Refined BPS invariants of 6d SCFTs fromanomalies and modularity

Minxin Huang

University of Science and Technology of China

Based on Jie Gu, MH, Amir-Kian Kashani-Poor, Albrecht Klemm,

arXiv:1701.00764.

Introductions

• Recently, there have been renewed interests in superconformal theoriesin 6d, with the amount of supersymmetry N = (2,0) or (1,0), i.e. 16or 8 real supercharges.C. Vafa’s talk in Strings 2016

• Some salient features: there is no effective Lagrangian due to tension-less strings; falls into discrete families, no continuous marginal defor-mation; obtain CFTs in lower dimensions by compactification; can beextended to “little string theory”.

• ADE classification of (2,0) theories: AN−1 theory from N M5 branes;type IIB on C2/Γ, where Γ is a discrete subgroup of SU(2) ∼= SO(3),due to McKay correspondence.

• F-theory classification of (1,0) theories: F-theory compactified on el-liptic Calabi-Yau 3-folds. D3 branes wrapping 2-cycles can lead to ten-sionless strings, while D7 branes wrapping 2-cycles provide bulk gaugesymmetry.

Minimal 6d SCFT

• These are the “atomic” building blocks for (1,0) theories, realized as

F-theory compactified in elliptic fibration over a base O(−n) → P1,

where n = 1,2, · · · ,8 and n = 12. For n > 2 the strings interact with

bulk gauge symmetry.

• In this talk: we consider the two simplest theories for n = 1,2, with no

bulk gauge symmetry. There are dual realizations in M-theory, known

as E-strings and M-strings. We would like to develop methods to

compute their elliptic genus, based on anomaly and modularity.

E, M-strings: brane configuration

0 1 2 3 4 5 6 7 8 9 10M9 X X X X X X · X X X XM5 X X X X X X · · · · ·M2 X X · · · · X · · · ·

• Some dualities from second string revolutions:

M-theory on a line = E8×E8 heterotic string theory (M5 branes appear

as instantons)

F-theory on K3 = E8 × E8 heterotic string theory on T2

F-theory on elliptic Calabi-Yau 3-folds = E8×E8 heterotic string theory

on K3

• M-strings: 6d (2,0) supersymmetry. Consider N M5-branes

E8 × E8 heterotic string theory on T4

= F-theory on K3× T2

= type IIB theory on C2/AN−1. (C2/A1∼= O(−2)→ P1)

• E-strings: 6d (1,0) supersymmetry.

“small instanton” in E8 × E8 heterotic string theory on K3

= F-theory on elliptic fibration over O(−1) → P1, known also as the

“half K3” Calabi-Yau.

Elliptic Genus

• In supersymmetric quantum mechanics (1d theory) with Hamiltonian

H. We can compute

partition function: Tre−βH

Witten index: Tr(−1)F e−βH = Tr(−1)F

Only ground states contribute to Witten index, encoding Euler number

of the target space.

• Consider 2d string world sheet theory with (n,0) supersymmetry. Com-

pactify the world-sheet on torus of modulus τ = τ1 + iτ2, q = e2πiτ . The

partition function is

Z(τ) = Tre−2πiτ1P−2πiτ2H = Tr qHLqHR

Elliptic genus: add (−1)F and more “fugacities”, contains more re-

fined topological information of the target space. It looks like a theta

functions of τ , due to absence of SUSY in right-moving sector.

Twisted background and fugacity

• We compactified 0,1 dimensions into T2 = S1 × S1. As we go around

the cycles, we twist the other dimensions R2,3,4,5 = (z1, z2), R7,8,9,10 =

(z3, z4) by action in R-symmetry

(z1, z2, z3, z4)→ (e2πiε1z1, e2πiε2z2, e

2πiε3z3, e2πiε4z4)

To preserve supersymmetry we have ε1 + ε2 + ε3 + ε4 = 0.

