Universidade de São Paulo
Instituto de Física
Supercordas e Aspectos da Correspondência
AdS/CFT
Renann Lipinski Jusinskas
ORIENTADOR: Prof. Dr. Victor de Oliveira Rivelles
Dissertação de mestrado apresentada
ao Instituto de Física para a obtenção
do título de Mestre em Ciências.
Banca Examinadora:
Prof. Dr. Victor de Oliveira Rivelles (Orientador - IFUSP)
Prof. Dr. Fernando Tadeu Caldeira Brandt (IFUSP)
Prof. Dr. Ricardo Iván Medina Bascur (UNIFEI)
FICHA CATALOGRÁFICA
Preparada pelo Serviço de Biblioteca e Informação
do Instituto de Física da Universidade de São Paulo
Jusinskas, Renann Lipinski
Supercordas e aspectos da correspondência AdS/CFT -
São Paulo, 2010.
Dissertação (Mestrado) – Universidade de São Paulo.
Instituto de Física, Departamento de Física Matemática.
Orientador: Prof. Dr. Victor de Oliveira Rivelles.
Área de Concentração: Física.
Unitermos: 1. Supergravidade; 2. Supersimetria;
3. Teoria de cordas; 4. Correspondência AdS/CFT.
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✈
Pró❧♦❣♦ ✈✐
❘❡s✉♠♦ ✈✐✐
❆❜str❛❝t ✈✐✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❈♦r❞❛ ❇♦sô♥✐❝❛ ✸
✷✳✶ ❆ ❆çã♦ ❞❛p✲❜r❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷✳✷ ❆ ❆çã♦ ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✸ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✹ ▼❛ss❛ ❡ ❊♥❡r❣✐❛ ❞❛ ❈♦r❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✺ ❖ ❈❛❧✐❜r❡ ❞♦ ❈♦♥❡ ❞❡ ▲✉③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✻ ◗✉❛♥t✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✻✳✶ ❊s♣❡❝tr♦ ❞❛ ❈♦r❞❛ ❆❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✻✳✷ ❊s♣❡❝tr♦ ❞❛ ❈♦r❞❛ ❋❡❝❤❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✼ ❉✉❛❧✐❞❛❞❡ ❚ ❡ ❉✲❜r❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼✳✶ ❈♦♠♣❛❝t✐✜❝❛çã♦ ♣❛r❛ ❈♦r❞❛s ❋❡❝❤❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼✳✷ ❈♦♠♣❛❝t✐✜❝❛çã♦ ♣❛r❛ ❈♦r❞❛s ❆❜❡rt❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✽ ❙✐♠❡tr✐❛ ❞❡ ❈❛❧✐❜r❡ U(N) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✾ ❈♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✸ ❙✉♣❡r❝♦r❞❛ ❘◆❙ ✶✼
✸✳✶ ❙✉♣❡rs✐♠❡tr✐❛ ▲♦❝❛❧ ❡ ❛ ❆çã♦ ❘◆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ ❈❛r❣❛s ❈♦♥s❡r✈❛❞❛s ❡ ●❡r❛❞♦r❡s ❞❡ ❙✉♣❡r ❱✐r❛s♦r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✹ ◗✉❛♥t✐③❛çã♦ ❈❛♥ô♥✐❝❛ ❈♦✈❛r✐❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✺ ❖ ❈❛❧✐❜r❡ ❞♦ ❈♦♥❡ ❞❡ ▲✉③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✻ ❆♥á❧✐s❡ ❞♦ ❊s♣❡❝tr♦ ❞❡ ▼❛ss❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✻✳✶ Pr♦❥❡çã♦ ●❙❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✻✳✷ ❊s♣❡❝tr♦ ❞❛ ❈♦r❞❛ ❆❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✻✳✸ ❊s♣❡❝tr♦ ❞❛ ❈♦r❞❛ ❋❡❝❤❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✻✳✹ ❙✉♣❡rs✐♠❡tr✐❛ ♥♦ ❊s♣❛ç♦✲❚❡♠♣♦❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✼ ❈♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
❙❯▼➪❘■❖ ✐✐
✹ ❙✉♣❡r❝♦r❞❛ ●❙ ✸✸
✹✳✶ ❙✉♣❡rs✐♠❡tr✐❛ ♥♦ ❊s♣❛ç♦✲❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ ❙✐♠❡tr✐❛ ❦❛♣❛ ❡ ❛ ❆çã♦ ●❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✸ ❈❛r❣❛s ❈♦♥s❡r✈❛❞❛s ❡ ●❡r❛❞♦r❡s ❞❡ ❙✐♠❡tr✐❛s ❞♦ ❊s♣❛ç♦✲❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✹ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ ●r❛✉s ❞❡ ▲✐❜❡r❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✺ ❖ ❈❛❧✐❜r❡ ❞♦ ❈♦♥❡ ❞❡ ▲✉③ ♥♦ ❋♦r♠❛❧✐s♠♦ ●❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✻ ◗✉❛♥t✐③❛çã♦ ❡ ❊s♣❡❝tr♦ ❋✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✻✳✶ ❙✉♣❡rs✐♠❡tr✐❛N = 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✻✳✶✳✶ ❈♦r❞❛s ❋❡❝❤❛❞❛s ❚✐♣♦II✲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✻✳✶✳✷ ❈♦r❞❛s ❋❡❝❤❛❞❛s ❚✐♣♦II✲❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹✳✻✳✷ ❙✉♣❡rs✐♠❡tr✐❛N = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹✳✻✳✷✳✶ ❈♦r❞❛s ❆❜❡rt❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✻✳✷✳✷ ❈♦r❞❛s ❋❡❝❤❛❞❛s ◆ã♦ ❖r✐❡♥t❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✼ ❉✉❛❧✐❞❛❞❡ ❚ ❡ ❛s ❙✉♣❡r❝♦r❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✼✳✶ ❈♦r❞❛s ❋❡❝❤❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✼✳✷ ❈♦r❞❛s ❆❜❡rt❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✼✳✸ ❈♦♥st❛♥t❡s ❞❡ ❆❝♦♣❧❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✽ ❈♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺ ❇r❛♥❛s ❡ ❙✉♣❡r❣r❛✈✐❞❛❞❡ ✹✾ ✺✳✶ p✲❜r❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✺✳✷ ❉p✲❜r❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✺✳✷✳✶ ❙✉♣❡rs✐♠❡tr✐❛ ❡ ❙✐♠❡tr✐❛ ❦❛♣❛ ❞♦ ❈❛♠♣♦ ❞❡ ❈❛❧✐❜r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✷✳✷ ❆çã♦ ❊❢❡t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✸ ❆çõ❡s ❞❛ ❙✉♣❡r❣r❛✈✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✸✳✶ ❚❡♦r✐❛ ▼ ❡ ❙✉♣❡r❣r❛✈✐❞❛❞❡D= 11 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✺✳✸✳✷ ❙✉♣❡r❣r❛✈✐❞❛❞❡II✲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✺✳✸✳✸ ❙✉♣❡r❣r❛✈✐❞❛❞❡II✲❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✺✳✹ ❈♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✻ ❆s♣❡❝t♦s ❞❛ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❆❞❙✴❈❋❚ ✻✶ ✻✳✶ N = 4 ❙✉♣❡r ❨❛♥❣✲▼✐❧❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
