac˜h_bd) + (˜h_abf˜h_{f cd}−4˜h_ab˜h_cd)

β¯^d∧α^c

−4e^2K(˜h_ah˜_bcd+ ˜h_b˜h_cda+ ˜h_ch˜_dab+ ˜h_d˜h_abc) ¯β^d∧α^c.

Remark 7.3.6 Note that the vanishing of the symmetric quartic tensor field Sabcdσ^a⊗σ^b⊗σ^c⊗σ^d

=−1 2

1 4³h²

3hτ(µνK^{τ τ0}hσρ)τ⁰. . .

. . .−12h_(µνh_σρ)+ 16h_(µh_νσρ)

!

dX^µ⊗dX^ν ⊗dX^σ⊗dX^ρ

=−1 2

1 4³h²

−12h_τ(µνh^{τ τ0}h_σρ)τ⁰. . . . . .−6h_(µνh_σρ)+ 16h_(µh_νσρ)

!

dX^µ⊗dX^ν ⊗dX^σ⊗dX^ρ

on the projective special K¨ahler manifold ( ¯M, gM¯, JM¯) is a necessary and suffi-cient condition for ( ¯M, gM¯) to be symmetric [CV].

In the following theorem, we use the notation from Section 7.1.

Theorem 7.3.7 The Sp(E)-curvature of manifolds in the image of the q-map can be written as

R˜^Γ_Γ0 =−1

2_αβC_Γ⁰_Γ⁰⁰f^αΓ∧f^βΓ00+C^ΓΓ0Ω_Γ⁰_Γ⁰_Γ⁰⁰_Γ⁰⁰⁰_αβf^αΓ00∧f^βΓ000, (7.103) where the non-vanishing components of the symmetric quartic tensor field Ω are given by

Ω_00˜0˜0 = 1

2, Ω_{0b˜0 ˜d}= 1

4δ_bd, Ω_ab˜cd˜= 1

4(δ_acδ_bd+δ_adδ_bc)−1

2e^2K˜h_abf˜h_{f cd}, Ω˜0bcd = Ω_0˜b˜cd˜=−i

2e^K˜h_bcd, Ω_abcd = Ω_˜a˜b˜cd˜=S_abcd (and symmetrization thereof ).

Proof: First, note that (_αβC_Γ⁰_Γ⁰⁰f^Γα∧f^Γ00β)^Γ=A,A˜

Γ⁰=B,B˜ = β^A∧β¯^B+ ¯α^A∧α^B β^A∧α¯^B˜−α¯^A∧β^B˜ α^A˜∧β¯^B−β¯^A˜∧α^B α^A˜∧α¯^B˜ + ¯β^A˜∧β^B˜

! .

Define

U^ΓΓ0 :=_αβf^Γα∧f^Γ0β = −β^A∧α¯^B+ ¯α^A∧β^B β^A∧β¯^B˜ + ¯α^A∧α^B˜

−α^A˜∧α¯^B−β¯^A˜∧β^B α^A˜∧β¯^B˜ −β¯^A˜∧α^B˜

!

and

R˚_Γ⁰_Γ⁰ :=C_Γ⁰_Γ( ˜R^Γ_Γ0 +1

2_αβC_Γ⁰_Γ⁰⁰f^Γα∧f^Γ00β).

Then

R˚AB˜ =− r^A_B+ 1

2(β^A∧β¯^B+ ¯α^A∧α^B)

=



























 1

2(β⁰∧β¯⁰+ ¯α⁰∧α⁰) +1

2(β^C ∧β¯^C+ ¯α^C ∧α^C)

1

2(β⁰∧β¯^b + ¯α⁰∧α^b) +ie^K˜h_bdeβ^e∧α¯^d

1

2(β^a∧β¯⁰+ ¯α^a∧α⁰)

−ie^Kh˜_adeα^d∧β¯^e

1

2δ_ab(β^C ∧β¯^C + ¯α^C∧α^C) +1

2(β^a∧β¯^b+ ¯α^a∧α^b)

+e^2Kh˜_adc˜h_ceb(α^d∧α¯^e+ ¯β^d∧β^e)





























=

















1

4(U^0˜0+U^˜00+U^CC˜ +U^CC˜ ) ¹₄(U^0˜b+U^˜b0)− ₂ⁱe^Kh˜_bdeU^de

1

4(U^a˜0+U^˜0a)− ₂ⁱe^K˜h_adeU^d˜˜e 1

4δ_ab(U^CC˜ +U^CC˜ ) + 1

4(U^a˜b+U^˜ba)

− e^2K 2

˜h_adf˜h_{f eb}(U^d˜e+U^de˜)















