Chern-Simons flows on Aloff-Wallach spaces and spin(7) instantons

Chern-Simons flows on Aloff-Wallach spaces and spin(7) instantons

Alexander S. Haupt,^1,*Tatiana A. Ivanova,^2,†Olaf Lechtenfeld,^1,3,‡and Alexander D. Popov^2,x

1Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, Appelstraße 2, 30167 Hannover, Germany

2Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia

3Centre for Quantum Engineering and Space-Time Research, Leibniz Universita¨t Hannover, Welfengarten 1, 30167 Hannover, Germany

(Received 4 May 2011; published 27 May 2011)

Because of their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications.

They carryG₂structures and admit theG₂-instanton equations, which are natural Bogomol’nyi-Prasad- Sommerfeld equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons–type functional. We consider the Chern-Simons flow between differentG₂instantons on Aloff-Wallach spaces, which is equivalent to spin(7) instantons on a cylinder over them. For a general SU(3)-equivariant gauge connection, the generalized instanton equations turn into gradient-flow equations on C³R², with a particular cubic superpotential. For the simplest member of the Aloff-Wallach family (with 3-Sasakian structure) we present an explicit instanton solution of tanh-like shape.

DOI:10.1103/PhysRevD.83.105028 PACS numbers: 11.15.Yc, 11.10.Kk, 11.27.+d, 12.10.g

I. INTRODUCTION AND SUMMARY

Yang-Mills theory in more than four dimensions natu- rally appears in the low-energy limit of superstring theory in the presence of D-branes. Also, heterotic strings yield heterotic supergravity, which contains supersymmetric Yang-Mills theory as a subsector [1]. Furthermore, natural Bogomol’nyi-Prasad-Sommerfield–type equations for gauge fields in dimension d >4, introduced in [2], also appear in superstring compactifications on spacetimes M_10dX^d as the condition for the survival of at least one supersymmetry in the low-energy effective field theory on M_10d [1]. These first-order Bogomol’nyi-Prasad- Sommerfield–type equations on X^d, which generalize four-dimensional Yang-Mills anti-self-duality, were considered e.g. in [3–9], and some of their solutions were found in [10–13].

In string/M-theory compactification, the most interest- ing dimensions seem to bed¼6, 7 or 8, and the corre- sponding generalized anti-self-duality equations are, respectively, called the Hermitian-Yang-Mills equations [4], the G₂-instanton equations [8,14], or the spin(7)- instanton equations [8,15]. Most work on the above- mentioned instanton equations has restricted its attention to Riemannian manifoldsX^dwith holonomy group SU(3) for d¼6, G₂ for d¼7, or spin(7) for d¼8, i.e. to integrable G structures. However, if one is interested in string compactification with fluxes [16], one should con- sidernonintegrableGstructures (weak holonomy groups) instead. The torsion of theGstructure, which measures the failure to be integrable, is identified with the three-form field (‘‘flux’’) of supergravity. Flux compactifications have

been investigated primarily for type II strings and M theory, but also in the heterotic theories, albeit to a lesser extent, despite their long history [17]. In particular, compactifications on Aloff-Wallach spaces [18,19]X_k;lof dimension seven and conesCðX_k;lÞover them were studied e.g. in [20–22]. The Yang-Mills equations on spin(7) mani- folds of topology RX_k;l with cylindrical and conical metric are the subject of the present paper.

For any coprime pair of integersðk; lÞ, the Aloff-Wallach space X_k;l is the coset SUð3Þ=Uð1Þ_k;l with Uð1Þ_k;l¼ fdiagðe^iðkþlÞ’; e^ik’; e^il’Þg [18,19]. It carries a G₂ structure defined by a torsion three-form c ^{with the} property that dc is proportional to the Hodge-dual four- form c^{. G}2 instantons extremize a Chern-Simons–type action functional onX_k;l. As an example, we describe the Abelian canonical connection on a line bundle over X_k;l. Next, we step up to eight dimensions via extendingX_k;lby a real line R. Our G₂ instantons are the end points of a gradient flow along this line, which is described precisely by the spin(7)-instanton equations [8] on RX_k;l. The most general SU(3)-equivariant connection on a rank-3 complex vector bundle is parametrized by three complex and two real functions on R. The spin(7)-instanton equa- tions reduce to gradient-flow equations for these functions, governed by a cubic superpotentialW with globalUð1Þ Uð1Þsymmetry. Interestingly, each function obeys alinear equation in the background of the others.

In order to be more explicit, we specialize to the case of k¼l¼1. We fix the metric moduli (up to a freedom of orientation) such that X_1;1 is 3-Sasakian and CðX_1;1Þ is hyper-Ka¨hler, i.e. its structure group reduces to Sp(2).