✻✳✶✳✶ ❖ ❈♦♥t❡ú❞♦ ❞❡ ❈❛♠♣♦s ❡ ❛ ❆çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✻✳✶✳✷ ❙✐♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✻✳✶✳✸ ▼✉❧t✐♣❧❡t♦s ❙✉♣❡r❝♦♥❢♦r♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✻✳✶✳✹ ❊①♣❛♥sã♦ 1
N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✻✳✷ ❙✉♣❡r❣r❛✈✐❞❛❞❡ II✲❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✻✳✷✳✶ ❙✐♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✻✳✷✳✷ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ ❙♦❧✉çõ❡s ❊①tr❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✻✳✷✳✸ ❆3✲❜r❛♥❛ ❡ ❛ ●❡♦♠❡tr✐❛AdS5×S5 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✻✳✸ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ▼❛❧❞❛❝❡♥❛ ❡ ♦ ❉✐❝✐♦♥ár✐♦ ❞❛ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✸✳✶ ❆❧❣✉♠❛s ❈♦♠♣❛r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✸✳✷ ❆ ❈♦♥❥❡❝t✉r❛ ❡ ♦ ❉✐❝✐♦♥ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✹ ❈♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✼ ❈♦♥❝❧✉sã♦ ✼✼
❙❯▼➪❘■❖ ✐✐✐ ❇ ❆❧❣✉♠❛s Pr♦♣r✐❡❞❛❞❡s ❞❡ ❊s♣✐♥♦r❡s ❡♠ ❱ár✐❛s ❉✐♠❡♥sõ❡s ✽✺ ❇✳✶ ▼❛tr✐③❡sΓ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺
❇✳✷ ❊s♣✐♥♦r❡s ❞❡ ▼❛❥♦r❛♥❛✱ ❲❡②❧ ❡ ▼❛❥♦r❛♥❛✲❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ❇✳✸ Pr♦♣r✐❡❞❛❞❡ ❊s♣❡❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ❈ ❙✉♣❡rs✐♠❡tr✐❛ ❡ ❘❡♣r❡s❡♥t❛çõ❡s ✾✶ ❈✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ❈✳✷ ❙✉♣❡rs✐♠❡tr✐❛ ❡♠D= 2❡ ❛ ❙✉♣❡r❝♦r❞❛ ❘◆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷
❈✳✸ ❙✉♣❡rs✐♠❡tr✐❛ ❡♠ ❉✐♠❡♥sõ❡s ❆r❜✐trár✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ❈✳✸✳✶ ❙✉♣❡rs✐♠❡tr✐❛ ❡♠D= 4❡ s✉❛s ❘❡♣r❡s❡♥t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹
❈✳✸✳✶✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ◆ã♦ ▼❛ss✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ❈✳✸✳✶✳✷ ❘❡♣r❡s❡♥t❛çõ❡s ▼❛ss✐✈❛s ❡ ♦ ✈í♥❝✉❧♦ ❇P❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ❈✳✸✳✷ ❙✉♣❡rs✐♠❡tr✐❛ ❡ ❙✉♣❡r❣r❛✈✐❞❛❞❡ ❡♠D= 10,11 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻
❉ ▼❛tr✐③❡s ❙✐♥❣✉❧❛r❡s ❡ ❛ ❙✐♠❡tr✐❛ ❦❛♣❛ ✾✾ ❉✳✶ ▼❛tr✐③❡s ❙✐♥❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ❉✳✷ ❙✐♠❡tr✐❛ ❦❛♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵
❊ ❙✐♠❡tr✐❛ ❈♦♥❢♦r♠❡ ✶✵✶
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ à ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ➚ ♠✐♥❤❛ ♠ã❡✱ ■♥ês✱ s❡♠♣r❡ tã♦ ❛t❡♥❝✐♦s❛ ❡ ❝❛r✐♥❤♦s❛✱ ❡ q✉❡✱ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✱ s♦✉❜❡ ♠❡ ♦✉✈✐r ❡ ❝♦♥❢♦rt❛r ❝♦♠ s❡✉ ❛♠♦r✳ ❆♦ ♠❡✉ ♣❛✐✱ ▲✉✐③✱ ❛❣r❛❞❡ç♦ ❡ss❡ ♠❡s♠♦ ❛♠♦r q✉❡✱ ♠❡s♠♦ ❝♦♥t✐❞♦✱ ♠❡ ♦r✐❡♥t♦✉ ❡ ❛❝♦♥s❡❧❤♦✉ s❛❜✐❛♠❡♥t❡✱ ❝♦♠ ♣❛❧❛✈r❛s q✉❡ ♥❡♠ s❡♠♣r❡ ❣♦st❡✐ ❞❡ ♦✉✈✐r✱ ♠❛s ♣r❡❝✐s❛♠❡♥t❡ ❛s q✉❡ ❡✉ ♥❡❝❡ss✐t❛✈❛ ❡♥t❡♥❞❡r✳ ➚ ♠✐♥❤❛ ✐r♠ã✱ ❋r❛♥❝✐♥❡✱ q✉❡✱ ❥✉♥t♦ ❛♦s ♠❡✉s ♣❛✐s✱ s❡♠♣r❡ ♠❡ ❛♣♦✐♦✉ ❡♠ t♦❞❛s ❛s ❞❡❝✐sõ❡s ❡ ♠❡ ❛❥✉❞♦✉ ❛ ✈❡r ♦ ♠✉♥❞♦ ❝♦♠♦ ❡✉ ♦ ✈❡❥♦ ❤♦❥❡ ❡ ❡♥t❡♥❞❡r ✉♠ ♣♦✉❝♦ ♠❛✐s ❛s ♣❡ss♦❛s✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s q✉❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♦✉ ❞❡ ♦✉tr❛✱ ❢❛③❡♠ ♣❛rt❡ ❞♦ q✉❡ s♦✉ ❝♦♠♦ ❢ís✐❝♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❡①✲♦r✐❡♥t❛❞♦r❡s ●✐❧❜❡rt♦ ▼❡❞❡✐r♦s ❑r❡♠❡r ❡ ▼✐❣✉❡❧ ❆❜❜❛t❡✱ ✈❡r❞❛❞❡✐r♦s ❡①❡♠♣❧♦s t❛♥t♦ ❝♦♠♦ ♣❡ss♦❛s q✉❛♥t♦ ❝♦♠♦ ❝✐❡♥t✐st❛s✱ ❝❛❞❛ ✉♠ ❛ s❡✉ ♠♦❞♦✱ ♠❛s ❛♠❜♦s ❢✉♥❞❛♠❡♥t❛✐s✳ ▲❛♠❡♥t♦ ♣r♦❢✉♥❞❛♠❡♥t❡ t❡r ♠❡ ❛❢❛st❛❞♦ t♦❞♦ ❡ss❡ t❡♠♣♦ ♠❛s ❝♦♥t♦ ❝♦♠ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♠✐③❛❞❡✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s ❞❛ s❛❧❛307✱ ▲❡❛♥❞r♦ ❡ P❡❞r♦✱ ✐❞❡❛❧✐③❛❞♦r❡s ❞♦ ❈❋❚✱ ♣♦r ✐♥ú♠❡r❛s
❡①♣❡r✐ê♥❝✐❛s✱ ❞✐s❝✉ssõ❡s✱ ❜r✐♥❝❛❞❡✐r❛s✱ ❞❡s❛❜❛❢♦s✱ ✐❞é✐❛s✳✳✳ P♦r ♦✉✈✐r❡♠ ♠❡✉s ❞❡✈❛♥❡✐♦s ❡ ❛❥✉❞❛r❡♠ ♥♦s ❛❥✉st❡s ✜♥♦s ❞❛ ❞✐ss❡rt❛ç❛♦✱ ❝♦♠ ❝♦♠❡♥tár✐♦s ❡ s✉❣❡stõ❡s ❞❡ ❣r❛♥❞❡ ✈❛❧♦r✳ ❊✉ ❡s♣❡r♦ s✐♥❝❡r❛♠❡♥t❡ ♣♦❞❡r♠♦s ❢❛③❡r ❛ ❞✐❢❡r❡♥ç❛ ❡ ❝♦♥tr✐❜✉✐r ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♣❛r❛ ❛ ❝✐ê♥❝✐❛✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❛ ♣ós✲❣r❛❞✉❛çã♦ ♥❛ ❯❙P ❡ ❛ t♦❞♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❢♦r♠❛ ♠❡♥♦s ❞✐r❡t❛✱ ♣♦r ♣r♦♣♦r❝✐♦♥❛r❡♠ ✉♠ ❛♠❜✐❡♥t❡ tã♦ ❢❛✈♦rá✈❡❧✱ s❡♠♣r❡ ❞✐s♣♦st♦s ❛ ❞✐s❝✉t✐r ❡ ❝♦♠♣❛rt✐❧❤❛r ❡①♣❡r✲ ✐ê♥❝✐❛s✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s r❡✈✐s♦r❡s ♦rt♦❣rá✜❝♦s✳ ❈♦♠ ♣❡q✉❡♥❛s ♦✉ ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s✱ ❞❡✜♥✐t✐✈❛♠❡♥t❡ t♦r♥❛r❛♠ ♦ t❡①t♦ ♠❛✐s ❛♣r❡s❡♥tá✈❡❧✳ ❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ✈❛✐ ♣❛r❛ ❛ ❈❛r♦❧ q✉❡✱ ♠❡s♠♦ ❞✐❛♥t❡ ❞❡ ✉♠ t❡①t♦ ❛✈❛♥ç❛❞♦✱ s❡ ❞✐s♣ôs ❛ ❧❡r ❡ ❝♦♥s❡rt❛r ♠✐♥❤❛s ❢❛❧❤❛s✳
P♦r ✜♠✱ ❛❣r❛❞❡ç♦ ❛♦ Pr♦❢✳ ❉r✳ ❱✐❝t♦r ❞❡ ❖❧✐✈❡✐r❛ ❘✐✈❡❧❧❡s ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ❛♦ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
Pró❧♦❣♦
❚♦♠♦ ❡st❡ ❡s♣❛ç♦ ♣❛r❛ ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ♣❡ss♦❛✐s ❡♠ r❡❧❛çã♦ ❛♦ t❡①t♦ ❞❛ ❞✐ss❡rt❛çã♦ ❡ ❛❧❣✉♠❛s ❞❛s ✐❞é✐❛s ♥❡❧❛ ❝♦♥t✐❞❛s✳
❆♣❡s❛r ❞❡ s❡r ✉♠ tr❛❜❛❧❤♦ ❞❡ r❡✈✐sã♦✱ ❡✉ ❛❝r❡❞✐t♦ ♥❛ ❝♦♥tr✐❜✉✐çã♦ q✉❡ ♦ t❡①t♦ ♣♦❞❡ ♦❢❡r❡❝❡r✱ ♠❡s♠♦ ♣❛r❛ ♦s ♠❛✐s ❡①♣❡r✐❡♥t❡s q✉❡ ❡✉ ♥♦ ❛ss✉♥t♦✳ ❚❡♥t❡✐ ❡s❝r❡✈ê✲❧♦ ❝♦♠♦ ❣♦st❛r✐❛ q✉❡ t✐✈❡ss❡♠ ❡s❝r✐t♦ ♣❛r❛ ♠✐♠✳ ❙ã♦ ❢r❡q✉❡♥t❡s✱ ♥❛ ✈❛st❛ ❜✐❜❧✐♦❣r❛✜❛ ❞❡ ❧✐✈r♦s ❡ ❛rt✐❣♦s s♦❜r❡ ♦ ❛ss✉♥t♦✱ ❝♦♠❡♥tár✐♦s ❝♦♠♦ ✏é ❢á❝✐❧ ✈❡r q✉❡✳✳✳✑✳ ❍á ♥♦ t❡①t♦ ♣❛ss❛❣❡♥s✱ ❞❡♠♦♥str❛çõ❡s rá♣✐❞❛s ❡ s✐♠♣❧❡s ✭❛❧❣✉♠❛s ♥❡♠ t❛♥t♦✮ ❞❡ ✐❞é✐❛s q✉❡✱ t❛❧✈❡③ tr✐✈✐❛✐s✱ t❛❧✈❡③ ✐rr❡❧❡✈❛♥t❡s✱ sã♦ ♦♠✐t✐❞❛s ♥❛s ✈ár✐❛s r❡❢❡rê♥❝✐❛s✳ ❈♦♥❝♦r❞♦ q✉❡ ✉♠❛ ✈❡③ q✉❡ ♦s ❝♦♥❝❡✐t♦s t❡♥❤❛♠ s✐❞♦ ❡♥t❡♥❞✐❞♦s✱ ✜❝❛ t✉❞♦ ♠❛✐s ❢á❝✐❧✳ ❊✉ ♠❡s♠♦ ❡s♣❡r♦ ♥ã♦ t❡r ❛❜✉s❛❞♦ ❞❡ ❝♦♠❡♥tár✐♦s ❞❡ss❡ ♥í✈❡❧ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ◆❡ss❡ ❝❛s♦✱ t♦♠♦ ♣❛r❛ ♠✐♠ t♦❞❛ ❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ♣♦r ♦♠✐ssõ❡s ❡ ♣♦ssí✈❡✐s ❡♥❝❛❞❡❛♠❡♥t♦s ❝♦♥❢✉s♦s✱ ♦✉ ❛té ❡rr❛❞♦s✱ ❞❡ ✐❞❡✐❛s✳ ❆❧é♠ ❞✐ss♦✱ t♦❞❛s ❛s ♣❛❧❛✈r❛s ❡①♣r❡ss❛♠ ♠❡✉ ❛♠❛❞✉r❡❝✐♠❡♥t♦ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❞♦✐s ❛♥♦s✳ ❚❡♥t❡✐✱ ♥ã♦ s❡✐ s❡ ❜❡♠ s✉❝❡❞✐❞♦✱ ❛❜♦r❞❛r ♦ ❛ss✉♥t♦ ❝♦♠ ✉♠❛ ❞❛s ❢❡rr❛♠❡♥t❛s ♠❛✐s ❜❡♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❡❧♦s ❢ís✐❝♦s✱ ❛ ❛♥á❧✐s❡ ❞❡ s✐♠❡tr✐❛s✳ ❊s♣❡r♦ ❞❡✐①❛r ❝❧❛r♦ q✉❡ t❛❧ ❛❜♦r❞❛❣❡♠ ♥ã♦ é ♦r✐❣✐♥❛❧✱ ❛♣❡♥❛s t❡♥t❡✐ r❡♣❛ssá✲❧❛ ❞❡ ❢♦r♠❛ ♠❛✐s ❛♠✐❣á✈❡❧✳ ❚❡♥❤♦ ♠✉✐t♦ ❛ ❛♣r❡♥❞❡r✳ ▼❡✉ ♦✉tr♦ ❝♦♠❡♥tár✐♦ é ♠❛✐s ✜❧♦só✜❝♦✱ ❡♥✈♦❧✈❡♥❞♦ ♦ ♠♦❞♦ ❝♦♠♦ ❡♥❝❛r♦ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❡ s❡✉ ♣♦t❡♥❝✐❛❧ ❞❡ ❞❡s❝r✐çã♦ ❞❛ ♥❛t✉r❡③❛✱ ❝♦♠♦ ✉♠❛ t❡♦r✐❛ ❞❡ ❣r❛♥❞❡ ✉♥✐✜❝❛çã♦✳
◗✉❛♥❞♦ ❞❡❝✐❞✐ ♣♦r ❡ss❡ ❝❛♠✐♥❤♦ ❝❡rt❛♠❡♥t❡ ♦✉✈✐ ❝rít✐❝❛s✳ ▼❛✐s ♣r❡♦❝✉♣❛♥t❡s✱ ❡♥tr❡t❛♥t♦✱ ❡r❛♠ s✉❛s ♦r✐❣❡♥s✳ ▼❡✉s ♣ró♣r✐♦s ♠❡♥t♦r❡s ❞✐③❡♥❞♦ q✉❡ ✐ss♦ ❡r❛ ✉♠ ❞❡s♣❡r❞í❝✐♦ ❞♦ ♠❡✉ ♣♦t❡♥❝✐❛❧✳ ❖✉✈✐✱ ❡ ❛✐♥❞❛ ♦✉ç♦✱ ❝♦♠❡♥tár✐♦s ❞❡ ❞❡s♣r❡③♦✱ ❝rít✐❝❛s ❢❡r♦③❡s ❞❡ ♣❡ss♦❛s q✉❡ ♥ã♦ ❝♦♥❤❡❝❡♠ ❛ ♠❡t❛❞❡ ❞❡ t✉❞♦ ♦ q✉❡ ❛♣r❡♥❞✐ ❛té ❛q✉✐ s♦❜r❡ ♦ ❛ss✉♥t♦ ✭♦ q✉❡ é ❛✐♥❞❛ ♠✉✐t♦ ♣♦✉❝♦✮✳ P❡r❣✉♥t❛s ♥❛t✉r❛✐s ❝♦♠♦ ✏❖♥❞❡ ❢♦r❛♠ ♣❛r❛r ❛s ♦✉tr❛s6❄✑✶ sã♦ r❡❝♦rr❡♥t❡s ♥♦ ♠❡✉ ❞✐❛✲❛✲❞✐❛ ❡ ❣❡r❛❧♠❡♥t❡ ❛♣♦♥t❛♠ ♣♦♥t♦s ❝rít✐❝♦s ❞❛ t❡♦r✐❛✳ ➱
❡✈✐❞❡♥t❡ q✉❡ t❛✐s q✉❡st✐♦♥❛♠❡♥t♦s sã♦ ❢✉♥❞❛♠❡♥t❛✐s ❡ ❝❡rt❛♠❡♥t❡ ❛ t❡♦r✐❛ ❡stá ✐♥❝♦♠♣❧❡t❛✳ ❊♥tr❡t❛♥t♦✱ é ✐♠♣♦rt❛♥t❡ q✉❡ ❛ ❛rr♦❣â♥❝✐❛ ❞♦ ❝❡t✐❝✐s♠♦ ♥ã♦ s✉❜❥✉❣✉❡ ❛ ♦✉s❛❞✐❛ ❞❛ ❝✉r✐♦s✐❞❛❞❡ ❡ ❞❛ ❞❡s❝♦❜❡rt❛✿ ❛ r✐q✉❡③❛ ♠❛t❡♠át✐❝❛ ❡ ❢ís✐❝❛ ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s é ✐♠♣r❡ss✐♦♥❛♥t❡ ✭❡st❛ ú❧t✐♠❛ ♣♦r ✈❡③❡s ✐❣♥♦r❛❞❛ ♠❛s s❡♠♣r❡ ♣r❡s❡♥t❡ ♥❛s ✐❞é✐❛s ❞♦s ✈❡r❞❛❞❡✐r♦s ❝✐❡♥t✐st❛s✮✳ ❉❡✐①❛♥❞♦ ❞❡ ❧❛❞♦ ❛ ❧❡✈✐❛♥❞❛❞❡ ❞❛ ♠❛✐♦r✐❛ ❞❛s ❝rít✐❝❛s ✭❡ ❞♦s ❝rít✐❝♦s✮ ❛✜r♠♦ ❛q✉✐ q✉❡ ♥ã♦ ♠❡ ❛rr❡♣❡♥❞♦ ❞❛s ❡s❝♦❧❤❛s q✉❡ ✜③✳ ❈♦♥❢♦r♠❡ ✉♠ ❣r❛♥❞❡ ♣❡sq✉✐s❛❞♦r ❞✐ss❡ ❝❡rt❛ ✈❡③✱ ♠❡s♠♦ q✉❡ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❡st❡❥❛ ❛✐♥❞❛ ❧♦♥❣❡ ❞❡ ✉♠❛ ❞❡s❝r✐çã♦ s❛t✐s❢❛tór✐❛ ❞❛ ♥❛t✉r❡③❛✱ ❡❧❛ ♣♦❞❡ ❝♦♥t❡r ♦s ✐♥❣r❡❞✐❡♥t❡s ❞❛ ♣♦ssí✈❡❧ ❡ tã♦ ❛❧♠❡❥❛❞❛ t❡♦r✐❛ ❢✉♥❞❛♠❡♥t❛❧✱ ❡ ✐ss♦ ❥á é r❛③ã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❡st✉❞á✲❧❛✳ ❊✱ ❛✜♥❛❧✱ s❡ ❛ t❡♦r✐❛ ❢♦r ❝♦♥s✐❞❡r❛❞❛ ✐♥❝♦♥s✐st❡♥t❡✱ ❛♣❡s❛r ❞❡ s✉❛ ❜❡❧❡③❛✱ t❡r❡♠♦s ❝♦♠♦ s❛❧❞♦ ✉♠ ❡♥♦r♠❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡✱ ❝❡rt❛♠❡♥t❡✱ ✉♠❛ ✐♥❞✐❝❛çã♦ ❞❡ q✉❛❧ ❞✐r❡çã♦ s❡❣✉✐r✱ ♠❡s♠♦ q✉❡ s❡❥❛ ❛♣❡♥❛s ✉♠❛ ♣❧❛❝❛ ❞✐③❡♥❞♦ ✏◆ã♦ ❊♥tr❡✦✑✳
✶P❛r❛ ❡♥t❡♥❞ê✲❧❛✱ ❜❛st❛ ❞❛r ✉♠❛ ♦❧❤❛❞❛ ♥❛ ♣á❣✐♥❛ ✷✼✳
❘❡s✉♠♦
❊st❡ é ✉♠ tr❛❜❛❧❤♦ ❞❡ r❡✈✐sã♦✳ ❖s ♣r✐♥❝✐♣❛✐s ❢♦r♠❛❧✐s♠♦s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s sã♦ ✐♥tr♦❞✉③✐❞♦s ❡ ❞✐s❝✉t✐❞♦s✿ ❛ ❝♦r❞❛ ❜♦sô♥✐❝❛ ❡ ❛s s✉♣❡r❝♦r❞❛s ♥♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❘❛♠♦♥❞✲◆❡✈❡✉✲❙❝❤✇❛r③ ❡ ♥♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ●r❡❡♥✲❙❝❤✇❛r③✳ ❙ã♦ ✐♥tr♦❞✉③✐❞❛s t❛♠❜é♠ ❛s ❛çô❡s ❡❢❡t✐✈❛s ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❞❛s ❉✲❜r❛♥❛s ✭❛çã♦ ❉❇■✮ ❡ ❞❛s t❡♦r✐❛s ❞❡ s✉♣❡r❣r❛✈✐❞❛❞❡ ✭D = 10,11✮✳ P♦r ✜♠✱ sã♦ ❛♥❛❧✐s❛❞♦s ❛❧❣✉♥s
❛s♣❡❝t♦s ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❆❞❙✴❈❋❚✱ ❡♥✉♥❝✐❛♥❞♦ ❛ ❝♦♥❥❡❝t✉r❛ ❡ ✐♥tr♦❞✉③✐♥❞♦ ❛❧❣✉♠❛s ❞❛s ❡♥tr❛❞❛s ❞♦ ❞✐❝✐♦♥ár✐♦✳ ❖s ❛♣ê♥❞✐❝❡s ❝♦♥tê♠ tó♣✐❝♦s ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ♣❛r❛ ♦ t❡①t♦ ❡ ♣♦❞❡♠ ❛❥✉❞❛r ❛ ❡s❝❧❛r❡❝❡r ✈ár✐♦s r❛❝✐♦❝í♥✐♦s ❡ ♣❛ss❛❣❡♥s ❛♦ ❧♦♥❣♦ ❞♦ ♠❡s♠♦✳
❆❜str❛❝t
❚❤✐s ✐s ❛ r❡✈✐❡✇ ✇♦r❦✳ ❚❤❡ ♠❛✐♥ ❢♦r♠❛❧✐s♠s ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ str✐♥❣ t❤❡♦r② ❛r❡ ✐♥tr♦❞✉❝❡❞ ❛♥❞ ❞✐s❝✉ss❡❞✿ t❤❡ ❜♦s♦♥✐❝ str✐♥❣ ❛♥❞ t❤❡ s✉♣❡rstr✐♥❣s ❢♦r♠❛❧✐s♠s ♦❢ ❘❛♠♦♥❞✲◆❡✈❡r✲❙❝❤✇❛r③ ❛♥❞ ●r❡❡♥✲ ❙❝❤✇❛r③✳ ❚❤❡ ❡✛❡❝t✐✈❡ ❛❝t✐♦♥s ✐♥ t❤❡ ❧♦✇ ❡♥❡r❣② ❧✐♠✐t ♦❢ t❤❡ ❉✲❜r❛♥❡s ✭❉❇■ ❛❝t✐♦♥✮ ❛♥❞ s✉♣❡r❣r❛✈✐t② t❤❡♦r✐❡s ✭D = 10,11✮ ❛r❡ ✐♥tr♦❞✉❝❡❞ ❛s ✇❡❧❧✳ ❋✐♥❛❧❧②✱ s♦♠❡ ❛s♣❡❝ts ♦❢ t❤❡ ❆❞❙✴❈❋❚ ❝♦rr❡s♣♦♥❞❡♥❝❡
❛r❡ ❛♥❛❧✐s❡❞✱ ❡♥✉♥❝✐❛t✐♥❣ t❤❡ ❝♦♥❥❡❝t✉r❡ ❛♥❞ ✐♥tr♦❞✉❝✐♥❣ s♦♠❡ ❡♥tr✐❡s ♦❢ t❤❡ ❞✐❝t✐♦♥❛r②✳ ❚❤❡ ❛♣♣❡♥❞✐❝❡s ❝♦♥t❛✐♥s s♦♠❡ t♦♣✐❝s t❤❛t ❛r❡ ❤✐❣❤❧② r❡❧❡✈❛♥t ❢♦r t❤❡ ✇❤♦❧❡ t❡①t ❛♥❞ ♠❛② ❤❡❧♣ ❝❧❛r✐❢② s❡✈❡r❛❧ ✐❞❡❛s ❛♥❞ ❛r❣✉♠❡♥ts t❤r♦✉❣❤ ✐t✳
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❆ t❡♦r✐❛ ❞❡ ❝♦r❞❛s é✱ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ✉♠❛s ❞❛s t❡♦r✐❛s q✉❡ ♠❛✐s ♦s❝✐❧♦✉ ❡♥tr❡ ♦ ❡sq✉❡❝✐♠❡♥t♦ ❡ ❛ ❣❧ór✐❛ ❡♠ t♦❞❛ ❛ ❤✐stór✐❛ ❞❛ ❢ís✐❝❛ t❡ór✐❝❛✳ ❈♦♠ ♠❛✐s ❞❡ 40 ❛♥♦s ❞❡ s❡✉s r✉❞✐♠❡♥t♦s✱ ❛ ❞ú✈✐❞❛✱ ❡ ♠❡s♠♦ ❛
❞❡s❝r❡♥ç❛✱ ❡stã♦ ♣r❡s❡♥t❡s ♥❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✳
❚❡♥❞♦ s✐❞♦ ✐♥tr♦❞✉③✐❞❛ ♥❛ ❞é❝❛❞❛ ❞❡ ✻✵✱ s❡✉ ♦❜❥❡t✐✈♦ ✐♥✐❝✐❛❧ ❡r❛ ❞❡s❝r❡✈❡r ❢♦rç❛s ♥✉❝❧❡❛r❡s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❡st❛s s❡r✐❛♠ ✈✐st❛s ❝♦♠♦ ♦❜❥❡t♦s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s q✉❡ ❧✐❣❛r✐❛♠ ♦s ❝♦♥st✐t✉✐♥t❡s ❞❛ ♠❛tér✐❛✳ ❙❡✉ ♣♦t❡♥❝✐❛❧✱ ❡♥tr❡t❛♥t♦✱ ❢♦✐ r❡❝♦♥❤❡❝✐❞♦ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛✱ ✉♠❛ ✈❡③ ✐❞❡♥t✐✜❝❛❞♦ ❡♠ s❡✉ ❡s♣❡❝tr♦ ✉♠ ❡st❛❞♦ ❛ss♦❝✐❛❞♦ ❛♦ ❣rá✈✐t♦♥✱ ❛ ♣❛rtí❝✉❧❛ ♠❡♥s❛❣❡✐r❛ ❞❛s ❢♦rç❛s ❣r❛✈✐t❛❝✐♦♥❛✐s✳
◆❡ss❡ ❝♦♥t❡①t♦ ♥❛s❝❡✉ t❛♠❜é♠ ❛ s✉♣❡rs✐♠❡tr✐❛✳ ❍❛✈✐❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♥❝♦r♣♦r❛r ✉♠❛ ❞❡s❝r✐çã♦ ❢❡r♠✐ô♥✐❝❛ ♥❛ t❡♦r✐❛✱ q✉❡ ❝♦♥t✐♥❤❛ ❛♣❡♥❛s ❜ós♦♥s✳ ❆ss✐♠✱ ♣♦ss✉✐♥❞♦ ❡♠ s❡✉ ❡s♣❡❝tr♦ ❝♦r❞❛s ❛❜❡rt❛s ❡ ❢❡❝❤❛❞❛s✱ ❡♥tr❡ ❜ós♦♥s ❡ ❢ér♠✐♦♥s✱ ❛ t❡♦r✐❛ ❣❛♥❤♦✉ ❞❡st❛q✉❡ ❝♦♠♦ ✉♠❛ ♣♦ssí✈❡❧ ✉♥✐✜❝❛çã♦ ❞❛s ❢♦rç❛s ❞❛ ♥❛t✉r❡③❛✳
❆♣ós ❡♠ ❝❡rt♦ ♣❡rí♦❞♦ ❞❡ ❡st❛❣♥❛çã♦✱ q✉❡ ✐♥❝❧✉✐✉ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ s✉♣❡r❣r❛✈✐❞❛❞❡✱ ❛ t❡♦r✐❛ ♣❛ss♦✉ ♣❡❧♦ q✉❡ é ❝❤❛♠❛❞♦ ❤♦❥❡ ❛ Pr✐♠❡✐r❛ ❘❡✈♦❧✉çã♦ ❞❛s ❈♦r❞❛s✳ ❊♠ ✉♠ ❜r❡✈❡ ♣❡rí♦❞♦✱ ❤♦✉✈❡ ❣r❛♥❞❡s ♠✉❞❛♥ç❛s ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s✳ ❖ ♣r✐♥❝✐♣❛❧ ❞❡❧❡s✱ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❝❛♥❝❡❧❛♠❡♥t♦ ❞❡ ❛♥♦♠❛❧✐❛s ♥❛s t❡♦r✐❛s ❞♦ t✐♣♦ ■✶✱ ❬✶❪✳
❉✉r❛♥t❡ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛ t❡♦r✐❛ ♣❛ss♦✉ ❛ ❝♦♥t❛r ❝♦♠ ❝✐♥❝♦ t✐♣♦s ❞✐❢❡r❡♥t❡s ❞❡ ❞❡s❝r✐çã♦✱ ♦ q✉❡✱ ❝♦♠♦ ❡s♣❡r❛❞♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ ✉♥✐✜❝❛çã♦✱ é ❜❛st❛♥t❡ ❞❡s❛❣r❛❞á✈❡❧✳ ❊♠ ✉♠ ❣r❛♥❞❡ tr❛❜❛❧❤♦ ❞❡ r❡✈✐sã♦✱ ❊✳ ❲✐tt❡♥ ❝♦♥s❡❣✉✐✉ ❝♦♥❡❝tá✲❧❛s ❡♠ ✉♠❛ sér✐❡ ❞❡ ❞✉❛❧✐❞❛❞❡s✱ ❛♣♦♥t❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ♣♦ssí✈❡❧ t❡♦r✐❛ ❡♠ 11❞✐♠❡♥sõ❡s q✉❡ ❞❛r✐❛ ♦r✐❣❡♠ à t♦❞❛s ❛s ❞❡s❝r✐çõ❡s ❝♦♥❤❡❝✐❞❛s ❞❛s ❝♦r❞❛s✱ ❛ t❡♦r✐❛
▼✳ ❊ss❛ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ❙❡❣✉♥❞❛ ❘❡✈♦❧✉çã♦ ❞❛s ❈♦r❞❛s✳
◆❡ss❡ ♠❡✐♦ t❡♠♣♦✱ ✉♠ ❣r❛♥❞❡ tr❛❜❛❧❤♦ ❞❡ ❏✳ P♦❧❝❤✐♥s❦✐✱ ❬✷❪✱ ❧❡✈♦✉ ♦s t❡ór✐❝♦s ❞❡ ❝♦r❞❛s ❛ ❝♦♥s✐❞❡r❛r❡♠ s❡r✐❛♠❡♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♦❜❥❡t♦sp✲❞✐♠❡♥s✐♦♥❛✐s✱ ❛s ❉✲❜r❛♥❛s✱ q✉❡ sã♦ ❤♦❥❡ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛
♥❛ ár❡❛✳
❙❡❣✉✐♥❞♦ ❛ ❡✈♦❧✉çã♦ ❞❛ t❡♦r✐❛✱ ♥♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ 90✱ ❏✳ ▼❛❧❞❛❝❡♥❛ ✐♥tr♦❞✉③✐✉ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛
❆❞❙✴❈❋❚ ❡✱ ❡♠ t❡r♠♦s ♣rát✐❝♦s✱ ❢✉♥❞♦✉ ✉♠ ♥♦✈♦ r❛♠♦ ❞❡ ♣❡sq✉✐s❛ ❡ ✐♠♣✉❧s✐♦♥♦✉ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s✱ tr❛③❡♥❞♦✲❛✱ ✉♠❛ ✈❡③ ♠❛✐s✱ ♣❛r❛ ❛ ✈❛♥❣✉❛r❞❛ ❞❛ ❢ís✐❝❛ t❡ór✐❝❛ ❡ ♣r♦♠♦✈❡♥❞♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❡♥tr❡ ✈ár✐❛s ár❡❛s ❞❡ ♣❡sq✉✐s❛✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ t❡①t♦ é ✐♥tr♦❞✉③✐r ♦s ❢♦r♠❛❧✐s♠♦s ♠❛✐s tr❛❞✐❝✐♦♥❛✐s ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❡ ❞❡s❝r❡✈❡r ✐♥tr♦❞✉t♦r✐❛♠❡♥t❡ ❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ▼❛❧❞❛❝❡♥❛✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s é ♠♦t✐✈❛❞❛ ♣♦r ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❛çã♦ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✱ ❝♦♠ ✉♠ ❡st✉❞♦ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♣❛rt✐❝✉❧❛r✐❞❛❞❡s✳ ❆ t❡♦r✐❛ q✉â♥t✐❝❛ é ♦❜t✐❞❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞♦ ❝❛❧✐❜r❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③ ❡ ♦ ❡s♣❡❝tr♦ ❞❡ ❡st❛❞♦s ❞❛ ❝♦r❞❛ ❛❜❡rt❛ ❡ ❞❛ ❝♦r❞❛ ❢❡❝❤❛❞❛ é ❛♥❛❧✐s❛❞♦✳ ❉✐s❝✉t❡✲s❡
