THE GAUGE PRINCIPLE 1. GENERAL STRUCTURES

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1 THE GAUGE PRINCIPLE IN THE 2D σ-model 1. GENERAL STRUCTURES RAFA L R. SUSZEK (KMMF WFUW) Exact Results in Quantum Theory & Gravity, IFT UW (11/1/2012) Based, in part, on joint work with K. Gawȩdzki, I. Runkel and K. Waldorf: 1. I.R. & R.R.S., Adv. Theor. Math. Phys. 13 (2009), K.G., R.R.S. & K.W., Commun. Math. Phys. 284 (2008), K.G., R.R.S. & K.W., Adv. Theor. Math. Phys. 15 (2011), in press. 4. R.R.S., Hamburger Beiträge zur Mathematik Nr. 360 (2011) [ [hep-th]]. 5. K.G., R.R.S. & K.W., Commun. Math. Phys. 302 (2010), R.R.S., Acta Phys. Pol. B Proc. Suppl. 4 (2011), K.G., R.R.S. & K.W., Gauging symmetries of 2d sigmamodels on world-sheets with defects, in writing. 8. R.R.S., Defects, dualities and the geometry of strings via gerbes, II. Generalised geometries with a twist, in writing. 1

2 I Introduction Setting: FIELD THEORY & ACTION FUNCTIONAL S[X] δs[x cl. ] = 0 (LAP) RIGID SYMMETRIES: F l G F F (g, σ, X i (σ)) (σ, g.x i (σ)), G Diff(F ), i.e. F is a G-space, and δs[g.x cl. ] = 0. Idea: RIGID GAUGING LOCAL : G g σ dep. g( ) G Σ 2

3 Problem: OBSTRUCTIONS GAUGE ANOMALIES LOCAL/INFINITESIMAL : non-invariance of S under infinitesimal gauge transformations ( homotopic to id) GLOBAL/TOPOLOGICAL : non-invariance of FEYNMAN AMPLITUDES A[X] = e S[X] under large gauge transformations (non-homotopic to id) Remark: The latter lead to destructive interferences of homotopy sectors within gauge orbits of states when gauge fields rendered dynamical. Ramifications: powerful selection rules for model building; effective field theories with F //G; exhaustive construction method for RCFT s; natural incorporation of BCFT s and CFT s over NON-ORIENTABLE WORLD-SHEETS (key rôle in susy extensions); T-DUALITY/MIRROR SYMMETRY. 3

4 II Requisites (IIσ) Rudiments of the field theory MONO-PHASE σ-model: & S σ,met. [X; γ] = 1 2 Σ g(dx, γ dx) (to start with) Problem: WEYL ANOMALY R µν ( L C (g)) Solution: H Z 3 (M) R µν ( L C (g)) = 1 4 H µαβ H νγδ (g 1 ) αγ (g 1 ) βδ, S σ [X; γ] = S σ,met. [X; γ] + S σ,wz [X], S σ,wz [X] = i Σ X d 1 H. Aside: WZ term consistent with STh Problem: It may happen that [H] dr 0 4

5 In general, we have MULTI-PHASE σ-model patches Σ i (M i, g i,...), combine into M = i (M i, g i,...); DEFECT LINES l i,j (Q i,j,...), combine into B = (i,j) (Q i,j,...), w/ ι 1, ι 2 Q M; DEFECT JUNCTIONS v i1,i 2,...,i n (T i1,i 2,...,i n,...), combine into J = n 3 (i1,i 2,...,i n )(T i1,i 2,...,i n,...), w/ π k,k+1 n T n Q. Relevance of defects: appear naturally in the orbifold σ-model (twisted sector); describe the most general CFT, incl. BCFT; capture symmetries/dualities of the σ-model. N.B. The missing bits (... ) come from 2-CATEGORY BGrb (F ). 5