We list all critical points and their Hessians and numeri- cally find an instanton solution whose shape is close to the tanh function. The corresponding gauge configuration in- terpolates between differentG₂instantons onX_1;1. It is not obvious how to establish the existence of further instanton

*Alexander.Haupt@itp.uni-hannover.de

†ita@theor.jinr.ru

‡Olaf.Lechtenfeld@itp.uni-hannover.de

xpopov@theor.jinr.ru

solutions. It would be interesting to extend and lift spin(7) instantons on a cylinder or a cone over Aloff- Wallach spaces to classical solutions of heterotic M theory.

II. ALOFF-WALLACH SPACES

Group SU(3).—Consider the group SU(3) with genera- torsfI_a; I_ig,a¼1;. . .;6,i¼7, 8, satisfying

½I_a;I_b ¼f^c_abI_cþfⁱ_abI_i; ½I_i;I_a ¼f_ia^bI_b; ½I_i;I_j ¼0;

(2.1) where the structure constants are

f₁₃⁵ ¼f₄₂⁵ ¼f₄₁⁶ ¼f⁶₃₂¼ 1 2 ffiffiffi

p3; f⁷₁₂¼f⁷₃₄¼ 1 2 ffiffiffi

p3; f₅₆⁷ ¼ 1

ffiffiffi3

p ; f₁₂⁸ ¼ f₃₄⁸ ¼ 1

2; (2.2)

plus those with cyclic permutations of indices in (2.2). The generators (2.1) of SU(3) can be chosen in the form

I₁¼ 1 2 ffiffiffi

p3

0 0 1 0 0 0 1 0 0 0 BB

@

1 CC

A; I₃¼ 1 2 ffiffiffi

p3

0 1 0 1 0 0 0 0 0 0

BB

@

1 CC A;

I₅¼ 1 2 ffiffiffi

3 p

0 0 0 0 0 1 0 1 0 0

BB

@

1 CC

A; I₂¼ 1 2 ffiffiffi

3 p

0 0 i 0 0 0 i 0 0 0 BB

@

1 CC A;

I₄¼ 1 2 ffiffiffi

p3

0 i 0 i 0 0 0 0 0 0 BB

@

1 CC

A; I₆¼ 1 2 ffiffiffi

p3

0 0 0 0 0 i 0 i 0 0 BB

@

1 CC A;

I₇¼ i 2 ffiffiffi

p3

0 0 0 0 1 0 0 0 1 0

BB

@

1 CC

A; I₈¼i 6

2 0 0 0 1 0 0 0 1 0

BB

@

1 CC A;

(2.3)

corresponding to the antifundamental representation.

The basis elements fI_a; I_ig of the Lie algebra suð3Þ can be represented by left-invariant vector fields fE_a; E_ig on the Lie group SU(3), and the dual basisfE^a; Eⁱgis a set of left-invariant one-forms which obey the Maurer-Cartan equations

dE^a¼ f_jb^aE^j^E^b1₂f_bc^aE^b^E^c; dEⁱ¼ ¹₂fⁱ_bcE^b^E^c; (2.4) where i,j¼7, 8 correspond to the Cartan subalgebra of suð3Þ.

Cosets SUð3Þ=Uð1Þ_k;l.—Let us consider a U(1) sub- group of SU(3) given by matrices of the form

h¼

expðiðkþlÞ’Þ 0 0

0 expðik’Þ 0

0 0 expðil’Þ

0

@

1

A; (2.5)

wherekandlare relatively prime integers and0’2.

Consider the coset space

X_k;l¼SUð3Þ=Uð1Þ_k;l; (2.6) where Uð1Þ_k;l is represented by matrices (2.5). For rela- tively prime integers k and l the coset spaces X_k;l are simply connected manifolds called Aloff-Wallach spaces [18,19].

The space SUð3Þ=Uð1Þ ¼:G=H consists of left cosets gH, g2G, and the natural projection g°gH is denoted by

: SUð3Þ !X_k;l (2.7) with fibers Uð1Þ_k;l. Over a contractible open subsetU of X_k;l, one can choose a map L: U!SUð3Þ such that L¼Id_U, i.e. L is a local section of the principal bundle (2.7). The pull-backs of fE^a; EⁱgbyLfrom SU(3) to X_k;l are denoted by fe^a; eⁱg which satisfy the same Maurer-Cartan equations

de^a¼ f^a_ibeⁱ^e^b1₂f_bc^a e^b^e^c; deⁱ¼ ¹₂fⁱ_bce^b^e^c (2.8) asfE^a; Eⁱg. Note that since all objects we consider will be invariant under some action of SU(3), it will suffice to do calculations just over the subsetU.

