A homotopy theorist looks at DFT

Transcription

1 Jim Stasheff and Andreas Deser 2 May 2014 Workshop on Non-associativty in Physics and Related Mathematical Structures Penn State U

2 literature But I ve gotta use words when I talk to you. T.S.Eliot a foreign tongue that was not sung at my cradle. Hermann Weyl Double, double toil and trouble. Shakespeare

3 the letter T Brought to you by the letter T T-duality T is for Torus T is for Target T is for Transformation T is for Theory

4 what is... What is Double Field Theory all about?

5 what is... Quoting from Double Field Theory: A Pedagogical Review by Aldazabal, Marqués and Núñez: what isdouble Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. Historically, T referred to Torus, but the full force of the approach is for T as the Target of a σ-model, as in string theory. That is to say, before doubling, the configuration space is of the form Maps(Σ, T ) or sections Γ(Σ, E) then the fiber T E M is to be doubled. Note I am looking at a doubled space topologically, in fact, cohomologically, prior to any geometry.

6 what is... Naively, doubling refers to doubling coordinates, meaning locally. Quoting from the work of several researchers here present: They review double field theory (DFT)... and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate.

7 problem 1 A major problem is to sort out a similarly mathematical global setting for Double Field Theory. Hohm and Zwiebach remarked in 2012 that It is not yet known how to construct a non-trivial patching of local regions of the doubled manifold leading to more general non-geometric configurations.

8 Generalized geometry The basic idea of the generalized geometry of Hitchin and Gualtieri is to combine vectors and one-forms into a single object. Formally, on a manifold M, one introduces a generalized tangent bundle E which is a particular extension of T by T 0 T M E π TM 0. (1) Locally over M, the bundle E looks like TM T M. Sections of E are called generalized vectors. Locally they can be written as X = x + ξ where x TM and ξ T M. In going from one coordinate patch U α to another U β, we first make the usual transition for vectors and one-forms, and then give a further twist describing how the algebra structure is fibered over TM.

9 Transition functions This gives [ ] x α + ξ α = a αβ x β + a T αβ ξ β i aαβ x β ω αβ, (2) where a αβ GL(d, R) while ω αβ is a two-form and a T = (a 1 ) T. If we then go from U α to another U γ, this gives [ ] x α + ξ α = a αγ x γ + aαγ T ξ γ i aαγxγ ω αγ, (3) whereas, if we compose the transitions going from U α to U β and then to U γ we arrive at: [ ] x α + ξ α = a αβ a βγ x γ + a T αβ [a T βγ ξ γ i aβγ) x γ) ω βγ ] i aαβ a βγ ω αγ. (4)

10 Cocycle condition Assuming a αβ is a 1-cocycle with values in GL(d, R), we are still left with i aαγx γ ω αγ versus i aβγ x γ ω βγ + i aαβ a βγ ω αγ. This is often described in terms of a failure of associativity; a more accurate description might be failure of the cocycle condition: g αβ g βγ = g αγ on U α U β U γ. or, yet again, as a failure to be a representation:

11 Cocycle condition If {U α } is an open covering, the disjoint union U α can be given a rather innocuous structure of a topological category U, i.e., Ob U = U α and Mor U = U α U β that is x y = x = y is defined iff x U α, y U β and x = y. Regarding the range of the transition functions as in a category with one object in the standard way, the one cocycle condition says that the transition functions define a functor/representation.

12 Up to homotopy Since these are 1-forms, it could be that the difference is an exact form dλ αβγ for some function λ αβγ on U α U β U γ. One could say the transition functions form a representation up to homotopy (RUTH). Alternatively, we have the data of a gerbe, but I would like to do better. In terms of spaces, one does not patch together such local data since the cocycle condition is not satisfied, but let us go instead to differential graded spaces.

13 dg manifold Roytenberg, among others, defines a dg manifold as a locally ringed space (M, O M ) (in dg commutative algebras over R), which is locally isomorphic to (U, O U ), where O U = C (U) S(V ) for {U} an open cover of M and V a dg vector space. Again the transition functions g αβ : U α U β G with respect to an open cover {U α } of M satisfy the cocycle condition.

14 dg and sh-manifolds Notice that manifold is irrelevant, a hold-over from the early days; at most a topological space with an open cover is needed. The homotopy generalization of the definition of a dg-manifold is straightforward, but not so the cocycle condition. Definition A generalized dg manifold or sh-manifold is a locally ringed space (M, O M ) (in dg commutative algebras over R), which is locally homotopy equivalent (as dcga s) to (U, O U ), where O U = C (U) S(V ) for {{U}} is an open cover of M and (S(V ), d) is a dcga.

15 Homotopy cocycle For sh-manifolds, the cocycle condition is more subtle. Given the local homotopy triviality O U = C (U) S(V ) and F := S(V ) with the implied differential d = d F, there are defined transition functions g αβ : U α U β H = H(F) where H is the dg monoid of homotopy equivalences of F to F. However, instead of the cocycle condition, one obtains only that g αβ g βγ is homotopic to g αγ as a map of U α U β U γ into H. Moreover, on multiple intersections, higher homotopies arise.

