First Year Seminar. Dario Rosa Milano, Thursday, September 27th, 2012

Transcription

1 Dipartimento di Fisica, Università degli studi di Milano Bicocca Milano, Thursday, September 27th, 2012

2 1 Holomorphic Chern-Simons theory (HCS) Strategy of solution and results 2

3 Strategy of solution and results HCS describes the dynamics of a stack of N 5-branes of topological B type (1 Γ = Ω Tr 2 A Z A A3) (1.1) X X is a Calabi-Yau threefold. A is the gauge eld: it is a (0, 1)-form with respect to the complex structure on X. Ω is the globally dened (3, 0)-form. (Z, Z) is a system of coordinates compatible with the complex structure.

4 Strategy of solution and results HCS describes the dynamics of a stack of N 5-branes of topological B type (1 Γ = Ω Tr 2 A Z A A3) (1.1) X X is a Calabi-Yau threefold. A is the gauge eld: it is a (0, 1)-form with respect to the complex structure on X. Ω is the globally dened (3, 0)-form. (Z, Z) is a system of coordinates compatible with the complex structure. The theory depends on two geometrical data: Ω and the complex structure on X, they are in correspondence with the closed moduli.

5 Strategy of solution and results To exhibit explicitly the dependence on the complex structure: Beltrami parametrization ( dz i = Λ i j dz j + µ j j dz j) Z µ i i (1.2) (z, z) is a xed system of coordinates. µ i is the Beltrami dierential.

6 Strategy of solution and results To exhibit explicitly the dependence on the complex structure: Beltrami parametrization ( dz i = Λ i j dz j + µ j j dz j) Z µ i i (1.2) (z, z) is a xed system of coordinates. µ i is the Beltrami dierential. The action rewrites as Γ 0 (Ω, µ) = X Ω ( 1 2 A A A3) (1.3)

7 Strategy of solution and results To exhibit explicitly the dependence on the complex structure: Beltrami parametrization ( dz i = Λ i j dz j + µ j j dz j) Z µ i i (1.2) (z, z) is a xed system of coordinates. µ i is the Beltrami dierential. The action rewrites as Γ 0 (Ω, µ) = X Ω ( 1 2 A A A3) (1.3) The dependence of the theory on the closed moduli is now captured by Ω and µ. They are like coupling constants for the model

8 Strategy of solution and results Symmetries (1) Original action (1.1): invariant under Ω-preserving holomorphic reparametrizations and under gauge transformations s A = Z c [A, c] + s c = c 2 (1.4)

9 Strategy of solution and results Symmetries (1) Original action (1.1): invariant under Ω-preserving holomorphic reparametrizations and under gauge transformations s A = Z c [A, c] + s c = c 2 (1.4) The coupling of A to the background µ i promotes holomorphic reparametrizations into a local symmetry, µ i transforms as a gauge eld: s di µ i = ξ i + ξ j j µ i j ξ i µ j (1.5) ξ i is the ghost of Ω-preserving local dieomorphisms

10 Strategy of solution and results Symmetries (2) After the coupling to the background gauge invariance is preserved if and only if F i µ i µ j j µ i = 0 (1.6) ˆ Ω Ω + i µ i Ω = 0 (1.7) both equations are well-known: the rst is the Kodaira-Spencer equation, the second is the holomorphicity of Ω

11 Strategy of solution and results Symmetries (2) After the coupling to the background gauge invariance is preserved if and only if F i µ i µ j j µ i = 0 (1.6) ˆ Ω Ω + i µ i Ω = 0 (1.7) both equations are well-known: the rst is the Kodaira-Spencer equation, the second is the holomorphicity of Ω These are the equations of motion (e.o.m.) of the closed topological string theory

12 Strategy of solution and results Symmetries (2) After the coupling to the background gauge invariance is preserved if and only if F i µ i µ j j µ i = 0 (1.6) ˆ Ω Ω + i µ i Ω = 0 (1.7) both equations are well-known: the rst is the Kodaira-Spencer equation, the second is the holomorphicity of Ω These are the equations of motion (e.o.m.) of the closed topological string theory Invariance only if backgrounds are on-shell, i.e. if they satisfy the equations imposed by the closed topological sector of the theory

13 Strategy of solution and results Main results of the paper Gauge invariance only on-shell

14 Strategy of solution and results Main results of the paper Gauge invariance only on-shell The presence of branes modies the closed equations of motion and puts the backgrounds o-shell

