Exponentially Suppressed Cosmological Constant with Gauge Enhanced - PowerPoint PPT Presentation

Exponentially Suppressed Cosmological Constant with Gauge Enhanced Symmetry in Heterotic Interpolating Models Sota Nakajima (Osaka City University) with Hiroshi Itoyama (OCU, NITEP) Based on arXiv: 1905.10745 @ YITP, 8/2, 2019

Int ntroduc uction When a top-down approach from string theory is considered, there are two choices depending on where SUSY breaking scale is ; 1.SUSY is broken at low energy in supersymmetric EFT ; 2.SUSY is already broken at high energy like string/Planck scale. In this talk, the second one is focused on, and non-supersymmetric string models are considered. In particular, the 𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟕 model is a unique tachyon-free non-supersymmetric string model in ten-dimensions. [Dixon, Hervey, (1986)]

Int ntroduc uction Considering non-supersymmetric string models, however, we face with the problem of vacuum instability arising from nonzero dilaton tadpoles; 𝑊 𝜚 : dilaton tadpole 𝑊(𝜚) ∝ Λ Λ: cosmological constant (vacuum energy) 𝜚 At 1-loop level, ∝ The desired model is a non-supersymmetric one whose cosmolosical constant is vanishing or as small as possible. Interpolating models have the possibility of such properties. [Itoyama, Taylor, (1987)]

Ou Outlin ine 1. Introduction 2. Heterotic Strings 3. 9D Interpolating models 4. 9D Interpolating models with Wilson line 5. Summary

Outlin Ou ine 1. Introduction 2. Heterotic Strings 3. 9D Interpolating models 4. 9D Interpolating models with Wilson line 5. Summary

Ide dea a of of Het eter erot otic ic S Strin ings Heterotic strings are hybrid closed strings of bosonic string in 26D and superstrings in 10D. L R Adopting the lightcone coordinates, the worldsheet contents are Right mover: 10d superstring 16d on torus Left mover: 26d bosonic string out of which internal 16d realize rank 16 current algebra 10=(8+2)d 10=(8+2)d [Gross, Hervey, Martinec, Rohm, (1985)]

The he o one ne-loo oop p pa partit itio ion n fu func nc. & & St Stat ate e Co Coun untin ing  The one-loop partition function is the trace over string Fock space:  𝑎 𝜐 counts #(states) at each mass level as coeff. in 𝑟 ത 𝑟 expansion. 𝒃 𝒏𝒐 denotes #(bosons) minus #(fermions) at mass levels ( 𝒏, 𝒐 ) In the string model with spacetime SUSY, 𝑏 𝑛𝑜 = 0 for all ( 𝑛, 𝑜 ) because of fermion-boson degeneracy. for supersymmetric string models.  In order for the string model to be consistent, 𝑎(𝜐) has to be invariant under modular transformation:

Cha Charac acter ers  𝑎(𝜐) is written in terms of 𝑇𝑃 2𝑜 characters 𝑃 2𝑜 , 𝑊 2𝑜 , 𝑇 2𝑜 , 𝐷 2𝑜 and the Dedekind eta function 𝜃(𝜐) , e.g, 𝑇𝑃 32 hetero: 𝐹 8 × 𝐹 8 hetero: 𝑇𝑃 16 × 𝑇𝑃(16) hetero: , , the Jacobi’s abstruse identity:

SU SUSY SY br brea eaki king ng by by Co Comp mpac actif ific icat atio ion  Compactification on a circle The translation operator for 𝑌 9 satifies identify 𝑌 9 This comp. affects bosonic and fermionic states in the same way . SUSY is NOT broken . 0 2𝜌𝑆  Compactification on a twisted circle The translation operator for 𝑌 9 satifies 𝑌 7 𝑌 8 identify with 𝟑𝝆 rot. on the 7-8 plane This comp. affects bosonic and fermionic states in the different way . It induces the mass splitting 𝑌 9 between bosonic and fermionic states. 0 2𝜌𝑆 SUSY is broken [Rohm, (1984)]

Ou Outlin ine 1. Introduction 2. Heterotic Strings 3. 9D Interpolating models 4. 9D Interpolating models with WL 5. Summary

Interp rpolat ation betwe ween SU SUSY SY an and non-SU SUSY Y mo models An interpolating model is a lower dimensional string model relating two different higher dimensional string models continuously. Model 𝑵 𝟐 Model 𝑵 𝟑 10 dim. non-SUSY SUSY Comp. on a T-dual twisted circle non-SUSY Interpolating model 9 dim. ∞ 0 Radius 𝑆 In the large 𝑆 (small 𝑏 ) region, the cosmological constant is SUSY breaking : # of massless fermions, bosons If 𝒐 𝑮 = 𝒐 𝑪 , the cosmological constant is exponentially suppressed. [Itoyama, Taylor, (1987)]

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) Interp rpolat ation betwe  The one-loop partition function where the sum is taken over • 𝑆 → ∞: contribution from the zero winding # only • 𝑆 → 0: contribution from the zero momentum only

