#### (2, 2) superconformal bootstrap in two dimensions

Received: April
(2; 2) superconformal bootstrap in two dimensions
Ying-Hsuan Lin 0 3 5
Shu-Heng Shao 0 4 5
Yifan Wang 0 1 2
Xi Yin 0 5
Cambridge 0
MA 0
U.S.A. 0
Cambridge 0
MA 0
U.S.A. 0
Princeton 0
NJ 0
U.S.A. 0
0 Open Access , c The Authors
1 Center for Theoretical Physics, Massachusetts Institute of Technology
2 Joseph Henry Laboratories, Princeton University
3 Walter Burke Institute for Theoretical Physics, California Institute of Technology
4 School of Natural Sciences, Institute for Advanced Study
5 Je erson Physical Laboratory, Harvard University
We nd a simple relation between two-dimensional BPS N = 2 superconformal blocks and bosonic Virasoro conformal blocks, which allows us to analyze the crossing equations for BPS 4-point functions in unitary (2; 2) superconformal theories numerically with semide nite programming. We constrain gaps in the non-BPS spectrum through the operator product expansion of BPS operators, in ways that depend on the moduli of exactly marginal deformations through chiral ring coe cients. In some cases, our bounds on the spectral gaps are observed to be saturated by free theories, by N = 2 Liouville theory, and by certain Landau-Ginzburg models. ArXiv ePrint: 1610.05371
Conformal Field Theory; Extended Supersymmetry; Field Theories in Lower
1 Introduction
2 The N = 2 superconformal algebra and its representations
2.1 Unitary representations 2.2 2.3
N = 2 selection rules
Spectral ow
2.4 The minimal gap in the chiral-chiral channel 3
4 Bounding the gaps in the OPE of BPS operators 4.1 Semide nite programming 4.2 Some comments on the details of the numerics
5 (2; 2) theories with exactly marginal deformations
5.1 (2,2) Landau-Ginzburg models with 3 < c < 4
6 Dependence on chiral ring data 6.1 The c = 3; q = 1=3 case 6.1.1
T 2=Z3 CFT saturating the bootstrap bound
Varying the chiral-chiral gap
The c = 6; q = 2=3 case
6.3 The c = 9; q = 1 case
7 Summary and outlook
A T 2=Z3 free orbifold CFT
A.1 Chiral-chiral channel
A.2 Chiral-antichiral channel
B Elliptic genus of the T 2=Z3 orbifold CFT
C Free fermion OPEs at the kinks
D The quintic threefold
D.1 Chiral ring coe cient of the Kahler moduli space
D.3 Gepner points of one-parameter Calabi-Yau models
The conformal bootstrap is based on the idea that a conformal eld theory may be
determined entirely by conformal symmetry, associativity of operator product expansion,
unitarity, and certain basic assumptions on the spectrum of operators and on the structure
of OPE. The method has been surprisingly successful in solving a variety of CFTs in various
(SCFT) in two dimensions using the bootstrap method, extending the results of [20].
(2; 2) SCFTs play a central role in the study of two-dimensional CFTs and string
compacti cations [21{23]. Typical constructions of such theories are based on
supersymmetric nonlinear sigma models on Calabi-Yau manifolds [21, 24, 25], Landau-Ginzburg
models [26{32], and orbifolds [33, 34]. They often admit exactly marginal deformations [35{37],
and the generic points on their moduli spaces are expected to give irrational theories [38].
While the BPS operator spectra and their OPEs in (2; 2) SCFTs have been extensively
studied [24, 27{29, 32, 39, 40], much less is known about the non-BPS spectrum, known
to exhibit highly nontrivial moduli dependence [41, 42] and control the massive spectrum
in models of string compacti cations. With the available analytic methods, the non-BPS
spectrum is accessible only at special solvable points in the moduli space [22, 23, 33, 43],
and through conformal perturbation theory [37] at the vicinity of these points or in the
large volume (weak coupling) limit [44].
The goal of this paper is to constrain the non-BPS spectrum across the entire moduli
space of (2; 2) SCFTs. There are two known (computable) ways to encode the moduli
dependence in the CFT data: through the chiral ring relations [32, 40], and through the
spectrum of boundary states (D-branes) [45]. Here we consider the former, since the chiral
ring relations can be straightforwardly incorporated into the conformal bootstrap based
on sphere 4-point functions. Imposing the crossing equation, while assuming unitarity
(reality of OPE coe cients), we will be able to constrain the scaling dimensions of
nonBPS operators that appear in the OPE of BPS operators through the chiral ring data.
algebra (SCA) are known as chiral or anti-chiral primaries that saturate the BPS bound
and depending on whether these representations are chiral or anti-chiral, are referred to
as (c; c) and (c; a), as well as their Hermitian conjugate, (a; a) and (a; c), operators. The
BPS operators of the same type have non-singular OPEs, and form a ring with respect to
products at coincident points, known as the (c; c) ring or the (c; a) ring [27]. The set of
(c; c) and (c; a) operators are exchanged under mirror symmetry, which amounts to ipping
ow symmetry, that are described by supersymmetric nonlinear sigma models on
CalabiYau threefolds, where the (c; c) ring and (c; a) ring capture the geometry of the quantum
Kahler and complex structure moduli spaces, respectively [27].
In this paper, we focus on BPS operators of the (c; c) type and their Hermitian
conjugate (a; a) operators, and investigate the non-BPS spectra in their OPEs. Of course the
exactly same analysis may be applied to (c; a) and (a; c) operators, but we do not consider
OPE of (c; c) with (c; a) operators here. The reason is that it is more di cult to incorporate
the chiral ring data in analyzing 4-point functions of a mixture of (c; c) and (c; a) operators.
Thus, without further speci cation, we will refer to (c; c) operators as \chiral primaries"
and (a; a) operators as \anti-chiral primaries". We will also restrict our attention to BPS
operators of equal left and right U(1) R-charge, although the generalization to cases with
unequal left and right R-charges would be straightforward.
an (a; a) primary. The OPE
contains the identity representation as well as R-charge
OPE as the chiral-antichiral (CA) channel, and denote by
gCaAp the scaling dimension
of the lowest non-BPS superconformal primaries appearing in this OPE.
On the other hand, in the
OPE, the lightest operator is a (c; c) primary 2q of twice
the R-charge of . We denote by
the coe cient of 2q in the OPE
are respectively normalized with unit two-point functions.
will be referred to as the chiral
ring coe cient. We will refer to the
OPE as the chiral-chiral (CC ) channel, and de ne
gCaCp to be the gap in the scaling dimensions between 2q and the lightest operator in the
OPE that does not belong to a (c; c) multiplet. The operators appearing in the CC channel
may be 12 -BPS, 14 -BPS (that is, BPS on the left, non-BPS on the right, or vice versa), or
non-BPS (that is, non-BPS on both left and right). Furthermore, non-BPS representations
that carry nonzero R-charges in a suitable range may be degenerate [39, 48, 49].1 Note
that in the CC channel, the lightest state in a non-BPS representation that appears on the
left or right of either a 14 -BPS operator or a non-BPS operator, is always a superconformal
descendant, rather than a primary (see subsection 2.2 for the selection rules in the OPE of
BPS operators).
The BPS four-point function
(z; z) (0) (1) (1) can be decomposed in terms of
of the three channels are
= FvCaAc(1
(CCA;s)2F C;As(1
while the third one comes from the OPE channel (z; z) (1). The functions F
The subscripts vac, (c; c), and ( ; s) indicate respectively the vacuum, (c; c), and a generic
representation ( 14 -BPS or non-BPS) labelling a superconformal primary of dimension
and spin s.
is the chiral ring coe cient as already mentioned, while CCC;s and CCA;s
are the OPE coe cients for the other representations in the CC and CA channels. In a
unitary theory, the latter OPE coe cients can be taken to be real (by a choice of phase
of the operators in question), hence so are their squares appearing in (1.1). By exploiting
1The role of such short (degenerate) but non-BPS representations will be clari ed in the next section.
the non-negativity of the coe cients (CCC;s)2 and (CCA;s)2, we can constrain the allowed set
of values for ( ; s) in the CC and CA channels, in a way that depends on the value of ,
which in turn varies over the moduli space of exactly marginal deformations of the SCFT.
The simplest example of such a constraint is an upper bound on the gap in the spectrum,
e.g. an upper bound on
gCaAp as a function of
Constraints on the spectrum of this sort can be found numerically through semide nite
precision. While the bosonic Virasoro conformal blocks can be e ciently computed using
Zamolodchikov's recurrence relation [51], the analogous formula for the general N
= 2
blocks are not yet available.2
Fortunately, there exists a simple relation between BPS
c = 3(k+2) , and bosonic Virasoro blocks of central charge c = 13+6k + k6 with appropriately
k
shifted weights on the external as well as internal primaries. We will derive this relation by
the result at low levels with computer algebra.
Our numerical investigation of the OPE spectrum will focus on two cases. The rst
case involves a marginal BPS operator
(which is necessarily exactly marginal [35, 36]),
Without making any assumption on the chiral ring coe cients or the CC channel operator
already bound the gap among the R-charge neutral non-BPS operators in the CA channel.
We will determine numerically an upper bound on
9. Interestingly, for several values of c that lie between 3 and 158 , the bound is
marginal deformation, and are conveniently described by Landau-Ginzburg models), and
we conjecture that the bound on
gCaAp is linear in c in this range.
The second case of our investigation concerns the OPE of BPS operators with
Rthe chiral ring coe cient
gCaAp as a function of
gCaAp as a function of central charge c,
conformal manifold, for all possible values of
OPE of marginal BPS operators in a Calabi-Yau threefold sigma model, yielding
nontrivial moduli dependent constraints on the mass spectrum of string compacti cation in the
out in [52{54].
the non-BPS channels. This is in contrast with modular bootstrap, where the analogous statement does
not hold for Virasoro characters.
4Such constraints are particularly nontrivial when non-BPS degenerate representations are present.
5Note that for this value of external R-charge, the internal chiral primary in the CC channel may be
related by (diagonal) spectral
ow to an anti-chiral primary with the opposite R-charge as the external
primary. In the analysis of the crossing equation, however, we do not make use of nor assume spectral ow
quantum regime that have been uncomputable with known analytic methods. We compare
our bounds on the gaps with the OPE of Kahler moduli (which belong to the (c; c) ring)
operators in the quintic threefold model, and the OPE of twist elds in the Z-manifold
to the kinks on the boundary of the allowed domain in the space of OPE gaps
and the chiral ring coe cient . The gap below the continuum of states that arise in the
its T-dual cigar SCFT) [41, 59], appears to saturate our bound in the asymptotic region
of large . Various Gepner models and free orbifolds are seen to satisfy the bounds but
do not lead to saturation. Much of the allowed domain of our superconformal bootstrap
analysis remains unexplored, and we will comment on the future perspectives at the end
of the paper.
energy tensor T (z), the superconformal currents G (z), and the U(1)R current J (z). Their
Fourier modes in radial quantization obey the commutation relations
[Lm; Ln] = (m
n)Lm+n +
[Lm; Gr ] =
[Lm; Jn] =
fGr+; Gs g = 2Lr+s + (r
s)Jr+s +
fGr+; Gs+g = fGr ; Gs g = 0 ;
[Jn; Gr ] =
[Jm; Jn] =
where r; s are integers in the R sector and half-integers in the NS sector.
Unitary representations
From now on we will focus on the NS sector. An irreducible highest weight representation
its primary operator. A representation is unitary provided that one of the following two
conditions is satis ed [39, 48, 49]:
gr(h; q) = 0 ;
gr+sgn(r)(h; q) < 0
and f1;1(h; q)
for some r 2 Z + : (2.3)
Here the functions gr(h; q) and fm;n(h; q) are de ned as
fm;n(h; q)
A unitary representation is called non-degenerate if
1 m + 2ni2 ; m; n 2 Z 0 : (2.4)
gr(h; q) > 0 ;
and degenerate otherwise.
some r 6=
the global supercharges.
q=2. The
tation. A non-BPS representation, on the other hand, refers to one that is generated either
there is generally a gap between the chiral primary and non-BPS primaries of the same
R-charge. We will come back to this when we discuss the gap in the chiral-chiral channel
in section 2.4.
Based on our de nition of BPS and non-BPS representations, independently in the
left and right sector, there are four di erent types of superconformal primaries. A 12 -BPS
primary involves BPS representations on both left and right. A 14 -BPS primary involves a
BPS representation on the left, and a non-BPS representation on the right, or vice versa.
A non-BPS primary involves non-BPS representations on both left and right.
N = 2 selection rules
We now describe the selection rules for the OPE of a pair of BPS primaries q1 and q2 of
R-charges q1 and q2, which can be derived from superconformal Ward identities on three
point functions along the lines of [20, 60]. These selection rules will apply independently
to the left and right moving sectors. Here we shall denote by
q a BPS primary of
Rcharge q, and by Oq a non-BPS one. Without loss of generality, it su ces to consider three
distinct cases:
(a) q1 > 0, q2 > 0, and q1 + q2 > 1. In this case, the only multiplets that can appear in
the OPE are those that contain either a chiral primary q1+q2 (of R-charge q1 + q2) or
a non-BPS primary Oq1+q2 1 (of R-charge q1 + q2
appear in the OPE would be the chiral primary
1). The operators that actually
q1+q2 itself or the level- 12
descendescendants of the same R-charge.
Lowest weight operators in OPE
q1+q2 , G+1=2Oq1+q2 1
q1+q2 , G+1=2 q1+q2 1, G+1=2Oq1+q2 1
(b) q1 > 0, q2 > 0, with q1 + q2 < 1. In this case, in addition to the multiplets that
appear in (2.2), another BPS multiplet that contains an anti-chiral primary q1+q2 1
may also appear in the OPE. The actual operators in the OPE that belong to this
descendants with the same R-charge.
(c) q1 > 0, q2 < 0. In this case, the only multiplets that can appear in the OPE are
those of an (anti)chiral primary q1+q2 and of a non-BPS primary Oq1+q2 .
The rules in cases where q1 < 0, q2 < 0 are similar to those of (a) and (b). These selection
rules are summarized in table 1.
Spectral ow
Ln ! Ln + Jn + 2 c
states. A chiral primary q with U(1)R charge q
0 is annihilated by G
1 spectral ow takes q to an anti-chiral primary of R-charge q
non-positive. This is guaranteed by the aforementioned unitarity bound f1;1
0 in (2.3)
c , which must be
which implies jqj
not be a symmetry of the SCFT. Calabi-Yau models admit independently left and right
spectral ow symmetries by integer ; in particular, the
= 1 spectral ow maps the
idensuperconformal algebra, and put strong additional restrictions on the unitary
representations [61, 62] (in particular, on the possible R-charges of the superconformal primaries);
they played an important role in the modular bootstrap analysis of [63, 64].
In our analysis of the OPE through the crossing equation, however, the spectral ow
symmetry does not play a signi cant role, due to the already existing selection rule on the
R-charge of the internal primaries. Unless otherwise stated for speci c models, we will not
assume the spectral ow symmetry in this paper.
The minimal gap in the chiral-chiral channel
In the OPE of a pair of identical chiral primaries
q, there is generally a nonzero gap
resentation.6 In this subsection, we will describe a lower bound on
gCaCp between the scaling dimensions of the chiral primary 2q and of the lightest operator
(necessarily a level 12 descendant, rather than a primary) that belongs to a di erent
repgCaCp that follows from
external R-charge q. Later when analyzing the crossing equation, this lower bound on
will be assumed.
degenerate multiplets,
A nontrivial lower bound on
gCaCp exists when the unitarity bound (2.5) for the
non1) = 2h
is stronger than h > q
12 (assuming q > 0). For central charges c > 3, this occurs when
central charge c
3. Firstly, note that when c < 6, the internal chiral primary of charge
the gap above this internal non-unitary R-charge 2 chiral primary (which is absent from
with r = 32 ; 52 ;
; r0 in the gap between the allowed range of non-degenerate non-BPS
. In particular, when
present in the CC channel. The lowest weight operator in the CC channel is the level- 12
6, there is no lower
h 1(z1) 1(z2)G 1 Or= 32 (z3)i
2
to be consistent with the existence of null states in the relevant non-BPS degenerate
rep6This de nition allows for a smooth limit of the CC channel superconformal block when the gap is taken
consideration of the three-point function.
r = 52
r = 32
r = 12
r = 32
r = 12
r = 12
r = 72
r = 52
r = 32
r = 12
c = 3
4 ≤ c < 4.5
4.5 ≤ c < 6
6 ≤ c
1 = 1. The internal chiral
primary is shown in dashed lines for 3
c < 6 because it violates the unitarity bound j2qj
and is not present in unitary CFTs. The gray shaded region corresponds to the continuum of
non-degenerate multiplets. Note there is necessarily a gap in the weight above the chiral primary
G 3=2 + J 1G 1=2 + L 1G 1=2
Or= 32 :
h 1(z1) 1(z2) (z3)i = 0 ;
we arrive at a di erential equation on h 1(z1) 1(z2)G 1 Or= 32 (z3)i which itself is a
three2
point function of Virasoro primaries.8 It turns out that this equation is trivially satis ed
consistent with the selection rule.
Therefore, for 3
3c and R-charge 1. The actual
operator that appears in the OPE is the level 12 descendant with R-charge 2. It follows
c < 6, and hgCaCp = 0 if 6
Finally, we need to combine the holomorphic and antiholomorphic weights to determine
the gap in the scaling dimension. Let us examine the possibility of a primary that is
point function (2.11) with two chiral primaries. More speci cally, we used
z=z3 2 i
in the three-point function (2.11). See for instance [20] for more details.
occur for 3
gCaCp = 2hgCaCp.
the range of 3
6. The actual operator that appears in the OPE is a level ( 12 ; 0)
3c , h = 1 and R-charge q = q = 2. In the
OPE between two identical scalars 1(z; z), only even spin Virasoro primaries are allowed.
Hence the above level ( 12 ; 0) descendant can appear only when 2
c < 6. Hence we may take the lower bound on the CC dimension gap to be
giving a gap
gCaCp = 2=3 may be
is discussed in detail in appendix A.
We conclude this subsection by recording the minimal values of
gCaCp allowed by the
superconformal bootstrap analysis later on:
c = 3 ;
c = 6; 9 ;
c < 6 ; q = 1 ;
q = 1 ;
q =
q =
gCaCp = 4
gCaCp = 0 ;
gCaCp =
gCaCp = 0 :
external BPS primaries of R-charge
q, with either BPS or non-BPS internal states.9 In
and the bosonic Virasoro block of a di erent central charge, generalizing the results of [20].
We will start with the superconformal blocks with either a non-BPS internal
representation. There are two distinct cases as discussed in section 2.2. The rst one is the
chiral-chiral (CC) block, where two chiral primaries of R-charge q fuse into descendants
of a non-BPS primary of R-charge 2q
1. The second one is the chiral-antichiral (CA) block, where a chiral and an anti-chiral primary of R-charge q and q fuse into a R-charge neutral non-BPS primary and its descendants. The CC block will be denoted by
primary of R-charge 2q
1, and z is the cross ratio of the four external vertex operators.
9For a technical simpli cation, the external BPS primaries will be taken to have R-charges of the same
non-BPS as well as BPS internal representations.
We emphasize here again that only the descendants of charge 2q actually appear in the
OPE. The CA block will be denoted by
where h is the weight of the R-charge-neutral internal non-BPS primary. The vacuum
block can be obtained as a limit of the non-BPS block,
F q;q;q; qjvac(z) = F q;q;q; qjh=0(z):
The CC block with an internal chiral primary with charge 2q can be obtained from a
limit of the non-BPS block,
h = 12
In the case 0 < q < 12 , there is another possible internal antichiral primary of weight
q and R-charge 2q
1 in the CC channel (see section 2.2 for the selection rule).
Its CC block can also be obtained as a limit of the non-BPS block,
using computer algebra.10
These limits of the CA and CC blocks are summarized in gure 2. We checked (3.3), (3.4)
10That is, we work with the oscillator representation of the descendant operators, and computing their
OPE coe cients with the external primaries and the relevant Gram matrices, order by order in the conformal
q can be
related to the bosonic Virasoro conformal blocks of di erent central charges. To understand
j;m;m, that descend from bosonic SL(2)k+2 primaries, of left and right weights and
h =
q =
j(j + 1) + m2
; h =
q =
j(j + 1) + m2
The quantum numbers m; m are subject to the constraints m
There is a set of normalizable states that correspond to certain discrete real values of j,
among which the (anti)chiral primaries are of the form
j;m;m with m = m, j = jmj
If we assume that k is a positive integer, the condition m + m 2 kZ may be relaxed to
acts by rotation along the circle direction of the cigar.
The correlation functions of operators of the form
j;m;m that conserve the total m
and m quantum numbers can be computed directly from the bosonic SL(2)k+2 WZW
model, by factoring out the U(1) part of the vertex operators. The correlators of SL(2)
primaries can further be related to those of a bosonic Liouville theory of central charge
of the former coincides with the bosonic Virasoro conformal block decomposition of the
blocks and Virasoro conformal blocks. For the CC block, we have
CC;c= 3(k+2)
z) k2 q(1 q)FcV=ir13+6k+ k6
where FcVir is the Virasoro block with central charge c, and
h ; h ; h+; h+; h +
h+ =
For the CA block,
CA;c= 3(k+2)
F q;q;q; qjkh (z) = (z(1
z)) k2 q(1 q)FcV=ir13+6k+ k6
The vacuum and the BPS blocks are also related to Virasoro conformal blocks
via (3.3), (3.4) and (3.5).
We summarize these relations between the N
= 2 Virasoro
blocks and the bosonic Virasoro blocks in gure 3. The relations (3.7) and (3.9) have been
checked by brute-force computations of (super)conformal blocks to the z4 order.
soro blocks, we can now compute the former to high precision e ciently. This is achieved
h ; h+; h+; h ; h +
Virasoro block with c=13+6k+ k6
= (z(1 − z)) k2 q(1−q)
= z k2 q2(1 − z) k2 q(1−q)
h + 12 + kq(1 − q)
Virasoro block with c=13+6k+ k6
Virasoro blocks.
through Zamolodchikov's recurrence relation [66, 67], which computes the (bosonic)
Virasoro block as a series expansion in the \nome" q(z), de ned as
exp(i (z));
F (z) = 2F1(1=2; 1=2; 1jz):
The holomorphic Virasoro block for a four-point function hO1(z)O2(0)O3(1)O4(1)i with
central charge c, external weights hi, and internal weight h has the following representation
[ 3(q(z))] c 81 4(h1+h2+h3+h4)H( i2; hjq(z)):
If we de ne
then H( i2; hjq(z)) satis es Zamolodchikov's recurrence relation
c = 1 + 6Q2;
Q = b + ;
hm;n =
m;n =
H( i2; hjq(z)) = 1 +
[q(z)]mnRm;n(f ig) H( i2; hm;n + mnjq(z));
Rm;n(f ig) = 2
where hm;n are the conformal weights of degenerate representations of the Virasoro algebra,
and Rm;n(f ig) are given by
The product of (r; s) is taken over
and the product of (k; `) is taken over
r =
s =
m + 1; m + 3;
n + 1; n + 3;
k =
` =
m + 1; m + 2;
n + 1; n + 2;
Bounding the gaps in the OPE of BPS operators
Our objective is to constrain the spectrum of non-BPS operators in the OPE of a pair
of BPS primaries, either of the form
q(0) (CA channel), or
q(z; z) q(0) (CC
channel), by analyzing the N
point function
q(0) q(1) q(1) . The latter can be decomposed in either the
chiral-chiral channel or the two chiral-antichiral channels. The equivalence of these
decompositions gives the following set of crossing equations,
CA
= j j2jF CqC; q;q;qjchiral(1
z)j2 + X(ChC;hC )2jF q; q;q;qjh(1
CC
BPS multiplets.
in the chiral-antichiral channel.
As discussed in section 2.2, the sum in the chiral-antichiral channels includes only the
nongCaAp is de ned as the scaling dimension of the lowest non-BPS primary
On the other hand, the spectrum in the chiral-chiral channel is more involved. When
q and R-charge 1
2q < 0 can also
contribute.11 We de ne
gCaCp to be the gap between the scaling dimension of the lightest
operator that does not belong to a (c; c) multiplet,12 and that of a charge 2q (c; c) primary.
Using the positivity of the coe cients (CCC)2 and (CCA)2, we will obtain numerical
upper bounds on
gCaAp. The bound will depend on the chiral ring coe cient
and the gap
gCaCp in the q q OPE, the chiral-chiral channel.
do not have to single out its contribution from the crossing equation (4.1) as we did for the internal (c; c)
12That is, the lightest operators in the second term of the second line in (4.1).
We now describe the method of using semide nite programming to generate numerical
upper bounds on the gap. Our rst task is to write the crossing equations in a form that is
convenient for the implementation of semide nite programming. By de ning (the operators
are placed in the order z; 0; 1; 1)
CA
the crossing equations can be packaged as [68]
0 =
CA; (z)CC + X(ChC;hC )2 B GhC;Ch; (z) C ;
where the sum includes the vacuum multiplet in the CA channel and the charge 2q chiral
multiplet in the CC channel. Next we act by a vector linear functional ~ with three
components, which we write as a sum ~ o + ~ e where
to put the crossing equations into the form
mi;n@zm@znjz=z=1=2;
mi;n@zm@znjz=z=1=2;
0 =
CA = f
= 0 or
CC = f
= 2q or
and nd the lowest b gCaAp and b gCaCp that can be ruled out to obtain the most stringent bound
on the gaps. Such a problem can be solved using the method of semide nite programming.
A hypothetical spectrum in the CA and CC channels can be ruled out by unitarity if
we can nd an ~ satisfying
hypotheses of the form
CC are the sets of scaling dimensions and spins for the superconformal
multiplets in the CA and CC channels, respectively. In particular, we aim to rule out
Some comments on the details of the numerics
We implement semide nite programming using the SDPB package [50]. In practice, to
obtain an upper bound on the gaps, we need to truncate our basis of linear functionals at
nite total derivative order N
in @z, @z. The most stringent upper bound on the gaps is
then bound by extrapolating to N
! 1. We must also truncate the set of spins on which
to impose positivity (4.6), and approximate the superconformal block in Zamolodchikov's
representation by truncating (3.13) to a
nite series in the nome q(z). The largest spin
considered and the order of the q(z)-series are denoted by smax and dq, respectively. We
would like to emphasize here that whereas the truncations in the spins and q(z)-orders are
(controlled) approximations, the truncation in derivative orders N
always yields rigorous
bounds (for su ciently high smax and dq).
