lectures on d-brane model building & conformal field theory€¦ · & conformal field...

Lectures on D-Brane Model Building& Conformal Field Theory

Gabriele Honecker

Cluster of Excellence PRISMA & Institut fur Physik, JG|U Mainz

String Pheno & String Cosmo 2016, Chengdu, July 2016

Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory

Outline

Lecture 1:

I particle physics from D-brane intersections

I limitations of geometric engineering

I string states from CFT ↔ dim. reduction of 10D SUGRA

Lecture 2:

I 1-loop partition functions

I vacuum amplitudes on T 6/Γ & stringy consistencyI 1-loop gauge thresholds

I vector-like spectrum & gauge group enhancement to USp/SOI 1-loop corrections to gauge couplings

Outlook


Lecture 1:

Geometric Approach


Lecture 1: Geometric Approach to D-brane Model Building

I basic idea in Type II string theory:

U(N)

U(M)

(M,N) (AdjN)I open strings:

I gauge groups U(N)I enhancement to USp(2N) or SO(2N) in special cases w/ ΩI U(1)’s generically massive by GS mechanism,

but massless linear combinations possibleI model building options:

U(3)a×

U(2)

USp(2)

a

× U(1)c × U(1)dGS−→SU(3)a × SU(2)b × U(1)Y

(×U(1)B−L × U(1)3 or 2

massive

)SU(3)× SU(2)L × SU(2)R × U(1)B−L

SU(4)× SU(2)L × SU(2)R

SU(5)(×U(1)

)Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory

I open strings:

I matter in (M,N) of U(M)× U(N):I SM, L-R symmetric, Pati-Salam, SU(5) XI no spinor of SO(10), no exceptional groups no SO(10) or E6 GUTs in perturbative Type II

compare w/ Inaki Garcıa Etxebarria’s lectures on F-theory

I closed strings:I gravityI if on R1,3 × CY3: Kahler & complex structure moduli

I naively: gauge sector (on Dp-branes) & gravity decoupledI but:

I longitudinal d.o.f. of U(1)massive is axionic partner of modulusI gauge & Yukawa couplings depend on moduliI mixing of closed & open string scalars beyond tree-level

I need for careful discussion / fine tuning of gstring,Mstring, . . .


Here: focus on D6-branes & O6-planes in Type IIA/ΩRWhy:

I O-planes needed for stability: N = 1 SUSY in 4D

I D6a & O6 wrap R1,3 × Πa & O6

special Lagrangian: JKahler|Π = 0with calibration: Re(Ω3)|Π > 0, Im(Ω3)|Π = 0

I Πa Πb = multiplicity of (Na,Nb) localised at points in CY3

I massless chiral spectrum determined by topology:

rep. mult. rep. mult.

(Na,Nb) Πa Πb (Antia) ΠaΠ′a+ΠaΠO6

2

(Na,Nb) Πa Π′b (Syma) ΠaΠ′a−ΠaΠO6

2

with Π′a = R(Πa) and R anti-holomorphic involution on CY3

I USp(2Na) or SO(2Na) groups if Πa = Π′aI type of enhancement cannot be derived from pure geometry

( CFT methods later)


I @ tree-level:

H

L

u

L

L

uR

U

C

T

I 1ga2 ∝ Vol(Πa) from DBI action ⊃

∫R1,3×Πa

trFa2

I Yabc ∝ e−Areaabc from worldsheet instantons

I advantage of geometric engineeringI topological intersection # quick to computeI stringy consistency conditions linear:

I RR tadpole cancellation:∑a

Na

(Πa + Π′a

)= 4 ΠO6

I K-theory constraint: ΠprobeSU(2)'USp(2)

∑a

NaΠa = 0 mod 2

I known for T 6 and T 6/Γ examples,can in principle be generalised to CY3 - problem: sLagcondition??


