Supersymmetric solutions withG2 and Spin(7) structure
Kazuki Hinoue(D1)
Yukinori Yasui
Osaka City University
Research Group for Mathematical Physics
日露共同研究Mini-workshop

2013/3/23

Contents
1. Introduction
2. Construction of SUSY solution with G2-structure
3. Construction of SUSY solution with Spin(7)-structure
4. Summary

Generating solutions of type II supergravity

1. We find first order equations which are consistent toequations of motion for SUGRA by imposing someconditions. This equations generate SUSY solutions.
2. We solve the first order equations explicitly.

There is a way to lift the solution from 7-dim or 8-dim abelian
heterotic SUGRA to 10-dim type II SUGRA.
D.Martelli and J.Sparks (arXiv:1010.4031v2[hep-th])So, in order to generate type II solutions we will study
7 and 8 dimensional abelian heterotic supergravity.

The reason why we select 7 dimension and 8 dimension is that
7-dim manifolds may have G2 –structure and 8-dim manifolds
may have Spin(7)-structure.
Then, G2 and Spin(7) structures give strong restrictions so that
we may obtain the solution explicitly.
So, we will study 7-dim case and 8-dim case.

7-dim or 8-dim abelian heterotic SUGRA
Lagrangian

Where,Equations of motion

and.

Killing spinor equations

Bianchi identity in abelian heterotic supergravity

J.P.Gauntlett, D.Martelli, etal, CMP vol.247,issue2,pp421-445

・Defining equations of the G-structure

・Bianchi identity

Contents
1. Introduction
2. Construction of SUSY solutions with G2-structure
3. Construction of SUSY solutions with Spin(7)-structure
4. Summary

3

Define fundamental 3-form on R7 Ω0 = d𝑎123 − ∑𝑑𝑎𝑖 ∧ 𝛽𝑖Where,(𝑎1 , 𝑎2 , 𝑎3 ) is fiber coordinate on ∧− 2 𝑹4β𝑖 ∈ Γ(∧2− 𝑹𝟒 ) = 𝛼 ∈ Γ(∧2 𝑹𝟒 ) ∗ 𝛼 = −𝛼}Explicitly 𝛽𝑖 are given by
β1 = 𝑑𝑦12 − 𝑑𝑦34 , 𝛽2 = 𝑑𝑦42 − 𝑑𝑦13 , 𝛽3 = 𝑑𝑦14 − 𝑑𝑦23Thus Ω0 (3) is written by choosing basis on R73

Ω0 = 𝑑𝑥123 + 𝑑𝑥147 + 𝑑𝑥165 + 𝑑𝑥257 + 𝑑𝑥246 + 𝑑𝑥354 + 𝑑𝑥367
Exceptional Lie group G2 is defined as subgroup of GL(7;R)

M7 ; 7-dim manifold
Frame bundle over M7 ;Definition of G2-structure :G2-structure on M7 is principal subbundleofframe bundlewith fibre G2 .
Here,frame bundle
is principal fiber bundle
over M7 with fibre.

By choosing a suitable base it takes the following form:; orthonarmal base on tangent space on M7 .

More explicitly, G2 invariant 3-form
exists on M7
which has one to one correspondence with G2 –structure.
It is enough to study G2 invariant.

Our purpose :1. find first order equations of abelian heterotic theory.
2. find SUSY solutions.

We start with number 1.

When we impose（on conditions ;; affine connection with 3-form torsion ）we have

Here,Lee form

is defined by

When there is no torsion and Lee-form, the condition represents
G2 holonomy and the correspondence metric is Ricci-flat. So theabove condition is a generalization by torsion.
Such a generalization is natural because there is a candidate
for the torsion, 3-form flux.

We identify 3-form flux with 3-form torsion ;and exterior derivative of dilaton with Lee form ;First order equations

Generalized self dual equation

Consistency condition between flux
and Maxwell field and this isobtained from definition of H(3) ;If we solve first order equations under this identification, we canobtain SUSY solutions.

In order to solve them, we do cohomogeneity one ansatz .

first order equations result in ODEs.

