Candidates for Inflation in Type IIB/F-theory Flux Compactifications

Candidates for Inflation in Type IIB/F-theory Flux Compactifications Irene Valenzuela IFT UAM/CSIC Madrid Geometry and Physics of F-Theory, Munich 2015 Garcia-Etxebarria,Grimm,Valenzuela [hep-th/1412.5537] Ibáñez,Valenzuela [hep-th/1404.5235] Ibáñez,Marchesano,Valenzuela [hep-th/1411.5380]

Inflationary models: Effective approach V = V 0 ( )+ i c i O i ( ) UV sensitivity: Mp 4 i Radiative corrections: m 2 & H 2! & 1 Higher order dimensional operators: Problematic for large field inflation >M p Need: Underlying symmetry protecting the potential UV completion String Theory

Large field inflationary candidates Scalar with an axion-like shift symmetry in the absence of a scalar potential Type IIB Zero modes of RR,NSNS fields } D7-brane Wilson lines Axions [Kim,Nilles,Peloso] [Dimopoulos et al.] [Grimm] [Silverstein,Westphal] [McAllister et al.] [Palti,Weigand] [Franco et al.]... Complex structure moduli } D7-brane position moduli Approximate (geometrical) axions Other open string moduli [Marchesano et al.] [Hebecker et al.] [Blumenhagen et al.] [Hayashi et al.]...

Large field inflationary candidates Scalar with an axion-like shift symmetry in the absence of a scalar potential Type IIB F-theory (CY 3 ) (CY 4 ) Garcia-Etxebarria,Grimm,Valenzuela [hep-th/1412.5537] Complex structure moduli} Complex structure moduli space of D7-brane position F-theory Ibáñez,Valenzuela [hep-th/1404.5235] Ibáñez,Marchesano,Valenzuela [hep-th/1411.5380]

Axions at Special Points of CY manifolds Where do axions arise? Close to special points which admit discrete monodromy symmetries Shift symmetry geometric origin Monodromy transf. of infinite order Kahler potential: Z K cs = log i X n ^ (invariant) Continuos shift symmetry is broken to a discrete shift as we move away from the special point.

Axions at Special Points of CY manifolds 1) Identify special points: Approximate axion in the effective theory with shift symmetric kinetic terms 2) Induce a scalar potential: Turning on background fluxes in Type IIB orientifolds and W = F-theory compactifications. Z X n G n ^ +... breaks shift symmetry Good candidates for F-term axion monodromy inflationary models

Complex str. moduli space of CY manifolds In more detail... z k : h 2,1 (X n ) complex structure moduli Periods: i = Effective theory: Z A i K = log(i i ij j )+... W = N i i +... holomorphic (n,0)-form Symplectic integral basis A i for H n (X n, Z) N i flux quanta A i \ A j = ij Special points = loci at which some i become singular Monodromy transformation: 0i (z) =T i j [z s ] j (z) G mon Sp(2(h 2,1 + 1))

Complex str. moduli space of CY manifolds In more detail... z k : h 2,1 (X n ) complex structure moduli If T [z s ] is of infinite order T [z s ] n 6= T [z s ] logarithmic behaviour of some periods log j (z) Axion in the effective theory = Arg(z) Special points = loci at which some i become singular Monodromy transformation: 0i (z) =T i j [z s ] j (z) G mon Sp(2(h 2,1 + 1))

Complex str. moduli space of CY manifolds Compute the periods close to the special points. Picard-Fuchs equation: L (z) =0 1) One-parameter Calabi-Yau threefolds constructed as mirrors of complete intersections P n [d 1...d k ]! 4Y L (z) = 4 z ( + i ) (z) =0 i=1 [Hosono,Klemm,Theisen,Yau] [Greene,Lazaroiu] z =0 z =1 z = 1 2) Elliptically fibered Calabi-Yau manyfolds constructed as mirrors of genus one fibrations [Alim,Scheidegger] [Klemm,Manschot,Wotschke]

Results for Calabi-Yau threefolds Different inflationary potentials arise at different points in moduli space Large complex structure point K lcs = log apple i Conifold point 1 6 K(t t) 3 +2ĉ W lcs = 1 6 N 4Kt 3 1 2 N 3Kt 2 + Ñ2t + Ñ1 K con = log apple 1 K t c 2 log t c +... W con = N 4 Kt c log(t c )+ 1 2 N 3Kt 2 c + Ñ2t c + N 1 Polynomial scalar potential Periodic cosinetype potential Small complex structure point K scs = W scs = X i log a u 2 apple log u +... Ni e u + X i (log(u)+...)+ Nj 0e u j +... j } All Two i i i different equal No axion Periodic All equal Polynomial