I limitations of geometric engineering:I only chiral spectrumI can classify if enhancement U(N)→ USp/SO(2N) arises,

but not determine which oneI only “maximal possible set” of K-theory constraintsI no access to SU(2)L/R ' USp(2) model building

I 4D effective action:I only dimensional reduction of 10D SUGRA & (p + 1)-D DBII only tree levelI no matter interactions


Complementarity of CFT Methods @ Orbifold Point

Conformal Field Theory

I uses quantised strings: possible only on simple geometriesI bosonic string:

Xµ(τ, σ) = XµL (σ+) + Xµ

R (σ−) XµL/R(σ

+−) ∼

∑n 6=0

1

nαµne

−inσ+−

with (αµn)† = αµ−n and

[xµ, pν ] = i ηµν [αµm, ανn] = m δm+n η

µν [αµm, ανn] = 0

I fermionic string:

Ψµ(τ, σ)NS/RL/R ∼

∑r∈Z+φ

ψµr e−i r σ+−

with φ =

12 NS0 R

and ψµr , ψνs = δr+s,0 ηµν


I so far: stringy e.o.m. solved for non-compact directions

I new on orbifolds: twisted sectors

X i (τ, σ + 2π) = hX i (τ, σ)h−1 with h ∈ Γ

here T 6 = ⊗3i=1T

2(i), and X i , X

iare complex coordinates

I from here on: restrict to Γ = ZN or ZN × ZM

can write X i θk−→ e2πikviX i with

~v ~v ~w

1N (1, 1,−2) with N = 3, 4, 6 1

N (1,−1, 0) 1M (0, 1,−1)

1N (1, 2,−3) with N = 6′, 7, 8 with (N,M) = (2, 2), (3, 3), (6, 6)

18 (1, 3,−4) (2, 3), (2, 6), (3, 6)

112 (1, 4,−5) and 1

12 (1, 5,−6) 12 (1,−1, 0) 1

6 (−2, 1, 1)

allowed by cristallographic action on T 6


I can solve e.o.m. for twisted strings

X iL/R(σ

+−) ∼

∑n 6=0

1

nαi

n−+φi

e−i(n−+φi )σ

+−

analogously ψi

r−+φi

withI φi = kvi for θk -twisted closed string sectorI πφi : relative angle among D-branes for open strings

I can now construct string states explicitly

I remember GSO projection: 1+(−1)F

2

I start from |s0, ~s〉 with all s0, si ∈ Z + 12 + φ

I twisted states |p0, ~p〉 = |s0, ~s−+φi 〉

I mass formula α′

4 m2 = 12p

2 + ET − 12

with ET = 12

∑i |φi |(1− |φi |)

I orbifold projection: |s0, ~s〉θ−→ e2πi~v ·~s |s0, ~s〉

I orientifold projection:I closed strings: |s0,~s〉L|s0, ~s〉R

ΩR−→ (±)NSR |s0,−~s〉L|s0,−~s〉R

I open strings: (i) end points, (ii) gauge charges


I matching with dimensional reduction of 10D SUGRA fields:I bosonic 10D fields & parity under Ω:

G(+)MN , Φ

(+)dilaton, B

(−)MN , C

(−)M , C

(+)MNP

I integrate over 2-cycles πi and 3-cycles ΠK with R parities:

vi =

∫π

(−)i

JKahler, bi =

∫π

(−)i

B2, ξK =

∫ΠK

(+)

C3, Ai =

∫π

(+)i

C3,

and cK from∫

ΠK(+)

Ω3 remember (Gij)CY3 → JKahler,Ω3

I so far: only untwisted sector from CFT

I but: geometric method also @ singularities, e.g. π(−)i = e

(i)αβ

I here: twisted sector at Z(i)2 fixed point (αβ) on T 4

(i)

I twisted states in CFT, e.g. Z(1)2 sector:

|00 + +〉|00−−〉NSNS

|00−−〉|00 + +〉NSNS

=(v , b)

(1)αβ

(|00 + +〉|00 + +〉+ |00−−〉|00−−〉

)NSNS

=c(1)αβ

I either (v , b)(1) or c(1) exist for Z2 × Z2:

discrete torsion (Z(2,3)2 ) phase η = ±1


I so far: closed string statesI open string states at angles πφi analogously

I e.g. at angle π(0, φ,−φ)

