From Loop Groups To 2-Groups

Transcription

1 From Loop Groups To 2-Groups John C. Baez Joint work with: Aaron Lauda Alissa Crans Danny Stevenson & Urs Schreiber 1 G f 1 f D 2 More details at: 1

2 Hiher Gaue Theory Ordinary aue theory describes how point particles transform as we move them alon 1-dimensional paths. It is natural to assin a roup element to each path: since composition of paths then corresponds to multiplication: while reversin the direction of a path corresponds to takin inverses: 1 and the associative law makes this composite unambiuous: In short: the topoloy dictates the alebra! 2

3 Hiher aue theory describes the parallel transport not only of point particles, but also 1-dimensional strins. For this we must cateorify the notion of a roup! A 2-roup has objects: and also morphisms: We can multiply objects: f multiply morphisms: 1 f f 2 2 and also compose morphisms: Various laws should hold... f In fact, we can make this precise and cateorify the whole theory of Lie roups, Lie alebras, bundles, connections and curvature. But for now, let s just look at 2-roups and Lie 2-alebras. f 3

4 2-Groups A roup is a monoid where every element has an inverse. Let s cateorify this! A 2-roup is a monoidal cateory where every object x has a weak inverse : x y = y x = I and every morphism f has an inverse: f = f = 1. A homomorphism between 2-roups is a monoidal functors. A 2-homomorphism is a monoidal natural transformation. The 2-roups X and X are equivalent if there are homomorphisms f : X X f : X X that are inverses up to 2-isomorphism: f f = 1, ff = 1. Theorem. 2-roups are classified up to equivalence by quadruples consistin of: a roup G, an abelian roup H, an action α of G as automorphisms of H, an element [a] H 3 (G, H). 4

5 Lie 2-Alebras To cateorify the concept of Lie alebra we must first treat the concept of vector space : A 2-vector space L is a cateory for which the set of objects and the set of morphisms are vector spaces, and all the cateory operations are linear. We can also define linear functors between 2-vector spaces, and linear natural transformations between these, in the obvious way. Theorem. The 2-cateory of 2-vector spaces, linear functors and linear natural transformations is equivalent to the 2-cateory of: 2-term chain complexes C 1 chain maps between these, d C 0, chain homotopies between these. The mor- The objects of the 2-vector space form C 0. phisms f : 0 x form C 1, with df = x. 5

6 A Lie 2-alebra consists of: a 2-vector space L equipped with: a functor called the bracket: [, ]: L L L, bilinear and skew-symmetric as a function of objects and morphisms, a natural isomorphism called the Jacobiator: J x,y,z : [[x, y], z] [x, [y, z]] + [[x, z], y], trilinear and antisymmetric as a function of the objects x, y, z, such that: the Jacobiator identity holds: the followin diaram commutes: [[[w,x],y],z] [J w,x,y,z] J [w,x],y,z [[[w,y],x],z]+[[w,[x,y]],z] [[[w,x],z],y]+[[w,x],[y,z]] J [w,y],x,z +J w,[x,y],z [J w,x,z,y]+1 [[[w,y],z],x]+[[w,y],[x,z]] [[w,[x,z]],y] +[w,[[x,y],z]]+[[w,z],[x,y]] +[[w,x],[y,z]]+[[[w,z],x],y] [J w,y,z,x]+1 J w,[x,z],y +J [w,z],x,y +J w,x,[y,z] [[[w,z],y],x]+[[w,[y,z]],x] [[[w,z],y],x]+[[w,z],[x,y]]+[[w,y],[x,z]] +[[w,y],[x,z]]+[w,[[x,y],z]]+[[w,z],[x,y]] +[w,[[x,z],y]]+[[w,[y,z]],x]+[w,[x,[y,z]]] [w,j x,y,z ]+1 6

7 We can also define homomorphisms between Lie 2-alebras, and 2-homomorphisms between these. The Lie 2-alebras L and L are equivalent if there are homomorphisms f : L L f : L L that are inverses up to 2-isomorphism. Theorem. Lie 2-alebras are classified up to equivalence by quadruples consistin of: a Lie alebra, an abelian Lie alebra (= vector space) h, a representation ρ of on h, an element [j] H 3 (, h). Just like the classification of 2-roups, but with Lie alebra cohomoloy replacin roup cohomoloy! Let s use this classification to find some interestin Lie 2-alebras. Then let s try to find the correspondin Lie 2-roups. A Lie 2-roup is a 2-roup where everythin in siht is smooth. 7