If we denote by fe^{a^}g, a^¼1;. . .;7, an orthonormal coframe onUX_k;l[basis forTðX_k;lÞoverU] then

e^{a^} ¼e^a for a¼1;. . .;6;

e^{^7} ¼ 1

ðkþlÞe⁷2

ðklÞe⁸ (2.9)

with fe^a; eⁱg obeying the Maurer-Cartan equations (2.8) and

e^{^8} ¼ 1

2ðklÞe⁷þ1

ðkþlÞe⁸ (2.10) is a canonical connection one-form in the bundle (2.7).

Here

:¼ 1 2 ffiffiffi

p3; ² :¼2ðk²þl²Þ: (2.11) Then as generators of SU(3) we have

I_a^ ¼I_a; I_^7 ¼¹ððkþlÞI₇ ffiffiffi 3

p ðklÞI₈Þ;

I_^8 ¼¹ 1

ffiffiffi3

p ðklÞI₇þ ðkþlÞI₈

;

(2.12)

so that

e^aI_aþeⁱI_i¼e^{a^}I_a^þe^{^8}I_^8; (2.13) andI_^8is the generator of the groupUð1Þ_k;lgiven by (2.5).

Let us now rescale matrices (2.12) as

~I₁¼¹&₁I₁; I~₂¼¹&₁I₂; I~₃¼¹&₂I₃;

~I₄¼¹&₂I₄; I~₅¼¹&₃I₅; I~₆¼¹&₃I₆;

~I₇¼ ðÞ¹I_^7; I~₈¼ ðÞ¹I_^8;

(2.14)

such that

e^aI_aþeⁱI_i¼e~^{a^}I~_a^ þe~⁸I~₈; (2.15) and therefore the rescaled coframe fields f~e^{a^}g and the rescaled connection one-form~e⁸ have the form

~

e¹¼&¹₁ e¹; e~²¼&¹₁ e²; ~e³¼&¹₂ e³; e~⁴¼&¹₂ e⁴; e~⁵¼&¹₃ e⁵; ~e⁶¼&¹₃ e⁶;

~

e⁷¼e^{^7}; ~e⁸¼e^{^8}:

(2.16)

Here we introduced real parameters

&₁; &₂; &₃; 2R: (2.17) As a metric onX_k;lwe take

d~s²₇ ¼_a^b^e~^{a^}~e^{b^}: (2.18) One can show that for any given relatively prime integersk, lone can choose parameters&and(¼1, 2, 3) such that the metric (2.18) will be Einstein for a connection with a torsion 3-form

c ^{¼ 1}

3!c_a^b^c^e~^{a^}^~e^{b^}^e~^{c^} (2.19) having the following nonvanishing components:

c135¼c425¼c416¼c326¼c127¼c347¼c567¼1:

(2.20) Furthermore, this connection has the holonomy groupG₂ and the 3-form (2.19) defines aG₂structure onX_k;l[18,19].

For more details on the geometry of Aloff-Wallach spaces see e.g. [18–21].

Complex basis on TðX_k;lÞ.—Note that X_1;1 can be fibered over the homogeneous manifold F3¼ SUð3Þ=Uð1ÞUð1Þwith fibers

Uð1Þ^?¼expð~I₇Þ (2.21) parametrized by an angle02. So, fork¼l¼1 we have a projection

X_1;1!F3; (2.22)

whose fibers Uð1Þ^? are orthogonal complements of Uð1Þ ¼Uð1Þ_1;1 from (2.5), (2.6), and (2.7) in the torus T²ffiUð1Þ Uð1Þ [the Cartan subgroup of SU(3)]. This case is very special sinceX_1;1 is an Einstein-Sasaki mani- fold and therefore the coneCðX_1;1Þwith the metric

ds²₈ ¼dr²þr²ds²_X1;1 (2.23) is a Calabi-Yau 4-conifold with the holonomy group¹ SUð4Þ spinð7Þ. Furthermore, onX_1;1there exists a metric such that X_1;1 becomes a 3-Sasakian manifold with a hyper-Ka¨hler structure Sp(2) on the coneCðX_1;1Þ.

Recall that F3 is fibered over the projective plane CP² ffiSUð3Þ=Uð2Þ,F3 !CP², and the same is true for X_k;l with any k and l. One can show that fibers of the projection X_k;l!CP² are lens spaces S³=Zp with p¼ jkþlj. For clarity, let us combine all the above fibrations into one diagram

whereX_k;lcan also be fibered overF3 ifk¼l¼1.

