Index for Ohashi Lecture series
Lecture I: Domain walls and Vortices(a) The simplest soliton: domain wall(b) Axially symmetric vortex in Abelian-Higgs theory(c) Asymptotic behaviors and inter-soliton forces
i. type I and type II vortex(d) Derrick’s theorem
Lecture II: Supersymmetry and superﬁelds(a) Supersymmetry algebra(b) chiral superﬁeld
i. Kähler potential and Kähler trf.
ii. super potential(c) vector superﬁeld
i. super gauge trf. and WZ gauge
ii. FI term(d) supersymmetric gauge-Higgs modeli. D-term and F -term conditions for vacua
Lecture III: Non-linear sigma model (NLσM) on Kähler manifold M(a) Strong coupling limit of SUSY gauge-Higgs theory(b) Target manifold M and vacua in the Higgs phase(c) Projective space CP Nf −1 and Grassmannian manifold GrNf ,N (C)i. Patches, transition functions and Kähler trf.
ii. Fubini-Study metric for M = CP Nf −1iii. Baryonic ﬁelds B and Plücker relations
iv. Kähler potential K(B, B † )v. Duality between U (N ) and U (Nf − N )(d) Target space M for G = SO(N ), U Sp(2N )i. ‘Meson’ M as the GC invariant
ii. Kähler potential K(M, M † )1

Lecture IV: BPS domainwalls in U (N ) gauge Higgs theory(a) Discrete vacua with non-degenerate mass terms(b) Bogomol’nyi bound, BPS equations and Killing spinors(c) The moduli matrix and the master equationi. All exact solutions at the strong coupling limit
ii. Complexiﬁed gauge trf.(V -trf)iii. GrNf ,N (C) as a moduli space for BPS domainwalls(d) Examples for U (1)i. Center of mass as a Goldstone modeii. Single wall solution
iii. Position moduli for a multi-wall solution in CP Nf −1
iv. Topological sector and cell decomposition of CP Nf −1(e) Example for U (N )i. patches for the manifold GrNf ,N (C)ii. walls for Gr3,1 Gr4,2
iii. domainwall solutions and string theory perspective
Lecture V: BPS vortices(a) A vacuum and BPS equations(b) The moduli matrix and constraints on the GC invariant(c) Semilocal vortex for G = U (1) and lump in NLσM on CP Nf −1
i. π2 [M] and lump solutions
ii. scale moduli and small lump singularity(d) Vortex moduli space Mv for G = U (N )i. Fixing V -trf and coordinate patches of Mv
ii. Transition functions(e) Local vortex for G = U (N ), Nf = N
i. Positions of vortices
ii. orientational moduli CP N −1 as Goldstone modes(f) Vortices (lumps) for G = U (1) × SO(N )i. Local vortex with a strong condition and orientational moduli
ii. fractional vortices

2Lecture VI: Vortex moduli space and string theory perspective(a) Hannany-Tong model from string theory perspective
i. D-term condition and a moduli space M′v(b) Kähler quotient construction of the vortex moduli spacei. half-ADHM relation
ii. Topological isomorphism between Mv and M′viii. Axially symmetric vortex and Young tableau
Lecture VII: Composite solitons as 1/4 BPS states(a) Composite solitons in 5D
i. BPS equations and instanton charge
ii. Moduli matrixiii. Intersection of vortices
iv. Instantons in a vortex(b) Composite solitons in 4D with non-degenerate mass terms
i. BPS equations and monopole chargeii. Moduli matrixiii. Junctions of walls and vortices
iv. Monopoles in a vortex
Lecture VIII: Formula for eﬀective actions on Solitons(a) Mass gap, higher derivative expansion and an eﬀective action(b) Eﬀective action on BPS domainwalls
i. Eﬀective action formula in superspace
ii. Exact examples with the strong coupling limit(c) Eﬀective action on BPS vortices
i. Eﬀective action formula in superspace
ii. Eﬀective actions for a single local vortex
iii. Semilocal vortex and non-normalizable modes
iv. Eﬀective action for a well separated vortices

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