Rethinking Gauge Theory through Connes Noncommutative Geometry

Transcription

1 Rethinking Gauge Theory through Connes Noncommutative Geometry Chen Sun Virginia Tech October 24, 2015 Chen Duke Gauge Theory through NCG October 24, / 44

2 Work with Ufuk Aydemir, Djordje Minic, Tatsu Takeuchi: Phys. Rev. D 91, (2015) [arxiv: ], Pati-Salam Unification from Non-commutative Geometry and the TeV-scale W R boson [arxiv: ], Review of NCG in preparation. For background of NCG, c.f. Chamseddine, Connes, et. al.: Nucl. Phys. Proc. Suppl. 18B, 29 (1991) Commun. Math. Phys. 182, 155 (1996) [hep-th/ ], Adv. Theor. Math. Phys. 11, 991 (2007) [hep-th/ ], and for superconnection, c.f. Neeman, Fairlie, et. al.: Phys. Lett. B 81, 190 (1979), J. Phys. G 5, L55 (1979), Phys. Lett. B 82, 97 (1979). Chen Duke Gauge Theory through NCG October 24, / 44

3 The quickest review of gauge theory Given ψ Ô element in rep space H, e.g. Dirac spinors, operator on H, e.g. /, we say the operator is covariant if under the transformation ψ uψ, the operator trasforms as Ô uôu 1, since that gives us Ôψ uôψ. At the end, a theory built with L ψ Ôψ is invariant under the transformation. Chen Duke Gauge Theory through NCG October 24, / 44

4 The quickest review of gauge theory -Cont d When we localize the transformation u, things sometimes change Ô uôu 1 + local terms. Therefore, we need to come up with another operator that transforms as  uâu 1 local terms, so that the combination of the two gives Ô +  u(ô + Â)u 1. Then we have made the combo operator Ô +  a covariant operator, denoted Ô A. Chen Duke Gauge Theory through NCG October 24, / 44

5 Example: U(1) from global to local We have L = ψi / ψ. Invariant under global U(1): ψ e iθ ψ, L L = L. When we localize the U(1) symmetry, i.e. θ = θ(x), ψ e iθ(x) ψ, L L = L θψψ. Chen Duke Gauge Theory through NCG October 24, / 44

6 Example: U(1) from global to local -Cont d Therefore we come up with a U(1) gauge field A, which transforms as A uau 1 + θ. and modify the Lagrangian as L = ψ(i / + /A)ψ. All together, we acquire an invariant theory. Chen Duke Gauge Theory through NCG October 24, / 44

7 Description in spectral triple Suppose we have The unitary transformations are Under transformations u, we have A =C (M), H =Γ(M, S), D =i /. {u A u u = uu = 1}. ψ uψ, Dψ Duψ = udψ + [D, u]ψ. Chen Duke Gauge Theory through NCG October 24, / 44

8 Under transformations u, we have ψ uψ, Dψ Duψ = udψ + [D, u]ψ, The theory built with ψdψ is invariant ψdψ ψu Duψ = ψdψ + ψu [D, u]ψ. [D, u] = 0, (u) = 0, u is a global symmetry. Chen Duke Gauge Theory through NCG October 24, / 44

9 Under transformations u, we have ψ uψ, Dψ Duψ = udψ + [D, u]ψ, The theory built with ψdψ is invariant What if [D, u] 0? ψdψ ψu Duψ = ψdψ + ψu [D, u]ψ. [D, u] = 0, (u) = 0, u is a global symmetry. Chen Duke Gauge Theory through NCG October 24, / 44

10 Under transformations u, we have ψ uψ, Dψ Duψ = udψ + [D, u]ψ, The theory built with ψdψ is invariant What if [D, u] 0? ψdψ ψu Duψ = ψdψ + ψu [D, u]ψ. [D, u] = 0, (u) = 0, u is a global symmetry. Old trick: use a gauge field to absorb the extra term. What should the gauge look like? Chen Duke Gauge Theory through NCG October 24, / 44

