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String Theory on Calabi-Yau manifolds: Prospects and applications Hefei CUST 21 September 2015 Albrecht Klemm arXiv:1501.04891, with Minxin Huang and Sheldon Katz

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P Why String Theory? ¶ · ¸ ¹ º »

Physics at the Planck scale Expectations form the Standard Model Problems with pert. Quantum Gravity The black hole riddle Compactifications and effective theories The power of dualities

P Toplogical String and state counting. ¶ Topological String Theory on CY 3-folds · The result and its interpretation

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P Jacobi forms ¶ Definition of Jacobi forms · The ring of weak Jacobi forms ¸ Witten’s wave function and weak Jacobi-forms P Elliptically fibred CY- manifolds ¶ Global fibration over P2 · Singular fibers P Conclusions

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P Why String Theory? G c ~

Newton’s gravitational constant speed of light Planck’s constant h QM

?

QFT NM

NG

SR GR

1 c

G

NM NG SR QM QFT GR  

?





Newtonian mechanics Newtonian gravity (G) special relativity (c) quantum mechanics (~) quantum field theory (c, ~) general relativity (c, G) contains quantum gravity (G, c, ~)

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When does it become relevant? If characteristic scales become comparable ~ MG λC = ≈ = RS 2 Mc c Compton length : characteristic scale of quantum mechanics

r ⇒

M ≈ MP =

Schwarzschild radius : characteristic scale of gravity

~c ≈ 10−5 g G

Planck mass

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• Scales when QG becomes relevant - Planck energy: - Planck length: - Planck time:

EP = c2MP ≈ 1019 GeV q −33 lP = G~ ≈ 10 cm c3 tP =

lP c

≈ 5 · 10−44 s

This scales might be practically remote. 

• However finding ? understanding of 





will be essential for the

ä the conception of space-time

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ä the origin the standard model ä the early history our universe ä new/future cosmological data ä black hole physics In the following I will argue for the possibility: 

?







=

String Theory

What would be the expectations on this theory?

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Expectations from the standard model Physics: • Matter Spectrum: leptons (e, νe,...), quarks (u, d,...). • Interactions: electro-magnetism (γ), weak (W ±, Z 0), strong (g a), gravitation (g νµ). Standard model: ê The gauge principle unifies electro-magnetism, weak and strong interaction into a field theory with local

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G = U1 × SU2 × SU3 invariance. ê Together with the Higgs sector the standard model explains high energy experiments as precise as they can be done. Success of: ê gauge principle ê local quantum field theory

However the standard model

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1. can not incorporate gravity Ý gravity is not renormalizable in perturbative QFT approach Ý (entropy ∼ area) in black hole physics. This is very unnatural in QFT Ý QFT theory seems for principal reasons inadequate to accomplish the quantization of general relativity 2. It does not explain : Ý the masses

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Ý the couplings

∼ 20 free parameters

Ý choice of gauge group Ý choice of representations

∞ many possibilities

Ý the number of generations 3.

QCD coupl. strong at 100 MeV. How to understand:

Ý confinement? Ý chiral symmetry breaking? Ý mass gap?

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4. Further challenges within the SM: Ý low mass of higgs Higgs? Ý naturality of scales (hierarchy problem) ? Ý supersymmetry ?

Does string theory lifts some of the drawbacks of the standard model discussed above?

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Prospects: ad 1. String theory as quantum gravity - ultraviolet finiteness in the perturbative approach - black hole thermodynamics and holography ad 2. Effective actions and compactifications - the string correspondence principle - supersymmetric compactifications - the power of dualities - quantum geometry and topological string

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Applications: ad 3. Geometric engineering of gauge theories: - non-perturbative calculations in supersymmetric gauge theories - large N limit of gauge theory

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What is the problem with perturbative quantum gravity? â Start with Einstein’s equations 1 8πG Rµν − gµν R − Λgµν = 4 Tµν . 2 c â Linearize them in small perturbation δgµν around fixed 0 metric gµν 8πG 0 gµν = gµν + 4 δgµν . c â Try perturbative quantization via Feynman diagrams with graviton as one particle excitation

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Problem: ê the dimensionless effective coupling is GE 2

