Flux Compactifications An Overview Sandip Trivedi Tata Institute

Flux Compactifications: An Overview Sandip Trivedi Tata Institute of Fundamental Research, Mumbai, India Korea June 2008

• Introduction & Motivation • A Toy Model • IIB String Theory with Fluxes • IIA/I String Theory with Fluxes • The Landscape And Conclusions

Flux Compactifications • Curl Up Extra Dimensions. • Turn on Fluxes Along These Directions. • Fluxes are generalisations of Magnetic Flux in Maxwell Theory.

Internal directions Non-compact directions

Introduction • Compactifications without Flux: Unsatisfactory. • Have unwanted flat directions. Called Moduli. • These are absent in Flux compactifications. With interesting consequences.

Moduli Stabilisation • Typically Many Flat Directions in String Compactifications. (~100) • Different Sizes and Shapes. Physical Parameters e. g. , G_N, alpha, vary along these directions

String Theory: Typically lead to run-away situations. Not stable vacua.

Introduction Turning on Fluxes: Leads To Controlled Stabilisation of Moduli. A mimimum which lies in some region of field space where approximations are valid.

Flux Compactifications ØImportant In Phenomenology: a) Calculate Standard Model Couplings b) Supersymmetry Breaking

ØImportant In Cosmology Positive Vacuum Energy: De. Sitter Universe Slowly Varying Potential: Inflation

Flux Compactifications Another Advantage: • Concentrated Flux gives rise to large Warping. • Natural way to constructed models of Randall Sundrum (or large extra dimension) type.

A Toy Model Why Does Flux Help? Any Value of R 1, R 2 Allowed: Moduli R 2 R 1 Torus Is Flat, Curvature Vanishes.

Flux Compactifications Size Modulus: Shape Modulus:

Turn On Magnetic Field R 2 R 1 Dirac Quantisation : Extra Cost In Energy: E A A=R 1 R 2

Toy Model Continued • Lesson: Flux tends to expand the size of directions along which it extends. • Also it tends to contract the size of directions in which it does not extend. • Balancing these leads to moduli stabilisation.

Type IIB String Theory: Promising Corner to Begin Giddings, Kachru, Polchinski Fluxes: Three-Forms: Five-Form: Branes: D 3 (fill 3+1 dimensions), D 7, 03, 07.

Type IIB String Theory: Promising Corner to Begin Giddings, Kachru, Polchinski Fluxes: Three-Forms: Five-Form: Branes: D 3 (fill 3+1 dimensions), D 7, 03, 07. (N 0 5 -Branes/Planes)

Type IIB String Theory: must be closed. • Such closed and non-trivial fluxes lie in a vector space. It’s dimensionality is a topological invariant, . • Fluxes are also quantised.

More On Fluxes Total Number of allowed Fluxes: Exponential in is finite, determined by tadpole condition:

More on Fluxes • For reasonably big the total number of allowed fluxes can be very large. • is quite common. • This gives rise to an exponentially large number of vacua.

More on Moduli Stabilisation • The moduli of interest are size and shape deformations of the Calabi-Yau space. • These get a mass, , • where, , is the Radius of compactification. • Thus the lifting of these moduli can be studied in a 4 dim. Effective field theory.

Shape Moduli Stabilisation • A superpotential arises at tree-level. • This depends on the shape moduli and the axion-dilaton. • Generically this fixes all these moduli.

Shape Moduli Stabilisation And is the holomorphic-three form on the Calabi Yau, which depends on the shape moduli. Gukov, Vafa, Witten; Giddings, Kachru, Polchinski

Size Moduli Stabilisation And Susy Breaking Kachru, Kallosh, Linde and Trivedi (KKLT) • Non-perturbative Corrections to Superpotential can also arise. • These are dependent on Size moduli and can stabilise them.

This can stabilise the size moduli Giving rise to a Vacum with negative Cosmological Constant.

Breaking Susy • Susy Breaking can be introduced, e. g. due to Anti-D 3 Branes. • The resulting vacua can then have a positive cosmological constant.

I) II)

Spectrum: String Modes KK Modes Shape Moduli Size Modulus gravitinio

Mixed Anomaly Moduli Mediation (Choi, Nilles, et. Al. ) • The F component of the size modulus:

• The resulting moduli mediated contribution to soft masses: • This can be comparable to the anomaly mediated contribution • For

Flavour Violations Might be Suppressed • Flavour structure related to shape moduli. • Susy breaking related to size moduli. • In this way the origin of flavour and susy breaking are naturally segregated, and flavour violation in soft susy breaking terms can be small. (Choi et. Al. , Conlon)

Variations on the Theme • Use Higher Derivative corrections to stabilise Size Moduli. • Balasubramanium, Conlon, Quevedo. • Etc

Type I Theory • Use Open String Fluxes to stabilise some of the moduli. • In Type I for example Kahler moduli can be stabilised in this way. • Also (on Torus) complex structure moduli. (Bacchas, Antoniadis, Maillard, Kumar…)

Type IIA String Theory • Both Open and Closed String Moduli can be stabilised at Tree-Level. (Derendinger, Kounnas, Petropoulos, Zwirner; de. Wolfe, Giryavets, Kachru, Taylor ) • Fluxes:

Fluxes in IIA String Theory • Superpotential: • Depends on both size and shape moduli

Type IIA Continued • Taking some fluxes to be large the volume can be stabilised at a large value, and dilaton at a small value.

IIA With Fluxes • The Manifolds are not Calabi Yau any more. • Instead they are half-flat manifolds. Need to be understood better.

Some More Recent Developments • Use Fluxes To Study Field Theory Models of Dynamical Susy Breaking. • Fluxes result in Geometrizing some aspects. (Diaconescu et. al, Kachru et. al, Verlinde et. al. )

Recent Developments • Most of our knowledge is restricted to when the volume is big and warping is small. • Attempts to go beyond are underway. Compute corrections to Kahler potential and superpotential (if any). (Giddings, Maharana, Douglas et. al. )

Recent Developments • Do not start with a Calabi Yau Manifold. • Instead consider a manifold with negative curvature, e. g. Nil Manifold. • This can lead to simpler constructions of d. S vaccua. • (Silverstein).

Landscape • Many many different vacuua • Exponential large number • Large number arise because starting with a given compactification one can turn on many different kinds of fluxes. Third Betti number

Landscape Bousso, Polchinski Susskind Many different vacua. Many different directions Varying cosmological constants. Transitions between them are possible.

Landscape: Many Questions: • Is String Theory Predictive? • Who ordered all the other vacua? • How do we find the Standard Model vacuum? • Should we give up on Naturalness? • The Anthropic Principle?

Landscape: My Views: • Anthropics: Should be the last resort. Conventional explanations have testable consequences.

Landscape: • Too early to conclude that string theory not predictive. By inputing some data( ) we might be able to predict a lot. • Key Question: In coupling constant space how closely spaced are the standard modellike vacua. We don’t know enough about theory to answer this yet. • Also, understanding time, the initial singularity etc might help.

Landscape • What is clear though is that at our present level of understanding, String Theory is more akin to a general framework than a specific UV completion of the standard model. • So we should use it as a framework for model building and for understanding gauge theories. • This might well be its best use as we lead up to the LHC.

Landscape • Statistics: Much maligned. • My main worry : don’t know enough about string theory to make reliable estimates. • An efficient way to zero in on small cosmological constant vaccua would be more useful. Don’t know how to do this yet.

Ashok, Douglas The number distribution of vacua for a small cosmological constant is flat: