paper

Analysis of $C P^{N - 1}$ sigma models via projective structure

arXiv:1010.2183

Abstract

In this paper, we study rank-1 projector solutions to the completely integrable Euclidean $C P^{N - 1}$ sigma model in two dimension and their associated surfaces immersed in the $s u (N)$ Lie algebra. We reinterpret and generalize the proof of A.M. Din and W.J. Zakzrewski [1980] that any solution for the $C P^{N - 1}$ $s i g mam o d e l d e f in e d o n t h e R i e mann s p h er e w i t h f ini t e a c t i o n c anb e w r i tt e na s a r ai s in g o p er a t or a c t in g o nah o l o m or p hi co n e, or a l o w er in g o p er a t or a c t in g o naan t ih o l o m or p hi co n e . O u r p r oo f i s f or m u l a t e d in t er m so f r ank - 1 H er mi t ian p r o j ec t or sso i t i se x pl i c i tl y g a ug e in v a r ian t an d g i v es n e w r es u l t so n t h es t r u c t u r eo f t h ecor r es p o n d in g se q u e n ceo f r ank - 1 p r o j ec t or s . N e x t, w e ana l y z es u r f a ces a ssoc ia t e d w i t h t h e$ CP^{N-1} $m o d e l s d e f in e d u s in g t h e G e n er a l i z e d W e i er s t r a ss F or m u l a f or imm er s i o n, in t r o d u ce d b y B . K o n o p e l c h e nk o [1996] . W es h o w t ha tt h es u r f a ces a r eco n f or ma l l y p a r am e t er i z e d b y t h e L a g r an g ian d e n s i t y w i t h f ini t e a r e a e q u a l t o t h e a c t i o n o f t h e m o d e l an d e x p r essse v er a l o t h er g eo m e t r i c a l c ha r a c t er i s t i cso f t h es u r f a ce in t er m so f t h e L a g r an g ian d e n s i t y an d t o p o l o g i c a l c ha r g e d e n s i t y o f t h e m o d e l . W e d e m o n s t r a t e t ha t an y s u c h s u r f a ce m u s t b eor t h o g o na l t o t h ese q u e n ceo f p r o j ec t or s d e f in e d b y r e p e a t e d a ppl i c a t i o n o f t h er ai s in g an d l o w er in g o p er a t or s . F ina l l y, w e p r o v i d e n ecess a r y an d s u f f i c i e n t co n d i t i o n s t ha t a s u r f a ce b er e l a t e d t o a$ CP^{N-1}$ sigma model.

Analysis of CPN−1 sigma models via projective structure

Abstract

Analysis of $C P^{N - 1}$ sigma models via projective structure