ψ3−1/2+φ|0〉

(tw)NS

|0〉(tw)R

(−−+)Z2×Z2

ψ2−1/2+φ|0〉

(tw)NS

ψµ0 ψ10 |0〉

(tw)R

(−+−)Z2×Z2

I scalars of N = 2 sector on T 6 or T 6/Z(1)2 : vector-like

I two N = 1 sectors if Z(2 and/or 3)2 ⊂ Γ: chiral

I Chan-Paton labels and fractional cycles:

I one Z2 symmetry: Πfrac = 12

(Πbulk + ΠZ2

)∗ Z2 eigenvalue (−1)τ

Z(1)2 = ±1

∗ displacements & Wilson lines σ2,3, τ2,3 ∈ 0, 1

I Z2 × Z2 symmetry: Πfrac = 14

(Πbulk +

∑3k=1 ΠZ(k)

2)

(−1)τZ(1)

2= ±1, (−1)τ

Z(2)2

= ±1, σ1,2,3 ∈ 0, 1, τ1,2,3 ∈ 0, 1


I Chan-Paton labels designed for bulk branes:

Πbulka =

4∑i=1

Πfrac,ia and Gbulk =

∏4i=1 U(N i

a)

I associated Gamma matrices:

γZ(1)2

= diag(1,1,-1,-1), γZ(2)2

= diag(1,-1,1,-1), γZ(3)2

= diag(1,-1,-1,1)

I Chan-Paton labels decompose as

λ =

(N1

a,N1b) (N1

a,N2b) (N1

a,N3b) (N1

a,N4b)

(N2a,N

1b) (N2

a,N2b) (N2

a,N3b) (N2

a,N4b)

(N3a,N

1b) (N3

a,N2b) (N3

a,N3b) (N3

a,N4b)

(N4a,N

1b) (N4

a,N2b) (N4

a,N3b) (N4

a,N4b)

I massless states: e.g. ψ3

−1/2+φ|0〉(tw)NS , ψ2

−1/2+φ|0〉(tw)NS


Lecture 2:

1-Loop Amplitudes


Lecture 2: Stringy Consistency from Vacuum Amplitudes

I partition functionI Σ (all possible string excitations)I per given sector (NS, R, untwisted . . . )I for given worldsheet topology

I RR tadpole cancellation conditions:I 1-loop divergences of RR states cancel outI ⇔ topological charge vanishes along compact dims.

I worldsheets @ 1-loop:

+ + +

closed strings open strings


I closed strings

+

T +K = 4c

∫ ∞0

dt

t3TrU+T

(1 + ΩR2

PorbPGSO(−1)Se−2πt(L0+L0))

I open strings

+

A+M = c

∫ ∞0

dt

t3Tropen

(1 + ΩR2

PorbPGSO(−1)Se−2πtL0

)I RR tadpoles read off from tree channel (`→∞):

t =1

κ`with κ =

4 K2 A8 M


I Jacobi theta & Dedekind eta functions (q ≡ e2πiτ ):

ϑ

[α

β

](ν, τ) =

∑n∈Z

q(n+α)2

2 e2πi(n+α)(ν+β), η(τ) = q1

24

∞∏n=1

(1− qn)

I useful for string vacuum amplitudes (− 12< α 6 1

2):

ϑ[αβ

](ν, τ)

η(τ)= e2πiα(ν+β) q

α2

2− 1

24

∞∏n=1

(1 + e2πi(ν+β) qn+α− 1

2

)(1 + e−2πi(ν+β) qn−α− 1

2

)I in particular:

* ν: from Porb

* spin structure (untw.): (α, β) ∈ (0, 0), (0, 12), ( 1

2, 0), ( 1

2, 1

2)

ϑ1 ≡ −ϑ

[1/2

1/2

], ϑ2 ≡ ϑ

[1/2

0

], ϑ3 ≡ ϑ

[0

0

], ϑ4 ≡ ϑ

[0

1/2

]

* twisted sectors / at angles: α = φ+

1/2

0

I modular transformation: twist sector ↔ projector insertion


I modular transformation: twist sector ↔ projector insertion

η(τ) =1√−iτ

η(−1

τ)

ϑ1234

(ν, τ) =(i×)1e−iπν

2/τ

√−iτ

ϑ1432

(ν

τ,−1

τ)

