Flops of any length, Gopakumar-Vafa invariants, and 5d Higgs Branches

The conifold is a basic example of a noncompact Calabi-Yau threefold that admits a simple flop, and in M-theory, gives rise to a 5d hypermultiplet at low energies, realized by an M2-brane wrapped on the vanishing sphere. We develop a novel gauge-theoretic method to construct new classes of examples that generalize the simple flop to so-called length l=1,...,6. The method allows us to naturally read off the Gopakumar-Vafa invariants. Although they share similar properties to the beloved conifold, these threefolds are expected to admit M2-bound states of higher degree. We demonstrate this through our computations of the GV invariants. Furthermore we fully characterize the associated Higgs branches by computing their dimensions and flavor groups. With our techniques we extract more refined data such as the charges of the hypers under the flavor group.

D Mathematica code for computing the zero modes 49

Introduction
Singular noncompact Calabi-Yau threefolds are interesting objects both from a mathematical as well as a physics point of view. In physics, by defining string theory or M-theory on them, they can provide the framework for modeling effective supersymmetric gauge theories in four and five dimensions, with gravity decoupled. This can happen via the branes at singularities paradigm, whereby a bunch of D3-branes probes such a singularity, or via the geometric engineering paradigm, whereby the field theory content emanates from membranes wrapping the vanishing cycles of the singularity. We will focus on the latter. 1 The prototypical example, the conifold, is a threefold that can be described as a hypersurface in C 4 . It admits a so-called small resolution, which means that the singularity is replaced by a P 1 (unlike other singularities where the exceptional locus is of complex dimension two). This sphere can flop to negative volume, and a homologically inequivalent sphere of positive area grows.
The conifold is but the first in a series of threefolds that admit simple flops. Here, 'simple' refers to the fact that only one sphere grows after resolution (as opposed to a chain of spheres). The conifold is known as a simple flop of length one. Here, the length of a flop refers to a multiplicity that can be attributed to the exceptional P 1 . If one pulls back the skyscraper sheaf corresponding to the singular point of the conifold w.r.t. the blow-down map, one will get the structure sheaf of the exceptional curve. (1.1) It has been known for decades that other lengths are possible, up to l = 6, such that Even though the above definition may appear rather obscure, the length has a very concrete physical meaning: as we will see later, when we geometrically engineer Mtheory on simple threefold flops described as deformed ADE singularities, we find that the length of the simple flop is nothing but the maximal flavor charge of the 5d hypermultiplets arising from M2 branes wrapped on the exceptional P 1 . We will get into the details of these aspects in later sections. In order to construct such higher length flops, the strategy in [1] has been to consider families of deformed ADE surfaces. In other words, fibering local K3's over the deformation parameter space. In general, the total space of the families may or may not be singular. Resolving the possible singularity inflates part of the exceptional P 1 's associated to the simple roots of the ADE algebra associated with the family. Given such a family, which will give rise to an n-fold, one can always cut out a complex three-dimensional subvariety, which will be guaranteed to give a Gorenstein threefold.
In this paper, following [2,3], we develop a new method to construct simple flops spanning all the possible values of the length of the exceptional P 1 (up to the maximal value l = 6), based on a gauge-theoretic approach. To this end, we consider M-theory on an ALE surface with an ADE singularity. The 7d effective theory is a SYM theory with three adjoint scalars and group G = A, D, E with N = 1 supersymmetry. One can break half of the supercharges by switching on a BPS vev for the scalar fields in the following way: • Take the 7d spacetime to be R 5 × C. Call w the local coordinate on C.
• Take a holomorphic vev for Φ that depends on a coordinate w, transverse to the local K3.
This preserves only a 5d Poincaré group. The 7d gauge group is broken to the commutant of Φ. The 7d N = 1 vector multiplet was made up of the gauge field A M (M = 0, 1, ..., 6) and the three scalars φ i (i = 1, 2, 3); after giving a vev to Φ, only a 5d N = 1 vector multiplet (A µ , φ 3 ) in the subalgebra H that commutes with Φ survives (these vector multiplets still propagate in 7d, and are then seen as background vector multiplets from the 5d point of view). As we will see in detail in the main text, there are also localized Φ-modes at w = 0 that organize in hypermultiplets charged under H and that propagate in 5d. The fields Φ and φ 3 can be given two physical interpretations: In the A or D series, the local K3 has a C * -fibration structure; we can then reduce M-theory to IIA, and regard them as the three adjoint scalars or a stack of D6-branes. For the E series, we can consider F-theory on an elliptically fibered K3, giving rise to 8d SYM. Physically, Φ is the adjoint Higgs on the stack of non-perturbative sevenbranes. By further compactifying on a circle, we arrive at 7d SYM with exceptional gauge group.
