Quantum Orientifold Effective Actions Marcus Berg Karlstad University, Sweden M.B., Haack, Kang, arxiv:1112.5156 M.B., Haack, Kang, Sjörs 12... Cambridge, June 3 2012
Why quantum effective action (one-loop)? Stabilization: calculate the complete modulidependence of the effective action to one-loop order (relevant in e.g. KKLT Wnonpert): can end up being classical or quantum stabilization Soft supersymmetry breaking terms: field space curvature of Kähler metrics (e.g. large volume scenario)
Why orientifolds? Sagnotti 87... Some of the reasons: Supersymmetry reduced (e.g. Type IIB to Type I) In compact models: consistency conditions (D-branes) Wide range (toroidal, Calabi-Yau, F,...) free CFT
General D=4, N=1 effective theory of gravity+moduli+gauge fields open and closed string moduli L eff = 1 2κ 2 R K Φ i Φ j µ Φ i µ Φ j 1 g 2 (Φ) tr af 2 V (Φ) + corrections Can t always think negligible even when numerically small at a point in moduli space (i.e. at fixed ). Φ i
String perturbation theory: two expansions (semi-)classical quantum V V 4 = +g s +g 2 s +... E 2 α 0 V V 4 = V
dipole charged (semi-)classical, string tree level
quantum, string one-loop
Simple models for extra dimensions Orbifold: Identify under spatial rotation Z 3 Θ makes cone Orientifold: Identify under worldsheet reflection Ω = 1 ± Ω 2 unoriented eigenstate
Review: one-loop effective action sample N =1 orientifolds: T 6 /Z 6 T 6 /(Z 2 Z 2 ) ΘZ 1 = e 2πiv 1 Z 1 ΘZ 2 = e 2πiv 2 Z 2 ΘZ 3 = e 2πiv 3 Z 3 U 1 U 2 U 3 D3 D3 φ 1 φ 2 φ 3 D3 D7 wraps D7 wraps Z 6 :(v 1,v 2,v 3 )= 1 D7 6, 1 2, 1 3
A first look at corrections: f Θ Θ 2 N=2 sector sample partially twisted N =1 T 6 /Z 6 orientifolds: T 6 /(Z 2 Z 2 ) N=1 sector completely twisted ΘZ 1 = e 2πiv 1 Z 1 ΘZ 2 = e 2πiv 2 Z 2 ΘZ 3 = e 2πiv 3 Z 3 U 1 U 2 U 3 D3 D3 φ 1 φ 2 φ 3 D3 D7 wraps D7 wraps Z 6 :(v 1,v 2,v 3 )= 1 D7 6, 1 2, 1 3
Singularities of T 6 /Z 6 A A a π 2 π 4 π 6 π 1 π 3 π 5 orientifold singularity orbifold singularity
Review: one-loop effective action L eff = 1 2κ 2 R K Φ i Φ j µ Φ i µ Φ j 1 g 2 (Φ) tr af 2 V (Φ) + corrections partially twisted: g, K corrections completely twisted: g, K corrections
A first look at corrections: f threshold corrections 1 loop = β ln M string 1 g 2 ph (φi ) µ + (φ,u) correct
f: partially twisted strings A a µ ( N=2 sectors ) U Dixon, Kaplunovsky, Louis 91... M.B., Haack, Körs 04 φ D3 A a µ D7 φ
f: partially twisted strings A a µ ( N=2 sectors ) U Dixon, Kaplunovsky, Louis 91... M.B., Haack, Körs 04 φ D3 A a µ D7 φ ln z 2 = ln z + ln z = 1 2Re ln z
f corrections A a µ U φ D3 A a µ D7 Plays a role if φnonperturbative W: Ganor 96 M.B., Haack, Kors 04 = a 1/N D7
f corrections A a µ U φ D3 A a µ D7 φ Moral: these are string loop corrections, but without them, Wnp doesn t depend on D3-brane scalar φ at all. So they are not negligible in any real sense.