✶◆♦♠❡♥❝❧❛t✉r❛ ❛ s❡r ❡s❝❧❛r❡❝✐❞❛ ❞✉r❛♥t❡ ♦ t❡①t♦✳
✷ t❛♠❜é♠ ❛ ❞✉❛❧✐❞❛❞❡ ❚ ✱ ✐♥❢❡r✐♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❛s ❉✲❜r❛♥❛s✳
❊♠ s❡❣✉✐❞❛✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❛ ❛çã♦ ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛ é ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ ♠♦❞♦ ❛ ✐♥❝❧✉✐r ❛ s✉♣❡rs✐♠❡tr✐❛ ❧♦❝❛❧ ♥❛ ❢♦❧❤❛✲♠✉♥❞♦✳ ❆ q✉❛♥t✐③❛çã♦ ❞❛ t❡♦r✐❛ é ❢❡✐t❛ ❝♦✈❛r✐❛♥t❡♠❡♥t❡ ♣❡❧♦ ♣r♦❝❡❞✐♠❡♥t♦ ❝❛♥ô♥✐❝♦ ❡ t❛♠❜é♠ ♥♦ ❝❛❧✐❜r❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③✳ ❆ ♣❛rt✐r ❞❡ ✉♠❛ ❛♥á❧✐s❡ ❞♦ ❡s♣❡❝tr♦✱ ❛ s✉♣❡rs✐♠❡tr✐❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ é s✉❣❡r✐❞❛✱ s❡♥❞♦ ✐♥tr♦❞✉③✐❞❛ ♥♦ ❝❛♣ít✉❧♦ ✹✱ ❝♦♠ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ●r❡❡♥✲❙❝❤✇❛r③ ❞❛s s✉♣❡r❝♦r❞❛s✳ ◆❡st❡ ♣♦♥t♦✱ ❛ ❡①t❡♥sã♦ ❞❛ ❛çã♦ ❜♦sô♥✐❝❛ ♥ã♦ é ❡✈✐❞❡♥t❡ ❡ ❛ s✐♠❡tr✐❛ ❦❛♣❛ é ✐♥tr♦❞✉③✐❞❛ ♣❛r❛ ❜❛❧❛♥❝❡❛r ♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❜♦sô♥✐❝♦s ❡ ❢❡r♠✐ô♥✐❝♦s✳ ❖ ❡s♣❡❝tr♦ ❞❛ t❡♦r✐❛ é ♦❜t✐❞♦ ♥♦ ❝❛❧✐❜r❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③ ❡ ❞✐s❝✉t✐❞♦ ❡①t❡♥s✐✈❛♠❡♥t❡✱ ❛♥❛❧✐s❛♥❞♦✱ ✐♥❝❧✉s✐✈❡✱ ♦s ❡❢❡✐t♦s ❞❛ ❞✉❛❧✐❞❛❞❡ ❚ ♥♦ s❡t♦r ❢❡r♠✐ô♥✐❝♦✳
❖ ❝❛♣ít✉❧♦ ✺ ✐♥tr♦❞✉③ ❛s ❛çõ❡s ❡❢❡t✐✈❛s ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❛ss♦❝✐❛❞❛s ❛ t❡♦r✐❛ ❞❡ s✉♣❡r❝♦r❞❛s✳ ❆ ❛çã♦ ❞❡ ❉✐r❛❝✲❇♦r♥✲■♥❢❡❧❞ é ✐♥tr♦❞✉③✐❞❛✱ ♦❜t❡♥❞♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ s✉♣❡rs✐♠❡tr✐❛ ❡ s✐♠❡tr✐❛ ❦❛♣❛ ❞♦ ❝❛♠♣♦ ❞❡ ❝❛❧✐❜r❡ ❞❛ ❉✲❜r❛♥❛✳ ❆s ❛çõ❡s ❞❛ s✉♣❡r❣r❛✈✐❞❛❞❡ sã♦ ✐♥tr♦❞✉③✐❞❛s✱ ❝♦♠ ✉♠❛ rá♣✐❞❛ ♠♦t✐✈❛çã♦ ❞❛ t❡♦r✐❛ ▼✳
P♦r ✜♠✱ ♦ ❝❛♣ít✉❧♦ ✻ ❛♣r❡s❡♥t❛ ❛❧❣✉♥s ❛s♣❡❝t♦s ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❆❞❙✴❈❋❚✳ ❆ t❡♦r✐❛ ❞❡N = 4❙✉♣❡r
❨❛♥❣✲▼✐❧❧s é ✐♥tr♦❞✉③✐❞❛ ❡ ❞✐s❝✉t✐❞❛✳ ❆ s✉♣❡r❣r❛✈✐❞❛❞❡II✲❇ é ❞❡s❡♥✈♦❧✈✐❞❛✱ ❝♦♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ s✉❛s
s♦❧✉çõ❡s ❡①tr❡♠❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦ ❝❛s♦ ❞❛3✲❜r❛♥❛✳ ❖ ❡♥✉♥❝✐❛❞♦ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ▼❛❧❞❛❝❡♥❛ é ❛♣r❡s❡♥t❛❞♦
❡♠ s❡✉s ✈ár✐♦s ♥í✈❡✐s ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❥✉♥t♦ ❛ ❛❧❣✉♠❛s ❡♥tr❛❞❛s ❞♦ ❞✐❝✐♦♥ár✐♦ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✳
❈❛♣ít✉❧♦ ✷
❈♦r❞❛ ❇♦sô♥✐❝❛
◆❡st❡ ❝❛♣ít✉❧♦ é ✐♥tr♦❞✉③✐❞❛ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❜♦sô♥✐❝❛s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s✳ ❆ t❡♦r✐❛ q✉â♥t✐❝❛ é ♦❜t✐❞❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞♦ ❝❛❧✐❜r❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③ ❡ ♦ ❡s♣❡❝tr♦ ❞❡ ❡st❛❞♦s ❞❛ ❝♦r❞❛ ❛❜❡rt❛ ❡ ❞❛ ❝♦r❞❛ ❢❡❝❤❛❞❛ é ❛♥❛❧✐s❛❞♦✳ ❉✐s❝✉t❡✲s❡ t❛♠❜é♠ ❛ ❞✉❛❧✐❞❛❞❡ ❚ ✱ ✐♥❢❡r✐♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❛s ❉✲❜r❛♥❛s ❡ ❞✐s❝✉t✐♥❞♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ✐♠♣❧✐❝❛çõ❡s✳ ❆ ❜❛s❡ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ ❝❛♣ít✉❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✱ ✹✱ ✺✱ ✻❪✳
✷✳✶ ❆ ❆çã♦ ❞❛
p
✲❜r❛♥❛
❆♦ ❡st❡♥❞❡r ♦ ❡st✉❞♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❡ ♦❜❥❡t♦s ♣♦♥t✉❛✐s ✭❞✐♠❡♥sã♦p= 0✮ à ♦❜❥❡t♦s ❞❡ ❞✐♠❡♥sã♦p≥1
♦ ♣r♦❝❡❞✐♠❡♥t♦ ♠❛✐s ♥❛t✉r❛❧ é ❛♥❛❧✐s❛r ♣♦ssí✈❡✐s ❣❡♥❡r❛❧✐③❛çõ❡s ❞❛ ❛çã♦S0 ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✳
❆ ❛çã♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s✉❛ ❧✐♥❤❛✲♠✉♥❞♦✱ s✉❛ tr❛✲ ❥❡tór✐❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❚r❛❥❡tór✐❛s ❞❡ ♦❜❥❡t♦s ❞❡ ❞✐♠❡♥sã♦p✭♣✲❜r❛♥❛s✮ ❣❡r❛♠ s✉♣❡r❢í❝✐❡sp+ 1❞✐♠❡♥✲
s✐♦♥❛✐s ❛♦ s❡ ♣r♦♣❛❣❛r❡♠✳ P♦rt❛♥❞♦ é ♥❛t✉r❛❧ ❝♦♥str✉✐r s✉❛ ❛çã♦Sp ♥❛ ❢♦r♠❛
Sp∝
ˆ q
−det(gµν∂αX
µ∂βXν)dp+1σ,
♥❛ q✉❛❧ ♦ ✐♥t❡❣r❛♥❞♦ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡p+ 1❞✐♠❡♥s✐♦♥❛❧✱σα✭❝♦♠α= 0,1, ..., p✮ é ❛ ♣❛r❛♠❡tr✐③❛✲
çã♦ ❡s❝♦❧❤✐❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ✈♦❧✉♠❡✲♠✉♥❞♦ ❡Xµ(σ)✭❝♦♠ µ= 0,1, ..., D−1✮ sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❛
♣✲❜r❛♥❛ ♥♦ ❡s♣❛ç♦ ❡♠ q✉❡ s❡ ♠♦✈❡✱ ❝❤❛♠❛❞♦ ❡s♣❛ç♦✲❛❧✈♦ ✭❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✲❛❧✈♦ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ❞✐♠❡♥sã♦D❞♦ ❡s♣❛ç♦✲t❡♠♣♦✮✳
■♥tr♦❞✉③✐♥❞♦ ✉♠ t❡♥s♦r ♠étr✐❝♦ ❛✉①✐❧✐❛r hαβ ♥♦ ✈♦❧✉♠❡✲♠✉♥❞♦✱ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠❛ ❛çã♦ Sσ
❝❧❛ss✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ àSp✱ ♦✉ s❡❥❛✱ q✉❡ r❡♣r♦❞✉③ ❛s ♠❡s♠❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦✿
Sσ=
ˆ √
−h
−T2phαβ∂αX·∂βX+ Λp
dp+1σ. ✭✷✳✶✮
❖ t❡r♠♦ ❝♦s♠♦❧ó❣✐❝♦Λp ♥ã♦ é ❛r❜✐trár✐♦✳ ■ss♦ é ✈❡r✐✜❝❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
δSσ
δhαβ = 0⇒Tp
∂αX·∂βX−
1 2hαβ(h
ργ∂
ρX·∂γX)
+ Λphαβ= 0.
❚♦♠❛♥❞♦ ♦ tr❛ç♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✿
Tp hαβ∂αX·∂βX
= Tp
2 (p+ 1) h
αβ∂
αX·∂βX
−Λp(p+ 1).
