A = C ⊕ C By analogy, can calculate the ‘distance’ between the two points in A = C ⊕ C . Introduce the third element, the generalization of Dirac operator, � � 0 m D = . m 0 The distance formula is d ( x , y ) = sup {| a ( x ) − a ( y ) | : a ∈ A , � [ D , a ] � ≤ 1 } . Distance between the two points d ( p 1 , p 2 ) = sup {| a ( p 1 ) − a ( p 2 ) |} a ∈A , � [ D , a ] �≤ 1 | λ − λ ′ | = sup ( λ,λ ′ ) ∈A , � [ D , a ] �≤ 1 = 1 | m | . The generalized Dirac operator encodes the distance information! Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 16 / 44
A = C ⊕ C A = C ⊕ C , H = C N ⊕ C N , � � M † 0 D = . M 0 For a ∈ A = ( λ, λ ′ ) , the ‘differential’ is ∼ ( λ − λ ′ ) : � � − M † 0 [ D , a ] = ( λ − λ ′ ) , M 0 By analogy with ǫ → 0 ( f ( x + ǫ ) − f ( x )) d x µ df = ∂ µ f d x µ = lim ǫ . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 17 / 44
A = C ⊕ C A = C ⊕ C , H = C N ⊕ C N , � � M † 0 D = . M 0 For a ∈ A = ( λ, λ ′ ) , the ‘differential’ is ∼ ( λ − λ ′ ) : � � − M † 0 [ D , a ] = ( λ − λ ′ ) , M 0 By analogy with ǫ → 0 ( f ( x + ǫ ) − f ( x )) d x µ df = ∂ µ f d x µ = lim ǫ . The ‘integral’ is ∼ ( λ + λ ′ ) : Tr ( a ) = λ + λ ′ . By analogy with � f ( x ) d x . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 17 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . It is natural to equip the spectral triple ( A , H , D ) with another object, γ , the grading operator. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . It is natural to equip the spectral triple ( A , H , D ) with another object, γ , the grading operator. For example, In the case of ( A , H , D ) = ( C ∞ ( M ) , Γ( M , S ) , i / ∂ ) , γ = γ 5 . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . It is natural to equip the spectral triple ( A , H , D ) with another object, γ , the grading operator. For example, In the case of ( A , H , D ) = ( C ∞ ( M ) , Γ( M , S ) , i / ∂ ) , γ = γ 5 . � � M † 0 In the case of ( A , H , D ) = ( C ⊕ C , C N ⊕ C N , ) , we can choose M 0 the grading operator to be γ = diag (1 , ..., 1 , − 1 , ..., − 1 ) � �� � � �� � N copies N copies Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . It is natural to equip the spectral triple ( A , H , D ) with another object, γ , the grading operator. For example, In the case of ( A , H , D ) = ( C ∞ ( M ) , Γ( M , S ) , i / ∂ ) , γ = γ 5 . � � M † 0 In the case of ( A , H , D ) = ( C ⊕ C , C N ⊕ C N , ) , we can choose M 0 the grading operator to be γ = diag (1 , ..., 1 , − 1 , ..., − 1 ) � �� � � �� � N copies N copies A device that helps us distinguish one part from the other. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
Grading Physically, we are specifically interested in the type of algebra A = A 1 ⊕ A 2 . e.g. the model with U (1) Y × SU (2) L , or SU (2) R × SU (2) L , etc. They correspond to a representation space ∼ H L ⊕ H R , or H f ⊕ H f . It is natural to equip the spectral triple ( A , H , D ) with another object, γ , the grading operator. For example, In the case of ( A , H , D ) = ( C ∞ ( M ) , Γ( M , S ) , i / ∂ ) , γ = γ 5 . � � M † 0 In the case of ( A , H , D ) = ( C ⊕ C , C N ⊕ C N , ) , we can choose M 0 the grading operator to be γ = diag (1 , ..., 1 , − 1 , ..., − 1 ) � �� � � �� � N copies N copies A device that helps us distinguish one part from the other. D , A = A 1 ⊕ A 2 ∼ two sheets structure. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 18 / 44
