Principal -bundles Theory and applications

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1 Principal -bundles Theory and applications Contents Urs Schreiber Notes for a talk, May 2012 reportin on joint work [NSS] with Thomas Nikolaus and Danny Stevenson with precursors in [NiWa, RoSt, SSS, S] 1 Motivation 1 2 Hiher eometry 2 3 G-principal -bundles 3 4 Associated and twisted -bundles 4 5 Selected examples 5 6 Outlook: -Geometric Prequantization 6 1 Motivation Classical fact. For a manifold and G a topoloical/lie roup, rearded as a sheaf of roups C(, G) on, there is an equivalence: (x, i) 4 alebraic data on { deree-1 nonabelian sheaf cohomoloy H 1 (, G) (x, j) x (x, k) ij (x) cocycle ik (x) BG jk (x) / { P G p 1 P eometric data on isomorphism classes of G-principal bundles over GBund() pullback G-principal bundle classifyin map EG G p 1 EG BG universal bundle G-actions total spaces quotient spaces Problem. In hiher differential eometry, for instance in Strin-eometry [SSS], Lie roups G are replaced by rouplike smooth A -spaces: by -roups (examples below in 5). Need to eneralize the above classical fact to this case. / 1

2 2 Hiher eometry We need eometry + homotopy theory = hiher eometry -topos theory. Here is a way to think of the above classical fact that will eneralize: let C := SmthMfd be the cateory of all smooth manifolds (or some other site, here assumed to have enouh points); Sh(C) be the cateory of roupoid-valued sheaves over C, for instance = {, BG = { G Sh(C); Ho Sh(C) the homotopy cateory obtained by universally turnin the stalkwise roupoidequivalences into isomorphisms. Fact: H 1 (, G) Ho Sh(C) (, BG). To eneralize, let Cateories N Groupoids QuasiCateories SimplicialSets N KanComplexes ssh(c) lfib Sh(C, sset) be the (stalkwise Kan) simplicial sheaves; be the Kan complexes, aka -roupoids inside all quasi-cateories aka -cateories L W ssh(c) lfib the simplicial localization obtained by universally turnin stalkwise homotopy equivalences into homotopy equivalences. Definition/Theorem. This is the -cateory theory analo of the sheaf topos over C, the -stack -topos: H := Sh (C) L W ssh(c) lfib. Example. Smooth Grpd := Sh (SmthMfd) is the -topos of smooth -roupoids / smooth -stacks. Example. For A a sheaf of abelian roups, B n+1 A := DoldKan(A[n + 1]) ssh(c) is the moduli n-stack of B n A-principal bundles (details in a moment). Proposition. Every object in Smooth Grpd is presented by a simplicial manifold, but not necessarily by a locally Kan simplicial manifold (see below). Definition A roup in the -topos is a G H equipped with a roupal A -alebra structure: coherently homotopy associative product with coherent homotopy inverses. Example. In Smooth Grpd this is a smooth -roup: for instance a Lie roup, or a Lie 2-roup, or a differentiable roup stack, or a sheaf of simplicial roups on SmthMfd. Fact. (Milnor-Lurie) There is an equivalence { roups in H loopin Ω deloopin B { pointed connected objects in H Proposition. Let C have a terminal object. For every -roup G Grp(Sh (C)) there is a sheaf of simplicial roups presentin it under Sh (C) L W ssh(c); and every -action : P G P is presented by a correspondin simplicial action. 2

3 3 G-principal -bundles Definition. A G-principal bundle over H is a morphism P ; with an -action : P G P ; such that P is -quotient P P//G () principality : P G n (p 1,) P P Theorem. There is equivalence of -roupoids GBund() 1. hofib sends a cocycle BG to its homotopy fiber; hofib lim H(, BG), where 2. lim sends an -bundle to the map on -quotients P//G //G BG. In particular, G-principal -bundles are classified by the intrinsic cohomoloy of H GBund()/ H 1 (, G) := π 0 H(, BG). Proof. Repeatedly apply two of the ()Giraud-Rezk-Lurie axioms that characterize -toposes: 1. every -quotient is effective; 2. -colimits are preserved by -pullbacks.. P G G P G p 1 P. G G G G- -actions total objects G-principal -bundle -pullback BG universal -bundle quotient objects cocycle This ives a eneral abstract theory of principal -bundles in every -topos. We also have the followin explicit presentation. Definition For G Grp(sSh(C)), and ssh(c) lfib, a weakly G-principal simplicial bundle is a G-action over such that the principality morphism (, p 1 ) : P G P P is a stalkwise weak equivalence. Theorem. Nerve weakly G-principal simplicial bundles over GBund(). Example. For terminal over C and restricted to cohomoloy classes, this is [JL]. Remark. We need more than that, notably = BG itself, see next pae. Example. For C = we have ssh(c) lfib = KanComplexes. Classical theory considers strictly principal simplicial bundles [Ma]. Proposition. Strictly principal simplicial bundles over C = do present the cohomoloy H 1 (, G), but not in eneral the full cocycle space H(, BG). For C nontrivial they do in eneral not even present H 1 (, G). Proposition. For G a simplicial Lie roup, which is CartSp-acyclic (e.. Strin), every G- principal -bundle over a smooth manifold is presented by a locally Kan simplicial smooth manifold. 3

4 4 Associated and twisted -bundles Observation. By the above theorem, every G- -action : V G G has a classifyin map: V V//G Proposition. This is the universal -associated V -bundle. BG Observation. Sections σ of the associated -bundle are lifts of the cocycle throuh ; and these locally factor throuh V : P G V V//G V//G V V//G σ BG σ BG U σ U Hence sections are cocycles in -twisted ΩV -cohomoloy relative : Γ (P G V ) H /BG (, ). BG Theorem. Equivalently this classifies P -twisted -bundles: twisted G-equivariant ΩV - bundles on P :. Q P -twisted ΩV -principal -bundle P V G-principal -bundle σ V//G BG section of -associated V - -bundle { sections of -associated V - -bundle { -twisted ΩV -cohomoloy relative { ΩV - -bundles twisted by P First example. Associated connected-fiber -bundles are -erbes. A (nonabelian/giraud-)erbe on is a connected 1-truncated object in H / (a connected stack on ). A (nonabelian/giraud-breen) -erbe over is a connected object in H /. A G- -erbe is an Aut(BG)-associated -bundle. Its band is the underlyin Out(G)- principal -bundle. Observation. G- -erbes bound by a band are classified by (BAut(BG) BOut(G))- twisted cohomoloy. 4

5 5 Selected examples extension / -bundle of coefficients B 2 ker(g) GL(d)/O(d) O(d)\O(d, d)/o(d) V V//G BG BAut(BG) BOut(G) BO(d) BGL(d) B(O(d) O(d)) BO(d, d) BU BP U dd B 2 U(1) twistin -bundle / twistin cohomoloy -associated V - -bundle band (lien) tanent bundle eneralized tanent bundle circle 2-bundle / bundle erbe twisted -bundle / twisted cohomoloy section nonabelian (Giraud-Breen) G- -erbe orthoonal structure / Riemannian eometry eneralized (type II) Riemannian eometry twisted vector bundle / twisted K-cocycle / bundle erbe module [NSS] B n U(1) B n U(1)//Z 2 J n 1 BZ 2 double cover hiher orientifold / n = 2: Jandl bundle erbe [FSSb] [SSW] V BSpin ν n+1 BStrin BFivebrane ν int n+1 B n U(1) BSpin 1 2 p 1 B 3 U(1) V B(T T ) c 1 c 1 B 3 U(1) BStrin 1 6 p 2 B 7 U(1) circle n-bundle circle 3-bundle / bundle 2-erbe circle 3-bundle / bundle 2-erbe circle 7-bundle smooth interal Wu structure twisted Strin 2-bundle twisted T-duality structure twisted Fivebrane 6-bundle [FSSb] [SSS] [FSSa] [SSS] [FSSa] B n U(1) B n U(1) curv dr B n+1 U(1) curvature (n + 1)-form circle n-bundle with connection 5

6 6 Outlook: -Geometric Prequantization Observation. There is a canonical -action γ of Aut H/BG () on the space of -sections Γ (P G V ). Claim. Since Sh (SmthMfd) is cohesive, there is a notion of differential refinement of the above discussion, yieldin connections on -bundles. Example. Let C C//U(1) BU(1) be the canonical complex-linear circle action. Then conn : BU(1) conn classifies a circle bundle with connection, a prequantum line bundle of its curvature 2-form; Γ (P U(1) C) is the correspondin space of smooth sections; γ is the exp(poisson bracket)-roup action of preqantum operators, containin the Heisenber roup action. Example. Let BU BPU B 2 U(1) be the canonical 2-circle action. Then conn : B 2 U(1) conn classifies a circle 2-bundle with connection, a prequantum line 2-bundle of its curvature 3-form; Γ (P BU(1) BU) is the correspondin roupoid of smooth sections = twisted bundles; γ is the exp(2-plectic bracket)-2-roup action of 2-plectic eometry, containin the Heisenber 2-roup action [RoSc]. References [FSSa] D. Fiorenza, U. Schreiber, J. Stasheff. Čech-cocycles for differential characteristic classes, ariv: [FSSb] D. Fiorenza, H. Sati, U. Schreiber, The E 8 -moduli 3-stack of the C-field, ariv: [JL] J.F. Jardine, Z. Luo, Hiher order principal bundles, K-theory 0681 (2004) [Ma] P. May, Simplicial objects in alebraic topoloy University of Chicao Press (1967) [NSS] T. Nikolaus, U. Schreiber, D. Stevenson, Principal -bundles General theory and presentations (2012) [NiWa] T. Nikolaus, K. Waldorf, Four equivalent versions of non-abelian erbes, ariv: [RoSt] D. Roberts, D. Stevenson, Simplicial principal bundles in parametrized spaces (2012) ariv: D. Stevenson, Classifyin theory for simplicial parametrized roups (2012) ariv: [RoSc] C. Roers, Hiher symplectic eometry, PhD (2011) ariv: , C. Roers, U. Schreiber, -Geometric prequantization, ncatlab.or/schreiber/show/infinity-eometric+prequantization [SSS] H. Sati, U. Schreiber, J. Stasheff, Twisted differential strin- and fivebrane structures, Communications in Mathematical Physics (2012) ariv: U. Schreiber, Differential cohomoloy in a cohesive topos ncatlab.or/schreiber/show/differential+cohomoloy+in+a+cohesive+topos [SSW] U. Schreiber, C. Schweiert, K. Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics, Volume 274, Issue 1 (2007) 6