Eric Sharpe - Examples of homological projective duality in physics

Q vs Q :
To explain the difference, it’s helpful to look at a
NLSM lagrangian on S 2 :
g i¯j ∂ m φ i ∂ m ¯φ¯j − ig i¯j ¯ψ¯j γ m D m ψ i + g i¯j F i ¯F ¯j − F i ( 1 2 g i¯j,¯k ¯ψ¯j ¯ψ¯k
− W i )
− ¯F ī( 1 2 g jī,kψ j ψ k − ¯Wī) − 1 2 W ijψ i ψ j
− 1 2 ¯Wī¯j ¯ψī ¯ψ¯j + 1 4 g i¯j,k¯lψ i ψ k ¯ψ¯j ¯ψ¯l
− 1
4r 2 g i¯jX i X ¯j +
i
4r 2 K iX i −
i
4r 2 K īXī −
i
2r g i¯j ¯ψ¯j ∇ j X i ψ j
(B. Jia, 2013,
to appear)
r = radius of S
2
X = holomorphic Killing vector
(defines Q of previous slide)
Specific to S 2
Constraints:
2W = −iX i ∂ i W
so if W ≠ 0 then X ≠ 0 -- important for GLSM