 ,

R˚A˜B˜ =− s^A_B+1

2(β^A∧α¯^B−α¯^A∧β^B)

=













−β⁰∧α¯⁰ −¹₂(β⁰∧α¯^b−α¯⁰∧β^b)

−¹₂(β^a∧α¯⁰ −α¯^a∧β⁰) − 1

2(β^a∧α¯^b−α¯^a∧β^b) +e^2K˜h_abf˜h_{f de}β^d∧α¯^e

−ie^K˜h_abc(β⁰∧β¯^c+ ¯α⁰∧α^c) + 2S_abcdα^c∧β¯^d













=

















1

2U^{00 1}₄(U^0b+U^b0)

1

4(U^a0 +U^0a) 1

4(U^ab+U^ba)−e^2K 2

˜h_abf˜h_{f de}U^de

− i 2

h˜abc(U^0˜c+U^˜c0) +SabcdU^˜cd˜















 ,

and ˚R_AB = ˚RA˜B˜, ˚R_AB˜ =−R˚AB˜ .

Eq. (7.58) is equivalent to ˚R_ΓΓ⁰ = Ω_ΓΓ⁰_Γ⁰⁰_Γ⁰⁰⁰U^Γ00Γ000. Now ˚R˜00 = Ω˜00Γ⁰⁰Γ⁰⁰⁰U^Γ00Γ000

implies Ω˜00CD = Ω˜00 ˜CD˜ = 0, Ω_{˜000 ˜d}= Ω˜00c˜0 = 0 and Ω˜000˜0 = 1

2, Ω˜00cd˜= 1 4δ_cd.

R˚_˜a0 = Ω_a0Γ˜ ⁰⁰_Γ⁰⁰⁰U^Γ00Γ000 implies Ω_a0CD˜ = 0, Ω_{˜a00 ˜d}= 0, Ω_a0c˜ d˜= 0 and Ω_˜a0˜cd˜=−i

2e^Kh˜_acd.

R˚_˜ab = Ω_abΓ˜ ⁰⁰_Γ⁰⁰⁰U^Γ00Γ000 implies Ω_˜abC˜D˜ = 0, Ω_˜abcd = 0, Ω_˜ab˜0d = 0 and Ω_˜ab˜cd = 1

4(δ_abδ_cd+δ_bcδ_ad)−e^2K 2

˜h_acf˜h_{f db}. R˚˜0b = Ω˜0bΓ⁰⁰Γ⁰⁰⁰U^Γ00Γ000 implies Ω˜0b˜0˜0 = 0, Ω˜0b˜0d= 0 and

Ω˜0bcd =−i

2e^Kh˜_bcd.

It remains to determine the components of the form Ω_ABCD and ΩA˜B˜C˜D˜: R˚A˜B˜ = ΩA˜BΓ˜ ⁰⁰Γ⁰⁰⁰U^Γ00Γ000 implies Ω_{˜0 ˜B}C˜D˜ = 0 and

Ω_˜a˜b˜cd˜=S_abcd.

UsingU^AB =U^A˜B˜ andU^AB˜ =−U^AB˜ , we find that ˚R_AB = ˚RA˜B˜ = Ω_ABΓ⁰⁰_Γ⁰⁰⁰U^Γ00Γ000 implies Ω_0BCD = 0 and

Ω_abcd =S_abcd.

7.4 Example: A series of inhomogeneous com-plete quaternionic K¨ ahler manifolds

In this section, we show that the members of a certain series of complete quater-nionic K¨ahler manifolds constructed from the q-map are not locally homoge-neous. This is done by calculating the pointwise norm of the Riemann tensor and showing that it is a non-constant function on the quaternionic K¨ahler mani-fold. We leave out the details of the calculation and just show some intermediate steps and the final result. Note that some simplifications of formulas were done using computer algebra software.

We will again leave out summation symbols and employ Einstein’s summation convention.

Forn ∈N, we consider the following series of projective special real manifolds:

H={h= 1, x > 0} ⊂Rⁿ, h:=x(x²−

n−1

X

i=1

(y_i)²). (7.104) The projective special real manifold (H, g_H) is a closed subset of Rⁿ and thus complete according to Theorem 6.2.8 (which was proven in [CNS]). Due to Theorems 6.2.6 and 6.3.3 (which were proven in [CHM]), the corresponding projective special K¨ahler manifold obtained from the supergravity r-map and the quaternionic K¨ahler manifold obtained from the q-map are complete as well.