Note thatS³=Zpis anS¹-fiber bundle overCP¹and one can consider complex forms which span CP² andCP¹ in X_k;las seen from (2.24). Namely, let us introduce complex one-forms²

~¹:¼~e¹þi~e²; ~²:¼e~³þi~e⁴; ~³¼~e⁵þi~e⁶ ~¹:¼~e¹i~e²; ~²:¼e~³i~e⁴; ~³¼~e⁵i~e⁶;

(2.25) plus reale~⁷,~e⁸ and matrices

~I₁ :¼¹₂ð~I₁iI~₂Þ; I~₂ :¼¹₂ðI~₃iI~₄Þ; I~₃ :¼¹₂ðI~₅i~I₆Þ;

~I^þ₁ :¼¹₂ð~I₁þiI~₂Þ; I~^þ₂ :¼¹₂ðI~₃þiI~₄Þ; I~^þ₃ :¼¹₂ðI~₅þi~I₆Þ;

iI~₇;iI~₈; (2.26) which form a basis of the complex Lie algebraslð3;CÞ ¼ suð3Þ C. Their explicit form is

I~₁ ¼&₁ 0 0 0 0 0 0 1 0 0 0 BB

@ 1 CC

A; I~^þ₁¼&₁

0 0 1 0 0 0 0 0 0 0 BB

@

1 CC A;

I~₂ ¼&₂ 0 1 0 0 0 0 0 0 0 0 BB

@ 1 CC

A; I~^þ₂¼&₂

0 0 0 1 0 0 0 0 0 0 BB

@

1 CC A;

I~₃ ¼&₃ 0 0 0 0 0 0 0 1 0 0 BB

@ 1 CC

A; I~^þ₃¼&₃

0 0 0 0 0 1 0 0 0 0 BB

@

1 CC A;

iI~₇¼ 2

lk 0 0 0 l 0 0 0 k 0

BB

@

1 CC

A; iI~₈¼ 2 ffiffiffi

3 p

kþl 0 0 0 k 0 0 0 l 0

BB

@

1 CC A: (2.27)

1Recall that the coneCðX_k;lÞover the general Aloff-Wallach spaceXk;lhas the holonomy group spin(7).

2Here, ~^1;2 span the CP² base in (2.24) and ~³ spans the CP¹,!S³=Zp, and the choice of the sign in~³is such that an associated almost complex structure on a six-dimensional sub- bundle of the tangent bundle, defined by~,¼1, 2, 3, will be integrable. For ~³¼e~⁵þi~e⁶ it will never be integrable. For k¼l¼1 our choice corresponds to a Ka¨hler structure onF3

and~³¼~e⁵þi~e⁶ corresponds to a nearly Ka¨hler structure on F3 [9,23].

We have the commutation relations

½i~I_j;I~ ¼C~_jI~; ½i~I_j;I~^þ ¼C~_jI~^þ

;

½I~;I~ ¼C~I~; ½I~^þ;~I^þ

¼C~

I~^þ;

½I~;I~^þ ¼C~^jði~I_jÞ þC~I~ þC~I~^þ;

(2.28)

with

C~¹₃₂¼ &₂&₃

&₁ ¼ C~¹₂₃; C~²₃₁¼&₃&₁

&₂ ¼ C~₁₃² ; C~³₁₂¼&₁&₂

&₃ ¼C~³_{1 2}; C~¹₇₁¼ 2

ð2klÞ ¼ C~₇₁¹ ; C~²₇₂¼ 2

ð2lkÞ ¼ C~₇₂² ; C~³₇₃¼ 2

ðkþlÞ ¼ C~₇₃³ ; C~¹₈₁¼ 2

ffiffiffi3

p ðkþ2lÞ ¼ C~₈₁¹ ; C~²₈₂¼ 2

ffiffiffi3

p ð2kþlÞ ¼ C~₈₂² ; C~³₈₃¼ 2

ffiffiffi3

p ðklÞ ¼ C~₈₃³ ; C~⁷₁₁¼ &²₁ k;

C~⁷₂₂¼ &²₂

l; C~⁷₃₃¼ &²₃

ðkþlÞ; C~⁸₁₁¼ ffiffiffi3 p &²₁

l;

C~⁸₂₂¼ ffiffiffi3 p &²₂

k; C~⁸₃₃¼ ffiffiffi3 p &²₃

ðlkÞ:

(2.29)

Note that standard undeformed structure constants correspond to k¼l¼1, ¼¹, &₁ ¼&₂¼,

&₃¼ ffiffi

2

p , and they are given by C¹

32¼ 1 ffiffiffi2

p ¼ C¹

23; C²

31¼ 1 ffiffiffi2

p ¼ C²

13; C³₁₂¼ ffiffiffi

2 p

¼C³_{1 2}; C¹₇₁¼C²₇₂¼ C¹

71¼ C²

72¼; C³₇₃¼2¼ C³

73; C¹₈₁¼ 1 2¼ C¹

81; C²₈₂¼1

2¼ C₈₂² ; C⁷₁₁¼C⁷₂₂¼C⁷₃₃¼ 1 2; C⁸

11¼1 4¼ C⁸

22:

(2.30)

In the new basis the Maurer-Cartan equations (2.8) become

d~¼ iC~_je~^j^~¹₂C~~^~C~~^~; d~¼ iC~_je~^j^~¹₂C~~^~C~~^~;

d~e^j¼iC~^j~^~; (2.31) where we have used the structure constants from (2.29).