• The elliptic genus for n Heterotic strings, E-strings, M-strings with full

fugacity are

ZHetn (τ, ε1, ε2, ε3, ε4, ~mE8×E8

),

ZE-stringn (τ, ε1, ε2, ~mE8

), ZM-stringn (τ, ε1, ε2,m)

Here E-strings have one set of E8 mass fugacity, and M-string has an

extra parameter m from the mass of 5d adjoint hypermultiplet.

How to compute elliptic genus

• Use duality with topological strings (holomorphic anomaly, refined topo-

logical vertex, etc)

E-strings: topological strings on “half K3” Calabi-Yau.

M-strings: topological strings on elliptic fibration over C2/A1.

The number of strings = base degree

• Localization method. Construct the proper world-sheet theory, and

use supersymmetric localization to compute the path integral, involving

JK residues. Elliptic Genus of E-strings, Kim, Kim, Lee, Park, Vafa,

arXiv:1411.2324.

• This talk: modularity ansatz method. Use weak Jacobi forms with

the proper modular weight and index, and pole structures. There are

only finite number of unknown constants which can be fixed by other

methods. Del Zotto, Lockhart arXiv:1609.00310; Jie Gu, MH, Amir-

Kian Kashani-Poor, Albrecht Klemm, arXiv:1701.00764

Weak Jacobi Forms

• Consider a holomorphic function ϕ : H × C → C depend on a modular

parameter τ ∈ H, an elliptic parameter z ∈ C. They transform under

the modular group as

ϕ

(aτ + b

cτ + d,

z

cτ + d

)= (cτ + d)ke

2πimcz2cτ+d ϕ(τ, z), ∀

(a bc d

)∈ SL(2;Z)

and under translations of 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 m ∈ Z is called the index.

• Due to the periodicity, the function has a Fourier expansion

φ(τ, z) =∑n,r

c(n, r)qnyr, where q = e2πiτ , y = e2πiz

We use the weak Jacobi form, satisfying c(n, r) = 0 unless n ≥ 0.

• Some weak Jacobi forms can be constructed by theta functions

φ−2,1(τ, z) = −θ1(z, τ)2

η6(τ),

φ0,1(τ, z) = 4[θ2(z, τ)2

θ2(0, τ)2+θ3(z, τ)2

θ3(0, τ)2+θ4(z, τ)2

θ4(0, τ)2].

Here φ0,1(τ, z) is the elliptic genus of K3.

• A Theorem (Zagier et al):

A weak Jacobi form of given index m and even modular weight k is

a polynomial of E4(τ), E6(τ), φ0,1(τ, z), φ−2,1(τ, z) whose modular

weights and indices are 4,6,0,−2 and 0,0,1,1 respectively.

E-strings: previous works

• Hilbert space of n M2 branes are labelled by Young diagrams of size n.We formally define the M5, M9 branes domain walls

DM5µν = 〈ν|DM5|µ〉, DM9,L

ν = 〈ν|ψM9,L〉, DM9,Rν = 〈ν|ψM9,R〉,

where µ, ν are 2D Young diagrams.Haghighat, Iqbal, Kozcaz, Lockhart and Vafa, arXiv:1305.6322;Haghighat, Lockhart and Vafa, arXiv:1406.0850.

• The elliptic genus of M-strings, E-strings, heterotic strings are con-

structed by these “domain wall”

ZM−strn =∑|ν|=n

DM5∅ν DM5

ν∅ , ZE−strn =∑|ν|=n

DM9,Lν DM5

ν∅ ,

ZHetn =∑|ν|=n

DM9,Lν DM9,R

ν + · · · ,

where · · · denotes symmetrization with respect to permutations of

εi, i = 1,2,3,4.

• The M5, M9 branes domain walls can be determined by “reverse en-

gineering” from topological strings for M-strings and E-strings. We

applied this idea to compute the elliptic genus of 3 E-strings and 3

heterotic strings. W. Cai, MH and K. Sun, arXiv:1411.2801

• A nice check: the elliptic genus of multiple heterotic strings can be

also derived from that of one heterotic string by a Hecke transform.