The vev's of Φ and φ encode the complex structure and Kähler moduli of the local K3, respectively. When both are zero, the surface is singular. Activating φ blows up spheres of non-zero Kähler volume. However, activating Φ will switch on versal deformations of the local K3, which will unfold the singularity according to its Casimir invariants. From the geometric point of view, we will obtain a fibration of deformed ALE surfaces over the w-plane, giving rise to a threefold X that admits a simple flop. Reducing M-theory directly on X produces the 5d spectrum of propagating hypermultiplets and background vector multiplets outlined before.
Resolving the full fibration is known as a simultaneous resolution. Given the Dynkin diagram of the gauge algebra in question, a simultaneous resolution corresponds to a choice of simple roots. This selects, the subspace H of the Cartan subalgebra t of the original ADE algebra where φ 3 lives. The commutant of φ 3 tells us what is the form of a generic Φ producing the sought-after family. The M-theory threefold is then obtained by choosing the w-dependence of the Casimir invariants of Φ, which uniquely fix the defining equation of the threefold.
At this point, one can compute the number of hypermultiplets of the 5d effective theory, by counting the zero modes of the complex adjoint Φ that are localized around the singularity. Their number corresponds to the BPS index of M2-states wrapping the vanishing P 1 's, aka the Gopakumar-Vafa invariants. In this paper, we are able to compute the charge of each hypermultiplet with respect to the flavor group, which gives us not only the multiplicity of states, but the actual refined information of the GV invariants, which are organized by homology classes of the threefold. This furthermore allows us to compute detailed information about the 5d Higgs branch: Its dimension is equal to the sum of all GV invariants while the action of the flavor group is determined by the charges of the hypers.
The paper is organized as follows. In Section 2 we review the families of deformed ADE singularities. In Section 3 we explain how the choice of spheres that are blown up in the simultaneous resolution determines the vev for Φ; by using this Φ we then show how to extract the hypersurface equation of the threefold and its GV invariants. In Section 4 we apply our method to simple flops, by constructing explicitly threefolds with any length from 1 to 6. In Section 5 we draw our conclusions. We add four appendices: in Appendix A we outline the Katz-Morrison method to derive the threefold equation; in Appendix B we write some long formulae for the exceptional cases, necessary to compute the corresponding threefold equations; in Appendix C we illustrate a check of our computations by using the Milnor number of the considered singularities; in Appendix D we expose the Mathematica code we worked out to compute the zero modes once one chooses a specific Φ.

ADE families and simultaneous resolution
In this section, we briefly summarize the philosophy and method of [1] for building simultaneous resolutions, by fibering ALE surfaces over the Cartan torus of an ADE group. Consider an ALE surface in C 3 with an ADE singularity, defined by the singular hypersurface equation f (x, y, z) = 0. The cases in the ADE classification can be 4 presented as: One can deform these equations by adding a certain number of monomials: where the monomials g i belong to the ring: The coefficients µ i of the monomials g i are coordinates on the space of deformations. One can construct a space by fibering the deformed ALE space over the deformation space: at the origin (where all deformation coefficients are zero), the ALE space develops the ADE singularity. The total space of the fibration may or may not be singular, depending on the actual fibration structure.
To illustrate the setup in more detail, we focus on an A r−1 type ALE family. This machinery is well-developed in [1]. The equation defining the A r−1 singularity is 1 uv = z r . (2.4) The versal deformation is Promoting the σ i 's to coordinates, this equation describes a fibration with fiber given by the (deformed) A r−1 surface and base the space t/W of gauge invariant coordinates σ i (i = 2, ..., r) 2 on the Lie algebra sl(r), where t is the Cartan torus and W the Weyl 1 Obtained from (2.1) by the change of variables x = v−u 2 , y = v+u 2i . 2 For A r−1 , the invariant σ i is the i-th elementary symmetric polynomial in the eigenvalues of an 5 group.
The space (2.5), a fibration over t/W, is non-singular. However, by making a base change, one obtains a singular space whose resolution blows up a subset of the roots of the central ALE fiber. Let us consider the simplest case, i.e. r = 2. The equation (2.5) becomes which is perfectly smooth as a total space, even though the central fiber over σ 2 = 0 is a singular A 1 -surface. If we make the base change σ 2 = t 2 , i.e. we pull-back the ALE fibration w.r.t. the map: where, in this case, t ∼ = C, we obtain a threefold with a conifold singularity: The small resolution of the conifold blows up the simple root of A 1 in the central fiber. This is called a simultaneous resolution. The family is now fibered over the base t, whose t is a coordinate. For generic r one can make an analogous base change by pulling back the family w.r.t. to the map by taking σ k 's written as W-invariant functions of coordinates t i on t. This resolves all the simple roots of A r−1 in the central fiber. One can also choose a different base change where where W ⊂ W. In this case, the resolution of the family blows up the roots that are left invariant by W , in the central fiber. The base of the fibration is now parametrized by the r − 1 W invariants, that we call i (i = 1, ..., r − 1). The i 's can also be associated with W -invariant functions of coordinates t i on t. This line of reasoning can be generalized [1] to all the cases in the ADE classification. element of the Lie algebra.
