Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12
JHE
Heterotic sigma models on T 8 and the Borcherds
Sarah M. Harrison 0 1 3 6 7
Shamit Kachru 0 1 3 4 7
Natalie M. Paquette 0 1 3 4 7
Roberto Volpato 0 1 2 3 4 5 7
Max Zimet 0 1 3 4 7
0 Via Marzolo 8 , 35131 Padova , Italy
1 382 Via Puebla Mall , Stanford, CA 94305 , U.S.A
2 Department of Particle Physics and Astrophysics , SLAC
3 17 Oxford St. , Cambridge, MA 02138 , U.S.A
4 Department of Physics, Stanford University
5 Department of Physics and Astronomy, University of Padua
6 Department of Physics, Harvard University
7 2575 Sand Hill Road, Menlo Park, CA 94309 , U.S.A
We consider the spectrum of BPS states of the heterotic sigma model with (0; 8) supersymmetry and T 8 target, as well as its secondquantized counterpart. We show that the counting function for such states is intimately related to Borcherds' automorphic
Superstring Vacua; Superstrings and Heterotic Strings

HJEP10(27)
form
12, a modular form which exhibits automorphy for O(2; 26; Z). We comment on
possible implications for Umbral moonshine and theories of AdS3 gravity.
1 Introduction
2 The Borcherds modular form
12
3 T 8 sigma models
3.1
Basic connection
3.2 Second quantization
4 Discussion
z !
e iTr ( )
10( )
=
Q Q Q P !
Q P P P
=
z
{ 1 {
where
( 1)Q P +1D(Q; P ) =
parametrizes the Siegel upper halfspace of degree 2, and
1
Introduction
II string theory on K3
duality group
Studies of the entropy of supersymmetric black holes in string theory have led to the
discovery of beautiful and unexpected relations between basic objects in the theory of
automorphic forms, algebraic geometry, and indices of supersymmetric sigma models. A basic
example is the formula of Dijkgraaf, Verlinde and Verlinde [
1
], capturing the degeneracy
of 1/4BPS dyons in the N = 4; d = 4 string theory obtained from compactifying type
T 2 or, equivalently, the heterotic string on T 6. This theory has
SL(2; Z)
SO(6; 22; Z);
where the electric and magnetic charges are given by 28dimensional vectors Q; P 2
and the SL(2; Z) factor is the electricmagnetic Sduality group of the theory.
The degeneracy of 1/4BPS dyons, D(Q; P ), with charges (Q; P ) is then
6;22,
encapsulates the three Tduality invariants of the black hole charges. The beautiful function
in the denominator of the integrand is the Igusa cusp form
10( ), a weight 10 Siegel
automorphic form for the modular group SP (2; Z). For large charges this formula for the
degeneracies has asymptotic growth
D(Q; P )
e
pQ2P 2 (Q P )2 ;
S =
pQ2P 2
(Q P )2:
reproducing the expected result for the BekensteinHawking entropy of these black holes,
The automorphic function
10 also has a connection to elliptic genera of symmetric
powers of the K3 surface, derived in [
2
] by considering the D1D5 system on K3
S1, which
of 1= 10;1 upon compacti cation and rederive the result of [
1
].
The purpose of this note is to report analogous formulae governing BPS state counts
in the heterotic sigma model with T 8 target. This is the model which would naturally arise
on the worldsheet of heterotic string compacti cations preserving halfmaximal
supersymmetry in two spacetime dimensions. We will focus in this note on the physics of the 2d
eld theory and its supersymmetrypreserving excitations, and mostly limit any discussion
of possible spacetime interpretations to the concluding section. This work was largely
motivated by trying to develop an understanding of BPS counts at Niemeier points in the
moduli space of compacti cations to 3 and 2 dimensions, and their possible interpretation
in light of Mathieu and Umbral moonshine [6{8], in the picture advocated in [9].
2
The Borcherds modular form
12
The hero of our story will be the Borcherds modular form
12 [10]. A nice description of
the relevant aspects of this form can be found in the work of Gritsenko [11], from which
we borrow heavily.
modular form
with this property.