The result agrees with that from the domain wall method by some

non-trivial identities of theta functions of E8 algebra.

E-strings: some results

• The elliptic genus for a single E-string is given by the torus partitionfunction for eight bosons compactified on an internal E8 lattice andfour spacetime bosons:

ZE-str1 = −

(ΘE8

(τ ; ~mE8,L)

η8

)η2

θ1(ε1)θ1(ε2)

• Two E-strings

ZE-str2 = −

N (~mE8,L, ε1, ε2)

θ1(ε1)θ1(ε2)θ1(ε1 − ε2)θ1(2ε1)η12

−N (~mE8,L, ε1, ε2)

θ1(ε1)θ1(ε2)θ1(ε2 − ε1)θ1(2ε2)η12

where N and N are E8 Jacobi forms, can be fixed by known results

from topological strings.

• This agrees with localization calculations.

How to make an ansatz?

• One may make an ansatz for the elliptic genus directly, without the

domain wall blocks.

• unrefined case: ε1 + ε2 = 0. One observes that the denominator for n

E-strings can be written as

n∏k=1

θ1(kε)2 ∼n∏

k=1

φ−2,1(kε). (1)

We conjecture the elliptic genus is the ratio of a weak Jacobi form

divided by the above denominator, up to some powers of η functions.

• Zeroes of θ1(z) are z = m1 +m2τ , with m1,m2 ∈ Z. This is consistent

with the poles appearing in Gopakumar-Vafa expansion in topological

strings θ1(z) ∼ sin(πz).

Compact models

• Our work (MH, S. Katz and A. Klemm, arXiv:1501.04891, [HKK15])

in 2015 applied the idea to compact elliptic Calabi-Yau three-folds.

• Localization method only works on non-compact Calabi-Yau spaces.

We can apply the ansatz method to general compact elliptic Calabi-

Yau spaces.

• Combined with the B-model method of holomorphic anomaly equation

and boundary conditions, we can solve topological strings to very high

base degree (for all fiber degrees and all genera), or very high genus

(for all base and fiber degrees).

• However, the previous work [HKK15] only considers the unrefined case

ε1 + ε2 = 0. Here we will use a ansatz for refined theory, but only works

on non-compact models (including E-strings and M-strings).

Refined case

• In general, the denominator of elliptic genus of n E-strings has possible

factors θ1(k1ε1 + k2ε2), with k1, k2 ∈ Z and |k1| + |k2| ≤ n. If one

includes all such factors in the ansatz, the modular weight and index

of the numerator are too large. It seems there is no hope for fixing the

ansatz.

• A small puzzle: The Gopakumar-Vafa expansions of topological strings

F =∑

gL,gR≥0

∑β

∑m>0

nβgL,gR(−1)gL+gR[2 sin(mπε−)]2gL[2 sin(mπε+)]2gR

4m sin(mπε1) sinh(mπε2)em(β·t),

It seems we only have poles at θ1(mε1)θ1(mε2), for m ≤ n. So the other

factors sin(π(k1ε1 + k2ε2)) cancel when we expand the theta functions.

• Initially I didn’t suspect anything interesting. Quite surprisingly, last

summer when I visited Bonn, Prof. Albrecht Klemm explained that

actually these extra factors θ1(k1ε1 + k2ε2) may not be necessary. If so

this would greatly simplify the ansatz for refined case.

Our ansatz

Zβ =

( √q

η(τ)12

)−β·KB φk,n+,n−,β(τ,m, ε+, ε−)∏ri=1

∏βis=1

[φ−1,12

(τ, sε1)φ−1,12(τ, sε2)

] ,

• Some explanations:

Here β is the base class. For E-strings and M-strings, only one class in

the base r = 1, and β = n, the number of strings.

φkm(m) a polynomial of appropriate weight and index in ε± = 12(ε1±ε2),

and the E8 Weyl invariant Jacobi forms.

The η(τ) dependent prefactor is argued by a shift of base Kahler class.

Zβ formally have vanishing weight.