Let us consider the family fibered over t. At the origin of t, the surface develops the corresponding ADE singularity. At a generic point of t, the singularity is deformed: the surface admits n non-vanishing S 2 intersecting in the same pattern as the nodes of the Dynkin diagram and whose volume is measured by the holomorphic (2,0)-form. In the deformed ALE surface, the holomorphic (2,0)-form is along an element t of the Cartan torus, such that vol We choose a set of coordinates t i of t such that the volumes of the simple roots are given by It is useful to state the explicit form of the versal deformations of the singularities (2.1) as fibrations of deformed ADE surfaces over the base t (with simultaneous resolution of all the simple roots): t i y = 0 E 6 : x 2 + z 4 + y 3 + 2 yz 2 + 5 yz + 6 z 2 + 8 y + 9 z + 12 = 0 E 7 : x 2 + y 3 + yz 3 +˜ 2 y 2 z +˜ 6 y 2 +˜ 8 yz +˜ 10 z 2 +˜ 12 y +˜ 14 z +˜ 18 = 0 E 8 : x 2 + y 3 + z 5 +ˆ 2 yz 3 +ˆ 8 yz 2 +ˆ 12 z 3 +ˆ 14 yz +ˆ 18 z 2 +ˆ 20 y +ˆ 24 z +ˆ 30 = 0, (2.14) where the i ,˜ i ,ˆ i are known functions of the parameters t i ∈ t (see [1] for the explicit expression of i ,˜ i and an algorithm to computeˆ i ). The forms (2.14) admit the blow up of a collection of P 1 s in the central fiber dual to all the simple roots of the corresponding ADE algebra. Exactly as for the A r -families, one can make a simultaneous resolution of a subset of the simple roots, by choosing a subgroup W ⊂ W that leaves these roots invariant. The corresponding family is fibered over the space parametrized by the coordinates i of t/W . 7 3 Threefolds as ALE families, GV invariants and Φ The families of ALE surfaces described in the previous section are a natural starting point for constructing CY three-folds: the coordinates i ∈ t/W can be chosen so as to depend (linearly) 3 on a single complex parameter w. Such a threefold has then the structure of an ALE fibration over the complex plane C w and it will be defined by an equation such as where F (x, y, z, 1 , ..., r ) = 0 is the defining equation of the (r + 2)-dimensional family (r is the rank of the ADE algebra). We will restrict on threefolds X that have isolated singularities, whose exceptional locus does not contain compact divisors. 4 In the following we are going to show how to build a threefold family of deformed ADE surfaces by starting from the requirement that the ALE fibration over C w presents a specific partial simultaneous resolution, i.e. a choice of resolved simple roots, say α 1 , ...α . As we will see, this method can actually produce in principle the full ADE family over t/W .

The Higgs field from the simultaneous resolution
The ADE families over t/W and the corresponding threefolds have been derived and studied using several methods, such as [1,[4][5][6][7][8]. We are going to use a different (more physical) approach to construct these spaces that is based on the choice of an element Φ of the ADE algebra, depending on the coordinate w. Our starting point is M-theory reduced on an ALE surface with an ADE singularity of type g. M-theory on an ADE singularity gives rise to a N = 1, d = 7 gauge theory with gauge group G ADE and three adjoint scalars φ i (i = 1, 2, 3). With a choice of a complex structure, a vev for Φ ≡ φ 1 + iφ 2 corresponds geometrically to a deformation of the singularity, while a vev for φ 3 corresponds to a resolution, with [ Φ , φ 3 ] = 0. In particular the coefficients of the deforming monomials are associated with the Casimir invariants of Φ.
We fiber this background over the C w -plane by switching on a non-zero w-dependent vev for the complex adjoint scalar Φ. Geometrically this corresponds to deforming the singularity differently at different values of w, obtaining a threefold X that is an ALE fibration over C w .
We now want to select a vev for Φ that allows the (simultaneous) resolution at the origin only of a choice of simple roots α 1 , ...α of g. Rephrased in the 7d field theory language, Φ must be compatible with switching on a vev for φ 3 along the subalgebra Since [ Φ , φ 3 ] = 0, then Φ must live in the commutant of H, that we call L. Such a subalgebra, that is the commutant of diagonal elements, is called a Levi subalgebra of g. 5 Summing up, the choice of the blown up simple roots select an abelian subalgebra H ⊂ g. This defines a Levi subalgebra L ⊂ g, that is of the form with L h simple Lie algebras. The Higgs field 6 Φ corresponding to the ALE fibration should be a generic element of L.
