Anomaly cancellations on heterotic 5-branes

( 前編 )

矢田　雅哉

contents

• Introduction• NS5-brane• Small instanton’s configuration• Type-I Heterotic duality• Summary

introductionHeterotic string naturally has internal gauge symmetry.

→This is preferable structure in phenomenology.

・ Anomaly free・ There are some S.M.like structure with compactification.

→5-brane couples to the dual,B6,of the 2-form B

Heterotic string has 2-form B in NS-NS sector

NS5-braneHeterotic string effective action for the bosonic field

SUSY 　 transformation 　 for 　 the 　 fermionic 　 fields

※Dirac matrix ※indices

=0…9(curved) =0…9(flat)

=6…9(curved)=0…5(curved)

[A.Strominger, J.A.Harvey, C.G.Callan Jr.,Nucl.Phys.B359(1991)]

Connection is given by

and

We will consider instanton solutions.So, we set

For simplicity we consider only the self-dual case.

Gauge solution [A.Strominger, Nucl.Phys.B343(1990)]

where

→ ・ Self-dual・ SU(2) subgroup of SO(4)

is a YM-instanton of scale size

This solution is valid when

Neutral solutionThis solution corresponding to a size instanton

is an integer.Only solution is reached as a limit of Gause solution as

⇒ later, we will explain…Near NS5-brane,the solution become wormhole throat.

→we can embed the connection in the gauge group.

⇒

This is SU(2) matrix that belongs to subgroup of SO(4)

generalized spin connection is

Now we calculate the connection…

And we recall the gauge field in gauge solutions…

Symmetric solution this solution embed the spin connection in the gauge group

Since the generalized connection is an SU(2) connection,the gauge field must lie in an SU(2) subgroup of E8 or SO(32).

⇒gauge symmetries spontaneous break

Wormhole throatFour-dimensional part of the metric

where

when , 1st term is not dominant

⇒wormhole throat

…Using spherical coordinates

Under the coordinate transformation

Small instanton’s configurationHere we consider instanton size → 0 case.In this case , gauge group is enhanced.

[E.Witten, Nucl.Phys.B460(1996)]

・ The Heterotic string on R6×K3

Supermultiplets are

(i) Graviton multiplet:(ii) Maxwell multiplet:(iii)Antisymmetric tensor multiplet:(iv)Hypermultiplet:

[P.K.Townsent ,Phys.lett.139B,num.4(1984)]

→Bosonic part of the hypermultiplet corresponding to instantons.

・ Hypermultiplet is on quaternionicmanifold →

・ Hypermultiplet originally has SU(2)R symmetry.

→

※The hypermultiplets transform as (2k,2) of

And the Gauge group that remains by Higgs mechanism

We treat the bosonic part of the hypermultiplet as…

※Here, the gauge group G generators are

We define the D-fields

the scalar potential

!The classical moduli space of vacua is obtained by setting V=0 anddividing the gauge group G.

→

…The gauge group G is remaining moduli space.

This　deg

ree　of　fr

eedom　wa

s

eaten　by

　the　massi

ve　vecto

r.

The moduli space will be singular when The unbroken gauge symmetry G is enhanced.

If there are k hypermultiplets,the dimension of moduli space is

We define the dimension of G is d:

One instanton Moduli spaceWhen an instanton shrinks to zero size…→we can simply think about the one instanton problem on R4

⇒ symmetric solution

We consider vacua for which the instanton are embedded in an SO(N)Subgroup of SO(32).

※ instanton’s position and scale size are embedded in SU(N)

The subgroup of SO(32) left unbroken by the instantons is SO(32-N)

unbroken

In symmetric solution, the instanton really has structure group K=SU(2)

⇒

The moduli space of these instantons has the dimension

center of mass decouples from the singularities...

The moduli space of instantons

Commute　with　K

Instanton’s　moduli　space

!known fact ; instanton shrinks to zero ⇒Full SO(N) is restored

if there are k hypermultiplets and the gauge group G has dimension d...

the most obvious way to obey the condition …

[C.G.Callan,Jr.,J.A.Harvey,A.Strominger, Nucl.Phys.B367(1991)]

・ k=N hypermultiplets form a representation of SO(N)

・ d=3 ⇒ G=SU(2) 　

This choice of gauge group and massless hypermultiplets lead to the correct moduli space.

small instantons have gauge symmetry

Type-I Heterotic duality

Instanton size:

Strong coupling with Heterotic stringS-duality:

↓↑

Weak coupling with type-I string

・ Mode expansion

N-N condition

D-D condition

N-D condition

・ Mass shell condition

NN,DD DN,NDBoson 1/24 -1/48Fermion(NS) 1/48 -1/24Fermion(R) -1/24 1/48

In light cone gauge…

Hyper multiplet

NS sector

We consider D5-D9 system

Using Fermion zero mode dm0(m=6 ～ 9) ⇒2 of SU(2)

→Scalar boson ; D=6 Lorentz group

R sector

Using Fermion zero mode dμ0(μ=2 ～ 5) ⇒2 of SU(2)

→Weyl spinor;D=6 Lorentz group

・ Gauge symmetry is introduced by Chan-Paton factor

summary

・ There instanton on the NS5-brane.

・ The isntanton has structure group SU(2).

・ An extra SU(2) gauge symmetry appears, when the instanton shrinks to size zero.

・ gauge group is enhanced in type-I string on D5-brane.