1/2-maximal consistent truncations of EFT and the K3 / Heterotic duality

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1 1/2-maximal consistent truncations of EFT and the K3 / Heterotic duality Emanuel Malek Arnold Sommerfeld Centre for Theoretical Physics, Ludwig-Maximilian-University Munich. Geometry and Physics, Schloss Ringberg, 24th November 2016 EM arxiv: , arxiv: Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 1 / 27

2 Introduction to EFT E d(d) invariant extension of supergravities. Similar to generalised geometry TM T M, TM Λ 2 T M, but with extra coordinates (in representation of E d(d) ). E d(d) (Z) is toroidal U-duality group. Does it play the same role here? No! E d(d) is linear symmetry group. DoFs of 11-dimensional supergravity organise into E d(d) reps, c.f. generalised geometry. GL(11) GL(7) GL(4) GL(7) SL(5) R + g ij, C ijk, C ijklmn,... M ab (x, Y ) SL(5)/USp(4) g µi, C µij, C µijklm,... A µ ab (x, Y ) In general, two-forms, etc. appear too. 10 extended coordinates Y ab. Why extra coordinates? Extra coordinates when on CY? Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 2 / 27

3 Consistent truncations EFT useful for consistent truncations to maximal gsugra, e.g. Lee, Strickland-Constable, Waldram ; Godazgar 2, Krüger Nicolai, Pilch arxiv: ; Hohm, Samtleben arxiv: ; EM, Samtleben arxiv: Makes sense: E d(d) scalar coset. What about partial SUSY-breaking consistent truncations? e.g. N = 2 in 4-d? Start easy: SL(5) EFT for N = 2 in 7-d with n vector multiplets, e.g. M-theory on K3. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 3 / 27

4 Action plan Understand N = 2 in SL(5) EFT. Express SL(5) EFT in terms of N = 2 variables. Perform N = 2 consistent truncation. Deformation to Heterotic DFT. K3 / Heterotic duality. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 4 / 27

5 N = 2 in 7 dimensions Fermions in SL(5) EFT USp(4) reps: i = 1,..., 4, Ω ij symplectic N = 2 two well-defined spinors θ α,i (symplectic Majorana) N = 2 spinors ( θ α,i ) = Ωij ɛ α βθ β,j, α = 1, 2 label SU(2) R. θ α i θ β j Ω ij = κɛ α β. κ is density of weight 1 5 Generalised tangent bundle has SU(2)-structure. Coimbra, Strickland-Constable, Waldram arxiv: Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 5 / 27

6 Spinor bilinears Form spinor bilinears in SL(5) R + reps: a = 1,..., 5 SU(2) SL(5) R + structure Generalised SU(2)-structure globally well-defined κ 1, A a 5, A a 5, B u,ab 10, u = 1, 2, 3, satisfying B u,ab A b = 0, A a A a = κ2 2, B u,abb v,cd ɛ abcde = 4 2δ uv κa e. Compatibility conditions follow from definition as spinors. These implicitly define a generalised metric! Generalised vector V u ab = 1 κ ɛabcde B u,cd A e, V u ab B v ab = 4 2δ u v κ 2. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 6 / 27

7 Express in terms of N = 2 variables (κ, A a, A a, B u,ab ) instead of generalised metric M. SU(2) S connection ab κ = ab A c = ab A c = ab B u,cd = 0. always exists but may not be torsion-free. Intrinsic torsion corresponds to fluxes. Scalar potential, SUSY variations,... in terms of intrinsic torsion. Truncation: intrinsic torsion embedding tensor. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 7 / 27

8 Intrinsic torsion Torsion is the tensor part of the connection. Recall: EFT has generalised Lie derivative. L Λ V a L Λ V a = 1 2 τ bc,d a V d Λ bc. Torsion: τ ab,c d. Torsion depends on choice of connection. Independent of connection: Intrinsic torsion W int = W /ImT. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 8 / 27

9 Intrinsic torsion Rep theory: W int = 2 (1, 1) 2 (1, 3) (3, 1) (3, 3) 3 (2, 2) (2, 4). Intrinsic torsion: independent of SU(2) connection can find expressions without using connection. Look for tensors involving one derivative, e.g. ( S = κ 1 A a ab A b = κ 1 A a ab A b + 1 ) 2 Aa A b c τ ca,b = κ 1 A a A b τ c ca,b. Singlets Use the three generalised vectors V u ab. W int = 2 (1, 1) 2 (1, 3) (3, 1) (3, 3) 3 (2, 2) (2, 4). T = 1 12κ 2 ɛ uvwv u,ab L V v B w ab, S = κ 1 A a ab A b. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 9 / 27

10 Intrinsic torsion - triplets W int = 2 (1, 1) 2 (1, 3) (3, 1) (3, 3) 3 (2, 2) (2, 4). (1, 3) T u = 2κA a L Vu ( Aa κ 4), S u = 2κ 6 L Vu κ 5. (3, 1) (3, 3) T ab = 1 12κ T u ab = 1 12κ ɛuvw ( δ cd ab 1 2 2κ 2 Bv abv cd v + 4 ) κ 2 A [aa [c d] δ b] L Vu B u cd. ( δ cd ab 1 2 2κ 2 Bv abv cd v + 4 ) κ 2 A [aa [c d] δ b] L Vu B w,ab. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 10 / 27

