With the traceless matrices 14 m m nt m m 1 n a tr m

Info icon This preview shows pages 27–30. Sign up to view the full content.

with the traceless matrices 14 : M + M NT = M + M - 1 N a tr { M + M } , MM + NT = MM + - 1 N b tr { MM + } . The scalar product (4.15) in Ω 2 D ( A ) is calculated using, besides the trace theorems, the identities: 1 2 d/ 2 tr c ( c ( ρ ( k ) )) + c ( ρ ( k ) = k ! ( ρ ( k ) i 1 ...i k ) * ρ ( k ) j 1 ...j k δ i 1 j 1 ...δ i k j k = g - 1 ( ρ ( k ) * ; ρ ( k ) ) . In terms of the Hodge dual , defined by g - 1 ( ρ ( k ) * ; ρ ( k ) ) ω = ( ρ ( k ) ) * ρ ( k ) , the scalar product reads : π D ( G ); π D ( G ) 2 ,D = 1 2 π N a S 2 ρ (2) aaa * ρ (2) aaa + N b S 2 ρ (2) bbb * ρ (2) bbb + tr { MM + } S 2 ρ (1) ab * ρ (1) ab + tr { M + M } S 2 ρ (1) ba * ρ (1) ba + tr [ M + M ] 2 NT S 2 ρ (0) aba * ρ (0) aba + tr [ MM + ] 2 NT S 2 ρ (0) bab * ρ (0) bab . (4.21) 14 Note that when N a = N b = N and M is a scalar matrix, these traceless matrices M + M NT and MM + NT vanish and there is no ρ (0) α term in π D ( G ). Physically this implies that, in order to have a Higgs mechanism, a nontrivial mass spectrum is necessary! 27
Image of page 27

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

4.1 The Yang-Mills-Higgs action The universal connection in M , given by the matrices (( A αβ ( x,y ))) of (3.6), is represented in Ω (1) D ( A ) by an operator of the form (4.10) where the differential forms σ ( k ) ··· are matrix-valued. σ (1) a ( x ) = α a ( x ) e k,y (( ω aa ( x,y ))) | y = x θ k x , σ (1) b ( x ) = α b ( x )(( P b ( x ))) e k,y ω b ( x,y ) | y = x θ k x (( P b ( x ))) , σ (0) ab ( x ) = | Φ ab ( x,x ) ν ( x ) | , σ (0) ba ( x ) = | ν ( x ) Φ ba ( x,x ) | . (4.22) The monopole connection (3.13) also implements a differential one-form : μ b ( x ) = ( e k,y m b ( x,y )) | y = x θ k x = ν ( x ) | d | ν ( x ) , = 1 / 2 1 + | ν ( x ) | 2 ν ( x ) * d ν ( x ) - ν ( x )d ν ( x ) * . (4.23) It is also convenient to introduce the Higgs field doublets : | η ab ( x ) = | H ab ( x,x ) = | Φ ab ( x,x ) + | ν ( x ) , η ba ( x ) | = H ab ( x,x ) | = Φ ba ( x,x ) | + ν ( x ) | . (4.24) The hermiticity of the connection (3.9) yields : α a + = - α a , ( α b ) * = - α b , ( μ b ) * = - μ b , η ba | = | η ab + . (4.25) From (3.17) it follows that, under an active gauge transformation, the differ- ential forms (4.22) and (4.23) behave as : (( α U a )) = (( U a )) - 1 (( α a ))(( U a )) + (( U a )) - 1 d(( U a )) α U b = ( u b ) - 1 α b ( u b ) + ( u b ) - 1 d u b = α b + ( u b ) - 1 d u b μ U b = ( u b ) - 1 μ b ( u b ) = μ b | η U ab = (( U a )) - 1 | η ab u b , η U ba | = ( u b ) - 1 η ba | (( U a )) . On the other hand, under a passive gauge transformation H B H A , accord- ing to (3.10) and (3.14), they transform as : (( α A a )) = (( α B a )) , α A b = α B b , | η A ab = | η B ab ( c AB ) - n/ 2 , η A ba | = ( c AB ) + n/ 2 η B ba | , μ A b = μ B b + ( c AB ) + n/ 2 d( c AB ) - n/ 2 = μ B b - ( n/ 2)( c AB ) - 1 d c AB . 28
Image of page 28
This means that the Higgs fields | η ab are actually Pensov fields of spin- weight n/ 2 and that the monopole potential cannot be represented by a globally defined one-form on the sphere, but adquires the inhomogeneous term - ( n/ 2)( c AB ) - 1 d c AB in H B H A . The canonical projection (4.14) induces a Ω (1) D ( A )-valued connection in M : D : M → M ⊗ A Ω (1) D ( A ) : X → ∇ D X, defined by D = 1 M π D ◦ ∇ . From (3.11) and (4.22) it follows that : | ( D X ) aa = - ic d | f a + (( α a )) | f a , | ( D X ) ab = | η ab f b - | f a γ 3 M + , | ( D X ) ba = | ν η ba | f a - f b γ 3 M, | ( D X ) bb = | ν - ic d f b + ( α b + μ b ) f b . The curvature (3.15) is represented in Ω (2) D ( A ) by π D ( R ) of the form (4.20), where the differential forms ρ ( k ) ...
Image of page 29

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 30
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern