Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops Andre Lukasa1Fabian Ruehlebc2 aRudolf Peierls Centre for Theoretical Physics University of Oxford

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Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops

Andre Lukasa,1,Fabian Ruehleb,c,2

aRudolf Peierls Centre for Theoretical Physics, University of Oxford

Parks Road, Oxford OX1 3PU, UK

bDepartment of Physics & Department of Mathematics, Northeastern University

360 Huntington Avenue, Boston, MA 02115, United States

cThe NSF AI Institute for Artiﬁcial Intelligence and Fundamental Interactions

Boston, MA, United States

Abstract

Calabi-Yau threefolds with inﬁnitely many ﬂops to isomorphic manifolds have an extended K¨ahler

cone made up from an inﬁnite number of individual K¨ahler cones. These cones are related by reﬂection

symmetries across ﬂop walls. We study the implications of this cone structure for mirror symmetry,

by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such

isomorphic ﬂops across facets of the K¨ahler cone boundary give rise to symmetry groups isomorphic

to Coxeter groups. In the dual Mori cone, non-ﬂopping curve classes that are identiﬁed under these

groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant

under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For

some cases, these functions can be expressed in terms of theta functions whose appearance can be

linked to an elliptic ﬁbration structure of the Calabi-Yau manifold.

1andre.lukas@physics.ox.ac.uk

2f.ruehle@northeastern.edu

arXiv:2210.09369v2 [hep-th] 9 Feb 2023

Contents

1 Introduction 2

2 A simple warm-up example 4

2.1 Single-ﬂop cases with Picard rank two . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Exampleforasingleﬂop.................................... 6

3 Inﬁnitely many ﬂops for Picard rank two 7

3.1 Groupstructure......................................... 8

3.2 Cones .............................................. 9

3.3 Theprepotential ........................................ 10

3.4 Example with inﬁnitely many ﬂops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Manifolds with h1,1(X)>2 and two symmetry generators 12

4.1 Groupstructure......................................... 13

4.2 Theprepotential ........................................ 14

4.3 Examples ............................................ 17

5 Coxeter groups, triangle groups, and reﬂection symmetries 18

6 Conclusions 21

A Appendix: Special symmetry groups and elliptic ﬁbrations 22

A.1 Type1.............................................. 22

A.2 Type2.............................................. 23

1 Introduction

Topology change is an interesting and characteristic feature of string theory which has been studied

for some time [1–3], mostly in the context of string compactiﬁcations on Calabi-Yau (CY) threefolds.

There are two main types of topology changing transitions for CY threefolds: a milder form known as

ﬂop transitions and a more severe form, the conifold transitions. In the paper, we will be interested in

the former.

Flop transitions of CY threefolds Xare known to leave the Hodge numbers h:= h1,1(X) and h2,1(X)

unchanged but they can change more reﬁned topological invariants, such as the intersection form and

the second Chern class of the tangent bundle. Recently, it has been noted [4,5] that topology-preserving

ﬂop transition X→X0between two isomorphic CY threefolds Xand X0are by no means rare. Such

isomorphic ﬂops, as we will call them, and their implications are the main topic of this paper.

Suppose an isomorphic ﬂop arises at a boundary facet of the K¨ahler cone Kof X. Then, there is an

involution, which, relative to a suitable basis (Di) of divisor classes, can be described by an h×hmatrix

M(satisfying ˜

M2=h×h). If two divisors D=kiDiand D0=k0iDiare related by this involution,

that is, k0=˜

Mk, then it turns out that their associated linear systems have the same dimension [6], so

h0(OX(D)) = h0(OX(D0)). Isomorphic ﬂops can arise across more than one facet of the K¨ahler cone

boundary. In this case, we have multiple involutions and corresponding matrices ˜

M1,..., ˜

Mkwith

1=··· =˜

k=h×h, generating groups ˜

G. Such groups generated by reﬂections were introduced

by Coxeter [7] and further studied by Tits and Vinberg [8]. Especially the latter studies reﬂections

along the walls of polyhedral cones, and shows that these correspond to Coxeter groups. This structure

gives rise to inﬁnite sequences of isomorphic ﬂops generated by repeatedly reﬂecting the K¨ahler cone

along a ﬂop wall and its reﬂection images under ˜

Mi. The union Kext =∪αKαof their K¨ahler cones is

referred to as the extended K¨ahler cone and is mirror-dual to the complex structure moduli space of

the mirror of X[2]. It turns out, the zeroth cohomology of line bundles on Xis invariant under the

entire group ˜

Gand this fact can be immensely helpful for deriving formulae for cohomology [6]. In the

context of inﬁnite ﬂop chains acting on divisors, the authors of [9] recently studied Euclidean D3 branes

and noted that theta functions also appear in the non-perturbative superpotential of Type IIB.

In the present paper we are interested in the implications of this symmetry for Gopakumar-Vafa (GV)

invariants [10, 11] and the instanton prepotential for K¨ahler moduli. To this end, we introduce a (dual)

basis (Ci) of curve classes and represent arbitrary classes Cby h-dimensional integer vectors dsuch

that C=diCi. Their GV invariants are denoted by nd. On these curve classes, the involutions act via

the matrices Ma=˜

aand the entire group via the dual Gof ˜

G, generated by the matrices Ma. Our

main observation is that classes dwhich do not ﬂop at any of the facets of the (possibly inﬁnite sequence

of) CYs Xα, the GV invariants are unchanged under the action of G, so nd=ngd for all g∈G. This

implies that a part of the instanton prepotential for the K¨ahler moduli (speciﬁcally, the part associated

to non-ﬂopping curve classes) is ˜

G-invariant and can be expressed in terms of ˜

G-invariant functions

ψG

d(T) = X

g∈G

e2πi(gd)·T=X

˜g∈˜

e2πid·(˜gT )⇒ψG

d(˜gT ) = ψG

d(T)∀˜g∈˜

G , (1.1)

where T=χ+it are the complexiﬁcations of the K¨ahler parameters ti. As far as we are aware, these

functions, invariant under certain representations of Coxeter groups, have not been introduced and

studied in this context. We will show that, for certain special cases, depending on the underlying CY

manifold X, they can be expressed in terms of Jacobi theta functions whose appearance can be traced

to an elliptic ﬁbration structure of X. For explicit examples, we will work with complete intersection

CYs in products of projective spaces (CICYs) [12].

Note added: On the day we submitted our paper to arxiv, a revised version of [13] appeared on the

arxiv, which included a discussion of Coxeter groups in the context of the Hulek-Verrill manifold [14].

The plan of the paper is as follows. In Section 2, we start with a simple warm-up example, a CY manifold

with h1,1(X) = 2 and only a single ﬂop boundary, leading to a ﬁnite symmetry ˜

G=h˜

M1i∼

=Z2. The

insight from this example is used in Section 3 to study CY manifolds with h1,1(X) = 2 and two ﬂop

boundaries, with the associated groups ˜

G=h˜

M1,˜

M2iisomorphic to universal Coxeter groups with two

generators. In Section 4 we generalize the discussion to manifolds with h1,1(X)>2 but still with two

isomorphic ﬂop boundaries and group ˜

G=h˜

M1,˜

M2i. As we will show, this case exhibits new features,

compared to the h1,1(X) = 2 case, and, in particular, we ﬁnd that the ˜

Ginvariant functions ψG

dcan

sometimes be expressed in terms of Jacobi theta functions. The general case with arbitrary h1,1(X)

and arbitrary number of ﬂop boundaries, and its relation to Coxeter groups, is discussed in Section 5.

We present our conclusions in Section 6. Appendix A contains details about the relation of CYs with

inﬁnitely many ﬂops, the occurrence of Jacobi theta functions in the prepotential, and the presence of

elliptic ﬁbrations.

2 A simple warm-up example

We are mostly interested in cases where the group Gis of inﬁnite order. However, there are two

complications, related to the structure of G, in analyzing such cases. First, it is diﬃcult to express

elements of Coxeter groups with more than two generators in a suﬃciently systematic way in terms

of the generators, which we rely on to facilitate computations such as working out the sums in (1.1).

Secondly, even for Coxeter groups with only two generators, where writing group elements in terms of

generators is relatively simple, the actual matrices in Gbecome complicated. For this reason, we start

our discussion with the simple single-ﬂop case where G∼

=Z2, thereby cutting out the above-mentioned

complications, and leaving us to focus on other issues, such as the relevant cone structures and properties

of GV invariants. Cases with inﬁnite order groups Gwill be discussed in the subsequent sections.

2.1 Single-ﬂop cases with Picard rank two

Consider a CY manifold Xwith h:= h1,1(X) = 2, a basis (D1, D2) of divisor classes generating

the K¨ahler cone, corresponding K¨ahler parameters t= (t1, t2) and K¨ahler forms JX=tiDi, where

ti≥0. So, relative to our chosen divisor basis, the K¨ahler cone is then the positive quadrant, K=

{t1e1+t2e2|ti≥0}(where eidenote the standard unit vectors). We assume that {t2= 0}is a boundary

of the eﬀective cone1while {t1= 0}is a boundary with an isomorphic ﬂop. Such a structure can be

easily detected from the triple intersection numbers λijk =Di·Dj·Dkby introducing the quantities

m1=2λ122

λ222

, m2=2λ112

λ111

.(2.1)

An isomorphic ﬂop at {t1= 0}occurs (for generic complex structure choice) if m1is ﬁnite and integral,

and {t2= 0}ends the eﬀective cone if m2does not satisfy these requirements. In this case, passing

through t1= 0 leads to an isomorphic CY X0and an involution, ˜

G=h˜

M1i∼

=Z2generated by

M1= −1 0

m11!.(2.2)

We reserve the symbol Mfor the 2 ×2 reﬂection matrix and later, when h1,1(X)>2, use the symbol

Mfor higher-dimensional matrices.

The K¨ahler cone K0=R+(˜v1,˜v2) of X0is generated by

˜v1=˜

M1e1= −1

m1!,˜v2=˜

M1e2=e2,(2.3)

and the two K¨ahler cones are exchanged by the involution, K0=˜

M1K. In particular, since the boundary

along e1is mapped into the one along v1the latter just as the former must be a boundary of the eﬀective

cone, so that Keﬀ =K ∪ K0={teﬀ

1˜v1+teﬀ

2e2|teﬀ

i≥0}.

1The discussion would not change if it was just the boundary of the extended K¨ahler cone.

Homology classes C=d1C1+d2C2are labeled by integer vectors d= (d1, d2), relative to a dual basis

(Ci), so Ci·Dj=δi

j, and their GV invariants are denoted by nd. Curve classes are acted on by the

dual group G=hM1i∼

=Z2, with generator

M1=˜

1= −1m1

0 1 !,(2.4)

and the dual (Mori) cones2

M={c1e1+c2e2|ci≥0},M0={c0

1v1+c0

2v2|c0

i≥0}, v1= m1

1!, v2=−e1(2.5)

are exchanged by the action of M1. We call their intersection Mrestr =M∩M0={crestr

1v1+

crestr

2e2|crestr

i≥0}the restricted cone.

To discuss what happens to GV invariants under the action of G, we should distinguish between ﬂopping

and non-ﬂopping curve classes.

Non-ﬂopping curve classes are those for which all curves retain a ﬁnite volume under a ﬂop transition,

that is, all curve classes that are not in the codimension 1 facet of the Mori cone dual to the ﬂop wall of

the K¨ahler cone. These must have G-invariant GV invariants, ngd =ndfor all g∈G, since the number

of curves in such a class is not aﬀected by the ﬂop. Non-ﬂopping classes split into two groups, namely

those in the restricted cone Mrestr and those outside. Under the action of G, non-ﬂopping classes in

d∈ M \ Mrestr are mapped outside the Mori cone of Xand consequently their GV invariants must

vanish. In other words, all non-ﬂopping classes with positive GV invariants reside in the (G-invariant)

restricted cone. Within the restricted cone, we can identify a fundamental region of the G-action,

Mf⊂ Mrestr. For the present case, this can be chosen as Kf={cf

1w+cf

2e2|cf

i≥0}. We illustrate this

cone structure for the example (to be discussed in more detail in the following sub-section) in Figure 1.

Flopping curve classes are classes with nd>0 that contain curves shrinking to zero volume, diti→0,

at the ﬂop locus. In the present case, the ﬂop wall is {t1= 0}, so they must be of the form d= (d1,0).

In fact, among those, the only classes which can have non-zero GV invariants [4] are (1,0) and (2,0),

so n(d1,0) = 0 for d1>2. We collect all ﬂopping classes in a set B, which in this example is simply

B ⊂ {(1,0),(2,0)}. The GV invariants for these ﬂopping classes are not G-invariant: The image of a

curve class dwith nd6= 0 is M1(d1,0)T= (−d1,0)T, which is outside the cone of eﬀective curves M

and, hence, its GV should vanish. Of course, from the point of view of the manifold X0with K¨ahler

parameter t0

1=−t1the situation is reversed and its GV invariants satisfy n0

(d1,0) = 0 for all d1>0 and

(−d1,0) >0 for d1∈ {1,2}.

Now we have enough information to discuss the implications of the symmetry Gfor the instanton

prepotential [15]

Finst =X

ndLi3(e2πid·T).(2.6)

Splitting the sum into ﬂopping and non-ﬂopping classes and introducing G-orbits for the latter we have

Finst =

X

d∈B

nd+X

d∈Mrestr

nd

Li3(e2πid·T) = X

d∈B

ndLi3(e2πid·T) + X

d∈Mf

ndΨG

d(T) (2.7)

2The Mori cones are denoted by M,M0, etc., while reﬂections are denoted by M1,M2, etc. Despite this slight clash

of notation, it should hopefully be clear which object is meant in any given context.

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SymmetriesofCalabi-YauPrepotentialswithIsomorphicFlopsAndreLukasa;1,FabianRuehleb;c;2aRudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxfordParksRoad,OxfordOX13PU,UKbDepartmentofPhysics&DepartmentofMathematics,NortheasternUniversity360HuntingtonAvenue,Boston,MA02115,UnitedStatescTheNSFAIInstitu...

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Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops Andre Lukasa1Fabian Ruehlebc2 aRudolf Peierls Centre for Theoretical Physics University of Oxford

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