1. Masato Arai Czech Technical University in Prague
Walls in 3-d supersymmetric Kähler nonlinear sigma models on SO(2N)/U(N) and Sp(N)/U(N)
Masato Arai
Czech Technical University in Prague
Based on collaboration with Sunyoung Shin

2. Introduction – progress on domain wall
Soliton - Playing an important role in physics
Domain wall solution Brane world scenario
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3. Introduction – progress on domain wall
Soliton - Playing an important role in physics
Domain wall solution Brane world scenario
Massive hyper-Kähler nonlinear sigma model on
D=5 N=1 SUSY U(N) gauge theory coupled to M massive flavors including the Fayet-Iliopoulos term with infinite gauge coupling limits
number of discrete vacua
MA, Nitta, Sakai, 2003
Domain wall solutions interpolating them are derived based on the moduli matrix approach
Isozumi, Nitta, Ohashi, Sakai, 2004
The moduli space of the wall: Grassmannian
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4. Introduction : one of compact Hermitian symmetric spaces (HSS)
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5. Introduction : one of compact Hermitian symmetric spaces (HSS)
Question: How about a NLSM on ( is the HSS except )?
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6. Introduction : one of compact Hermitian symmetric spaces (HSS)
Question: How about a NLSM on ( is the HSS except )?
The moduli matrix approach can be used in SUSY gauge theory.
Need to describe a NLSM on as a SUSY gauge theory (with infinite gauge coupling limit)
Difficult to construct a SUSY gauge theory corresponding to a NLSM on
other than , for example,
(cf. MA, Nitta, 2006, MA, Lindström, Kuzenko, 2007)
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7. Introduction Simplifying setup
Observation: Cotangent part is irrelevant for vacua and domain walls in
Isozumi, Nitta, Ohashi, Sakai, 2004
The cotangent part can be dropped for investigation of vacua and walls.
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8. Introduction Simplifying setup
Observation: Cotangent part is irrelevant for vacua and domain walls in
Isozumi, Nitta, Ohashi, Sakai, 2004
The cotangent part can be dropped for investigation of vacua and walls.
The result is respected as one of the massive Kähler NLSM on
The situation would be the same for other NLSMs on ( is the HSS).
Base manifold part
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9. Purpose of our work
Domain walls in massive kähler NLSM on the complex quadric surface
in 3-dimensional space-time.
MA, Lee, Shin, 2009
Solutions of the BPS equations were obtained by the moduli matrix approach.
: Kähler potential of
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10. Purpose of our work
Domain walls in massive kähler NLSM on the complex quadric surface
in 3-dimensional space-time.
MA, Lee, Shin, 2009
Solutions of the BPS equations were obtained by the moduli matrix approach.
We consider domain walls for massive Kähler NLSM on
formulated by gauge theory.
: Kähler potential of
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11. Setup Starting with the massless NLSM on , in 4 dimensions.
Higashijima, Nitta, 1999
4D N=1 U(N) gauge theory coupled to 2N flavors including FI term (with )
: vector rep. of SO(2N) or Sp(N)
: symmetric (SO(2N))/anti-symmetric (Sp(N)) rep.
No kinetic term for gauge part and , whose eqs. of motion give constraints
(D-term)
(F-term)
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12. Setup Dimensional reduction Getting a non-trivial scalar potential
Cartan subalgebra of
Giving rise to discrete vacua
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13. Vacua
Vacuum condition
with constraints
by gauge trans.)
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14. Vacua Vacuum condition with constraints
Ex. SO(4)/U(2) & Sp(2)/U(2) case
by gauge trans.)
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15. Vacua Vacuum condition with constraints
Ex. SO(4)/U(2) & Sp(2)/U(2) case for SO(2N)/U(N) & Sp(N)/U(N)
by gauge trans.)
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16. BPS equation Bogomol’nyi completion BPS equation
Supposing a non-trivial configuration along direction,
with constraints
BPS equation
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17. Solving the BPS equation – moduli matrix approach
Rewriting the equations
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18. Solving the BPS equation – moduli matrix approach
Rewriting the equations
Solution
: integration constants called the moduli matrix
Constraints
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19. Solving the BPS equation – moduli matrix approach
Moduli matrix – key issue in solving the equation
It includes information of vacua, boundary conditions and positions of walls.
The rest of work is to choose them so that they satisfy
and appropriate boundary conditions at
determines
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20. Solution - N=2 case
SO(4)/U(2) case
4 discrete vacua
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21. Solution - N=2 case
SO(4)/U(2) case
4 discrete vacua
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22. Solution - N=2 case
SO(4)/U(2) case
4 discrete vacua
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23. Solution - N=2 case
SO(4)/U(2) case
Single walls
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24. Solution - N=2 case
SO(4)/U(2) case
Single walls
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25. Solution - N=2 case Expected result
Massive nonlinear sigma model on discrete vacua and 1 domain wall
Our result : 4 discrete vacua and 2 domain walls, double structure
(O(4) constraint)
Massless limit: (vacuum moduli)
are identified since P is a symmetry of the theory.
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26. Solution - N=2 case Expected result
Massive nonlinear sigma model on discrete vacua and 1 domain wall
Our result : 4 discrete vacua and 2 domain walls, double structure
(O(4) constraint)
Massless limit: (vacuum moduli)
are identified since P is a symmetry of the theory.
Massive case: P is not a symmetry mass term (Cartan subalgebra of SO(2N))
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27. Solution - N=2 case
Sp(2)/U(2) case
4 discrete vacua
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28. Solution - N=2 case
Sp(2)/U(2) case
Single wall
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29. Solution - N=2 case
Sp(2)/U(2) case
Double walls
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30. Solution - N=2 case
Sp(2)/U(2) case
Double walls
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31. Solution - N=2 case
Sp(2)/U(2) case
Double walls
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32. Solution - N=2 case
Sp(2)/U(2) case
Double walls
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33. Solution - N=2 case
Sp(2)/U(2) case
Double walls
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34. Solution - N=2 case Sp(2)/U(2) case Double walls Penetrable wall
Result is consistent
with
case (our previous work)
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35. Solution - N=3 case SO(6)/U(3) case
: Expected 4 discrete vacua discrete vacua
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36. Solution - N=3 case
SO(6)/U(3) case
single walls
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37. Solution - N=3 case SO(6)/U(3) case single walls Double structure:
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38. Solution - N=3 case
SO(6)/U(3) case
Double walls
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39. Solution - N=3 case Sp(3)/U(3) case 8 discrete vacua, 12 single walls
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40. Solution - N=3 case
Sp(3)/U(3) case
Double, triple walls
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41. Solution - N=3 case Sp(3)/U(3) case Double, triple walls
Penetrable like
Sp(2)/U(2) case
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42. Summary & Discussion
Investigating domain walls in massive Kähler NLSM on SO(2N)/U(N) and Sp(N)/U(N) in 3-dimensional space-time.
Showing discrete vacua in both models.
BPS wall solutions
Deriving BPS domain wall solutions up to N=3 case
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