C-spaces, Gerbes and Patching DFT

Transcription

1 C-spaces, Gerbes and Patching DFT George Papadopoulos King s College London Bayrischzell Workshop April Germany 23 April 2017 Material based on GP: arxiv: ; arxiv: PS Howe and GP arxiv:

2 Main Question Patching problem in DFT: Expectations and Obstacles Emphasis on global properties Can consistent patching be achieved? If it can, what is the cost?

3 Main Question Patching problem in DFT: Expectations and Obstacles Emphasis on global properties Can consistent patching be achieved? If it can, what is the cost?

4 Main Question Patching problem in DFT: Expectations and Obstacles Emphasis on global properties Can consistent patching be achieved? If it can, what is the cost?

5 Main Question Patching problem in DFT: Expectations and Obstacles Emphasis on global properties Can consistent patching be achieved? If it can, what is the cost?

6 Main Question Patching problem in DFT: Expectations and Obstacles Emphasis on global properties Can consistent patching be achieved? If it can, what is the cost?

7 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

8 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

9 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

10 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

11 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

12 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

13 Criteria Some of the criteria that can be used for consistent patching are The final theory should produce both local and global properties of T-duality as described by the Buscher rules The theory must exhibit a O(d, d) symmetry The patching of theory and its associated space requires for consistency the Dirac quantisation property of the 3-form flux The final theory and its associated space satisfies the topological geometrisation condition The theory and its associated space can be constructed for all backgrounds with 3-form flux Generalised geometry emerges naturally

14 Buscher Rules Amongst the criteria the first one on Buscher rules is perhaps the most conservative. Given a common sector background with an isometry X = θ, the geometry ds 2 = V 2 (dθ + q i dx i ) 2 + g ij dx i dx j, B = (dθ + q i dx i ) p j dx j b ijdx i dx j. transforms under T-duality to d s 2 = V 2 (d θ + p i dx i ) 2 + g ij dx i dx j, B = (d θ + p i dx i ) q j dx j b ijdx i dx j, e 2 Φ = e 2Φ V 2, θ is a new angular coordinate.

15 Buscher Rules Spacetime M has coordinates (θ, x i ) while the T-dual M has coordinates ( θ, x i ). θ is the T-dual coordinates of θ. M and M may have different geometries M and M may have different topology The T-dual coordinate θ may be topologically twisted over M (or vice versa). The transition functions are given by some of those of B-field.

16 Buscher Rules Spacetime M has coordinates (θ, x i ) while the T-dual M has coordinates ( θ, x i ). θ is the T-dual coordinates of θ. M and M may have different geometries M and M may have different topology The T-dual coordinate θ may be topologically twisted over M (or vice versa). The transition functions are given by some of those of B-field.

17 Buscher Rules Spacetime M has coordinates (θ, x i ) while the T-dual M has coordinates ( θ, x i ). θ is the T-dual coordinates of θ. M and M may have different geometries M and M may have different topology The T-dual coordinate θ may be topologically twisted over M (or vice versa). The transition functions are given by some of those of B-field.

18 Buscher Rules Spacetime M has coordinates (θ, x i ) while the T-dual M has coordinates ( θ, x i ). θ is the T-dual coordinates of θ. M and M may have different geometries M and M may have different topology The T-dual coordinate θ may be topologically twisted over M (or vice versa). The transition functions are given by some of those of B-field.

19 KK paradigm Let us review the construction of Kaluza-Klein (KK) space. Consider a closed 2-form F 2, df 2 = 0, and a good cover {U α } α I on spacetime M. Then consider the Čech-de Rham decomposition of F 2. F 2 = da 1 α, U α A 1 α + A 1 β = da 0 αβ, U αβ = U α U β a 0 βγ a0 αγ + a 0 αβ = 2πn αβγ, U αβγ = U α U β U γ If all n αβγ Z, then 1 2π [F2 ] H 2 (M, Z)

20 KK space The construction of KK space ˆM can be done by introducing an angular coordinate θ α at every open set U α and imposing the patching conditions θ α + θ β = a 0 αβ mod 2πZ Consistency at U αβγ requires that a 0 βγ a0 αγ + a 0 αβ = 0 mod 2πZ which is satisfied iff n αβγ Z and so 1 2π [F2 ] H 2 (M, Z). ˆM is a circle bundle over M with curvature F 2 and c 1 ( ˆM) = 1 2π [F2 ].