The Casimir invariants of the Higgs field Φ tell us how the ALE fiber is deformed. Since at the origin of C w the fiber presents the full ADE singularity, we expect that all the Casimir invariants of Φ vanish at w = 0. This means that at w = 0 the Higgs field should be a non-zero nilpotent element of L. In particular, when restricted on each summand L h of the Levi subalgebra, Φ must be in the corresponding principal nilpotent orbit, if we do not want terminal singularities. 7 We call the Higgs at the origin X + . The generic Higgs field has then the following form where Y ∈ L can in principle depend on w; in this paper we mainly consider cases with constant Y . The Higgs field Φ must deform the singularity outside the origin. At least in a neighborhood of w = 0, we demand that the ALE fiber does not develop any singularity. 5 At the level of the effective 7d theory, the vev Φ breaks the gauge algebra to its commutant H. It is important that H is a subspace of the Cartan subalgebra of g. In fact, if the preserved algebra contained a simple factor g , the vev Φ would not deform the ADE singularity completely, but the fiber over generic w would have a singularity of type g , i.e. X would have a non-isolated singularities. 6 For ease of notation, from now on we call Φ the vev of the field corresponding to a given family of ADE deformed singularities. 7 If this does not happen, there will be neutral zero modes of Φ, that, as we will see, will correspond to hypermultiplets that cannot get mass by going to the Coulomb branch; this corresponds to nonresolvable singularities.
This happens when Φ restricted on each summand of L is a non-zero semisimple element of that summand. For L h = A m−1 , the generic form of Φ, up to gauge transformations is in the form of a reconstructible Higgs [9].
withσ j (j = 2, ..., m) the Casimirs of Φ| A m−1 (we called them the partial Casimir of g). There are analogous canonical forms when the summand is a different Lie algebra.
Collecting the Casimirsσ j 's for each summand L h and the coefficient deformations along H one obtains the set of invariant coordinates i that span the base of the family with simultaneous partial resolution. The total Casimirs of Φ (that appear as coefficients of the deforming monomials in the versal deformation of the ADE singularity) can be written as functions of the i 's. Notice that by formulae like (3.5), we give Φ as a function of the partial Casimirs i . The choice of a dependence of i on w produces a threefold. We finally notice that the ADE Lie Algebra g can be decomposed into representations of the Levi subalgebra: where the irreps R L p include the terms in the decomposition (3.3). This will turn out useful in the following.
Let us see how the choice of Φ works in a simple example: a family of deformed A 3 singularity where only the central simple root α 2 is simultaneously resolved at the origin (see Figure 1). We then have H = α * 2 and W = Z 2 × Z 2 . The commutant of H is where A (i) is generated by the triplet e ±α i , h i normalized s.t. the commutation relations are h i = [e α i , e −α i ] and [h i , e ±α i ] = ±2e ±α i . Following the prescription (3.5) for each A 1 summand we have is the only Casimir of the s 2 algebra A (i) 1 . Φ can also have a component along the Cartan α * 2 , with generic coefficient 2 . 1 , 2 , 3 are the partial Casimirs relative to the chosen partial simultaneous resolution, i.e. they are invariant under the Weyl subgroup W . In matrix form, the chosen Higgs field vev is: One can then take i = i (w) in order to obtain a fibration over C w .

The threefold equation from Φ
In order to obtain the explicit equation of the threefold we first derive the expression of the family of ADE surfaces fibered over t/W with coordinates i . We then let these coordinates depend on w ∈ C. Katz and Morrison [1] developed a method for obtaining the hypersurface equation for an ADE family with a given simultaneous resolution, which we review in Appendix A. In the following, instead, we propose a novel method to read off the explicit equation of the simple flops in the ADE classification using the Higgs field Φ. 8 Although the resulting expressions match those of [1], this approach tackles the problem from a different perspective, unifying the physical description via the Higgs field Φ and the algebraic expressions of the deformed ADE singularities.
We first consider the adjoint quotient map, that associates an element g of the ADE algebra g of rank r with r independent polynomials χ i , whose value gives a point in t/W: (3.10) The polynomials χ i (g) are the Casimirs of g and are defined in the following way. By choosing a representation of the Lie algebra g, g can be put in a matrix form. Its Casimirs are then given by specific invariant polynomials of this matrix: For the A and D series, the canonical Casimirs of g are (with g a matrix in the fundamental/vector representation): For the exceptional algebras E r , one takes g in the following representations: 27 for E 6 , 133 for E 7 and 248 for E 8 . One then defines the Casimirs of g as [10,11]: and i = 1, ..., r.
