Spectrum of CHL Dyons (II) Tata Institute of Fundamental Research First Asian Winter School Phoenix...

Spectrum of CHL Dyons (II)

Tata Institute of Fundamental Research

First Asian Winter School Phoenix Park

Atish Dabholkar

15 JAN 2007 Korea 2

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?


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


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


String-String Triality

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


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.


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.


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.


String junction tension balance


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 Qe+Qm string.


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.


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.


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.


M-lift of String Webs


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.


• 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.


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.


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.


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.


Microscopic Derivation

For orbifolds, requred twisted determinants

can be explicitly evaluated using orbifold

techniques (N=1,2 or k=10, 6) to obtain


S-duality Invariance

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


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.


S-duality

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


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.


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.


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.


• 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.


Negative discriminant states

• Consider a charge configuration

• Degeneracy d(Q) = N


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


Two centered solution

• One electric center with

• One magnetic center with

• Field angular momentum is N/2


Supergravity Analysis

• The relative distance between the two

centers is fixed by solving Denef’s

constraint.

• Angular momentum quantization gives

(2 J +1) » N states in agreement with the

microscopic prediction.

• Intricate moduli dependence.


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.

o T is a shift along the circle, X ! X + R

so twisted states are massive.

o Fermions not affected so N =4 susy.


Z2 Orbifold

• Bosonic realization of E8 £ E8 string

• Orbifold action flips X and Y.


Prym periods

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

• Prym periods


Twisted determinants

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


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)


Orbifold = Circle

•

•

•


• 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.


Schottky Relations


• 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


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.


• 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.


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.

Spectrum of CHL Dyons (II) Tata Institute of Fundamental Research First Asian Winter School Phoenix...

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