The metric onX_k;lin terms of~ande~⁷ is

d~s²₇ ¼~~þ ð~e⁷Þ²; (2.32) i.e. we have

g ¼¹₂; g₇₇¼1: (2.33) Coset space X_1;1.—It is known (see e.g. [19]) that for k¼l¼1the Aloff-Wallach space admits a metric such that the cone CðX_1;1Þ over it admits metrics with the holonomy group SUð4Þ spinð7Þ (Calabi-Yau 4-fold) andSpð2Þ SUð4Þ spinð7Þ(hyper-Ka¨hler 4-fold). This means that in the Calabi-Yau case onCðX_1;1Þthere exists a closed (1,1)-form !^1;1 (Ka¨hler form) and in the hyper- Ka¨hler case onCðX_1;1Þthere exist three Ka¨hler forms:

!₃¼!^1;1; !₁; and !₂; (2.34) i.e. besides the closed form !^1;1, we also have a closed (2,0)-form!^2;0 :¼!₁þi!₂.

For the general metric (2.23) onCðX_1;1Þ, one can intro- duce the (1,1)-form as

!^1;1:¼i

2r²ð~^~ þ~⁴^~⁴Þ; (2.35) where

~⁴ :¼dr

r i~e⁷; ~⁴:¼dr

r þi~e⁷: (2.36) We obtain

2id!^1;1¼ ð2&²₁Þ~¹¹^rdrþð2&²₂Þ~²²^rdr þð1&²₃Þ~³³^2rdr

þr² &₂&₃

&₁ þ&₃&₁

&₂ &₁&₂

&₃

ð~¹²³~^{1 23}Þ;

(2.37) where ~:¼~^~, etc. From (2.37) it follows that

!^1;1 is closed if

&²₁¼&²₂¼2&²₃ ¼2²; ¼ 1

² (2.38)

for any real number.

SU(4)- and Sp(2)-holonomy on CðX_1;1Þ.—Note that the closure of the form !^1;1 means that the holonomy group of the cone CðX_1;1Þ reduces to the group U(4) (Ka¨hler structure). For having on CðX_1;1Þ a Calabi-Yau structure [SU(4) holonomy] one should impose an additional con- dition of closure of the (4,0)-form

^4;0 :¼r⁴~¹^~²^~³^~⁴: (2.39) By differentiating (2.39), from the condition d^4;0 ¼0 one obtains

¼ þ1 or ¼ 1 (2.40)

that fixes a Calabi-Yau metric on CðX_1;1Þ. Both from (2.40) correspond to the same metric onX_1;1and the choice of different sign ofcorresponds to the choice of different orientation on X_1;1.

Now we want to check whether this metric allows fur- ther reduction of the structure group SU(4) to the group

Spð2Þ SUð4Þ spinð7Þ, i.e. allows an introduction of a hyper-Ka¨hler structure on CðX_1;1Þ. On the Calabi-Yau spaceCðX_1;1Þ, we consider the (2,0)-form

!^2;0¼r²ð~¹^~²þ~³^~⁴Þ; (2.41) where is a complex number. Then from the equation d!^2;0 ¼0we obtain

¼¼ 1: (2.42)

Therefore, the metric with¼ 1from (2.40) allows a hyper-Ka¨hler structure³ on the cone CðX_1;1Þ and a 3-Sasakian structure onX_1;1.

III. SPIN(7) INSTANTONS

G₂instantons and gradient flows.—Consider the Chern- Simons–type functional onX_k;l,

S¼ 1 4 Z

Xk;l

trðF^FÞ^c

¼ 1 4 Z

Xk;l

trðA^dAþ2

3A^A^AÞ^dc 1

4 Z

X_k;l

dðtrðA^dAþ2

3A^A^AÞ^c^{Þ; (3.1)} where A is a connection on a rank-3 complex vector bundle over X_k;l [we will specialize to the gauge group SU(3) in a moment] andF ¼dAþA^A is its cur- vature. For the variation of (3.1) we have

S

A

^ a

¼1

2 ðdc ^{^}FÞ_a^ ¼1

2c_a^b^c^Fb^c^; (3.2) where is the Hodge operator and is some coefficient which can be calculated. Here, we used the fact that dc c ^{) d}c c ^{on X}k;l. Therefore, the equations of motion are

dc^{^}F ¼0,c^cF ¼0,ca^b^c^Fb^c^ ¼0: (3.3) Note that (3.3) are exactly G₂-instanton equations on X_k;l. Now we can define the Chern-Simons gradient-flow equations

A_ a^ ¼¹

S

A

^ a

¼1

2ca^b^c^Fb^c^; (3.4) whose stable pointsA_ :¼^dA_d ¼0areG₂instantons onX_k;l.

Spin(7)-instanton equations on RX_k;l.—On the one hand, (3.4) is the flow equation. On the other hand, it is exactly the spin(7)-instanton equation

F0 ^a¼¹₂ca^b^c^Fb^c^ (3.5) on the spaceRX_k;l,2R, in the gaugeAA0¼0, wherex⁰,d¼e~⁰.

So, let us considerA,F 2suð3Þ, and Eq. (3.5) on the spaceRX_k;l. The SU(3)-equivariantAnsatzforAis

A¼X_a^e~^{a^}þI~₈~e⁸ ¼Y~þY~þX₇e~⁷þI~₈~e⁸;

A0 ¼0; (3.6)

with the following restrictions which guarantee the SU(3) equivariance:

½iI~₈;Y ¼C~₈Y; ½i~I₈;Y ¼C~₈ Y; ½I~₈;X₇ ¼0:

(3.7) Here

Y₁:¼¹₂ðX₁iX₂Þ; Y₂ :¼¹₂ðX₃iX₄Þ;

Y₃:¼¹₂ðX₅iX₆Þ; Y ¼ Y^y

(3.8) are some33complex matrices depending on2R,, ¼1, 2, 3.

For (3.6) and (3.7) we have

F¼Y_e~⁰^~þY_e~⁰^~þX_₇e~⁰^~e⁷þ¹₂ð½Y;Y C~YÞ~^~þð½Y;YC~

YC~

Y þiC~⁷X₇þiC~⁸I~₈Þ~^~þ¹₂ð½Y;Y C~YÞ~^~þð½Y;X₇þiC~₇YÞ~^e~⁷ þð½Y;X₇þiC~₇ YÞ~^e~⁷; (3.9) whereY_:¼dY=d, etc. We get

F0¼Y_; F0 ¼Y_; F07¼X_₇;

F¼ ½Y;YC~Y; F¼ ½Y;YC~Y; F¼ ½Y;YC~YC~YþiC~⁷X₇þiC~⁸I~₈;

F7¼ ½Y;X₇þiC~₇Y; F7 ¼ ½Y;X₇þiC~₇ Y: (3.10) Reduction to matrix equations.—Note that

c^¼1

3!c_a^b^c^e~^{a^b^c^}

¼c123~¹²³þc1 23~^{1 23}þ1

2!c7e~⁷^~; (3.11) and therefore

3Comparing with the standard expression for the symplectic form in Darboux coordinates, the careful reader might notice an unconventional relative sign appearing in (2.41) for the choice ¼ 1. To arrive at the standard expression!^2;0¼~¹^~²þ ~³^~⁴, which is unique up to an overall rescaling, one needs to absorb the sign by replacing ~⁴ with minus itself in the definition (2.36). This has no further consequences except for an irrelevant overall sign flip in (2.39) corresponding to the change of orientation.

c123¼c1 23¼ ¹₂; c711¼c722¼i

2; c733¼ i 2: (3.12) Thus, from (3.5) we have

2F0¼ 2cFþcFþ2c7 F7; (3.13) 2F0 ¼2c Fþc Fþ2c7 F7 ; (3.14) 2F07¼c7 ^ab^F_a^b^ ¼ 2c7F: (3.15) Substituting (3.10), (3.11), and (3.12) into (3.13), (3.14), and (3.15), we obtain the following matrix equations:

2 _Y₁¼ ðC~¹₂₃þC~₇₁¹ ÞY₁þ ½iX₇; Y₁ ½Y₂; Y₃; (3.16) 2 _Y₂¼ ðC~²₃₁þC~₇₂² ÞY₂þ ½iX₇; Y₂ ½Y₃; Y₁; (3.17) 2 _Y₃¼ ðC~³_{1 2}C~₇₃³ ÞY₃ ½iX₇; Y₃ ½Y₁; Y₂; (3.18) 2 _X₇ ¼C~⁸I~₈þC~⁷X₇ið½Y₁; Y₁ þ ½Y₂; Y₂ ½Y₃; Y₃Þ;

(3.19) where

C~⁸:¼C~⁸₁₁þC~⁸₂₂C~⁸₃₃; C~⁷ :¼C~⁷₁₁þC~⁷₂₂C~⁷₃₃: (3.20) All structure constants in (3.16), (3.17), (3.18), (3.19), and (3.20) can be taken from (2.29). The above matrix equations can be written concisely by means of a ‘‘super- potential’’Wvia