12. Consider the split
Let 2;26 denote the (unique) even unimodular lattice of signature (2; 26). The Borcherds
12 is of weight 12 with respect to O+( 2;26). It is the unique cusp form
Of great interest for us will be the following explicit multiplicative lift formulae for
2;26 =
1;1
1;1
number of the root system R associated with N , where we set h(R) = 1 when N is the
Leech lattice. To each possible choice of N , we can associate the re ned lattice theta series
N ( ; ) =
X e i ( ; )+2 i( ; ) :
2N
Here, (v; w) denotes the lattice inner product of v; w 2 N , and
12( ; ; ) = qArB~ pC
(1
qnr pm)f(mn; )
Y
n;m2Z
(n; 2;mN)>0
p
e2 i ;
r = e2 i( ; ):
In the above, we have de ned A
n > 0, or m = n = 0 and
< 0. Furthermore,
> 0 (or < 0) means that
48
1 P
2N f (0; )( ; ),2 and used the notation (n; ; m) > 0 to mean m > 0, or m = 0 and
positive (respectively, negative) scalar product with a reference vector x 2 N
2 N has
R. The
24
2N f (0; ); B~
2
vector x must be chosen so that (x; ) 6= 0 for all
related to one each other by automorphisms in O( 2;26).
2 N , and di erent choices of x are
More precisely, the Niemeier and Leech points de ne cusps in the domain of de nition
of 12. These formulae should be thought of as expansions of the modular form around
the cusps.
To make contact with the earlier work [9], it is useful to specialize the chemical
potentials as follows. Choose a xed lattice vector
2 N . Then we can de ne
N; ( ; z) =
X q ( 2; ) y( ; )
2N
2In the interpretation of 12 as a denominator for the fake Monster Lie algebra, one views (A; B~ ; C) as
a Weyl vector; see [10, 11] for details and section 4 for further comments on potential applications of the
algebraic structure to physics.
{ 3 {
with y = e2 iz. One now obtains a Jacobi form of weight 0 and index ( ; )=2:
F N; ( ; z) =
N; ( ; z)
24( )
=
X
n;l2Z
f (n; l)qnyl
f (n; l) =
f (n; ) :
X
2N;
( ; )=l
where
as well:
These Jacobi forms, for suitable choices of , are the BPS counting functions discussed
in [9]. That is, they control the coe cients in the expansion of a certain \F 4" term in the
lowenergy e ective action of heterotic string compacti cation to three dimensions, when
the moduli are deformed a slight distance away from a point with Niemeier symmetry (the
enhanced symmetry point itself having singular couplings).
A specialized form of 12 can be obtained as a lift of these BPS counting functions
HJEP10(27)
1N2; ( ; z; ) = qAyBpC
(1
qnylpm)f (mn;l)
Y
n;m;l2Z
(n;l;m)>0
with the prefactors A and C as above and B
automorphic form on the Siegel upper halfspace
2
>0 f (0; )( ; ). This object is an
f( ; z; ) 2 C3 j =( )=( )
=(z)2 > 0; =( ) > 0g
for the group SO+(L ), where L is the lattice of signature (2,3) and with quadratic form
B0 1
1 0
( ; ) 0 0C
0
0
0
0
T 8 sigma models
Basic connection
The Narain moduli space [12] of compacti cations of heterotic strings on T 8 is the double
8;24(M). The worldsheet
bosons propagating on the relevant lattice.
M = O(8; 24; Z)nO(8; 24)=O(8)
O(24) :
This structure also arises in the nonperturbative description of heterotic strings on T 7 [13].
One can think of M as parametrizing even unimodular lattices of signature (8; 24),
eld theory at a given point in moduli space consists of 2d
Let us consider this theory on a toroidal worldsheet. At any point in moduli space,
there are 24 abelian currents of conformal dimension (1; 0). One can consider coupling
{ 4 {
these to background chemical potentials (\Wilson lines"). In this setup, the parameter
of section 2 can be considered as the modular parameter of the torus, while the 24 complex
chemical potentials
should be thought of as these Wilson line degrees of freedom. The
parameter p will emerge upon secondquantization.