21 KK space Some properties are The construction of KK space requires the Dirac quantisation condition as 1 2π [F2 ] H 2 (M, Z) KK space satisfies the topological geometrisation condition θ α + θ β = a 0 αβ mod 2πZ = dθ α A 1 α = dθ β A 1 β and so dθ A 1 is globally defined on ˆM. Moreover and so F 2 is exact on ˆM. F 2 = d(dθ A 1 )

22 Čech-de Rham The Čech-de Rham decomposition of a NS-NS closed 3-form flux H 3, dh 3 = 0, is H 3 α = db 2 α, U α B 2 α + B 2 β (δb2 ) αβ = da 1 αβ, U αβ a 1 βγ a1 αγ + a 1 αβ (δa1 ) αβγ = da 0 αβγ, U αβγ a 0 βγδ a0 αγδ + a0 αβδ a0 αβγ (δa0 ) αβγδ = 2πn αβγδ, U αβγδ δ is the Čech cohomology differential, δ 2 = 0. Again if n αβγδ Z, then 1 2π [H3 ] H 3 (M, Z).

23 Čech-de Rham B 2 α, a 1 αβ and a0 αβγ are not unique but subject to the gauge transformations B 2 α B 2 α + du 1 α, a 1 αβ a1 αβ + δu1 αβ + du0 αβ a 0 αβγ a0 αβγ + δu0 αβγ + c αβγ where c are constants. If δa 1 = 0, then H is exact If δa 1 = 0, then B 2 α = B 2 α + d( γ ρ γ a 1 αγ) is a globally defined 2-form B 2 α + B 2 β = da1 αβ d( γ ρ γ (a 1 αγ a 1 βγ )) = da1 βγ d( γ ρ γ a 1 αβ ) = 0 But H 3 = db 2 = d B 2 and so it is exact. {ρ α } α I partition of unity.

24 DFT DFT is described on doubled space where the spacetime coordinates x m are doubled to X M = (x m, x n ) where x n are thought to describe the dual coordinates. The theory is formulated after imposing the strong section condition on the fields and infinitesimal symmetry parameters A B x m + B A x m x m = 0, x m 2 A x m x m = 0 which effectively restricts the fields and parameters to depend on either x m or x n. It is thought that different choices of splitting X M as above correspond to identifying different dual pairs. Question: how do the coordinates (x m, x n ) patch?

25 DFT transformations There are 2 main approached to these which depend on whether the transformations of the dual coordinates x n are related to the gauge transformations of the B field. In particular depend In the approach of [Hohm, Zwiebach; Hohm, Lüst, Zwiebach] the coordinates of the doubled space transform as x i = x i (x j ), x i = x i v i (x), under finite transformations, and the induced transformation on the B field is B ij(x ) = xk x i x l ( v l x k v k x l ) ) (B x j kl (x) ( x k v j x i x k xk v ) i x j x k.

26 DFT transformations In the proposal of [Rey, Sakatani] a closed 2-form b is introduced, db = 0 which transforms as b ij(x ) = xk x i x l x j ((b kl + k v l l v k )(x)), while B := B b is taken to transform tensorially: B ij(x ) = xk x i x l x j B kl(x) The B-field transforms as b. Doubled space coordinate transformations are x i = x i (x j ), x i = x i + v i (x)

27 In the proposal of [Hull], the doubled space coordinates transform as x i = x i (x j ), x i = x i The B-field transforms as B ij(x ) = xk x i x l x j ( B kl (x) + ( k v l l v k ) (x) ) the spacetime coordinates transform with local diffeomorphisms while the dual coordinates remain inert the v-transformation is considered as a gauge transformation instead of a coordinate one There are other proposals which either apply to special cases [Blumenhagen, Hassler, Lüst; Hassler] or are related to those three above [Cederwall].

28 In the proposal of [Hull], the doubled space coordinates transform as x i = x i (x j ), x i = x i The B-field transforms as B ij(x ) = xk x i x l x j ( B kl (x) + ( k v l l v k ) (x) ) the spacetime coordinates transform with local diffeomorphisms while the dual coordinates remain inert the v-transformation is considered as a gauge transformation instead of a coordinate one There are other proposals which either apply to special cases [Blumenhagen, Hassler, Lüst; Hassler] or are related to those three above [Cederwall].

