WZW terms in a cohesive -topos
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1 WZW terms in a cohesive -topos Talk at Representation Theoretical and Categorical Structures in Quantum Geometry and CFT 2011 Urs Schreiber November 1, 2011
2 With Domenico Fiorenza Hisham Sati Details and references at cohomology+in+a+cohesive+topos
3 Outline Motivation Higher WZW bundles Higher WZW connections Example Addendum
4 Goal: understand geometry of Chern-Simons models and their Wess-Zumino-Witten models.
5 Goal: understand geometry of Chern-Simons models and their Wess-Zumino-Witten models. in the spirit of Carey-Johnson-Murray-Stevenson-Wang (math/ ), Waldorf ( )
6 Goal: understand geometry of higher Chern-Simons models and their higher Wess-Zumino-Witten models.
7 I Motivation
8 I Motivation Please! No need.
9 The Holographic Principle of quantum field theory (QFT):
10 The Holographic Principle of quantum field theory (QFT): There are pairs consisting of an (n + 1)-dimensional topological QFT (TQFT);
11 The Holographic Principle of quantum field theory (QFT): There are pairs consisting of an (n + 1)-dimensional topological QFT (TQFT); an n-dimensional conformal QFT (CFT).
12 The Holographic Principle of quantum field theory (QFT): There are pairs consisting of an (n + 1)-dimensional topological QFT (TQFT); an n-dimensional conformal QFT (CFT). such that...
13 Holographic principle states of TFT n+1 identify with correlators of CFT n
14 Two realizations known:
15 Two realizations known: AdS n+1 /CFT n ; supergravity on asymptotic anti-de-sitter spacetime
16 Two realizations known: AdS n+1 /CFT n ; supergravity on asymptotic anti-de-sitter spacetime CS n+1 /CFT n Chern-Simons theory in 3d or higher dim abelian
17 best understood example: ChernSimons 3 /WZW 2. ordinary 3d Chern-Simons / Wess-Zumino-Witten model (WZW)
18 Witten (hep-th/ ): also AdS 5 /CFT 4 AdS 7 /CFT 6 governed by their higher Chern-Simons subsystems
19 Goal: understand higher CS models and their higher WZW models
20 Goal: understand higher CS models and their higher WZW models Strategy: 1. internalize construction in higher topos theory
21 Goal: understand higher CS models and their higher WZW models Strategy: 1. internalize construction in higher topos theory 2. unwind what the machinery spits out
22 II Higher WZW bundles
23 -topos theory: pairs homotopy theory with geometric structure
24 -topos theory: pairs homotopy theory with geometric structure Running example: H := Sh (SmthMfd) smooth -groupoids / smooth -stacks
25 -topos theory: pairs homotopy theory with geometric structure here we have long fiber sequences for smooth higher bundles...
26 G Start with an -group object: a grouplike A -space internal to the -topos H.
27 In running example: G is a smooth -group. G
28 BG The moduli stack of G-principal bundles.
29 X g BG A classifying map.
30 P X g BG The corresponding G-principal bundle.
31 P X g BG (All squares here and in the following are homotopy pullback squares.)
32 P X g BG c B n+1 U(1) Consider a characteristic map.
33 P X g BG c B n+1 U(1) Classifying a circle n + 1-bundle / bundle n-gerbe on BG.
34 P X g BG BG c B n+1 U(1) K(Z, n + 2) c Lifting a topological cohomology class [c] H n+2 (BG, Z).
35 P X g c(p) BG c B n+1 U(1) If the obstruction class [c(p)] vanishes...
36 P X ĝ? BG c B n+1 U(1)... then g lifts...
37 P X ĝ BĜ BG c B n+1 U(1)... to the extension Ĝ G classified by c.
38 P? X ĝ BĜ BG c B n+1 U(1) On the total space of P this is, by the pasting law,...
39 P B n U(1) X ĝ BĜ BG c B n+1 U(1)...a circle n-bundle...
40 G P B n U(1) x X ĝ BĜ BG c B n+1 U(1)...whose restriction to any fiber...
41 G P Ωc B n U(1) x X ĝ BĜ BG c B n+1 U(1)...is the looping of c.
42 G P B n U(1) x X ĝ BĜ BG c B n+1 U(1) This classifies...
43 WZW G P B n U(1) x X ĝ BĜ BG c B n+1 U(1) the WZW circle n-bundle / bundle (n 1)-gerbe induced by c...
44 Ĝ G P B n U(1) x X ĝ BĜ BG c B n+1 U(1)... which is Ĝ itself (all by the pasting law).
45 Next: add connections
46 Next: add connections same idea of looping but now with a differential twist
47 III Higher WZW connections
48 cohesive -topos theory: pairs homotopy theory with differential structure
49 cohesive -topos theory: pairs homotopy theory with differential structure Fact: our example H = Sh (SmthMfd) is cohesive.
