From Flux Compactification to Cosmology

Transcription

1 From Flux Compactification to Cosmology Xin Gao Virginia Tech & ITP-CAS CTP-SCU, 3 July, 2014

2 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

3 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

4 The Road to Unification I Physics, in Greek, meaning nature, is the study of matter and its motion through spacetime and all that derives from these, such as energy and force.

5 The Road to Unification I Physics, in Greek, meaning nature, is the study of matter and its motion through spacetime and all that derives from these, such as energy and force. How to describe the nature?

6 The Road to Unification I Physics, in Greek, meaning nature, is the study of matter and its motion through spacetime and all that derives from these, such as energy and force. How to describe the nature? Experiment Aspects: Epoch of precise measurement at a very high energy

7 The Road to Unification I Physics, in Greek, meaning nature, is the study of matter and its motion through spacetime and all that derives from these, such as energy and force. How to describe the nature? Experiment Aspects: Epoch of precise measurement at a very high energy Particle physics: LHC (14 TeV), 100 TeV Colider in China?... Cosmology: PLANCK, BICEP2,...

8 The Road to Unification II High Energy Theoretical Aspects:

9 The Road to Unification II High Energy Theoretical Aspects: Microcosmic structure: Quantum Field Theory (Algebra) Electric force, Weak force, Strong force.

10 The Road to Unification II High Energy Theoretical Aspects: Microcosmic structure: Quantum Field Theory (Algebra) Electric force, Weak force, Strong force. Spacetime structure: General Relativity (Geometry) Gravity force.

11 The Road to Unification II High Energy Theoretical Aspects: Microcosmic structure: Quantum Field Theory (Algebra) Electric force, Weak force, Strong force. Spacetime structure: General Relativity (Geometry) Gravity force. Problems in Standard Model? Gauge hirachy problem? Dark Matter and Dark Energy? UV complete of particle theory? Cosmological constant?

12 The Road to Unification II High Energy Theoretical Aspects: Microcosmic structure: Quantum Field Theory (Algebra) Electric force, Weak force, Strong force. Spacetime structure: General Relativity (Geometry) Gravity force. Problems in Standard Model? Gauge hirachy problem? Dark Matter and Dark Energy? UV complete of particle theory? Cosmological constant? Unification? Quantum Gravity Algebraic Geometry

13 The Road to Unification III Superstring Theory is the leading candidate for a fundamental unifying theory of all known forces in nature, including gravity.

14 The Road to Unification III Superstring Theory is the leading candidate for a fundamental unifying theory of all known forces in nature, including gravity. Perturbative finiteness of the theory Microscopic description of black hole entropy Remarkable way to describe cosmology Applications of Gauge/Gravity duality Incorporates the key ingredients of all interactions in nature

15 The Road to Unification III Superstring Theory is the leading candidate for a fundamental unifying theory of all known forces in nature, including gravity. Perturbative finiteness of the theory Microscopic description of black hole entropy Remarkable way to describe cosmology Applications of Gauge/Gravity duality Incorporates the key ingredients of all interactions in nature However, a consistent superstring theory is 10 dimension. From string to the real wold: 10D 4D String Compactification

16 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

17 Reference I Green, Schwarz, Witten, Superstring Theory, 1987 J.Polchinski, String theory 1998 Becker,Becker, Schwarz, String theory and M-theory: A modern introduction, 2007 Ibanez, Uranga, String theory and particle physics: An introduction to string phenomenology,2012 Blumenhagen, Lüst, Theisen, Basic concepts of string theory, 2013

18 General Properties Properties of Bosonic String Spacetime dimension D = 26 (Conformal anomaly free) Massless spectrum including graviton Tachyon m 2 < 0 No Fermion

19 General Properties Properties of Bosonic String Spacetime dimension D = 26 (Conformal anomaly free) Massless spectrum including graviton Tachyon m 2 < 0 No Fermion Properties of Supersting Include Fermion Spacetime dimension D = 10 To remove tachyon, one need GSO-projection SUSY

20 Worldsheet action String is a field of dimensional world-sheet: X µ (τ, σ) : world sheet Σ spacetime M with µ = 1,..., D

21 Worldsheet action String is a field of dimensional world-sheet: X µ (τ, σ) : world sheet Σ spacetime M with µ = 1,..., D From the simplest relativistic free particle action, we can end up with a Lorentz covariance manifested, square-root free string require, called Polyakov action (Bose): S P = T dσdτ hh αβ g µν (X ) α X µ β X ν 2 Σ

22 Worldsheet action String is a field of dimensional world-sheet: X µ (τ, σ) : world sheet Σ spacetime M with µ = 1,..., D From the simplest relativistic free particle action, we can end up with a Lorentz covariance manifested, square-root free string require, called Polyakov action (Bose): S P = T dσdτ hh αβ g µν (X ) α X µ β X ν 2 Σ Poincare invariant: X µ = Λ µ νx ν + a µ World sheet reparameterization invariant: X µ (τ, σ ) = X µ (τ, σ) τ d σ with word sheet metric h ab = c τ b σ h a cd Weyl Invariance in 2D: X µ (τ, σ) = X µ (τ, σ) with h ab = e2ω h cd (τ, σ) (z, z) CFT with X (z, z) 26 scalars X µ cancels the conformal anomaly D = 26

23 Quantization I-Boundary Condition ) S P = T d 2 z g µν (X ) ( X µ X ν + α Σ 2 (ψµ ψ ν + ψ µ ψ ν ) Superconformal anomaly is canceled in the presenc of 10 fields and thier superpartners D = 10 Ramond-Neveu-Schwarz formulism

24 Quantization I-Boundary Condition ) S P = T d 2 z g µν (X ) ( X µ X ν + α Σ 2 (ψµ ψ ν + ψ µ ψ ν ) Superconformal anomaly is canceled in the presenc of 10 fields and thier superpartners D = 10 Ramond-Neveu-Schwarz formulism General boundary condition z = e τ+iσ Neumann: X r σ=0,2π = 0 for r = 0,..., p Dirichlet: δx s σ=0,2π = 0 for s = p + 1,..., 9 Ramond: ψ µ (z + 2π) = +ψ µ (z) Neveu-Schwarz: ψ µ (z + 2π) = ψ µ (z)

25 Quantization I-Boundary Condition ) S P = T d 2 z g µν (X ) ( X µ X ν + α Σ 2 (ψµ ψ ν + ψ µ ψ ν ) Superconformal anomaly is canceled in the presenc of 10 fields and thier superpartners D = 10 Ramond-Neveu-Schwarz formulism General boundary condition z = e τ+iσ Neumann: X r σ=0,2π = 0 for r = 0,..., p Dirichlet: δx s σ=0,2π = 0 for s = p + 1,..., 9 Ramond: ψ µ (z + 2π) = +ψ µ (z) Neveu-Schwarz: ψ µ (z + 2π) = ψ µ (z) Closed string require periodicity boundary condition X µ (τ, σ + 2π) = X µ (τ, σ) EOM X µ (z, z) = X µ L (z) + X R ν( z) Chiral and anti-chiral fermion: ψ µ (z, z) = ψ µ (z), ψ µ (z, z) = ψ µ ( z) Gluing left-moves and right-moves together: R-R, R-NS, NS-R, NS-NS sectors

26 Quantization II-Modes Expansion Diff b.c Diff mod expansion Neumann: X µ L (z) = xµ 0 2 i α 2 pµ 0,L ln(z) + i α 2 0 n Z αµ n n z n Ramond: ψ µ L (z) = n Z d µ n z n 1 2 Neveu-Schwarz: ψ µ L (z) = r Z+ 1 b n µ z r 1 2 2

27 Quantization II-Modes Expansion Diff b.c Diff mod expansion Neumann: X µ L (z) = xµ 0 2 i α 2 pµ 0,L ln(z) + i α 2 0 n Z αµ n n z n Ramond: ψ µ L (z) = n Z d µ n z n 1 2 Neveu-Schwarz: ψ µ L (z) = r Z+ 1 b n µ z r Canonical Quantization of closed string Bose: [α m, µ αn] ν = [ α m, µ α n] ν = mδ m+n,0 η µν, [x µ 0, pν 0 ] = iηµν Ramond: {d m, µ dn ν } = { d m, µ d n ν } = δ m+n,0 η µν Neveu-Schwarz: {b µ r, b ν s } = { b µ r, b ν s } = δ r+s,0 η µν

28 Quantization II-Modes Expansion Diff b.c Diff mod expansion Neumann: X µ L (z) = xµ 0 2 i α 2 pµ 0,L ln(z) + i α 2 0 n Z αµ n n z n Ramond: ψ µ L (z) = n Z d µ n z n 1 2 Neveu-Schwarz: ψ µ L (z) = r Z+ 1 b n µ z r Canonical Quantization of closed string Bose: [α m, µ αn] ν = [ α m, µ α n] ν = mδ m+n,0 η µν, [x µ 0, pν 0 ] = iηµν Ramond: {d m, µ dn ν } = { d m, µ d n ν } = δ m+n,0 η µν Neveu-Schwarz: {b µ r, b ν s } = { b µ r, b ν s } = δ r+s,0 η µν Spectrum Dynamical degrees of freedom 8 transverse directions µ = 2,..., 9 Light-cone gauge NS ground state is tachyonic, R ground state is degenerate GSO-projection

29 GSO-Projection Quantization III-GSO Projection G = { ( 1) F L +1 NS sector Γ 11 ( 1) F L R sector Keep states with G NS = 1 F L : Number of left-moving fermion NS-sector: Tachyon is projected out, ground state trans as 8 v R-sector: Choose a chirality G R = ±1 for an 8 s,c -component Majorana-Weyl spinor grounds states a, ȧ Consistent with Modular Invariant of Scattering Amplitude, Superconformal anomaly free in critical dimension 10gD.

30 GSO-Projection Quantization III-GSO Projection G = { ( 1) F L +1 NS sector Γ 11 ( 1) F L R sector Keep states with G NS = 1 F L : Number of left-moving fermion NS-sector: Tachyon is projected out, ground state trans as 8 v R-sector: Choose a chirality G R = ±1 for an 8 s,c -component Majorana-Weyl spinor grounds states a, ȧ Consistent with Modular Invariant of Scattering Amplitude, Superconformal anomaly free in critical dimension 10gD. G NS G R G R Left-moving Right-moving Type IIA Type IIB

31 Type II Spectrum NS-NS bµ 0 1 NS b ν 0 1 NS dilaton φ, B-field B µν, Graviton g µν v 8 v R-R a b IIB: C 0, C 2, C 4 8 s 8 s C 4 self-dual a ȧ IIA: C 1, C 3 8 s 8 c NS-R b µ NS a dilatino, gravitino 8 v 8 s

32 Type IIB Effective Action In Einstein frame (g E µν = e φ/2 g s µν) ( S (10D) 1 IIB = 2 R dφ dφ + 1 ) 4 e φ H 3 H 3 (e 2φ dc 0 dc 0 + e φ F 3 F ) F 5 F C 4 H 3 F 3 H 3 = d ˆB 2, F 3 = dc 2 C 0 db 2 F 5 = dc db 2 C B 2 dc 2. where the self-dual condition F 5 = F 5 is inposed at EOM level.

33 Other Superstring Theory SUSY of Real Supercharge Heterotic SO(32) N = 1 SUSY in 10D 16 Heterotic E 8 E 8 N = 1 SUSY in 10D 16 Type I string N = 1 SUSY in 10D 16 Type IIA IIB N = 2 SUSY in 10D 32 M-theory N = 1 SUSY in 11D 32

34 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

35 Reference II Blumenhagen, Kors, Lüst, Stieberger, Phys.Rept. 445 (2007) 1-193, Grana, Flux compactifications in string theory: A Comprehensive review, Phys.Rept.423 (2006) Douglas, Kachru, Flux compactification, Rev.Mod.Phys.79 (2007) Denef, Les Houches Lectures on Constructing String Vacua, Fortsch.Phys.53 (2005) Kachru,Kallosh,Linde,Trivedi, De Sitter vacua in string theory,phys.rev.d68 (2003) Balasubramanian, Berglund, Conlon, Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactification, JHEP 0503(2005) 007

36 Two Aspects of String Phenomenology Two longstanding problems of realistic string model building Global Issues: Complete compact Calabi-Yau compactification with, moduli stabilization, tadpole cancellation and Freed-Witten anomaly cancellation, SUSY breaking, realizing ds solution and cosmology... Local Issues: Local sets of lower dimensional D-branes, which are localised in some area of the Calabi-Yau and reproduce chiral spectrum, tree-level Yukawa couplings, gauge couplings... However, it is fair to say that models developed so far are still far from being realistic.

37 Flux Compactification I From string to the real wold: 10D 4D What we want: N = 1 Supersymmetry with chiral spectrum Best under control: N = 1 Flux Compactification

38 Flux Compactification I From string to the real wold: 10D 4D What we want: N = 1 Supersymmetry with chiral spectrum Best under control: N = 1 Flux Compactification Background Flux (in Type II): Neveu-Schwarz flux: H 3 = db 2, dh 3 = 0. Ramond flux: F p+1 = dc p, df p+1 = 0. Metric flux: F k ij from T-dual of H ijk. Non-geometric flux: T-duality with Buscher rules. H ijk T k Fij k T j Q i jk T i R ijk.

39 Flux Compactification II M 4 X : ds 2 = e A(y) ḡ µν (x)dx µ dx ν + ḡ mn (y)dy m dy n where µ, ν = 1,..., 4. m, n run away the inner coordinates

40 Flux Compactification II M 4 X : ds 2 = e A(y) ḡ µν (x)dx µ dx ν + ḡ mn (y)dy m dy n where µ, ν = 1,..., 4. m, n run away the inner coordinates 1. Find a compactified space X, such as M 4 satisfy: Maximal Symmetry, i.e. M 4 = {ds 4, AdS 4, Minks} Chiral N = 1 SUSY in 4D, i.e. 4 real supercharge

41 Flux Compactification II M 4 X : ds 2 = e A(y) ḡ µν (x)dx µ dx ν + ḡ mn (y)dy m dy n where µ, ν = 1,..., 4. m, n run away the inner coordinates 1. Find a compactified space X, such as M 4 satisfy: Maximal Symmetry, i.e. M 4 = {ds 4, AdS 4, Minks} Chiral N = 1 SUSY in 4D, i.e. 4 real supercharge 2. Find light perturbative around background, i.e. Moduli

42 Flux Compactification II M 4 X : ds 2 = e A(y) ḡ µν (x)dx µ dx ν + ḡ mn (y)dy m dy n where µ, ν = 1,..., 4. m, n run away the inner coordinates 1. Find a compactified space X, such as M 4 satisfy: Maximal Symmetry, i.e. M 4 = {ds 4, AdS 4, Minks} Chiral N = 1 SUSY in 4D, i.e. 4 real supercharge 2. Find light perturbative around background, i.e. Moduli 3. Get the effective theory of the chiral spectrum and moduli

43 Flux Compactification II M 4 X : ds 2 = e A(y) ḡ µν (x)dx µ dx ν + ḡ mn (y)dy m dy n where µ, ν = 1,..., 4. m, n run away the inner coordinates 1. Find a compactified space X, such as M 4 satisfy: Maximal Symmetry, i.e. M 4 = {ds 4, AdS 4, Minks} Chiral N = 1 SUSY in 4D, i.e. 4 real supercharge 2. Find light perturbative around background, i.e. Moduli 3. Get the effective theory of the chiral spectrum and moduli 4. Based on some concrete model study the particle phenomenology and cosmology

44 Flux Compactification III Four dimensional N = 1 supersymmetry flux compactification: Het string on CY 3 Type IIA/B on CY 3 with orientifold (include Type I = Type IIB orientifold with O9 plane) F-theory on CY 4 M-theory on CY 3 S 1 /Z 2 or on M 7 with G 2 holonomy

45 Flux Compactification III Four dimensional N = 1 supersymmetry flux compactification: Het string on CY 3 Type IIA/B on CY 3 with orientifold (include Type I = Type IIB orientifold with O9 plane) F-theory on CY 4 M-theory on CY 3 S 1 /Z 2 or on M 7 with G 2 holonomy Calabi-Yau threefold CY 3 or fourfold CY 4. Extra massless spectrum in four dimensions. Contradict with fifth force experiments and the standard paradigm of early cosmology, i,e. interfering with Big Bang Nucleosynthesis. Q: Where is these massless spectrum comes from?

46 Moduli The existence of deformations of the underlying geometry (Moduli) The size and shape of the internal manifold is dynamically determined by the vacuum expectation values of moduli. Moduli of Calabi-Yau X : g g + δg s.t. R m n (g + δg) = 0. For Kähler manifold, under proper gauge (δg) = 0, it decouples Kähler moduli: δg m n = iv i ( ˆD i ) m n, i = 1,..., h 11 (X ) Complex moduli: δg mn = i Ω 2 Ūa ( χ a ) Ω p q m p q n, a = 1,..., h 12 (X )

47 Type IIB orientifolds Focus on Type IIB orientifold. Why?

48 Type IIB orientifolds Focus on Type IIB orientifold. Why? D-brane provide non-abelian gauge symmetry and chiral matter. As a localized sources ends the no-go theorems which prevents turn on background fluxes. In some situation, Complex, dilaton moduli decoupled with Kähler moduli. Background fluxes fix complex moduli and dilaton at tree level. Kähler moduli can be stabilized by non-perturbative effects (KKLT, LARGE Volume scenario). There can exists more flux than moduli. The theory are more controlled. Realise inflationary model (moduli inflation). Gukov, Vafa, Witten, Kachru, Kallosh, Linde, Trivedi, Blumenhagen, Lüst, Becker, Haack, Louis, Balasubramanian, Berglund, Conlon, Quevedo

49 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

50 Calabi-Yau Space Q: What is Calabi-Yau and Why it appears?

51 Calabi-Yau Space Q: What is Calabi-Yau and Why it appears? Calabi-Yau n-folds is a complex n-dimentional compacted Kähler Manifold satisfied: Its first chern class vanish, i.e c 1 (M) = 0 H 2 (M, Z). The normal bundle K M = n T (1, 0)(M) is trivial since c 1 (K M ) = c 1 (M) There exist a unique no where vanishing holomophic n-form, Ω n Ω n,0 (M), dω n = 0 The Ricci tensor vanish, i.e. R mn = 0 The holonomy group of M is SU(n)

52 The Original of Calabi-Yau I Under M 4 X 6, the Lorentz group SO(1, 9) splits into SO(1, 9) SO(1, 3) SO(6) The corresponding spinor representation 16 SO(1, 9) splits into: 16 (2, 4) ( 2, 4) where 4 and 4 is the weyl spinor of SO(6) while 2, 2 SO(1, 3) However, no singlet in SO(6) no SUSY

53 The Original of Calabi-Yau I Under M 4 X 6, the Lorentz group SO(1, 9) splits into SO(1, 9) SO(1, 3) SO(6) The corresponding spinor representation 16 SO(1, 9) splits into: 16 (2, 4) ( 2, 4) where 4 and 4 is the weyl spinor of SO(6) while 2, 2 SO(1, 3) However, no singlet in SO(6) no SUSY Q: What is the consequence if we require SUSY?

54 The Original of Calabi-Yau I Under M 4 X 6, the Lorentz group SO(1, 9) splits into SO(1, 9) SO(1, 3) SO(6) The corresponding spinor representation 16 SO(1, 9) splits into: 16 (2, 4) ( 2, 4) where 4 and 4 is the weyl spinor of SO(6) while 2, 2 SO(1, 3) However, no singlet in SO(6) no SUSY Q: What is the consequence if we require SUSY? < δ ɛ Fermion > = < M ɛ > + <..., Bosonic, H 3, >= 0. < M ɛ > M ɛ = 0 Global well defined Killing spinor, s.t. M ɛ = 0 means ω

55 The Original of Calabi-Yau II ɛ(x, y) = ξ(x) η(y)

56 The Original of Calabi-Yau II ɛ(x, y) = ξ(x) η(y) µ ξ = 0 [ µ, ν ]ξ = 1 4 R µνρσγ ρσ ξ = 0 By the requirement of Maximal Symmetry, R µνρσ = R 12 (g µρg νσ g µσ g ρν ) R = 0, i.e. Minkowski spacetime m η = 0 [ m, n ]η = 1 4 R mnpqγ pq η = 0 Multiply by Γ p get R mn Γ n η = 0 R mn = 0, i.e. Ricci Flat

57 The Original of Calabi-Yau II ɛ(x, y) = ξ(x) η(y) µ ξ = 0 [ µ, ν ]ξ = 1 4 R µνρσγ ρσ ξ = 0 By the requirement of Maximal Symmetry, R µνρσ = R 12 (g µρg νσ g µσ g ρν ) R = 0, i.e. Minkowski spacetime m η = 0 [ m, n ]η = 1 4 R mnpqγ pq η = 0 Multiply by Γ p get R mn Γ n η = 0 R mn = 0, i.e. Ricci Flat Define (1,1)-form J and (3,0)-form Ω through η: η ± γmn η ± = ± i 2 Jmn, η γmnp η + = i 2 Ωmnp, η + γmnp η = ± i 2 Ω mnp, dj = 0, dω = 0 J is Kähler form and X is Calabi-Yau threefold

58 The Original of Calabi-Yau III From representations, it equivalent to find a inner spacetime with holonomy group SU(3) SO(6) = SU(4) 4 SO(6) = SU(4) split under SU(3) as SO(6) singlet η(y) is also no where vanishing covariant constant spinor Calabi-Yau

59 The Original of Calabi-Yau III From representations, it equivalent to find a inner spacetime with holonomy group SU(3) SO(6) = SU(4) 4 SO(6) = SU(4) split under SU(3) as SO(6) singlet η(y) is also no where vanishing covariant constant spinor Calabi-Yau Similarly Compactified with holonomy SU(2), N = 2 in 4D Compactified on torus T 6 N = 4 in 4D Start from N = 2, compactified on T 6 N = 8 in 4D Start from N = 2, compactified on K 3 T 2 N = 4 in 4D, compactified on CY 3 N = 2 in 4D

60 Bulk Spectrum Under R 3,1 X 6, the EOM of scalar φ satisfied 10 φ = ( ) φ = ( 4 + m 2) φ = 0. the number of 4D massless field is determined by the harmonic form of X 6, i.e. the Zero modes of 6. In fact, all the massless field in 4D is determined by harmonic (p, q)-form, which in CY 3 case, is H p,q (X ).

61 Bulk Spectrum Under R 3,1 X 6, the EOM of scalar φ satisfied 10 φ = ( ) φ = ( 4 + m 2) φ = 0. the number of 4D massless field is determined by the harmonic form of X 6, i.e. the Zero modes of 6. In fact, all the massless field in 4D is determined by harmonic (p, q)-form, which in CY 3 case, is H p,q (X ). Hodge -duality H p,q (X ) = H 3 p,3 q (X ) Complex conjugate H p,q = H q,p (M) h 0,0 h 1,0 h 0,1 h 2,0 h 1,1 h 0,2 h 3,0 h 2,1 h 1,2 h 0,3 h 3,1 h 2,2 h 1,3 h 3,2 h 2,3 h 3,3 = h 1,1 0 1 h 1,2 h 1,2 1 0 h 1,

62 Moduli Revisit Moduli of Calabi-Yau X : g g + δg s.t. R m n (g + δg) = 0. Kähler moduli: δg m n = iv i ( ˆD i ) m n, i = 1,..., h 1,1 (X ) In Type II theory, complexfied Kähler form J c = J + ib 2 = t i ˆD i Kähler moduli space {t i } special Kähler manifold M K h 1,1 Kähler form can be written as J = ig m n dy m dȳ n. g + δg positive definite Kähler cone condition: C J > 0, S J J > 0, X J J J > 0

63 Moduli Revisit Moduli of Calabi-Yau X : g g + δg s.t. R m n (g + δg) = 0. Kähler moduli: δg m n = iv i ( ˆD i ) m n, i = 1,..., h 1,1 (X ) In Type II theory, complexfied Kähler form J c = J + ib 2 = t i ˆD i Kähler moduli space {t i } special Kähler manifold M K h 1,1 Kähler form can be written as J = ig m n dy m dȳ n. g + δg positive definite Kähler cone condition: C J > 0, S J J > 0, X J J J > 0 Complex moduli: δg mn = i Ω 2 Ūa ( χ a ) Ω p q m p q n, a = 1,..., h 12 (X ) Complex moduli space special Kähler manifold M cs h 1,2 At tree level, we have M = M cs h 1,2 M K h 1,1

64 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

65 Orientifold I Q: Why orientifold? Break half SUSY to get N = 1. Want to introduce open string modes, if only D-brane.

66 Orientifold I Q: Why orientifold? Break half SUSY to get N = 1. Want to introduce open string modes, if only D-brane. Two category: { Ωp σ with σ (J) = J, σ (Ω 3 ) = Ω 3, O5/O9 O = ( ) F L Ω p σ with σ (J) = J, σ (Ω 3 ) = Ω 3, O3/O7 Ω p : World-sheet parity F L : Left-moving space-time fermion number σ: Z 2 involution on Calabi-Yau

67 Type of orientifold Orientifold II Dim of fixed point locus in X O-plane D-brane σ (Ω 3 ) = Ω 3 2 O5-plane D5-brane 6 O9-plane D9-brane σ (Ω 3 ) = Ω 3 0 O3-plane D3-brane 4 O7-plane D7-brane

68 Type of orientifold Orientifold II Dim of fixed point locus in X O-plane D-brane σ (Ω 3 ) = Ω 3 2 O5-plane D5-brane 6 O9-plane D9-brane σ (Ω 3 ) = Ω 3 0 O3-plane D3-brane 4 O7-plane D7-brane φ, B 2, g µν, C 0, C 2, C 4, two dilatino, two gravitino Q: What is the spectrum left under oreintifold?

69 Type of orientifold Orientifold II Dim of fixed point locus in X O-plane D-brane σ (Ω 3 ) = Ω 3 2 O5-plane D5-brane 6 O9-plane D9-brane σ (Ω 3 ) = Ω 3 0 O3-plane D3-brane 4 O7-plane D7-brane φ, B 2, g µν, C 0, C 2, C 4, two dilatino, two gravitino Q: What is the spectrum left under oreintifold? ( ) F L : NS-NS (+), NS-R (+), R-NS (-), R-R (-) Ω : NS-NS, only sym part of tensor product is even under parity transformation, φ, g µν R-R, two dilatino, two gravitino combined to one dilatino and one gravitino, provide = 64 fermion d.o.f. φ, g µν provide Bose d.o.f C 2 left, 28 d.o.f

70 Orientifold III Four dimensional orientifold invariant closed string fields. ( ) F L Ω p σ φ C 0 + g µν B 2 + C 2 + C 4 + J = t α ˆD α, α = 1,... h 1,1 + (X ) C 2 = c a ˆD a, B 2 = b a ˆD a a = 1,... h 1,1 (X ) C 4 = Q α 2 ˆD α + V α α α + U α β α + ρ α D α D α and D a is a basis of H 1,1 + (X ) and H 1,1 (X ). (α α, β α ) is a real symplectic basis of H+(X 3 ).

71 Closed Spectrum of Type IIB on CY 3 /σ h 2,1 Ūã chiral multiplets h 1,1 + (t α, ρ α) h 1,1 (b a, c a ) 1 (φ, C 0) vector multiplet h 2,1 + V α gravity multiplet 1 g µν N = 1 massless bosonic spectrum of Type IIB Calabi Yau orientifold with O3/O7 plane.

72 Low Energy Effective Action I Supergravity form: Kähler potentail K, Holomorphic superpotential W, holomorphic gauge-kinetic coupling function f. V = e K (K I J D I WD J W 3 W 2 ) (Re f ) 1ab D a D b with the tree-level superpotential, Gukov, Vafa, Witten... W = G 3 Ω 3. X K I J is the inverse of Kähler metric K I J = I JK(Φ, Φ), Covariant derivative D I W = I W + W I K. [ ] D a = K I + W I W (T a ) IJ Φ J with T a here the gauge generator.

73 Low Energy Effective Action II Moduli spectrum from dimension reduction are not necessarily the good Kähler coordinate appearing in SUGRA formalism. τ = C 0 + ie φ, Uã = uã + iv ã, G a = c a τ b a, T α = 1 2 κ αβγ t β t γ + i (ρ α 12 ) κ αab c a b b 1 4 eφ κ αab Ḡ a (G + Ḡ) b. Becker, Jockers, Louis, Grimm...

74 Low Energy Effective Action II Moduli spectrum from dimension reduction are not necessarily the good Kähler coordinate appearing in SUGRA formalism. τ = C 0 + ie φ, Uã = uã + iv ã, G a = c a τ b a, T α = 1 2 κ αβγ t β t γ + i (ρ α 12 ) κ αab c a b b 1 4 eφ κ αab Ḡ a (G + Ḡ) b. Becker, Jockers, Louis, Grimm... k αab = k αβγ = X X ˆD α ˆD a ˆD b, ˆD α ˆD β ˆD γ. Again, ˆD α H 1,1 (X ). It is Poincare dual is the codimension one hypersurface in CY 3, called divisor D α. Its imagine under orientifold denotes D α.

75 Low Energy Effective Action III Expand the Kähler form J in a basis { ˆD i } of H 1,1 (X ) as J = t i ˆD i the Calabi-Yau volume V measured with Einstein frame metric gµν E = e φ/2 gµν s and in units of l s = 2π α is: V = 1 J J J = κ αβγt α t β t γ X To leading order, the low-energy tree-level Kähler potential is: K = ln ( S + S ) ( ) ln i Ω 3(U) Ω 3(Ū) 2 ln (V (T α)) X h 1,1 i=1 Complex structure deformations do not mix with the other scalars M = M cs h 12 M K h 11 +1

76 Tadpole Cancellation in Type IIB orientifolds D7-brane tadpole cancellation: a N a( ˆD a + ˆD a) = 8Ô7

77 Tadpole Cancellation in Type IIB orientifolds D7-brane tadpole cancellation: a N a( ˆD a + ˆD a) = 8Ô7 Freed-Witten anomaly: c 1 (L) i B c 1(K D a ) H 2 (D a, Z) When the divisor D a wrapped by a D7 brane is non-spin,i.e, c 1 (K D a ) 0 mod 2, K D is the canonical bundle of D a.

78 Tadpole Cancellation in Type IIB orientifolds D7-brane tadpole cancellation: a N a( ˆD a + ˆD a) = 8Ô7 Freed-Witten anomaly: c 1 (L) i B c 1(K D a ) H 2 (D a, Z) When the divisor D a wrapped by a D7 brane is non-spin,i.e, c 1 (K D a ) 0 mod 2, K D is the canonical bundle of D a. D3-brane tadpole cancellation: N D3 + N flux 2 + N gauge = N O3 4 + χ(d O7) + 12 a N a χ o(d a) + χ o(d a) 48 with N flux = 1 (2π) 4 α 2 H3 F 3, N gauge = a 1 8π 2 D a trf 2 a

79 Tadpole Cancellation in Type IIB orientifolds D7-brane tadpole cancellation: a N a( ˆD a + ˆD a) = 8Ô7 Freed-Witten anomaly: c 1 (L) i B c 1(K D a ) H 2 (D a, Z) When the divisor D a wrapped by a D7 brane is non-spin,i.e, c 1 (K D a ) 0 mod 2, K D is the canonical bundle of D a. D3-brane tadpole cancellation: N D3 + N flux 2 + N gauge = N O3 4 + χ(d O7) + 12 a N a χ o(d a) + χ o(d a) 48 with N flux = 1 (2π) 4 α 2 H3 F 3, N gauge = a 1 8π 2 D a trf 2 a D5-brane tadpole cancellation: If H (X 2 ) 0 with some non-trivial gauge-flux turned on, for all ω H (M), 2 ω (trf a D a + trf a D a ) = 0 a M

80 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

81 Overview of Moduli Stabilisation The problem of giving masses to these unwanted scalars is called Moduli Stabilisation.

82 Overview of Moduli Stabilisation The problem of giving masses to these unwanted scalars is called Moduli Stabilisation. Marvelous, if

83 Overview of Moduli Stabilisation The problem of giving masses to these unwanted scalars is called Moduli Stabilisation. Marvelous, if What is the correct vacuum? and Why? Is it picked out by some special mathematical property, or just an environmental accident of our particular corner of the Universe?

84 Overview of Moduli Stabilisation The problem of giving masses to these unwanted scalars is called Moduli Stabilisation. Marvelous, if What is the correct vacuum? and Why? Is it picked out by some special mathematical property, or just an environmental accident of our particular corner of the Universe? : Best understood in Type IIB orientifold with LARGE Volume Scenario (LVS). Becker, Becker, Haack, Louis, Balasubramanian, Berglund, Conlon, Quevedo

85 LARGE Volume Scenario 1. Stabilize complex moduli and dilaton modulus by perturbative contribution, i.e, flux generated superpotential. Gukov, Vafa, Witten 2. The scalar potentail generated by GVW-superpotential for the Kähler moduli is still flat due to the no-scale structure. 3. Stabilize Kähler moduli by all possible perturbative and non-perturbative corrections.

86 LARGE Volume Scenario 1. Stabilize complex moduli and dilaton modulus by perturbative contribution, i.e, flux generated superpotential. Gukov, Vafa, Witten 2. The scalar potentail generated by GVW-superpotential for the Kähler moduli is still flat due to the no-scale structure. 3. Stabilize Kähler moduli by all possible perturbative and non-perturbative corrections. KKLT: Non-perturbative correction to superpotential Fine-tune tree level superpotential W 0. Kachru, Kallosh, Linde, Trivedi LVS: Perturbative α 3 correction to Kähler potential Break no-scale structure. Non-perturbative contribution to superpotentail, i.e, gaugino condensation or Euclidean D3 brane instantons W 0 O(1) naturely. For τ = ReT, V e aτ with τ 1. δv α δv np O( 1 V 3 ) Becker, Becker, Haack, Louis, Balasubramanian, Berglund, Conlon, Quevedo

87 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

88 Dilaton and Complex Moduil Stabilisation I Start with Gukov-Vafa-Witten Fluxed-superpotential W τ,u = G 3 Ω, G 3 = F 3 τh 3 X A scalar potential : { V = e K K τ τ D τ WD τ W + K UŪ D U WD W Ū + K α β D αwd β W 3 W 2} D αw = W T α + W K T α W α + WK α, D β W = W T + β W K T β W β + W K β

89 Dilaton and Complex Moduil Stabilisation II { V = e K K τ τ D τ WD τ W + K UŪ D U WD W ( ) Ū + K i j K i K j 3 W 2} No-scale structure: ( ) 1 2 K tree K tree K tree T i T j T i T j = 3 A scaler potential dose not dependent on Kähler moduli.

90 Dilaton and Complex Moduil Stabilisation II { V = e K K τ τ D τ WD τ W + K UŪ D U WD W ( ) Ū + K i j K i K j 3 W 2} No-scale structure: ( ) 1 2 K tree K tree K tree T i T j T i T j = 3 A scaler potential dose not dependent on Kähler moduli. D τ W = D U W = 0 to minimize the scalar potential. Stabilise dilaton and complex moduli at tree level

91 Dilaton and Complex Moduil Stabilisation II { V = e K K τ τ D τ WD τ W + K UŪ D U WD W ( ) Ū + K i j K i K j 3 W 2} No-scale structure: ( ) 1 2 K tree K tree K tree T i T j T i T j = 3 A scaler potential dose not dependent on Kähler moduli. D τ W = D U W = 0 to minimize the scalar potential. Stabilise dilaton and complex moduli at tree level After integrate out these heavy modes, the no-scale structure still hold. The direction of Kähler moduli is still flat at tree level.

92 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

93 Quantum Correction I N = 1 non-renormalisable theorem, superpotential only get non-perturbative correction. For scalar potential: K = K tree + K p + K np, W = W tree + W np. δv = δv α + δv np.

94 Quantum Correction I N = 1 non-renormalisable theorem, superpotential only get non-perturbative correction. For scalar potential: K = K tree + K p + K np, W = W tree + W np. δv = δv α + δv np. W np : D-brane instanton (D3-brane wrapping on the internal four-cycle divisor E a ) Gaugino condensation (D7-brane wrapping on the internal four-cycle divisor D a )

95 Quantum Correction I N = 1 non-renormalisable theorem, superpotential only get non-perturbative correction. For scalar potential: K = K tree + K p + K np, W = W tree + W np. δv = δv α + δv np. W np : D-brane instanton (D3-brane wrapping on the internal four-cycle divisor E a ) Gaugino condensation (D7-brane wrapping on the internal four-cycle divisor D a ) W = G 3 Ω + X E = W 0 + W np. A E (Uã, G a, F E,...) e aα E Tα

96 Quantum Correction II NOT all D-brane instanton and gaugino condensation can contribute superpotential. (Zero modes) The instanton must have exact two zero modes (O(1)-instanton). Mathematically, the divisor D a wrapped by the D-brane should be rigid. Necessary condition: 1 = χ 0 (D a ) := 3 p=0 ( 1)p h p,0 (D a ) Sufficient condition: h 0,0 (D a ) = 1, h 0,p (D a ) = 0 p 1 The same condition for gaugino condensation

97 Zero modes for O(N) instanton Quantum Correction III Universal zero-modes: X µ, θ α and τ α Deformation zero-modes: b 1 = dimh 1 (E; O): # of Wilson-line moduli. b 2 = dimh 2 (E; O): # of normal deformations. Charged (matter) zero-modes: Arising from the intersection of D-brane and instanton E-brane

98 Quantum Correction IV For contributions to the holomorphic superpotential, the anti-holomorphic τ α zero modes should be removed, i,e. O(1)-instanton.

99 Quantum Correction IV For contributions to the holomorphic superpotential, the anti-holomorphic τ α zero modes should be removed, i,e. O(1)-instanton. Given a CFT, one can distinguish O/SP instantons. Assume SP(N) on D-brane: Placing an E3 instanton right on top of an O7-plane gives an SP-instanton. For a divisor E which is parallel to the O7 plane, in the sense E O7 =, the respective instanton is expected to be SP. If the divisor E intersects the O7-plane in a curve, we have four additional Neumann-Dirichlet type boundary conditions, which will changes the former SP- to an O-projection. Thus, we have an O-instanton. Gimon, Polchinski

100 Quantum Correction V Kähler potential: K np K p. The leading correction is α 3 correction, which comes from R 4 -term in supergravity. ( ) K = 2 ln V + ξ = 2 ln V ξ (1/V V + O 2) χ(x )ζ(3) 2g 3/2 s g 3/2 s with ξ =, χ(x ) = 2 (h 2(2π) 3 1,1 h 2,1 ) the Euler characteristic of Calabi-Yau, Riemann zeta function ζ(3) k=1 1/k3 1.2.

101 Quantum Correction V Kähler potential: K np K p. The leading correction is α 3 correction, which comes from R 4 -term in supergravity. ( ) K = 2 ln V + ξ = 2 ln V ξ (1/V V + O 2) χ(x )ζ(3) 2g 3/2 s g 3/2 s with ξ =, χ(x ) = 2 (h 2(2π) 3 1,1 h 2,1 ) the Euler characteristic of Calabi-Yau, Riemann zeta function ζ(3) k=1 1/k e K 1/V 2, in large volume limit, the expansion of α equivalent to large volume expansion, i.e. δv α δv np.

102 Quantum Correction V Kähler potential: K np K p. The leading correction is α 3 correction, which comes from R 4 -term in supergravity. ( ) K = 2 ln V + ξ = 2 ln V ξ (1/V V + O 2) χ(x )ζ(3) 2g 3/2 s g 3/2 s with ξ =, χ(x ) = 2 (h 2(2π) 3 1,1 h 2,1 ) the Euler characteristic of Calabi-Yau, Riemann zeta function ζ(3) k=1 1/k e K 1/V 2, in large volume limit, the expansion of α equivalent to large volume expansion, i.e. δv α δv np. D-term (V D O( 1 ) ) will set to be zero if h 1,1 V 2 = 0 or by choosing proper orientifold odd field b 2.