GE

GE 2

2

∼ G2

Z

∞

dE 0E 0

Ý divergences with powers of E cannot be remove by finite number of counter terms

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Ý problem: interactions coincide in position space Ý hope: canonical quantization of GR: this could be an artifact of perturbation theory The GSW model of weak interaction cures a similar pathology of Fermi’s theory by introducing the massive W W G

G

finite

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• The idea of introducing heavy particles to regularize Fermi’s interaction can be generalized to gravity. • In position space one removes the divergencies by softening the world line to a world-sheet of a string. 01

l s closed strings l s open strings

r generic choice :

ls ∼ lp =

Gh −33 = 1.6 × 10 cm 3 c

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point particle

string

(world line)

(world sheet)

â String theory provides a covariant cut-off for the UV

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divergencies of quantum gravity and defines a consistent perturbation expansion â However, like in QCD the coupling might be strong in some interesting regions of the theory â One would need therefore a non-perturbative definition of string theory → dualities

Building blocks of string theory â open and closed vibrating strings 01

l s closed strings l s open strings

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â elementary particles = harmonic excitations of string â open string ground state ⊃ spin-1 field: gauge boson â closed string ground state ⊃ spin-2 field: graviton â infinitely many massive (∼ Planck mass) excitations ~ m = nT , c ~1 T = , 2 c ls 2

n = 1, 2, . . . string tension [mass/length]

â String length is the only free parameter

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String coupling

D-Branes

gs = ehΦi

Vev of dilaton

â D-branes are solitonic extended objects (hyperplanes) â open strings with Dirichlet boundaries end on D-branes and transfer momentum to them

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1 0

1 0 0 11 1 00 0 1

1 0

1 0 0 1

11 00 00 11

1 0 0 1

1 0

1 0

â D-branes are therefore dynamical objects â very heavy ∼

1 gs

for weak string coupling

Effective gauge theories and gravitation â the dynamic of the endpoints of open string is described by a gauge theory localized on the brane. E.g.

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1 0

1 0 0 11 1 00 0 1

1 0

1 0 1 0

11 00 00 11

1 0 1 0

1 0

U(2)

1 0

â the gauge coupling of the effective theory is a derived quantity α = gslsp−3, (d = p + 1) â gravity is carried by the closed strings and not localized â the gravitational coupling is also derived G = gs2lsD−2 â all parameters of the effective theory are related to ls

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New theories as reconciliation of contradictions within established ones. Galilean-Newtonian mechanics Newtonian gravity

electro magnetism special relativity

general relativity

thermo dynamics quantum mechanics

quantum field theory

? It there a direct conceptual contradiction as clue for the unknown step ?

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Yes: Black hole thermodynamics: Classically a black hole cannot radiate: escape velocity > speed of light c. and is specified by mass, charge and angular momentum Quantum mechanically this is impossible:

hermiticity of Hamiltonian absorbtion matrix element

emission matrix element

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Bekenstein-Hawking Black hole radiate at temperature: 1 ~c3 T = M 8πGk Black holes have entropy: SBH

A kc3 . = 4 Gh

With thermodynamic relations (dE = T dS/dE) dM = T dSBH

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and dSBH ≥ 0 . We would interprete this entropy as usual as SBH = log(# BH quantum states)

How can it be proportional to the area not to the volume?

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In a local QFT the total number of states N would be

N  nV number of states in Plan kian Volume

total number of states

infrared uto nite size

⇒

ultraviolett uto by Plan k s ale

S ∼ V log(n) ≥ V

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Quantum field theory describtion

Black hole thermodynamics

What is the interpretation these quantum states ?

• For certain supersymmetric black holes Strominger and Vafa showed that the number of states is given by

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counting states in a D-brane configuration. D0 branes N-> & open strings 8

Black hole

1111111111111111111111 0000000000000000000000 0000000000000000000000 1111111111111111111111 000 111 0000000000000000000000 1111111111111111111111 000 111 000 111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 deformation

x x x x x x x x x xx x x x x x x x x x x x x x xx x xx x x x x x x x x x x x x x x x x x x x x x xx x xx x x x x x x x x xx x x x x xxxxx x x x xx x x x xx x x x x x

does not change the number of states

SBH

A kc3 = . 4 Gh

• Fact that entropy ∼ area is called holography.

