arXiv: (hep-th) Toshiaki Fujimori (Tokyo Institute of Technology) Minoru Eto, Sven Bjarke Gudnason, Kenichi Konishi, Muneto Nitta, Keisuke Ohashi and Walter Vinci Constructing Non-Abelian Vortices with Arbitrary Guage Group 1

1. Introduction 2 Recently, there has been significant progress in the understanding of BPS vortices in gauge theory. Vortices in abelian-Higgs model has been extensively studied for a long time. [ Abrikosov,1958, Nielsen-Olesen,1973] Many interesting phenomena has been clarified by using the knowledge of the moduli space of the BPS vortices. Extension to more general gauge group Powerful and general framework for the studies of vortices in gauge theories with

2. BPS Vortices gauge theory Field : U(1) gauge field : gauge fields : Hypermultiplet scalars charged under U(1) and in fundamental ( - dim) representation. ・ ・ ・ flavor index 3 : simple Lie group (3+1) dimensional (8 SUSY) ( : color-flavor mixed matrix ) with hypermultiplets

Lagrangian potential : gauge coupling constant for U(1) : gauge coupling constant for : Fayet-Iliopoulos parameter 4 vacua Gauge symmetry is broken.

boundary condition (pure gauge) 5 Ansatz static configuration without dependence Magnetic flux

6 (single valued) : center of Topological charge (non-contractible loop)

Bogomol’nyi bound 7 ・・・ Tension of BPS vortices ( )

BPS equations for vortices Configurations satisfying these BPS equations preserve the half of the supersymmetry. Minimum energy condition for a given boundary condition 8 × 1/2 = 4 SUSY ・・・ (I) ・・・ (II) ・・・ (III) 8

First BPS equation (I) Complexified gauge group : matrix holomorphic in 9 3. Method of Moduli matrix “ moduli matrix ”

Second and third BPS equations Master equations for vortices symmetry of master equation Physical fields are invariant 10

Procedure for solving BPS equations 11 Pick up a moduli matrix Solve the master equations with respect to and and are obtained from and up to gauge transformation The physical fields are obtained through the relations Solution of the BPS equations

12 Moduli space of BPS vortex solutions We assume the existence and uniqueness of the solution of the master equations for a given There are at least two circumstantial evidences (i) Index theorem (number of zero modes) one to one correspondence BPS vortex solution All the information of the moduli parameters are encoded in the moduli matrix ・・・ equivalence relation (ii) Strong coupling limit (NLSM limit)

13 boundary condition boundary condition for Holomorphic in and are related by the complexified gauge transformation. : rank invariant tensor of the group : invariant under Invariant constructed from ( )

14 Examples of moduli space for single vortex VEV of at infinity ・・・ color-flavor locked vacuum color-flavor diagonal symmetry matrix : ( ) vortex position Internal orientation

15 Examples of moduli space for single vortex broken by vortex Orientational moduli (Goldstone zero mode)

16 4. Summary The BPS vortices in gauge theory are studied. All the necessary tools to construct the vortex solution and the moduli space are given. We have examined the moduli space of single vortex. Our method can be generalized to 1/4 BPS configuration. Monopoles in the Higgs phase U(N) case [D.Tong, 2003] Monopole = Kink of vortex moduli

17 Monopole in the Higgs phase Mass and adjoint scalar mass matrix (generator of flavor symmetry) adjoint scalar in vector multiplet Potential on moduli space : Killing vector associated with BPS state (fixed points)

18 Mass of the monopole Magnetic charge Magnetic flux of vortices Monopole in the Higgs phase = 1/2 BPS kink on the vortex string 8 × 1/2 × 1/2 = 2 SUSY

19 Monopole in the Higgs phase (SO(N), Usp(2M) case) Non-degenerate masses Monopole mass root vectors

20 (i) Index theorem (number of zero modes) Linearized BPS equations gauge condition : infinitesimal fluctuations Condition for zero mode Index of the linear operator vanish for BPS background Dimension of moduli space

21 (II) Strong coupling limit (non-linear sigma model limit) : metric of : coordinates of BPS configuration in non-linear sigma model : ・・・ holomorphic map from -plane to 2-cycle of Master equations algebraic equations All the exact solution in the strong coupling limit can be obtained from the data of the moduli matrix exactly solvable

22 Future works study of non-Abelian monopole and non-Abelian duality by using the vortices. Kahler quotient construction of the moduli space. moduli space of vortices = half dimensional submanifold of moduli space of instantons Are there such kind of relations for general group ? case