15 Strategy of solution and results Main results of the paper Gauge invariance only on-shell The presence of branes modies the closed equations of motion and puts the backgrounds o-shell A formulation of HCS for o-shell backgrounds is important to explore the coupling between open and closed sector

16 Strategy of solution and results Main results of the paper Gauge invariance only on-shell The presence of branes modies the closed equations of motion and puts the backgrounds o-shell A formulation of HCS for o-shell backgrounds is important to explore the coupling between open and closed sector A formulation of HCS valid for o-shell backgrounds is obtained. It is invariant under gauge transformation and under chiral dieomorphisms (the extension of dieomorphisms valid o-shell), i.e. s tot = s + s di A twisted N = 2 supersymmetric structure is also discovered. Index of the (semi)topological character of the model

17 Strategy of solution and results O-shell action Standard method to go o-shell Introduction of new elds acting as Lagrange multipliers for the closed e.o.m., with gauge transformation properties such that the action is gauge invariant for generic o-shell backgrounds

18 Strategy of solution and results O-shell action Standard method to go o-shell Introduction of new elds acting as Lagrange multipliers for the closed e.o.m., with gauge transformation properties such that the action is gauge invariant for generic o-shell backgrounds Following this method the action becomes Γ = 1 [ ( 2 Ω Tr A A A3) + Ω ( )] B + F i C i X B and C i are the Lagrange multipliers. To achieve gauge invariance their BRST transformations must start with s B = Tr ( c A ) (1.8) s C i = Tr ( c i A + i c A ) (1.9)

19 Strategy of solution and results Symmetries of the o-shell action The introduction of B and C i extends greatly the gauge symmetry of the theory new ghost elds (called d and f i )

20 Strategy of solution and results Symmetries of the o-shell action The introduction of B and C i extends greatly the gauge symmetry of the theory new ghost elds (called d and f i ) This new ghost elds are however redundant in many respects a ghost-for-ghost eld is necessary (e eld)

21 Strategy of solution and results Symmetries of the o-shell action The introduction of B and C i extends greatly the gauge symmetry of the theory new ghost elds (called d and f i ) This new ghost elds are however redundant in many respects a ghost-for-ghost eld is necessary (e eld) Final BRST gauge transformations s A = c [A, c] + s c = c 2 s B = Tr (c A) F i f i d s C i = Tr ( c i A + i c A) ˆ f i + i d s d = Tr (A c 2 ) e s f i = Tr (c i c) + i e s e = 1 3 Tr c3 (1.10)

22 Strategy of solution and results Twisted N = 2 supersymmetric structure (1) In topological theories (TFT) the derivative is BRST-trivial µ = {s, G µ } (1.11) This twisted SUSY structure proves that correlators of local observables are space-time independent TFT

23 Strategy of solution and results Twisted N = 2 supersymmetric structure (1) In topological theories (TFT) the derivative is BRST-trivial µ = {s, G µ } (1.11) This twisted SUSY structure proves that correlators of local observables are space-time independent TFT HCS is semi-topological: it does not depend on the full space-time metric but only on the Beltrami dierential µ i

24 Strategy of solution and results Twisted N = 2 supersymmetric structure (1) In topological theories (TFT) the derivative is BRST-trivial µ = {s, G µ } (1.11) This twisted SUSY structure proves that correlators of local observables are space-time independent TFT HCS is semi-topological: it does not depend on the full space-time metric but only on the Beltrami dierential µ i Consequently it is conceivable that a holomorphic version of the relation (1.11) holds for HCS: ˆ ī = {s, Gī} (1.12)

25 Strategy of solution and results Twisted N = 2 supersymmetric structure (2) For each eld of the theory (and also coupling constant) Φ, a corresponding anti-eld Φ is introduced (BV-formalism)

26 Strategy of solution and results Twisted N = 2 supersymmetric structure (2) For each eld of the theory (and also coupling constant) Φ, a corresponding anti-eld Φ is introduced (BV-formalism) Up redenitions all elds (and coupling constants), together with anti-elds, sit in complete superelds, i.e. superelds with components of every anti-holomorphic form degrees (as usual in TFT)