6 (IIg) Rudiments of gerbe theory 0. BUNDLE GERBE WITH CONNECTIVE STRUCTURE: (L, L, µ L ) G π L Y [2] F pr 1 pr 2 (YF, B) CURVATURE π YF (F, H) curv(g) = H Z 3 (F, 2πZ); CURVING B Ω 2 (YF ) π YF H = db; CONNECTION L on L, with curv( L ) = B [2] B [1] ; GROUPOID STRUCTURE µ L L [1,2] L [2,3] L[1,3]. Upshot: A WZ [X] Hol G (X) = [X G] Ȟ2 (Σ, U(1)) U(1) 1. GERBE 1-ISOMORPHISM: Φ G 1 G 2, (E, E, α) π E YY 1,2 F π YY1,2 F Y 1 F F Y 2 F = Y 1,2 F pr 1 pr 2 Y 1 F Y 2 F π Y1 F π Y2 F F CONNECTION E on E, with curv( E ) = π 2 B 2 π 1 B 1; COHERENT ISO α E [1,2] L 2 [2,4] L 1,[1,3] E [3,4] 6

7 2. GERBE 2-ISOMORPHISM: ϕ G 1 i.e. COHERENT ISO φ E 1 [1] E 2 [2] over YY 1,2 Y 1,2 F Φ 1 ϕ Φ 2 G 2, π YY 1,2 Y1,2 F Y 1 Y 1,2 F Y1,2 F Y 2 Y 1,2 F = Y 1,2 Y 1,2 F pr 1 pr 2 Y 1 Y 1,2 F Y 2 Y 1,2 F π Y 1 Y1,2 F Y 1,2 F π Y 2 Y1,2 F These compose STRICT MONOIDAL 2-CATEGORY BGrb (F ) 0-cells: gerbes; 1-cells: gerbe 1-isos (incl. canonical Id G G G), with assoc. HORIZONTAL COMPOSITION Φ 1,2 Φ 2,3 Φ 2,3 Φ 1,2 G 1 G 2 G 3 Hom BGrb (F ) (G 1, G 2 ) is a CATEGORY with morphisms 2-cells: gerbe 2-isos (incl. canonical id IdG Id G Id G ), with assoc. COMPOSITIONs: ϕ 2 ϕ 1 G 1 Φ 1,2 ϕ 1 Φ 1,2 G 2 Φ 1,2 Φ 2,3 ϕ 2 Φ 1,2 G 3 (HORIZONTAL) ; ϕ 2 ϕ 1 G 1 ϕ Φ 1 1,2 ϕ 2 Φ 1,2 G 2 subject to INTERCHANGE LAW. (VERTICAL). 7

8 Xtras: there exist distinguished TRIVIAL GERBEs 2-forms: I ω = (F, ω, F C, m), m ((x, z) (x, z )) = (x, z z ) smooth maps f F 1 F 2 give rise to PULLBACK (2- )FUNCTORS f BGrb (F 2 ) BGrb (F 1 ). neat description in terms of Deligne hypercohomology H 2 (F, D(2) F ), i.e. cohomology of Čech-extended DELIGNE COMPLEX D(2) 1 i 0 U(1) d log Ω 1 (F ) d Ω 2 (F ) F there exists TRANSGRESSION FUNCTOR we have τ H 2 (F, D(2) F ) H1 (LF, D(1) LF ) Proposition 1. The following holds true in BGrb (F ): (1) Given G α, α = 1, 2 with curv(g 1 ) = curv(g 2 ), there exist G 2 D G1, curv(d) = 0, and G 1 G 2 iff D I 0. (2) Given Ψ α G H, α = 1, 2, there exist Ψ 2 Bun 1 (D) Ψ 1, curv(d) = 0, where Bun Bun 0 (F ) End(I 0 ) is the canonical equivalence of categories, and Φ 1 Φ 2 iff D J 0. (3) Given ψ α Ψ Ξ, α = 1, 2, there exists ψ 2 = d ψ 1, d C (π 0 (F ), U(1)). 8

9 Definition 2. STRING BACKGROUND B = (M, B, J ): (1) TARGET M = (M, g, G) with smooth metric TARGET SPACE (M, g), and bundle gerbe G ; (2) G-BI-BRANE B = (Q, ι 1, ι 2, ω, Φ) with smooth G-BI-BRANE WORLD-VOLUME Q, G-BI-BRANE CURVATURE ω Ω 2 (Q), smooth maps ι 1, ι 2 Q M, and Φ ι 1G ι 2G I ω ; (3) collection J = (J 3, J 4, J 5,...) of n-valent (G, B)-INTER- BI-BRANES J n = (T n ; ε 1,2 n, ε 2,3 n,..., εn n 1,n, ε n,1 n ; πn 1,2, πn 2,3,..., πn n 1,n, πn n,1 ; ϕ n ) consisting of: (a) smooth (G, B)-IBB WORLD-VOLUMES T n ; (b) ORIENTATION MAPS ε k,k+1 n T n { 1, 1}; (c) smooth maps π k,k+1 n ι εk 1,k n 2 πn k 1,k T n Q subject to constraints = ι εk,k+1 n 1 πn k,k+1 = πn k, k Z/nZ, where ι +1 1 = ι 1, ι +1 2 = ι 2, ι 1 1 = ι 2 and ι 1 (d) for Φ k,k+1 n = (πn k,k+1 ) Φ εk,k+1 n ε k,k+1 n (π k,k+1 n ) ω, Gn 3 I ω 1,2 2 = ι 1;, Gn k = (πn) k G, and ω k,k+1 n +ωn 2,3 n = G 2 n I ω 1,2 n Φ 2,3 n id ϕ n Φ 3,4 n id Φ 1,2 n Φ n,n+1 n id id Gn 1 Gn 1 I ω 1,2 n +ωn 2, ωn n,1 F = M Q T is termed TARGET SPACE of B. 9

10 Upshot: The many uses of B: determines FEYNMAN AMPLITUDE A [(X Γ); γ] = exp ( 1 2 Σ g (dx, γ dx)) Hol G,Φ,ϕn (X Γ) as 2-DECORATED HOLONOMY for NETWORK-FIELD CONFIGURATIONS (X Γ). through transgression, induces PRE-QUANTUM BUNDLE C L B P (N) σ, [curv( LB )] dr = [Ω (N) σ ] dr, for (pre-)symplectic FORM Ω (N) σ on (N-TWISTED) PHASE SPACE P (N) σ (derived in GKST formalism). 10

11 III Rigid symmetries of the multi-phase σ-model (IIIi) The infinitesimal picture Consider F K = ( M K, Q K, Tn K ) X(F ) subject to ALIGNMENT ι α Q K = M K ια (Q), πn k,k+1 T n K = Q K π k,k+1 n (T n ). Under their (local) flows ξ t F F, A [(X Γ); γ] 1 d dt t=0 A [(ξ t X Γ); γ] = 1 2 Σ ( L M K g) (dx, γ dx) + i Σ X ( M K H) + i Γ (X Γ ) ( Q K ω). Proposition 3. F K engenders RIGID SYMMETRY iff L M K g = 0, M K H = dκ, κ Ω1 (m), Q K ω + Qκ = dk, k Ω 0 (Q) Tn k = 0, where Q = ι 2 ι 1 and T n = n k=1 ε k,k+1 n (π k,k+1 n ). The above combine into σ-symmetric SECTION K = ( M K κ, Q K k, Tn K ) Γ σ (EF ) of GENERALISED TANGENT BUNDLES EF = (TM T M) (TQ (Q R)) TT F, on which there exists (H, ω; Q )-TW. BRACKET STRUCTURE (EF,, (H,ω; Q), (, ), α TF ), generalising (TF, [, ]) and CLOSING ON Γ σ (EF ). 11

12 (H, ω; Q )-TW. BRACKET (of V i = ( M V i υ i, Q V i f i, Tn V i )) V 1, V 2 (H,ω; Q) M = [ M V 1, M V 2 ] ( L V1 υ 2 L V2 υ d(m V 1 υ 2 M V 2 υ 1 ) + ı V1 ı V2 H), V 1, V 2 (H,ω; Q) Q = [ Q V 1, Q V 2 ] ( L V1 f 2 L V2 f 1 + Q V 1 Q V 2 ω (Q V 1 Q υ 2 Q V 2 Q υ 1 )), V 1, V 2 (H,ω; Q) Tn = [ Tn V 1, Tn V 2 ]. Physical interpretation: On P (N) σ, (pre-)symplectic form reads Ω (N) σ = pr T C (S 1 (N),M) (δθ T C (S 1 (N),M) + π T C (S 1 (N),M) S 1 (N) and we find NOETHER MAP ev H) + N k=1 ε k pr Q (k) ω, N Γ σ (EF ) Γ H (E (1,0) P (N) σ ) K X K h K for can. lifts X K of α TF (K) and NOETHER HAMILTONIANS h K C (P (N) σ, R) X K Ω (N) σ = δh K. The latter are in INVOLUTION with respect to Ω (N) σ -TW. VINO- GRADOV BRACKET [ X 1 f 1, X 2 f 2 ] Ω(N) σ V = [X 1, X 2 ] (X 1 (f 2 ) X 2 (f 1 ) + X 1 X 1 Ω (N) σ ). Proposition 4. N (Γ σ (EF ),, (H,ω; Q) ) (Γ (E (1,0) P (N) σ is a homomorphism of Lie algebras over R. Comments: (H, ω; Q )-twisted bracket structure ), [, ] Ω(N) σ V ) restricts to Courant algebroid H-twisted à la Ševera Weinstein; admits a natural gerbe-theoretic interpretation. 12