✹ ❈♦♠♦Λp ✐♥❞❡♣❡♥❞❡ ❞❡Xµ✱ ❛ ú♥✐❝❛ s♦❧✉çã♦ ♣♦ssí✈❡❧ é∂αX·∂βX ∝hαβ✳ ❙✉❜st✐t✉✐♥❞♦ ❡ss❡ r❡s✉❧t❛❞♦ ♥❛
❡q✉❛çã♦ ❛❝✐♠❛✱ ♦❜t❡♠✲s❡Λp∝T2p(p−1)✳
✷✳✷ ❆ ❆çã♦ ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛
❆ ❛çã♦ ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛ ✭✶✲❜r❛♥❛✮ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛s ❛çõ❡s ❞✐s❝✉t✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳ ❆♦ s❡ ♣r♦♣❛❣❛r✱ ❛ ❝♦r❞❛ ❢♦r♠❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❛ ❢♦❧❤❛✲♠✉♥❞♦✱ ❡ sã♦ ♥❡❝❡ssár✐♦s ❞♦✐s ♣❛râ♠❡tr♦s ♣❛r❛ ❞❡s❝r❡✈ê✲❧❛✿ σ0
≡τ ❡ σ1
≡σ✳ ❉❡ ❢♦r♠❛ ❣❡r❛❧✱ τ ❡st❛rá ❛ss♦❝✐❛❞♦ ❛ ✉♠ t❡♠♣♦
♥❛ ❢♦❧❤❛✲♠✉♥❞♦ ❡ σ ∈[0, π]✱ t❛❧ q✉❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡✜♥✐❞♦ ❝♦❜r❡ ✉♥✐✈♦❝❛♠❡♥t❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❛ ❝♦r❞❛
♣❛r❛ ✉♠ ❞❛❞♦ τ✳ P❛r❛ p = 1✱ Λp ❛♥✉❧❛✲s❡ ❡♠ Sσ ❡✱ ❛❧é♠ ❞✐ss♦✱ ✉♠❛ ♣♦ssí✈❡❧ ❞✐♥â♠✐❝❛ ❞❛ ♠étr✐❝❛ é
❧✐♠✐t❛❞❛✱ ✉♠❛ ✈❡③ q✉❡ ♦ t❡r♠♦ ❞❡ ❊✐♥st❡✐♥ é ✉♠ ✐♥✈❛r✐❛♥t❡ t♦♣♦❧ó❣✐❝♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱ ♣r♦♣♦r❝✐♦♥❛❧ à ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❛ s✉♣❡r❢í❝✐❡ ❞❡s❝r✐t❛✶✳ ❆ ❛çã♦ ❞❛ ❝♦r❞❛ é✱ ♣♦rt❛♥t♦✷✿
Sb=−
T1
2
ˆ √
−hhαβ∂
αX·∂βX d2σ. ✭✷✳✷✮
P♦r ✉♠❛ s✐♠♣❧❡s ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♣❛râ♠❡tr♦T1t❡♠ ❞✐♠❡♥sã♦[T1] =M/T✳ ❊♠ ✉♠❛
❛♥❛❧♦❣✐❛ ✐♥❣ê♥✉❛ ❝♦♠ ❛ ❛çã♦ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✱T1=λc✱ ❡♠ q✉❡λé ✉♠❛ ❞❡♥s✐❞❛❞❡ ❧✐♥❡❛r ❞❡ ♠❛ss❛
❡c é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦✳ P♦ré♠ t❛❧ ✐♥t❡r♣r❡t❛çã♦ ♥ã♦ é ❝♦♠♣❛tí✈❡❧ ♣♦r ❞✉❛s r❛③õ❡s ♣r✐♥❝✐♣❛✐s✿
❝♦♠♦ ✉♠❛ t❡♦r✐❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ♥❛t✉r❡③❛✱ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❞❡✈❡ ❞❡s❝r❡✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❜ós♦♥s ❞❡ ❝❛❧✐❜r❡✱ q✉❡ ♥ã♦ ♣♦ss✉❡♠ ♠❛ss❛❀ ❡ ❛s s♦❧✉çõ❡s ❝❧áss✐❝❛s ✐♠♣❧✐❝❛♠ q✉❡ ♦s ❡①tr❡♠♦s ❞❛ ❝♦r❞❛ ❞❡✈❡♠ ♠♦✈❡r✲s❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡c✳ ❈♦♠♦ ♦ ♦❜❥❡t♦ ❡♠ q✉❡stã♦ é ✉♠❛ ❝♦r❞❛✱T1=T /c✱ ❡♠ q✉❡T é ✐♥t❡r♣r❡t❛❞♦
❝♦♠♦ s✉❛ t❡♥sã♦✳ ◆♦ s✐st❡♠❛ ♥❛t✉r❛❧ ❞❡ ✉♥✐❞❛❞❡s ✭~=c = 1✮✱ T ❡stá ❛ss♦❝✐❛❞♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ls ❞❛
❝♦r❞❛ ♣♦rT = 1
πl2
s✳
❈♦♠♦ ❡♠ q✉❛❧q✉❡r ♦✉tr❛ t❡♦r✐❛ ❢ís✐❝❛✱ ❛ ❛♥á❧✐s❡ ❞❛s s✐♠❡tr✐❛s ❞❛ ❛çã♦ ✭✷✳✷✮ é ❜❛st❛♥t❡ út✐❧✱ ♣❡r♠✐t✐♥❞♦ ✜①❛r ❞❡ ♠♦❞♦ ❝♦♥✈❡♥✐❡♥t❡ ❛ ♠étr✐❝❛ ❛✉①✐❧✐❛r✳ Sb ❛♣r❡s❡♥t❛ ❛s s❡❣✉✐♥t❡s s✐♠❡tr✐❛s✿
❼ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré
❆ s✐♠❡tr✐❛ ❞❡ P♦✐♥❝❛ré é ❡✈✐❞❡♥t❡ ❡ ♣♦❞❡ s❡r r❡s✉♠✐❞❛ ♣♦r δXµ =aµ
νXν+bµ✳ ❖ ♣❛râ♠❡tr♦ ❞❡
tr❛♥s❧❛çã♦ ébµ✱ ❡♥q✉❛♥t♦ ♦ ♣❛râ♠❡tr♦ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ éaµ ν❀
❼ ❘❡♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❋♦❧❤❛✲▼✉♥❞♦ ✭❉✐❢❡♦♠♦r✜s♠♦✮ ❉❡ ❢♦r♠❛ ❣❡r❛❧✿
σa →
fα(τ, σ) =σ′α
hαβ(τ, σ) = ∂f ρ ∂σα
∂fγ
∂σβhργ(τ′, σ′)
❼ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧
❆ s✐♠❡tr✐❛ ❞❡ ❲❡②❧ ❝♦rr❡s♣♦♥❞❡✱ ❜❛s✐❝❛♠❡♥t❡✱ ❛ ✉♠ r❡❡s❝❛❧♦♥❛♠❡♥t♦ ❞❛ ♠étr✐❝❛ ♥❛ ❢♦❧❤❛✲♠✉♥❞♦✿
hαβ→λ(τ, σ)hαβ✳
❚♦♠❛♥❞♦ ✉♠❛ ✈❛r✐❛çã♦ ❞❛ ❛çã♦Sb ❡♠ r❡❧❛çã♦ ❛ ♠étr✐❝❛ ❛✉①✐❧✐❛rhαβ✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ❡ss❛ s✐♠❡tr✐❛
✐♠♣❧✐❝❛ q✉❡ ♦ tr❛ç♦ ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ✭Tαβh
αβ✮ é ♥✉❧♦✳
❉❡✈✐❞♦ às ❞✉❛s ú❧t✐♠❛s s✐♠❡tr✐❛s✱hαβ♣♦❞❡ s❡r ✜①❛❞❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ✭❞✉❛s s✐♠❡tr✐❛s ❞❡ r❡♣❛r❛♠❡tr✐③❛çã♦
❡ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❲❡②❧ ✜①❛♥❞♦ ❛s três ❝♦♠♣♦♥❡♥t❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❛ ♠étr✐❝❛ ❛✉①✐❧✐❛r✮✳
✶❉❡ ❢❛t♦✱ ❛♦ ❝♦♥s✐❞❡r❛r ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❣❡♥ér✐❝♦✱ ♦ ♣❛♣❡❧ ❞♦ t❡r♠♦ ❞❡ ❊✐♥st❡✐♥ é ✐♠♣♦rt❛♥t❡ ♣♦✐s ❞á ♦ ❛❝♦♣❧❛♠❡♥t♦
❝♦♠ ✉♠ ❞♦s ❝❛♠♣♦s ❞❡ ❢✉♥❞♦✳ ▼❛✐s ❞❡t❛❧❤❡s sã♦ ❢♦r♥❡❝✐❞♦s ♥❛ s❡çã♦ ✷✳✻✳✷✳
✷❉❡ss❡ ♣♦♥t♦ ❡♠ ❞✐❛♥t❡ s❡rá ✉t✐❧✐③❛❞❛ ❛ ♠étr✐❝❛ ♣❧❛♥❛ηµν = (−1,1, ...,1)✱ ❛ ♥ã♦ s❡r q✉❡ s❡❥❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞✐t♦ ♦
✺
✷✳✸ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ ❙♦❧✉çõ❡s
❆ ❛çã♦ Sb ❢♦r♥❡❝❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦✳ ❯♠ ❛ss♦❝✐❛❞♦ à ♠étr✐❝❛ ❛✉①✐❧✐❛r✱
∂αXµ∂βXµ−
1 2hαβh
γρ∂
γXµ∂ρXµ= 0,
❡ ♦✉tr♦ ❛ss♦❝✐❛❞♦ ❛♦s ❝❛♠♣♦sXµ(τ, σ)✱
∂α
√
−hhαβ∂βXµ
= 0
P❛r❛ ♦ ❝❛s♦ ❞❡ ❢♦❧❤❛s✲♠✉♥❞♦ ❞❡s❝r✐t❛s ♣❡❧❛s ❝♦r❞❛s ❛❜❡rt❛s ❡ ❢❡❝❤❛❞❛s ❧✐✈r❡s ✭❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s✉♣❡r❢í❝✐❡s s❡♠ ❡①❝❡♥tr✐❝✐❞❛❞❡s t♦♣♦❧ó❣✐❝❛s✮✱ é ♣♦ssí✈❡❧ ❡s❝♦❧❤❡r ♦ ❝❛❧✐❜r❡ ❝♦♥❢♦r♠❡✱ ❡♠ q✉❡
hαβ=ηαβ=
−1 0
0 1
,
♦✉ s❡❥❛✱ ❛ ♠étr✐❝❛ ♣❧❛♥❛✳ ❉❡✜♥✐♥❞♦Pµ
a ≡T ∂aXµ✱ ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛Xµé s✐♠♣❧✐✜❝❛❞❛ ♣❛r❛
∂αPµ
α = 0. ✭✷✳✸✮
❡✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♠étr✐❝❛ ❛✉①✐❧✐❛r ❢♦✐ ✜①❛❞❛✱ s✉❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡✈❡ s❡r ✐♠♣♦st❛ ❝♦♠♦ ✈í♥❝✉❧♦✿
(Pτ±Pσ)2 = 0. ✭✷✳✹✮
❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ♣♦❞❡♠ s❡r r❡❡s❝r✐t❛s ♥❛s ❝♦♦r❞❡♥❛❞❛sσ±=τ±σ✳ ❈♦♠♦∂±= 12(∂τ±∂σ)✱
