For complex ϕ in U(1) gauge theory,
|ϕ1|2+|ϕ2|2+⋯|ϕN|2=r
This equation
|ϕ|2=r, describes sphere
S2N−1. Dividing the space of this solution by the gauge group
U(1) we obtain that the moduli space for
ϕ which is
CPN−1
This procedure is based on the explanation in Witten's paper of "Phase of N=2 theories in two dimensions". (Above situation corresponds to N=2 supersymmetric U(1) gauge theory with N chiral superfields. Here i solve the equation for minimizng potential energy.)
Here what i want to extend this idea to following equations,
(This situation corresponds to N=2 supersymmetric U(1) gauge theory with N chiral superfields and N anti-chiral superfields. )
For same complex ϕ in U(1) gauge theory, we have
|ϕ1|2+|ϕ2|2⋯+|ϕN|2−|ϕN+1|2−|ϕN+2|2⋯−|ϕ2N|2=r
The results for this moduli space is known as T∗CPN−1 where T∗ represents cotangent bundle.
Here i want to know why this space is T∗CPN−1.
Can anyone give some explanation about this?
This post imported from StackExchange Physics at 2014-12-23 15:10 (UTC), posted by SE-user phy_math