Intersecting Solitons, Amoeba and Tropical Geometry Norisuke Sakai - PowerPoint PPT Presentation

Intersecting Solitons, Amoeba and Tropical Geometry Norisuke Sakai (Tokyo Woman’s Christian University) , In collaboration with Toshiaki Fujimori , Muneto Nitta , Kazutoshi Ohta , and Masahito Yamazaki arXiv:0805.1194(hep-th) , 2008.7.28-8.1, “ 量子場理論と弦理論の発展 ” workshop at YITP Contents 1 Introduction 2 2 Vortices and Instantons 3 Webs of Vortex Sheets on (C ∗ ) 2 3 5 4 Instantons inside Non-Abelian Vortex Webs 9 5 Conclusion 11

1 Introduction Solitons in Yang-Mills-Higgs theory in the Higgs phase (with 8 SUSY) Elementary solitons: Vortex and Domain wall ( Kink ) Vortices and Domain walls preserve 1 / 2 of SUSY : 1/2 BPS solitons Composite solitons in the Higgs phase : 1 / 4 BPS solitons Webs of domain walls , Magnetic monopoles with vortices , Instantons inside a Vortex ( Web of Vortices ) (Scherk-Schwarz twisted) dimensional reduction : Web of Vortices → all other 1 / 4 BPS composite solitons Web of Vortices is most important among composite BPS solitons Our purpose: Study configurations of instantons and vortex sheets ( webs of vortices ) In 8 SUSY U ( N C ) gauge theory with N F = N C Higgs scalars On R t × (C ∗ ) 2 ∼ R 2 , 1 × T 2 ( 5 dimensions) → Dim. reduction By using Moduli Matrix formalism Use amoeba and tropical geometry to describe Webs of vortices 2

Results 1. Vortex sheets : zeros of a polynomial in the Moduli matrix Instanton positions : common zeros with another polynomial 2. Mathematical language of amoeba and tropical geometry are useful to visualize the web of vortices and to evaluate physical quantities. 3. Moduli matrix approach plays a crucial role to describe web of vor- tices. 2 Vortices and Instantons SUSY U ( N C ) Gauge Theory with N F Higgs fields Higgs fields H as an N C × N F matrix, µ, ν = 0 , 1 , 2 , 3 , 4 2 g 2 F µν F µν + D µ H ( D µ H ) † − g 2 − 1 [ ] 4 ( HH † − c 1 N C ) 2 L = Tr Gauge coupling g for U ( N C ) , Fayet-Iliopoulos (FI) parameter c Coordinates of (C ∗ ) 2 : ( x 1 , y 1 , x 2 , y 2 ) , z 1 ≡ x 1 + iy 1 , z 2 ≡ x 2 + iy 2 Higgs Phase : Walls , Vortices are the only elementary solitons 3

Instantons , monopoles , junctions are composite solitons Energy lower bound of static field configurations Tr F ∧ ω = 8 π 2 − 1 ∫ ∫ E ≥ Tr ( F ∧ F ) − c g 2 I + 2 πc V g 2 ω ≡ i ahler form on (C ∗ ) 2 2 ( dz 1 ∧ d ¯ z 1 + dz 2 ∧ d ¯ z 2 ) : the K¨ Total instanton charge I , Instanton charge density I 1 ∫ ∫ ∫ I ≡ I ≡ − Tr ( F ∧ F ) = ch 2 8 π 2 Vortex charge V , Vortex charge density V − 1 ∫ ∫ ∫ V ≡ V ≡ Tr F ∧ ω = c 1 ∧ ω 2 π Lower bound is saturated if the BPS equations are satisfied z 2 ) = g 2 2 ( HH † − c 1 N C ) F ¯ z 2 = 0 , D ¯ z i H = 0 , − 2 i ( F z 1 ¯ z 1 + F z 2 ¯ z 1 ¯ BPS equations contain at least instantons and intersecting vortex sheets solutions to BPS eqs. preserve 1 / 4 of SUSY → 1/4 BPS states 4

Solution of BPS equations z i = − iS − 1 ∂ ¯ F ¯ z 2 = 0 : integrability condition for D ¯ z i W ¯ z i S z 1 ¯ Solution of the first 2 equations: H = S − 1 H 0 with ∂ ¯ z i H 0 = 0 N C × N F matrix H 0 should be holomorphic : Moduli Matrix c H 0 H † Remaining BPS eq.( Master eq. ): Ω ≡ SS † , Ω 0 ≡ 1 0 z 2 (Ω ∂ z 2 Ω − 1 ) = − g 2 c z 1 (Ω ∂ z 1 Ω − 1 ) + ∂ ¯ 1 N C − Ω 0 Ω − 1 ) ( ∂ ¯ 4 We consider N C = N F = N case Meissner effect in the Higgs phase (Higgs VEV): Magnetic flux can penetrate superconducting (Higgs) phase as Vortices (Partial) restoration of gauge symmetry at the core of vortex Vortex sheet in z 1 , z 2 ∈ (C ∗ ) 2 can be defined by det H 0 ( z 1 , z 2 ) = 0 Webs of Vortex Sheets on (C ∗ ) 2 3 Web of Vortices on (C ∗ ) 2 ≅ R 2 × T 2 : y i ∼ y i + 2 πR i , i = 1 , 2 zi ∑ a n 1 ,n 2 u n 1 1 u n 2 P ( u 1 , u 2 ) ≡ det H 0 = 2 , u i ≡ e Ri ( n 1 ,n 2 ) ∈ Z 2 5

