Manifestly Supersymmetric Effective Actions on Walls and on Vortices - PowerPoint PPT Presentation

Manifestly Supersymmetric Effective Actions on Walls and on Vortices Keisuke Ohashi with M.Eto, Y.Isozumi, M.Nitta, N.Sakai Tokyo Institute of Technology based on hep-th/0502***, hep-th/0405194, hep-th/0404198

§ 1. Introduction & Motivation • brane world scenario 5D bulk It is interesting and important to study the fat brane scenario in a case that a codimension is 1(2), hidden sector that is, domain walls (vortices). our world ⇓ It is natural to consider domain walls (vortices) which are realized as 1/2 BPS states in a 5D(6D) SUSY theory. Therefore, it is important to investigate effective theories on domain walls (vortices), preserving the half super symmetry.

• Moduli Spaces Moduli spaces for 1/2 BPS states in non-Abelian gauge theory were determined by codim. instantons 4 ADHM monopoles 3 Nahm vortices 2 Hanany-Tong (domain-)walls 1 INOS • Effective actions on walls and on vortices We obtain formulas for effective actions on walls and on vortices in superfield formulation by Manton’s method. moduli parameters φ α → massless superfields on solitons φ α ( x µ , θ )

Contents § 1. Introduction & Motivation § 2. 1/2 BPS Wall Solutions and Their Moduli § 3. Manifestly Supersymmetric Effective Actions on Walls ( § 4. Manifestly Supersymmetric Effective Actions on Vortices ) § 5. Summary& Discussion

§ 2 1/2 BPS Wall Solutions and Their Moduli Space Phys.Rev.Lett 93(2004)161601[hep-th/0404198], hep-th/0405194 • Our model: 5D SUSY U ( N C ) gauge theory with N F ( > N C ) fundamental hypermultiplets Field contents (bosonic part): ( M, N = 0 , 1 , 2 , 3 , 4) Vector multiplet: gauge field W M , adjoint scalar Σ, Hyper multiplets: complex N C × N F matrix ( H i ) rA ≡ H irA , SU (2) R i = 1 , 2 , color r = 1 , · · · , N C , flavor A = 1 , 2 , · · · N F Our Lagrangian (bosonic part) � bosonic = − 1 2 g 2 Tr[( F MN ( W )) 2 ] + 1 g 2 Tr[( D M Σ) 2 ] � L � iAr D M H irA − V pot + ( D M H ) †

The scalar potential of this model V pot = g 2 �� � 2 � c a − ( σ a ) j i H i H † + H † iAr [(Σ − m A ) 2 ] r s H isA 4 Tr j Fayet-Illiopoulos parameter: c a = (0 , 0 , c > 0) non-degenerate masses m A : If m 1 > m 2 > · · · > m N F , then SU ( N F ) → U (1) N F − 1 • color-flavor locking vacua Vacua are labeled by � A 1 , A 2 , · · · , A N C � H 1 rA = √ cδ A r A , H 2 rA = 0 , Σ = diag( m A 1 , · · · , m A N C ) N F ! #vacua = N C !( N F − N C )! where U (1) N F − 1 → broken

For example, three vacua with N C = 2 , N F = 3 vacuum � 1 , 2 � � � � � H 1 = √ c 1 0 0 0 m 1 , Σ = 0 1 0 0 m 2 vacuum � 1 , 3 � � � � � H 1 = √ c 1 0 0 m 1 0 Σ = , 0 0 1 0 m 3 vacuum � 2 , 3 � � � � � H 1 = √ c 0 1 0 0 m 2 , Σ = 0 0 1 0 m 3

• Bogomol’nyi bound for walls with boundaries � A � at y = ∞ , and � B � at y = −∞ , E = (l.h.s of BPS eqs.) 2 + T wall   � ∞ N C N C � �  > 0 ≥ T wall = dy Tr[ ∂ y ( c Σ)] = c m A r − m B r  −∞ r =1 r =1 • 1/2 BPS equations for walls We find a set of BPS equations: ( M ) AB ≡ m A δ AB 0 = D y H 1 + Σ H 1 − H 1 M 0 = D y Σ − g 2 2 ( c − H 1 H 1 † ) we assume that solutions depend on only a coordinate x 4 = y , and for lorentz symmetry along the walls, W µ = 0 , ( µ = 0 , 1 , 2 , 3) .