• A simpler model: massless limit, no mass dependence. For simplicity in

the followings I consider massless model. The E8 mass can be included

similarly.

Index from anomaly

• Modular forms are generated by Eisenstein series E4, E6. The secondEisenstein series E2 is not exactly a modular form. We usually call thehomogeneous polynomials of E2, E4, E6 “quasi-modular” forms, and thedependence on E2 “modular anomaly”.

• The weak Jacobi forms can be expanded in terms of quasi-modularforms

φ−2,1(z, τ) = −z2 +E2z

4

12+−5E2

2 + E4

1440z6 +O(z8),

φ0,1(z, τ) = 12− E2z2 +

E22 + E4

24z4 +O(z6),

and they satisfy the modular anomaly equation

∂E2φ−2,1(z, τ) = −

z2

12φ−2,1(z, τ), ∂E2

φ0,1(z, τ) = −z2

12φ0,1(z, τ).

• Therefore we can deduce the index of a weak Jacobi form from itsmodular anomaly.

• E-strings: The refined modular anomaly equation for topological strings

on half K3 was proposed in Huang, Klemm and Poretschkin, arXiv:1308.0619,

based on earlier works on unrefined case.

• M-strings: The refined anomaly equation is derived from refined topo-

logical vertex calculations. Haghighat, Iqbal, Kozcaz, Lockhart and

Vafa, arXiv:1305.6322

• In both case the anomaly is a quadratic symmetric polynomial of ε1,2.

So we can determine the modular index for ε±

n+ =

nb3 (n2

b + 3nb − 4) for the E-string,nb6 (2n2

b + 9nb − 5) for the M-string,

and

n− =

nb3 (n2

b − 1) for the E-string,nb6 (2n2

b − 3nb + 1) for the M-string.

Index from anomaly polynomial

• Quantum field theory and supergravity with parity violation in even di-

mension n usually has gauge anomaly and gravitational anomaly. The

contributions of chiral fermions to anomaly can be encoded by a dif-

ferential n+ 2 form, known as “anomaly polynomial”. For a consistent

quantum theory, the anomaly must vanish or cancelled by some mech-

anism, e.g. Green-Schwarz mechanism, anomaly inflow for 6d SCFT.

• The anomaly polynomial for 6d SCFT and their 2d world-sheet theories

are proposed. Shimizu and Tachikawa, Anomaly of strings of 6d N =

(1,0) theories, arXiv:1608.05894.

• One can also deduce the modular index of elliptic genus from the

anomaly polynomials.

How to fix the ansatz

• A geometric bound (known as Castelnuovo bound): For a given Kahler

class, the Gopakumar-Vafa invariants vanish for sufficiently large genus.

• We impose the constrains on the ansatz, and call the remaining am-

biguity “restricted ansatz”. We want to argue the restricted ansatz is

zero, i.e. we can completely fix the ansatz by Castelnuovo bound.

• First consider unrefined case. Denote x = (2 sin(z2))2, z ≡ gs, then

the topological free energy at a given base degree is a power series of

q = e2πiτ and x. For a power series φ(q, x), we organize the expansion

as

φ(q, x) =∞∑n=0

qnfn(x).

We call the series a “geometrically bounded” series if f(x) is a polyno-

mial, a “Gopakumar-Vafa series” if xf(x) is a polynomial.

• The single cover topological free energy for curve class β comes from

the ansatz for partition function at curve class β, corrected by con-

tributions from lower curve classes including multi-cover contributions,

which are not necessarily Gopakumar-Vafa series.

• Assuming the geometric picture is correct, then there should exist at

least some ansatz at curve class β such that the total single cover

contribution is a Gopakumar-Vafa series. So the restricted ansatz are

simply the sub-family of the initial ansatz which are Gopakumar-Vafa

series.

• Introducing some notations A(τ, z) = φ−2,1, B(τ, z) = φ0,1, and the

polynomials due to Zagier

Pd(τ, z) =∏k|dA(τ, kz)µ(d/k),

where µ(n) is the Mobius function.