11 [D, u] 0 In this case, D transforms as D u(d + u [D, u])u. Apparently it is not covariant. It is perturbed during the transformation, with the extra term is of the form u [D, u]. We want to absorb the extra term into D, with the hope the overall operator is recovered covariant. Therefore we define another operator as A = a i [D, b i ], where a i, b i A. We can immediately tell the extra term is nothing but of the form of A, thus can be absorbed. Chen Duke Gauge Theory through NCG October 24, / 44

12 D u(d + u [D, u])u = u(d + A 0 )u. With transformation u: D D + A 0. Need A A A 0. Chen Duke Gauge Theory through NCG October 24, / 44

13 Using the language we are familiar with, we have (up to order one condition) ψ uψ, D u(d + u [D, u])u, A u(a u [D, u])u, D + A u(d + A)u. Formally, D works similarly to a differential operator as in W = W µ dx µ, and A works like the gauge field. In this way, we can define the new differential one forms as elements in { } Ω 1 = ai [D, b i ] a i, b i A. Chen Duke Gauge Theory through NCG October 24, / 44

14 Using the language we are familiar with, we have (up to order one condition) ψ uψ, D u(d + u [D, u])u, A u(a u [D, u])u, D + A u(d + A)u. Formally, D works similarly to a differential operator as in W = W µ dx µ, and A works like the gauge field. In this way, we can define the new differential one forms as elements in { } Ω 1 = ai [D, b i ] a i, b i A. Define the perturbed D A to be the combination of the two D A = D + A. Chen Duke Gauge Theory through NCG October 24, / 44

15 Generalization As it is shown above, (A, H, D) = (C (M), Γ(M, S), i / ) gives us a U(1) gauge theory. But, what for? With a few modifications, we can build a generalized gauge theory. Chen Duke Gauge Theory through NCG October 24, / 44

16 A = C C According to Gelfand-Naimark, if we study all the algebra in C (M), we can get all the information of the geometry M. where f C (M). f : M C, p f (p), By analogy: Consider changing A = C (M) to A = C C, a A, we denote a = (λ, λ ). This is the map, a : {p 1, p 2 } C, p 1 a(p 1 ) = λ, p 2 a(p 2 ) = λ. Similar to C (M) M, roughly, we have C C {p 1, p 2 }, a two point space. Chen Duke Gauge Theory through NCG October 24, / 44

17 A = C C At this point, there is no relation for the two points space. In A = C (M), the distance is d(x, y) = inf ds, γ d 2 s = g µν dx µ dx ν. How to extract this information from the algebra, if Gelfand-Naimark is correct? Chen Duke Gauge Theory through NCG October 24, / 44

18 A = C C At this point, there is no relation for the two points space. In A = C (M), the distance is d(x, y) = inf ds, γ d 2 s = g µν dx µ dx ν. How to extract this information from the algebra, if Gelfand-Naimark is correct? d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Chen Duke Gauge Theory through NCG October 24, / 44

19 d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Chen Duke Gauge Theory through NCG October 24, / 44

20 Distance formula: d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Translate: f x x 1 x 2 x Chen Duke Gauge Theory through NCG October 24, / 44

21 Distance formula: d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Translate: f x x 1 x 2 x Chen Duke Gauge Theory through NCG October 24, / 44

22 Distance formula: d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Translate: f x x 1 x 2 x Chen Duke Gauge Theory through NCG October 24, / 44

23 Distance formula: d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Translate: f x x 1 x 2 x Chen Duke Gauge Theory through NCG October 24, / 44

24 Distance formula: d(x, y) = sup{ f (x) f (y) : f C (M), f (x) 1}. Translate: f x x 1 x 2 x Chen Duke Gauge Theory through NCG October 24, / 44

25 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, The distance formula is D = [ 0 m m 0 ]. 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 A, [D,a] 1 { a(p 1 ) a(p 2 ) } = sup (λ,λ ) A, [D,a] 1 λ λ = 1 m. Chen Duke Gauge Theory through NCG October 24, / 44