First order equations

Equations of motion
consistent

Cohomogeneity one ansatz

Ordinary differential equations

SUSY solutions

Cohomogeneity one manifold admitting G2 -structure

A.Dancer and M.Y.Wang

We select case of
because the other cases are
similar to this one. So we consider the manifold ;We consider the follwing metric ansatz ;t : coordinate on R

We assume that F(2) is invariant underThus components of F(2) are dependent only t.
F(2) and H(3) are restricted byisometry .
.

andHere, the orthonormal base {𝒆𝝁 } is defined byThus first order equations consist of ODEs and algebraic equations.

First order equations reduce to ordinary differential equations

algebraic equations

It is shown numerically that regular solutions doesn’t exist , so
we consider ansatz,and.

After all, the equations reduce toSolutions of these equations automatically solves
equations of motionand Killing spinor exists on the back ground.

We impose boundary conditions at t=0 ;Then we have the following expansions near t=0 ;Each coefficients are determined by method of series expansion.
The series includes two free parameters, p and q.

If the expansion has analytic continuation up to large t region,
the expansion gives a regular metric.

singularity

SU(2) ×SU(2)S3 Bolt

SU(2) ×SU(2)tt00

Resolution of singularity at t=0

The left figure illustrates conical metrics and it has a singularity at t=0.
We resolve the singularity of the conical metric;

Figure illustrates regular metrics corresponding to the above
expansion; the metric collapses into S3 metric at t=0.
This is the reason that we imposed the boundary condition.

We give two examples by numerical calculation.

ALC(Asymptotically Locally Conical)

AC metric :ALC metric :AC(Asymptotically Conical)
forforWhen 3-form flux, Maxwell filed and dilaton exist,
AC solution doesn’t exist.

AC metric is a Ricci-flat metric with G2 holonomy.

We find the following exact solutions ;32

is a constant which represents radius of S3.

C is a positive constant .

General solution

We investigate general solutions by using ‘flow method’. Because
it is difficult to see the flows going to infinity, it is convenience tocompactify configuration space ;So, we will study flows on S4 (4-dimensional sphere).
In this purpose, we rewrite first ODEs.

For simplicity, we rewrite these equations the following ;Note thatis homogeneous and it means that

For arbitraryWe putand.
whereNote that

because of homogeneity.

We write.

The first order equations are regarded as the flow equations on S4by using the variable.

Then flow equations on S4 are（where,.

）AC

Ricci flat case

()

ACALC
Starting point

ALC(This figure represents the solutions as flows on S4.
・Arrows represent flows on S4. Stating point is given by zerothorder Taylor expansion.
・ALC solutions have two free parameters, and these parameters
give directions of flows.
・AC solutions have one parameter, and their flows are one
dimensional.
・Final points are stationary points of the flow equation.

)

The equation has discrete symmetry;
There are eight starting points, so AC solutions are totally 16.
Eight starting points on S4ACContents
1. Introduction
2. Construction of SUSY solution with G2-structure
3. Construction of SUSY solution with Spin(7)-structure
4. Summary

We will construct SUSY solutions of 8-dim abelian heterotic
SUGRA by using the similar way to case of 7 dimension.
First order equations

metric on;; principal SO(3) fiber bundle on M4
global curvature 2-form on

global 1-form on;;

4-form which has one to one correspondence with Spin(7)-structure :F(2) and H(3) are restricted by;

The first order equations reduce toWhere, we imposedin order to get regular solution.

solution,andsatisfy first order equations when there are no matter fields.

hypergeometric function

Also, matters areContents
1. Introduction
2. Construction of SUSY solution with G2-structure
3. Construction of SUSY solution with Spin(7)-structure
4. Summary

G2Spin(7)22stationary point on S42(up to discreetsym)

2(up to discreetsym)

AC metrics exist only
Ricci-flat case○○○(numerical)○(analytic)

Free parameters

Exact solution

summary
We constructed SUSY solutions on 7 and 8 dimensional
manifold;
They are G2 structure and Spin(7) structure, respectively.

・We would like to challenge to solve general solution in G2 caseas in Spin(7) case whose solution is written by Hyper-geometric function.

・We would like to understand the G2-structure associated withsupergravity from a half-flat SU(3)-structure.
・We would like to construct supersymmetric solutions in
non-abelian heterotic theory.