16 Homotopy cocycle In the situation we are considering where the first homotopy is given by λ αβγ which is just a function, there is no room for a higher homotopy; the combination (g a b, λ αβγ ) satisfies the respective cocycle condition. However, some care is necessary to construct the global object from the homotopy cocycle data. In the case of fibrations, Wirth s patching/glueing/ recollement (mapping cylinder) theorem, which is of fundamental importance in more general fibration theories, is essential.

17 Doubling - beyond generalized geometry There is an extensive literature using generalized geometry to study T-duality. However, this does not give the right number of local coordinates as expressed in Double Field Theory. When I first came to Double Field Theory, I found the fields depend on local coordinates X M = ( x µ, ỹ m, x µ, y m ) of some unspecified space. I thought it would be good to know how to interpret these coordinates globally, e.g is there a bundle or at least a fibration here?

18 Doubling - beyond generalized geometry In his thesis but independent of DFT, Roytenberg suggests what I think is an appropriate setting, very much consistent with the motivation for doubling. In fact, ironically, his motivation was the study of doubles in a different sense. There is an elegant way to express the structure relations of a Lie bialgebra by embedding it into a larger space endowed with a canonical Poisson superalgebra structure. By viewing this construction from an appropriate angle we shall be able to generalize it to Lie bialgebroids, obtain a new notion of the Drinfeld double and recover the old one. Lie bialgebroids by Mackenzie and Xu.

19 Lie bialgebroid First, the tangent bundle TM M is the prototype of a Lie algebroid A M, which I will assume we all (almost all of us?) know. The Lie bracket on the sections Γ(A) and its action on C (M) extends to the Schouten bracket [, ] A on all of Γ(Λ A. In particular, for X Γ(Λ p+1 A), there is the derivation of degree p given by [X, ] A. In fact, the Schouten bracket gives us a Gerstenhaber algebra. Dually, we have a derivation d A on the graded commutative algebra Γ(Λ A ) given by the Cartan-Chevalley-Eilenberg formula. The prototype example of a Lie bialgebroid is the pair (TM, T M) with familiar structures.

20 Lie bialgebroid Definition A Lie bialgebroid consist of a pair (A, A ) of Lie algebroids in duality as vector bundles such that the induced differential d A is a derivation of the Schouten bracket [, ] A on Γ( A ).

21 superization For a Lie bialgebroid (A, A ), Roytenberg s idea is to embed Γ(Λ(A A )) into a Poisson superalgebra. Compare Hull s description: T -duality acts by changing from one subspace of the doubled space Ẽ to another, not all of which may be bundles. The superization refers to the operator Π on a vector bundle that changes the parity of the fibre coordinates. If they are ungraded to begin with, Π will assign them degree (or weight) ±1. The exterior algebra Γ(ΛA ) of differential forms can be seen as the algebra of functions on the supermanifold ΠA.

22 superization Start with the "undoubled" manifold M with coordinates x i. For its tangent bundle TM, we have the corresponding local vector (fields) i and for T M we have momenta, p i = dx i. Then, using T. Voronov s notation, we have correspondingly u i for linear coordinates on the fibers of TM and u i on the fibers of T M, so that a typical vector is written u i i and a typical 1-form as u i dx i. To superize" TM, Roytenberg applies the functor Π to TM which acts by suspending (reversing the parity) of the fibre coordinates: ξ i := Πu i and similarly θ i := Πu i. These coordinates are generators of respective Grassman algebras.

23 possible identification?? Look at the cotangent bundle T ΠTM of ΠTM. It now has coordinates x i, ξ i, v i, w i where v i is the momentum coordinate corresponding to x i and w i is the momentum coordinate corresponding to ξ i. So perhaps we can identify ξ i with x i and w i with ṽ i.

24 Courant bracket A characteristic feature of both generalized geometry and Double Field Theory is the extension of the Lie bracket of vector fields to smooth sections of TM T M as a Courant bracket. Definition For vector fields X and Y on a manifold M and 1-forms ω and η, the Courant bracket on Γ(TM T M) of X + ω and Y + η is defined to be [X + ω, Y + η] = [X, Y ] + L X η L Y ω 1/2 d(i X η i Y ω) (5) where L X is the Lie derivative along the vector field X, d is the exterior derivative and i is the interior product.

25 The essential features of this bracket lead to the formal definition of a Courant algebroid. Definition A Courant algebroid is a pseudo-euclidean vector bundle (E, <, >) over a manifold M, together with a bilinear operation on Γ(E) and a bundle map a : E TM (the anchor), satisfying certain properties. In its skew-symmetric form, is written as a bracket but does not satisfy the Jacobi identity, but a Courant algebra is a Lie algebra up to homotopy. were originally introduced by Liu, Weinstein and Xu in their investigation of doubles AHA! of Lie bialgebroids.