The conformal blocks are computed numerically via Zamolodchikov's recurrence
relation that was reviewed in section 3. The blocks are computed separately for each value
of the central charge, so that all inputs to the recurrence relation except for the internal
weight h are numerical numbers. Since the conformal block for arbitrary internal weight h
is a combination of H( i2; hm;n + mnjq(z)) for m; n
1 via the recurrence relation (3.13),
an e cient way to compute the general conformal block is to rst compute H at these
special values of the internal weight. Moreover, in order to compute the general conformal
block to O([q(z)]dq ), we only need H( i2; hm;n + mnjq(z)) for mn
column vector that contains this
nite set of H as entries, the recurrence relation (3.13)
dq. Denoting by H~ the
implies a matrix equation of the form
where I is the identity matrix, ~1 is a column vector with every entry equal to 1, and M is
a matrix with elements
M)H~ = ~1 + O([q(z)]dq+1);
[M](p;q);(m;n) =
It is then straightforward to invert I
M to obtain H~ .13
For a given derivative order N , the dependence of the bound on smax and dq has
the following behavior: when the truncation order is small, an ~ satisfying (4.6) always
exists even when the hypothetical gaps ( b gCaAp; b gCaCp) are set to zero, thereby ruling out any
hypothesis of the form (4.7); as the truncation order exceeds some minimum, a bound on
( b gCaAp; b gCaCp) starts to exist and stabilize as we go to higher truncation orders. We adjust
the truncation order to make sure that the bound has stabilized to within the desired
+ 4 usually
su ces, sometimes higher truncation orders are needed, for example when the chiral ring
coe cient
is sent to in nity, or when the central charge is close to 3.
The bottleneck for the speed of the numerical computation is the truncation order
of the q(z)-series. This is because in the Zamolodchikov representation of the conformal
13This inversion is performed by writing H~ and I
M both as series in q(z), and matching the coe cients
order by order. A direct matrix inversion would be extremely ine cient and unnecessary since (4.8) is only
accurate to a nite order.
block, the coe cients in the q(z)-expansion have denominators that are higher and
higherdegree polynomials in h, and the degree of the polynomial is a key factor a ecting the
computation speed. This imposes a limit on the highest derivative order N
we can go
to, since as mentioned in the previous paragraph, the derivative order must be somewhat
lower than the q(z)-expansion order dq. We have chosen to only consider dq up to 28, and
up to 24 or less.
(2; 2) theories with exactly marginal deformations
In this section, we study constraints on the R-charge neutral non-BPS spectrum of (2; 2)
SCFTs with exactly marginal deformations, by considering the OPE of a pair of BPS
primaries of R-charge
1 (on both left and right), in theories whose central charges lie in
the range 3
9. The G 1=2G 1=2 descendants of these primaries generate N = (2; 2)
preserving exactly marginal deformations. When there are more than one modulus for the
only one of them.
Let us comment on the chiral ring coe cient , which controls the contribution from
this case) in the chiral-chiral channel. For c < 6, an R-charge 2 chiral primary would be
forbidden by the unitarity bound, and thus
= 0. For c
can be nonzero, and we
into the crossing equation, the bootstrap bounds will be strictly stronger than
internal representation may be viewed as a limiting case of superconformal blocks with a
non-BPS internal representation, as we have seen in section 3. Thus, we will simply set
and its orbifolds. In this case the crossing equation can be trivially solved as follows. The
CA block with external q =
1 BPS primaries and a R-charge neutral internal non-BPS
primary of weight h has the following closed form expression,14
F 1;1;1; 1jh(z) =
It turns out that crossing symmetry constrains the four-point function with BPS primaries
1 of R-charge q =
uum block,15
1(z; z) 1(0) 1(1)
CA;c=3
1(1)i = jF 1;1;1; 1jh=0(z)j2 =
14This expression is checked by computer algebra up to z6 order.
15For example, the four-point function of the fermion bilinears
(z) e (z) in the T 2 CFT or its orbifolds can be readily computed to be (5.2).
tion (2.13) on the gap in the CC channel
orders 4; 8; 12; 16 (from green to red). Right: the same plot zoomed into 3
gCaCp, as a function of the central charge c, at derivative
gCaAp, with the minimal
assumpTo see this, note that for a xed real z 2 (0; 1), the di erence between the (2; 2)
superconformal block in two CA channels related by crossing,
F 1;1;1; 1jh(z)F C1A;1;c;=1;3 1jh(z)
CA;c=3
z) =
= h + h)
is of a de nite sign for all positive
(and vanishes for
ing the two CA channels, which involves a sum of such terms with non-negative coe cients,
can be satis ed only if all coe cients for
> 0 vanish, hence the claim.
We now proceed to more general central charges. Figure 4 shows the numerical upper
gCaAp on the gap in the CA channel for 3
constraints on the CC channel gap
theory demands the CC channel gap to be no smaller than 4
not restrict the CC gap for c
9, taking into account the unitarity
c < 6, and does
Speci c examples of four-point functions that saturate the bounds to within
numerical precision are marked in black.
Towards the left, we have certain tensor
products of N
145 ; 1 ; as well as the free point (3; 0). These tensor
prodpoint (6; 2), that is realized by a four-point function of fermion bilinears. In [20], by
extrapolating to in nite derivative order, it was found that
suggesting that the numerical saturation at (6; 2) is exact.
For 3 < c < 3:3, the numerics do not stabilize even when we truncate the q(z)-series
up to the maximum order 28 that we consider. Nonetheless, saturation of the bounds by
the tensor products of minimal models as well as the free theory suggests that the bounds
could be given by the exact formula
gCaAp is likely to be exactly 2,
in the range 3
gCaAp =
For small central charges 3 < c < 4, we can construct (2; 2) Landau-Ginzburg (LG) models
de ned by quasi-homogeneous superpotentials that possess nontrivial exactly marginal
models. See [69] for examples of such (2; 2) SCFTs.
It is easy to classify such LG models with up to 3 chiral super elds. They are of the
following types17
X3 + Y 3n + aXY 2n;
X4 + Y 2n + aX2Y n;
X4 + Y 8 + aX2Y 4 + bXY 6;
X5 + Y 5 + aX3Y 2 + bX2Y 3;
n = 3; 5;
c =
c =
c =
c =
X3 + Y 3n with n
where the superconformal moduli spaces are parametrized by the coe cients a; b.
models (D.21). For example for the N
3, the lowest non-chiral superconformal primary in the CA channel is
gCaAp =
( 1k;=0;30n) =
In the CC channel, the gap between the lowest non-chiral superconformal primary whose
gCaCp = 2 h( k1=3
2 ; 12 ; 12 ) + h( kn2=+31n; n2 ; n2 ) + 1
1 =
type in singularity theory [70].
the LG description (see table 2).
Note that the lowest operator appearing in the CC channel here is the product of the
level- 12 descendants of
similar manner and we summarize them in table 2. In particular we see that all of these LG
models saturate the lower bound
23c on the CC gap from N = 2 representation
theory. Moreover, the c = 130 ; 7 ; 158 ; 145 models18 sit on the numerical CA gap bound along
2
16Marginal deformations in (2; 2) SCFTs are exactly marginal [35].
17The polynomials that de ne such superpotentials are known to be of the unimodal quasi-homogeneous
orders 4, 8, 12, 16, 20 (green to red). Right: the width of the peak plotted over inverse derivative
gCaAp for the gap in the CC channel
order. The width is de ned as r
r is the value of
edge, and l is the value of
gCaAp = 1:5 on the left edge.
gCaAp = 1 on the right
gCaAp in the c = 9, q = 1 case. The green
to red curves are the numerical bounds obtained from conformal bootstrap, in increasing derivative
CFT, in the absence of B- eld. The blue circle dot marks the maximal
gCaAp in this case. The blue
square marks the large volume limit of the quintic that sits at the kink of the bounding curve. The
blue diamond (buried in the black dots) marks the 35 Gepner model, which has
gCaCp = 65
1:55532. In this Gepner model,
gCaAp = 45 is well within the bootstrap
bounds if we assume its value of the gap in the CC channel,
gCaCp = 65
. A discussion of
Gepner points that lie inside the moduli space of one parameter Calabi-Yau sigma models
can be found in appendix D.3.
We can also compare our bounds with the twist eld OPE of the Z-manifold, i.e. the
as a function of the gap in the CC channel
gCaCp, in the limit of in nite chiral ring coe cient
gCaAp from the numerical bootstrap,
= 1 and
= 1,
realized at ( ; gCaAp)
but with generic moduli for the Z3 invariant T 6). The chiral ring coe cient is given by (6.6)
with n = 3, and the CA gap
CFT in the absence of B- eld are shown as black dots in gure 12, with the maximal gap
(1:26419; p2). When a nonzero B- eld is turned on, all values of
general nonzero B- eld, despite having numerically sampled over a large set of points over
our bootstrap bound on the CA gap, for any value of .
Figure 13 shows the bound on
gCaAp in the limit of in nite chiral ring coe cient
= 1,
with dependence on
gCaCp. For Calabi-Yau models, the in nite
conifold point. A continuum of operators with dimension gap 12 is expected to develop in
limit corresponds to the
appendix D.2). Indeed, our bound on
= 1 is likely saturated by the N = 2
Liouville theory. So far, we have been unable to optimize the bounds of gure 13 by a
reliable extrapolation to in nite derivative order, due to the limitation of computational
power. Unlike at
nite , where the bounds stabilize at dq = N
+ 4, at in nite
found empirically that at least dq = N
+ 8 is required. Note that there appears to be
gCaAp( = 1) decreases from 12 , and vanishes as
a transition at
gCaCp exceeds 2.
1:4, above which
Summary and outlook
We began with the known knowns: the chiral ring data, whose moduli dependence is
understood, and constrained the known unknowns: the spectrum of non-BPS operators in
Landau-Ginzburg or Calabi-Yau models at generic points on their moduli spaces. We have
also probed the unknown unknowns: the spectra of general (2; 2) SCFTs that admit exactly
marginal deformations, by constraining the OPE content of marginal BPS operators.