Results for Calabi-Yau threefolds Different inflationary potentials arise at different points in moduli space Large complex structure point K lcs = log apple i 1 6 K(t t) 3 +2ĉ W lcs = 1 6 N 4Kt 3 1 2 N 3Kt 2 + Ñ2t + Ñ1 (T (0) I) 4 =0 Polynomial scalar potential Conifold point K con = (T (1) I) 2 =0 apple 1 log K t c 2 log t c +... W con = N 4 Kt c log(t c )+ 1 2 N 3Kt 2 c + Ñ2t c + N 1 Periodic cosinetype potential Small complex structure point K scs = W scs = X i log a u 2 apple log u +... Ni e u + X i (log(u)+...)+ Nj 0e u j +... j (T (1) k I) m =0 All Two i i } i different equal No axion Periodic All equal Polynomial

Results for elliptic fibrations (Calabi-Yau mirrors of genus one fibrations) Large volume limit of the base One-parameter model PF operator reduces to that of the fiber: L = 2 z( + 1 )( + 2 ) = z @ @z i depend on the type of torus fiber: E 8,E 7,E 6,D 5 F-point: small volume of the torus fiber (small cs point in the mirror) Fiber E 8,E 7,E 6 Fiber D 5 : : 1 6= 2 1 = 2 T [1] T [1] finite order infinite order no axion axion Monodromy matrix: 1 a 1 T [1] = a 1 a = # of sections Existence of axions depends on the number of sections N>3 ( N 1= rank Mordell-Weil group)

Remarks Dualities ex. Conifold point: [Vafa] [Vafa,Heckman] [Aganagic,Beem,Kachru] cs modulus with flux induced W conifold transition kahler modulus with W from gaugino condensate Corrections in 0 and g s Aligned or N-Inflation models? Mirror symmetry, localization techniques... [Mayr] [Klemm et al.]... [Benini et al.][doroud et al.]... For a D7 position modulus 0 corrections from DBI Moduli stabilization mixing with other moduli, backreaction... see talk of Hebecker [Blumenhagen et al.] [Hayashi et al.] [Hebecker et al.]

Remarks Dualities ex. Conifold point: [Vafa] [Vafa,Heckman] [Aganagic,Beem,Kachru] cs modulus with flux induced W conifold transition kahler modulus with W from gaugino condensate Corrections in 0 and g s Aligned or N-Inflation models? Mirror symmetry, localization techniques... [Mayr] [Klemm et al.]... [Benini et al.][doroud et al.]... For a D7 position modulus 0 corrections from DBI [Ibáñez,Marchesano,Valenzuela] Moduli stabilization mixing with other moduli, backreaction... see talk of Hebecker [Blumenhagen et al.] [Hayashi et al.] [Hebecker et al.]

D7 position modulus as the inflaton N=1 supergravity approach In toroidal orientifolds or CY orientifolds (large cs point): K = log[(s S )(U U 1 ) 2 ( ) 2 ] 3log(T T ) Tree level in 0 [Hebecker et al.] Dirac-Born-Infeld + Chern-Simons effective action S DBI = S CS = µ 7 Z µ 7 Z d 8 STr apple e q d 8 STr [ C 6 ^ B 2 + C 8 ] det (P [E µ ]+ F µ ) D7 position: z =2 0 h i Exact in 0 But no information about other closed moduli...

D7 position modulus as the inflaton Expand DBI+CS in the presence of ISD background G3-fluxes Z S = keeping all terms in (large field) D7 position: d 4 x STr f(, )D µ D µ V (, ) z =2 0 h i f(, ) = 1 + (V 4µ 7 g s ) 1 2 V (, ) Non-canonical kinetic terms Scalar potential: V = g s 2 Tr G S 2 G G (0,3) S G (2,1) field redefinition V! Ĝ ' linear for large field 0 corrections Flattening of the potential

Kaloper-Sorbo lagrangian KS lagrangian: L KS = 1 2 Z d 4 x[(@ ) 2 + F 4 2 µ 4 F 4 ] [Kaloper,Sorbo] [Dvali] [Dudas] F 4 non dynamical induce scalar potential V Gauge invariance protects V from dangerous UV corrections V = O c n n M n 4 UV V = V 0 V0 M 4 UV Although >M UV, if V 0 <M UV UV corrections are under control. n From DBI: Z Z 1 1 µ 7 2 F 2 ^ 8 F 2 = µ 7 2 Z Z KS coupling F 4 2 Z F 4 2 F 6 ^ 8 F 6 + B 2 ^ F 6 +... d 4 x dc 3 2 R 1,3 g s d ZR 4 x 1,3 (G dc 3 S d C 3 )+c.c.