26 C-brackets in Double Field Theory In Double Field Theory. the usage is to refer to a C-bracket [, ] C which reduces to a Courant bracket under a certain constraint, but in fact [, ] C satisfies the Courant bracket relations if the double setting is that of a Lie bialgebroid.. Evaluating this bracket on X + ω and Y + η displays some terms unusual in generalized geometry: [X + ω, Y + η] C = [X, Y ] + L ω Y L η X 1 2 d(ι ωy ι η X (6) + [ω, η] + L X η L Y ω 1 2 d(ι X η ι Y ω). (7) Because of the doppelgängers hiding in X, ω, Y and η and also in d, ι and L, these unusual terms make sense. Andreas has worked this out in detail.

27 C-brackets in Double Field Theory For comparison, in Roytenberg s thesis we see: Suppose now that both A and A are Lie algebroids over the base manifold M, with anchors a and a respectively. Let E denote their vector bundle direct sum: E = A A. On E, there exist two natural nondegenerate bilinear forms, one symmetric and another antisymmetric: (X 1 + ξ 1, X 2 + ξ 2 ) ± = ( ξ 1, X 2 ± ξ 2, X 1 ). (8) On Γ(E), we introduce a bracket by [e 1, e 2 ] = ([X 1, X 2 ] A + L A ξ 1 X 2 L A ξ 2 X d A (e 1, e 2 ) )+ + ([ξ 1, ξ 2 ] A + L A X 1 ξ 2 L A X 2 ξ d A(e 1, e 2 ) ), (9) where e 1 = X 1 + ξ 1 and e 2 = X 2 + ξ 2.

28 H-flux Bakas lured me into the depths of Double Field Theory by his work with Dieter Lüst on flux, especially H-flux where Bakas recognized homotopy associativity lurking. Here is an attempt to provide a still more formal treatment of 3-cocycles. Let A be the graded commutative algebra R[x 1,, x n ] Λ[dx 1,, dx n ], where, for the time being, dx i is just a name/symbol. For the famous H-flux 3-form model, we have H 3 = H dx 1 dx 2 dx 3. (10) Now introduce an ungraded commutative algebra A = R[x 1,, x n ; p 1,, p n ] and deform this algebra to [x i, p j ] = iδ ij, [p i, p j ] = 0. (11)

29 H-flux Next introduce a superficially dual algebra à = R[y 1 = x 1,, y n = x n ; q 1 = p 1,, q n = p n ] with the analogs of the above relations. For the H-flux 3-form model, add the relation [y 1, y 2 ] = Hq 3. There are also other flux models in which the additional relations are instead [x 1, y 2 ] = fq 3 for the f -model, for the Q-model and [x 1, x 2 ] = Qq 3 [x 1, x 2 ] = Rq 3 for the R-model. Notice this last model does not really need the duals.

30 We have a string of dualities of fluxes H abc f a bc Qab c R abc, numerically equal and generically called F. The raising and lowering can be interpreted without resort to a metric. For example, f a bc a dx b dx c (12) Qc ab a b dx c (13) R abc a b c (14) where the variables x i may denote coordinates x i or dual coordinates y i.

31 These can be interpreted as structure functions of various kinds of algebras if we have a suitable Poisson bracket. For example, f a bc a dx b dx c (15) describes [ b, c ] = f a bc a. (16) From there, we can demonstrate vividly not only the non-commutative, but the non-associative aspects of all H, f, Q, R-deformed algebras. In fact, H can be used to twist the Courant bracket and provide a homotopy for the Jacobiator. Again Roytenberg and Deser are both could sources on this material.

32 afterthoughts Here are some remarks generated by other talks and discussions at the Workshop. Barton mentioned field dependent gauge transformations which reveal the structure of an L -algebra. The first occurrence of this of which I am aware is in the work of Behrends, Burgers and van Dam in physics followed by my work with Barnich, Fulp and Lada just as mathematics. Barton reminded me that the Courant homotopy triple bracket [[,, ]] is determined by the initial bracket [[, ]] and the inner product whereas the L -algebra in Barton s CSFT (closed string field theory) can have terms of arbitrarily high order.

33 afterthoughts Lada reminded me that OCSFT gave rise to OCHAs (open-closed homotopy algebras) in which an L -algebra acts on an A algebra A constraint surface S which is not closed with respect to a nice structure on the full space may acquire a homotopy nice structure by projection from the full space; e.g. is this possibly how a gerbe arises rather than a bundle in generalized geometry or Double Field Theory? Thinking of the doubled space ˆM = M M with coordinates X i, X i, might we consider not an extension of bundles over M, but rather over M M or even over M N, where N is some dual manifold. The trivial extension is then the external product TM T N. Klimčik and Ševera have some interesting ideas on how to construct N dynamically.

34 back to the future Fire burn and cauldron bubble. Shakespeare