50 cohesive -topos theory: pairs homotopy theory with differential structure so we have long fiber sequences for smooth higher bundles with connection...
51 BG First, cohesion induces coefficients for flat G-connections.
52 X BG In that morphisms into it are flat G-principal connections on X.
53 X BG BG Canonically equipped with a map to the underlying G-bundles.
54 BG BG The homotopy fiber of this...
55 dr BG BG BG...is the coefficient for flat G-valued differential forms.
56 X A dr BG BG BG In that morphisms into it are flat L -algebra valued forms A Ω flat (X, g).
57 dr BG BG BG The pasting law gives a universal g-valued form...
58 G θ dr BG BG BG... on G itself. The -Maurer-Cartan form θ.
59 G θ dr BG BG BG Proceeding by similar constructions...
60 G θ dr BG BG BG conn...one finds coeffiecients for non-flat G-connections.
61 G θ dr BG BG BG conn The flat connections sit inside.
62 G θ dr BG BG BG conn B n+1 U(1) conn For G = B n U(1), this object classifies circle (n + 1)-connections / n-gerbes with connection.
63 G θ dr BG BG BG conn CS c B n+1 U(1) conn Characteristic maps c : BG B n+1 U(1) may lift to differential characteristics CS c.
64 G θ dr BG BG BG conn CS c B n+1 U(1) conn The homotopy fiber of the total map...
65 G B n U(1) θ conn dr BG BG BG conn CS c B n+1 U(1) conn... is the coefficient for circle n-connections with curvature given by c(θ).
66 G B n U(1) θ conn dr BG BG BG conn CS c B n+1 U(1) conn By universality, θ factors through this...
67 G WZW c B n U(1) θ conn dr BG BG BG conn CS c B n+1 U(1) conn... and this defines the WZW circle n-connection on G induced by CS c.
68 From these maps CS c and WZW c on moduli stacks one obtains
69 From these maps CS c and WZW c on moduli stacks one obtains higher CS functional exp(is CSc ( )) : [Σ n+1, BG conn ] CS c [Σ n+1, B n+1 U(1) conn ] Σ U(1)
70 From these maps CS c and WZW c on moduli stacks one obtains higher CS functional exp(is CSc ( )) : [Σ n+1, BG conn ] CS c [Σ n+1, B n+1 U(1) conn ] G-connections CS Lagrangian Σ volume holonomy U(1)
71 From these maps CS c and WZW c on moduli stacks one obtains higher CS functional: exp(is CSc ( )) : [Σ n+1, BG conn ] CS c [Σ n+1, B n+1 U(1) conn ] higher WZW functional: Σ U(1) exp(is WZWc ( )) : [Σ n, G] WZWc [Σ n, B n U(1) conn ] Σ U(1)
72 From these maps CS c and WZW c on moduli stacks one obtains higher CS functional: exp(is CSc ( )) : [Σ n+1, BG conn ] CS c [Σ n+1, B n+1 U(1) conn ] higher WZW functional: Σ U(1) exp(is WZWc ( )) : [Σ n, G] WZWc [Σ n, B n U(1) conn ] Σ U(1) maps to G WZW Lagrangian surface holonomy
73 IV Example
74 Theorem. Let G be a compact, simply connected Lie group. Then...
75 Theorem. Let G be a compact, simply connected Lie group. Then the canonical topological class c : BG K(Z, 4) has unique smooth lift c : BG B 3 U(1) to smooth moduli -stacks the Chern-Simons 2-gerbe
76 Theorem. Let G be a compact, simply connected Lie group. Then the extension it classifies is the smooth String(G)-2-group Ĝ String(G).
77 Theorem. Let G be a compact, simply connected Lie group. Then BG conn is moduli stack of G-connections; there is a differential refinement CS c : BG conn B 3 U(1) conn.
78 Theorem. Let G be a compact, simply connected Lie group. Then exp(is CSc ) is ordinary CS-functional; exp(is WZWc ) is ordinary WZW functional (topological term).
79 Next: consider the same for higher groups and higher differential classes.
80 Next: consider the same for higher groups and higher differential classes. But not today.
81 view Addendum End.
82 Addendum Pasting law and looping
83 Consider two squares in an -category: A B C D E F
84 Consider a pasting diagram of two squares in an -category: A B C D E F Pasting law A): If both squares are homotopy pullbacks, then so is the total rectangle.
85 Appl.: long fiber sequence Define loop space objects ΩA of pointed objects A: ΩA. A
86 Appl.: long fiber sequence for any f : A B we get ΩA Ωf ΩB hofib(f ) A f B
87 Back to first occurence of pasting.
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