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N (V ) ≤ e

A/4

• Especially there is a case where we have a more precise map between the bulk theory — string theory on Anti de Sitter space— and the theory on its boundary — conformal gauge theory.

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Low energy effective theory and compactification The string correspondence principle One can verify ls → 0 String Theory general rel. + gauge theory =⇒ How? Calculate the Scattering amplitude of the vibrating String. A = +

and take the ls → 0 limit.

+

+

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ß In the low energy limit one gets for the massless excitations: Z 10 √ 2 S = d x g|R + (F + ...) {z } | {z } universal

depends on theory

ß + . . . includes also correction to general relativity, such as R2 terms. ß quantum consistency, i.e. absence of anomalies, requires string theory to be formulated in 10 spacetime dimensions.

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ß the absence of tachyons requires a QI : boson → fermion symmetry, called supersymmetry ß the five consistent string theories in D=10 Name Gauge group Type I SO(32) Type IIA −− Type IIB U (1) heterotic E8 × E8 heterotic SO(32)

Supersymmetry N =1 N =2 N =2 N =1 N =1

are different phases in the parameter space of an

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underlying yet unknown microscopic theory, called Mtheory. ß M theory includes the string coupling as an extra dimension. ß One gets in D=10(11) the classified (unique) supergravity + gauge theories

From D = 10(11) → D = 4 via compactification

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General Picture: 10

M =R(3,1)x M

Compactifying space M

lc~ l s Minkowski Space R(3,1)

How does the physics in R(3, 1) depend on M ? â The number of covariant constant spinors on the compactification manifold M determines the number of surviving supersymmetries QI in D = 4 â These special holonomy manifolds are classified by

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mathematicians â Light spectrum and interactions depend on the topology of the M . E.g. the number of light generations is half of the Euler number of M â For a suitable choice of compactification one gets spectrum and interactions very close to the N = 1 supersymmetric standard model Is relation between M and the string physics unique?

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No! One calls string compactifications with different M , which lead to the same string physics (target-space) dual. Duality: In general two different mathematical models describe exactly identical physics This is very wellknown in quantum mechanics: Particle wave duality in QM: Momentum and position space describtion of QM are exactly equivalent by Fourier-transformation. The power of dualities String theory exhibits a lot of

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dualities: R ↔

ls2 R

duality: Consider closed string theory

on a circle M = S 1

2 ls /R

R

mass spectrum :

n~ m = + (wRT ) . RC 2

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n∈Z

from momentum quantization on S1

w∈Z

from strings winding w times S1

Ý this spectrum in invariant under R ↔ ls2/R

n↔w

Ý in fact, the full theory is invariant under this transformation Ý the coupling in the non-linear σ model is ∼ R1 , i.e. duality maps a perturbative regime of the NLSM to an non-perturbative regime

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This is a much desired property of duality: It maps the perturbative regime of one description to the non-perturbative regime of another and vice versa! Perturbative expansion in B non−perturbative in A

physics non−perturbative in B

perturbative expansion in A

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Mirror duality: Consider string theory on a torus R2 R1

R2

R2 R1 " Kahler structure

Volume: R1 R 2 Shape :

R2/R1

complex structure

1/R1 " Kahler structure

Volume: R /R 2

Shape :

1

R1 R 2

complex structure

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Ý R ↔ 1/R duality applied to one circle Ý the exchange of size and shape is mirror duality Ý mathematically it is the exchange of complexified K¨ahler structure and the complex structure of M Ý both are complex parameters, in particular the “size” gets complexified by an antisymmetric two form B. Ý these geometric parameters of M appear in the low energy action as vacuum expectation values of complex scalars, called moduli.

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P Topological String and state counting ¶ Topological String Theory: Classically string theory is defined by map x : Σg → M × R3,1 from a 2d world-sheet Σg of genus g into a target space M × R3,1. Σg is equipped with a 2d super diffeomorphism invariant action S, of type II. The partition function of the first quantized string is formally Z i S(x,h,φ,ferm,G,B) , Z(G, B) = DxDhDferm e ~

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with the bosonic part of the action ( ∼ areaC). Z √ 2π 2 i j αβ SB (x, h, φ, M ) = 0 hGij + iαβ Bij ) dσ ∂αx ∂β x (h α ΣgZ 1 − φ R(2) ← topol. term φ(2 − 2g) 2π Σg where metric G and B field on M × R3,1 are background parameters. R

• For dcrit = 10 the variational integral RDh over the w-s P metric becomes a discrete integral g Mg dµg over the 3g − 3 dimensional moduli space of cmpl. str. on Σg .

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• If M is a Calabi-Yau 3-fold ∃ (J11, Ω30) & c1(T M ) = 0 ê S has (2, 2) super conformal symmetry with four ¯ ±. Vector- and axial U (1) nilpotent operators Q±, Q allow to define twisted nilpotent scalar operators QA and QB , which define two inequivalent cohomological topological string theories, called A and B model. ê In the A-model, super symmetric localisation localizes to δSB = 0, i.e. maps with minimal area called G (j, J) holomophic maps xhol , so that Z Z X X β DxDh → cvir = r g,β g g,β∈H2(M,Z)

Mg,β

g,β∈H2(M,Z)

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localises to a sum over finite dimensional Z dimC(Mg,β ) =

c1(TM ) + (dim(M ) − 3)(1 − g) Cβ

integrals over the moduli space of xhol : Σg → [Cβ ] ∈ M . The latter are mathematically defined sympletic invariants called Gromow-Witten invariants. We get a mathematical definitionRof the topol. sector of string theory in a large vol = [Ca] J expansion   ∞ X Z = exp  gs2g−2Fg (z(t)) , g=0

Fg =

X β∈H2(M,Z)

rgβ Qβ

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β

a

P

R

with Q = exp(2πi a taβ ) and ta = [Ca](B + iJ). Here rgβ ∈ Q are the Gromov-Witten invariants. While each coefficient of F = log(Z) is mathematically well defined and Fg (Q) is a convergent series in Q for |Q| < c, the sum over g yields ony an asymptotic expansion for all values of Q. E.g. for Q = 0 the constant map contribution to each genus is for g > 1 Fg (0) = (−1)g

χ(M ) 2

Z Mg

c3g−1 = (−1)g

χ |B2g B2g−2| . 2 (2g(2g − 2)!

This is a divergent sum because of the growth of the Bernoulli numbers Bm. ê After resumation and motivic refinement F (gs, t) has

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an even more interesting intepretation as generation function for the dimensions Njβljr of vector spaces of BPS representations. F (1, 2, t) =

X jl ,jr m≥1β∈H2

Njβljr

[jl ]s[jr ]r Qβm m 2

m (q − 1

m −m q1 2 )(q22

−

−m q2 2 )

Here jl , jr ∈ 12 N are spin representations of the BPS x2j−1−x−2j−1 2πik state and we define [j]x = and q = e , k −1 x−x q √ q1 k = 1, 2, s = q2 and r = q1q2 for the parameters labelling the spin content.

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β Njljr

Mathematically ∈ N is the dimension of vectors spaces that correspond to an sll (2) × slr (2) Lefshetz decompostion of the cohomology of the moduli of stable pairs. Choi, Katz, AK : 1210.4403. Example 1: Huang, Poretschkin, AK: 1308.0619 local del Pezzo Calabi-Yau space O(−KS ) → S with S = d8P2. dK For the BPS states Njl,jr at dK = 2 one gets: 2jl\2jr 0 1 2

0

1 3876

2

3

248 1

dK = 2 It is obvious that the adjoint represention 248 of

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E8 appears as the spin N 11, 3 , which decomposes into 2 2 two Weyl orbits with the weights w1 + 8w0, further 3876 = 1 + 3875, where the latter decomposes in the Weyl orbits of w1 + 7w8 + 35w0. Example 2: Katz, Pandharipande, AK: 1407.3181 S = K3 For the BPS states Njdl,jr at d = 3 one gets: 2jL\2jR 0 1 2 3

0 1981

1

2 1

3

252 1

21 1

d=3 Now 1981 = 2 · 990 + 1 and 252 are representations

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of the Mathieu group M24 ∈ S24, which is one of sporadic finite groups

of order |M24| = 244823040. ê The topological string limit of the refined theory is

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1 = −2 =

gs 2π .

In this limit

F T S (gs, t) =

X g≥0 m≥1β∈H2

nβg  mgs 2g−2 βm 2 sin Q . m 2

g nβ

here ∈ Z are the Gopakumar-Vafa invariants. The are indices in the Hilbert space of BPS invariants, which are invariant under complex deformations of M . Note that F T S (gs, t) has poles at gs = 2πQ from the genus zero sector.

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ê Z captures in particular integer symplectic invariants, the DT- or the PT- invariants, physically related to the unrefined BPS invariants nβg ∈ Z as can be seen from the product formula   2g−2 !   ∞ ∞ Y Y Y g+l 2g−2 β nβ g−l−1 β (−1) k β kn0 g l (1 − y Q ) (1 − y q ) Z= ,  β

m=k

g=1 l=0

where y = exp(gs). ê For non-compact toric Calabi-Yau spaces O(−KS ) → S the invariants can be calculated using localization, the topological vertex, Matrix model techniques. The real challenge is for compact Calabi-Yau spaces

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ê Physics: These geometric invariants determine parts of the spectrum of string, M- and F-theory compactifications. A direct physical motivation is to calculate the BPS saturated correlations functions in the effective 4d (6d) N = 2 field theory F := F0  KXL 2iImF ImF X −2 IK IL ê gauge coupl: gIJ = Im F¯IJ + ImF XKXL KL

ê BPS ê grav

2 K 2 masses: MnE ,nM = e |nE tE + nM FM | R 4 g 2g−2 2 ¯ couplings: d x F (t, t)F+ R+.

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· The result: Let M be an elliptically fibred 3-fold over a 2d fano surface S2 E −→ M −→ S2 To be concrete we consider here the simplest case S2 = P2 and call τ and tB be the K¨ahler parameters of the elliptic fiber E and a line l ⊂ P2, q = exp(2πiτ ) and Q = exp(2πitB ). We expand Z in terms of the base degrees dB as  Z(t, gs) = Z0(τ, λ) 1 +

∞ X dB =1

 ZdB (τ, gs)QdB  .

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Then ZdB >0 is a quotient of even weak Jacobi forms of the following form HKK’15 ϕdB (τ, z) ZdB (τ, z) = . Q d B η 36dB (τ ) k=1 ϕ−2,1(τ, kz)

(1)

Here η(τ ) is the Dedekind function and ϕdB (τ, z) is an even weak Jacobi form of index 13 dB (dB − 1)(dB + 4) and weight 16dB .

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P Jacobi forms ¶ Definition of Jacobi forms Jacobi forms ϕ : H × C → C depend on a modular parameter τ ∈ H and an elliptic parameter z ∈ C. They transform under the modular group (Eichler & Zagier) aτ + b z τ 7→ τγ = ,z→ 7 zγ = with cτ + d cτ + d



a b c d

 ∈ SL(2; Z) =: Γ0

as 2πimcz 2 k cτ +d

ϕ (τγ , zγ ) = (cτ + d) e

ϕ(τ, z)

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and under quasi periodicity in the elliptic parameter as ϕ(τ, z + λτ + µ) = e

−2πim(λ2τ +2λz)

ϕ(τ, z),

∀ λ, µ ∈ Z .

Here k ∈ Z is called the weight and Bm ∈ Z>0 is called the index of the Jacobi form. The Jacobi forms have a Fourier expansion X φ(τ, z) = c(n, r)q ny r , where q = e2πiτ , y = e2πiz n,r

Because of the quasi peridicity one has c(n, r) =: C(4nm − r2, r), which depends on r only

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modulo 2m. For a holomorphic Jacobi form c(n, r) = 0 unless 4mn ≥ r2, for cusp forms c(n, r) = 0 unless 4mn > r2, while for weak Jacobi forms one has only the condition c(n, r) = 0 unless n ≥ 0.