The scalar curvature of the corresponding projective special K¨ahler manifold ¯M in the image of the supergravity r-map can be calculated to be (see Theorem 3 in⁷ [CDL] for the general formula)

scalM¯ =−2n²+n−2hh_αβγh^αα0h^ββ0h^γγ0h_α⁰_β⁰_γ⁰

=−2n(n+ 1) + 1

32h²h_αβγK^αα0K^ββ0K^γγ0h_α⁰_β⁰_γ⁰

=−n·(2n−1) + 3h· n−2

h−4x³ + 36x³h²

(h−4x³)³. (7.105) We find the following expression for the squared pointwise norm of the quartic tensor field B_µνσρ :=h_µνκK^κκ0h_κ⁰_ρσ on ¯M:

BµνσρK^µµ0K^νν0K^σσ0K^ρρ0Bµ⁰ν⁰σ⁰ρ⁰ = 4096h⁴ (h−4x³)⁶ ·

h⁶(n−1)(n+ 3)

−4h⁵(n+ 3)(5n−7)x³ + 4h⁴(n(41n+ 98)−159)x⁶

−64h³(n(11n+ 43)−75)x⁹+ 128h²(n(13n+ 73)−78)x¹²

−2048h(n(n+ 7)−3)x¹⁵+ 1024n(8 +n)x¹⁸

. (7.106)

The squared pointwise norm of the Riemann tensor of the projective special K¨ahler manifold is (see Theorem 3 in [CDL] for the general formula for the

7Note that compared to [CDL] we scaled the projective special K¨ahler metric gM¯ by a factor of ¹₂, which leads to a scaling of the scalar curvaturescalM¯ by a factor of 2.

Riemann tensor)

kRM¯k² = 16Rµνσ¯ ρ¯K^µµ0K^νν0K^σσ0K^ρρ0Rµ⁰¯ν⁰¯σ⁰ρ⁰

=−32scalM¯ −32n(n+ 1) + 1

4⁴h⁴B_ρσµνK^ρρ0K^σσ0K^µµ0K^νν0B_ρ⁰_σ⁰_µ⁰_ν⁰

= 16

(h−4x³)⁶

h⁶(n(3n−8) + 9)−4h⁵(n(17n−46) + 57)x³ + 4h⁴(n(161n−382) + 537)x⁶−64h³(n(51n−97) + 99)x⁹ + 128h²(n(73n−107) + 78)x¹²−2048h(n(7n−8) + 3)x¹⁵ + 1024n(9n−8)x¹⁸

. (7.107)

For the squared pointwise norm of the tensor field Sµνσρ=−1

2e^2K

(˜hbcf˜hf ad−4˜hbc˜had) + (˜hacf˜hf bd−4˜hac˜hbd) + (˜h_abf˜h_{f cd}−4˜h_ab˜h_cd)

+ 4˜ha˜hbcd+ 4˜hb˜hcda+ 4˜hc˜hdab+ 4˜hd˜habc

on ¯M (see the definition ofS in Proposition 7.3.5), we find:

S_µνσρK^µµ0K^νν0K^σσ0K^ρρ0S_µ⁰_ν⁰_σ⁰_ρ⁰ (7.108)

= 3x⁶ (h−4x³)⁶

h⁴(n(n+ 16) + 207)−16h³(n−2)(n+ 9)x³ + 96h² n²+n−6

x⁶−256h(n−2)nx⁹+ 256(n−2)nx¹² . The squared pointwise norm of the quaternionic Weyl tensor is

1

64kWk² = Ω_ΓΓ⁰_Γ⁰⁰_Γ⁰⁰⁰C^Γ∆C^Γ0∆0C^Γ00∆00C^Γ000∆000Ω_∆∆⁰_∆⁰⁰_∆⁰⁰⁰

= 2Ω_ABCDΩA˜B˜C˜D˜ −8Ω_ABCD˜ΩA˜B˜CD˜ + 6Ω_ABC˜D˜ΩA˜BCD˜

= 2Ω_abcdΩ_˜a˜b˜cd˜−8Ω_abc˜0Ω_a˜˜b˜c0+ 6(Ω_00˜0˜0)²+ 24Ω_0b˜0 ˜dΩ˜0˜b0d+ 6Ω_ab˜cd˜Ω_˜a˜bcd

= 2S_abcdS_abcd+ 2n(n+ 1) +scalM¯ + 3

2(n+ 1) + 6( 1

4³kRM¯k²+ 1

4scalM¯ +n²+n 8 )

= 2S_abcdS_abcd+1

4(11n+ 6)(n+ 1) + 3

32kRM¯k²+5 2scalM¯

= 3

2 (h−4x³)⁶

h⁶n(n+ 1)−4h⁵(n+ 1)(5n−2)x³+ 8h⁴(n(21n+ 37) + 112)x⁶

−256h³(n(3n+ 10)−11)x⁹+ 256h²(n(8n+ 33)−20)x¹²

−1024h(n(3n+ 11) + 2)x¹⁵+ 2048(n+ 1)(n+ 2)x¹⁸ + 3n

4 (n+ 1). (7.109)

By evaluating the above function in different points, one can check that it is non-constant for n >1. This gives the following proposition:

Proposition 7.4.1 For n >1, the series of manifolds obtained from the com-plete projective special real manifolds in Eq. (7.104) via the q-map consists of complete quaternionic K¨ahler manifolds that are not locally homogeneous.