Now, let us make a precise connection to the discussion of section 2. Consider a point
in M where
where N is again one of the 24 even unimodular positivede nite lattices, and 8;0 the E8
lattice. We will call this CFT VN .
One obtains BPS states in the heterotic sigma model by putting rightmovers in their
ground state and performing a trace counting leftmoving excitations.
We restrict the
counting to BPS states carrying no rightmoving momentum, i.e. carrying zero charge with
respect to the rightmoving E8 lattice:
TrVN ( 1)F qL0 qL0 y a Ja
(with Ja the 24 leftmoving abelian currents). A subtlety arises because of the extra
rightmoving zero modes of this string background; without inserting extra factors of the
rightmoving fermion number operator FR in the trace, the computation will formally
vanish due to the existence of fermion zero modes. One can circumvent this vanishing by
inserting powers of F in the trace to soak up zero modes, as in [14]; however, this will alter
the modularity properties and spoil the connection we intend to make to
12. Instead, we
consider, following [15], a `twisted' computation where we trace over leftmoving excitations
but consider the ground states of the Z2 orbifold of the rightmoving sector  that is, we
consider the rightmovers to live in the Ramond sector of the Conway module constructed
by Duncan [16].3 The orbifold action kills the extra zero modes, but leaves the leftmoving
sector untouched. The groundstate degeneracy of the rightmovers gives an overall factor
of 24, as in [15]. Including this degeneracy, the result (for a given N ) is
BPS counting function(V~N )
TrV~N
( 1)F qL0 qL0 y a Ja
= 24
(where V~N is the twisted Hilbert space described above). Specializing chemical potentials
as before by choosing a
2 N , we see that the BPS counting function is F N; ( ; z).
We note here that the need to consider the twisted index computation to nd a nonzero
answer plays well with the structure of 12. While the heterotic compacti cation without
the twist has a moduli space obtained by choosing left and right moving sublattices of
8;24, after twisting the E8 lattice is rigidi ed. Therefore, compactifying on an additional
T 2 to compute the index, one expects a moduli space based on
2;26. This makes it natural
to expect a connection to an automorphic form for the Narain modular group O(2; 26; Z),
such as 12.
3This is related by modular invariance to computations with a Z2 insertion in the trace, which give
character valued indices of the original model  hence, we view it as a trick to extract certain BPS
degeneracies of the original theory.
{ 5 {
Now, let us consider the secondquantized version of this counting function. To do this,
consider the heterotic sigma model based on the conformal eld theory Symn(V~N ). (V~N
appears because again, one can count BPS states without additional insertions to absorb
fermion zero modes by quotienting by the natural Z2 action on the E8 lattice, before taking
the symmetric product and putting right movers into their ground state). We still restrict
ourselves to BPS states carrying zero charge with respect to the rightmoving U(1) currents.
Following the logic of Dijkgraaf, Moore, Verlinde and Verlinde [
2
], we see that
1
n=0
log
X pnF Symn(V~N )( ; z)
= 24 log
!
N; ( ; z; )
1N2; ( ; z; )
!
:
Here, F Symn(V~N ) is the specialized BPS counting function for the CFT based on the nth
symmetric product of the (twisted) heterotic sigma model, and
N; =
ph(R) 24( )
Y
2R+
1( ; z( ; ))
( )
where R+ is the set of positive roots of the lattice N , and the sign depends on the choice
of the set of positive roots. The factor of 24 again arises from the rightmoving ground
state degeneracy, as in [15].
correction of) 1
10
We see that there is a precise analogy to the ndings in [
1, 2
]: just as an (automorphic
governs the BPS states of the sigma models on the Hilbert scheme of
K3 surfaces, (an automorphic correction of) 1
sigma models Symn(V~N ). The role of the factor
12
governs the BPS states of the heterotic
N; is super cially similar to that of the
automorphic correction
10;1 in the former story. As mentioned earlier, this factor achieves
an interpretation in the 4d/5d lift [4, 5], and it would be interesting to give a precise similar
interpretation to
N; .
