1. Spectrum of CHL Dyons (II) Tata Institute of Fundamental Research First Asian Winter School Phoenix Park Atish Dabholkar

2. 15 JAN 2007Korea2 Dyon Partition Function Recall that dyon degeneracies for Z N CHL orbifolds are given in terms of the Fourier coefficients of a dyon partition function We would like to understand the physical origin and consequences of the modular properties of  k

3. 15 JAN 2007Korea3 Dyon degeneracies

4. 15 JAN 2007Korea4 The complex number ( , , v) naturally group together into a period matrix of a genus-2 Riemann surface   k is a Siegel modular form of weight k of a subgroup of Sp(2, Z) with

5. 15 JAN 2007Korea5 Sp(2, Z) 4 £ 4 matrices g of integers that leave the symplectic form invariant: where A, B, C, D are 2£ 2 matrices.

6. 15 JAN 2007Korea6 Genus Two Period Matrix Like the  parameter of a torus transforms by fractional linear transformations

7. 15 JAN 2007Korea7 Siegel Modular Forms  k (  ) is a Siegel modular form of weight k and level N if under elements of a specific subgroup G 0 (N) of Sp(2, Z)

8. 15 JAN 2007Korea8 Three Consistency Checks All d(Q) are integers. Agrees with black hole entropy including sub-leading logarithmic correction, log d(Q) = S BH d(Q) is S-duality invariant.

9. 15 JAN 2007Korea9 Questions 1) Why does genus-two Riemann surface play a role in the counting of dyons? The group Sp(2, Z) cannot fit in the physical U-duality group. Why does it appear? 2) Is there a microscopic derivation that makes modular properties manifest?

10. 15 JAN 2007Korea10 3) Are there restrictions on the charges for which genus two answer is valid? 4) Formula predicts states with negative discriminant. But there are no corresponding black holes. Do these states exist? Moduli dependence? 5) Is the spectrum S-duality invariant?

11. 15 JAN 2007Korea11 String Webs Quarter BPS states of heterotic on T 4 £ T 2 is described as a string web of (p, q) strings wrapping the T 2 in Type-IIB string on K3 £ T 2 and left-moving oscillations. The strings arise from wrapping various D3, D5, NS5 branes on cycles of K3

12. 15 JAN 2007Korea12 Heterotic $ Type-II Duality Heterotic on T 4 $ IIA on K3 With T 6 = T 4 £ T 2 Then the T-duality group that is The part acting on the T 2 factor is

13. 15 JAN 2007Korea13 String-String Triality The three groups are interchanged for heterotic, Type-IIA and Type-IIB

14. 15 JAN 2007Korea14 Type-IIB The S field in heterotic gets mapped to the T field in Type-IIB which is the complex structure modulus of the T 2 in the IIB frame.The S-duality group thus becomes a geometric T-duality group in IIB. Description of dyons is very simple. Electric states along a-cycle and magnetic states along the b-cycle of the torus.

15. 15 JAN 2007Korea15 Half-BPS states For example, a half-BPS electric state corresponds to say a F1-string or a D1- string wrapping the a cycle of the torus. The dual magnetic state corresponds to the F1-string or a D-string wrapping the b- cycle of the torus. A half-BPS dyon would be a string wrapping diagonally.

16. 15 JAN 2007Korea16 Quarter-BPS Quarter-BPS dyons are described by (p, q) string webs. The basic ingredient is a string junction where an F1-string and a D1-string can combine in to a (1, 1) string which is a bound state of F1 and D1 string. More general (p, 0) and (q, o) can combine into a (p, q) string.

17. 15 JAN 2007Korea17 String junction tension balance

18. 15 JAN 2007Korea18 Supersymmetry Such a junction is quarter-BPS if tensions are balanced. More generally we can have strands of various effective strings for example K3- wrapped D5 or NS5 string with D3-branes dissolved in their worldvolumes. A Q e string can combine with Q m string into a Q e +Q m string.

19. 15 JAN 2007Korea19 String Web String junction exists in non-compact space. We can consider a string web constructed from a periodic array of string junctions. By taking a fundamental cell we can regard it as a configuration on a torus. The strands of this configuration can in addition carry momentum and oscillations. Lengths of strands depend torus moduli.

20. 15 JAN 2007Korea20

21. 15 JAN 2007Korea21 Effective Strings Note that the 28 charges in heterotic string arise from wrapped D3-branes, D5, NS- branes etc. A general strand can be a D5 brane or an NS-5 brane with fluxes turned on two cycles of the K3 which corresponds to D3- brane charges. In M-theory both D5 and NS lift to M5.

22. 15 JAN 2007Korea22 M-theory $ IIB M-theory on a is dual to IIB on S 1 Type-IIB has an SL(2, Z) B in ten dimensions under which NS5 and D5 branes are dual. It corresponds to the geometric SL(2, Z) M action on the M- torus. So NS5 in IIB is M5 wrapping a cycle and D5 is M5 wrapping the b cycle of M-torus.

23. 15 JAN 2007Korea23 M-lift of String Webs

24. 15 JAN 2007Korea24 Partition Function To count the left-moving fluctuations of the string web, we evaluate the partition function by adding a Euclidean circle and evolving it along the time direction. This makes the string web into a Euclidean diagram which is not smooth in string theory at the junctions but is smooth in M-theory.

25. 15 JAN 2007Korea25 Genus-2 world-sheet is worldvolume of Euclidean M5 brane with various fluxes turned on wrapping K3 £ T 2. The T 2 is holomorphically embedded in T 4 (by Abel map). It can carry left-moving oscillations. K3-wrapped M5-brane is the heterotic string. So we are led to genus-2 chiral partition fn of heterotic counting its left- moving BPS oscillations.