26 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, The distance formula is D = [ 0 m m 0 ]. 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 A, [D,a] 1 { a(p 1 ) a(p 2 ) } = sup (λ,λ ) A, [D,a] 1 λ λ = 1 m. The generalized Dirac operator encodes the distance information! Chen Duke Gauge Theory through NCG October 24, / 44

27 A = C C A = C C, H = C N C N, [ ] 0 M D =. M 0 For a A = (λ, λ ), the differential is (λ λ ): By analogy with [ ] [D, a] = (λ λ 0 M ), M 0 df = µ f dx µ = lim ɛ 0 (f (x + ɛ) f (x)) dx µ ɛ. Chen Duke Gauge Theory through NCG October 24, / 44

28 A = C C A = C C, H = C N C N, [ ] 0 M D =. M 0 For a A = (λ, λ ), the differential is (λ λ ): [ ] [D, a] = (λ λ 0 M ), M 0 By analogy with df = µ f dx µ = lim ɛ 0 (f (x + ɛ) f (x)) dx µ ɛ. The integral is (λ + λ ): By analogy with Tr(a) = λ + λ. f (x)dx. Chen Duke Gauge Theory through NCG October 24, / 44

29 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 Duke Gauge Theory through NCG October 24, / 44

30 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 Duke Gauge Theory through NCG October 24, / 44

31 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 Duke Gauge Theory through NCG October 24, / 44

32 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 Duke Gauge Theory through NCG October 24, / 44

33 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. [ ] In the case of (A, H, D) = (C C, C N C N 0 M, ), we can choose M 0 the grading operator to be γ = diag(1,..., 1, 1,..., 1) }{{}}{{} N copies N copies Chen Duke Gauge Theory through NCG October 24, / 44

34 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. [ ] In the case of (A, H, D) = (C C, C N C N 0 M, ), 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 Duke Gauge Theory through NCG October 24, / 44

35 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. [ ] In the case of (A, H, D) = (C C, C N C N 0 M, ), 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 Duke Gauge Theory through NCG October 24, / 44

36 A = C H A toy model A = C H, H = C 2 C 2, [ ] 0 M D =. M 0 Chen Duke Gauge Theory through NCG October 24, / 44

37 A = C H A toy model A = C H, H = C 2 C 2, [ ] 0 M D =. M 0 How do we fit this with our particle spectrum? Flavor space: ν R = 0, e R = 0, ν L = 1, e L = λ For any a A, a = To give mass terms out of ψ Dψ, let M = λ α β [ ] mν 0. 0 m e β. α Chen Duke Gauge Theory through NCG October 24, / 44

38 The unitary transformations are {u A u u = uu = 1}. This implies u = e iθ e iθ α β β s.t. α 2 + β 2 = 1. 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 L 0 0 Chen Duke Gauge Theory through NCG October 24, / 44

39 When we make the U(1) R SU(2) L transformation, L =Ψ DΨ Ψ u DuΨ = Ψ DΨ + 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 = i a i [D, b i ]. L =Ψ (D + A)Ψ Ψ DΨ + Ψ u [D, u]ψ + Ψ AΨ Ψ u [D, u]ψ =Ψ (D + A)Ψ Chen Duke Gauge Theory through NCG October 24, / 44

40 Demanded to be Hermitian, this gauge field is [ ] M A = Φ, ΦM [ ] φ1 φ Φ = [φ 1 φ 2 ] = 2 φ 2 φ 1 Chen Duke Gauge Theory through NCG October 24, / 44

41 Demanded to be Hermitian, this gauge field is [ ] M A = Φ, ΦM [ ] M D + A = (Φ + 1). (Φ + 1)M Chen Duke Gauge Theory through NCG October 24, / 44