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· The ring of weak Jacobi forms A weak Jacobi form of given index m and even modular weight k is freely generated over the ring of modular forms of level one, i.e. polynomials in Q = E4(τ ), R = E6(τ ) and A = ϕ0,1(τ, z), B = ϕ−2,1(τ, z) as weak Jk,m =

m M

Mk+2j (Γ0)ϕj−2,1ϕm−j 0,1 .

j=0

The generators are the Eisenstein series E4, E6 1 Ek (τ ) = 2ζ(k)

X m,n∈Z (m,n)6=(0,0)

∞ X (2πi)k 1 n = 1+ σ (n)q , k−1 k (mτ + n) (k − 1)!ζ(k) n=1

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as well as 2

θ1(τ, z) A=− 6 , η (τ )



2

2

2

θ2(τ, z) θ3(τ, z) θ4(τ, z) B=4 + + 2 2 θ2(0, τ ) θ3(0, τ ) θ4(0, τ )2



To summarize generators and quantitites defining the tpological string partition function Q R A B weight k: 4 6 -2 0 index m: 0 0 1 1

ϕdB ZdB (τ, z) 16dB 0 dB (dB −3) 1 d (d − 1)(d + 4) B 3 B B 2

Since the numerator in ϕdB (τ, z) . ZdB (τ, z) = QdB 36d η B (τ ) k=1 ϕ−2,1(τ, kz)

.

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is finitly generated, we can get for each dB the full genus answer based on a finite number of data. Using as boundary data • the conifold gap condition and reguarity at the orbifold Huang, Quakenbush, A.K. hep-th/0612125 • the involution symmetry on M I : Ω 7→ iΩ ↔ fibre modularity • the parametrization of Z in terms of weak Jacobi-Forms we can solve the compact elliptic fibration over P2 to

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dB = 20 for all dE and all genus or to genus 189 for all classes dB and dE .

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¸ Witten’s wave function and weak Jacobi-forms Witten gave a wave function interpretation the topological string partition function, which implies   ∂ i 2 b¯b c¯ c D D + gs Ca¯¯b¯cg g Z(gs, τ, tB ) = 0 , 0 a ¯ b c ∂(t ) 2 Dt Dt and summarizes all holomorphic anomaly equations If we apply this equation to Z with (t0)a¯ = τ¯ and Qβ = e2πidB tB , we get in the large base limit, because of the special from of the intersection matrix of elliptically fibered Calabi-Yau 3 folds only derivatives in the base

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direction tB for tb and tc. Identifying gs with 2πiz and ˆ2 this using the fact that the only τ¯ dependence is in E becomes 



dB (dB − 3) 2 ∂Eˆ2 + z ZdB (τ, z) = 0, 24

which is solved by a weak Jacobi form of index dB (dB −3) m= as we argue below. 2 Because of modularity and quasiperiodicity given a weak Jacobi form ϕk,m(τ, z) one can always define modular

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form of weight k as follows ϕ˜k (τ, z) = e

π 2 mz 2E (τ ) 2 3

ϕk,m(τ, z) .

It follows that the weak Jacobi forms ϕk,m(τ, z) have a Taylor expansion in z with coefficients that are quasi-modular forms as in Eichler and Zagier 1.

ϕk,m

1

    0 0 2 00 mξ1(τ ) m ξ0 (τ ) ξ2(τ ) ξ0(τ ) mξ0(τ ) 2 + z + + + z 4+O(z 6) . = ξ0(τ )+ 2 k 24 2(k + 2) 2k(k + 1)

2

E.g. φ−2,1(τ, z) = −z +

E2 z 4 12

+

−5E22 +E4 6 1440 z

+

35E23 −21E2 E4 +4E6 8 z 362880

+ O(z 10).

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Moreover one has   2 mgs ∂E2 + ϕk,m(τ, z) = 0 . 12 In particular A and B are quasi-modular forms that satisfy the modular anomaly equation gs2 ∂E2 A = − A, 12

gs2 ∂E2 B = − B . 12

(2)

We can write this as the holomorphic anomaly equation   2 mgs 2 ¯ 2πiIm (τ )∂τ¯ − ϕˆk,m(τ, z) = 0 . (3) 4

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P Elliptically fibred CY- manifolds ¶ Global fibration over P2 The formalism leads to a series of all genus predictions of BPS invariants for low base degree HKK’15. E.g. for db = 1 and db = 2 the numerator is Q(31Q3 + 113P 2) , ϕ1 = − 48 which leads to the following prediction of BPS invariants