11 Intrinsic torsion - doublets (2, 2) W int = 2 (1, 1) 2 (1, 3) (3, 1) (3, 3) 3 (2, 2) (2, 4). S a = κ 2 ( ab (A b κ 2) 2A a A b bc A c), T a = 1 12κ 3 ɛuvw B u,ab V v bc L Vw A c, U a = κ 2 B u,ab L V ua b. (2, 4) T u a = κ 2 (δ a b δ u v + ) 2 3κ Bu cb acv v ɛ vwx B w,bc L Vx A c. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 11 / 27

12 Scalar potential Scalar potential = part of EFT action involving only ab derivatives. Rewrite the scalar potential in terms of the intrinsic SU(2) torsion. Intrinsic torsion involves one build 2 terms using (intrinsic torsion) 2. Build metric on the (2, 2) and on (1, 3) (3, 1) M ab = κ 3 ɛ uvw B u acb v bdv w,cd, N [ab],[cd] = 1 κ ɛ abcdea e. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 12 / 27

13 The scalar potential of SL(5) EFT can be written as V ɛ i = 1 2 jk ki ɛ j 1 2 jk jk ɛ i ik jk ɛ j. Coimbra, Strickland-Constable, Waldram, arxiv: Use N = 2 spinors ɛ i = θ i αɛ α and integrate by parts. V = jk θ α i ik θ j α 1 2 jkθ α i jk θ i α ik θ α i jk θ j α. Explicit calculation shows V = 1 2 S T ST 3 64 T ut u T us u 3 64 S us u 2T ab T ab T u abt u ab S as a 1 3 S at a U as a. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 13 / 27

14 Consistent partially SUSY-breaking truncation Want to define consistent truncation to 7-d N = 2 gsugra. truncate on generalised SU(2) manifold. expand A a, A a, B u,ab in finite basis of tensors depending on Y ab. N = 2 theory no doublets of SU(2) S. For (κ, A a, A a, B u,ab ) under SL(5) SU(2) S SU(2) R Truncation basis 5 (1, 1) (2, 2), 10 (1, 3) (3, 1) (2, 2). Finite basis of truncation: (ρ, n a, n a, ω M,ab ) ω M,ab = (ω I,ab, ω A,ab ), ω M,ab n b = 0, M = 1,..., n + 3, 3 sections of (1, 3) and n sections of (3, 1) bundle of SU(2) S SU(2) R. n a n a = 1, ɛ abcde ω M,ab ω N,cd = 4η MN n e, η MN is O(3, n) metric. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 14 / 27

15 Truncation Ansatz Scalar Truncation Ansatz κ(x, Y ) = k(x) ρ(y ), A a (x, Y ) = 1 2 a(x) k(x) ρ(y ) n a (Y ), A a (x, Y ) = a(x) k(x) ρ(y ) n a(y ), B u,ab (x, Y ) = a(x) k(x) b u M (x) ρ(y ) ω M,ab (Y ). Coefficients depend on external 7-d coordinates x µ and give 7-d gsugra scalars. Compatibility condition B u,ab B v,cd ɛ abcde = 4 2κA e b u M b v N η MN = δ uv. Identify b u M related by SU(2) R symmetry. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 15 / 27

16 Scalar coset space Recall: a(x), k(x), b u M (x) with b u M b v N η MN = δ uv, identify by SU(2) R. Scalar degrees of freedom b u M (x) has 3n = 3n degrees of freedom. b u M b u,n = 1 2 a(x): dilaton, k(x): g 7. M scalar = ( η MN H MN) O(3, n) O(3) O(n). O(3, n) O(3) O(n) R+ R +. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 16 / 27

17 Truncation Ansatz for other fields 7 d vector fields A µ ab (x, Y ) A µ M (x) ρ(y ) ω M,ab (Y ), (1) n + 3 vector fields of 1/2-max 7-d gsugra coupled to n vector multiplets. 2-form B µν,a (x, Y ) = B µν (x) ρ 2 (Y ) n a (Y ), (2) 3-form C µνρ a (x, Y ) = C µνρ (x) ρ 3 (Y ) n a (Y ). (3) Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 17 / 27

18 Consistent truncation Conditions for consistent truncation involve generalised Lie derivative. Use generalised vectors: ω ab M = ɛ abcde ω M,cd n e. N = 2 no doublets! Intrinsic torsion: 3 (2, 2) (4, 2) = 0. Doublet conditions for consistent truncation n a L ωm ω N,ab = 0, ab (n b ρ 3) = ρ 3 n a n b bc n c. Closure condition for consistent truncation ω M,ab must form closed set under generalised Lie derivative: L ωm ω N ab = 1 4 ( ) L ωm ω N cd ω cd P ω P ab. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 18 / 27