We now need to provide the definition of Slodowy slices. Consider a nilpotent element x ∈ g belonging to some nilpotent orbit O: the Jacobson-Morozov theorem ensures that there exists a standard triple {x, y, h} of elements in g satisfying the su(2) algebra relations. 9 Now, we define the Slodowy slice through the point x as those Lie algebra elements satisfying: For our purposes, we are interested in the so-called subregular nilpotent orbit O subreg , and in the Slodowy slice passing through it. 10 Let us immediately clarify the definitions given so far with a simple example. Consider the A 3 Lie algebra, and a nilpotent element x lying in its subregular nilpotent orbit. The element x and its Slodowy slice are 14) The crucial fact for us is the following: Given an algebra g in the ADE classification, and an element x ∈ O subreg , the intersection of the Slodowy slice through x with the fiber of the adjoint quotient map (3.10) is isomorphic to the versal deformation of the corresponding ADE singularity, namely: where u is a point in t/W C r with coordinates u i . In particular, the intersection of S subreg and the nilpotent cone 11 N = χ −1 (0) is isomorphic to the ADE singularity. The isomorphism (3.15) is telling us that the coefficients of the monomials in the versal deformation can be written in terms of the coordinates u i related to the polynomials χ i . In order to see this explicitly, let us come back to the A 3 example. In the A cases, the Casimir invariants χ i are given in (3.11). Considering the elements in the Slodowy slice (3.14) through the subregular nilpotent orbit, we find: We now compute S subreg ∩ χ −1 (u): 13 that matches with (2.5) (up to an invertible redefinition of the coordinates in t/W). 12 We now want to describe the ALE families of deformed ADE singularities over C w . We have constructed them by the choice of a Higgs field Φ(w), that through its Casimirs is telling us how the deformation is performed on top of each point of C w . In other words, take w ∈ C w ; to see which is the deformed ALE surface over w, we pick Φ(w) and compute its Casimirs χ i (Φ). Their values select a specific point u ∈ t/W and then a specific deformed ALE surface (with precise volumes of the non-holomorphic spheres (2.13)). The equation of the threefold is obtained imposing the relations with χ i (g) defined in (3.11) and (3.12), and substituting the resulting In general, if we keep Φ(w) unspecified, (3.19) gives us the versal deformations of the ADE singularity, with deformation parameters depending on the Casimirs of Φ. We have worked out these relations for all the ADE algebras. For the A r and D r singularities, one obtains (up to coordinate redefinition) the compact and known form A r : and D r : For the E r singularities the expression of the deformation parameters i ,˜ i ,ˆ i (appearing in (2.14)) in terms of the Casimirs (3.12) are given in Appendix B. Hence, given a Higgs field Φ(w), one obtains the threefold equation by simply computing its Casimirs and then inserting them into the relations (3.20) and (3.21) for the A and D cases, or using the formulae in Appendix B for the E cases. 12 The match works for In order to make the procedure explicit, let us apply it to the example of A 3 . Take the Higgs field (3.9) and compute its Casimirs: One can check that the same equation can be obtained using directly (3.20). Later we will be interested into the threefold given by 1 = − 3 = w and 2 = 0, that leads to

Gopakumar-Vafa invariants of X from zero modes of Φ
In the non-compact Calabi-Yau threefolds studied in this paper, we have isolated singularities whose exceptional locus is given by a bunch of P 1 's. These (genus-zero) holomorphic curves are rigid and the BPS M2-branes wrapped on them generate massless hypermultiplets in the 5d theory coming from reduction of M-theory in X. The genus-zero and degree d = (d 1 , ..., d ) Gopakumar-Vafa invariants n g=0 d 1 ,...,d count such states. In particular, we say that n g=0 d = # 5d hypers with charges (d 1 , ..., d ) under the flavor group generated by H .
(3.24) These numbers are reproduced by counting the zero modes of the Higgs field Φ, that are localized at w = 0.
Given a vev for Φ, the zero modes are the deformation ϕ ∈ g of the Higgs field up to the (linearized) gauge transformations (3.25) We take Φ = X + + w Y as explained at page 9, where X + ∈ L is in the principal nilpotent orbit of each L h . We need to work out which components of the deformation ϕ can be set to zero by a gauge transformation (3.25). One then tries to solve the with unknown g ∈ g. As we will see, at special points in C w , there can be components of ϕ that cannot be gauge-fixed to zero: these directions in the Lie algebra g support zero modes.
Since the irreducible representations R L of L are invariant under the action of Φ, we implement the decomposition (3.6) and we solve the equation (3.26) in each representation R L at a time, where now g, ϕ ∈ R L ⊂ g. We can write more explicitly the representation where R L h is an irreducible representation of the simple summand L h and (q 1 , ..., q ) are the charges under the U (1) group generated by H. If there are n 5d modes in the representation R L , there will be other n 5d modes in the conjugate representationR L ; together these generate n massless hypermultiplets in the 5d theory localized at the singularity. By using the correspondence (3.24), we can then say that the GV invariant with degrees (q 1 , ..., q ) is Let us describe our algorithm to compute the number and the charges of zero modes. For each representation R L of L with dimension d R , we choose a basis e 1 , ..., e d R of R L . In this basis, the equation (3.26) becomes where ρ and φ are the d R -column vectors of coefficients of g and ϕ in the given basis and A, B are the constant d R × d R matrices representing the linear operators X + and Y respectively. If A + w B is invertible, then there exists a vector ρ (i.e. a g ∈ R L ) that completely gauge fixes ϕ ∈ R L to zero at generic w. At the values of w where the rank of A + w B decreases, there will be vectors φ that cannot be set to zero, leaving a zero mode localized at that points.
With the chosen X + , we immediately see that such a special point is (by construc-tion) the origin w = 0. Here the matrix A+w B reduces to the nilpotent matrix A, that has non-trivial kernel. 13 In the following we only use the fact that A has rank r < d R ; hence, our conclusions are valid also when A is not necessarily nilpotent. What we are going to say of course applies also for a nilpotent A. We choose the basis e 1 , ..., e d R of R L such that A is in the Jordan form. If the rank of A is r, we then have d R − r rows of zeros and d R − r columns of zeros. We can rearrange rows and columns such that A takes the block diagonal form Doing the same operations on B, we obtain The equations (3.29) now read is invertible (at least in the vicinity of w = 0), from the first set of equations we see that we can always gauge fix the φ u components to zero, 14 by setting Substituting in the second set of equations we obtain We see that the components φ d cannot be fixed identically to zero: at w = 0 there can be a remnant, i.e. a localized mode. Said differently, the best we can do is to cancel from φ d its dependence on w, leaving a constant entry (instead of a generic polynomial in w). This is possible for all components of φ d only when the matrix B d has maximal rank, i.e. rank equal to d R − r. In this case, the number of zero modes is We can then rewrite the equation (3.34) as This has the same form as (3.29), so we can again change the basis such that A ≡ B d is in the Jordan form and write the equations in this basis. We will obtain a set of equations in the form (3.32) where we have to substitute The matrix A will now have rank r < d R − r. There will then be r components of φ that can be gauge fixed to zero; we correspondingly have r zero modes along the corresponding components of φ d . If the matrix B d has maximal rank (i.e. d R − r − r ), then the other d R − r − r components of φ d will be of the form a + bw and hence each hosts two zero modes. In this case the number of zero modes is On the other hand, if B d has rank r < d R − r − r , then we have to iterate once more the algorithm above and, provided B d has maximal rank (i.e. d R − r − r − r ) we obtain We now have the factor "3" because the d R − r − r − r directions of φ d are of the form a + bw + cw 2 , i.e. they host three zero modes each.
In conclusion, let us assume that the algorithm stops at the N -th step and let us call r (k) the rank of the matrix A at the step k, then the number of zero modes is where r (0) = r.
If there are other values of w, say w = w 0 , where the rank of A+wB is not maximal, one can shift w → w + w 0 ending out with the same situation as above, where the new A is now A + w 0 B. Applying the algorithm that we have just outlined, one computes the zero modes localized at w = w 0 . In this case the matrix at w = w 0 is not necessarily nilpotent.
Notice that this algorithm could never end. This is the case for example when the A + wB matrix is identically zero at one step. The corresponding directions of ϕ cannot be gauge fixed at any order in w, leaving a zero mode that lives in 7d.
In conclusion, in this section we have shown that one can reduce the problem of finding the zero modes to a simple exercise in linear algebra. These computations are algorithmic and can be done by a calculator in a reasonable amount of time. In Appendix D we describe the implementation of the algorithm in Mathematica, that we used for our computations.
4 Simple flops with l = 1, ..., 6 and their GV invariants One-parameter deformations of ADE surfaces admitting a small simultaneous resolution blowing up a single P 1 are known as simple threefold flops. From a mathematical point of view, they can be classified according to a variety of invariants.
A rich classification of simple flops can be obtained employing the so-called length invariant, first introduced in [12], that we defined in Section 1. It was proven (see [1]) that the length of a simple flop can only assume discrete values ranging from 1 to 6, and that examples of any length indeed exist.
