Спинорное расслоение

В дифференциальной геометрии, спинорное расслоение — локально тривиальное расслоение специального вида над (псевдо)римановым многообразием. Сечение спинорного расслоения, называемое спинорным полем, моделирует в физике фермионное поле в произвольном пространстве.

Определение

Данное ниже определение обобщается естественным образом на случай псевдориманова многообразия произвольной сигнатуры. Пусть $(M,g)$ — ориентируемое риманово многообразие, $\pi :\mathrm {F} _{\mathrm {SO} }(M)\to M$ — расслоение ортонормированных реперов, $\rho :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)$ — двулистное накрытие. Спинорной структурой называют пару $(\mathrm {F} _{\mathrm {Spin} }(M),\varphi )$ , где $\pi _{s}:\mathrm {F} _{\mathrm {Spin} }(M)\to M$ — $\mathrm {Spin} (n)$ -главное расслоение над $M$ , $\varphi :\mathrm {F} _{\mathrm {Spin} }\to \mathrm {F} _{\mathrm {SO} }$ — эквивариантное двулистное накрытие такое, что

\pi \circ \phi =\pi _{s},\qquad \phi (pq)=\phi (p)\rho (q)\quad

для всех

p\in \mathrm {F} _{\operatorname {Spin} }

q\in \operatorname {Spin} (n)

Расслоение допускает спинорную структуру тогда и только тогда, когда второй класс Штифеля — Уитни w₂(M) ∈ H²(M, Z₂) обращается в ноль.

Пусть на $(M,g)$ задана спинорная структура, тогда спинорным расслоением называют ассоциированное c $\mathrm {F} _{\mathrm {Spin} }(M)$ расслоение с типичным слоем $V=\mathbb {C} ^{n}$ с заданным спинорным представлением $\mathrm {Spin} (n)\to \mathrm {GL} (V)$ . Его сечения называют спинорными полями.

См. также

Литература

Lawson, H. Blaine. Spin Geometry / H. Blaine Lawson, Marie-Louise Michelsohn. — Princeton University Press, 1989. — ISBN 978-0-691-08542-5.
Friedrich, Thomas. Dirac Operators in Riemannian Geometry. — American Mathematical Society, 2000. — ISBN 978-0-8218-2055-1.
Karoubi, Max. K-Theory. — Springer, 2008. — P. 212–214. — ISBN 978-3-540-79889-7.
Greub, Werner. On the lifting of structure groups // Differential Geometrical Methods in Mathematical Physics II / Werner Greub, Herbert-Rainer Petry. — Springer-Verlag, 2006. — Vol. 676. — P. 217–246. — ISBN 9783540357216. — doi:10.1007/BFb0063673.
Scorpan, Alexandru. 4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of // The wild world of 4-manifolds. — American Mathematical Society, 2005. — P. 174–189. — ISBN 9780821837498.

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