I remember: NS(-NS): 3, 4, R(-R): 1, 2I tree channel: ` = 1

κt and t = iτ

I RR tadpoles from `→∞:

ϑ2(0, τ)

η3(τ)τ→i∞−→ 2,

ϑ3

ϑ4(ν, τ)

τ→i∞−→ 1,ϑ2

ϑ1(ν, τ)

τ→i∞−→ cot(πν)


For D6-branes wrapping factorisable cycles on (T 2)3):

I (−1)2(α+β) - from spin structure

Iϑ[αβ](0,2i`)

η3(2i`)- fermionic + bosonic oscillators along transversal

direction in R1,3 (light-cone gauge)I per (complex) compact direction T 2

(i):

i D6a ↑↑ D6b & 1I in loop channel: V(i)ab L

(i)A,ab(`)

ϑ[αβ](0,2i`)

η3(2i`)

L(i)A,ab = sum over KK and winding models

ii D6a ↑↑ D6b & Z2 in loop channel: δσiab,0

δτ iab,0

ϑ[α+1/2β ]

ϑ[ 01/2]

(0, 2i`)

iii D6a

πφ(i)ab

∠ D6b & 1I in loop channel: I(i)ab

ϑ[αβ]ϑ[1/2

1/2](φ

(i)ab , 2i`)

I(i)ab cot(πφ

i)ab) = V

(i)ab

iv D6a

πφ(i)ab

∠ D6b & Z2 in loop channel: IZ2,(i)ab

ϑ[α+1/2β ]

ϑ[ 01/2]

(φ(i)ab , 2i`)

I i & iii contribute to untwisted RR tadpolesI ii & iv contribute to twisted RR tadpoles

I . . . can be extended to ΩR insertions & closed strings . . .


I K +M+A `→∞= 0 can be viewed as[

4 ΠO6 −∑a

Na(Πa + Π′a)]

︸︷︷︸ ∗[4 ΠO6 −

∑a

Na(Πa + Π′a)]

︸︷︷︸ = 0

= 0 = 0

with symmetric composition Πa∗Πb ∼ Vab

I prefactors of Mobius strip amplitude already implicitly encodeenhancement U(N)→ USp(2N) or SO(2N)

I traditionally written for bulk branes in front of open stringamplitudes:

I AZ2 insertionab : (trγZ2

a )(tr(γZ2b )−1)

I M: tr((γΩR

a′ )−TγΩRa

) compute action on Chan-Paton labels λaa′

I not suitable for fractional D-branes / realistic model building


Beyond RR Tadpole Cancel.: 1-Loop Gauge Thresholds

I trick: (magnetically) gauge non-compact directions:

πqa∂2

∂B2Bϑ[αβ

]ϑ[1/2

1/2

](arctan(πqaB)

π, τ)∣∣∣B=0

=− π2q2a

(1

3+

1

6E2(τ)

)ϑ[αβ

](0, τ)

η3(τ)+

1

2π2

ϑ′′[αβ

](0, τ)

η3(τ)

i Σ (spin structure)

iiϑ[αβ](0,τ)

η3(τ) identical to vacuum amplitudesRR tcc 0

iii identities of Jacobi theta fct.s of the form ϑ′′ · ϑ3 + . . . give

1

π

3∑i=1

ϑ′1ϑ1

(φ(i), τ) =3∑

i=1

cot(πφ(i)) + 4∞∑

n,k=1

sin(2πφ(i)k) qnk

1

π

[ϑ′1ϑ1

(φ(2), τ)+∑i=1,3

ϑ′4ϑ4

(φ(i), τ)]= cot(πφ(2))+ 4

∞∑n,k=1

(sin(2πφ(2)k) qnk+

∑i=1,3

sin(2πφ(i)k) q(n− 12

)k)

with 1I and Z2 insertion, respectively, in loop channelGabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory

I 1-loop magnetically gauged amplitudes continued:iv perform

∫∞0

d` `ε(. . .)

dimensionally regularised

result: (finite)×∫ ∞

0

d`+(1

ε+ γ − ln 2

)︸︷︷︸+ ∆

↓ ↓ ↓

RR tcc= 0 ln

(Mstring

µ

)2

finite

with finite 1-loop corrections to gauge couplings ∆,which lead to kinetic mixing (application e.g. to dark photons)