42 Demanded to be Hermitian, this gauge field is [ ] M A = Φ, ΦM [ ] M D + A = (Φ + 1). (Φ + 1)M The perturbation of D 2 derived from the (spectral) action: Tr ( (D + A) 2 D 2) Tr ( (D + A) 2 D 2) Tr ( (MM ) 2) ( Φ ) 2. Chen Duke Gauge Theory through NCG October 24, / 44

43 Demanded to be Hermitian, this gauge field is [ ] M A = Φ, ΦM [ ] M D + A = (Φ + 1). (Φ + 1)M The perturbation of D 2 derived from the (spectral) action: Tr ( (D + A) 2 D 2) Tr ( (D + A) 2 D 2) Tr ( (MM ) 2) ( Φ ) 2. This gives us a Mexican-hat-shaped potential. A field expanded at the minimum 0. By counting d.o.f, we have = 4 real degrees, i.e. Φ is a pair of complex numbers. Chen Duke Gauge Theory through NCG October 24, / 44

44 Demanded to be Hermitian, this gauge field is [ ] M A = Φ, ΦM [ ] M D + A = (Φ + 1). (Φ + 1)M The perturbation of D 2 derived from the (spectral) action: Tr ( (D + A) 2 D 2) Tr ( (D + A) 2 D 2) Tr ( (MM ) 2) ( Φ ) 2. This gives us a Mexican-hat-shaped potential. A field expanded at the minimum 0. By counting d.o.f, we have = 4 real degrees, i.e. Φ is a pair of complex numbers. SSB now has a reason: D + A gives a VEV shift. Chen Duke Gauge Theory through NCG October 24, / 44

45 By analogy, local twist e iθ µ e iθ = µ θ u [D, u] ω A µ d µ x ai [D, b i ] = ΦM basis d µ x M comp A µ Φ M θ (d + A) (d + A) Tr ( (D + A) 2 D 2) F µν µ A ν + [A µ, A ν ] DA + A 2 S F µν F µν d 4 x (Tr ( (D + A) 2 D 2) ) 2 M Φ Chen Duke Gauge Theory through NCG October 24, / 44

46 Product geometry Consider the algebra: This corresponds to a geometry A =C (M) C (M) C (M) (C C). F =M M, M {p 1, p 2 }. Chen Duke Gauge Theory through NCG October 24, / 44

47 Product geometry Consider the algebra: This corresponds to a geometry A =C (M) C (M) C (M) (C C). F =M M, M {p 1, p 2 }. Combining continuous part with C H, a double-layer structure. A = C (M) (C H). Chen Duke Gauge Theory through NCG October 24, / 44

48 The Dirac operator of the product geometry: The gauge field: D x = i / + γ 5 D. Chen Duke Gauge Theory through NCG October 24, / 44

49 The Dirac operator of the product geometry: The gauge field: D x = i / + γ 5 D. A x f i [/, g i ] + a i [D, b i ] } {{ } } {{ } A [1,0] A [0,1] Chen Duke Gauge Theory through NCG October 24, / 44

50 The Dirac operator of the product geometry: The gauge field: D x = i / + γ 5 D. A x f i [/, g i ] }{{} A [1,0] [ B Φ Φ W ]. + a i [D, b i ] } {{ } A [0,1] Chen Duke Gauge Theory through NCG October 24, / 44

51 Highlights: A two sheet structure. A gauge field in between. SSB feature out of box. Chen Duke Gauge Theory through NCG October 24, / 44

52 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 Duke Gauge Theory through NCG October 24, / 44

53 Color sector In order to reproduce SM, color sector must be involved. Introduce the color space. H = C C 3, with basis l = 0, r = 0, g = 1, b = A = C M 3 (C), with a A, λ a = Symmetry group is m 11 m 21 m 12 m 22 m 13 m. 23 m 31 m 32 m 33 {u A u u = uu = 1}, together with the unimodularity condition, det u = 1. a = e iθ e iθ/3 m, m SU(3). Chen Duke Gauge Theory through NCG October 24, / 44