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g\dE g=0 1 2 3 4 5 6

dE = 0 3 0 0 0 0 0 0

1 -1080 -6 0 0 0 0 0

2 143370 2142 9 0 0 0 0

3 204071184 -280284 -3192 -12 0 0 0

4 21772947555 -408993990 412965 4230 15 0 0

5 1076518252152 -44771454090 614459160 -541440 -5256 -18 0

6 33381348217290 -2285308753398 68590330119 -820457286 665745 6270 21

Table 1: Some BPS invariants ng(dE ,1) for base degree dB = 1 and g, dE ≤ 6.

4

2

B Q ϕ2



=

3

31Q + 113R

2

2 +

23887872 3

5

1 3 7 3 4 3 [2507892B AQ R + 9070872B AQ R 1146617856 2

2

9

2

2

6

2

2

2

3

+2355828B AQR + 36469B A Q + 764613B A Q R − 823017B A Q R 2

2

6

3

8

3

5

3

3

2

4

+21935B A R − 9004644BA Q R − 30250296BA Q R − 6530148BA Q R 4

10

+31A Q

4

7

2

4

4

4

4

6

+ 5986623A Q R + 19960101A Q R + 4908413A QR ] ,

5

(4)

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g\dE g=0 1 2 3 4 5 6 7 8

dE = 0 6 0 0 0 0 0 0 0 0

1 2700 15 0 0 0 0 0 0 0

2 -574560 -8574 -36 0 0 0 0 0 0

3 74810520 2126358 20826 66 0 0 0 0 0

4 -49933059660 521856996 -5904756 -45729 -132 0 0 0 0

5 7772494870800 1122213103092 -47646003780 627574428 -453960 -5031 -18 0 0

6 31128163315047072 879831736511916 -80065270602672 3776946955338 -95306132778 1028427996 -771642 -7224 -24

Table 2: Some BPS invariants for ng(dE ,2) · Checks from algebraic geometry: Using the definition of BPS states as Hodge numbers of the BPS moduli space, we get vanishing conditions, from

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the Castelnouvo bounds, as well as explicite results for non singular moduli spaces: genus 40

dB=5

dB=4

dB=3

dB=2

dB=1

30

20

10

5

10

15

20

25

30

35

dE

Figure 1: The figure shows the boundary of non-vanishing curves for the values of dB = 1, 2, 3, 4, 5.

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Computing the Euler characteristic of the BPS moduli space, we obtain for these values on the edges of the figure dE dB −(3d2B −dB −2)/2 ndE ,dB

dE dB −(1/2)(3d2B +dB −4)

= (−1)   2 3dB +dB −6 3 dE dB − . 2

which perfectly matches the predction of the weak Jacobi forms.

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P Conclusions: 2 String theory provides an perturbative definition of quantum gravity. 2 The D-brane picture of black holes explains the area low of the entropy. 2 dualities in string theory can lead to non-perturbative description of the physics 2 The topological string calculates exact nonperturbative terms in super symmetric theories including

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gravity 2

ϕdB (τ, z) ZdB (τ, z) = . Q d B ϕ−2,1(τ, kz) η 36dB (τ ) k=1

(5)

2 Since the elliptic argument z of the Jacobi forms is identified with the string coupling gs = 2πiz this expression captures all genus contributions for a given base class.

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2 From the transformation properties of weak Jacobiforms it follows that the depence of Z on string the coupling is coupling is quasi periodic. 2 Since (1) has poles only at the torsion points of the elliptic argument f in , + Z ZdB (τ, z) = Zdpol d B B

where the finite part f in ZdB (τ, z)

=

X l∈Z/2mZ

hl (τ )θm,l (τ, z)

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has an expansion in terms of mock modular forms hl (τ ). 2 The latter fact can be used to check the microscopic entropy of 5d N = 2 spinning black holes and the wall crossing behaviour of 4d BPS states. Some partial results have been obtained by Vafa et. al. arXiv:1509.00455 2 We can make infinitly many checks from algebraic geometry for those curves which have smooth moduli spaces, as seen above. But e.g. for dB = 1 one can confirm the formulas for all classes Jim Bryan et. all work in progress