103 Kähler Moduli Stabilisation e.g. h 1,1 = 0, W np = A i e a i T i τ i = Re T i. From N = 1 supergravity V = V F + V D : i V F = e K ( K I JD I WD J W 3 W 2 ), (1)

104 Kähler Moduli Stabilisation e.g. h 1,1 = 0, W np = A i e a i T i τ i = Re T i. From N = 1 supergravity V = V F + V D : i V F = e K ( K I JD I WD J W 3 W 2 ), (1) V F ( K τ τ D τ WD τ W + K UŪD U WD W ) ( Ū Ae 2aτ i + V 2 V Be aτ i W 0 V 2 ) + C W0 2. V 3 D τ W = D U W = 0 τ and U stabilised at O( 1 V 2 ) 0.1 W in general parameter space C α 3. If C > 0, the minimal condition for the second term: V e aτ i with τ i 1

105 Moduli Inflations I Requirements and subsequent attempts for a (semi)realistic inflationary model in string theoretic framework Moduli stabilization (and AdS to ds uplifting) Looking for available flat-directions UV sensitivity: η problem (protecting flatness against higher order operators) Consistent realization of cosmological observables from the point of view of present/future experimental constraints; No. of e-foldings, N e O(60) Almost scale invariant power spectrum, n s 1 Signatures of non-gaussianities, f NLlocal O(1) tensor-to-scalar ratio, r

106 Moduli Inflations II Usually, Moduli inflation get a small tensor-to scalar ratio r, which is consistent with PLANCK result but BICEP2. One type of moduli inflation can get r 0.1 in the proper n s region in a nature parameter space Axion-like inflation. C 0, c a, ρ α

107 r Moduli Inflations II Usually, Moduli inflation get a small tensor-to scalar ratio r, which is consistent with PLANCK result but BICEP2. One type of moduli inflation can get r 0.1 in the proper n s region in a nature parameter space Axion-like inflation. C 0, c a, ρ α 0.20 f 4 f f 6 f 7 f 8 f f 10 f 12 f n S Figure: n s r plot for axion-like inflation

108 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

109 Reference III Bouchard, Lectures on complex geometry, Calabi-Yau manifolds and toric geometry, hep-th/ Hartshorne, Alebraic Geometry, 1997 Griffiths, Harris, Principles of Algebraic Geometry, 1978 Cox, Little, Toric Varieties, 2011 Hori, Katz, Klemm, Pandharipande, Thomas, et al. Mirror symmetry

110 Toric Geometry I The topological quantity of the internal compactified geometry is crucial for physical model building, for which algebraic geometry is the proper tools to analyse these quantities. bundle-valued cohomology on the hypersurface, Chern characters, intersection number as well as Mori and Kähler cone... These topological quantities describe the no. of generations for the particle spectrum, the number of moduli and the survive of non-perturbative correction..., which should be calculated through computational algebraic geometry.

111 Toric Geometry I The topological quantity of the internal compactified geometry is crucial for physical model building, for which algebraic geometry is the proper tools to analyse these quantities. bundle-valued cohomology on the hypersurface, Chern characters, intersection number as well as Mori and Kähler cone... These topological quantities describe the no. of generations for the particle spectrum, the number of moduli and the survive of non-perturbative correction..., which should be calculated through computational algebraic geometry. Proper String Vacuum Proper Geometry Configuration

112 Toric Geometry II Calabi-Yau Huge number (Hundreds of Millions) Too abstract to get any information

113 Toric Geometry II Calabi-Yau Huge number (Hundreds of Millions) Too abstract to get any information Toric Geometry: Cox, Witten 92 Guage Linear Sigma Model (GLSM)

114 Calabi-Yau Toric Geometry II Huge number (Hundreds of Millions) Too abstract to get any information Toric Geometry: Cox, Witten 92 Guage Linear Sigma Model (GLSM) Compacted Calabi-Yau sub-spaces of toric varieties Geometric Data combinatorial terms of homogeneous coordinates x i. Divisor D i {x i = 0} Linebundle L Da

115 Calabi-Yau Toric Geometry II Huge number (Hundreds of Millions) Too abstract to get any information Toric Geometry: Cox, Witten 92 Guage Linear Sigma Model (GLSM) Compacted Calabi-Yau sub-spaces of toric varieties Geometric Data combinatorial terms of homogeneous coordinates x i. Divisor D i {x i = 0} Linebundle L Da Building Block Vertex in (dual-)polytopy lattice GLSM charge Resolve singularity on the (dual-)polytopy Blow-up Triangulations the (dual-)polytopy Stanley-Reisner Ideal

116 Goal Toric Geometry III Algorithmic and tools for calculation of the bundle-valued cohomology. Large-scale computation on ambient space and consequence hypersurfaces. Maximal Triangulations, SR-ideal, Kähler cone, Intersecting number... WHY Linebundle cohomology? Chiral spectrum in Type II string and Het Stinrg. H (C, L a L b K 1/2 C ), H (M, p V), H (M, p V ), H (M, V p V q ). D-brane instanton contribution to superpotential χ 0(D) = 3 p=0 ( 1)p h p,0 (D) = 1. Stable vector bundle on c 1 (V) = 0 manifold H 0 (M, p V) = 0.

117 Algorithm for Linebundle Cohomology on Hypersurface One algorithm in cohomcalg. Blumenhagen, Jurke, Rahn, Roschy Only take account dimentions in cohomology sequence. Ignore the map in cohomology sequence. Complicated hypersurface: No Result.

118 Algorithm for Linebundle Cohomology on Hypersurface One algorithm in cohomcalg. Blumenhagen, Jurke, Rahn, Roschy Only take account dimentions in cohomology sequence. Ignore the map in cohomology sequence. Complicated hypersurface: No Result. New algorithm based on cohomcalg. Take account representations in cohomology sequence. Construct the map in cohomology sequence. Complicated hypersurface: Exact Result Two intersecting hypersurface: Exact Result Checking stability of vector-bundle: Possible

119 Computational Tools Scanning tools of toric geometry Combined scanning package for toric ambient space in a phtyon language. It combines some virtue and get rid the shortcoming of PALP (a Package for Analyzing Lattice Polytopes) SAGE (System for Algebra and Geometry Experimentation) SINGULAR (computer algebra system for polynomial computations) cohomcalg

120 The Road to Unification Overview of String Theory Outline Flux Compactifications Calabi-Yau and its Moduli Space N = 1 Type IIB Orientifold Fluxed Compactification Moduli Stabilisation Dilaton and Complex Moduil Stabilisation Quantum Correction and Kähler Moduil Stabilisation Underline Mathematics Outlook

121 There are many things we didn t mentions. Some of them are crucial. Particle Physics from intersection D-branes is not discussed here. Here it is the starting point to study string cosmology in Type IIB framework. How about other string theory? Non-geometric flux case? More mathematical issues....

122 Thanks!