Spectrum of CHL Dyons II

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Spectrum of CHL Dyons (II)
Atish Dabholkar

• Tata Institute of Fundamental Research

First Asian Winter School
Phoenix Park
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Dyon Partition Function

• Recall that dyon degeneracies for ZN 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

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Dyon degeneracies
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• 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

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Sp(2, Z)

• 4 4 matrices g of integers that leave the
  symplectic form invariant
• where A, B, C, D are 2 2 matrices.

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Genus Two Period Matrix

• Like the ? parameter of a torus
• transforms by fractional linear
  transformations

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Siegel Modular Forms

• ?k(?) is a Siegel modular form of weight k and
  level N if
• under elements of a
  specific subgroup G0(N) of Sp(2, Z)

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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.

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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?

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• 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?

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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

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Heterotic Type-II Duality

• Heterotic on T4 IIA on K3
• With T6 T4 T2
• Then the T-duality group that is
• The part acting on the T2 factor is

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String-String Triality

• The three groups are interchanged for heterotic,
  Type-IIA and Type-IIB

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Type-IIB

• The S field in heterotic gets mapped to the T
  field in Type-IIB which is the complex structure
  modulus of the T2 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.

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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.

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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.

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String junction tension balance
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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 Qe string can combine with Qm string into a
  QeQm string.

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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.

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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.

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M-theory IIB

• M-theory on a is dual to IIB on S1
• 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.

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M-lift of String Webs

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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.

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• Genus-2 world-sheet 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
  we are led to genus-2 chiral partition fn of
  heterotic counting its left-moving BPS
  oscillations.

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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 S1
• K3-wrapped M5 brane is dual to the heterotic
  string.

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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.

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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.

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Microscopic Derivation

• For orbifolds, requred twisted determinants
  can be explicitly evaluated using orbifold
  techniques (N1,2 or k10, 6) to obtain

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S-duality Invariance

• The physical S-duality group can be embedded into
  the Sp(2, Z)

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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.

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S-duality

• Different expansion for different charges.
  Consider a function with Z2 symmetry.

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S-dual Prescription

• Here the meaning of Z2 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.

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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.
  

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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.

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• 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.

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Negative discriminant states

• Consider a charge configuration
• Degeneracy d(Q) N

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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

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Two centered solution

• One electric center with
• One magnetic center with
• Field angular momentum is N/2

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Supergravity Analysis

• The relative distance between the two centers is
  fixed by solving Denefs constraint.
• Angular momentum quantization gives
• (2 J 1) N states in agreement with the
  microscopic prediction.
• Intricate moduli dependence.

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CHL Orbifolds D4 and N4

• 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.

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Z2 Orbifold

• Bosonic realization of E8 E8 string
• Orbifold action flips X and Y.

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Prym periods

• Prym differentials are differentials that are odd
  across the branch cut
• Prym periods

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Twisted determinants

• We have 8 bosons that are odd. So the twisted
  partition function is

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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)

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Orbifold Circle

• 
  

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• 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.

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Schottky Relations

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• 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

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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.

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• 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.

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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.