Y_ ¼ @W

@Y; X_₇¼ @W

@X₇: (3.21) The explicit form of the superpotential WðY₁; Y₂; Y₃; Y₁; Y₂; Y₃; X₇Þ,

W¼¹₂trfðC~₂₃¹ þC~₇₁¹ÞY₁Y₁þðC~²₃₁þC~₇₂² ÞY₂Y₂ þ ðC~³_{1 2}C~₇₃³ÞY₃Y₃½Y₂;Y₃Y₁½Y₃;Y₁Y₂ þið½X₇;Y₁Y₁þ½X₇;Y₂Y₂½X₇;Y₃Y₃Þ

þC~⁸I~₈X₇þ¹₂C~⁷ðX₇Þ²g; (3.22) follows by inspection of (3.16), (3.17), (3.18), (3.19), and (3.20). It can also be obtained directly by inserting the Ansatz(3.6) into the Chern-Simons–type action (3.1).

Reduction to equations on scalar fields of.—The SU (3)-equivariance conditions (3.7) are solved by

Y₁¼¹I~₁; Y₂¼²I~₂; Y₃¼³I~₃; Y₁¼¹~I₁; Y₂¼²I~₂; Y₃¼³I~₃; X₇¼ ⁷I~₇þ ⁸I~₈; (3.23) where(¼1, 2, 3) are complex scalar fields depend- ing on, and ⁱ(i¼7, 8) are real scalar fields of.

Substituting (3.23) into (3.16), (3.17), (3.18), and (3.19), we obtain

2 _¹¼ ðC~₂₃¹ þC~₇₁^{1 7}C~₇₁^{1 8}C~₈₁¹ Þ¹C~₂₃^{1 23}; 2 _²¼ ðC~²₃₁þC~₇₂^{2 7}C~₇₂^{2 8}C~₈₂² Þ²C~²₃₁¹³; 2 _³¼ ðC~³_{1 2}C~₇₃³ þ ⁷C~₇₃³ þ ⁸C~₈₃³ Þ³C~³_{1 2}¹²;

2 _⁷¼C~^{7 7}C~⁷₁₁j¹j²C~⁷₂₂j²j²þC~⁷₃₃j³j²; 2 _⁸¼C~⁸þC~^{7 8}C~⁸₁₁j¹j²C~⁸₂₂j²j²þC~⁸₃₃j³j²;

(3.24) whereC~⁷andC~⁸are given in (3.20). The superpotentialW becomes

2W¼ &²₁ðC~₂₃¹ þC~₇₁¹ Þj¹j²&²₂ðC~²₃₁þC~₇₂² Þj²j²

&²₃ðC~³_{1 2}C~₇₃³ Þj³j²þ&₁&₂&₃ð¹²³þ¹²³Þ þð&²₁C~₇₁¹ j¹j²þ&²₂C~₇₂²j²j²&²₃C~₇₃³ j³j²Þ ⁷ þð&²₁C~₈₁¹ j¹j²þ&²₂C~₈₂²j²j²&²₃C~₈₃³ j³j²Þ ⁸ C~⁸K_8i ⁱ¹₂C~⁷K_ij ^{i j}; (3.25) where K is the Killing metricKðI; JÞ ¼ trðI; JÞ for the rescaled generators (2.27). The necessity to introduceKis due to the fact that ~I₇ andI~₈ are not mutually orthogonal for general values of k and l. The explicit form of K is given by

K¼ trðI~I~Þ ¼&² ðno sum overÞ;

K₇₇¼ trðI~₇I~₇Þ ¼8ðk²klþl²Þ ²² ; K₈₈¼ trðI~₈I~₈Þ ¼8ðk²þklþl²Þ

3²² ; K₇₈¼ trðI~₇I~₈Þ ¼ 4ðklÞðkþlÞ

ffiffiffi3 p ²²

(3.26)

with all other components vanishing. We are now in a position to express the first-order equations (3.24) in terms of the superpotentialW,

_ ¼ K @W

@; _ⁱ¼ K^ij@W

@ ^j: (3.27) The nonvanishing components of the inverse Killing metric are given by

K ¼&² ; K⁷⁷¼ ⁴ 4 K₈₈; K⁸⁸¼ ⁴

4 K₇₇; K⁷⁸¼⁴ 4 K₇₈;

(3.28)

such thatKK¼,KK¼, andK^ijK_jk¼ⁱ_k. Equation (3.24) is a complicated set of coupled, non- linear first-order ordinary differential equations and finding the general solution is a formidable task. Instead, one may

consider simplifications of these equations by setting some of the fields to zero and hoping to find explicit solutions for these special cases. Indeed, Eq. (3.24) admits a particularly simple yet important special solution, namely,

¹¼²¼³ ¼ ⁷ ¼0;

8ðÞ ¼ 8>

<

>: ^C~_~⁸

CþAexp C~⁷

2

; ifC~⁷ 0;

C~⁸

2 þB; ifC~⁷ ¼0;

(3.29)

whereA,B2Rare constants of integration. ForC~⁷0, A¼0, this solution is stationary and corresponds to the Abelian (rescaled, if C~⁸ 0) canonical connection on a line bundle overX_k;l. This is arguably the simplest example for a G₂ instanton on Aloff-Wallach spaces. A similar conclusion also holds forC~⁷ ¼C~⁸¼0with the rescaled canonical connection corresponding to the caseB0.

Before specializing to k¼l¼1, we briefly mention that the second-order equations of motion and the potential V for the scalar fields can be obtained straightforwardly from the above first-order equations by simply applying another time derivative to (3.27). The result can be written as

€¼K @V

@; €ⁱ¼K^ij@V

@ ^j: (3.30) The potential V is determined by the usual formula in terms of the superpotential

V¼KWWþ¹₂K^ijW_iW_j; (3.31) where we introduced the shorthand notation W¼

@W=@, W¼@W=@, and W_i¼@W=@ ⁱ. Computing the gradient ofVyields

V¼KðWWþWWÞ þK^ijW_iW_j;

V_i¼KðW_iWþW_iWÞ þK^jkW_ijW_k: (3.32) From this and (3.31) we can read off that critical points of the superpotential are both zeros and critical points of the potential. On the other hand, the critical points of V fall into two categories: zero-energy ones (V¼0) and positive-energy ones (V >0). The former are precisely the critical points ofW, which will be studied further for the special case k¼l¼1 in the remainder of this section.

However, the positive-energy critical points of V do not correspond to critical points of W. Instead, for them the gradient ofWis a ‘‘zero eigenvector’’ of the Hessian ofW. They will not play a role in our analysis.

Specialization to k¼l¼1.—For the special case of X_1;1, with&andgiven in (2.38), from (3.24) we obtain

2 _¹¼ ð1þ ⁷ ffiffiffi 3

p 8Þ¹²³; 2 _²¼ ð1þ ⁷þ ffiffiffi

3

p 8Þ²¹³; _³¼ ðþ1 ⁷Þ³¹²;

2 _⁷¼ ⁷þ j¹j²þ j²j² j³j²; 2 _⁸¼ ⁸ ffiffiffi

3

p j¹j²þ ffiffiffi 3 p j²j²;

(3.33)

with ¼ 1 for the 3-Sasakian structure on X_1;1. The Killing metric in this case becomes diagonal with nonzero components

K₁₁¼K₂₂¼2K₃₃¼K₇₇¼K₈₈¼2: (3.34) The superpotential simplifies to

W¼ ð1Þðj¹j²þj²j²Þð1þÞj³j² þ¹₂ðð ⁷Þ²þð ⁸Þ²Þþð¹²³þ¹²³Þ ðj¹j²þj²j²j³j²Þ ⁷þ ffiffiffi

3

p ðj¹j²j²j²Þ ⁸

(3.35) and (3.33) may be written as

2 _¹¼ W₁; 2 _² ¼ W₂; 2 _³¼ 2W₃; 2 _⁷¼ W₇; 2 _⁸¼ W₈: (3.36) The superpotential (3.35) is invariant under globalUð1Þ Uð1Þtransformations of the form

ð¹; ²; ³Þ°ðeⁱ¹¹; eⁱ²²; eⁱ³³Þ with

₁þ₂₃ ¼0 mod 2: (3.37) Note that the phases of theonly enter in the cubic terms ð¹²³þ¹²³Þin the superpotential, which are thus proportional to cosðarg¹þarg²arg³Þ. The super- potential is extremized whenarg¹þarg²arg³ ¼0 or and, together with (3.37), this allows us to consider purely real fields when searching for extrema ofW. After fixing2R, there is a residual symmetry which acts by flipping the sign of any two of the three complex functions . Therefore, we can restrict ourselves not only to real fields but also take, for example,¹ and² non-negative when searching for extrema ofW.

In addition, there is a Z₂ symmetry which acts by interchanging¹and²accompanied by a sign flip of ⁸ ð¹; ²; ⁸Þ°ð²; ¹; ⁸Þ: (3.38) Explicit solutions fork¼l¼1.—We begin by finding the extrema of the superpotential (3.35). Making use of the argument given at the end of the previous section, we take all fields to be real and¹,²non-negative. We then need to solve the following equations in five real variables: