CC BY
38
UDC 530.1; 539.1
Instantons and Chern-Simons flows in six and seven dimensions
O. Lechtenfeld
Institut für Theoretische Physik and Riemann Center for Geometry and Physics, Leibniz Universität Hannover
E-mail: [email protected]
The existence of K-instantons on a cylinder M7 = RT x K over a homogeneous nearly Kahler 6-manifold H requires a conformally parallel or a cocalibrated G2-structure on M7. The generalized anti-self-duality on M7 implies a Chern-Simons flow on KH which runs between instantons on the coset. For K-equivariant connections, the torsionful Yang-Mills equation reduces to a particular quartic dynamics for a Newtonian particle on C. We obtain kink- or bounce-type solutions for generic values of the torsion. When the latter corresponds to the conformally parallel or cocalibrated G2-structure on M7, the dynamics follows from a gradient or hamiltonian flow, respectively, and we encounter Yang-Mills instantons.
Keywords: instantons, Chern-Simons Sow, special geometry, G-structures, nearly-Kä,hler manifolds.
1 Introduction
Yang-Mills instantons exist dimensions d larger than four only when there is additional geometric structure on the manifold Md (besides the Riemannian one). In order to formulate generalized first-order anti-selfduality conditions which imply the second-order Yang-
Md
equipped with a so called G-structure, which is a globally defined but not necessarily closed (d-4)-form E,
Md
Instanton solutions in higher dimensions are rare in the literature. In the mid-eighties, Fairlie and Nuyts and also Fubini and Nicolai discovered the Spin(7)-instanton on R8. Eight years later, a similar G2-instanton on R7 was found by Ivanova and Popov and also by Günaydin and Nicolai. Our recent work shows that these so called octonionic instantons are not isolated but embedded into a whole family living on a class of conical non-compact manifolds [1].
The string vacua in heterotic flux compactifications contain non-abelian gauge fields which in the supergravity limit are subject to Yang-Mills equations with torsion H determined by the three-form flux. Prominent cases admitting instantons are AdSio-d x Md, Md G G
ing SU(3), G2 or Spin(7) for d = 6, 7 or 8, respectively. Homogeneous nearly Kâhler 6-manifolds H (iterated) cylinders and (sine-)cones over them provide simple examples, for which all K-equivariant Yang-Mills connections can be constructed [2,3]. Natural choices for the gauge group are K or G.
Clearly, the Yang-Mills instantons discussed here serve to construct heterotic string solitons, as was first done in 1990 by Strominger for the gauge five-brane. It is therefore of interest to extend our new instantons to solutions of (string-corrected) heterotic supergravity
and obtain novel string/brane vacua [4-6].
In this talk, I present the construction for the simplest case of a cylinder over a compact homogeneous nearly Kahler coset H which allows for a conformally parallel or a cocalibrated ^-structure. I display a family of non-BPS Yang-Mills connections, which contain two instantons at distinguished parameter values corresponding to those G2-structures. In these two cases, anti-self-duality implies a Chern-Simons flow on K.
Finally, I must apologize for the omission - due to page limitation - of all relevant literature besides my own papers on which this talk is based. The reader can find all references therein.
2 Self-duality in higher dimensions
The familiar four-dimensional anti-self-duality condition for Yang-Mills fields F may be generalized to dM
*F = —E A F with E G Ad-4(M)
2 ( ) (1) for F = dA + A A A G A2(M) ,
if there exists a geometrically natural (d—4)-form E on M. Applying the gauge-covariant derivative D = d + [A, •] it follows that
D*F + dE A F = 0 Yang-Mills with torsion H = *dE G A§(M) .
(2)
This torsionful Yang-Mills equation extremizes the action
SYM + Scs = I tr{F A *F + (—)d-3E A F A F}
JM
= i tr{F A *F + 2 dE A (A dA + § A3)} Jm
d G E cases
6 SU(3) w Kahler
6 SU(3) w nearly Kahler
7 G2 G2
7 G2 G2
7 G2 G2
8 Spin(7) E parallel Spin (7)
example
structure
CP §
g6 ___ G2
SU(3)
RT x nearly Kahler Xk.
SU(§)
U(1)fc,£ cone(nearly Kahler)
dw = 0
dw ~ ImQ, dReQ ~ w2
d-0 ~ ^Adr, d*-0 ~ —*-0Adr d-0 ~ ^ d*-0 = 0 d-0 = 0 = d*-0
RT x parallel G2 dE = 0, *E = E
Table 1: Examples of G-structure manifolds at d = 6, 7, 8.
Related to this generalized anti-self-duality is the gradient Chern-Simons flow on M,
— = —r$cs = *(dS A F) ~ *dS j F . (4)
dr oA
In fact, this equation follows from generalized anti-selfduality on the cylinder M = RT xM over M (in the AT =0 gauge).
The question is therefore: Which manifolds admit a global (d-4)-form? And the answer is: G-structure manifolds, i.e. manifolds with a weak special holonomy. Some of the key cases we shall encounter are listed in Table 1. For this talk I shall consider (reductive non-symmetric) coset spaces M=K in d=6 as well as cylinders and cones over them. In all these cases, the gauge group is chosen to be K.
3 Six dimensions: nearly Kahler coset spaces
All known compact nearly Kahler 6-manifolds M6 are nonsymmetric coset spaces K/H:
S6 = G2 Sp(2) SU(3) CT T(9)yQTT(9)
S = SU(3) ’ Sp(1)xU (1) , U(1)xU (1) , D U(2)xDU(2) .
(5)
The coset structure H < K implies the decomposition
Lie(K) = k = h ® m , h = Lie(H) , [h, m] C m . (6)
Interestingly, the reflection automorphism of symmetric spaces gets generalized to a so called tri-symmetry automorphism S : K ^ K with S3 = id implying s : k ^ k with
s|h = 1 and s|m = -^ ^ J = exp{21J} , (7)
effecting a 231 rotation on TM6. I pick a Lie-algebra basis
{1a=1,...,6 , 1i=7,...4mG} , [1a,16]= fOlb1i + fab^c , (8)
involving the structure constants fb. The Cartan-Killing form then reads
(-, •){ = -tr{(ad(0 oad(^) = 3 (-, % = 3 (•, •)m = 1 .
1also known as ‘hermitian Yang-Mills equations’
(9)
Expanding all structures in a basis of canonical one-forms ea framing T*(G/H),
g = Sab eaeb , W = 1 Jab ea A eb ,
2
Q = — yg (f + iJf ) abc ea A eb A e° ,
(Jab)
structure constants fabc rule everything.
Nearly Kahler 6-manifolds are special in that the torsion term in (2) vanishes by itself I What is more, this property is actually equivalent to the generalized anti-self-duality condition (1):
*F = — w A F 0 = dw A F ~ ImQ A F
(H)
^ DUY equations ,
where the Donaldson-Uhlenbeck-Yau (DUY) equations 1 state that
F2’0 = F0’2 = 0 mid wjF = 0 . (12)
Another interpretation of this anti-self-duality con-
F
eigenspace of the endomorphism *(w A •) with eigenvalue — 1, which contains the part of F1,1 orthogonal to w. The equations (11) imply also ReQ A F = 0 and the (torsion-free) Yang-Mills equations D*F = 0. Clearly, they seperately extremize both Sym and Ses in (3), but of course yield only BPS-type classical solutions. In components the above relations take the form
2 ^abcdef Fef J[abFcd] 0 fabcFbc (13)
^ WabFab = 0 , (Jf )abcFbc =0 , DaFab = 0 .(14)
I notice that each Chern-Simons flow Aa ~ fabcFbc on M6 ends in an instanton.
KA on M6. If I restrict their value to h the answer is
H
cal connection
Acan = ei Ii ^ Fcan = - 2 fab eaAeb Ii , (15)
where ei = eaea. Generalizing to ‘K-instantons’, I extend to the ansatz
A = ei Ii + ea $ab /b with ($ab) =: $ = (1I + (2 J ,
(16)
with one complex parameter ( = ( + i(2. This is in fact general for G2 invariance on S6. Its curvature is readily computed to
F _ F 1,1 | f2,0©0,2
Fab = Fab 2+ F“b. _2 (17)
= (|$|2-1) fab /i + [($2-$) f]abc /c
and displays the tri-symmetry under $ ^ exp{ 21 J}$• The solutions to the BPS conditions (11) are
$2 = $ ^ $ = 0 or $ = exp{ 2^ J} (18)
for k = 0,1, 2, which yields three flat K-instanton connections besides the canonical curved one,
A(k) = ei /i + ea (sk/)a and Acan = ei /i . (19)
4 Seven dimensions: cylinder over nearly
Kahler
Let me step up one dimension and consider 7-manifolds M7 with weak G2 holonomy associated with a G2-structure three-form Here, the 7 gener-
F
-1 eigenspace of *(^ A •), which is 14-dimensional and
G2
*F = -^ A F ^ A F = 0 ^ ^jF = 0 ,
(20)
providing an alternative form of the condition. In components, it reads
2 ^abcde/gFfg ^[abcFde] 0 ^abcFbc .
(21)
G2
vious accident (11) recurs,
d-0 ~ ^ d^AF = 0 ^ D*F = 0 ,
(22)
and the torsion decouples. Note that on a general weak G2-m^ifold there are two different flows (a G R),
dA^ = *d^ j F(a^d dA^ = ^ j f(a) , (23) da da
which coincide in the nearly parallel case. The second
M7
In this talk I focus on cylinders M7 = RT x H over nearly Kahler cosets, with a metric g = (dr)2 + 0ab eaeb. Note that these do not carry a parallel or G2
parallel one. I will study the Yang-Mills equation with a torsion given by
*% = 3K dw A dr ^ Tabc = K fabc (24)
with a real parameter k. We shall see that for special
K
D*F + 1 k dw A dr A F = 0 (25)
descends from an anti-self-duality condition (20).
Taking the Ao=0 gauge and borrowing the ansatz (16) from the nearly Kahler base, I write
Aa = ea/i + [$(T) /]„ ^ Foa = [$ /]„
O • — o ( )
and Fab =(|$|2-1) fab /i + [($2-$) f]abc /c
which depends on a function ((t) = ^1(r)+i(2(T) G C. Sticking this into (25) and computing for a while, one arrives at
( = (k-1)^ - (k+3)</>2 + 4((2 =: 1 I— . (27)
Nicely enough, I have obtained a (4 model with an action
S[(] ~ /dr {3|(|2 + V(()} for
Jr.
V(() = (3-k) + 3(k-1)|(|2 - (3+k)((3+(3) + 6|(|4
(28)
K=-3
ing tri-symmetry in the complex plane. It leads me to a mechanical analog problem of a Newtonian particle on C in a potential -V. I obtain the same action byplugging (26) directly into (3) with d£ = *% from (24).
For the case of K = S6 = gG(23); equation (27) proG
on RT x H- On sp(S1)xu(1) u(1)xu(1) ’ however, the
G
respective three complex functions of t. The corre-
C2 C3
of similar type but constrained by the conservation of Noether charges related to relative phase rotations of the complex functions [3].
Figure 1: Contour plots of the potential and straight kink trajectories.
5 Seven dimensions: solutions
Finite-action solutions require Newtonian trajectories between zero-potential critical points </>. With two exotic exceptions, dV(<) = 0 = V(<) yields precisely the BPS configurations on K:
• < = e2nik/3 with V(<) = 0 for all values of k and k = 0,1,2
• < = 0 with V(<) = 3—k vanishing only at k=3
Kink solutions will interpolate between two different critical points, while bounces will return to the critical starting point. Thus for generic k values one may have kinks of ‘transversal’ type, connecting two third roots of unity, as well as bounces. For k=3 ‘radial’ kinks, reaching such a root from the origin, may occur as well. Numerical analysis reveals the domains of existence in k:
Let me first discuss k=+3. For this value I find that
3¿ = % with
W
(29)
which is a gradient flow with a real superpotential W,
as
V = 6|f^|2 for K
+3 . (30)
It admits the obvious analytic radial kink solution,
((t) = e23k(2 ± 2 tanh ^75) . (31)
What is the interpretation of this gradient flow in terms of the original Yang-Mills theory? Demanding
G2
*% = d0, I am led to
0 = K w A dr + aim Q
=>
d0 ~ k toQ A dr
0 A dr (32)
k interval (-<x, -3] (-3, +3) +3 (+3, +5) a
types of trajectory radial bounce transversal kink radial kink radial bounce 0
Magic happens at three special values of k: At k=—3 rotational symmetry emerges; this is a degenerate situation. At k=—1 and at k=+3, displayed in Figure 1 above, the kink trajectories are straight lines, indicating integrability. Indeed, behind each of these two cases lurks a first-order flow equation, which originates from anti-self-duality and hence a particular G2-structure 0.
where a is undetermined. This is a conformally parallel G2-structure, and (20) quantizes the coefficients to and k=3, fixing (with eT = r)
0 = w A dr + ImQ = (r2w A dr + r3ImQ) = ^3 0cone ,
(33)
where I displayed the conformai relation to the parallel G2-structure on the cone over H.
Alternatively, with this G2-structure the 7 anti-selfduality equations (20) turn into
With the ansatz (26). the first relation is automatic, and the second one indeed reduces to (29). As a consistency check, one may verify that
i tr {w A F A F} « W(<) + 3 .
(35)
I now come to the other instance of straight trajec-K= 1
3<= f with
H - Uj~3 ' ^3
ô (< +< ) —
(36)
which is a hamiltonian flow (note the imaginary multiplier!), running along the level curves of the function H, that is identical to W. It has the obvious analytic transverse kink solution,
with coefficients k and a to be determined. It has not appeared in my table in Section 2, but obeys d * 0 = 0, which is known as a cocalibrated G2-structure. But can it produce the proper torsion,
d0AF ~ («ImQAdr+2awAw)AF = —ImQAdrAF ?
(39)
Employing the anti-self-duality with respect to 0,
*0 A F = 0 ^ w A w A F = 2 ImQ A dr A F , (40)
it works out, adjusting the coefficients to k=3 and a=—1. Hence, the cocalibrated G2-structure
0
w A dr — ReO
(41)
(37)
and its images under the tri-symmetry action.
G2
here? Let me try the other obvious choice,
is responsible for the hamiltonian flow.
To see this directly, I import (41) into (20) and get
0
and
A a
[J f ]abc Fbc . (42)
0 = K wAdr+aReO ^
d0 ~ KimQAdr+2a wAw , (38)
Again, the ansatz (26) fulfils the first relation, and the second one nicely turns into (36).
6 Partial summary
Let me schematically sum up the construction.
0 A F = —* F on R x K
Aa ^ fabc Fbc
j ansatz A = e®/j + ea[$ I ]a
^2< = ± TW
0 A F = — * F
Aa ^ [ Jf ]abc Fbc
j
W = 3 (<3 + <3) — |<|2 = H
F (t ) = dT Aea [$ /]a + 1 e“Ae^ (|$|2-1) fab /i + [($ 2-$) f ] abc /c }
are G2-instantons for "V'ang-Mills with torsion D *F + (*%) A F = 0 from S[A] = /Rx k tr{F A *F + 1 kw A dT A F A F} with k =+3 or -1
and obey gradient or hamiltonian flow equations for fK tr {w A F A F} « W(() + 3
2
Acknowledgement
The author thanks the organizors of ‘QFTG’12’ for a pleasant atmosphere. This work was partially sup-
ported by the Deutsche Forschungsgemeinschaft, the Cluster of Excellence EXC 201 ‘QUEST’, the Research Training Group GRK 1463, the Heisenberg-Landau program and Russian Foundation for Basic Research.
References
[1] Gemmer K. P., Lechtenfeld 0., Nölle C, Popov A. D. 2011 JHEP 1109 103.
[2] Harland D., Ivanova T. A., Lechtenfeld 0., Popov A. D. 2010 Commun. Math. Phys. 300 185.
[3] Bauer I., Ivanova T. A., Lechtenfeld 0., Lubbe F. 2010 JHEP 1010 44.
[4] Lechtenfeld 0., Nölle C., Popov A. D. 2010 JHEP 1009 074.
[5] Chatzistavrakidis A., Lechtenfeld 0., Popov A. D. 2012 JHEP 1204 114.
[6] Gemmer K. P., Haupt A., Lechtenfeld 0., Nölle C, Popov A. D., arXiv:1202.5046 [hep-th].
Received 01.10.2012
О. Лехтенфелъд
ИНСТАНТОНЫ И ПОТОКИ ЧЕРНА-САЙМОНСА В ШЕСТИ И СЕМИ
РАЗМЕРНОСТЯХ
Существование К-инстантонов на цилиндре M7 = RT х а ^ по однородному почти Кэлерову G-многообразию H требует конформно параллельной пли коградуируемой Сг-структуры на M7. Обобщенная антиавтодуальность на M7 предполагает поток Черна-Саймонса на Hg, который проходит между пнстантонамп на смежном классе. Для К-эквивариантных связностей, торсионные уравнения Янга-Миллса сводятся к уравнениям движения ньютоновской частицы в четвертичном потенциале на C. Мы получаем решения для общих значений кручения. Когда последний соответствует конформно параллельной пли коградуируемой Сг-структуре на M7, динамика следует из градиента или гамильтонова потока, соответственно, и мы сталкиваемся с инстантонами Янга-Миллса.
Ключевые слова: инстантоны, поток Черна-Саймонса, специальная геометрия, G-етруктуры, почти Кэлеровы многообразия.
Лехтенфельд О., профессор.
~Университет Лейбница, Институт теоретической физики г. Ганновер.
30167, Hannover, Германия.
E-mail: [email protected]