Inflationary cosmology A a µ A a µ φ PLANCK satellite (taking data now) This gives a contribution to the inflaton potential in D-brane inflationary cosmology. Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan 06...
Particle phenomenology Berg, McAllister, Marsh, Pajer 10 de Alwis 12 M. B., Conlon, Marsh, Witkowski, in progress Analogous contributions to the effective action could also impact sequestering, which is one strategy to deal with the flavor problem of gravity-mediated supersymmetry breaking.
K: partially twisted strings brane at arbitrary position φ M.B., Haack, Körs, 05 τ = Re T τ τ ττ = Möbius strip Annulus τ τ τ Kähler adapted vertex operators τ Klein bottle
K: partially twisted strings brane at arbitrary position φ M.B., Haack, Körs, 05 τ = Re T τ τ ττ = Möbius strip Annulus τ τ τ Kähler adapted vertex operators τ TIE Fighter (Star Wars) Klein bottle
K: partially twisted strings M.B., Haack, Körs, 05 integrate one-loop corrected Kähler metric to get oneloop corrected Kähler potential: K = ln (S + S)(T + T )(U + Ū) ln 1 1 N i (φ i + φ i ) 2 8π (T + T )(U + Ū) 1 128π 6 i i E 2 (φ i, U) (S + S)(T + T ) sum over images of E 2 (φ i, U) E 2 (φ,u)= (n,m)=(0,0) Re(U) 2 n + mu 4 exp 2πi φ(n + mū)+ φ(n + mu) U + Ū
partially twisted strings: summary Efforts by many people: moduli-dependent gauge coupling: one-loop order g(φ, Φ) =g(s, S,T, T,U,Ū,φ, φ) Kähler potential: one-loop order K(Φ, Φ) =K(S, S,T, T,U,Ū,φ, φ)
What about completely twisted? Less is known. Are they not interesting? [completely twisted strings] effectively force the fields in the path integral to lie near some fixed point. The path integral is therefore insensitive to the shape of the spacetime torus and so is independent of the untwisted moduli. If on the other hand [they are only partially twisted], then the amplitude can depend on the moduli. Polchinski, Vol. 2, p. 299
A a µ A a µ g ph : completely twisted strings ( N=1 sectors ) Branes at angles: Completely twisted strings give moduli dependence too 0 d ϑ 1(v) ϑ 1 (v) = π 2 log e 2γ Ev Lüst, Stieberger 03 Γ(1 v) Γ(1 + v) angle (some fine print here)
A a µ g ph : completely twisted strings ( N=1 sectors ) Branes at angles: Completely twisted strings give moduli dependence too A a µ 0 d ϑ 1(v) ϑ 1 (v) = π 2 log e 2γ Ev Riemann 1859 Lüst, Stieberger 03 Γ(1 v) Γ(1 + v) angle (some fine print here)
K: completely twisted strings φ i ( N=1 sectors ) Kähler metric of D-brane scalars φī K φ φ Bain, M.B. 00 A k=1,5 99 (φ 3, φ 3 ) = δe3 ē 3 96π 2 tr(γk 9 λ 1 λ 2 )tr(γk 9 ) 0 dt η(it) 3(1 δ) t 2 ϑ 1 (a, it) t 0 3 ( 2 sin πkv j ) i=1 dν e πδν2 /t ϑ 1(iν + a, it) ϑ 1 (iν,it) 1 δ
K: completely twisted strings φ i ( N=1 sectors ) A k=1,5 99 (φ 3, φ 3 ) = δe3 ē 3 96π 2 tr(γk 9 λ 1 λ 2 )tr(γk 9 ) state of the art until recently: Kähler metric extracted field theory anomalous of D-brane dimensions, scalars but could not perform integrals φī no explicit result for one-loop Kähler metric even without angles or flux 0 dt η(it) 3(1 δ) t 2 ϑ 1 (a, it) t 0 K φ φ 3 ( 2 sin πkv j ) i=1 Bain, M.B. 00 dν e πδν2 /t ϑ 1(iν + a, it) ϑ 1 (iν,it) 1 δ