From a Lie-algebraic point of view, the length of a simple flop corresponds to the dual Coxeter label of the node of the Dynkin diagram that is being resolved by the small simultaneous resolution. Given an ADE algebra g of rank r, a set of simple roots α i , with i = 1, . . . r, and the highest root θ, the dual Coxeter label of a node is the multiplicity of the corresponding simple root in the decomposition of the highest root. In other words, given a node corresponding to a simple root α i 0 and the decomposition of the highest root then c i 0 is the dual Coxeter label of the node. As we will use this fact extensively in the explicit constructions of simple threefold flops of length up to 6, it is useful to report the Dynkin diagrams of all the ADE cases, along with the dual Coxeter labels of their  Figure 2, where we have highlighted in black the nodes that are being resolved in the simple threefold flops that we will analyze in the following sections. The classification of simple threefold flops based on the length can be further refined introducing the Gopakumar-Vafa (GV) invariants [13][14][15]. These invariants can be used to distinguish between simple flops of the same length. 15 In the following, we apply the method discussed in Section 3 to construct threefolds with a simple flop. The threefold will be obtained from a family of deformed ADE singularities in which only the black node in Figure 2 is simultaneously resolved. Let us call it α c . The subalgebra H is then generated by α * c , i.e.
and the Higgs field will correspondingly be chosen in the commutant L of H, i.e. the Levi subalgebra corresponding to the chosen partial simultaneous resolution. From Figure 2, we see that the simple summands L h of L are of A-type. The Higgs Φ restricted on these spaces is then of the form (3.5), and collecting theσ's from each summand L h gives the partial Casimirs i 's that parametrize the base t/W . The threefold is obtained by We will construct threefold with different values of length from 1 to 6. For each manifold we give the Higgs field Φ that produces the desired simple flop threefold X. This allows us to build the 5d theory realized from reducing M-theory on X.
In particular the flavor group will always be the U (1) group generated by α * c . The number of hypermultiplets and their charges under the U (1) flavor group, namely the GV invariants of X and their degrees, will be derived by counting the zero modes of Φ.

Simple flop with length 1
The simplest Example: the Conifold The Conifold threefold is given by This is actually a family of deformed A 1 surfaces over C w , with the simultaneous resolution of the exceptional P 1 at w = 0. It can be constructed following the previous sections in the following way: A 1 has only one simple root α. We require it to be blown up by the simultaneous resolution (the only other choice is to blow up no sphere, that would produce a non-singular threefold). The Levi subalgebra is now simply The partial Casimir is the coefficient along the Cartan α * . Choosing = w, the Higgs field is and the threefold equation is easily checked to be (4.3). This is the simplest example of simple flop, where the flavor group (i.e. the preserved 7d gauge group) is U (1).
Zero modes. Notoriously, M-theory on the conifold gives a free 5d hypermultiplet (localized at w = 0). This can be checked by computing the zero modes of Φ. This computation has already been shown in [3], by explicitly using the linearized equations of motion in holomorphic gauge, 16 as explained in [9]. In order to illustrate the method outlined in Section 3.3, we apply it to the conifold case to reproduce the result of [3].
The decomposition (3.6) of the A 1 algebra in representations of the Levi subalgebra Let us consider each representation individually. Remember that the matrices A and B in Section 3.3 are the restriction of X + and Y on the considered representation, where 1 0 : Φ restricted to this representation is zero. Hence, the two 'matrices' A and B vanish, nothing is gauge fixed and then there is one 7d mode.
1 + : Φ| 1 + = 2w, so A = 0 and it has rank zero, but B = 2 has rank one; then d R − r = 1 − 0 = 1 and this gives one localized mode at w = 0. Threefolds with a simple flop of length 1: generic case We now generalize the conifold case, by starting from the Lie algebra A k−1 . The simple roots are now α 1 , ..., α k−1 . We require that the only root that is simultaneously resolved in the threefold is α p for a given choice of p ∈ {1, ..., k − 1} (without loss of generality, we can take p ≥ k 2 ). Consequently, we have H = α * p . Its commutant is The Higgs field at w = 0 is (in the principal nilpotent orbit when restricted on the simple summands of L) X + = e α 1 + · · · + e α p−1 + e α p+1 + · · · + e α k−1 . (4.8) We choose the w-dependence of the partial Casimirs such that the Higgs restricted on each block is 17 (4.10) The equation of the threefolds is read form (3.20), by using the chosen Φ = X + + w Y : When k = 2n is even and p = n, we have the Reid Pagoda of degree n (whose Dynkin diagram for the simultaneous resolution is depicted in Figure 2).
Zero modes. Let us perform the zero mode computation in the case p = n = 2, i.e. for the Reid Pagoda with degree 2. The Higgs field is actually given by the A 3 example studied before (see (3.9)), where one chooses the following dependence of i on w: The A 3 algebra decomposes in the following way in representations of the Levi subalgebra L = s 2 ⊕ α * 2 : Let us consider each Levi representation R L individually.