G.H., Ripka, Staessens ‘12

v compare 1-loop beta function coefficient with QFT:

bSU(Na) =Na

(ϕAdja − 3

)︸︷︷︸+Na

2

(ϕSyma + ϕAntia

)︸︷︷︸+

(ϕSyma − ϕAntia

)︸︷︷︸+∑b 6=a

Nb

2

[ϕab + ϕab′

]︸︷︷︸

= bAaa + bAaa′ + bMaa′ +∑b 6=a

Nb

2

[bAab + bAab′

]and read off multiplicities of massless states

I determine chiralities from sgn(Πa Πb)


I 3 complimentary ways to obtain massless spectrum:

1 Πa Πb for chiral matter2 bSU(Na) from gauge thresholds: (chiral)±? + vector-like3 explicit construction of states & Chan-Paton labelsI 1 + 2: suitable for extensive computer scans

Z2 × Z′6: G.H., Ripka, Staessens ‘12; Z2 × Z6: Ecker, G.H., Staessens ‘14-‘15

I 3 needed to compute interactions (e.g. localisation on (T 2)3 required)

I in case of gauge group enhancement:

bUSp/SO(2Mx ) = Mx

(ϕSymx + ϕAntix − 3

)+(ϕSymx − ϕAntix∓1

)+∑a 6=y

Na

2ϕxa

I can now classify of all USp(2) probe branesI K-theory constraintXI SU(2)L/R ' USp(2) models

I gaugino condensation in strongly coupled hidden sector* requires dim(Adjh) large* as little (better: no) charged matter as possibile: bhidden < 0


back to 1-loop gauge thresholds ∆:

vi evaluate 1-loop corrections to (hol.) gauge couplings:I from D-branes at angles: constants (∠-dependent)

I from D-branes ↑↑ along some T 2(i): ∝ bAab Λ∆τ i

ab,∆σiab

(vi ) with

Λτ,σ(v) =− 1

4π×

ln(η(iv)

)(τ, σ) = (0, 0)

ln(e−πσ

2v/4 ϑ1( τ−iσ v2 ,iv)

η(iv)

)6= (0, 0)

v→∞−→

v48 (τ, σ) = (0, 0)

- v48 (0, 1)

v24 −

ln 24π (1, 0)

- v48 (1, 1)

G.H., Ripka, Staessens ‘12

I negative contributions possible, can compensate tree-levelresult for highly unisotropic tori

1

g2a,tree

∝ Vol(Πa) ∝√v1v2v3

example=√

104 · 102 · 102 = 104

↑ ↑

I new possibilities for: large volume & low Mstring @ weak gstring


Outlook


Outlook

CFT techniques:I here: vacuum amplitudes generalise to scattering

amplitudes with vertex operator insertionsI here: 1-loop in perturbation theory can be used for

D-brane instantonsI corrections to Kahler potential some results for bulk branes only

Berg, Haack, Kang, (Sjors) ‘12, ‘14

I largely unexplored arena (despite occasional contrary claims)

Physical implications: . . . see SPSC Workshop talk

I value of Mstring can significantly change @ 1-loopI Yukawa couplings for particle physics models: prefactor not

known (might even vanish )

I Higgs-axion potentialI kinetic mixing of U(1)Y and dark photonI . . .


Some Textbooks

I Green, Schwarz, Witten: String Theory, Vol. 1 & 2,Cambridge University Press 1987

I Blumenhagen, Lust, Theisen: Basic Concepts of StringTheory, Springer 2013

Extension of Lectures on String Theory, Springer 1989

I Polchinski: String Theory, Vol. 1 & 2, Cambridge UniversityPress 1998

I Zwiebach: A First Course in String Theory, CambridgeUniversity Press 2004

I Becker, Becker, Schwarz: String Theory and M-Theory - AModern Introduction, Cambridge University Press 2007

I Ibanez, Uranga: String Theory and Particle Physics,Cambridge University Press 2012


lectures on d-brane model building & conformal field theory€¦ · & conformal field...

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