27 Strategy of solution and results Twisted N = 2 supersymmetric structure (2) For each eld of the theory (and also coupling constant) Φ, a corresponding anti-eld Φ is introduced (BV-formalism) Up redenitions all elds (and coupling constants), together with anti-elds, sit in complete superelds, i.e. superelds with components of every anti-holomorphic form degrees (as usual in TFT) Working on this supereld formulation of HCS, it is easily found the action of the operator Gī which satises on dynamical elds ˆ ī = {s, Gī}

28 Few words about 4D, N = 2 gauged SUGRA We consider 4D, N = 2 SUGRA, coupled to n vector multiplets and m hypermultiplets, with gaugings of some of the isometries of the scalar manifolds

29 Few words about 4D, N = 2 gauged SUGRA We consider 4D, N = 2 SUGRA, coupled to n vector multiplets and m hypermultiplets, with gaugings of some of the isometries of the scalar manifolds Field content Gravity multiplet: a graviton eµ a, a pair of gravitini (Weyl) ψ iµ i = 1, 2 and a vector (graviphoton) A 0 µ Vector multiplets: each has a complex scalar Z l, a pair of gaugini (Weyl) λ il i = 1, 2 and a vector A l µ, l = 1... n Hypermultiplets: each has 4 real scalar q u u = 1 4m (hyperscalars) and a pair of hyperini ζ α α = 1 2m (Weyl)

30 Few words about 4D, N = 2 gauged SUGRA We consider 4D, N = 2 SUGRA, coupled to n vector multiplets and m hypermultiplets, with gaugings of some of the isometries of the scalar manifolds Field content Gravity multiplet: a graviton eµ a, a pair of gravitini (Weyl) ψ iµ i = 1, 2 and a vector (graviphoton) A 0 µ Vector multiplets: each has a complex scalar Z l, a pair of gaugini (Weyl) λ il i = 1, 2 and a vector A l µ, l = 1... n Hypermultiplets: each has 4 real scalar q u u = 1 4m (hyperscalars) and a pair of hyperini ζ α α = 1 2m (Weyl) Z l are coordinates on a Special Kähler manifold q u are coordinates on a quaternionic manifold

31 SUSY transformation and supersymmetric congurations One is often interested to nd supersymmetric, bosonic, congurations, i.e. congurations in which all the fermions go to zero and which are invariants under SUSY transformations

32 SUSY transformation and supersymmetric congurations One is often interested to nd supersymmetric, bosonic, congurations, i.e. congurations in which all the fermions go to zero and which are invariants under SUSY transformations Imposing that fermions go to zero (classical solutions) SUSY variations of bosons automatically vanish.

33 SUSY transformation and supersymmetric congurations One is often interested to nd supersymmetric, bosonic, congurations, i.e. congurations in which all the fermions go to zero and which are invariants under SUSY transformations Imposing that fermions go to zero (classical solutions) SUSY variations of bosons automatically vanish. One obtains the system [ δ ɛ ψ iµ = D µ ɛ i + T µνε + ij 1 2 S x η µν ε ik (σ x ) k j δ ɛ λ il = i DZ l ɛ i + ] γ ν ɛ j c = 0 [( G l + + W l ) ε ij + i 2 W l x (σ x ) i k εkj ] ɛ j c = 0 δ ɛ ζ α = iu αi u Dq u ɛ i + N i αɛ i c = 0 (2.13) W l, S x, N i α due to gaugings, U αiu given by the quaternionic metric

34 Aim of my work One searches bosonic congurations which admit at least a solution of (2.13)

35 Aim of my work One searches bosonic congurations which admit at least a solution of (2.13) System (2.13) is a system of spinorial dierential equations (ɛ i are spinors) it is very dicult to handling

36 Aim of my work One searches bosonic congurations which admit at least a solution of (2.13) System (2.13) is a system of spinorial dierential equations (ɛ i are spinors) it is very dicult to handling The aim is to obtain an equivalent system of dierential equations which contain only dierential forms and external calculus drastically simpler to treat than (2.13)

37 Aim of my work One searches bosonic congurations which admit at least a solution of (2.13) System (2.13) is a system of spinorial dierential equations (ɛ i are spinors) it is very dicult to handling The aim is to obtain an equivalent system of dierential equations which contain only dierential forms and external calculus drastically simpler to treat than (2.13) Even if this is a very natural task, a partial result was obtained only recently. That result has to be considered partial because it assumes that the Killing vector constructed from SUSY spinors is timelike everywhere. We will see the diculties that arise when one tries to remove this assumption