26. 15 JAN 2007Korea26 Genus one gives electric states Electric partition function is just a genus- one partition function of the left-moving heterotic string because in this case the string web are just 1-dimensional strands of M5 brane wrapped on K3 £ S 1 K3-wrapped M5 brane is dual to the heterotic string.

27. 15 JAN 2007Korea27 Genus-two gives dyons Genus-2 determinants are complicated. One needs determinants both for bosons and ghosts. But the total partition function of 26 left-moving bosons and ghosts can be deduced from modular properties.

28. 15 JAN 2007Korea28 Theta function at genus g Here are g-dimensional vectors with entries as (0, ½). Half characteristics. There are 16 such theta functions at genus 2. Characteristic even or odd if is even or odd. At genus 2, there are 10 even and 6 odd.

29. 15 JAN 2007Korea29 Microscopic Derivation For orbifolds, requred twisted determinants can be explicitly evaluated using orbifold techniques (N=1,2 or k=10, 6) to obtain

30. 15 JAN 2007Korea30 S-duality Invariance The physical S-duality group can be embedded into the Sp(2, Z)

31. 15 JAN 2007Korea31 Invariance From the transformation properties its clear that  k is invariant because Furthermore the measure of inverse Fourier transform is invariant. However the contour of integration changes which means we have to expand around different points.

32. 15 JAN 2007Korea32 S-duality Different expansion for different charges. Consider a function with Z 2 symmetry.

33. 15 JAN 2007Korea33 S-dual Prescription Here the meaning of Z 2 invariance is that the Laurent expansion around y is the same as the Laurent expansion around y -1 The prescription is then to define the degeneracies by the Laurent expansions for `primitive charges’. For all other charges related to the primitive charges by S-duality.

34. 15 JAN 2007Korea34 Higher genus contributions For example if then Now genus three contribution is possible. The condition gcd =1 is equivalent to the condition Q 1 and Q 5 be relatively prime.

35. 15 JAN 2007Korea35

36. 15 JAN 2007Korea36 Dual graph Face goes to a point in the dual graph. Two points in the dual graph are connected by vector if they are adjacent. The vector is equal in length but perpendicular to the common edge. String junction goes to a triangle.

37. 15 JAN 2007Korea37 If one can insert a triangle at a string junction then the junction can open up and a higher genus web is possible. Adding a face in the web is equivalent to adding a lattice point in the dual graph. If the fundamental parellelogram has unit area then it does not contain a lattice point. is the area tensor. Unit area means gcd of all its components is one.

38. 15 JAN 2007Korea38 Negative discriminant states Consider a charge configuration Degeneracy d(Q) = N

39. 15 JAN 2007Korea39 Big Black Holes Define the discriminant which is the unique quartic invariant of SL(2) £ SO(22, 6) Only for positive discriminant, big black hole exists with entropy given by

40. 15 JAN 2007Korea40 Two centered solution One electric center with One magnetic center with Field angular momentum is N/2

41. 15 JAN 2007Korea41 Supergravity Analysis The relative distance between the two centers is fixed by solving Denef’s constraint. Angular momentum quantization gives (2 J +1) » N states in agreement with the microscopic prediction. Intricate moduli dependence.

42. 15 JAN 2007Korea42 CHL Orbifolds D=4 and N=4 Models with smaller rank but same susy. For example, a Z 2 orbifold by {1,  T}  flips E 8 factors so rank reduced by 8. oT is a shift along the circle, X ! X +  R so twisted states are massive. oFermions not affected so N =4 susy.

43. 15 JAN 2007Korea43 Z 2 Orbifold Bosonic realization of E 8 £ E 8 string Orbifold action flips X and Y.

44. 15 JAN 2007Korea44 Prym periods Prym differentials are differentials that are odd across the branch cut Prym periods

45. 15 JAN 2007Korea45 Twisted determinants We have 8 bosons that are odd. So the twisted partition function is

46. 15 JAN 2007Korea46 X ! –X and X ! X +  R Boson X » X + 2  R at self-radius Exploit the enhanced SU(2) symmetry (J x, J y, J z ) = (cos X, sin X,  X) X ! –X (J x, J y, J z ) ! (J x, -J y, -J z ) X! X +  R (J x, J y, J z ) ! (-J x, -J y, J z )

47. 15 JAN 2007Korea47 Orbifold = Circle

48. 15 JAN 2007Korea48 Express the twisted determinant in terms of the untwisted determinant and ratios of momentum lattice sums. Lattice sums in turn can be expressed in terms of theta functions. This allows us to express the required ratio of determinants in terms of ratio of theta functions.

49. 15 JAN 2007Korea49 Schottky Relations

50. 15 JAN 2007Korea50 Multiplying the untwisted partition fn with the ratios of determinants and using some theta identities we get Almost the right answer except for the unwanted dependence on Prym

51. 15 JAN 2007Korea51 Odd Charges and Prym In the orbifold, there are no gauge fields that couple to the odd E 8 charges. Nevertheless, states with these charges still run across the B 1 cycle of the genus two surface in that has no branch cut. Sum over the odd charges gives a theta function over Prym that exactly cancels the unwanted Prym dependence.

52. 15 JAN 2007Korea52 Orbifold partition function obtained from string webs precisely matches with the proposed dyon partition function. The expression for  6 in terms of theta functions was obtained by Ibukiyama by completely different methods. Our results give an independent CFT derivation.

53. 15 JAN 2007Korea53 Conclusions M-lift of string webs illuminates the physical and mathematical properties of dyon partition fn manifest. It makes that makes the modular properties manifest allowing for a new derivation. Higher genus contributions are possible. Physical predictions such as negative discriminant states seem to be borne out.