QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

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QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

(vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann [4] first order formalism Palatini formalism Riemann F () Heller [5] Newton 1 1 E-mail address: hfukaya[at]het.phys.sci.osaka-u.ac.jp ([at] @ ) 2

2 [2, 3] Nash & Sen [2] Riemann F E R 4 F F F G G F G U(1) x ϕ(x) F g(x) G ϕ (x) = g(x)ϕ(x), (1) U i = S 4 G = SU(2) S 3 F F G P G F E P F/G G P u T u (P ) V u (P ), H u (P ) u P T (P ) = V (P ) H(P ) 2 2 P P T (P ) 3

- P u = (x, g) [x R 4, g G] - ω = g 1 dg + g 1 Ag, A = A a µ(x)t a dx µ, (2) A a µ(x) T a G T (P ) X H(P ) ω, X = 0 (3) ω G ω T u (P ) H u (P ) ω ( ) g hg A hdh 1 + hah 1, (4) 2- ω Ω = dω + ω ω = g 1 (da + A A)g = g 1 F g, (5) F 2- F E ω 3 QCD QCD Euclid SU(3) P ρ ρ S θ = θ 4 TrF F, (6) θ ρ = exp(is θ ) 3 QCD θ 3 4

g µν = diag(1, 1, 1, 1) Hodge : F µν = 1 2 F αβg αγ g βδ ϵ γδµν. (7) S g = 1 4g 2 TrF F +, (8) P : Q = P F/G. (9) F Gauge Dirac D S q = d 4 x g qdq, (10) F QCD P Q ρ = exp( S g + is θ S q ) ρ G Lüscher U(1) [6] 4 solder 1- GL(4, R) F () 5

4 + 4 2 = 20 x T x R 4 GL(4, R) V R 4 v V t T x e v a = e a µt µ. (11) 4 1- e = (e 1 µdx µ, e 2 µdx µ, e 3 µdx µ, e 4 µdx µ ) solder 1- (solder 1-form) solder 1- solder 1-1- solder 1- e solder 1- e 1- solder 1- θ F () 1- θ = g 1 e (12) g hg e he θ F () u = (x, g) 4 θ F () 1- x u X X V u (F ()) θ, X = 0 5 (3) 1- H u (F ()) X T u (F ()) ω, X = 0 θ, X = 0 X = 0, (14) X 0 ω, θ X ω, θ ( ) 4 (12) u = (x, g) solder 1- F () X θ, X = e, π (X), (13) π π : F () ( ) π : T (F ()) T () e θ F () 1-5 X V u (F ()) π (X) = 0 6

ω 4 4 = 16, θ 4 F () 20 ω, θ F () X F () F () 1- solder 1- F () 2- (torsion 2-form) Θ = dθ + ω θ, (15) F () GL(4, R) 4 2 4 = 64 4 4 = 16 80 2 (reduction) GL(4, R) C GL(4, R) = O(4) C C O(4) O(4) Euclid Riemann Affine g µν = e a µe b νη ab, η ab = diag(1, 1, 1, 1), (16) Γ λ µν = [ A A ] a ν b η cae b µe c σg σλ + ( ), (17) ( ) A A ν O(4) a, b O(4) QCD QCD g µν (x)x µ (x)y ν (x) (18) g µν (metricity condition) ρ g µν g µν x ρ g µσ Γ σ νρ g νσ Γ σ µρ = 0, (19) 7

2 O(4) (reduction) 6 4 = 24, g µν 10, g µν O(4) 6 40 10 4 = 40 80 40 40 = 0 40 [D ν e µ ] a = ( ν δ a b + [A ν] a b )eb µ = 0, (20) A ν GL(4, R) D ν A S ν (10 4 = 40 ) A A ν (6 4 = 24 ) A ν = A S ν + A A ν ( ν δ a b + [AA ν ] a b )eb µ = [A S ν ] a b eb µ, (21) e a µ 16 AA ν 24 40 AS ν 40 A S A A ν O(4) GL(4, R) O(4) Γ ρ µν = [A S ν ] a b eb µ[e 1 ] ρ a, (22) 21 vierbein postulate [ D ν e µ ] a ( ν δ a b + [AA ν ] a b )eb µ = Γ ρ µνe a ρ. (23) D ν O(4) x ρ (g µν ) = x ρ (e a µe b νη ab ) = [ D ρ e µ ] a e b νη ab + e a µ[ D ρ e ν ] b η ab = Γ λ µρg λν + Γ λ νρg µλ, (24) (19) (20) GL(4, R) 80 40 = 40 Affine Γ λ µν = [e 1 ] λ a[ D ν e µ ] a = [ A A ] a ν b η cae b µe c σg σλ + ( ν e a µ)η ca e c σg σλ, (25) 8

Γ λ µν = Γ λ νµ T λ µν = Γ λ µν Γ λ νµ (23) [ D ν e µ ] a [ D µ e ν ] a = 0, (26) 2-15 2- F () Θ 24 O(4) g µν (10 ) g µν O(4) 6 16 16 A µ QCD ( 6) g µν ( 10) Γ λ µν Christoffel 2 θ S θ = θ 4 TrF F, (27) GL(4, R) QCD iπ ( ) GL(4, R) F = 0 GL(4, R) O(4) θ Hirzebruch Signature σa b = (e 1 ) µ a[d ν e µ ] b dx ν Tr [σ σ F ], (28) (21) (21) GL(4, R) O(4) g µν UV S Λ = Λpl 2 e a e b e c e d ϵ abcd, (29) S EH = 2 pl e a e b [ DA A ] c d ηde ϵ abce, (30) 9

Einstein-Hilbert pl DA A QCD 2 Einstein-Hilbert S m = d 4 x ψg µν γ a e a µ( ν + [A ν ] b cη cd γ b γ d )ψ(x), (31) (γ a 4 4 ) S = S Λ +S EH +S m A A µ (26) e a µ Einstein 6 2 2: GL(4, R) [A µ ] a b, [e µ] a 80 ( ) [D ν e µ ] a = 0 O(4)+ [A A µ ] a b, [e µ] a (g µν ) 40 ( ) [ D ν e µ ] a [ D µ e ν ] a = 0 O(4)+ O(4) [e µ ] a (g µν ) 16 g µν 10 5 QCD [7] g µν 6 (25) Riemann Rµνρ λ = [e 1 ] λ a[ D ρ A A ν ] a b e b µ 10

g µν 2 2 QCD g µν QCD 2 0, 1/2, 1 [8] (29) (30) Einstein-Hilbert g µν QED QED SU(2) Higgs W Witten [9] Chern-Simons 2 GL(4, R) 11

6 Affine (Einstein-Hilbert) 1 3 Chern-Simons QCD [1],, (1987) (ISBN-10: 4000050400). [2] C. Nash and S. Sen, Topology and Geometry for Physicists, Dover Books on athematics Reprint (2011) (ISBN-10: 0486478521). [3] ( ), I, II, (2000,2001) (ISBN-10: 4894711656, 4894714264). [4],, (1989) (ISBN-10: 4785310588). [5]. Heller, Evolution of Space-Time Structure, Concepts of Physics 3, 2006, pp. 119-133. [6]. Luscher, Nucl. Phys. B 549, 295 (1999) doi:10.1016/s0550-3213(99)00115-7 [heplat/9811032]. [7],, Volume1 (2009). [8]. Srednicki, Quantum Field Theory, (2007) Cambridge University Press (ISBN- 10: 0521864496). 12

[9] E. Witten, Nucl. Phys. B 311, 46 (1988) doi:10.1016/0550-3213(88)90143-5. 13

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