4
Discussion
In this note, we've described how the (inverse of the) Borcherds modular form
12 serves
as a generating function for the BPS state degeneracies of heterotic sigma models on T 8
(after suitably twisting to kill rightmoving zero modes), and their symmetric products.
We conclude with several comments and possible avenues for further development.
In light of AdS3/CFT2 duality, it is natural to conjecture that these heterotic
conformal eld theories, Symn(VN ), (at least at large central charge, i.e. large n) are dual to
AdS3 gravity theories with discrete symmetry groups corresponding to those of the
associated Niemeier lattice  i.e. the Umbral groups and Conway's largest sporadic
group. A criterion was developed in [17], using BPS degeneracies of 2d SCFTs to test
for a possible large radius gravity dual. By modifying and checking this criterion for
the case at hand, it has been found that these theories will have (at best) `stringy'
gravity duals  they will not achieve a (parametric) separation between the inverse
AdS radius and Mstring as the central charge c ! 1 [18].
{ 6 {
The 2d heterotic compacti cations governed by the CFTs VN are the dimensional
reductions of the Niemeier points studied in [9] in relation to moonshine. It is tempting
to try and connect Borcherds' modular form
12 to Umbral moonshine. Making these
thoughts precise is di cult because the notion of counting BPS states in gravity
theories in d
3 at dimensions is fraught with subtlety; charged or gravitating particles
have strong infrared e ects in low dimensions. Possibly, nding an interpretation
of the present formulae in the setting of the supersymmetryprotected amplitudes
studied in [19] is a route forward.
While the models VN depend on the choice of a Niemeier lattice N , the function
counting the second quantized BPS states is the same in all cases. Indeed, these
counting functions are just Fourier expansions at di erent cusps of the same
automorphic function 12. This is nontrivial: the points where the lattice 8;24 splits into
an E8 lattice plus a Niemeier lattice are isolated points in the Narain moduli space
of perturbative heterotic strings on T 8. The basic reason behind this phenomenon
is that these isolated points are the di erent decompacti cation limits in the Narain
moduli space of heterotic strings on T 9, with the condition that the Narain lattice
9;25 splits as an orthogonal sum of E8 and
1;25. This suggests that there might be
an interpretation of
12 in terms of compacti cation of the heterotic string to one
dimension, possibly along the lines suggested in [15].
It is known that 1= 10 is related to the square of the denominator of a generalized
KacMoody (GKM) algebra. There is a beautiful story relating this algebra to
wallcrossing of 1/4BPS dyons in N = 4; d = 4 string theory [20]. As we mentioned
earlier, 1= 12 is also the denominator of a GKM algebrathe fake monster Lie algebra.
It would be interesting to explore a similar story relating this algebra to the BPS
states in the theories we discuss.
The GromovWitten theory of K3
T 2 has recently been conjectured to be governed
by the Igusa cusp form [21]. This follows from the role of 10 in black hole entropy
counts, and string duality. Given that the heterotic string on T 8 is dual to type IIA
on K3
T 4, it is tempting to think that the appearance of
12 in BPS counts in the
present setting presages a similar role for
12 in ( avored) enumerative geometry,
perhaps of K3
Acknowledgments
We are grateful to N. Benjamin and A. Tripathy for discussions. S.M.H. is supported
by a Harvard University Golub Fellowship in the Physical Sciences and DOE grant
DESC0007870. The research of S.K. was supported in part by the National Science Foundation
under grant NSFPHY1316699. N.M.P. is supported by a National Science Foundation
Graduate Research Fellowship under grant DGE114747. R.V. is supported by a grant
from `Programma per giovani ricercatori Rita Levi Montalcini' 2013. M.Z. is supported by
the Mellam Family Fellowship at the Stanford Institute for Theoretical Physics.
{ 7 {
Open Access.
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Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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