29 B-dependent patching Suppose that the coordinates x are patched as x α + x β = v αβ, a 1 αβ = α(v αβ) α some linear map. Then δv αβγ = 0 = δa 1 αβγ = 0 a 1 αβ is a Čech cocycle. H is exact! Also if B = B + b with db = 0 and B to transform as a tensor H = db = db + db = db is exact.

30 B-dependent patching Suppose that the coordinates x are patched as x α + x β = v αβ, a 1 αβ = α(v αβ) α some linear map. Then δv αβγ = 0 = δa 1 αβγ = 0 a 1 αβ is a Čech cocycle. H is exact! Also if B = B + b with db = 0 and B to transform as a tensor H = db = db + db = db is exact.

31 B-independent patching To investigate the third proposal, consider the T-dual pair S 3 with N-units of H-charge and LN 3 with 1-unit of H charge. Arises after using the Buscher rules with isometry along the Hopf fibre of S 3. The Hopf fibre of S 3 twists over the Lens space LN 3 P S 1 S 1 LN 3 S 3 S S 2 The S 1 fibre will twist over L 3 N iff the circle bundle P L3 N is topologically non-trivial

32 To investigate this note that the cohomology of S 3 and L 3 N are H 0 (S 3, Z) = H 3 (S 3, Z) = Z, H 1 (S 3, Z) = H 2 (S 3, Z) = 0, H 0 (L 3 N, Z) = H 3 (L 3 N, Z) = Z, H 1 (L 3 N, Z) = 0, H 2 (L 3 N, Z) = Z N. As H 2 (S 3, Z) = 0, P is a topologically trivial bundle over S 3 and so P = S 3 S 1. In particular H 2 (P, Z) = 0 Suppose now that P = S 1 LN 3, the Künneth formula for computing the cohomology of a topological product would have implied that H 2 (P, Z) = H 2 (L 3 N, H 0 (S 1, Z)) = H 2 (L 3 N, Z) = Z N This is a contradiction as H 2 (P, Z) = 0.

33 To investigate this note that the cohomology of S 3 and L 3 N are H 0 (S 3, Z) = H 3 (S 3, Z) = Z, H 1 (S 3, Z) = H 2 (S 3, Z) = 0, H 0 (L 3 N, Z) = H 3 (L 3 N, Z) = Z, H 1 (L 3 N, Z) = 0, H 2 (L 3 N, Z) = Z N. As H 2 (S 3, Z) = 0, P is a topologically trivial bundle over S 3 and so P = S 3 S 1. In particular H 2 (P, Z) = 0 Suppose now that P = S 1 LN 3, the Künneth formula for computing the cohomology of a topological product would have implied that H 2 (P, Z) = H 2 (L 3 N, H 0 (S 1, Z)) = H 2 (L 3 N, Z) = Z N This is a contradiction as H 2 (P, Z) = 0.

34 The Buscher T-dual angular coordinates may topologically twist over the spacetime If the DFT dual coordinates do not patch with B-field transformations, the theory does not reproduce the global aspects of Buscher T-duality rules In particular the DFT dual coordinates cannot be identified with the T-dual Buscher angular coordinates

35 The Buscher T-dual angular coordinates may topologically twist over the spacetime If the DFT dual coordinates do not patch with B-field transformations, the theory does not reproduce the global aspects of Buscher T-duality rules In particular the DFT dual coordinates cannot be identified with the T-dual Buscher angular coordinates

36 The Buscher T-dual angular coordinates may topologically twist over the spacetime If the DFT dual coordinates do not patch with B-field transformations, the theory does not reproduce the global aspects of Buscher T-duality rules In particular the DFT dual coordinates cannot be identified with the T-dual Buscher angular coordinates

37 C-spaces Let H be a closed 3-form with transition function a 1 αβ, a0 αβγ and n αβγδ. Introduce coordinates y 1 α and angular coordinates θ αβ and impose the patching conditions (δy 1 ) αβ + dθ αβ = a 1 αβ, U αβ (δθ) αβγ = a 0 αβγ mod2πz, U αβγ The compatibility of the first condition of triple overlaps is implied by the second The compatibility of the second on 4-fold overlaps requires that n αβγδ Z and so 1 2π [H] H3 (M, Z) C-spaces are independent from the choice of representative for 1 2π [H]

38 C-spaces and topological geometrization condition The construction of C-spaces requires the Dirac quantisation condition! The C-spaces satisfy the topological geometrisation condition. From the first equation dy 1 α B α = dy 1 β B β and H = d(dy 1 B) ie it is exact! Generalised geometry emerges from C-spaces

