Spectrum of CHL Dyons I

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Spectrum of CHL Dyons (I)
Atish Dabholkar

• Tata Institute of Fundamental Research

First Asian Winter School
Seoul
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Motivation

• Exact BPS spectrum gives valuable information
  about the strong coupling structure of the
  theory. The quarter-BPS Dyons are not weakly
  coupled in any frame (naively), hence more
  interesting.
• Nontrivial information about the bound states of
  KK, NS5, F1, P

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Comparison with black holes.

• When string coupling is large the dyonic state
  will gravitate and form a BPS black hole.
• If we know the exact spectrucm beyond the leading
  order we can carry out precise comparison with
  black hole entropy including higher derivative
  corrections to black hole entropy. Useful
  diagnostics..

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Black Hole Entropy.

• Macroscopic (effective action)
• Microscopic (counting microstate)
• 
  

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Striking Agreement ..

• The macroscopic side can be analyzed using the
  entropy function formalism described earlier in
  Sens lectures.
• Surprisingly, precise counting is possible for
  these dyons with N4 supersymmetry.
• There is impressive agreement even for
  subleading terms between Wald entropy and
  statistical entropy.

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Plan

• The partition function for the CHL dyons has nice
  modular properties under group Sp(2, Z) which
  cannot be accommodated in the physical duality
  group.
• We want to understand the origin and
  consequence of modular properties and duality
  invariance of the resulting spectrum.

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• We will motivate these results by analogy with
  half-BPS states..
• Represent dyons as string webs in Type-II. It
  will allow for a new, more geometric derivation
  of the partition function that makes the modular
  properties manifest.
• This gives more insight into the physical content
  of the partition function resolving questions
  about the range of validity.

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Dyon Degeneracies

• Recall that dyon degeneracies for ZN CHL
  orbifolds are given in terms of the Fourier
  coefficients of a dyon partition function
• In previous lecture it was constructed
  algebraically but here we would like to explore
  its modular properties.

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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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• Siegel modular forms have a rich mathematical
  structure. We would like to explain these
  mathematical concepts and understand the
  underlying physics.
• Their modular properties are of physical interest
  to derive black hole entropy and to show
  S-duality invariance.
• We focus mainly on the case N1 with k10 and
  later on N2 with k6 which illustrates all the
  main points.

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Heterotic on T4 T2

• Total rank of the four dimensional theory is
• 16 ( ) 12 (
  ) 28
• N4 supersymmetry in D4
• Duality group

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Massless fields

• There are 28 vectors transforming linearly under
  the T-duality group. The S-duality group
  exchanges electric magnetic.
• Axion-dilaton field
• where ? is the 4d dilaton and a is axion.
• This lives on the coset
• SL(2, Z)\SL(2, R)/SO(2).

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S-duality Group

• Electric-magnetic duality
• Acts on the axion dilaton field by

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Half-BPS states

• Consider Heterotic on T4 S1 S1
• A heterotic state (n, w) with winding w and
  momentum n. Two charges in four dimensions.
  
• It is BPS if the right-moving oscillators are in
  the ground state.

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• It can carry arbitrary left-moving oscillations
  subject to Virasoro contraint
• Here NL is the number operator of 24 left-moving
  bosons. BPS formula M PR .

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• Distribute energy N -1 among oscillations of
  different frequencies. Find all possible sets of
  integers min such that
• Partition function

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• Consider single oscillator with frequency n
• Altogether,
• ?24 is the Jacobi discriminant, modular form
  of weight 12.

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• Partition function
• One loop partition function of chiral bosonic
  string, 24 left-moving bosons

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• Modular property and asymptotics
• Follows from the fact that ?24(q) is a modular
  form of weight 12 ?24(q) q, for small q.
  Ground state energy is -1.

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• Microscopic degeneracy
• Large N asymptotics governed by high temperature,
  ? ! 0 limit
• Evaluate by saddle point method.

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• The saddle occurs at
• The degeneracy goes as
• For general charge vector Qe in the Narain
  lattice,
• in precise agreement with entropy of small
  black holes.

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Half-BPS States

• A general electric state is specified by a charge
  vector
• in the Narain Lattice which is an even
  integral lattice in 28 dimensional Lorentzian
  space of signature (22, 6). Even means the
  length-squared of all charges is even.

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Chemical Potential

• The degeneracy d(Q) of a given state of charge Q
  depends only the T-duality invariant combination
• There is a chemical potential ? conjugate to
  this integer.

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Electric Partition function

• For a given ZN CHL orbifold the the electric
  partition function is
• Electric degeneracies given in terms Fourier
  coefficients by

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Modular Properties

• One finds that ?k(?) is a modular form of a
  subgroup of SL(2, Z) of weight k
• Note that SL(2,Z) Sp(1, Z) is the modular
  group of a genus one Riemann surface.

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Cusp form

• N1
• is the well-known Jacobi-Ramanjuan function.
  Its a unique cusp form of weight 12 of Sp(1,Z)
• N2

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Quarter BPS States

• A dyon is specified by charge vector
• that is a vector of SO(22,6) and a doublet of
  SL(2).
• Define T-duality invariant integers

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Chemical Potentials

• There are three chemical potentials
• (?, ?, v) conjugate the three integers
• Write a partition fn depending on these chemical
  potentials. Degeneracies d(Q) of dyons are
  given by the Fourier coefficients of the dyon
  partition function.

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Dyon Partition function

• The partition function can be written as
• For other values of N one has ?k(?) instead of
  ?10 with

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Igusa Cusp form

• Just as ?24 ?12 was the unique cusp form of
  weight 12 of Sp(1, Z), here ?10 is the unique
  Siegal modular form of weight 10 of the group
  Sp(2, Z).
• For N2 we will encounter ?6
• Both have explicit representation in terms of
  theta functions.

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Dyon degeneracies
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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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Duality Invariance

• Note that T-duality invariance is assumed in this
  proposal and is built in because the
  degeneracies depend only on invariant
  combinations.
• S-duality invariance on the other hand is
  nontrivial and is not obvious. Modular properties
  of Siegel form will be crucial to demonstrate it.

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Genus-2 Riemann Surface

• The objects ? and Sp(2, Z) are naturally
  associated with genus-2.
• Consider a genus-g surface. Choose A and B-cycles
  with intersections

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Period matrix

• Holomorphic differentials
• Higher genus analog of on a torus

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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

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Boson Partition Function

• Period matrix arise naturally in partition fn of
  bosons on circles or on some lattice.
• is the quantum fluctuation determinant.

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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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1) Why genus-two?

• Dyon partition function can be mapped by duality
  to genus-two partition function of the
  left-moving heterotic string on T6 or on CHL
  orbifolds.
• Makes modular properties under subgroups of Sp(2,
  Z) manifest.
• Suggests a new derivation of the formulae.

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2) Microscopic Derivation

• Using the string web picture, the dyon partition
  function can be shown to equal of the genus-2
  partition function of the left-moving heterotic
  string.
• This perturbative computation can be explicitly
  performed and the determinants can be evaluated
  resulting in a microscopic derivation. N1, 2

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?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.

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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 I1.
• In general I1 genus will contribute.

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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.

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5. S-duality Invariance

• By embedding the physical S-duality group into
  Sp(2, Z) one can demonstrate S-duality
  invariance.
• There can be moduli dependence which is not
  evident from the formulae.