1. Heterotic strings and fluxes Based on: K. Becker, S. Sethi, Torsional heterotic geometries, to appear. K. Becker, C. Bertinato, Y-C. Chung, G. Guo, Supersymmetry breaking, heterotic strings and fluxes, to appear

2. Generic string compactifications involve both non-trivial metrics and non-trivial background fluxes. Flux backgrounds have mostly been studied in the context of type II string theory and generically involve NS-NS and R-R fluxes. The analysis of these backgrounds can be done in the supergravity approximation since the size is not stabilized by flux at the perturbative level. However, quantum corrections are hard to describe since the string coupling constant is typically of order 1....

3. In this lecture we will be discussing the construction of flux backgrounds for heterotic strings. Such compactifications involve only NS fields: namely the metric and 3-form H. The dilaton is never stabilized. Spaces with non-vanishing H are called torsional spaces. The heterotic string is a natural setting for building realistic models of particle phenomenology. The approach taken in the past is to specify a compact Kaehler six-dimensional space together with a holomorphic gauge field. For appropriate choices of gauge bundle it has been possible to generate space time GUT gauge groups like E 6, SO(10) and SU(5). Candelas, Horowitz, Strominger,Witten, '85 Braun,He, Ovrut, Pantev, 0501070 Bouchard, Donagi, 0512149

4. Progress in this direction has been slow. On the one side we do not have algebraic geometry techniques available to describe torsional space. On the other there has been a lack of concrete examples. orientifolds of K3xT 2 quotients of nilmanifolds orbifolds It is natural to expect that many of the interesting physics and consequences for string phenomenology which arise from type II fluxes (and their F-theory cousins) will also be found in generic heterotic compactifications. M. Becker, L-S. Tseng, Yau, 0807.0827 Dasgupta, Rajesh, Sethi, 9908088 Fernandez, Ivanov, Ugarte, Villacampa, 0804.1648

5. The number of flux backgrounds for heterotic strings is of the same order of magnitude than the number of flux backgrounds in type II theories... Type IIB: M: Calabi-Yau 3-fold, Heterotic strings: M: complex and M

6. torsional background Complex manifold with hermitian metric Hull+Witten, '85 Strominger, '86

7. The goal of this talk: construct new classes of torsional heterotic geometries using a generalization of string-string duality in the presence of flux. We will work with a concrete metric. use the intuition gained from duality to obtain a more general a set of torsional spaces. comments about phenomenology: susy breaking and gauge symmetry breaking.

8. Method of Construction

9. M-theory In the presence of flux the space-time metric is no longer a direct product warp factor coordinates of the CY 4 Kaehler form of CY 4

10. The warp factor satisfies the differential equation 8-form which is quartic in curvature Since X 8 is conformal invariant (Chern+Simons, '74) this is a Poisson equation. number of M2 branes Euler ch. of CY 4 The data specifying this background are the choice of CY 4 and the primitive flux. As long as the total charge vanishes a solution exists. A solution exists if

11. Additional restrictions: the CY4 is elliptically fibred so that we can lift the 3d M-theory background to 4d. By shrinking the volume of the fiber to zero this can be converted in a 4d compactification of type IIB with coupling constant determined by the complex structure of the elliptic fiber. The G 4 flux lifts to. together with gauge bundles on 7-branes. Susy requires

12. the CY 4 admits a K3 fibration. The heterotic string appears by wrapping the M5 brane on the K3 fibers. The M5 brane supports a chiral 2-form. Since the number of self-dual 2-forms of K3 is 3 there appear 3 compact scalars parameterizing a T 3. In the F-theory limit this T 3 becomes a T 2. G 4 flux can be included and the choice of flux will determine how the T 2 varies over the base. Schematically the metric of the space on which the heterotic string is propagating is This is almost an existence proof for the heterotic torsional geometries...

13. Moreover, we will assume that the CY 4 has a non-singular orientifold locus so that the elliptic fibration is locally constant. We are left with type IIB with constant coupling...

14. The orientifold locus Type IIB on an elliptic CY 3 with constant coupling Symmetry base coordinates

15. The metric for an elliptic CY space Away from the singular fibers there is a U(1)xU(1) isometry... In the semi-flat approximation (stringy cosmic string):

16. 5-form flux warp factor 3-form flux Type IIB background

17. Orientifold action Quotient of type IIB on CY 3 by Decompose the forms The forms which are invariant under Z 2 are where

18. Decompose integral harmonic 1-forms of the fiber Under an transformation Locally we can write use this connection to twist the torus fiber of the CY space. the 2-form transforms as

19. The conditions for unbroken susy are a repackaging of the susy conditions on the type IIB side. In particular any elliptic CY 3 space gives rise to a torsional space.. Type I background The dilaton is

20. Torsional heterotic geometry Dictionary between type I and heterotic

21. The Bianchi identity

22. Type IIB Bianchi identity...giving rise to the type I Bianchi identity T-duality maps type I flux

23. In the large radius limit on the het side, which corresponds to large base and small fiber in type IIB, this is a non-linear differential equation for the dilaton (of Monge-Ampere type if the base is K3). which we can solve in the large radius expansion, which means for a large base and a large fiber. So the existence of a solution is guaranteed. Using the type IIB Bianchi identity becomes the differential equation for the warp factor

24. Topology change

25. When the fiber is twisted the topology of the space changes. A caricature is a 3-torus with NS-flux Using the same type of argument one can show that the twisting changes the betti numbers of the torsional space compared to the CY3 we started with. Topology change is the `mirror' statement to moduli stabilization in type IIB. Some fields are lifted from the low-energy effective action once flux is included....

26. Breaking E 8 anti-self dual (1,1) form Gauge fields with components along K3 K. Becker, M. Becker, J-X. Fu, L-S. Tseng, S-T Yau,0604137

27. SUSY

28. susy N=2 N=1 N=1' N=0 choices of (1,1) ASD (2,0) SD (0,2) SD (2,0)+(0,2) All of these backgrounds satisfy the e.o.m. in sugra. To solve the Bianchi identity gauge fields along the fiber have to be included.

29. Generalized solutions

30. A further generalization is possible Here and is the B field in the two fiber directions. The low energy effective action is invariant under

31. Conclusion We have seen how to construct a torsional space given the data specifying an elliptic CY space. There are many more torsional spaces than CY's. Many properties which are generic for compactifications of heterotic strings on Kaehler spaces may get modified once generic backgrounds are considered.

32. The End