19 Embedding tensor Embedding tensor L ωm ω N ab ω P,ab has only three irreps: Additionally, define L ω[m ω N ab ω P],ab = f MNP, L ωm ω (N ab ω P),ab n a L ωm n a η NP = f M η NP, L ω(m ω N),ab ω P ab L ωp ρη MN = ξ P η MN. Θ = n a ab n b. These are the O(3, n) embedding tensor and singlet deformation parameter Θ. Section condition closure of algebra of generalised Lie derivative f MNP, f M, ξ M, Θ satisfy quadratic constraints. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 19 / 27

20 Intrinsic torsion and T -tensor Intrinsic torsion components become the T -tensor of 1/2-max gsugra. S k a 2 Θ, T k a ɛ uvw b u M b v N b w P f MNP, T u k a b u M (3ξ M f M ), S u k a b u M ξ M, T ab k ω M,ab (η MN + H MN) (ξ N f N ), ( T u ab k ω M ab η MN + H MN) ɛ uvw b P v b Q w f NPQ,.. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 20 / 27

21 Scalar potential Scalar potential We obtain the potential of the half-maximal gauged supergravity ( V = k2 1 a 12 f MNPf QRS H MQ H NR H PS f MNPf QRS H MQ η NR η PS 1 6 f MNPf QRS η MQ η NR η PS ) + (f M ) 2 + (ξ M ) 2 + Θ Similarly, for kinetic terms, topological term, SUSY variation. All Y -dependence appears only through embedding tensor: f MNP, f M, ξ M, Θ. Constant f MNP, f M, ξ M, Θ consistent truncation. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 21 / 27

22 Heterotic DFT as a deformation What if we allow coefficients to depend on Y ab as well? Heterotic Deformation Ansatz κ(x, Y ) = k(x, Y ) ρ(y ), A a (x, Y ) = 1 2 a(x, Y ) k(x, Y ) ρ(y ) n a (Y ), A a (x, Y ) = a(x, Y ) k(x, Y ) ρ(y ) n a(y ), B u,ab (x, Y ) = a(x, Y ) k(x, Y ) b u M (x, Y ) ρ(y ) ω M,ab (Y ), But no doublet derivatives n a ab k = n a ab a = n a ab b u M = 0. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 22 / 27

23 Heterotic doubled space Impose n a ab k = n a ab a = n a ab b u M = 0. Assume (1, 3) (3, 1) can be expanded in ω M,ab : ab = 1 4 ωm abω M cd cd. Define D M = 1 2 ω M ab ab. [D M, D N ] = 1 2 f MN P D P ω N [ab ab ω M cd] cd. Impose f MN P D P = 0 and section condition on ω M ab. Can write D M = M 3 + n coordinates of Heterotic DFT. f MNP encodes gauge group of Heterotic string Graña, Marques arxiv: f MN P P = 0 is required in Heterotic DFT Siegel hep-th/ ; Kwak, Hohm arxiv: Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 23 / 27

24 Heterotic generalised Lie derivative Heterotic generalised Lie derivative Take V ab = ρ(y ) ω ab M (Y ) V M (x, Y ), Λ ab = ρ(y ) ω ab M (Y ) Λ M (x, Y ) ω M abl Λ V ab = Λ N N V M V N N Λ M + η MN η PQ V P N Λ Q + f M NPV N Λ P. We obtain the Heterotic generalised Lie derivative. Section condition becomes η MN M N = 0. Restricts dependence to 3 coordinates. Action becomes Heterotic O(n, 3) DFT action with 7 external directions. b u M frame-like formulation of DFT Siegel, hep-th/ ; Hohm, Hull, Zwiebach arxiv: ; Kwak, Hohm arxiv: Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 24 / 27

25 M-theory / Heterotic duality Consider consistent truncation on K3 SU(2)-structure. Dependence only on Y i5, i = 1,..., 4. n 5 = n 5 = 1, n i = n i = 0. ω M,ij are 22 harmonic forms. Doublet constraints dω M = 0. All gaugings vanish: ungauged 7-D SUGRA. Duality is change of section. Interpret Y i5 as section: four M-theory coordinates K3 truncation of M-theory. Interpret 3 out of Y ij as section: Deformation to Heterotic DFT with gauge group U(1) 22 T 3 reduction of Heterotic SUGRA. K3 lives in the extended space for the heterotic string! Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 25 / 27

26 Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 26 / 27

27 Conclusions and Outlook Studied consistent 1/2-SUSY truncations of SL(5) EFT. These give 1/2-maximal 7-d SO(3, n) gsugra. Embedding tensor is found from generalised Lie derivative. Full gaugings are obtained, including singlet deformation Θ. SU(2)-structure in extended space gives rise to heterotic DFT: embedding tensor defines gauge group. K3 gauge enhancement at orbifold singularities? Non-geometric SU(2)-backgrounds? Uplift 7-d stable desitter gsugra? Contains Θ 0. Dibitetto, Fernández-Melgarejo, Marqués arxiv: Lower dimensions, less SUSY. Emanuel Malek EFT on K3 & Heterotic DFT Schloss Ringberg 27 / 27