- #1
niss
- 7
- 2
- Homework Statement
- To show the winding number is invariant under continuous deformation
- Relevant Equations
- ##\delta \nu\propto \int d\theta_1 d\theta_2 d\theta_3\epsilon ^{ijk}Tr\partial _ig^{-1}\partial_jg\partial_k\delta T##
I'm reading Coleman's "Aspects of symmetry" chap 7.
On the topic of the SU(2) winding number on ##S^3##on page 288, three parameters on ##S^3## are defined ##\theta_1,\theta_2,\theta_3##. Afterwards, it defines the winding number and to show it's invariant under continuous deformation of gauge field it claims
$$\delta \nu\propto \int d\theta_1 d\theta_2 d\theta_3\epsilon ^{ijk}Tr\partial _ig^{-1}\partial_jg\partial_k\delta T\qquad (3.37)$$
But I don't get how to execute integ by parts. To integ by parts and get vanishing surface terms, I think the boundary of ##\theta_i## s should be connected like the usual ##\phi## of polar coordinates ##(r,\theta,
\phi)## and not like ##\theta## which varies from 0 to ##\pi##. Then I'm wondering how to define ##\theta_i## s globally.
On the topic of the SU(2) winding number on ##S^3##on page 288, three parameters on ##S^3## are defined ##\theta_1,\theta_2,\theta_3##. Afterwards, it defines the winding number and to show it's invariant under continuous deformation of gauge field it claims
$$\delta \nu\propto \int d\theta_1 d\theta_2 d\theta_3\epsilon ^{ijk}Tr\partial _ig^{-1}\partial_jg\partial_k\delta T\qquad (3.37)$$
But I don't get how to execute integ by parts. To integ by parts and get vanishing surface terms, I think the boundary of ##\theta_i## s should be connected like the usual ##\phi## of polar coordinates ##(r,\theta,
\phi)## and not like ##\theta## which varies from 0 to ##\pi##. Then I'm wondering how to define ##\theta_i## s globally.
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