38 Bilinears method It is a method which allows us to extract from one or more spinors dierential forms. Given ɛ 1 and ɛ 2 we consider the expression (in d dimensions) ɛ 1 ɛ 2 = d k=0 1 2 d 2 k! (ɛ 2γ Mk...M 1 ɛ 1 )γ M 1...M k (2.14) This is a collection of spinor bilinears that can be mapped to forms via Cliord map

39 Bilinears method It is a method which allows us to extract from one or more spinors dierential forms. Given ɛ 1 and ɛ 2 we consider the expression (in d dimensions) ɛ 1 ɛ 2 = d k=0 1 2 d 2 k! (ɛ 2γ Mk...M 1 ɛ 1 )γ M 1...M k (2.14) This is a collection of spinor bilinears that can be mapped to forms via Cliord map It is a useful method to convert spinorial dierential equations in dierential equations on forms

40 Bilinears method It is a method which allows us to extract from one or more spinors dierential forms. Given ɛ 1 and ɛ 2 we consider the expression (in d dimensions) ɛ 1 ɛ 2 = d k=0 1 2 d 2 k! (ɛ 2γ Mk...M 1 ɛ 1 )γ M 1...M k (2.14) This is a collection of spinor bilinears that can be mapped to forms via Cliord map It is a useful method to convert spinorial dierential equations in dierential equations on forms Together with an analysis of the geometry dened by SUSY spinors, and with the concept of intrinsic torsions, it makes possible to reformulate (2.13) in terms of forms

41 Geometry dened by ɛ 1 and ɛ 2 Single spinor ɛ i : one obtains a real vector Z i and a complex vector W i which satisfy Z 2 i = W 2 i = W 2 i = 0 W i Wi = 2 (2.15) In general one can not extract from these 6 vectors a vierbein introduction of e i with property (e i ) 2 = 0 and Z i e i = 2 The spinors ɛ i and e i ɛ c i dene two basis for the spinors of given chirality (ɛ 1 and ɛ 2 does not in general)

42 Geometry dened by ɛ 1 and ɛ 2 Single spinor ɛ i : one obtains a real vector Z i and a complex vector W i which satisfy Z 2 i = W 2 i = W 2 i = 0 W i Wi = 2 (2.15) In general one can not extract from these 6 vectors a vierbein introduction of e i with property (e i ) 2 = 0 and Z i e i = 2 The spinors ɛ i and e i ɛ c i dene two basis for the spinors of given chirality (ɛ 1 and ɛ 2 does not in general) If Z 1 + Z 2 is timelike (the assumption used in literature): (Z 1, W 1, W1, Z 2 ) dene a vierbein Crucial simplication!!

43 Geometry dened by ɛ 1 and ɛ 2 Single spinor ɛ i : one obtains a real vector Z i and a complex vector W i which satisfy Z 2 i = W 2 i = W 2 i = 0 W i Wi = 2 (2.15) In general one can not extract from these 6 vectors a vierbein introduction of e i with property (e i ) 2 = 0 and Z i e i = 2 The spinors ɛ i and e i ɛ c i dene two basis for the spinors of given chirality (ɛ 1 and ɛ 2 does not in general) If Z 1 + Z 2 is timelike (the assumption used in literature): (Z 1, W 1, W1, Z 2 ) dene a vierbein Crucial simplication!! Two spinors: by considering the bilinears dened by ɛ 1 and ɛ 2 together, one obtains a scalar λ, a complex vector v and a self-dual two-form ω

44 Intrinsic torsions Return to (2.13): rst equations are dierent from the others. They contain derivatives of spinors Intrinsic torsions

45 Intrinsic torsions Return to (2.13): rst equations are dierent from the others. They contain derivatives of spinors Intrinsic torsions Given a spinor ɛ i one can expand its derivative on a base D µ ɛ i = p iµ ɛ i + q iµ e i ɛ c i (2.16) The coecients p iµ and q iµ are called intrinsic torsions