Anomalous dimensions, simply W = 1 6 λabc ijkφ i aφ j b Φk c (γ j i ) a b = 1 16π 2 2g 2 C 2 (R a )δ j i δ a b λ acd ikl λ jkl bcd e.g. Z3, with two fields: 1 g 2 γ Φ = 2 7 4 +2 1 2 143 24 + 1 24 1 g 2 γ χ = 2 1 2 130 12 +4 1 24 8=3 11 = 3/2
K: completely twisted strings φ i ( N=1 sectors ) M.B., Haack, Kang 11 φī labels spin structures S = α dt Z α dep. on t dν 1 dν 2 V φ iv φī α dep. on ν s and t
K: completely twisted strings Type IIA D-brane moduli Kähler modulus along brane transverse to brane Φ i T i φ i A i V A g o V φ g o Σ Σ dτ A α τ X α +fermions dτ φ a σ X a +fermions along and transverse depend on brane pick adapted coordinates Hassan 03
U D θ L convenient metric ds 2 = L 2 (dx α ) 2 + D 2 (dx a ) 2
Method of images v = θ π θ 0 π Ψ(2π) =e 2i(θ π θ 0 ) Ψ(0) unphysical θ 0 θ π θ 0 0 π 2π unphysical 0 π 2π work on covering torus with twist of holomorphic fields cf. Bertolini, Billò, Lerda, Morales, Russo 05
What string states can run in the loop? Completely twisted states are localized at intersections of brane images I =7
Lüst, Stieberger 03 Gmeiner, Honecker, 07, 08, 09 I = n 2 + n +1=7, 13, 21, 31, 43,...
Integrate external states annulus Möbius strip τ A = it 2 1/2 1 τ M = it 2 + 1 2 Burgess, Morris 87... 1/2 1
K: completely twisted strings α Z α V Φ iv Φī α = e δ G B(ν,τ) G F 1/2 1/2+v (ν, τ) = e δ G B(ν,τ) G F 1/2 1/2+v (ν, τ) π cot πν +e δ G B(ν,τ) π cot πν δ = p 1 p 2 poles at 0 and 1 cancel remove δ
Möbius strip similar!
UV finiteness + UV +
Finite part vanishes (!) extend 0-1 integral to entire contour τ 0 1
Summary completely twisted strings: K 1 loop Φ i Φī =0
K: completely twisted for closed strings no angles yet! brane at arbitrary position φ M.B., Haack, Kang, Sjörs 12 τ = Re T τ τ ττ = Möbius strip Annulus τ τ τ Kähler adapted vertex operators τ Klein bottle
K: completely twisted for closed strings no angles yet! IIB amplitude involves involution F σ d 2 ν 1 d 2 ν 2 2 G γ B (ν 12) G F,γ α (ν 12)G F,γ=0 α (ν 12 ) perform spin structure sum, lift to covering torus using of covering torus ν2 ν1 G γ B (ν 1 I σ (ν 2 )) G F,γ α (ν 1 I σ (ν 2 ))G F,γ=0 α (ν 1 I σ (ν 2 )) σ f(w)+f I σ (w) = perform torus integral with zeta function regularization T I σ orbifold twist f(w). R δ +c.c.. T d 2 ν G F (1/2,1/2+γ) (ν, τ) νg F (1/2,1/2+γ)(ν, τ)
K: completely twisted for closed strings gives derivative of use reflection formula for Epstein zeta function finally perform integral over worldsheet modulus I = 0 dt t 2 G F (γ,it/2) = π 4 24 sin 2 πγ π2 12 ψ (γ) (finite part)
Conclusions cf. Anastasopoulos, Antoniadis, Benakli, Goodsell, Vichi 11 string nonrenormalization theorem for Kähler metric of D-brane moduli computed finite constant additions for Kähler metric of closed string moduli M.B., Haack, Kang, Sjörs 12 developed/consolidated many techniques for amplitude calculations with D-branes and O-planes implications: currently none known, next set of calculations can add moduli dependence