We live in a 4 dimensional space-time with 3 spactial and 1 time dimension. Being ten dimensional theories, if superstring theories are to describe our world, then the other six dimensions must be effectively explained to so compact that we cannot perceive them. They are believed to be compact, so small and curved that in order to detect them we need extremely high energies. Therefore, to describe the real world we compactify the remaining 6 dimensions. Compactification is usually done on Calabi Yau manifolds. Calabi Yau manifolds are complex manifolds of vanishing first chern class. Compactifications of superstring theories lead to moduli space problem, that is it gives rise to undetermined parameters. A compactification on a Calabi–Yau three-fold with fluxes can be called flux compactification. The fluxes can be understood as generalization of electric or magnetic fluxes. Compactifications with fluxes generates the potential required to fix the undetermined parameters and hence solves the moduli space problem. In this presentation I have discussed compactification of the six dimensions of the type IIB superstring theory with fluxes.

1. A Presentation on
Flux Compactifications of
Type IIB Superstring Theory
by
Rayyan Abdullah
Under supervision of
Dr Abbas Ali
Associate Professor
Department of Physics
Aligarh Muslim University

2. String Theory: A Brief Overview
• Basic idea: Particles are strings with different vibration modes representing
different particles.
• Bosonic String theory lives in 26 dimensions but its unrealistic (does not contain
fermions)
• Fermions are included through supersymmetry, resulting theory is ten
dimensional superstring theory.
• Five superstring theories: Type IIA and Type IIB, Type I and two heterotic
strings.
• These theories are connected through dualities.
• Five superstring theories are different corners of a more fundamental theory.
• Superstrings reduce to their corresponding supergravity theories at low energy.

3. Type IIB Superstring Theory
• It is 10 dimensional theory.
• It has N=1 supersymmetry.
• It is chiral.
• Tachyon is projected out through GSO projection.
Field Content
• Graviton 𝐺𝜇𝜈
• Anti-symmetric tensor 𝐵𝜇𝜈
• Dilaton Φ
• Axion 𝐶0, 2-form 𝐶𝑀𝑁, 4-form 𝐶𝑀𝑁𝑃𝑄 with self dual five-form 𝐹5.
It reduces to 10 dimensional SUGRA at low energy.

4. Calabi-Yau Manifolds
• Calabi-Yau manifolds are complex manifolds.
• They are compact manifolds.
• Calabi-Yau three folds have six real (or three complex) dimensions.
• Calabi-Yau four folds have four complex (or 8 real) dimensions.

5. Compactification
• Original idea: Kaluza-Klien Theory (5 dimensional
theory, 5th dimension compactified on a circle)
• Remaining six dimensions of superstrings are compact and unobservable.
• Compactification of superstrings done on six torus is far from reality.
• Compactifications on Calabi-Yau manifolds can may be phenomenological.
• We compactify 10 dimensional SUGRA on Calabi-Yau manifolds.
𝑀10 = 𝑀4 × 𝑀6
𝑀4 → 4𝐷 𝑀𝑖𝑛𝑘𝑜𝑤𝑠𝑘𝑖 𝑠𝑝𝑎𝑐𝑒𝑡𝑖𝑚𝑒
𝑀6 → 𝐶𝑎𝑙𝑎𝑏𝑖 − 𝑌𝑎𝑢 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑚𝑎𝑛𝑖𝑓𝑜𝑙𝑑

7. Flux Compactification
What are fluxes?
• Fluxes are generalization of Magnetic flux.
• Source of these fluxes are Dp-branes, which are higher dimensional objects.
• They generate warped metrics.
Why we need compactification with fluxes?
• Moduli space problem: Compactifications without
fluxes give scalar fields called moduli with undetermined
vevs.
• To fix vevs of moduli, we need potential.
• Flux generate that potential.

8. • The ten dimensional metric is
𝑑𝑠10
2
= 𝐺𝑀𝑁𝑑𝑥𝑀
𝑑𝑥𝑁
𝑀, 𝑁 = 0,1, … 9
• Compactified metric without fluxes
𝑑𝑠10
2
= 𝐺𝑀𝑁𝑑𝑥𝑀
𝑑𝑥𝑁
= 𝜂𝜇𝜈𝑑𝑥𝜇
𝑑𝑥𝜈
+ 𝑔𝑚𝑛(𝑦)𝑑𝑦𝑚
𝑑𝑦𝑛
𝑥𝜇 (𝜇 = 0,1,2,3) are coordinates of 𝑀4 and 𝑦𝑚 (𝑚 = 5,6 … .9) are
coordinates of compact manifold 𝑀
• Compactified metric with fluxes
𝑑𝑠10
2
= 𝐺𝑀𝑁𝑑𝑥𝑀𝑑𝑥𝑁 = 𝛥 𝑦 −1𝜂𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 + 𝛥 𝑦 1 2𝑔𝑚𝑛 𝑦 𝑑𝑦𝑚𝑑𝑦𝑛
where Δ(y) is the warp factor

9. Type IIB Flux Compactification
• No-go theorem under quite general conditions rules out type IIB
warped compactifications to Minkowski spaces in the presence of
fluxes and if no brane sources are included.
• No-go theorem can be evaded if only D7-branes and one-form flux
involved.
• Ten dimensional supergravity action
𝑆 =
1
2𝜅2 ∫ 𝑑10
𝑥 −𝐺 𝑅 −
𝜕𝜏
2 𝐼𝑚 𝜏 2 +
𝐺3
2
2 𝐼𝑚 𝜏
−
𝐹5
2
4
−
1
8𝑖𝜅2
𝐶4 ∧ 𝐺3 ∧ 𝐺3
⋆
𝐼𝑚 𝜏

10. • The compactified warped metric have the following form
𝑑𝑠10
2
= 𝐺𝑀𝑁𝑑𝑥𝑀
𝑑𝑥𝑁
= 𝑒2𝐴 𝑦
𝜂𝜇𝜈𝑑𝑥𝜇
𝑑𝑥𝜈
+ 𝑒−2𝐴 𝑦
𝑔𝑚𝑛(𝑦)𝑑𝑦𝑚
𝑑𝑦𝑛
• Consistency conditions
 Conditions on fluxes
 Tadpole cancellation condition
Superpotential
• Fluxes generate superpotential which stabilizes all the moduli fields.
• All moduli are stabilized in type IIB flux compactification.
• This results in moduli stabilized and compactified supergravity theory.

11. Thank You