1. Anomaly cancellations on heterotic 5-branes ( 前編 ) 矢田 雅哉

2. contents Introduction NS5-brane Small instanton’s configuration Type-I Heterotic duality Summary

3. introduction Heterotic 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,B 6,of the 2-form B Heterotic string has 2-form B in NS-NS sector

4. NS5-brane Heterotic 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)]

5. Connection is given by and We will consider instanton solutions. So, we set For simplicity we consider only the self-dual case.

6. 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

7. Neutral solution This 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.

8. →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…

9. 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

10. Wormhole throat Four-dimensional part of the metric where when, 1 st term is not dominant ⇒ wormhole throat …Using spherical coordinates Under the coordinate transformation

11. Small instanton’s configuration Here we consider instanton size → 0 case. In this case, gauge group is enhanced. [E.Witten, Nucl.Phys.B460(1996)] ・ The Heterotic string on R 6 ×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.

12. ・ 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…

13. ※ 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 and dividing the gauge group G. →

14. …The gauge group G is remaining moduli space. This degree of freedom was eaten by the massive vector. 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:

15. One instanton Moduli space When an instanton shrinks to zero size… →we can simply think about the one instanton problem on R 4 ⇒ 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

16. 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

17. !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

18. Type-I Heterotic duality Instanton size: Strong coupling with Heterotic string S-duality: ↓↑ Weak coupling with type-I string

19. ・ Mode expansion N-N condition D-D condition N-D condition ・ Mass shell condition

20. NN,DDDN,ND Boson1/24-1/48 Fermion(NS)1/48-1/24 Fermion(R)-1/241/48 In light cone gauge…

21. Hyper multiplet NS sector We consider D5-D9 system Using Fermion zero mode d m 0 (m=6 ～ 9) ⇒ 2 of SU(2) →Scalar boson ; D=6 Lorentz group R sector Using Fermion zero mode d μ 0 (μ=2 ～ 5) ⇒ 2 of SU(2) →Weyl spinor;D=6 Lorentz group ・ Gauge symmetry is introduced by Chan-Paton factor

22. 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.