46 Intrinsic torsions Return to (2.13): rst equations are dierent from the others. They contain derivatives of spinors Intrinsic torsions Given a spinor ɛ i one can expand its derivative on a base D µ ɛ i = p iµ ɛ i + q iµ e i ɛ c i (2.16) The coecients p iµ and q iµ are called intrinsic torsions It is clear that these equations, expanded on a base, give rise to corresponding equations on p 1, p 2, q 1 and q 2 One has to see what dierential forms are necessary and sucient to fully reconstruct the intrinsic torsions

47 Intrinsic torsions Return to (2.13): rst equations are dierent from the others. They contain derivatives of spinors Intrinsic torsions Given a spinor ɛ i one can expand its derivative on a base D µ ɛ i = p iµ ɛ i + q iµ e i ɛ c i (2.16) The coecients p iµ and q iµ are called intrinsic torsions It is clear that these equations, expanded on a base, give rise to corresponding equations on p 1, p 2, q 1 and q 2 One has to see what dierential forms are necessary and sucient to fully reconstruct the intrinsic torsions Other equations in (2.13) are simpler: it is sucient to expand them on a base

48 Results Holomorphic Chern-Simons theory (HCS) The analysis of the dierential forms dened by ɛ 1 and ɛ 2 (considering also e 1 and e 2 ) shows that intrinsic torsions are fully reconstructed (in a slightly redundant way) by considering µ Z 2ν + ν Z 1µ d v d( v) d λ d ω d( λ)(2.17) plus some components of the external derivatives of the following bilinears e 1 ɛ 1 ɛ 2 e 1 ɛ 1 ɛ c 2 ɛ 1 ɛ 2 e 2 ɛ 1 ɛ c 2 e 2 (2.18) In this manner one obtains a system of dierential equations which is completely equivalent to (2.13) but in terms of dierential forms.

49 In ten-dimensional type II supergravity a system of dierential equations which classies solutions was already found

50 In ten-dimensional type II supergravity a system of dierential equations which classies solutions was already found It is well-known that compactication of Type II supergravities on a six-dimensional manifold which admits a pair of SU(3) structure, can gives rise to 4D, N = 2 gauged SUGRA

51 In ten-dimensional type II supergravity a system of dierential equations which classies solutions was already found It is well-known that compactication of Type II supergravities on a six-dimensional manifold which admits a pair of SU(3) structure, can gives rise to 4D, N = 2 gauged SUGRA To this end ten-dimensional SUSY spinors decompose as ɛ (10) 1 = ɛ 1 +(x) η 1 +(x, y) + ɛ 1 (x) η 1 (x, y) ɛ (10) 2 = ɛ 2 +(x) η 2 (x, y) + ɛ 2 (x) η 2 +(x, y) (2.19) y are the internal coordinates and η± i spinors are internal, chiral

52 In ten-dimensional type II supergravity a system of dierential equations which classies solutions was already found It is well-known that compactication of Type II supergravities on a six-dimensional manifold which admits a pair of SU(3) structure, can gives rise to 4D, N = 2 gauged SUGRA To this end ten-dimensional SUSY spinors decompose as ɛ (10) 1 = ɛ 1 +(x) η 1 +(x, y) + ɛ 1 (x) η 1 (x, y) ɛ (10) 2 = ɛ 2 +(x) η 2 (x, y) + ɛ 2 (x) η 2 +(x, y) (2.19) y are the internal coordinates and η± i are internal, chiral spinors It is interesting to see if it is possible to obtain, from this reduction, a system of equations equivalent to that obtained in four dimensions

53 Results (Until Now) The system of ten-dimensional equations was rewritten in a manner appropriate for the compactication that we are considering

54 Results (Until Now) The system of ten-dimensional equations was rewritten in a manner appropriate for the compactication that we are considering This is useful in order to see if a particular four-dimensional solution can be lifted to a ten-dimensional solution

55 Results (Until Now) The system of ten-dimensional equations was rewritten in a manner appropriate for the compactication that we are considering This is useful in order to see if a particular four-dimensional solution can be lifted to a ten-dimensional solution We are not sure (until now) that the resulting system is completely equivalent to the system that one obtains in four dimensions

56 Results (Until Now) The system of ten-dimensional equations was rewritten in a manner appropriate for the compactication that we are considering This is useful in order to see if a particular four-dimensional solution can be lifted to a ten-dimensional solution We are not sure (until now) that the resulting system is completely equivalent to the system that one obtains in four dimensions This is an important question that could give us general conditions about the possibility of lifting solutions from four dimensions to ten dimensions