(a) Newton polytope (b) Amoeba Figure 1: An example of amoeba; P ( u 1 , u 2 ) = a 0 , 0 + a 1 , 0 u 1 + a 2 , 0 u 2 1 + a 3 , 0 u 3 1 + a 0 , 1 u 2 + a 1 , 1 u 1 u 2 + a 2 , 1 u 2 1 u 2 + a 3 , 1 u 3 1 u 2 + a 0 , 2 u 2 2 + a 1 , 2 u 1 u 2 2 + a 2 , 2 u 2 1 u 2 2 . Newton polytope ∆( P ) ⊂ R 2 of a Laurent polynomial P ( u 1 , u 2 ) { ( n 1 , n 2 ) ∈ Z 2 � } ∆( P ) = conv . hull � a n 1 ,n 2 ̸ = 0 � a n 1 ,n 2 : moduli parameters for the webs of vortices Amoeba of P : a projection of generic webs of vortices on x 1 , x 2 {( } � P ( u 1 , u 2 ) = 0 ∈ R 2 � ) A P = R 1 log | u 1 | , R 2 log | u 2 | Tenticles : asymptotic regions extending to infinity 6

Normals to the Newton polytope: semi-infinite cylinders of vortices Internal lattice points of Newton polytope: holes ( vortex loops ) Relation with Tropical Geometry − − − − − − − − − − → R → 0 (a) Amoeba (b) Tropical variety Figure 2: An example of the amoeba and corresponding tropical variety. Amoeba is smooth even in the thin wall limit l = 1 /g √ c → 0 Tropical limit : R 1 = R 2 = R → 0 with fixed r n 1 ,n 2 ≡ R log | a n 1 ,n 2 | Amoeba degenerates into a set of lines (“spines”), called “tropical variety” Skeleton (spine) of amoeba in R → 0 : position of domain walls 7

Figure 3: Intersection of one tropical variety and its shift and Newton polytope ∆( P ) (below). Number of intersection points is given by 2Area(∆( P )) . 8

Intersection charge density becomes complex Monge-Amp` ere measure I 1 1 ∫ ∫ ∫ I intersection = dd c log | P |∧ dd c log | P | = dd c log | P | 8 π 2 4 π X X Regularization : P → P 1 , P 2 associated with the same Newton polytope 1 dd c log | P 1 | ∧ dd c log | P 2 | = 1 ∫ 2#( X 1 · X 2 ) = Area(∆) 8 π 2 4 Instantons inside Non-Abelian Vortex Webs ( 1 b ( u 1 , u 2 ) ) H 0 = 0 P ( u 1 , u 2 ) ∑ a n 1 ,n 2 u n 1 1 u n 2 ∑ b n 1 ,n 2 u n 1 1 u n 2 P ( u 1 , u 2 ) = 2 , b ( u 1 , u 2 ) = 2 Vortex sheets are localized at P ( u 1 , u 2 ) = 0 Instanton number: computed from Ω with the correct boundary conditions ( ) 1 + | b | 2 bP Ω ≡ Ω ∗ −| P | 2 1+ | b | 2 + | P | 2 P b 9

∗ ) = − g 2 c ¯ ∗ ) + ¯ z 1 (Ω ∗ ∂ z 1 Ω − 1 z 2 (Ω ∗ ∂ z 2 Ω − 1 4 (1 − | P | 2 Ω − 1 ∂ ¯ ∂ ¯ ∗ ) ( Ω becomes a solution of the master equation if b is a constant) 1 ∫ ( dd c log | P | ∧ dd c log(1 + | b | 2 ) − dd c log | P | ∧ dd c log | P | ) I = 8 π 2 = I instanton − I intersection 1 (C ∗ ) 2 dd c log | P |∧ dd c log(1+ | b | 2 ) = 1 ∫ ∫ dd c log(1+ | b | 2 ) I instanton = 8 π 2 4 π X X : zero locus of P corresponding to the vortex sheets instanton number is given by the degree of the map b | X : X → C P 1 Distribution of topological charge: Small instanton limit: b n 1 ,n 2 → ∞ with fixed b n 1 ,n 2 /b ˜ n 1 , ˜ n 2 dd c log(1 + | b | 2 ) → dd c log | b | 2 : delta function on b ( u 1 , u 2 ) = 0 Instantons are localized at common zeros of b ( u 1 , u 2 ) and P ( u 1 , u 2 ) 10

5 Conclusion 1. Vortex sheets : zeros of a polynomial in the Moduli matrix Instanton positions : common zeros with another polynomial 2. Mathematical language of amoeba and tropical geometry are useful to visualize the web of vortices and to evaluate physical quantities. 3. Moduli matrix approach plays a crucial role to describe web of vor- tices. 11