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 Z N 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 G 0 (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 T 4 £ T 2

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 T 4 £ S 1 £ S 1 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 N L is the number operator of 24 left- moving bosons. BPS formula M = P R.

Distribute energy N -1 among oscillations of different frequencies. Find all possible sets of integers m i 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 Q e 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 Z N 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) = S BH 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 T 6 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 Q i e and Q i m 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.