A = C ⊕ H – A toy model A = C ⊕ H , H = C 2 ⊕ C 2 , � � M † 0 D = . M 0 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 19 / 44
A = C ⊕ H – A toy model A = C ⊕ H , H = C 2 ⊕ C 2 , � � M † 0 D = . M 0 How do we fit this with our particle spectrum? ‘Flavor’ space: 1 0 0 0 0 1 0 0 ν R = , e R = , ν L = , e L = . 0 0 1 0 0 0 0 1 λ λ For any a ∈ A , a = . α β − β α � � m ν 0 To give mass terms out of ψ † D ψ , let M = . 0 m e Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 19 / 44
The unitary transformations are { u ∈ A| u † u = uu † = 1 } . This implies e i θ e − i θ | α | 2 + | β | 2 = 1 . u = , s.t. α β − β α which automatically fulfills det u = 1 . This is the symmetry U (1) R × SU (2) L . The U (1) R charge is |↑� |↓� 2 R 1 − 1 2 L 0 0 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 20 / 44
When we make the U (1) R × SU (2) L transformation, L =Ψ † D Ψ �→ Ψ † u † Du Ψ = Ψ † D Ψ + Ψ † u † [ D , u ]Ψ . � �� � the ‘local’ twist In general [ D , u ] � = 0 , therefore, this demands for a ‘gauge’ field to absorb the local twist, in the discrete direction. According to our recipe, we do have a gauge field between the two sheets, � A = a i [ D , b i ] . i L =Ψ † ( D + A )Ψ �→ Ψ † D Ψ + Ψ † u † [ D , u ]Ψ + Ψ † A Ψ − Ψ † u † [ D , u ]Ψ =Ψ † ( D + A )Ψ Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 21 / 44
Demanded to be Hermitian, this gauge field is � � M † Φ † A = , Φ M � φ 1 � φ 2 Φ = [ φ 1 φ 2 ] = − φ 2 φ 1 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 22 / 44
Demanded to be Hermitian, this gauge field is � � M † Φ † A = , Φ M � � M † (Φ † + 1) D + A = . (Φ + 1) M Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 22 / 44
Demanded to be Hermitian, this gauge field is � � M † Φ † A = , Φ M � � M † (Φ † + 1) D + A = . (Φ + 1) M The perturbation of ‘ D 2 ’ derived from the (spectral) action: � ( D + A ) 2 − D 2 � � ( D + A ) 2 − D 2 � � ( MM † ) 2 � ( | Φ + 1 | 2 − 1) 2 . ∼ Tr Tr Tr Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 22 / 44
Demanded to be Hermitian, this gauge field is � � M † Φ † A = , Φ M � � M † (Φ † + 1) D + A = . (Φ + 1) M The perturbation of ‘ D 2 ’ derived from the (spectral) action: � ( D + A ) 2 − D 2 � � ( D + A ) 2 − D 2 � � ( MM † ) 2 � ( | Φ + 1 | 2 − 1) 2 . ∼ Tr Tr Tr This gives us a Mexican-hat-shaped potential. A field expanded at the minimum � = 0 . By counting d.o.f, we have 4 + 4 − 4 = 4 real degrees, i.e. Φ is a pair of complex numbers. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 22 / 44
Demanded to be Hermitian, this gauge field is � � M † Φ † A = , Φ M � � M † (Φ † + 1) D + A = . (Φ + 1) M The perturbation of ‘ D 2 ’ derived from the (spectral) action: � ( D + A ) 2 − D 2 � � ( D + A ) 2 − D 2 � � ( MM † ) 2 � ( | Φ + 1 | 2 − 1) 2 . ∼ Tr Tr Tr This gives us a Mexican-hat-shaped potential. A field expanded at the minimum � = 0 . By counting d.o.f, we have 4 + 4 − 4 = 4 real degrees, i.e. Φ is a pair of complex numbers. SSB now has a reason: D + A gives a VEV shift. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 22 / 44
By analogy, e − i θ ∂ µ e i θ = ∂ µ θ u † [ D , u ] local ‘twist’ M † Φ † � a i [ D , b i ] = A µ d µ x ω Φ M M † d µ x basis M comp’ A µ Φ � ( D + A ) 2 − D 2 � θ ( d + A ) ∧ ( d + A ) Tr ∼ F µν ∼ DA + A 2 ∼ ∂ µ A ν + [ A µ , A ν ] � � ( D + A ) 2 − D 2 � F µν F µν d 4 x ) 2 S ( Tr Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 23 / 44
