Towards a cubic closed string field theory

Transcription

1 Towards a cubic closed string field theory Harold Erbin Asc, Lmu (Germany) Nfst, Kyoto 18th July 2018 Work in progress with: Subhroneel Chakrabarti (Hri) 1 / 24

2 Outline: 1. Introduction Introduction Hubbard Stratonovich action IBL action Conclusion 2 / 24

3 Motivations String field theory: QFT with infinite number of spacetime fields split string amplitudes into Feynman diagrams motivations: unified framework: off-shell amplitudes, renormalisation classical solutions consistency of string theory path integral non-perturbative effects 3 / 24

4 String field theory definitions CFT Hilbert space H, string field Ψ H (bosonic) classical action S = n=2 g n 2 s n! g s : coupling constant V n (Ψ n ) string vertex V n : H n C 4 / 24

5 String field theory definitions CFT Hilbert space H, string field Ψ H (bosonic) classical action S = n=2 g n 2 s n! V n (Ψ n ) = n=2 g n 2 s n! Ψ l n 1 (Ψ n 1 ) g s : coupling constant string vertex V n : H n C, string product l n : H n H BRST operator: l 1 = Q B 4 / 24

6 String field theory definitions CFT Hilbert space H, string field Ψ H (bosonic) classical action S = n=2 g n 2 s n! V n (Ψ n ) = n=2 g n 2 s n! Ψ l n 1 (Ψ n 1 ) g s : coupling constant string vertex V n : H n C, string product l n : H n H BRST operator: l 1 = Q B eom and gauge invariance (L structure) n=1 g n 1 s n! l n (Ψ n ) = 0, δψ = n=1 g n 1 s n! l n (Λ, Ψ n 1 ) 4 / 24

7 String field Fourier expansion Ψ = r d D k (2π) D ψ r(k) k, r ψr (k) : spacetime fields { k, r } : basis of H k : spacetime momentum (non-compact dim.) r : discrete modes (Lorentz indices, compact dim., etc.) classical closed string N gh (Ψ) = 2, L 0 Ψ = b 0 Ψ = 0 5 / 24

8 String field theory difficulties infinite number of interactions non-polynomial action definition of vertex V n : n holomorphic function {fn,i } (with constraints) subspace of moduli space V n M 0,n gluing with propagators and summing = single covering of M 0,n no explicit representation for n > 5 [hep-th/ , Belopolsky][hep-th/ , Moeller][hep-th/ , Moeller][ , Erler-Konopka-Sachs] 6 / 24

9 String field theory difficulties infinite number of interactions non-polynomial action definition of vertex V n : n holomorphic function {fn,i } (with constraints) subspace of moduli space V n M 0,n gluing with propagators and summing = single covering of M 0,n no explicit representation for n > 5 [hep-th/ , Belopolsky][hep-th/ , Moeller][hep-th/ , Moeller][ , Erler-Konopka-Sachs] almost nothing computable explicitly 6 / 24

10 String field theory difficulties infinite number of interactions non-polynomial action definition of vertex V n : n holomorphic function {fn,i } (with constraints) But: subspace of moduli space V n M 0,n gluing with propagators and summing = single covering of M 0,n no explicit representation for n > 5 [hep-th/ , Belopolsky][hep-th/ , Moeller][hep-th/ , Moeller][ , Erler-Konopka-Sachs] almost nothing computable explicitly general properties understood, used to prove consistency of string theory [Sen-Zwiebach 94, Sen 15-18,... ] expect progress from [ , Headrick-Zwiebach][ , Headrick-Zwiebach] 6 / 24

11 The open string miracle open string: truncation to cubic interactions [Witten 86] l 2 (A, B) A B, n > 2 : l n = 0 non-commutative -product 7 / 24

12 The open string miracle open string: truncation to cubic interactions [Witten 86] l 2 (A, B) A B, n > 2 : l n = 0 non-commutative -product open string cubic action S = 1 2 Ψ Q BΨ + g 3 Ψ Ψ Ψ 7 / 24

13 The open string miracle open string: truncation to cubic interactions [Witten 86] l 2 (A, B) A B, n > 2 : l n = 0 non-commutative -product open string cubic action S = 1 2 Ψ Q BΨ + g 3 Ψ Ψ Ψ many explicit computations: analytic and level-truncation solutions (tachyon condensation, brane decay, marginal deformations, defects, time-dependent backgrounds... ) effective actions etc. 7 / 24