8 The Lie 2-Alebra k Suppose is a finite-dimensional simple Lie alebra over R. To et a Lie 2-alebra havin as objects we need: a vector space h, a representation ρ of on h, an element [j] H 3 (, h). Assume without loss of enerality that ρ is irreducible. To et Lie 2-alebras with nontrivial Jacobiator, we need H 3 (, h) 0. By Whitehead s lemma, this only happens when h = R is the trivial representation. Then we have H 3 (, R) = R with a nontrivial 3-cocycle iven by: ν(x, y, z) = [x, y], z. Usin k times this to define the Jacobiator, we et a Lie 2-alebra we call k. In short: every simple Lie alebra admits a 1-parameter deformation k in the world of Lie 2-alebras! Do these Lie 2-alebras k come from Lie 2-roups? We should use the relation between Lie roup cohomoloy and Lie alebra cohomoloy. How is H 3 (G, U(1)) related to H 3 (, R)? 8

9 Suppose G is a simply-connected compact simple Lie roup whose Lie alebra is. We have H 3 (G, U(1)) Z R = H 3 (, R) So, for k Z we et a 2-roup G k with G as objects and U(1) as the automorphisms of any object, with nontrivial associator when k 0. Can G k be made into a Lie 2-roup? Here s the bad news: Theorem. Unless k = 0, there is no way to ive the 2-roup G k the structure of a Lie 2-roup for which the roup G of objects and the roup U(1) of endomorphisms of any object are iven their usual topoloy. However, all is not lost. k is equivalent to a Lie 2- alebra that does come from a Lie 2-roup! However, this Lie 2-alebra is infinite-dimensional! This is where loop roups enter the ame... 9

10 Theorem. For any k Z, there is a Fréchet Lie 2-roup P k G whose Lie 2-alebra P k is equivalent to k. An object of P k G is a smooth path f : [0, 2π] G startin at the identity. A morphism from f 1 to f 2 is an equivalence class of pairs (D, α) consistin of a disk D oin from f 1 to f 2 toether with α U(1): 1 G f 1 D f 2 For any two such pairs (D 1, α 1 ) and (D 2, α 2 ) there is a 3-ball B whose boundary is D 1 D 2, and the pairs are equivalent when ( ) exp 2πik ν = α 2 /α 1 B where ν is the left-invariant closed 3-form on G with ν(x, y, z) = [x, y], z and, is the smallest invariant inner product on such that ν ives an interal cohomoloy class. There s an obvious way to compose morphisms in P k G, and the resultin cateory inherits a Lie 2-roup structure from the Lie roup structure of G. 10

11 The Role of Loop Groups We can also describe the 2-roup P k G as follows: An object of P k G is a smooth path in G startin at the identity. Given objects f 1, f 2 P k G, a morphism l: f 1 f 2 is an element l Ω k G with p( l) = f 2 /f 1 where Ω k G is the level-k Kac Moody central extension of the loop roup ΩG: 1 U(1) Ω k G p ΩG 1 Note: p( l) is a loop in G. We can et such a loop with from a disk D like this: p( l) = f 2 /f 1 1 G f 1 D f 2 An element l Ω k G is an equivalence class of pairs [D, α] consistin of such a disk D toether with α U(1). 11

12 An Application to Topoloy For any simply-connected compact simple Lie roup G there is a topoloical roup Ĝ obtained by killin the third homotopy roup of G. When G = Spin(n), Ĝ is called Strin(n). Theorem. For any k Z, the eometric realization of the nerve of P k G is a topoloical roup P k G. We have π 3 ( P k G ) = Z/kZ When k = ±1, P k G Ĝ. This, and the appearance of the Kac Moody central extension of ΩG, suest that P k G will be an especially interestin Lie 2-roup for applications of hiher aue theory to strin theory. 12