Kaloper-Sorbo lagrangian KS lagrangian: L KS = 1 2 Z d 4 x[(@ ) 2 + F 4 2 µ 4 F 4 ] [Kaloper,Sorbo] [Dvali] [Dudas] F 4 non dynamical induce scalar potential V Gauge invariance protects V from dangerous UV corrections V = O c n n M n 4 UV V = V 0 V0 M 4 UV Although >M UV, if V 0 <M UV UV corrections are under control. n F-term axion monodromy inflation model [Marchesano,Shiu,Uranga] always? 4d effective Kaloper- Sorbo lagrangian Quantum corrections should come as powers of the potential

Higgs-otic Inflation If Inflaton = scalar from a hidden sector problems with reheating... Solution: Inflaton = MSSM Higgs boson (easy embedding in String Theory if it corresponds to an open string modulus) Non-abelian structure two-field inflation V = Z 2 g s 2 ( G S ) 2 h 2 +( G + S ) 2 H 2 SM higgs Heavy higgs + non-canonical kinetic terms Inflation comes along with EW symmetry breaking Constraints from } Particle Physics: Cosmology: Approx. massless SM higgs inflaton mostly Heavy higgs m H M SS ' 10 12 10 13 GeV Intermediate SUSY breaking scale!

Conclusions Systematic study of the special points in the complex str. moduli space at which axions appear. Different inflationary scalar potentials arise at different points of the moduli space (polynomial, sinusoidal...). Natural candidate for an axion in the limit of small fiber if the rank of the Mordell-Weil group is N-1 with N>3. Mirror symmetry and localization techniques can help to address the issue of higher order corrections. For a D7 position modulus of the potential. 0 corrections give rise to a flattening Pheno interesting model: Higgs-otic inflation.

Thank you!

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Periods around the special points Large complex structure point = 0 B @ K Conifold point =! 0 (z) 1 t 2 t2 + K 2 t + ˆb K 6 t3 + tˆb +ĉ 0 B @ 1 C A K 2 1 t c 1 2 i t2 c K 2 i t c log t c U 0 (z) =1 U 1 (z) = log(z) 1 2 i U 2 (z) = 1 2 log(z) + 3 log(z) + 8 b 2 2 i 2 2 i 8 U 3 (z) = 1 3 2 log(z) log(z) + log(z) 6 2 i 2 i 2 i K 2 1 2 i t c 1 C A (3b 44) 24 + (2b 8) 8 (2 i) 3 K (3). Small complex structure point U j (z) = A k (2 i) j P 5 [2, 4] : sin( k ) = 4i 2 u 1/2 3 j e i(j+1) k (applez) k + A 3 (2 i) j 0 B @ 1 4 B + i + log(u) 1 2 ( B log(u)) ( 3 = 4 ) ( BK 8 Klog(u)) 1 24 (24ˆb K)(B + log(u)) 3 j sin( 3 ) e i(j+1) 3 (applez) 3 (B (3 j) cot( 3 )+(j + 1)i + log(applez)) 1 C A + (1/4)6 4 4p u 1/4 0 B @ e 3 i/4 i p 2 1 2 Kei /4 i 3 p 2 (6ˆb K) 1 C A +...

Kaloper-Sorbo lagrangian F-term axion monodromy inflation model 4d effective Kaloper- Sorbo lagrangian From DBI: µ 7 Z 1 2 F 2 ^ 8 F 2 = µ 7 Z 1 2 2 F 6 ^ 8 F 6 + B 2 ^ F 6 +... S 4 admits a (2,0)-form A 5 = ic 3 ^! 2 i C 3 ^! 2 B 2 = g s 2i (G After integrating out F 4 : S )! 2 +c.c. V = g s 2 Tr G S 2 =! 2 Z d 4 x dc 3 2 R 1,3 g s d ZR 4 x (G dc 3 S d C 3 )+c.c. 1,3 KS coupling Z F 4 F 4 = dc 3