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) Interp rpolat ation betwe  The limiting case: 𝑺 → ∞ the one-loop partition function of SUSY 𝑇𝑃(32) heterotic model, which is vanishing SUSY is restored in 𝑺 → ∞ (𝒃 → 𝟏)

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) Interp rpolat ation betwe  The limiting case: 𝑺 → 𝟏 the one-loop partition function of 𝑇𝑃(16) × 𝑇𝑃(16) heterotic model non-SUSY SUSY 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) 𝑻𝑷(𝟒𝟑) SUSY 𝑇𝑃(32) model in 𝑆 → ∞ realizes non-SUSY 𝑇𝑃(16) × 𝑇𝑃(16) model in 𝑆 → 0 9D Int. model ∞ 0 Radius 𝑆

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) Interp rpolat ation betwe  Massless spectrum at generic R, massless states come from n=w=0 part Massless bosons • 9-dim. graviton, anti-symmetric tensor, dilaton: 𝟑 (𝟐) • Gauge bosons in adj rep of 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) × 𝑽 𝑯,𝑪 Massless fermions 𝑕 9𝜈 , 𝐶 9𝜈 • Fermions

Outlin Ou ine 1. Introduction 2. Heterotic Strings 3. 9D Interpolating models 4. 9D Interpolating models with Wilson line 5. Summary

Bo Boost st on mo mome mentum um lat attice • Considering 𝑒 -dimensional compactification, the boost in the momentum lattice corresponds to putting massless constant backgrounds , that is, adding the following term to the worldsheet action 𝑏 = 10 − 𝑒, ⋯ , 9 𝐵 = 𝑏, 𝐽 = 10 − 𝑒, ⋯ , 26 𝐷 𝑐𝑏 : metric and antisymmetric tensor, 𝐷 𝐽𝑏 : 𝑉(1) 16 gauge fields (WL) [Narain, Sarmadi, Witten, (1986)] • The 𝑒 -dimensional compactifications are classified by the transformation 𝑇𝑃(16+𝑒,𝑒) 𝑇𝑃 16+𝑒 ×𝑇𝑃(𝑒) , whose DOF agree with that of 𝐷 𝐵𝑏 . • In this work, we will consider one-dimensional compactification and put a single WL background 𝐵 = 𝐷 𝐽=1,𝑏=9 for simplicity.

Bo Boost st on mo mome mentum um lat attice 𝑏=9 and 𝑌 𝑆 𝑏=9 are changed as 𝐽=1 , 𝑌 𝑀 After turning on WL, the momenta of 𝑌 𝑀 boost and rotation 𝐽=1 𝑚 𝑀 is the left-moving momentum of 𝑌 𝑀 The effective change in the 1-loop partition function is introduction of WL

ǁ ǁ Th The fu fundam amental al re region of f mo modul uli sp spac ace NO! Do all the points in moduli space correspond to different models? 𝜐 as It is convenient to introduce a modular parameter ǁ 𝜐 2 (𝛽,𝛾) is invariant under the shift Momentum lattice Λ (𝛿,𝜀) The fundamental region of moduli space is 𝜐 1 − 2 2

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL  The one-loop partition function • 𝑆 → ∞: • 𝑆 → 0:

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL  The one-loop partition function  The limiting cases • 𝑆 → ∞: the 1st and 2nd lines survive non-SUSY SUSY • 𝑆 → 0: the 1st and 3rd lines survive 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) 𝑻𝑷(𝟒𝟑) For any WL 𝐵 , SUSY 𝑇𝑃 32 model in 𝑆 → ∞ non-SUSY realizes 9D Int. model ∞ 0 𝑇𝑃(16) × 𝑇𝑃(16) model in 𝑆 → 0 Radius 𝑆 + WL 𝑩

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL at generic R, massless states come from n=w=0 part  Massless spectrum Massless bosons • 9-dim. graviton, anti-symmetric tensor, dilaton: 𝟑 (𝟐) • Gauge bosons in adj rep of 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟓) × 𝑽(𝟐) × 𝑽 𝑯,𝑪 Massless fermions • 8

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL ∃  Massless spectrum a few conditions under which the additional massless states appear condition ① • two new massless states ： • two 𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL ∃  Massless spectrum a few conditions under which the additional massless states appear condition ② • two new massless states ： • two 𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟗 × 𝑻𝑷(𝟐𝟓)

ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi Interp rpolat atio ion betwe with WL  Summary of the conditions We have found the two conditions under which the additional massless states appear: condition ① condition ② Actually, there are only four inequivalent orbits in the fundamental region: 𝒐 1 = 𝟏 and 𝟑 (or −𝟑 ) 𝒐 𝟑 = −𝟐 and 𝟐 Condition 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) 𝑻𝑷(𝟐𝟗) × 𝑻𝑷(𝟐𝟓) Gauge gp 𝒐 𝑮 > 𝒐 𝑪 𝒐 𝑮 = 𝒐 𝑪