Product geometry Consider the algebra: A = C ∞ ( M ) ⊕ C ∞ ( M ) ∼ C ∞ ( M ) ⊗ ( C ⊕ C ) . This corresponds to a geometry F = M ⊕ M , ∼ M × { p 1 , p 2 } . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 24 / 44
Product geometry Consider the algebra: A = C ∞ ( M ) ⊕ C ∞ ( M ) ∼ C ∞ ( M ) ⊗ ( C ⊕ C ) . This corresponds to a geometry F = M ⊕ M , ∼ M × { p 1 , p 2 } . Combining continuous part with C ⊕ H , A = C ∞ ( M ) ⊗ ( C ⊕ H ) . ∼ a double-layer structure. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 24 / 44
The Dirac operator of the product geometry: ∂ + γ 5 ⊗ D . D x = i / The gauge field: Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 25 / 44
The Dirac operator of the product geometry: ∂ + γ 5 ⊗ D . D x = i / The gauge field: � � f i [ / A x ∼ ∂, g i ] + a i [ D , b i ] � �� � � �� � A [1 , 0] A [0 , 1] Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 25 / 44
The Dirac operator of the product geometry: ∂ + γ 5 ⊗ D . D x = i / The gauge field: � � f i [ / A x ∼ ∂, g i ] + a i [ D , b i ] � �� � � �� � A [1 , 0] A [0 , 1] � � Φ ∗ B ∼ . Φ W Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 25 / 44
Highlights: A two sheet structure. A gauge field in between. SSB feature out of box. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 26 / 44
Highlights: A two sheet structure. A gauge field in between. SSB feature out of box. ∼ Implies a Higgs as the discrete gauge, generated similarly as the continous gauge fields. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 26 / 44
Color sector In order to reproduce SM, color sector must be involved. Introduce the ‘color’ space. H = C ⊕ C 3 , with basis 1 0 0 0 0 1 0 0 ℓ = , r = , g = , b = . 0 0 1 0 0 0 0 1 A = C ⊕ M 3 ( C ) , with ∀ a ∈ A , λ m 11 m 12 m 13 a = . m 21 m 22 m 23 m 31 m 32 m 33 Symmetry group is { u ∈ A| u † u = uu † = 1 } , together with the ‘unimodularity’ condition, det u = 1 . e − i θ , m ′ ∈ SU (3) . a = e i θ/ 3 m ′ Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 27 / 44
e − i θ , m ′ ∈ SU (3) . a = e i θ/ 3 m ′ This gives the U (1) charge ℓ r g b 1 1 1 − 1 3 3 3 We recognize them as B − L charge, and this gives us the symmetry U (1) B − L × SU (3) C . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 28 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Denote the space as ( 2 R ⊕ 2 L ) ⊗ ( 1 ℓ ⊕ 3 C ) . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Denote the space as ( 2 R ⊕ 2 L ) ⊗ ( 1 ℓ ⊕ 3 C ) . Can identify the basis with SM particle spectrum, for example Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Denote the space as ( 2 R ⊕ 2 L ) ⊗ ( 1 ℓ ⊕ 3 C ) . Can identify the basis with SM particle spectrum, for example ν L = |↑� L ⊗ ℓ ∈ 2 L ⊗ 1 ℓ , Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Denote the space as ( 2 R ⊕ 2 L ) ⊗ ( 1 ℓ ⊕ 3 C ) . Can identify the basis with SM particle spectrum, for example d R , g = |↓� R ⊗ g Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) To combine the flavor sector with the color sector, let A = C ∞ ( M ) ⊗ ( C ⊕ H ⊕ M 3 ( C )) . Introduce the bimodule representation: |↑� R ℓ |↓� R r ⊗ . |↑� L g |↓� L b Denote the space as ( 2 R ⊕ 2 L ) ⊗ ( 1 ℓ ⊕ 3 C ) . Can identify the basis with SM particle spectrum, for example � r � d R = |↓� R ⊗ g ∈ 2 R ⊗ 3 C , b Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 29 / 44
Introduce J , charge conjugate, |↑� R ℓ ℓ |↑� R |↓� R r r |↓� R J ⊗ ∼ ⊗ , |↑� L g g |↑� L |↓� L b b |↓� L ∀ a ∈ A with left action on flavor space as before, JaJ − 1 is the right action on color space. Ready to combine the previous result on flavor space and color space. |↑� |↓� ℓ r g b 2 R 1 − 1 1 1 1 − 1 3 3 3 2 L 0 0 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 30 / 44
U (1) R : |↑� ⊗ 1 0 |↓� ⊗ 1 0 |↑� ⊗ 3 0 |↓� ⊗ 3 0 2 L 0 0 0 0 2 R 1 − 1 1 − 1 U (1) B − L : |↑� ⊗ 1 0 |↓� ⊗ 1 0 |↑� ⊗ 3 0 |↓� ⊗ 3 0 1 1 − 1 − 1 2 L 3 3 1 1 − 1 − 1 2 R 3 3 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 31 / 44
|↑� ⊗ 1 0 |↓� ⊗ 1 0 |↑� ⊗ 3 0 |↓� ⊗ 3 0 1 1 2 L − 1 − 1 3 3 4 − 2 2 R 0 − 2 3 3 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 32 / 44
Spectral Action According to Chamseddine et. al. (hep-th/9606001), one builds the action based on spectral action principle: The physical (bosonic) action only depends upon the spectrum of D. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 33 / 44
Spectral Action According to Chamseddine et. al. (hep-th/9606001), one builds the action based on spectral action principle: The physical (bosonic) action only depends upon the spectrum of D. S spec = Tr ( f ( D A / Λ)) . We can expand it as � L ( g µν , A ) √ g d 4 x . Tr ( f ( D A / Λ)) ∼ M Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 33 / 44
The bosonic action, S Bosonic = S Higgs + S YM + S Cosmology + S Riemann , � | D µ φ | 2 √ g d 4 x + − 2 af 2 Λ 2 + ef 0 � S Higgs = f 0 a | φ | 2 √ g d 4 x 2 π 2 π 2 � + f 0 b | φ | 4 √ g d 4 x , 2 π 2 S YM = f 0 µν ) 16 π 2 Tr ( F µν F � = f 0 µν i + g 2 µν i + 5 µν ) √ g d 4 x ( g 2 3 G i 2 W i 3 g 2 µν G µν W 1 B µν B 2 π 2 where the parameters are a = Tr ( M ∗ ν M ν + M ∗ e M e + 3( M ∗ u M u + M ∗ d M d )) ν M ν ) 2 + ( M ∗ e M e ) 2 + 3( M ∗ u M u ) 2 + 3( M ∗ b = Tr (( M ∗ d M d ) 2 ) c = Tr ( M ∗ R M R ) d = Tr (( M ∗ R M R ) 2 ) e = Tr ( M ∗ R M R M ∗ ν M ν ) , f n is the ( n − 1) th momentum of f . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 34 / 44
Output: Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , W = 1 2 + m i 2 + 3 m i 2 + 3 m i � 2 ) , M 2 i ( m i ν e u d 8 Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , W = 1 2 + m i 2 + 3 m i 2 + 3 m i � 2 ) , M 2 i ( m i ν e u d 8 Can be calculated from spectral action. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , W = 1 2 + m i 2 + 3 m i 2 + 3 m i 2 ) , � M 2 i ( m i ν e u d 8 Can be calculated from spectral action. Intuitively, Cont’ Disc’ ψ/ Ψ † D Ψ Fermion ∂ψ ∂ µ W ∂ µ W D 2 W 2 Boson Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , W = 1 2 + m i 2 + 3 m i 2 + 3 m i � 2 ) , M 2 i ( m i ν e u d 8 m H ≈ 170 GeV , Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Output: 2 = 5 g 2 3 = g 2 3 g 2 1 , � φ � � = 0 , W = 1 2 + m i 2 + 3 m i 2 + 3 m i � 2 ) , M 2 i ( m i ν e u d 8 m H ≈ 170 GeV , problematic, which is naturally saved by the left-right completion we propose. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 35 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � δ L δφ δφ + δ L δ S = δ∂φδ∂φ � δ L � δ L � δ L � � = δφ δφ + ∂ δ∂φδφ − ∂ δφ δ∂φ � δ L � � EOM = ∂ δ∂φδφ � � � ǫ δ L = ∂ δ∂φ Φ � = ∂ ( ǫ j ) � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . = Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . δ S = ∂ ( ǫ j ) = ∂ µ j µ = 0 a symmetry. ∂ µ ( ǫ j µ ) = 0 a global symmetry. ∂ µ ( ǫ j µ ) � = 0 , a local symmetry with a gauge. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . δ S = ∂ ( ǫ j ) = ∂ µ j µ = 0 a symmetry. ∂ µ ( ǫ j µ ) = 0 a global symmetry. ∂ µ ( ǫ j µ ) � = 0 , a local symmetry with a gauge. Ψ † D Ψ �→ Ψ † D Ψ + Ψ † [ D , ǫ i σ i ]Ψ . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . δ S = ∂ ( ǫ j ) = ∂ µ j µ = 0 a symmetry. ∂ µ ( ǫ j µ ) = 0 a global symmetry. ∂ µ ( ǫ j µ ) � = 0 , a local symmetry with a gauge. Ψ † D Ψ �→ Ψ † D Ψ + Ψ † [ D , ǫ i σ i ]Ψ . Ψ † [ D , ǫ i σ i ]Ψ by analogy with ∂ µ ( ǫ j µ ) . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . δ S = ∂ ( ǫ j ) = ∂ µ j µ = 0 a symmetry. ∂ µ ( ǫ j µ ) = 0 a global symmetry. ∂ µ ( ǫ j µ ) � = 0 , a local symmetry with a gauge. Ψ † D Ψ �→ Ψ † D Ψ + Ψ † [ D , ǫ i σ i ]Ψ . Ψ † [ D , ǫ i σ i ]Ψ by analogy with ∂ µ ( ǫ j µ ) . [ D , ǫ i σ i ] = 0 a ‘global’ symmetry in the discrete direction. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] is insensitive to local/global transformation w.r.t. M. φ �→ φ + δφ , with δφ = ǫ i σ i φ = ǫ i Φ i , � � � ∂ µ ( ǫ ) j µ + ǫ∂ µ j µ . δ S = ∂ ( ǫ j ) = ∂ µ j µ = 0 a symmetry. ∂ µ ( ǫ j µ ) = 0 a global symmetry. ∂ µ ( ǫ j µ ) � = 0 , a local symmetry with a gauge. Ψ † D Ψ �→ Ψ † D Ψ + Ψ † [ D , ǫ i σ i ]Ψ . Ψ † [ D , ǫ i σ i ]Ψ by analogy with ∂ µ ( ǫ j µ ) . [ D , ǫ i σ i ] = 0 a ‘global’ symmetry in the discrete direction. [ D , ǫ i σ i ] � = 0 a ‘local’ symmetry in the discrete direction, with a gauge. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 36 / 44
Other Fun Facts – ‘local twist’ [ D , u ] = 0 refers to Du = uD , D = uDu † . ⇔ In SM, this refers to the VEV shift is invariant under the transformation u . This describes the transformation of VEV shift, or the symmetry under which vacuum is invariant. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 37 / 44
Other Fun Facts – ‘local twist’ [ D , u ] = 0 refers to Du = uD , D = uDu † . ⇔ In SM, this refers to the VEV shift is invariant under the transformation u . This describes the transformation of VEV shift, or the symmetry under which vacuum is invariant. ∼ Remaining symmetry, Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 37 / 44
Other Fun Facts – ‘local twist’ [ D , u ] = 0 refers to Du = uD , D = uDu † . ⇔ In SM, this refers to the VEV shift is invariant under the transformation u . This describes the transformation of VEV shift, or the symmetry under which vacuum is invariant. ∼ Remaining symmetry, ∼ Breaking chain. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 37 / 44
Other Fun Facts – ‘local twist’ � � � � M † 0 0 m u In the simplest case, A = H ⊕ H , D = and M = . M 0 m d 0 Pictorially, the twist between ‘left sheet’ and ‘right sheet’. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 38 / 44
Other Fun Facts – ‘local twist’ � � � � M † 0 0 m u In the simplest case, A = H ⊕ H , D = and M = . M 0 m d 0 Pictorially, the twist between ‘left sheet’ and ‘right sheet’. But even we make same twists for left and right, we still have a local ‘twist term’, unless m u = m d , isospin-like. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 38 / 44
Other Fun Facts – The seperation Totally independent of the base manifold M. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 39 / 44
Other Fun Facts – The seperation Totally independent of the base manifold M. Extra dimension but discrete. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 39 / 44
Other Fun Facts – The seperation Totally independent of the base manifold M. Extra dimension but discrete. The separation introduces a second scale ∼ EW, from a i [ D , b i ] , different from the GUT scale which is led by the fluctuation in the continuous direction f i [ / ∂, g i ] . Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 39 / 44
Other Fun Facts – The seperation Totally independent of the base manifold M. Extra dimension but discrete. The separation introduces a second scale ∼ EW, from a i [ D , b i ] , different from the GUT scale which is led by the fluctuation in the continuous direction f i [ / ∂, g i ] . When the separation goes to ∞ , m f → 0 . This corresponds to the decouple of Higgs sector: left and right stop talking to each other, physically and geometrically. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 39 / 44
Back to the Left-Right Completion Different realizations. For example. NCG/ spectral triple is built using lattice, supersymmetric quantum mechanics operators, Moyal deformed space, etc. We have tried a specific realization using superconnection, su (2 | 1) , and the left-right completion of su (2 | 2) . Low energy emergent left-right completion, ∼ 4 TeV . (Ufuk Aydemir, Djordje Minic, C.S., Tatsu Takeuchi: Phys. Rev. D 91 , 045020 (2015) [arXiv:1409.7574]) Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 40 / 44
More About the Left-Right Completion Hints for left-right symmetry behind the scene. ( Pati-Salam Unification from NCG and the TeV-scale WR boson , [arXiv:1509.01606], Ufuk Aydemir, Djordje Minic, C.S., Tatsu Takeuchi ) Changing the algebra to ( H R ⊕ H L ) ⊗ ( C ⊕ M 3 ( C )) does not change the scale. 2 3 g 2 BL = g 2 2 L = g 2 2 R = g 2 3 . Through the mixing of SU (2) R × U (1) B − L into U (1) Y , we get g ′ 2 = 1 1 1 = 5 1 g 2 + g 2 . g 2 3 BL ∼ LR symmetry breaking at GUT. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 41 / 44
Myths and Outlooks So far it is a classical theory – only classical L is given. But it has a GUT feature! Without adding new d.o.f. If it just happens at one scale, how to accommodate Wilson picture. Quantization of the theory? Loops? Relation to the D-brane structure? Measure of the Dirac operator? ... Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 42 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. The dish: Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. The dish: Two sheets structure. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. The dish: Two sheets structure. An extra discrete direction. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. The dish: Two sheets structure. An extra discrete direction. Separation of the sheets ( m f → 0 , second scale, etc.) Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
Summary (of Fun Facts) Recipe to cook up a (generalized) gauge theory: ( A , H , D ) , the spectral triple. Take mass matrix as a derivative, trace as the integral. Generate the gauge field A = � a [ D , b ] . Spectral action, Tr ( f ( D / Λ)) ∼ DA + A 2 , as the gauge strength Generalized free fermion action, Ψ † D A Ψ , for the fermionic part. The dish: Two sheets structure. An extra discrete direction. Separation of the sheets ( m f → 0 , second scale, etc.) Higgs is a gauge in that direction. Chen Sun @ Duke Gauge Theory through NCG October 24, 2015 43 / 44
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