• (2 sin(nz2 ))2 can written as a polynomial of (2 sin(z2))2, related to the

Chebyshev polynomials, which we denote as −(2 sin(nz2 ))2 = an(x), so

that an(x) is the leading term of A(τ, nz) in q-series expansion. For

example we have a1(x) = −x, a2(x) = −x(4− x). We can also define a

more primitive polynomial similarly as

pn(x) =∏k|n

ak(x)µ(n/k).

For example p1(x) = −x, p2(x) = 4−x, p3(x) = (3−x)2, p4(x) = (2−x)2.

• For n > 2, the polynomial pn(x), Pn(τ, z) are always a perfect square,

due to double zeros at the primitive torsion points.

• We can prove the followings: For a general weak Jacobi form φ(τ, z),

if φ(τ,z)pn(x) with n = 1,2 is geometrically bounded, then φ(τ, z) must be

divisible by Pn(τ, z) as a polynomial of A(τ, z), B(τ, z), E4, E6. Similarly ifφ(τ,z)√pn(x)

for n > 2 is geometrically bounded, then φ(τ, z) must be divisible

by√Pn(τ, z).

• The upshot: The restricted ansatz has only A(τ, z) in the denominator

while the general ansatz’s denominator is∏nbk=1A(τ, kz).

• The modular index in the numerator of the restricted ansatz is actually

negative for nb > 1 since n− < −1, so must be zero.

The refined case

• We introduce the notation x± = (2 sin(ε±2 ))2, and call a power series in

q, x± organized in the form

φ(q, x+, x−) =∞∑n=0

qnfn(x+, x−)

geometrically bounded if the fn are polynomials in both x+ and x−.

• We can define the refined polynomial similarly as

Pd(τ, ε±) =∏k|d

[φ−1,12(τ, kε1)φ−1,12

(τ, kε2)]µ(d/k),

which is a polynomial of A(τ, ε+), B(τ, ε+), A(τ, ε−), B(τ, ε−), E4, E6 since

the possible poles from the denominator in its definition cancel.

• We denote the leading terms in the q-expansion of φ−1,12(τ, kε1)φ−1,12

(τ, kε2)

and Pk(τ, ε±) as ak(x±) = 4[sin2(kε+

2 )−sin2(kε−2 )] and pk(x±), which are

polynomials of x±. For example, we have explicitly

p1(x±) = x+ − x−, p2(x±) = 4− x+ − x−,p3(x±) = 9− 6x+ − 6x−+ x2

−+ x−x+ + x2+,

p4(x±) = 4− 4x− − 4x+ + x2−+ x2

+, · · ·

The difference with unrefined case is that now all pk(x±) and Pk(τ, ε±)

are simply the most primitive unfactorizable polynomials, without the

square in the case of k > 2.

• Our conjecture: For a weak Jacobi form φ(τ, ε±), if φ(τ,ε±)pn(x±) is a geomet-

rically bounded series, then φ(τ, ε±) must be divisible by Pk(τ, ε±).

• If this is true, we can in principle completely fix the elliptic genus of

E(M)-strings for all base degrees. In practice we calculate up to nb = 4

for massless case, nb = 3 for massive case.

Summary and Conclusion

• We compute the elliptic genus of E-strings and M-strings by making

a good ansatz using weak Jacobi forms, then fixing the ansatz by

geometric constrains. Although the results have been obtained before,

our method seems more efficient than previous works in certain aspects,

without introducing spurious poles in the ansatz.

• Our results agree with previous works, by some non-trivial identities of

Jacobi theta functions.

• It would be interesting to apply the method to other cases of minimal

6d SCFT, i.e. realized by F-theory on elliptic fibration of O(−n)→ P1

for n = 3,4,5,6,7,8,12. Del Zotto, Lockhart arXiv:1609.00310

• Our work may provide some lessons on the refinement for topological

strings on (elliptic) compact Calabi-Yau spaces.

Thank You