13 (IIIg) The global picture In what follows, we shall also ultimately deal with integrated rigid symmetries. Definition 5. LIE GROUPOID Gr = (Ob Gr, Mor Gr, s, t, Id, Inv, ) consists of smooth OBJECT SET Ob Gr; smooth ARROW SET Mor Gr; smooth STRUCTURE MAPS: SOURCE s Mor Gr Ob Gr (surj. subm.); TARGET t Mor Gr Ob Gr (surj. subm.); UNIT Id Ob Gr Mor Gr m Id m ; INVERSE Inv Mor Gr Mor Gr g g 1 Inv( g ); MULTIPLICATION Mor Gr s t Mor Gr Mor Gr ( g, h ) g h ; subject to consistency constraints: (i) s( g h ) = s( h ), t( g h ) = t( g ); (ii) ( g h ) k = g ( h k ); (iii) Id t( g ) g = g = g Id s( g ) ; (iv) s( g 1 ) = t( g ), t( g 1 ) = s( g ), g g 1 = Id t( g ), g 1 g = Id s( g ). N.B. A (Lie) groupoid is a small category, and a Lie group is a Lie groupoid with a singleton as object set. 13

14 Relation to the infinitesimal picture shall be established through Definition 6. LIE ALGEBROID Gr = (V, [, ], α TF ) over smooth BASE F consists of vector bundle π V V F ; Lie bracket [, ] on Γ(V ); ANCHOR (bundle map) α TF V TF with the following properties: (i) induced map Γ(α TF ) Γ(V ) Γ(TF ) is a Lie-algebra homomorphism; (ii) Leibniz identity (for all X, Y Γ(V ) and f C (F, R)) and [X, f Y ] = f [X, Y ] + Γ(α TM )(X)(f) Y. Definition 7. Let Gr = (Ob Gr, Mor Gr, s, t, Id, Inv, ) be a Lie groupoid, R g s 1 ({t( g )}) s 1 ({s( g )}) h R g ( h ) = h g, and X s inv (Mor Gr) = { V Γ(ker ds) dr (V ) = V } space of right Gr-invariant vector fields on Mor Gr. TANGENT ALGE- BROID of Gr is gr = (Id ker ds, [, ], α T(Ob Gr) ). Anchor α T(Ob Gr) induces map dt i between spaces of sections, defined in terms of canonical vector-space isomorphism i Γ(Id ker ds) X s inv (Mor Gr), and Lie bracket is the unique bracket on Γ(Id ker ds) for which i is an isomorphism of Lie algebras. 14

15 In the case of interest, we encounter ACTION GROUPOID with structure maps G F = (F, G F, pr 2, F l, Id, Inv, ) Id m = (e, m), Inv(g, m) = (g 1, g.m), (h, g.m) (g, m) = (h g, m). Its tangent algebroid, termed ACTION ALGEBROID, is with Lie bracket dim R g g F = ( C (F, R) R A, [, ] g F, α TF ), A=1 [λ A R A, µ B R B ] g F = f ABC λ A µ B R C + ( L λ A F K A µ B L µ A F K A λ B ) R B and anchor α TF (R A ) = F K A. 15

16 A taste of the stuff to come... Idea: F enter gauge field P F reduction of DOF count P G F F Σ for P Σ arbitrary principal G-bundle with principal G-connection A Ω 1 (P) g, g = Lie G Sketch of construction: Step 1. Finding consistent coupling between B and A. Problem: Minimal-coupling recipe fails in general. Step 2. Lifting geometric G-action from F to B. Problem: Obstructions to equivariantisation. Step 3. Descending coupled string background from P F to associated bundle P G F, and subsequently (whenever admissible) to coset (P G F )/G. Problem: None, miraculously! Guiding principle: The Principle of Categorial Descent. 16