Remark 7.4.2 The curvature tensor of the quaternionic K¨ahler manifolds discussed above splits as R = νRHPⁿ⁺¹ + W. Note that in our conventions for quaternionic K¨ahler manifolds obtained via the supergravity c-map from an 2n-dimensional projective special K¨ahler manifold manifold, the reduced scalar curvature is ν =−2. The squared pointwise norm of RHPⁿ⁺¹ is

kR_HPⁿ⁺¹k² = 20n²+ 44n+ 24 = 20(n+ 1)²+ 4(n+ 1). (7.110) Using computer algebra software, we have calculated the squared pointwise norm kRk² of the Riemann tensor for n = 2 and n = 3 and have checked that kRk² −4kRHPⁿ⁺¹k² agrees with the squared pointwise norm of the Weyl ten-sor in Eq. (7.109).

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I am particularly indebted to my supervisor Vicente Cort´es for introducing me to the c-map and to quaternionic K¨ahler geometry, and for his guidance and support during my time in Hamburg. Thanks a lot!

Thanks also go to Oliver Goertsches and Jan Louis for giving advice on the mathematical, respectively physical part of my work as part of my supervisory committee.

I would like to thank my coauthors Thomas Mohaupt, Dmitri Alekseevsky and David Lindemann for inspirational collaborations and enjoyable discussions.

I also want to thank my fellow PhD students Benedict Meinke, Marco Freibert, Lana Casselmann and Peter-Simon Dieterich in the differential geometry group who, apart from enriching my private life, took part in invaluable discussions and answered even my most stupid questions.

For further work-related discussions, I would like to thank Andrew Swann, An-driy Haydys, Sergei Alexandrov, Stefan Vandoren and Uwe Semmelmann.

I would also like to thank Andriy Haydys and Stefan Vandoren for taking the time and effort to grade my PhD thesis.

Special thanks go to Owen Vaughan. Apart from inviting me to parties, and from playing and watching football with me, he invited me to the University of Liverpool, and, during our time in Hamburg, he explained to me all the formulas and tricks that can be found in the physics literature on special geometry.

Many thanks to Giovanni Bazzoni, in particular for inviting me to Bielefeld University on several occasions.

This work was made possible by the financial support and stimulating atmo-sphere of the RTG 1670 “Mathematics inspired by String Theory and Quantum Field Theory”. I would like to thank all present and former members of the RTG 1670 and all colleagues at the Department of Mathematics in Hamburg for this fruitful and friendly environment. Particular thanks in this regard go to my office mate Sebastian Novak.

Thanks a lot to my family, to Paloma IV and to my friends from Schwerte for supporting me and giving my life meaning.

My deepest gratitude belongs to Charlotte for being in my life and for marrying me.

D.V. Alekseevsky, V. Cort´es, M. Dyckmanns and T. Mohaupt, Quaternionic K¨ahler metrics associated with special K¨ahler manifolds, J. Geom. Phys.92 (2015), 271–287.

This collaboration was part of my doctoral project. The formulation of the HK/QK correspondence in Section 4.1 is (as opposed to its proof) taken from this publication. The same is true for the account of the Swann bundle construction in Section 3.6. The application of the HK/QK correspondence to the c-map in Sections 5.1–5.4 is part of this publication. The results on the K¨ahler/K¨ahler correspondence from this publication did not enter this doctoral thesis.

V. Cort´es, M. Dyckmanns and D. Lindemann, Classification of complete projective special real surfaces, Proc. London Math. Soc. 109 (2014), no. 2, 423–445.

My contribution to this collaboration is mainly based on my M.Sc. thesis written in Hamburg under the supervision of Vicente Cort´es. In this doctoral thesis, the classification of projective special real surfaces is only mentioned in a short remark in Chapter 6. Chapter 7 makes some use of the curvature results for projective very special K¨ahler manifolds from this publication.

M. Dyckmanns,

A twistor sphere of generalized K¨ahler potentials on hyperk¨ahler manifolds, arXiv:1111.3893 (hep-th).

This preprint contains the results of my M.A. thesis written in Stony Brook under the supervision of Martin Roˇcek and is essentially unrelated to this doctoral thesis.

Im Dokument The hyper-Kähler/quaternionic Kähler correspondence and the geometry of the c-map (Seite 160-188)