54 a = e iθ e iθ/3 m, m SU(3). This gives the U(1) charge l r g b We recognize them as B L charge, and this gives us the symmetry U(1) B L SU(3) C. Chen Duke Gauge Theory through NCG October 24, / 44

55 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 Duke Gauge Theory through NCG October 24, / 44

56 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 Duke Gauge Theory through NCG October 24, / 44

57 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: Denote the space as R l R r L g. L b (2 R 2 L ) (1 l 3 C ). Chen Duke Gauge Theory through NCG October 24, / 44

58 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: Denote the space as R l R r L g. L b (2 R 2 L ) (1 l 3 C ). Can identify the basis with SM particle spectrum, for example Chen Duke Gauge Theory through NCG October 24, / 44

59 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: Denote the space as R l R r L g. L b (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 Duke Gauge Theory through NCG October 24, / 44

60 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: Denote the space as R l R r L g. L b (2 R 2 L ) (1 l 3 C ). Can identify the basis with SM particle spectrum, for example d R,g = R g Chen Duke Gauge Theory through NCG October 24, / 44

61 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: Denote the space as R l R r L g. L b (2 R 2 L ) (1 l 3 C ). Can identify the basis with SM particle spectrum, for example d R = R [ r g b ] 2 R 3 C, Chen Duke Gauge Theory through NCG October 24, / 44

62 Introduce J, charge conjugate, R l l R J R r r L g g R, 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. 2 R L 0 0 l r g b Chen Duke Gauge Theory through NCG October 24, / 44

63 U(1) R : L R U(1) B L : L R Chen Duke Gauge Theory through NCG October 24, / 44

64 L R Chen Duke Gauge Theory through NCG October 24, / 44

65 Spectral Action According to Chamseddine et. al. (hep-th/ ), one builds the action based on spectral action principle: The physical (bosonic) action only depends upon the spectrum of D. Chen Duke Gauge Theory through NCG October 24, / 44

66 Spectral Action According to Chamseddine et. al. (hep-th/ ), 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 Tr(f (D A /Λ)) L(g µν, A) g d 4 x. M Chen Duke Gauge Theory through NCG October 24, / 44

67 The bosonic action, S Bosonic =S Higgs + S YM + S Cosmology + S Riemann, S Higgs = f 0a 2π 2 D µ φ 2 g d 4 x + 2af 2Λ 2 + ef 0 π 2 φ 2 g d 4 x + f 0b 2π 2 φ 4 g d 4 x, S YM = f 0 16π 2 Tr(F µνf µν ) = f 0 2π 2 (g3 2 GµνG i µν i + g2 2 WµνW i µν i g 1 2 B µν B µν ) g d 4 x where the parameters are a =Tr(M ν M ν + M e M e + 3(M u M u + M d M d )) b =Tr((M ν M ν ) 2 + (M e M e ) 2 + 3(M u M u ) 2 + 3(M d M d ) 2 ) c =Tr(M RM R ) d =Tr((M RM R ) 2 ) e =Tr(M RM R M ν M ν ), f n is the (n 1)th momentum of f. Chen Duke Gauge Theory through NCG October 24, / 44

68 Output: Chen Duke Gauge Theory through NCG October 24, / 44

69 Output: g 2 3 = g 2 2 = 5 3 g 2 1, Chen Duke Gauge Theory through NCG October 24, / 44

70 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, Chen Duke Gauge Theory through NCG October 24, / 44

71 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, M 2 W = 1 8 i (mi 2 ν + m i 2 e + 3m i 2 u + 3m i 2 d ), Chen Duke Gauge Theory through NCG October 24, / 44

72 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, MW 2 = 1 i 8 (mi 2 ν + m i 2 e + 3m i 2 u + 3m i 2 d ), Can be calculated from spectral action. Chen Duke Gauge Theory through NCG October 24, / 44

