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Spectrum of CHL Dyons

Spectrum of CHL Dyons

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Spectrum of CHL Dyons

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  1. Spectrum of CHL Dyons Atish Dabholkar Tata Institute of Fundamental Research Indian Strings Meeting 2006 Puri

  2. A. D., Suresh Nampuri hep-th/0603066 • A. D., Davide Gaiotto hep-th/0612011 • A. D., Davide Gaiotto,Suresh Nampuri hep-th/0612nnn ISM06 PURI

  3. Plan • Proposal for dyon degeneracies in N=4 • Questions • Answers • Derivation ISM06 PURI

  4. Motivation • Exact BPS spectrum gives valuable information about the strong coupling structure of the theory. • Dyons that are not weakly coupled in any frame (naively), hence more interesting. • Counting of black holes including higher derivative corrections for big black holes. ISM06 PURI

  5. Heterotic on T4 £ T2 • Total rank of the four dimensional theory is 16 ( ) + 12 ( ) = 28 • N=4 supersymmetry in D=4 • Duality group ISM06 PURI

  6. CHL Orbifolds D=4 and N=4 • Models with smaller rank but same susy. • For example, a Z2 orbifold by {1,  T} •  flips E8 factors so rank reduced by 8. • T is a shift along the circle, X ! X +  R so twisted states are massive. • Fermions not affected so N =4 susy. ISM06 PURI

  7. Why CHL Orbifolds? • S-duality group is a subgroup of SL(2, Z) so counting of dyons is quite different. • Wald entropy is modified in a nontrivial way and is calculable. • Nontrivial but tractable generalization with interesting physical differences for the spectrum of black holes and dyons. ISM06 PURI

  8. S-duality group = 1(N) Because of the shift, there are 1/2 quantized electric charges (winding modes) This requires that c = 0 mod 2 which gives 0(2) subgroup of SL(2, Z). ISM06 PURI

  9. Spectrum of Dyons • For a ZN orbifold, the dyonic degeneracies are encapsulated by a Siegel modular form of Sp(2, Z) of level N and index k as a function of period matrices  of a genus two Riemann surface.. ISM06 PURI

  10. Sp(2, Z) • 4 £ 4 matrices g of integers that leave the symplectic form invariant: where A, B, C, D are 2£ 2 matrices. ISM06 PURI

  11. Genus Two Period Matrix • Like the  parameter at genus one ISM06 PURI

  12. Siegel Modular Forms • k() is a Siegel modular form of weight k and level N if under elements of G0(N). ISM06 PURI

  13. Fourier Coefficients • Define T-duality invariant combinations • Degeneracies d(Q) of dyons are given by the Fourier coefficients of the inverse of an image of . ISM06 PURI

  14. ISM06 PURI

  15. Big Black Holes • Define the discriminant which is the unique quartic invariant of SL(2) £ SO(22, 6) • For positive discriminant, big black hole exists with entropy given by ISM06 PURI

  16. Three Consistency Checks • All d(Q) are integers. • Agrees with black hole entropy including sub-leading logarithmic correction, log d(Q) = SBH • d(Q) is S-duality invariant under 1(N) ISM06 PURI

  17. 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? ISM06 PURI

  18. 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? ISM06 PURI

  19. 1) Why genus-two? • Dyon partition function can be mapped by duality to genus-two partition function of the left-moving heterotic string on CHL orbifolds. • Makes modular properties under subgroups of Sp(2, Z) manifest. • Suggests a new derivation of the formulae. ISM06 PURI

  20. 2) Microscopic Derivation Required twisted determinants can be explicitly evaluated using orbifold techniques (N=1,2 or k=10, 6) to obtain ISM06 PURI

  21. 3) Irreducibility Criteria • For electric and magnetic charges Qie and Qim that are SO(22, 6) vectors, define • Genus-two answer is correct only if I=1. • In general I+1 genus will contribute. ISM06 PURI

