Lectures on D-Brane Model Building & Conformal Field Theory€¦ · & Conformal Field Theory...
Transcript of Lectures on D-Brane Model Building & Conformal Field Theory€¦ · & Conformal Field Theory...
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
Lecture 1:
Geometric Approach
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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, . . .
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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)
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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??
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
I projections on open strings:
Z2 cZ2state
(γZ2,aλabγ
−1Z2,b
)|state〉ab
λab|state〉abΩR cΩR
state
(γΩR,aλabγ
−1ΩR,b
)T |state〉ba
I massless closed string spectrum:I invariant under Porb = 1
N
∑N−1k=0 θ
k
I invariant under PΩR = 1+ΩR2
I invariant under PGSO = 1+(−1)F
2
I e.g. in untwisted sector:(ψµ−1/2ψ
ν−1/2 + ψν−1/2ψ
µ−1/2
)|0〉NSNS Gµν + Φdilaton
ψi−1/2ψ
i−1/2|0〉NSNS, ψi
−1/2ψi−1/2|0〉NSNS (vi , bi )i=1,2,3
| − + + +〉| + + + +〉RR − | +−−−〉| − − − −〉RR ξ0
some models:(ψi−1/2ψ
i−1/2 + ψi
−1/2ψi−1/2
)|0〉NSNS ci
| +− + +〉| − − + +〉RR − | − +−−〉| + +−−〉RR ξiGabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
Lecture 2:
1-Loop Amplitudes
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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)
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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 . . .
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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)
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
Outlook
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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 . . .
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory
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
Gabriele Honecker Lectures on D-Brane Model Building & Conformal Field Theory