73 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, MW 2 = 1 i 8 (mi 2 ν + m i 2 e + 3m i 2 u + 3m i 2 d ), Can be calculated from spectral action. Intuitively, Cont Disc Fermion ψ / ψ Ψ DΨ Boson µ W µ W D 2 W 2 Chen Duke Gauge Theory through NCG October 24, / 44

74 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, M 2 W = 1 8 m H 170 GeV, i (mi 2 ν + m i 2 e + 3m i 2 u + 3m i 2 d ), Chen Duke Gauge Theory through NCG October 24, / 44

75 Output: g 2 3 = g 2 2 = 5 3 g 2 1, φ 0, M 2 W = 1 8 i (mi 2 ν + m i 2 e + 3m i 2 u + 3m i 2 d ), m H 170 GeV, problematic, which is naturally saved by the left-right completion we propose. Chen Duke Gauge Theory through NCG October 24, / 44

76 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, Chen Duke Gauge Theory through NCG October 24, / 44

77 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δl δl δs = δφ + δφ δ φ δ φ ( ) ( ) δl δl δl = δφ δφ + δ φ δφ δφ δ φ ( ) EOM δl = δ φ δφ ( = ɛ δl ) δ φ Φ = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. Chen Duke Gauge Theory through NCG October 24, / 44

78 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δs = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. µ j µ = 0 a symmetry. µ (ɛ j µ ) = 0 a global symmetry. µ (ɛ j µ ) 0, a local symmetry with a gauge. Chen Duke Gauge Theory through NCG October 24, / 44

79 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δs = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. µ j µ = 0 a symmetry. µ (ɛ j µ ) = 0 a global symmetry. µ (ɛ j µ ) 0, a local symmetry with a gauge. Ψ DΨ Ψ DΨ + Ψ [D, ɛ i σ i ]Ψ. Chen Duke Gauge Theory through NCG October 24, / 44

80 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δs = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. µ j µ = 0 a symmetry. µ (ɛ j µ ) = 0 a global symmetry. µ (ɛ j µ ) 0, a local symmetry with a gauge. Ψ DΨ Ψ DΨ + Ψ [D, ɛ i σ i ]Ψ. Ψ [D, ɛ i σ i ]Ψ by analogy with µ (ɛj µ ). Chen Duke Gauge Theory through NCG October 24, / 44

81 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δs = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. µ j µ = 0 a symmetry. µ (ɛ j µ ) = 0 a global symmetry. µ (ɛ j µ ) 0, a local symmetry with a gauge. Ψ DΨ Ψ DΨ + Ψ [D, ɛ i σ i ]Ψ. Ψ [D, ɛ i σ i ]Ψ by analogy with µ (ɛj µ ). [D, ɛ i σ i ] = 0 a global symmetry in the discrete direction. Chen Duke Gauge Theory through NCG October 24, / 44

82 Other Fun Facts local twist [D, u] is insensitive to local/global transformation w.r.t. M. φ φ + δφ, with δφ = ɛ i σ i φ = ɛ i Φ i, δs = (ɛj) = µ (ɛ)j µ + ɛ µ j µ. µ j µ = 0 a symmetry. µ (ɛ j µ ) = 0 a global symmetry. µ (ɛ j µ ) 0, a local symmetry with a gauge. Ψ DΨ Ψ DΨ + Ψ [D, ɛ i σ i ]Ψ. Ψ [D, ɛ i σ i ]Ψ by analogy with µ (ɛj µ ). [D, ɛ i σ i ] = 0 a global symmetry in the discrete direction. [D, ɛ i σ i ] 0 a local symmetry in the discrete direction, with a gauge. Chen Duke Gauge Theory through NCG October 24, / 44

83 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 Duke Gauge Theory through NCG October 24, / 44

84 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 Duke Gauge Theory through NCG October 24, / 44

85 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 Duke Gauge Theory through NCG October 24, / 44

86 Other Fun Facts local twist [ ] 0 M In the simplest case, A = H H, D = and M = M 0 Pictorially, the twist between left sheet and right sheet. [ ] 0 mu. m d 0 Chen Duke Gauge Theory through NCG October 24, / 44

