Spectrum of CHL Dyons (I)

Tata Institute of Fundamental Research

First Asian Winter School Seoul

Atish Dabholkar

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…

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..

Black Hole Entropy.

• Macroscopic (effective action)

• Microscopic (counting microstate)

Striking Agreement …..

• The macroscopic side can be analyzed using the entropy function formalism described earlier in Sen’s lectures.

• Surprisingly, precise counting is possible for these dyons with N=4 supersymmetry.

• There is impressive agreement even for subleading terms between Wald entropy and statistical entropy.

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.

• 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.

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.

• 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

Sp(2, Z)

• 4 £ 4 matrices g of integers that leave the symplectic form invariant:

where A, B, C, D are 2£ 2 matrices.

Genus Two Period Matrix

• Like the parameter of a torus

transforms by fractional linear transformations

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)

• 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 N=1 with k=10 and later on N=2 with k=6 which illustrates all the main points.

• Total rank of the four dimensional theory is

16 ( ) + 12 ( ) = 28

• N=4 supersymmetry in D=4

• Duality group

Heterotic on T4 £ T2

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).

S-duality Group

• Electric-magnetic duality

• Acts on the axion dilaton field by

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.

• 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 .

• Distribute energy N -1 among oscillations of different frequencies. Find all possible sets of integers mi

n such that

• Partition function

• Consider single oscillator with frequency n

• Altogether,

• 24 is the Jacobi discriminant, modular form of weight 12.

• Partition function

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

• 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.

• Microscopic degeneracy

• Large N asymptotics governed by high

temperature, ! 0 limit

• Evaluate by saddle point method.

• 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.

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.

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.

Electric Partition function

• For a given ZN CHL orbifold the the electric partition function is

• Electric degeneracies given in terms Fourier coefficients by

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.

Cusp form

• N=1

is the well-known Jacobi-Ramanjuan function. It’s a unique cusp form of weight 12 of Sp(1,Z)

• N=2

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

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.

Dyon Partition function

• The partition function can be written as

• For other values of N one has k() instead of 10 with

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 N=2 we will encounter 6

• Both have explicit representation in terms of theta functions.

Dyon degeneracies

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.

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.

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

Period matrix

• Holomorphic differentials

• Higher genus analog of on a torus

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

Boson Partition Function

• Period matrix arise naturally in partition fn of bosons on circles or on some lattice.

• is the quantum fluctuation determinant.

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?

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?

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.

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. N=1, 2

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.

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.

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.

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.