• Solutions of the 1/2 BPS Eqs. for walls Phys.Rev.Lett 93(2004)161601[hep-th/0404198], hep-th/0405194 Σ + iW y ≡ S − 1 ∂ y S, W µ = 0 H 2 = 0 H 1 ( y ) = S − 1 ( y ) H 0 e My , with an arbitrary constant N C × N F matrix H 0 , and an S ( y ) ∈ GL( N C , C). ‘Master equation’ for a gauge invariant quantity Ω ≡ SS † y Ω − ( ∂ y Ω)Ω − 1 ( ∂ y Ω) = g 2 � † � ∂ 2 c Ω − H 0 e 2 My H 0 Physical fields Σ , W y , H 1 can be obtained by given H 0 , H 0 → Ω( y ) → S ( y ) → Σ , W y , H 1 H 0 parametrize the moduli space for walls.

The simplest example with N C = 1 , N F = 2 and M = diag( m, − m ) A solution with H 0 = √ c (1 , 1) in the strong coupling limit g 2 → ∞ Σ + iW y = m tanh(2 my ) H 1 = S − 1 H 0 e My � � e − my e my = √ c , � � cosh(2 my ) cosh(2 my ) � √ c (1 , 0) : vacuum � 1 � at y → ∞ → √ c (0 , 1) : vacuum � 2 � at y → −∞

• Total Moduli Space The toatal moduli space of Walls is the deformed complex Grassmann manifold. SU ( N F ) M total wall = G N F , N C ≃ SU ( N C × SU ( N F − N C ) × U (1)) dim M total wall = 2 N C ( N F − N C )  N C ( N F − N C ) : positions of walls   = + N F − 1 : NG modes   + ( N C − 1)( N F − N C − 1) : QNG modes Let us promote moduli parameters φ α → massless superfields on the walls φ α ( x µ , θ ) and obtain an effective acton on the walls.

Manifestly Supersymmetric § 3. Effective Action on (Multi-) Walls hep-th/0502 ∗∗∗ To obtain the effective action with manifest supersymmetry, let us consider superfield formulation respecting the unbroken half super- symmetry on the BPS walls. superfield respecting configurations for walls � ˆ H 1 ( x, θ ) θ =0 = H 1 ( x ) , Hypermultiplet → chiral : � � ˆ H 2 ( x, θ ) θ =0 = H 2 ( x ) chiral : � � ˆ 5D vector multiplet → chiral : Σ( x, θ ) θ =0 = Σ( x ) + iW y ( x ) , � � V ( x, θ, ¯ ˆ vector : θ ) θγ µ θ = W µ ( x ) , (WZ gauge) � ¯

5D Action in superfield formulation A.Hebecker Nucl. Phys. B 632, 101 (2002) � L w = dy L = − T wall � 1 � � V ˆ V ) 2 + 2 c ˆ 2 g 2 ( e − 2 ˆ D y e 2 ˆ dyd 4 θ Tr + V � � H 2 � V ˆ H 1 + ˆ V ˆ H † 1 e − 2 ˆ H † 2 e 2 ˆ ˆ dyd 4 θ Tr + � 1 �� � H 2 † � W α ˆ H 1 − ˆ 4 g 2 ˆ W α + ˆ D y ˆ ˆ dyd 2 θ H 1 M + + c . c . where T wall = [Tr( c Σ)] ∞ −∞ covariant derivatives V = ∂ y e 2 ˆ V + ˆ V + e 2 ˆ V ˆ D y e 2 ˆ Σ e 2 ˆ ˆ Σ H 1 = ∂ y ˆ D y ˆ ˆ H + ˆ Σ ˆ H 1

Manton’s Method (slow moving approximation) ∂ y φ = O (1) φ, ∂ µ φ = O ( λ ) φ, λ ≪ 1 , µ = 0 , 1 , 2 , 3 ⇒ For consistency with SUSY, we have to take rules, dθ ∼ ∂ 1 ∂θ ∼ O ( λ 2 ) By use of these rules, we can set ansatz for wall configularations cosis- tently. H 1 ∼ O (1) , H 2 ∼ O ( λ ) ˆ ˆ � � ˆ ˆ Σ ∼ O (1) , V ∼ O (1) , W µ ∼ O ( λ ) � 1 � � � � H 2 � W α ˆ V ˆ H † 2 e 2 ˆ 4 g 2 ˆ ˆ dyd 2 θ ∼ O ( λ 4 ) , dyd 4 θ Tr ∼ O ( λ 4 ) ⇒ W α Omitting O ( λ 4 ) terms, N = 2 theory is broken into N = 1 ⇔

L w = − T wall � 1 � � V ˆ V ) 2 + 2 c ˆ V ˆ 2 g 2 ( e − 2 ˆ D y e 2 ˆ H † 1 e − 2 ˆ V + ˆ dyd 4 θ Tr H 1 + � � H 2 † � �� H 1 − ˆ ˆ D y ˆ ˆ dyd 2 θ H 1 M + + c . c . Equations of motion for auxiliary fields ˆ V , ˆ H 2 , V ) = g 2 � H 1 † � V ˆ V ˆ H 1 ˆ D y ( e − 2 ˆ D y e 2 ˆ c − e − 2 ˆ ˆ H 1 = ˆ D y ˆ ˆ H 1 M • the lowest components of these Eqs. → 1/2 BPS equations for walls • higher components of these Eqs. → equations for y -dependence of higher components All components of these equations are solved with a chiral fields ˆ S by Σ = ˆ ˆ S − 1 ∂ y ˆ S, H 1 = ˆ S − 1 ˆ ˆ H 0 e My ˆ H 0 : y -independent chiral fields

V ˆ S † are determined by Se 2 ˆ and the vector field ˆ Ω ≡ ˆ supersymmetric master equations Ω − 1 ˆ H 0 e 2 My ˆ ∂ y (ˆ Ω − 1 ∂ y ˆ Ω) = g 2 ( c − ˆ H 0 ) Solutions are obtained by use of the solution of the bosonic master eq. Ω = Ω sol ( H 0 , H † H † Ω = Ω sol ( ˆ ˆ H 0 , ˆ 0 ) → 0 ) By substituting these solution, we obtain � d 4 θK wall + O ( λ 4 ) L w = − T wall + which turns out to be an effective action on the walls. K¨ ahler potential of the effective action is given by, � 1 � � � Ω) 2 + c log ˆ Ω − 1 ˆ H 0 e 2 My ˆ 2 g 2 (ˆ Ω − 1 ∂ y ˆ Ω + ˆ H † � K wall = dy Tr 0 � ˆ Ω=ˆ Ω sol � �� � H 0 e 2 My ˆ Lagrangian for ˆ Ω with a source ˆ H † 0

• Example with SU ( N ) F × SU ( N ) F ′ , ( N F = 2 N C ≡ 2 N ) Hypermultiplets: H i = ( H i + , H i − ) U ( N ) C SU ( N ) F SU ( N ) F ′ mass ¯ m H i 1 N N + 2 ¯ − m H i N 1 N − 2 A moduli matrix for N -walls solution is H 0 = √ c (1 N , e φ ) where a moduli parameter φ is an complex N × N matrix. ⇓ φ → φ ( x, θ ): chiral field K¨ ahler potential of the effective action for arbitrary g : �� � 2 � c log( e φ e φ † ) + O ( λ 2 ) K wall = 4 m Tr We believe that this gives Skyrm model in superfield formulation.

§ 4. Effective Action on Vortices • 6D N = 1(8 SUSY) theory ( M = 0) in superfield formulation N. Arkani-Hamed, T. Gregoire and J. Wacker, JHEP 0203, 055 (2002) ⇓ Neglecting halves of N = 2 supermultiplets • 4D N = 1(4 SUSY) effective theory on BPS vortices � d 4 θK vortex + O ( λ 4 ) L v = − 2 πc k + � �� � tension of k vortices K¨ ahler potential of the effective action, � K vortex = 1 � dzdz ∗ L Ω � � ˆ 2 i Ω=ˆ Ω sol � 2 � Ω − 1 ¯ Ω − 1 H 0 H † g 2 (ˆ Ω − 1 ∂ ˆ Ω)(ˆ ∂ ˆ Ω) + c log ˆ Ω + ˆ L Ω = Tr + L WZW 0 with a Wess-Zumino-Witten term � � L WZW = 4 ∂ Φsinh L Φ − L Φ ¯ g 2 Tr ∂ Φ , L 2 Φ where Φ ≡ log ˆ Ω , L Φ X = [Φ , X ]