87 Other Fun Facts local twist [ ] 0 M In the simplest case, A = H H, D = and M = M 0 [ ] 0 mu. 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 Duke Gauge Theory through NCG October 24, / 44

88 Other Fun Facts The seperation Totally independent of the base manifold M. Chen Duke Gauge Theory through NCG October 24, / 44

89 Other Fun Facts The seperation Totally independent of the base manifold M. Extra dimension but discrete. Chen Duke Gauge Theory through NCG October 24, / 44

90 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 Duke Gauge Theory through NCG October 24, / 44

91 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 Duke Gauge Theory through NCG October 24, / 44

92 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, (2015) [arxiv: ]) Chen Duke Gauge Theory through NCG October 24, / 44

93 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: ], 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 2L = g 2 2R = g 2 3. Through the mixing of SU(2) R U(1) B L into U(1) Y, we get LR symmetry breaking at GUT. 1 g 2 = 1 g gbl 2 = g 2. Chen Duke Gauge Theory through NCG October 24, / 44

94 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 Duke Gauge Theory through NCG October 24, / 44

95 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 Duke Gauge Theory through NCG October 24, / 44

96 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 Duke Gauge Theory through NCG October 24, / 44

97 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 Duke Gauge Theory through NCG October 24, / 44

98 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 Duke Gauge Theory through NCG October 24, / 44

99 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 Duke Gauge Theory through NCG October 24, / 44

100 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 Duke Gauge Theory through NCG October 24, / 44

101 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. SSB has a reason. Chen Duke Gauge Theory through NCG October 24, / 44

102 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. SSB has a reason. Fit in all SM fermions and bosons. Chen Duke Gauge Theory through NCG October 24, / 44

103 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. SSB has a reason. Fit in all SM fermions and bosons. GUT without new degrees of freedom. Chen Duke Gauge Theory through NCG October 24, / 44

104 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. SSB has a reason. Fit in all SM fermions and bosons. GUT without new degrees of freedom. Mass relation. Chen Duke Gauge Theory through NCG October 24, / 44

105 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. SSB has a reason. Fit in all SM fermions and bosons. GUT without new degrees of freedom. Mass relation. Predicts Higgs mass. Chen Duke Gauge Theory through NCG October 24, / 44

106 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. SSB has a reason. Fit in all SM fermions and bosons. GUT without new degrees of freedom. Mass relation. Predicts Higgs mass. Local twist with different setting of D. Chen Duke Gauge Theory through NCG October 24, / 44

107 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. SSB has a reason. Fit in all SM fermions and bosons. GUT without new degrees of freedom. Mass relation. Predicts Higgs mass. Local twist with different setting of D. Minimally coupled gravity sector.... Chen Duke Gauge Theory through NCG October 24, / 44

108 Reference Chamseddine, Connes, et. al.: Nucl. Phys. Proc. Suppl. 18B, 29 (1991) Commun. Math. Phys. 182, 155 (1996) [hep-th/ ], Adv. Theor. Math. Phys. 11, 991 (2007) [hep-th/ ], JHEP 0611, 081 (2006) [hep-th/ ], Commun. Math. Phys. 186, 731 (1997) [hep-th/ ]. Phys. Rev. Lett. 99, (2007) [arxiv: [hep-th]], J. Geom. Phys. 58, 38 (2008) [arxiv: [hep-th]], Fortsch. Phys. 58, 553 (2010), [arxiv: [hep-th]]. Neeman, Fairlie, et. al.: Phys. Lett. B 81, 190 (1979), J. Phys. G 5, L55 (1979), Phys. Lett. B 82, 97 (1979). Ufuk Aydemir, Djordje Minic, C.S., Tatsu Takeuchi: Phys. Rev. D 91, (2015) [arxiv: ], Pati-Salam Unification from Non-commutative Geometry and the TeV-scale W R boson [arxiv: ]. Chen Duke Gauge Theory through NCG October 24, / 44