We carved out some allowed domains in the space of possible gaps in the CA and CC
OPE channels and the chiral ring coe cient. Let us recap some of the main results:
c = 3; 130 ; 27 ; 158 ; 145 .
An upper bound on the gap in the OPE of a marginal BPS operator and its conjugate
was computed for 3
9. Interestingly, the bound appears to be saturated by
31 , as a function of the chiral ring coe cient. The entire bounding curve
curve in the Kahler moduli space of the latter.
primaries (of R-charge
1), as a function of the chiral ring coe cient.
making any assumptions on the CC channel gap, we saw that a kink on the bounding
= p2 ,
of Kahler deformations of 1-parameter Calabi-Yau models, the kinks corresponds to
the large volume limit. The Kahler deformations of the quintic model only realizes
p2 . Smaller values of can be realized on other 1-parameter Calabi-Yau models.
In this case, we found that
gCaAp may exceed the free eld value, namely 2. It remains
to be seen whether this larger allowed gap can be realized in the quantum regime of
Calabi-Yau models.
within our bounds. The gap in the continuum that develops at the conifold point,
however, appears to saturate our bound in the
! 1 limit.
We observed various kinks on the boundary of the allowed domain in ( ; gCaCp;
some of which are saturated by OPEs of free
elds. Many of the features of this
plot remain unexplained, and it would be nice to understand whether all of it can be
realized by (2; 2) SCFTs.
The non-BPS spectrum in Calabi-Yau sigma models has also been constrained from
modular invariance of the torus partition function [63]. In that work, an upper bound
on the dimension of the lightest non-BPS operator in the entire spectrum (rather than
in speci c OPEs) is obtained numerically as a function of the total Hodge number. The
latter plays an analogous role as the chiral ring coe cient
in the crossing equation of
four-point functions. In particular, the authors nd that there is always a non-BPS primary
with dimension less than 2 for all values of the total Hodge number. On the other hand,
our bound (see gure 9) constrains the R-charge neutral non-BPS operator in the speci c
OPE between a pair of BPS primaries and depends on the conformal moduli through the
chiral ring coe cient . If we do not keep track of the moduli dependence by setting
= 0
gCaCp = 0, our bound
2:272 (at derivative order 24)26 appears to be a
weaker bound than that of [63] as far as the entire spectrum is concerned.
26When extrapolated to in nite derivative order, the upper bound on gCaAp is roughly 2.26.
Obvious generalizations of this work include studying the crossing equations for mixed
correlators, especially ones that involve simultaneously (c; c) and (c; a) ring operators.
For Calabi-Yau models, this is particularly important in that we wish to pin down the
point on both the complex and Kahler moduli spaces of the theory, and to constrain the
spectrum thereof. Further, one would like to extend our analysis to non-BPS 4-point
superconformal blocks, that is not yet available. Eventually, we wish to combine the
crossing equation for the sphere 4-point correlators with the modular crossing equation
for the torus partition function and 1-point functions [63, 64, 72{74]. Another potentially
fruitful route is to study the crossing equation for disc correlators, subject to boundary
conditions that respect spectral ow symmetry (spacetime-BPS D-brane boundary states
in the context of string compacti cation). We are hopeful that much more is to be learned
along these lines toward classifying and solving (2; 2) superconformal theories.
Acknowledgments
We would like to thank Kazuo Hosomichi, Zohar Komargodski, Juan Maldacena, David
Poland for discussions, and David Simmons-Du n for correcting a reference.
grateful to the Tata Institute of Fundamental Research, and the organizers of the workshops
Higher Spin Theory and Duality, MIAPP, Munich, Germany, Conformal Field Theories and
Renormalization Group Flows in Dimensions d > 2, Galileo Galilei Institute for Theoretical
Physics, Florence, Italy, NCTS Summer Workshop on Strings and Quantum Field Theory,
National Tsing Hua University, Hsinchu, Taiwan, and Strings 2016, YMSC, Tsinghua
University, Beijing, China, for their hospitality during the course of this work. This work
is supported by a Simons Investigator Award from the Simons Foundation, and in part by
DOE grant DE-FG02-91ER40654. YL is supported by the Sherman Fairchild Foundation
and the U.S. Department of Energy, O
ce of Science, O ce of High Energy Physics, under
Award Number DE-SC0011632. YL would also like to thank the hospitality the Berkeley
Center for Theoretical Physics during the course of this work. SHS is supported by the
National Science Foundation grant PHY-1606531. YW is supported by the NSF grant
PHY-1620059 and by the Simons Foundation Grant #488653. The numerical computations
in this work are performed using the SDPB package [50] on the Odyssey cluster supported
by the FAS Division of Science, Research Computing Group at Harvard University.
T 2=Z3 free orbifold CFT
In this appendix we will demonstrate that the four-point function of chiral and antichiral
We start by reviewing some basic facts about the torus orbifold CFT. Consider a torus
the torus by X(z; z) and X(z; z) with periodicity X
X + 2 R!. Here
in the twisted sector. For each xed point, there are N
1 twist elds with weights
X(e2 iz; e 2 iz) = !X(z; z) ;
X(e2 iz; e 2 iz) = !X(z; z) ;
h = h =
k = 1;
+(z; z) and
holomorphic fermions
(z) with R-charge q =
1 and q = 0, as well as their
and H~ (z) be the bosonization of the holomorphic and antiholomorphic fermions,
1. Let H(z) J (z) =
(z) = i@H(z) ;
J~(z) = ~+ ~ (z) = i@H~ (z) ;
where J (z) and J~(z) are the holomorphic and antiholomorphic U(1)R currents, respectively.
consider the OPE and the four-point function of the q = q = 13 (c; c) primary 1 (z; z) and
3
its (a; a) conjugate primary
h =
1 (z; z) = e 3i H(z)+ 3i H~ (z) +(z; z) ;
1 (z; z) = e 3i H(z) 3i H~ (z)
Note that the weights of 1 (z; z) and
two real moduli, the radius R and the B- eld b. Note that there is no complex structure
moduli because the shape of the torus is xed. We will normalize the B- eld to have
periodicity 1, i.e., b
in [33, 75]
For arbitrary moduli, the four-point function of
1 has been computed
= jz(1
z)j 2=3
w 21 (p+v=2+B v=2)2 w 12 (p v=2+B v=2)2 ;
27Recall that to be compatible with the identi cation on T 2, N will be restricted to 2; 3; 4; 6.
w(z) = exp
w(z) = exp
F (z) = 2F1
is the lattice for the original target space torus.28
a sublattice of
de ned as c = (1
is the dual lattice of . c is
is the rotation by
1 that restricts the winding
number v to live in
c but not the full .
In the following we will study the chiral-chiral channel
chiral-antichiral channel
1 (0; 0) of the four-point function (A.6). We will see
that this free orbifold theory saturates the numerical bootstrap bound on the gap in the
chiral-antichiral channel along certain loci on the moduli space M .
Chiral-chiral channel
channel:29
allowed by the N
Let us rst consider the chiral-chiral OPE channel between 1 and 1 . There are two types
of 12 -BPS primaries, two types of 14 -BPS primaries, and one type of non-BPS primary
The chiral ring coe cient, i.e. the OPE coe cient for 2 in the chiral-chiral channel,
has been computed in30 [33, 75]
: (A.10)
and h = h = 13 ,
(R; b) =
h = h = 16 .
2. The G+ 1 G~+ 1 descendant of an (a; a) primary31 0 1 (z; z) with q = q =
28The lattice
is normalized such that there is no factor of 2 . For example, for a square torus with
sides 2 R is normalized to be
= f(nR; mR) j n; m 2 Zg.
29As noted in section 2.4, all the non-BPS primaries in the chiral-chiral channel are in fact degenerate in
30Throughout this paper we adopt the
and external q = 1=3 are labeled by a half-integer r 6=
0 = 2 convention.
1=2 with weight given by h = r=3.
31We add a prime to distinguish this internal (a; a) primary from the external (a; a) primary
has the same charges and weights.
Quantum Numbers
of the Primary
q = q = 23 ; h = h = 13
q = q =
q = q =
q = 23 ; q =
q = q =
Level of the Operators
four-point function h 13
non-BPS, respectively. Note that the level ( 1 ; 1 ) descendant of an (a; a) primary minimizes the
gCaCp, which is de ned as the gap in the scaling dimensions of the operator that appears in the OPE
and of the (c; c) primary. We omitted the conjugates (i.e. (n; a) and (n; c)) of the 14 -BPS primaries
in the above table.
3. The G+ 1 G~+ 1 descendant of a 14 -BPS primary that is antichiral on the left and
nonBPS on the right, with q = q =
13 and h = 16 , h > 16 as well as their conjugates.
Calabi-Yau CFT). They are the BPS limits of the non-BPS operators discussed below.
4. The level G~+ 1 descendant of a di erent type of 14 -BPS primary that is chiral on the
left and non-BPS on the right, with q = 23 ; q =
13 and h = 13 , h > 16 , as well as their
charge q = q =
5. The level G+ 1 G~+ 1 descendant of a non-BPS operator on the left and right with
There is another constraint on the weights of the 14 -BPS primaries and the non-BPS
primaries. In the OPE between two identical scalars 1 (z; z), only even spin operators can
appear. This further constrains the (antichiral, non-BPS) 14 -BPS primary to have h
and the (chiral, non-BPS) 14 -BPS primary to have h
11 . Similar constraints apply to
their conjugates. In particular, this constraint on the spin forbids the G~+ 1 descendant of
13 , as well as its conjugate, to appear
in the chiral-chiral channel. We summarize the quantum numbers of the allowed internal
multiplets in the chiral-chiral channel in table 4.
The gap in the chiral-chiral channel.
gCaCp is de ned as the gap between
the dimension of the lightest operator that does not belong to a (c; c) multiplet, and that of a
charge 2q (c; c) primary 2q. In the current case,
gCaCp is the scaling dimension of this
lightest operator subtracted by 23 , the scaling dimension of the lowest dimensional operator
Note that this lightest operator is always a superconformal descendant while its primary
does not show up in the chiral-chiral channel due to R-charge conservation (see table 4).
We summarize the
level 12 ; 12 descendant
gCaCp for various internal channels in table 4. In particular, the
channel with
of an (a; a) primary 0 1 (z; z) with q = q =
13 minimizes the gap in the chiral-chiral
gCaCp =
(c; n) h =
; (a; n) h =
(n; n) h = h = ;
Note that the (a; a) primary 0 1 itself does not appear in the chiral-chiral OPE. We will
assume this minimal gap in the crossing equation when we do the numerical bootstrap.
Over a generic point on the moduli space M , the OPE coe cient of this (a; a) primary
0 1 (z; z) in the chiral-chiral channel is non-vanishing and hence
special points this OPE coe cient might vanish and
gCaCp = 23
. However, over
gCaCp would be bigger than 23 . The
OPE coe cient C(R; b) of this (a; a) primary 0 1 can be extracted from the subleading
terms in the large z expansion of the exact expression of the four-point function (A.6)
We can read o the low-lying multiplets in the chiral-chiral channel from the above
Here h and h denote the weights of the primaries, not the actual operators that appear
in the OPE. c; a; n stand for chiral, antichiral, and non-BPS primaries, respectively. For
the 14 -BPS primaries, their conjugates are also implicitly included. We see that all possible
primaries in (A.13) are of the above form.
As another consistency check, note that the lowest 14 -BPS operators that appear
tively. The latter is also related by a diagonal spectral ow to a 14 -BPS operator of (c; n)
of them are captured by the elliptic genus in appendix B.
gCaCp can jump to higher
value at some special points over the moduli space where the OPE coe cient C(R; b) for
the (a; a) primary 0 1 vanishes. From the next to leader term in the expansion of the
four-point function (A.6), we obtain the analytic expression for C(R; b)
C(R; b) =
1 9=2 R
3 R2j(v1 + !v2)j2)
which is proportional to @R (R; b). The OPE coe cient C(R; b) has zeroes at
R = p ;
R = p ;
b = 0 ;
b =
1.32 At these points, the gap in the chiral-chiral channel is saturated by the G+ 1 G~+ 1
be seen from the numerical bootstrap bound.
Chiral-antichiral channel
In the chiral-antichiral OPE channel between
exponential operator Op;v(z; z) in the untwisted sector,
Op;v(z; z) = N
X exp [ i(p + v=2)
1 , the internal primaries are the
XL(z) + i(p
where the sum in
over the Z3 images is to project to the Z3 invariant combinations. The
constant N is chosen such that the two-point function of Op;v is one. The exponential
h = 12 (p + v=2)2 and h = 12 (p
operator is labeled by the momentum p 2
and the winding v, with the weight given by
can be any lattice point in . However, a selection rule [33] in the chiral-antichiral channel
allows only those v 2
c to appear in the OPE between
We can parametrize the weights of these exponential operators more explicitly. Let us
write the metric ds2 = G
= 1; 2) of the target space torus as
x + 2 R. The B- eld background is
32The analytic expression for the gap in the chiral-antichiral channel is given in (A.19).
ds2 = (dx1 + !dx2)(dx1 + !2dx2) ;
= b R2
gCaAp =
with b normalized to have periodicity b
b + 1. The weight of the exponential operator
Op;v(z; z) with momentum p 2
and winding v 2
h =
h =
+ B )v R
with p ; v 2 Z. The selection rule in the chiral-antichiral channel that v 2 c
translated into the requirement that
with the restriction (A.20) is our
nal formula for the lowest dimension
BPS primaries in the chiral-antichiral channel of the four-point function h 13
gCaAp is not a continuous function of the moduli (R; b). This is because the
the momentum p
and winding number v that minimize the dimension h + h in (A.19)
gCaAp of
nonmight jump as we vary the moduli.
orbifold CFT can be computed at this point to be [76]
ZT 2=Z3 = (ZA2)3 =
1(Q; y2=3) !3
1(Q; y1=3)
Here we de ne Q = e2 i .
The NS sector elliptic genus is related by (diagonal) spectral ow,
ZTN2S=Z3(Q; y) = ZT 2=Z3(Q; yQ1=2)y1=2Q1=4 :
To see if there are 14 -BPS operators (BPS on the (anti)holomorphic side only) at a generic
q > 0 and r > 0, the characters are given by [77{79]
and the identity character
chr;q=1(Q; y) =
(1 + Qry)(1 + Qr+1y) FNS(Q; y) ;
chr;0<q<1(Q; y) =
ch0(Q; y) =
(1 + Q1=2y)(1 + Q1=2y 1) FNS(Q; y) ;
FNS(Q; y) =
Y1 (1 + Qk 1=2y)(1 + Qk 1=2y 1)
and similarly for characters with q < 0 and r < 0 (the representations are charge conjugate
to those with q > 0).
The twisted NS characters are de ned by
cfhr;q(Q; y) ( 1) rchr;q(Q; y 1) :
We thus have the decomposition
3cfh 52 ; 13 (Q; y) + 3cfh 72 ; 13 (Q; y) + cfh 32 ; 1(Q; y)
+ 3cfh 123 ; 31 (Q; y) + 3cfh 72 ; 23 (Q; y) + : : : ;
where the rst line comes from the BPS operators ((c; c) ring elements), while the second
and third lines are associated to 14 -BPS operators (in the non-BPS degenerate
representations on one side). As we have seen in the previous subsection, some of the latter operators
appear in the chiral-chiral channel of the four point function h 13
Free fermion OPEs at the kinks
In this appendix, we show that the bound (6.7) for OPEs of marginal BPS operators in
c = 9 SCFTs at
in nite volume point of the quintic). In the C3, T 6, or their orbifold CFTs, we have
three holomorphic fermions
q =
q =
1. Here ;
is some general complex 3 3 matrix. The operator 1(z; z) normalized to have
unit two-point function, h 1(z; z)
1(0; 0)i = 1=jzj4.
rst note that
gCaCp is 2, which is our assumption in the bootstrap bound in
gure 11, as is realized by operators of the form
Note that the dimension 3 operators M
jugate are descendant of 2 (de ned below).
(z) (z) ~ (z) ~ (z) and its complex
conp2p(Tr M M y)2
Tr M M yM M y
normalized such that it has unit two-point function. Combining (C.2) and (C.4), we have
computed the chiral ring coe cient
for 2 in the 1 1 OPE,
= p
gCaCp = 2.
Tr M M yM M y
Tr (M M y)2
By choosing di erent matrices M
, we will see that the four-point function of 1
saturates the bound (6.7) with
To start with, note that
> 2=p3 in the bootstrap bounding curve (6.7).
1 realizes
Next, in the chiral-antichiral channel, the lightest non-identity operator is
with scaling dimension 1. The operator O is a superconformal primary unless the matrix
= 2=p3 and the operator O is the
< 2=p3 in the bounding
R-current. It follows that the lightest non-BPS primary is replaced by the normal-ordered
operator : 1
1 :, which has dimension 2. This explains the peak at
= 2=p3 in the
bounding curve (6.7).
In summary, we see that by choosing di erent linear combinations of free fermion
bilinears, the numerical bootstrap bound is realized for all values of the chiral ring coe cient
in the case of
gCaCp = 2.33
The quintic threefold
the quintic Calabi-Yau threefold. In particular, we will review the exact formula of the
chiral ring coe cient and discuss various special points on the Kahler moduli space.
Let x0; x1; x2; x3; x4 be the homogeneous coordinates of P4. A quintic threefold M is
a hypersurface de ned by the vanishing locus of a quintic polynomial of xi's in P4. The
coe cients in the quintic polynomials, modulo linear rede nitions of the coordinates xi,
parametrize the complex structure moduli space. Hence the dimension of the complex
25 = 101. On the other hand,
i=1 Xi3. A
particular Z3 orbifold of the tensor product theory describes the mirror of the Z-manifold [80]. By taking
fusion rules (D.21), we see
gCaCp = 3
there is one parameter associated to the choice of the Kahler class, which can be thought
of as the size of P4, i.e. h1;1(M) = 1.
To construct its mirror, we consider a one-parameter family of the quintic M that is
p = x05 + x15 + x25 + x35 + x45
5 x0x1x2x3x4 = 0;
in P4. The mirror quintic W is obtained by performing the following Z35 orbifold action,
(x0; x1; x2; x3; x4) 7! ( 4x0; x1; x2; x3; x4) ;
(x0; x1; x2; x3; x4) 7! ( 4x0; x1; x2; x3; x4) ;
(x0; x1; x2; x3; x4) 7! ( 4x0; x1; x2; x3; x4) :
structure moduli space of the mirror quintic W is 5, since the replacement 7!
can be undone by coordinate rede nitions of xi. Hence the complex structure moduli space
of the mirror quintic can be taken to be the fundamental region 0
< 2 =5 on the
Chiral ring coe cient of the Kahler moduli space
The chiral ring coe cient and the metric on the Kahler moduli space of the quintic was
obtained in the seminal work of [46] using mirror symmetry. In this subsection we review
The Kahler potential K on the complex structure moduli space of the mirror quintic
W, or equivalently, on the Kahler moduli space of the quintic M, is given by [46],
where the functions $j( ) are de ned as
e K =
$0( ) = X
in the region j j > 1 of the fundamental domain on the the -plane, and
$j( ) =
2m (m=5)(5 j )m
in the region j j < 1. The coe cients bjrn are de ned in appendix B of [46]. The metric
on the complex structure moduli space is given by g
= @2K=@ @ . Going back to the
structure modulus
of the mirror quintic W by the mirror map,
t =
2$0 + $2
In the large volume limit of the quintic, the exactly marginal (c; c) primary operator
is the harmonic representative of the Kahler class. The chiral ring coe cient
for this (c; c) primary is given by the following combination
which is invariant under the coordinate and Kahler transformations. The latter is given
by rescaling the holomorphic three-form
of the mirror quintic by a holomorphic function
is thought of as a function of the Kahler modulus t through the inverse of
the mirror map (D.6). The \Yukawa coupling" is de ned as
= R
In gure 14 we present the contour plot of the chiral ring coe cient (t) of the Kahler
modulus of M in the t-coordinates. There are a few special points on the Kahler moduli
space that we will pay special attention to:
The gaps in the CC and CA channel at the large volume limit have been discussed
in appendix C.
structure point
simpli es to
The Kahler potential on the moduli space expanded around the large volume point is
where the : : : stand for the worldsheet instanton corrections that are powers of e 2 t2 .
The value of the chiral ring coe cient at the large volume point is the global minimum
on the whole moduli space:
e K =
(t = i1) = p
of the Kahler modulus for the quintic
M in the t-coordinates. The blue shaded region is the Kahler moduli space of the quintic. The
black curves are the constant
loci, with the values of
indicated at the ends of the curves. The
= 1. The
= 0 is shown in red.
The conifold point t
singular and the chiral ring coe cient diverges
! 1. The gaps at the conifold CFT
gCaAp = 1=2
The point t =
0:5 + 0:69i or
= 0 is where the 35 Gepner point
is located at on the Kahler moduli space. The 35 Gepner model, realized at a speci c
point on the Kahler and complex structure moduli space, is exactly solvable and is
value of the chiral ring coe cient at this point is a local minimum:
(t =
)) =
3 15=2
2 15=2
1 5=2
4 5=2
34The other point t
= 1 on the moduli space.
The gaps in the CC and the CA channel are
gCaCp = 6=5 and
gCaAp = 4=5.
Approaching the conifold point of the quintic moduli space, the (2; 2) Calabi-Yau sigma
model becomes singular and develops a continuum in the operator spectrum. The
conVj;m;m with quantum numbers
h =
q =
BPS representations with j = jmj
representations with
j = 2; 3; : : : .
12 + iR which are non-degenerate, discrete
CA channel, the gap is saturated by the bottom of the continuum representations with
q = 0 at j =
gCaAp = 2hgCaAp =
h j;j;j j;j;j 2j; 2j; 2j i = ut
gCaCp = 2hgCaCp = 2
We have written the gaps in terms of c because although the gaps were derived for integral
k, they are expected to hold for general k.35
Gepner points of one-parameter Calabi-Yau models
A simple class of one-parameter (i.e. with only one complex Kahler modulus) Calabi-Yau
manifolds generalizing the quintic are given by hypersurfaces in weighted projective space:
x51 + x25 + x35 + x45 + x55 = 0
x61 + x26 + x36 + x46 + x53 = 0
x81 + x28 + x38 + x48 + x52 = 0
x110 + x120 + x130 + x45 + x52 = 0:
2) : 35; 441; 64; 833.
Let us denote
the chiral ring generators by Xi.
Then the marginal chiral primaries are given by
Qi5=1 Xi; Q5
i=1 Xi; Q4
i=1 Xi2 and Q4
i=1 Xi2 respectively.
The chiral ring coe cients
are determined by the three point function coe cients in
N = 2 minimal models, which are given by
j;m=j;m=j [83]. Therefore we have
= 0
A-type modular invariants) by
j;m, with quantum numbers
h =
q =
minimal models (we will focus on the holomorphic side for simplicity) [84].
j1;m1 j2;m2 =
min(j1+j2;(k 2) j1 j2)
j=jj1 j2j
It is clear from the above fusion rules that the lightest operator in the OPE of a chiral
j;j;j with its conjugate is 1;0;0, with the following gap,
gCaAp =
( 1;0;0) =
Similarly, in the OPE of a pair of j;j;j , the lightest non-BPS superconformal primary
(whose descendant appears) is
k2 2j;2j k2 ;2j k2 , leading to a gap
gCaCp = 2
= 2
: (D.24)
We summarize the results in table 5.
m1 + m2 > j ;
m1 + m2 <
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
(2012) 110 [arXiv:1109.5176] [INSPIRE].
the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022
[arXiv:1203.6064] [INSPIRE].
(2014) 091 [arXiv:1307.6856] [INSPIRE].
Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical
Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].
Phys. Rev. D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE].
Dimensions, arXiv:1412.6549 [INSPIRE].
[INSPIRE].
3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
[arXiv:1510.03866] [INSPIRE].
Bootstrap of the K3 CFT, arXiv:1511.04065 [INSPIRE].
Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
Models, Nucl. Phys. B 296 (1988) 757 [INSPIRE].
Phys. Lett. B 199 (1987) 380 [INSPIRE].
[INSPIRE].
[hep-th/9301042] [INSPIRE].
Lett. B 218 (1989) 51 [INSPIRE].
Phys. B 324 (1989) 427 [INSPIRE].
[28] S. Cecotti, L. Girardello and A. Pasquinucci, Nonperturbative Aspects and Exact Results for
Int. J. Mod. Phys. A 6 (1991) 2427 [INSPIRE].
strings, World Scienti c, Singapore (1990).
Int. J. Mod. Phys. A 6 (1991) 1749 [INSPIRE].
Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
[INSPIRE].
B 303 (1988) 286 [INSPIRE].
Lett. B 220 (1989) 153 [INSPIRE].
otherwise, in Summer Workshop in High-energy Physics and Cosmology, Trieste Italy (1987).
Commun. Math. Phys. 246 (2004) 181 [hep-th/0203213] [INSPIRE].
Superconformal Algebras, Phys. Lett. B 210 (1988) 125 [INSPIRE].
[INSPIRE].
Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].
[hep-th/9903224] [INSPIRE].
Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
Math. Phys. 49 (2008) 032302 [hep-th/0612075] [INSPIRE].
B 477 (1996) 407 [hep-th/9606112] [INSPIRE].
an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].
Superconformal Algebras in Two-Dimensions or Exact Results on String Compacti cation,
Phys. Lett. B 172 (1986) 316 [INSPIRE].
[INSPIRE].
[50] D. Simmons-Du n, A Semide nite Program Solver for the Conformal Bootstrap, JHEP 06
[51] A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion
representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
[INSPIRE].
superconformal block, JHEP 03 (2007) 032 [hep-th/0611266] [INSPIRE].
Unreasonable E ectiveness of 6D SYM), JHEP 12 (2014) 176 [arXiv:1407.7511] [INSPIRE].
D black hole and N = 2 Liouville
theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE].
primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
[63] C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compacti cations, Commun.
[65] S. Ribault and J. Teschner, H+(3)-WZNW correlators from Liouville theory, JHEP 06
[66] A.B. Zamolodchikov, conformal symmetry in two-dimensions: an explicit recurrence formula
for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
[67] A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap
in Liouville eld theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
[70] V. Arnold, A. Varchenko and S. Gusein-Zade, Singularities of Di erentiable Maps: Volume I:
The Classi cation of Critical Points Caustics and Wave Fronts, Monographs in Mathematics,
STOC'98, Dallas U.S.A. (1998), pg. 10.
abstract), Proceedings of the thirtieth Annual ACM Symposium on Theory of Computing
Lett. B 233 (1989) 141 [INSPIRE].
Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
01 (2004) 025 [hep-th/0311141] [INSPIRE].
theories, JHEP 01 (2005) 027 [hep-th/0411041] [INSPIRE].
Compacti cations, Nucl. Phys. B 309 (1988) 295 [INSPIRE].
[1] A.A. Belavin , A.M. Polyakov and A.B. Zamolodchikov , In nite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl . Phys . B 241 ( 1984 ) 333 [INSPIRE].