  22. 4) Negative discriminant states • States with negative discriminant are realized as multi-centered configurations. • In a simple example, the supergravity realization is a two centered solution with field angular momentum • The degeneracy is given by (2J + 1) in agreement with microscopics. ISM06 PURI

  23. k is a complicated beast • Fourier representation (Maass lift) Makes integrality of d(Q) manifest • Product representation (Borcherds lift) Relates to 5d elliptic genus of D1D5P • Determinant representation (Genus-2) Makes the modular properties manifest. ISM06 PURI

  24. String Webs • Quarter BPS states of heterotic on T4£ T2 is described as a string web of (p, q) strings wrapping the T2 in Type-IIB string on K3 £ T2 and left-moving oscillations. • The strings arise from wrapping various D3, D5, NS5 branes on cycles of K3 ISM06 PURI

  25. String junction tension balance ISM06 PURI

  26. ISM06 PURI

  27. M-lift of String Webs ISM06 PURI

  28. Genus-2 worldsheet is worldvolume of Euclidean M5 brane with various fluxes turned on wrapping K3 £ T2. The T2 is holomorphically embedded in T4 by Abel map. It can carry left-moving oscillations. • K3-wrapped M5-brane is the heterotic string. So genus-2 chiral partition fn of heterotic counts its left-moving BPS oscillations. ISM06 PURI

  29. Genus one gives electric states • Degeneracies of electric states are given by the Fourier coefficients of the genus-one partition fn. In this case the string web are just 1-dimensional strands. ISM06 PURI

  30. Computation • 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. ISM06 PURI

  31. 10 (Igusa cusp form) is the unique weight ten cusp form of Sp(2, Z). • Just as at genus one, 24 (Jacobi-Ramanujan function) is the unique weight 12 form of Sp(1, Z) » SL(2, Z). Hence the one-loop partition function is ISM06 PURI

  32. Z2 Orbifold • Bosonic realization of E8£ E8 string • Orbifold action flips X and Y. ISM06 PURI

  33. Twisted Partition Function • We need to evaluate the partition function on a genus two surface with twisted boundary conditions along one cycle. • Consider a genus-g surface. Choose A and B-cycles with intersections ISM06 PURI

  34. Period matrix • Holomorphic differentials • Higher genus analog of on a torus ISM06 PURI

  35. Sp(g, Z) • Linear relabeling of A and B cycles that preserves the intersection numbers is an Sp(g, Z) transformation. • The period matrix transforms as • Analog of ISM06 PURI

  36. Boson Partition Function • Period matrix arise naturally in partition fn of bosons on circles or on some lattice. • is the quantum fluctuation determinant. ISM06 PURI

  37. Double Cover ISM06 PURI

  38. Prym periods • Prym differentials are differentials that are odd across the branch cut • Prym periods ISM06 PURI

  39. Twisted determinants • We have 8 bosons that are odd. So the twisted partition function is ISM06 PURI

  40. X ! –X and X ! X +  R • Boson X » X + 2 R at self-radius Exploit the enhanced SU(2) symmetry (Jx, Jy, Jz) = (cos X, sin X, X) • X ! –X (Jx, Jy, Jz) ! (Jx, -Jy, -Jz) • X! X +  R (Jx, Jy, Jz) ! (-Jx, -Jy, Jz) ISM06 PURI

  41. Orbifold = Circle ISM06 PURI

  42. 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. ISM06 PURI

  43. 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. ISM06 PURI

  44. Schottky Relations ISM06 PURI

  45. 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 ISM06 PURI

  46. Odd Charges and Prym • In the orbifold, there are no gauge fields that couple to the odd E8 charges. Nevertheless, states with these charges still run across the B1 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. ISM06 PURI

  47. 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. ISM06 PURI

  48. Higher genus contributions • For example if then • Now genus three contribution is possible. • The condition gcd =1 is equivalent to the condition Q1 and Q5 be relatively prime. ISM06 PURI

  49. ISM06 PURI

  50. 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. ISM06 PURI