39 C-spaces and generalized geometry The patching condition for y 1 can be solved to yield y 1 α = ỹ 1 α + γ ρ γ (dθ αγ a 1 αγ), where ỹ 1 α = ỹ 1 β transforms like a 1-form. This gives dy 1 α = (dy 1 α) i dxα. i Consider the bundle E with sections X α = Yα i x + (w α i α ) i (y where 1 α )i dy 1 αi, (y = 1 δi j. The patching conditions yield α )j Yα i = xi αβ x j Y j β, β (w α) i = xj β ( xαβ i (wβ ) j (B β ) jk Yβ) k Thus as in generalized geometry. 0 T M E TM 0 ỹ 1 can be identified with the doubled space coordinates x according to the third proposal.

40 Gerbes Consider an open cover {U α } of M which is not necessarily a good cover. A gerbe [Hitchin-Chatterjee] is an assignment of a circle bundle P αβ at each U αβ with P βα = P 1 αβ on triple overlaps U αβγ the circle bundle P αβ P βγ P γα admits a section g αβγ On 4-fold overlaps U αβγδ, it satisfies the condition g βγδ g 1 αγδ g αβδg 1 αβγ = 1 To make connection with C-spaces θ αβ are the fibre coordinates of the bundles and g αβγ = exp ia 0 αβγ Gerbes have a notion of equivalence under refinement. This allows to define a gerbe at any refinement of the original open cover Any open cover admits a good refinement

41 T-duality Consider S 3 with N units of H flux [Murray]. Cover S 3 with the stereographic cover S 3 = U 0 U 1. Then U 01 = U 0 U 1 is S 2 I where I is an open interval. The circle bundles over U 01 are classified by H 2 (U 01, Z) = H 2 (S 2, Z) = Z Consider the circle bundle P 01 with first Chern class N represented by the 2-form F 01. A representative of the 3-form flux Ĥ can be given using the Mayer-Vietoris construction as Ĥ 0 = dρ 1 F 01, on U 0 ; Ĥ 1 = dρ 0 F 01, on U 1 {ρ 0, ρ 1 } is a partition of unity subordinate to the cover {U 0, U 1 }. Ĥ is globally defined on S 3 as on U 01 Ĥ 0 + Ĥ 1 = d(ρ 1 + ρ 0 ) F 01 = d1 F 01 = 0.

42 T-dual coordinates Furthermore Stoke s theorem reveals that [H] = [Ĥ]. The rest of the compatibility conditions for the gerbe are trivially satisfied. The bundle space of P 01 restricted on S 2 is the lens space L 3 N which in turn is Buscher dual to S 3 with N units of H flux The Buscher dual angular coordinate θ is identified with the θ 01 coordinate of the gerbe in the C-space! The gebre space is the union of S 3 L 3 N.

43 General case Take M to be a circle bundle over Q, π : M Q, with 3-form flux H such that [H] = aw, where w H 2 (Q, Z) and H 1 (S 1, Z) = Z a. Consider a trivialization of M π 1 (W α ) = ϕ 1 α (W α S 1 ) and the cover of M with the two open sets U 0 = α ϕ 1 α (W α V 0 ), where S 1 = V 0 V 1 U 1 = α ϕ 1 α (W α V 1 ) Consider a representative F of the class w H 2 (Q) and its pull back to M with the projection map π. Take F 01 = F U01 and construct a representative Ĥ of the H flux as before The Buscher T-dual space M is the circle bundle over Q with first Chern class w. The T-dual angular coordinate θ is identified with the θ 01 angular coordinate of the gerbe in C-space.

44 General case The gerbe construction gives the same T-dual space under a variety of choice of open covers. This can be turned into a statement of covariance for the Buscher T-duality rules The gerbe construction does not need the isometries of the Buscher construction. So potentially T-dual spaces can be identified for manifolds without isometries Although the gerbe construction can always be done, it is not always obvious that it will lead to an identification of a T-dual space

45 Conclusion C-spaces proposal has all the characteristics required for the definition of manifestly T-duality covariant theory: Dirac quantisation of the H-flux, topological geometrisation condition, generalized geometry and O(d, d) covariance, and incorporates T-duality via the identification of T-dual angular coordinates with the gerbe circle fibres The gerbes open a window to understanding T-duality beyond the isometry set up of Buscher rules which may be useful in the context of mirror symmetry. It requires more coordinates than double of those of spacetime and the underlying space may not be a manifold.

46 Conclusion C-spaces proposal has all the characteristics required for the definition of manifestly T-duality covariant theory: Dirac quantisation of the H-flux, topological geometrisation condition, generalized geometry and O(d, d) covariance, and incorporates T-duality via the identification of T-dual angular coordinates with the gerbe circle fibres The gerbes open a window to understanding T-duality beyond the isometry set up of Buscher rules which may be useful in the context of mirror symmetry. It requires more coordinates than double of those of spacetime and the underlying space may not be a manifold.

47 Conclusion C-spaces proposal has all the characteristics required for the definition of manifestly T-duality covariant theory: Dirac quantisation of the H-flux, topological geometrisation condition, generalized geometry and O(d, d) covariance, and incorporates T-duality via the identification of T-dual angular coordinates with the gerbe circle fibres The gerbes open a window to understanding T-duality beyond the isometry set up of Buscher rules which may be useful in the context of mirror symmetry. It requires more coordinates than double of those of spacetime and the underlying space may not be a manifold.

48 Conclusion C-spaces proposal has all the characteristics required for the definition of manifestly T-duality covariant theory: Dirac quantisation of the H-flux, topological geometrisation condition, generalized geometry and O(d, d) covariance, and incorporates T-duality via the identification of T-dual angular coordinates with the gerbe circle fibres The gerbes open a window to understanding T-duality beyond the isometry set up of Buscher rules which may be useful in the context of mirror symmetry. It requires more coordinates than double of those of spacetime and the underlying space may not be a manifold.

49 What do all these mean for DFT? One option is to view DFT in terms only of generalized geometry. This will allow for a consistent O(d, d) formulation on the spacetime. However the O(d, d) covariance cannot be identified with T-duality. In such a formulation the doubled space is not essential. But adding new coordinates may lead to interesting algebraic structures The solutions of this theory will produce all the T-dual pairs, as the standard Einstein formulation, but smooth O(d, d) transformations will not relate different T-dual pairs If the incorporation Buscher T-duality rules is not negotiable, then an alternative way of formulating the theory must be found in which the θ αβ coordinates have an essential role

50 What do all these mean for DFT? One option is to view DFT in terms only of generalized geometry. This will allow for a consistent O(d, d) formulation on the spacetime. However the O(d, d) covariance cannot be identified with T-duality. In such a formulation the doubled space is not essential. But adding new coordinates may lead to interesting algebraic structures The solutions of this theory will produce all the T-dual pairs, as the standard Einstein formulation, but smooth O(d, d) transformations will not relate different T-dual pairs If the incorporation Buscher T-duality rules is not negotiable, then an alternative way of formulating the theory must be found in which the θ αβ coordinates have an essential role

51 What do all these mean for DFT? One option is to view DFT in terms only of generalized geometry. This will allow for a consistent O(d, d) formulation on the spacetime. However the O(d, d) covariance cannot be identified with T-duality. In such a formulation the doubled space is not essential. But adding new coordinates may lead to interesting algebraic structures The solutions of this theory will produce all the T-dual pairs, as the standard Einstein formulation, but smooth O(d, d) transformations will not relate different T-dual pairs If the incorporation Buscher T-duality rules is not negotiable, then an alternative way of formulating the theory must be found in which the θ αβ coordinates have an essential role

52 What do all these mean for DFT? One option is to view DFT in terms only of generalized geometry. This will allow for a consistent O(d, d) formulation on the spacetime. However the O(d, d) covariance cannot be identified with T-duality. In such a formulation the doubled space is not essential. But adding new coordinates may lead to interesting algebraic structures The solutions of this theory will produce all the T-dual pairs, as the standard Einstein formulation, but smooth O(d, d) transformations will not relate different T-dual pairs If the incorporation Buscher T-duality rules is not negotiable, then an alternative way of formulating the theory must be found in which the θ αβ coordinates have an essential role

53 What do all these mean for DFT? One option is to view DFT in terms only of generalized geometry. This will allow for a consistent O(d, d) formulation on the spacetime. However the O(d, d) covariance cannot be identified with T-duality. In such a formulation the doubled space is not essential. But adding new coordinates may lead to interesting algebraic structures The solutions of this theory will produce all the T-dual pairs, as the standard Einstein formulation, but smooth O(d, d) transformations will not relate different T-dual pairs If the incorporation Buscher T-duality rules is not negotiable, then an alternative way of formulating the theory must be found in which the θ αβ coordinates have an essential role