[2] R. Rattazzi , V.S. Rychkov , E. Tonni and A. Vichi , Bounding scalar operator dimensions in [3] V.S. Rychkov and A. Vichi , Universal Constraints on Conformal Operator Dimensions , [4] D. Poland and D. Simmons-Du n, Bounds on 4D Conformal and Superconformal Field [5] D. Poland , D. Simmons-Du n and A. Vichi , Carving Out the Space of 4D CFTs , JHEP 05 [6] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving [7] F. Kos , D. Poland and D. Simmons-Du n, Bootstrapping the O(N ) vector models , JHEP 06 [8] C. Beem , L. Rastelli and B.C. van Rees , The N = 4 Superconformal Bootstrap , Phys. Rev.
[9] C. Beem , M. Lemos , P. Liendo , L. Rastelli and B.C. van Rees , The N = 2 superconformal [10] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving [11] S.M. Chester , J. Lee , S.S. Pufu and R. Yacoby , The N = 8 superconformal bootstrap in three [12] S.M. Chester , J. Lee , S.S. Pufu and R. Yacoby , Exact Correlators of BPS Operators from the 3d Superconformal Bootstrap , JHEP 03 ( 2015 ) 130 [arXiv:1412.0334] [INSPIRE].
[13] S.M. Chester , S.S. Pufu and R. Yacoby , Bootstrapping O (N ) vector models in 4 < d < 6, [14] J.- B. Bae and S.-J. Rey , Conformal Bootstrap Approach to O(N ) Fixed Points in Five [15] S.M. Chester , S. Giombi , L.V. Iliesiu , I.R. Klebanov , S.S. Pufu and R. Yacoby , Accidental Symmetries and the Conformal Bootstrap , JHEP 01 ( 2016 ) 110 [arXiv:1507.04424] [16] L. Iliesiu , F. Kos , D. Poland , S.S. Pufu , D. Simmons-Du n and R. Yacoby, Bootstrapping [17] F. Kos , D. Poland , D. Simmons-Du n and A. Vichi , Bootstrapping the O(N ) Archipelago, [18] C. Beem , M. Lemos , L. Rastelli and B.C. van Rees , The ( 2 ; 0) superconformal bootstrap, [19] M. Lemos and P. Liendo , Bootstrapping N = 2 chiral correlators , JHEP 01 ( 2016 ) 025 [20] Y.-H. Lin , S.-H. Shao , D. Simmons-Du n , Y. Wang and X. Yin , N = 4 Superconformal [21] P. Candelas , G.T. Horowitz , A. Strominger and E. Witten , Vacuum Con gurations for [22] D. Gepner , Space-Time Supersymmetry in Compacti ed String Theory and Superconformal [23] D. Gepner , Exactly Solvable String Compacti cations on Manifolds of SU(N ) Holonomy , [24] T. Eguchi , H. Ooguri , A. Taormina and S.-K. Yang , Superconformal Algebras and String Compacti cation on Manifolds with SU(N ) Holonomy, Nucl . Phys . B 315 ( 1989 ) 193 [25] E. Witten , Phases of N = 2 theories in two-dimensions, Nucl . Phys . B 403 ( 1993 ) 159 [29] S. Cecotti , L. Girardello and A. Pasquinucci , Singularity Theory and N = 2 Supersymmetry , [30] E.J. Martinec , Criticality, catastrophes and compacti cations , in Physics and mathematics of [31] C. Vafa , Topological Landau -Ginzburg models, Mod. Phys. Lett. A 6 ( 1991 ) 337 [INSPIRE].
[32] S. Cecotti , N = 2 Landau-Ginzburg versus Calabi-Yau -models: Nonperturbative aspects , [33] L.J. Dixon , D. Friedan , E.J. Martinec and S.H. Shenker , The Conformal Field Theory of [34] C. Vafa , String Vacua and Orbifoldized L-G Models, Mod. Phys. Lett. A 4 (1989) 1169 [35] L.J. Dixon , Some world sheet properties of superstring compacti cations , on orbifolds and [36] N. Seiberg , Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys.
[37] D. Kutasov , Geometry on the Space of Conformal Field Theories and Contact Terms, Phys.
[38] S. Gukov and C. Vafa , Rational conformal eld theories and complex multiplication , [39] T. Eguchi and A. Taormina , On the Unitary Representations of N = 2 and N = 4 [40] S. Cecotti and C. Vafa , Topological antitopological fusion, Nucl . Phys . B 367 ( 1991 ) 359 [41] H. Ooguri and C. Vafa , Two-dimensional black hole and singularities of CY manifolds, Nucl .
[42] N. Seiberg and E. Witten , The D1=D5 system and singular CFT , JHEP 04 ( 1999 ) 017 [43] O. Lunin and S.D. Mathur , Correlation functions for M N = SN orbifolds, Commun . Math.
[44] M.R. Douglas , R.L. Karp , S. Lukic and R. Reinbacher , Numerical Calabi-Yau metrics , J.
[45] H. Ooguri , Y. Oz and Z. Yin , D-branes on Calabi-Yau spaces and their mirrors, Nucl . Phys.
[46] P. Candelas , X.C. De La Ossa , P.S. Green and L. Parkes , A Pair of Calabi-Yau manifolds as [47] E. Witten , Mirror manifolds and topological eld theory, hep-th/9112056 [INSPIRE].
[48] W. Boucher , D. Friedan and A. Kent , Determinant Formulae and Unitarity for the N = 2 [49] K. Hosomichi , N = 2 Liouville theory with boundary , JHEP 12 ( 2006 ) 061 [hep-th /0408172] [52] V.A. Belavin , N = 1 supersymmetric conformal block recursion relations, Theor . Math. Phys.
[53] L. Hadasz , Z. Jaskolski and P. Suchanek , Recursion representation of the Neveu -Schwarz [54] L. Hadasz , Z. Jaskolski and P. Suchanek , Elliptic recurrence representation of the N = 1 superconformal blocks in the Ramond sector , JHEP 11 ( 2008 ) 060 [arXiv:0810.1203] [55] A. Giveon and D. Kutasov , Little string theory in a double scaling limit , JHEP 10 ( 1999 ) [56] O. Aharony , B. Fiol , D. Kutasov and D.A. Sahakyan , Little string theory and heterotic/type-II duality, Nucl . Phys . B 679 ( 2004 ) 3 [hep -th/0310197] [INSPIRE].
[57] O. Aharony , A. Giveon and D. Kutasov , LSZ in LST, Nucl. Phys . B 691 ( 2004 ) 3 [58] C.-M. Chang , Y.-H. Lin , S.-H. Shao , Y. Wang and X. Yin , Little String Amplitudes (and the [59] K. Hori and A. Kapustin , Duality of the fermionic 2 [60] M. Baggio , J. de Boer and K. Papadodimas , A non-renormalization theorem for chiral [61] S. Odake , Extension of N = 2 Superconformal Algebra and Calabi-Yau Compacti cation , [64] D. Friedan and C.A. Keller , Constraints on 2d CFT partition functions , JHEP 10 ( 2013 ) 180 [68] F. Kos , D. Poland and D. Simmons-Du n, Bootstrapping Mixed Correlators in the 3D Ising [69] M. Bertolini , I.V. Melnikov and M.R. Plesser , Accidents in ( 0 ; 2) Landau-Ginzburg theories, [71] M. Ajtai , The shortest vector problem in L2 is NP-hard for randomized reductions (extended [72] S. Hellerman , A Universal Inequality for CFT and Quantum Gravity , JHEP 08 ( 2011 ) 130 [73] J.D. Qualls and A.D. Shapere , Bounds on Operator Dimensions in 2D Conformal Field Theories , JHEP 05 ( 2014 ) 091 [arXiv:1312.0038] [INSPIRE].
[74] S. Collier , Y.-H. Lin and X. Yin , Modular Bootstrap Revisited, arXiv:1608 .06241 [INSPIRE].
[75] E.J. Chun , J. Mas , J. Lauer and H.P. Nilles , Duality and Landau-Ginzburg Models , Phys.
[76] E. Witten , On the Landau-Ginzburg description of N = 2 minimal models , Int. J. Mod.
[77] V.K. Dobrev , Characters of the Unitarizable Highest Weight Modules Over the N = 2 Superconformal Algebras , Phys. Lett . B 186 ( 1987 ) 43 [INSPIRE].
[78] E. Kiritsis , Character Formulae and the Structure of the Representations of the N = 1, N = 2 Superconformal Algebras , Int. J. Mod. Phys. A 3 (1988) 1871 [INSPIRE].
[79] Y. Matsuo , Character Formula of C < 1 Unitary Representation of N = 2 Superconformal Algebra, Prog. Theor. Phys . 77 ( 1987 ) 793 [INSPIRE].
[80] P. Candelas , E. Derrick and L. Parkes , Generalized Calabi-Yau manifolds and the mirror of a rigid manifold, Nucl . Phys . B 407 ( 1993 ) 115 [hep-th /9304045] [INSPIRE].
[81] T. Eguchi and Y. Sugawara , Modular bootstrap for boundary N = 2 Liouville theory , JHEP [82] T. Eguchi and Y. Sugawara , Conifold type singularities, N = 2 Liouville and SL (2:R)/U(1) [83] J. Distler and B.R. Greene , Some Exact Results on the Superpotential from Calabi-Yau [84] G. Mussardo , G. Sotkov and M. Stanishkov , N = 2 superconformal minimal models, Int . J.