14 Cubic closed string field theory Question: Can one find a cubic closed string field theory? 8 / 24

15 Cubic closed string field theory Question: Can one find a cubic closed string field theory? No-go theorem [Sonoda-Zwiebach 90] There is no covariant BRST invariant single covering of M 0,4 built from a symmetric cubic vertex assuming standard plumbing fixture. 8 / 24

16 Cubic closed string field theory Question: Can one find a cubic closed string field theory? No-go theorem [Sonoda-Zwiebach 90] There is no covariant BRST invariant single covering of M 0,4 built from a symmetric cubic vertex assuming standard plumbing fixture. Way out: use auxiliary fields to decompose interactions 8 / 24

17 Cubic closed string field theory Question: Can one find a cubic closed string field theory? No-go theorem [Sonoda-Zwiebach 90] There is no covariant BRST invariant single covering of M 0,4 built from a symmetric cubic vertex assuming standard plumbing fixture. Way out: use auxiliary fields to decompose interactions 1. Hubbard Stratonovich transformation 2. IBL structure 8 / 24

18 Outline: 2. Hubbard Stratonovich action Introduction Hubbard Stratonovich action IBL action Conclusion 9 / 24

19 Hubbard Stratonovich transformation Hubbard Stratonovich (intermediate field) representation: originally: many-body physics (condensed matter and nuclear physics) decompose interactions to lower-order interactions through auxiliary fields optimal representation: cubic interactions, quadratic in the physical field, linear in the auxiliary fields 10 / 24

20 Hubbard Stratonovich transformation Hubbard Stratonovich (intermediate field) representation: originally: many-body physics (condensed matter and nuclear physics) decompose interactions to lower-order interactions through auxiliary fields optimal representation: cubic interactions, quadratic in the physical field, linear in the auxiliary fields (vector) φ 4, φ 2n [ , Rivasseau-Wang][ , Lionni-Rivasseau], matrix and tensor models [ , Lionni-Rivasseau] 10 / 24

21 Scalar φ 4 model φ 4 action S = ( ) 1 d d x 2 φkφ + λφ4 11 / 24

22 Scalar φ 4 model φ 4 action S = ( ) 1 d d x 2 φkφ + λφ4 introduce auxiliary field σ σ = φ 2 (on-shell) 11 / 24

23 Scalar φ 4 model φ 4 action S = ( ) 1 d d x 2 φkφ + λφ4 introduce auxiliary field σ σ = φ 2 (on-shell) Hubbard Stratonovich action ( ) 1 S HS = d d σ2 x φkφ 2λ + 2λ σφ2 2 2 Feynman graphs = + perms 11 / 24

24 Scalar φ 4 vector model vector φ 4 [Weinberg vol. 2] S = dk 1 dk 4 V i1 i 2 i 3 i 4 (k 1,..., k 4 )φ i1 (k 1 ) φ i4 (k 4 ) + 1 dk φ i (k)k ij (k)φ j ( k) 2 12 / 24

25 Scalar φ 4 vector model vector φ 4 [Weinberg vol. 2] S = dk 1 dk 4 V i1 i 2 i 3 i 4 (k 1,..., k 4 )φ i1 (k 1 ) φ i4 (k 4 ) + 1 dk φ i (k)k ij (k)φ j ( k) 2 introduce auxiliary field δs = ( ) dk 1 dk 4 V i1 i 4 (k 1,..., k 4 ) σ i1 i 2 (k 1, k 2 ) φ i1 (k 1 )φ i2 (k 2 ) ( ) σ i3 i 4 (k 3, k 4 ) φ i3 (k 3 )φ i4 (k 4 ) 12 / 24

26 Scalar φ 4 vector model vector φ 4 [Weinberg vol. 2] S = dk 1 dk 4 V i1 i 2 i 3 i 4 (k 1,..., k 4 )φ i1 (k 1 ) φ i4 (k 4 ) + 1 dk φ i (k)k ij (k)φ j ( k) 2 introduce auxiliary field δs = ( ) dk 1 dk 4 V i1 i 4 (k 1,..., k 4 ) σ i1 i 2 (k 1, k 2 ) φ i1 (k 1 )φ i2 (k 2 ) ( ) σ i3 i 4 (k 3, k 4 ) φ i3 (k 3 )φ i4 (k 4 ) Hubbard Stratonovich action S HS = dk 1 dk 4 V i1 i 4 σ i1 i 2 σ i3 i dk φ i K ij φ j dk 1 dk 4 V i1 i 4 σ i1 i 2 φ i3 φ i4 12 / 24

27 String field theory string field action to O(g 2 s ) S = 1 2 V 2(Ψ 2 ) + g s 3! V 3(Ψ 3 ) + g 2 s 4! V 4(Ψ 4 ) 13 / 24

28 String field theory string field action to O(g 2 s ) S = 1 2 V 2(Ψ 2 ) + g s 3! V 3(Ψ 3 ) + g 2 s 4! V 4(Ψ 4 ) introduce auxiliary field Σ H 2 δs = g 2 s 4! V 4(Σ Ψ Ψ, Σ Ψ Ψ) 13 / 24

29 String field theory string field action to O(g 2 s ) S = 1 2 V 2(Ψ 2 ) + g s 3! V 3(Ψ 3 ) + g 2 s 4! V 4(Ψ 4 ) introduce auxiliary field Σ H 2 δs = g 2 s 4! V 4(Σ Ψ Ψ, Σ Ψ Ψ) Hubbard Stratonovich action S HS = 1 2 V 2(Ψ 2 ) + g s 3! V 3(Ψ 3 ) g 2 s 4! V 4(Σ, Σ) + 2g 2 s 4! V 4(Σ, Ψ 2 ) 13 / 24

30 Properties new product m 2 : H 2 H 2 V 4 (A 1,..., A 4 ) = A 1 A 2 m 2 (A 3, A 4 ) A 1 m 2 (A 2, A 3 ) = l 3 (A 1, A 2, A 3 ) 14 / 24

31 Properties new product m 2 : H 2 H 2 equation of motion V 4 (A 1,..., A 4 ) = A 1 A 2 m 2 (A 3, A 4 ) A 1 m 2 (A 2, A 3 ) = l 3 (A 1, A 2, A 3 ) l 1 (Ψ) + g s 2 l 2(Ψ 2 ) + g 2 s 3! Ψ m 2(Σ) = 0 m 2 (Σ) = m 2 (Ψ 2 ) 14 / 24

32 Properties new product m 2 : H 2 H 2 equation of motion gauge invariance V 4 (A 1,..., A 4 ) = A 1 A 2 m 2 (A 3, A 4 ) A 1 m 2 (A 2, A 3 ) = l 3 (A 1, A 2, A 3 ) l 1 (Ψ) + g s 2 l 2(Ψ 2 ) + g 2 s 3! Ψ m 2(Σ) = 0 m 2 (Σ) = m 2 (Ψ 2 ) δ Λ Ψ = l 1 (Λ) + g s l 2 (Λ, Ψ) + g 2 s 2 Λ m 2(Ψ 2 ) δ Λ Σ = l 1 (Λ) Ψ + Ψ l 1 (Λ) 14 / 24

33 Properties new product m 2 : H 2 H 2 equation of motion gauge invariance V 4 (A 1,..., A 4 ) = A 1 A 2 m 2 (A 3, A 4 ) A 1 m 2 (A 2, A 3 ) = l 3 (A 1, A 2, A 3 ) l 1 (Ψ) + g s 2 l 2(Ψ 2 ) + g 2 s 3! Ψ m 2(Σ) = 0 m 2 (Σ) = m 2 (Ψ 2 ) δ Λ Ψ = l 1 (Λ) + g s l 2 (Λ, Ψ) + g 2 s 2 Λ m 2(Ψ 2 ) δ Λ Σ = l 1 (Λ) Ψ + Ψ l 1 (Λ) Fourier expansion entangled states Σ = r,s d D k 1 (2π) D d D k 2 (2π) D σ rs(k 1, k 2 ) φ r (k 1 ) φ s (k 2 ) 14 / 24

34 Worldsheet interpretation (1) action still not exactly cubic: cubic in the field but: V4 contains 4 local coordinates 15 / 24

35 Worldsheet interpretation (1) action still not exactly cubic: cubic in the field but: V4 contains 4 local coordinates introduce new vertices W 3 (A 2, B 1, C 1 ) = A 2 m 2 (B 1, C 1 ) W 2 (A 2, B 2 ) = A 2 m 2 (B 2 ) 15 / 24

36 Worldsheet interpretation (1) action still not exactly cubic: cubic in the field but: V4 contains 4 local coordinates introduce new vertices W 3 (A 2, B 1, C 1 ) = A 2 m 2 (B 1, C 1 ) W 2 (A 2, B 2 ) = A 2 m 2 (B 2 ) remarks on W 3 asymmetric vertex same properties as V 3 from [Sen-Zwiebach 94] twist sewing of special puncture reproduces V4 (with SL(2, C) local coordinates) 15 / 24

37 Worldsheet interpretation (1) action still not exactly cubic: cubic in the field but: V4 contains 4 local coordinates introduce new vertices W 3 (A 2, B 1, C 1 ) = A 2 m 2 (B 1, C 1 ) W 2 (A 2, B 2 ) = A 2 m 2 (B 2 ) remarks on W 3 asymmetric vertex same properties as V 3 from [Sen-Zwiebach 94] twist sewing of special puncture reproduces V4 (with SL(2, C) local coordinates) worldsheet: insert states of tensor product at different punctures merge them together (no singularities) V4 : glue two punctures with two punctures W 2 : glue one puncture with one puncture 15 / 24

38 Worldsheet interpretation (2) = 16 / 24

39 Generalisation Questions: how to generalize to higher orders? will encounter products with more outputs m 2 : two inputs and two outputs not a L product which structure will have the action? 17 / 24

40 Generalisation Questions: how to generalize to higher orders? will encounter products with more outputs m 2 : two inputs and two outputs not a L product which structure will have the action? = IBL algebra 17 / 24

41 Outline: 3. IBL action Introduction Hubbard Stratonovich action IBL action Conclusion 18 / 24

42 IBL algebra symmetric tensor algebra (implicit symmetrization) SH = H n n 1 hat: embedding map from H n to SH ˆl 1 = l l 1 + l / 24

43 IBL algebra symmetric tensor algebra (implicit symmetrization) SH = H n n 1 hat: embedding map from H n to SH ˆl 1 = l l 1 + l IBL products p m,n : H m H n ˆp = m,n 1 g m+n 2 s ˆp m,n [ , Cieliebak-Fukaya-Latschev] (see also [ , Muenster-Sachs][ , Muenster-Sachs]) 19 / 24

44 IBL algebra symmetric tensor algebra (implicit symmetrization) SH = H n n 1 hat: embedding map from H n to SH ˆl 1 = l l 1 + l IBL products p m,n : H m H n ˆp = m,n 1 g m+n 2 s ˆp m,n [ , Cieliebak-Fukaya-Latschev] (see also [ , Muenster-Sachs][ , Muenster-Sachs]) IBL relations ˆp ˆp = 0 = ˆp m1,n 1 ˆp m2,n 2 = 0 m 1 +m 2 =m+1 n 1 +n 2 =n+1 19 / 24

45 IBL algebra symmetric tensor algebra (implicit symmetrization) SH = H n n 1 hat: embedding map from H n to SH ˆl 1 = l l 1 + l IBL products p m,n : H m H n ˆp = m,n 1 g m+n 2 s ˆp m,n [ , Cieliebak-Fukaya-Latschev] (see also [ , Muenster-Sachs][ , Muenster-Sachs]) IBL relations ˆp ˆp = 0 = ˆp m1,n 1 ˆp m2,n 2 = 0 m 1 +m 2 =m+1 n 1 +n 2 =n+1 induces a BV algebra [ , Markl-Voronov] 19 / 24

46 IBL action string field Ψ = Ψ n, n 1 Ψ n H n 20 / 24

47 IBL action string field Ψ = Ψ n, n 1 Ψ n H n action ω = n 1 ω n, S = 1 0 ( ) dt ω Ψ, ˆp(e tψ ) ω n = ω n 1 + perms, ω 1 : H H C 20 / 24

48 IBL action string field Ψ = Ψ n, n 1 Ψ n H n action S = 1 0 ( ) dt ω Ψ, ˆp(e tψ ) ω = n 1 ω n, ω n = ω1 n + perms, ω 1 : H H C eom = Maurer Cartan equation ˆp(e Ψ ) = 0 20 / 24

49 IBL action string field Ψ = Ψ n, n 1 Ψ n H n action S = 1 0 ( ) dt ω Ψ, ˆp(e tψ ) ω = n 1 ω n, ω n = ω1 n + perms, ω 1 : H H C eom = Maurer Cartan equation gauge invariance ˆp(e Ψ ) = 0 δ Ψ = ˆp(Λ, e Ψ ), Λ = Λ n, n 1 Λ n H n 20 / 24

50 IBL action string field Ψ = Ψ n, n 1 Ψ n H n action S = 1 0 ( ) dt ω Ψ, ˆp(e tψ ) ω = n 1 ω n, ω n = ω1 n + perms, ω 1 : H H C eom = Maurer Cartan equation gauge invariance ˆp(e Ψ ) = 0 δ Ψ = ˆp(Λ, e Ψ ), Λ = Λ n, n 1 Λ n H n note: generalize at quantum level (new index g 0) 20 / 24

51 IBL string field theory Conjecture 1 The non-polynomial L closed string field theory is equivalent to a IBL string field theory on-shell. 21 / 24

52 IBL string field theory Conjecture 1 The non-polynomial L closed string field theory is equivalent to a IBL string field theory on-shell. Conjecture 2 There is a parametrization such that the action is cubic in the fields. 21 / 24

53 Comments the multi-output products are built from the L products p n,1 = ω 1 p n 1,2, p n,2 = ω 1 p n 1,3,... no new information 22 / 24

54 Comments the multi-output products are built from the L products p n,1 = ω 1 p n 1,2, p n,2 = ω 1 p n 1,3,... no new information auxiliary fields: must be fixed algebraically by the eom (kinetic terms involve only higher-order products, not Q B, no singularity in V n 3 [ , Pius-Sen]) 22 / 24

55 Comments the multi-output products are built from the L products p n,1 = ω 1 p n 1,2, p n,2 = ω 1 p n 1,3,... no new information auxiliary fields: must be fixed algebraically by the eom (kinetic terms involve only higher-order products, not Q B, no singularity in V n 3 [ , Pius-Sen]) recover (almost) the Hubbard Stratonovich action for p 1,1 = l 1, p 2,1 = l 2, p 3,1 = l 3, p 2,2 = m 2 and p n,m = 0 otherwise 22 / 24

56 Comments the multi-output products are built from the L products p n,1 = ω 1 p n 1,2, p n,2 = ω 1 p n 1,3,... no new information auxiliary fields: must be fixed algebraically by the eom (kinetic terms involve only higher-order products, not Q B, no singularity in V n 3 [ , Pius-Sen]) recover (almost) the Hubbard Stratonovich action for p 1,1 = l 1, p 2,1 = l 2, p 3,1 = l 3, p 2,2 = m 2 and p n,m = 0 otherwise IBL action: bigger gauge invariance how to gauge fix? 22 / 24

57 Comments the multi-output products are built from the L products p n,1 = ω 1 p n 1,2, p n,2 = ω 1 p n 1,3,... no new information auxiliary fields: must be fixed algebraically by the eom (kinetic terms involve only higher-order products, not Q B, no singularity in V n 3 [ , Pius-Sen]) recover (almost) the Hubbard Stratonovich action for p 1,1 = l 1, p 2,1 = l 2, p 3,1 = l 3, p 2,2 = m 2 and p n,m = 0 otherwise IBL action: bigger gauge invariance how to gauge fix? worldsheet interpretation? 22 / 24

58 Outline: 4. Conclusion Introduction Hubbard Stratonovich action IBL action Conclusion 23 / 24

59 Outlook Summary: path for constructing a cubic closed string field theory SFT with IBL algebra algebraic description of SFT with auxiliary fields If correct: classical solutions (consistent truncation of auxiliary fields) thermal effects (σ saddle point = mean field theory) explicit computations of vertices and amplitudes non-perturbative effects If not correct: learn more about the structure of SFT if holds up to some order, still useful for perturbative computations 24 / 24