♦❜t❡♠✲s❡✿
∂+∂−Xµ= 0 (∂+X)2= (∂−X)2= 0 ✭✷✳✺✮
P❡❧❛s ❡q✉❛çõ❡s ❛❝✐♠❛✱Xµ(τ, σ)♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❡♠ ❞✉❛s s♦❧✉çõ❡s q✉❡ s❡ ♣r♦♣❛❣❛♠ ❡♠ ❞✐r❡çõ❡s
♦♣♦st❛s ♥❛ ❢♦❧❤❛✲♠✉♥❞♦✱Xµ =Xµ
R(σ−) +XLµ(σ+)✳ ❊①♣❛♥❞✐♥❞♦ ❡♠ sér✐❡s ❞❡ ❋♦✉r✐❡r ❝♦♠♣❧❡①❛s✿
XRµ(τ, σ) = x
µ R
2 +a
µ
0(τ−σ) +
X
m6=0
1
ma
µ
me−im(τ−σ) ✭✷✳✻✮
XLµ(τ, σ) = x
µ L
2 + ˜a
µ
0(τ+σ) +
X
m6=0
1
m˜a
µ
me−im(τ+σ). ✭✷✳✼✮
❊♠❜♦r❛ ♥ã♦ t❡♥❤❛ s✐❞♦ ♠♦str❛❞♦ ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡r✐✈❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ❡♥✈♦❧✈❡ ❛ ✜①❛çã♦ ❞❡ t❡r♠♦s ❞❡ s✉♣❡r❢í❝✐❡✳ P♦rt❛♥t♦✱ ❛s s♦❧✉çõ❡s ❞❡ ✭✷✳✺✮ ❞❡✈❡♠ s❡r ✜①❛❞❛s ♣❡❧❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❡ ♣❡❧❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞♦ ♣r♦❜❧❡♠❛ ✭❛✈❛❧✐❛❞❛s ♥♦s ❡①tr❡♠♦s σ∗ = 0, π✮✳ ❆ ✜①❛çã♦ ❞❛s
❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s é ❜❛st❛♥t❡ s✐♠♣❧❡s ❡ ♥ã♦ ❛♣r❡s❡♥t❛ r❡❧❡✈â♥❝✐❛ ♥❛ ❞✐s❝✉ssã♦ s✉❜s❡q✉❡♥t❡✳ ❏á ❛ ✜①❛çã♦ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✱ q✉❡ sã♦ ♦❜t✐❞❛s ♣❡❧♦ t❡r♠♦ ❞❡ s✉♣❡r❢í❝✐❡
ˆ
dτ
∂Xµ
∂σ δXµ
σ=π
− ∂X
µ
∂σ δXµ
σ=0
= 0,
❞á ♦r✐❣❡♠ à ❞✉❛s ❢❛♠í❧✐❛s ❞❡ s♦❧✉çõ❡s✿ ✶✳ ❈♦r❞❛s ❋❡❝❤❛❞❛s
❚❛✐s s♦❧✉çõ❡s ♦❜❡❞❡❝❡♠ à ❝♦♥❞✐çã♦ ❞❡ q✉❛s✐✲♣❡r✐♦❞✐❝✐❞❛❞❡ Xµ(τ, σ+π) =Xµ(τ, σ) +Bµπ✳ P❡❧❛s
❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞❡✜♥✐❞❛s✱ ♦s ♠♦❞♦s ❞❡ ♦s❝✐❧❛çã♦ í♠♣❛r❡s sã♦ ♥✉❧♦s✳ ❆❧é♠ ❞✐ss♦˜aµ0−a
µ
✻ ❡ ❡ss❛ é ❛ ú♥✐❝❛ r❡❧❛ç❛♦ ❡♥tr❡ ♦s ♠♦❞♦s ❞✐r❡✐t♦s ✭❘✮ ❡ ❡sq✉❡r❞♦s ✭▲✮✸✳ ❉❡ ❢❛t♦✱ ❛ ❞❡♥♦♠✐♥❛çã♦
❝♦r❞❛ ❢❡❝❤❛❞❛ t❡♠ s❡♥t✐❞♦ ❛♣❡♥❛s ♣❛r❛ Bµ= 0✭❝♦♥❞✐çã♦ ❛ss♦❝✐❛❞❛ à s✐♠❡tr✐❛ ❞❡ tr❛♥s❧❛çã♦ ❡♠σ
❡ s✐♠❡tr✐❛ ❞❡ ♣❛r✐❞❛❞❡ σ↔ −σ♥❛ ❢♦❧❤❛✲♠✉♥❞♦✮ ♦✉ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝♦♠♣❛❝t✐✜❝❛çã♦✳ XRµ(τ, σ) = x
µ R
2 +lsα
µ
0(τ−σ) +i
ls
2
X
m6=0
1
mα
µ me−
2im(τ−σ) ✭✷✳✽✮
XLµ(τ, σ) = x
µ L
2 +lsα˜
µ
0(τ+σ) +i
ls
2
X
m6=0
1
mα˜
µ
me−2im(τ+σ). ✭✷✳✾✮
✷✳ ❈♦r❞❛s ❆❜❡rt❛s
✭❛✮ ❈♦♥❞✐çã♦ ❞❡ ❊①tr❡♠♦ ❋✐①♦
❆ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t✱ ♦✉ ❝♦♥❞✐çã♦ ❞❡ ❡①tr❡♠♦ ✜①♦✱ ✐♠♣❧✐❝❛ q✉❡ Pµ
τ (τ, σ∗) = 0✳ ❉❡ss❛
❢♦r♠❛✱ aµ
m = −a˜µm✳ ❚❛❧ ❝♦♥❞✐çã♦ q✉❡❜r❛ ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ P♦✐♥❝❛ré ✭♥ã♦ ❤á ❝♦♥s❡r✈❛çã♦ ❞❡
♠♦♠❡♥t♦✮✳ ❆❧é♠ ❞✐ss♦ é ✐♥❝♦♠♣❛tí✈❡❧ ♣❛r❛µ= 0✳ ◆❡ss❡ ❝❛s♦ X0 s❡r✐❛ ❡stát✐❝♦✱ ✐♥✈❛❧✐❞❛♥❞♦
s✉❛ ✐♥t❡r♣r❡t❛çã♦ ❝♦♠♦ ❝♦♦r❞❡♥❛❞❛ t❡♠♣♦r❛❧✳ ❊♠ ❣❡r❛❧✿
Xµ(τ, σ) =xµ+l2
swµσ+ils
X
m6=0
1
mα
µ
me−imτsin (mσ). ✭✷✳✶✵✮
✭❜✮ ❈♦♥❞✐çã♦ ❞❡ ❊①tr❡♠♦ ▲✐✈r❡
❆ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥✱ ♦✉ ❝♦♥❞✐çã♦ ❞❡ ❡①tr❡♠♦ ❧✐✈r❡✱ ✐♠♣❧✐❝❛ q✉❡ Pµ
σ (τ, σ∗) = 0 ❡✱ ❝♦♥s❡✲
q✉❡♥t❡♠❡♥t❡✱ aµ
m = ˜aµm✳ ❆tr❛✈és ❞♦s ✈í♥❝✉❧♦s ❞❛❞♦s ❡♠ ✭✷✳✹✮✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❡①tr❡♠♦ ❧✐✈r❡
✐♠♣❧✐❝❛ q✉❡Pτ(τ, σ∗)2 = 0✳ ❖✉ s❡❥❛✱ ♦s ❡①tr❡♠♦s ❧✐✈r❡s ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛ ❞❡s❝r✐t❛ ♣♦r ✭✷✳✷✮
♠♦✈❡♠✲s❡ à ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✳ ❆s s♦❧✉çõ❡s sã♦ ❞❛ ❢♦r♠❛✿
Xµ(τ, σ) =xµ+l2spµτ+ils
X
m6=0
1
mα
µ
me−imτcos (mσ). ✭✷✳✶✶✮
✭❝✮ ❈♦♥❞✐çã♦ ▼✐st❛
❆s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♠✐st❛s ♣♦ss✉❡♠ ✉♠ ❡①tr❡♠♦ q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✷❛ ❡ ✉♠ ❡①tr❡♠♦ q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✷❜✳ P♦r ❡①❡♠♣❧♦✱ ❛♦ ❝♦♥s✐❞❡r❛r ♦ ❡①tr❡♠♦σ= 0✜①♦ ❡ ♦ ❡①tr❡♠♦σ=π
❧✐✈r❡✱ ♦❜t❡♠✲s❡✿
Xµ(τ, σ) =xµ+ X
r∈Z+1 2
1
rα
µ
re−irτsin (rσ).
❈♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡ ❝♦r❞❛s ❢❡❝❤❛❞❛s ❝♦♠ Bµ6= 0s❡rã♦ ❞✐s❝✉t✐❞❛s ❛♣❡♥❛s ♥❛ s❡çã♦ ✷✳✼✳
❈♦♥❞✐çõ❡s ♠✐st❛s ♥ã♦ s❡rã♦ ❛❜♦r❞❛❞❛s✳
❆ ❝♦♥❞✐çã♦ ❞❡ r❡❛❧✐❞❛❞❡ ❞❡Xµ✐♠♣❧✐❝❛ q✉❡ ♦s ♠♦❞♦s ❞❡ ❋♦✉r✐❡raµ
ms❛t✐s❢❛③❡♠ àα µ
−m= (αµm)†✳ ❆❧é♠
❞✐ss♦✱ ♦s ♠♦❞♦s ③❡r♦ sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠ ♦ ♠♦♠❡♥t♦ ❞❡ tr❛♥s❧❛çã♦ ❞❛ ❝♦r❞❛pµ✳ P❛r❛ ❛ ❝♦r❞❛ ❢❡❝❤❛❞❛✱
αµ0 = ˜αµ0 = 1
2lspµ✳ P❛r❛ ❛ ❝♦r❞❛ ❛❜❡rt❛✱α
µ
0 =lspµ✳
❉❡ ♣♦ss❡ ❞❛s s♦❧✉çõ❡s ❝❧áss✐❝❛s✱ é ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r t♦❞❛s ❛s ❝♦rr❡♥t❡s ❝♦♥s❡r✈❛❞❛s ♥❛ ❢♦❧❤❛✲♠✉♥❞♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦❜t❡r ♦s ♦❜s❡r✈á✈❡✐s ❞❡ r❡❧❡✈â♥❝✐❛✱ ❝♦♠♦ ❛ ❡♥❡r❣✐❛ ❡ ♦ ❡s♣❡❝tr♦ ❞❡ ♠❛ss❛ ❝❧áss✐❝♦s✳
Pµ
α =T ∂αXµ → ❙✐♠❡tr✐❛ ❞❡ ❚r❛♥s❧❛çã♦✳
Jαµν =XµPαν−XνPαµ → ❙✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✳
Tαβ=Pα·Pβ−(tra¸co) → ❚❡♥s♦r ❊♥❡r❣✐❛✲▼♦♠❡♥t♦✳
✸❍á ❛✐♥❞❛ ✉♠ t✐♣♦ ❞❡ ♣r♦❥❡çã♦ q✉❡ ♣♦❞❡ r❡❧❛❝✐♦♥❛r ♦s ♠♦❞♦s ❞❛ t❡♦r✐❛ q✉❡ ❡stá ❛ss♦❝✐❛❞♦ à ✐♥✈❡rsã♦ ❞♦s ❡①tr❡♠♦s ❞❛
❝♦r❞❛✱ ❜❛s✐❝❛♠❡♥t❡ ♣❡❧❛ tr♦❝❛σ↔π−σ♥❛s s♦❧✉çõ❡s✳ ❈♦r❞❛s ❝✉❥❛s s♦❧✉çõ❡s ❝❧áss✐❝❛s sã♦ s✐♠étr✐❝❛s ♣♦r ❡ss❛ tr❛♥s❢♦r♠❛çã♦
✼ ❈❛❧❝✉❧❛♥❞♦ ❛s ❝❛r❣❛s ❝♦♥s❡r✈❛❞❛s✱ sã♦ ♦❜t✐❞♦s ♦s ❣❡r❛❞♦r❡s ❞❡ s✐♠❡tr✐❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡ ❛ ❤❛♠✐❧✲ t♦♥✐❛♥❛ H ❞❛ ❝♦r❞❛ ❜♦sô♥✐❝❛✳ ❆ s✐♠❡tr✐❛ ❞❡ tr❛♥s❧❛çã♦ é tr✐✈✐❛❧✳ ❖s ❣❡r❛❞♦r❡s ❞❡ ▲♦r❡♥t③ sã♦ ❞❛❞♦s
♣♦r✿
❼ ❈♦r❞❛ ❋❡❝❤❛❞❛
Jµν =xµpν
−xνpµ
−i ∞ X n=1 1 n α µ
−nανn−αν−nαµn+ ˜α µ
−nα˜νn−α˜ν−nα˜µn
.
❼ ❈♦r❞❛ ❆❜❡rt❛
Jµν =xµpν
−xνpµ
−i ∞ X n=1 1 n α µ
−nανn−αν−nαµn
.
✷✳✹ ▼❛ss❛ ❡ ❊♥❡r❣✐❛ ❞❛ ❈♦r❞❛
❖ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ Tαβ é ♣r♦♣♦r❝✐♦♥❛❧ à ✈❛r✐❛çã♦ ❞❛ ❛çã♦ Sb ❡♠ r❡❧❛çã♦ à ♠étr✐❝❛ ❛✉①✐❧✐❛r✳
❉❡ ❡s♣❡❝✐❛❧ ✐♥t❡r❡ss❡ sã♦ ❛s ❝♦♠♣♦♥❡♥t❡sT++ ❡T−−✱ ❝✉❥♦s ♠♦❞♦s ❞❡ ❋♦✉r✐❡r sã♦ ❞❛❞♦s ♣♦r✿
Lm=
1 2
∞
X
n=−∞
αm−n·αn L˜m=
1 2
∞
X
n=−∞ ˜
αm−n·α˜n. ✭✷✳✶✷✮
Lm❡L˜msã♦ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❣❡r❛❞♦r❡s ❞❡ ❱✐r❛s♦r♦ ❡ ♣♦ss✉❡♠ ❡str❡✐t❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ♦s ❣❡r❛❞♦r❡s
❞❛ s✐♠❡tr✐❛ ❝♦♥❢♦r♠❡ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✹✳
❯♠❛ ✈❡③ q✉❡ ♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❞❡✈❡ s❡r ♥✉❧♦✱ ❝❛❞❛ ✉♠ ❞♦s ❣❡r❛❞♦r❡s ❞❡ ❱✐r❛s♦r♦ ❞❡✈❡ s❡ ❛♥✉❧❛r ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♦✉ s❡❥❛✱ ♦s ✈í♥❝✉❧♦s ❞❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❛ ♠étr✐❝❛ ♠❛♥✐s❢❡st❛♠✲s❡ ❝♦♠♦Lm= 0❡L˜m= 0♥❛ t❡♦r✐❛ ❝❧áss✐❝❛✳ ❊ss❡s sã♦ ♦s ✈í♥❝✉❧♦s ❞❡ ❱✐r❛s♦r♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱L0= ˜L0= 0
✐♠♣❧✐❝❛ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ ♠❛ss❛ ❞❛ ❝♦r❞❛✱ ❥á q✉❡ ❡st❛ é ❝❛❧❝✉❧❛❞❛ à ♣❛rt✐r ❞♦s ♠♦❞♦s ③❡r♦ ❞❛ s♦❧✉çã♦ ❝❧áss✐❝❛✱M2=
−pµp µ✳
❼ ❈♦r❞❛ ❋❡❝❤❛❞❛
T++= 2l2s
∞
X
m=−∞ ˜
Lme−2im(τ+σ)
T−−= 2l2s
∞
X
m=−∞
Lme−2im(τ−σ)
H= 2L0+ ˜L0
M2= 4
l2
s
∞
X
n=1
(α−n·αn+ ˜α−n·α˜n)
❼ ❈♦r❞❛ ❆❜❡rt❛
T++ =l2s
∞
X
m=−∞
Lme−im(τ+σ)
T−−=l2s
∞
X
m=−∞
Lme−im(τ−σ)
H=L0
M2= 2
l2
s
∞
X
n=1
α−n·αn
❖ ❡s♣❡❝tr♦ ❞❡ ♠❛ss❛ ❞❛ t❡♦r✐❛ ❝❧áss✐❝❛ é ❝♦♥tí♥✉♦✱ ❡♥tr❡t❛♥t♦ é ❡✈✐❞❡♥t❡ q✉❡ ♥ã♦ é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ❆ t❡♦r✐❛ q✉â♥t✐❝❛ t❛♠❜é♠ ❡stá s✉❥❡✐t❛ ❛ t❛❧ ✐rr❡❣✉❧❛r✐❞❛❞❡✱ ❡♠❜♦r❛ s✉❛ ♠❛♥✐❢❡st❛çã♦ s❡❥❛ ❛tr❛✈és ❞❡ ❡st❛❞♦s ❞❡ ♥♦r♠❛ ♥❡❣❛t✐✈❛✳ ❉❡ ❢❛t♦✱ ♥ã♦ ❤á ✐♥❝♦♥s✐stê♥❝✐❛s✳ ❆♣❡s❛r ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❡①♣❧í❝✐t❛ ❞❛s s♦❧✉çõ❡s ❝❧áss✐❝❛s✱ ♥ã♦ ❤á ❣❛r❛♥t✐❛ ❛❧❣✉♠❛ ❞❡ q✉❡ s❛t✐s❢❛ç❛♠ ❛♦s ✈í♥❝✉❧♦s ❞❡ ❱✐r❛s♦r♦✳ ❍á ❛✐♥❞❛ ✉♠❛ s✐♠❡tr✐❛ ❡s❝♦♥❞✐❞❛ q✉❡ ♣❡r♠✐t❡ ❛ ✐♠♣♦s✐çã♦ ❞♦s ✈í♥❝✉❧♦s ❞❡ ❢♦r♠❛ ❜❛st❛♥t❡ s✐♠♣❧❡s✱ ❝♦♠♦ s❡rá ✈✐st♦ ❛ s❡❣✉✐r✳
✽
✷✳✺ ❖ ❈❛❧✐❜r❡ ❞♦ ❈♦♥❡ ❞❡ ▲✉③
❆♦ ❝♦♥s✐❞❡r❛r ❛s s✐♠❡tr✐❛s ❞❛ ❛çã♦Sb❢♦r❛♠ ❝✐t❛❞❛s ❛ s✐♠❡tr✐❛ ❞❡ r❡♣❛r❛♠❡tr✐③❛çã♦ ♥❛ ❢♦❧❤❛✲♠✉♥❞♦ ❡
♦ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ ❲❡②❧✳ ❯♠❛ ✈❡③ ✜①❛❞♦ ♦ ❝❛❧✐❜r❡ ❝♦♥❢♦r♠❡ é ♣♦ssí✈❡❧ ❛✐♥❞❛ ❢❛③❡r ✉s♦ ❞❡ t❛✐s s✐♠❡tr✐❛s ❞❡✐①❛♥❞♦ ❛ ♠étr✐❝❛ ❛✉①✐❧✐❛r ✐♥❛❧t❡r❛❞❛✳ P❛r❛ tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐sσ′
a =σa+ǫa❡ηab=ηab−Ληab✱
❜❛st❛ q✉❡
∂aǫb+∂bǫa= Ληab+O ǫ2,Λǫ
, ✭✷✳✶✸✮
♣❛r❛ q✉❡ ♦ ❝❛❧✐❜r❡ ❡s❝♦❧❤✐❞♦ s❡❥❛ ♠❛♥t✐❞♦✳ ❖✉ s❡❥❛✱ ❛s tr❛♥s❢♦r♠❛çõ❡sσ± →f±(σ±) sã♦ ✉♠❛ s✐♠❡tr✐❛
r❡s✐❞✉❛❧✳ ▲❡✈❛♥❞♦ ✐ss♦ ❡♠ ❝♦♥t❛ ❡ ❧❡♠❜r❛♥❞♦ q✉❡∂+∂−f±= 0✱ ❡s❝♦❧❤❡✲s❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ♠♦❞♦ q✉❡
X+(τ, σ) =x++l2
sp+τ✱ ❡♠ q✉❡
X±= X0±XD−1/√2.
❊ss❛ ❡s❝♦❧❤❛ é ❞❡♥♦♠✐♥❛❞❛ ❝❛❧✐❜r❡ ❞♦ ❝♦♥❡✲❞❡✲❧✉③✳ ❆❣♦r❛✱ ✐♠♣♦♥❞♦ ♦s ✈í♥❝✉❧♦s(Pτ±Pσ)
2
= 0✱ ♦❜t❡♠✲s❡✿
∂τX−±∂σX− =
1 2p+l2
s
∂τXi±∂σXi
2
.
❙✉❜st✐t✉✐♥❞♦ ❛s s♦❧✉çõ❡s✱ ♦s ♠♦❞♦s α−
m sã♦ ❡s❝r✐t♦s ❡♠ ❢✉♥çã♦ ❞♦s ♠♦❞♦s ❞✐t♦s tr❛♥s✈❡rs❛✐s✱ αim✱ ❝♦♠
i= 1, . . . , D−2✿
α−m=
1 2p+l
s
∞
X
n=−∞
αim−nαin
!
.
❉❡ss❛ ❢♦r♠❛✱ ♦s ✈í♥❝✉❧♦s ❞❡ ❱✐r❛s♦r♦ sã♦ s❛t✐s❢❡✐t♦s ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ❡ ♦ ❡s♣❡❝tr♦ ❞❡ ♠❛ss❛ t♦r♥❛✲s❡ ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✱ s❡♥❞♦ ❝♦♥st✐t✉í❞♦ ❛♣❡♥❛s ♣❡❧♦s ♠♦❞♦s tr❛♥s✈❡rs❛✐s✳ ❙ã♦ ❞❡✜♥✐❞♦s t❛♠❜é♠ ♦s ❣❡r❛❞♦r❡s ❞❡ ❱✐r❛s♦r♦ tr❛♥s✈❡rs❛✐sLT
m❡L˜Tm✿
LTm=
1 2
∞
X
n=−∞
D−2
X
i=1
αim−nαin L˜Tm=
1 2
∞
X
n=−∞
D−2
X
i=1
˜
αim−nα˜in. ✭✷✳✶✹✮
❘✉♠♦ à q✉❛♥t✐③❛çã♦ ❞❛ t❡♦r✐❛✱ é ✐♠♣♦rt❛♥t❡ ❛♥❛❧✐s❛r q✉❛✐s sã♦ ♦s ♣❛rê♥t❡s❡s ❞❡ P♦✐ss♦♥ ❛ss♦❝✐❛❞♦s ❛♦s ♠♦❞♦s ♥♦r♠❛✐sαµ
m✳
❇❛s❡❛♥❞♦✲s❡ ♥❛s r❡❧❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s
{P0µ(τ, σ), P0ν(τ, σ′)}P P ={X
µ(τ, σ), Xν(τ, σ′)}
P P = 0,
{P0µ(τ, σ), Xν(τ, σ′)}
P P =ηµνδ(σ−σ′),
✈❡r✐✜❝❛✲s❡ q✉❡
p+, x− P P = −1
pi, xj
P P = η ij
αi
m,α˜jn P P = 0 ✭✷✳✶✺✮
αi
m, αjn P P =
˜
αi
m,α˜jn P P = imδm+n,0η ij,