(3, 1) 0 : the operator X + is represented in the basis {−e α 1 , 1 2 h 1 , 1 2 e −α 1 } by the matrix that has rank r = 2. In the same basis Y is represented by (4.14) We plug them into the expression (3.29) and apply the algorithm: arranging the rows and columns to arrive to the expression (3.30) is equivalent to taking We can then read In particular B d − wB l (A u + w B u ) −1 B r vanishes identically. This means that at the second step A + wB = 0 and the corresponding zero mode left by the rank 2 matrix A is not localized at any w. We have found a 7d zero mode.
Hence, the number of localized modes at w = 0 is 4, that gives rise to two hypermultiplets with charge 1 with respect to the U (1) flavor group. This matches the results of [2,3], that were obtained with a different procedure.
For generic k and p, the zero mode counting proceeds analogously as for the Pagoda with n = 2. The A k−1 algebra decomposes in the following way in representations of the Levi subalgebra The first three representations host 7d modes, but no localized one. Let us concentrate on the charged representation (p, k − p) + of dimension p(k−p). With the choice p ≥ k 2 , we have p ≥ k − p. The matrix representing X + in this representation has kernel with dimension equal to k − p, then in our algorithm r = (p − 1)(k − p). With a bit of work, one can check that B d has rank k − p = d R − r, that gives then k − p modes localized at w = 0 with charge +1 with respect to the flavor U (1). The other charge representation hosts again k − p modes localized at w = 0 and with charge −1. In total we then have k − p charged hypermultiplets, i.e. the GV invariant is (4.18) When k = 2n and p = n, we obtain n hypers, that is the result we have found for the Reid's Pagodas in [2,3], i.e. n g=0 1 = n.

Simple flop with length 2
In this section we consider a family of flops of length 2 arising from a D 4 singularity deformed over the C w plane. The threefold is singular at the origin (where the fiber exhibits a D 4 singularity) and can only be partially resolved inflating a P 1 corresponding to the central root of the D 4 Dynkin diagram. As we can see from Figure 2 the central node has dual Coxeter label equal to 2, and thus its resolution yields a flop of length 2. In Figure 3 we show our conventions for the labeling of the simple roots. Since we wish to blow up only the central node, we have H = α * 2 . The Levi subalgebra L commuting with H is: where the A 1 algebras correspond to the white "tails" in picture 3, generated by the roots α 1 , α 3 and α 4 respectively.
Following the prescription (3.5) for each A 1 summand we have where i (i = 1, 3, 4) is the Casimir of the s 2 algebra A (i) 1 . Moreover Φ can have a component along α * 2 with coefficient 2 . Although not necessary for the employment of our machinery, we report for the sake of visual clarity the explicit matrix form of the adjoint Higgs field corresponding to the choice (4.19), employing the standard basis of [17]: The threefold is found by imposing where we take the c i (w)'s such that c i (0) = 0. Later we will simply choose the c i (w)'s to be constant in w.
The Higgs at the origin is then The threefold equation is simply obtained by taking the choice (4.22) and the expression of Φ (4.21) and plugging them into the formula (3.21): 18 Let's consider what happens when one of the c i 's vanishes. If c 2 = 0, the preserved gauge group after Higgsing is SU (2) instead of U (1). This says that the ALE fiber has an A 1 singularity for all values of w, i.e. the threefold has a non-isolated singularity. If c i = 0 with i = 1, 3, 4, then the preserved group is still U (1). However, the threefold equation has an A 1 singularity for generic w ∈ C w : in fact, the threefold equation is the same one would obtain by taking Φ| A (i) the "1" in (4.20)). Such a nilpotent vev for the Higgs field is called a T-brane [9].
Since we want to consider isolated singularities (with a simple flop), avoiding Tbrane configurations, we will take c i = 0 ∀i.
Zero modes. We now analyze the 5d zero modes arising from M-theory reduced on the flop of length 2 defined by (4.22). We keep the c i 's as generic constants.
As in the case of the flops of length 1, the first step consists in determining the decomposition of the algebra g = D 4 into irreps of the Levi subalgebra (4.19), obtaining: (1, 1, 1) 2 : X + is represented by a 1-dimensional matrix that, in the basis e α 1 +2α 2 +α 3 +α 4 , reads We also have: As a result we find that B has maximal rank, i.e. 1, and so we obtain one localized 5d zero-mode with U (1) charge 2. Analogously, the representation (1, 1, 1) −2 yields one 5d zero-mode of U (1) charge −2.
Rearranging rows and columns to get to the form (3.30) we obtain: Notice that the rank of B d , which is surely non-maximal, depends on the precise choice of the partial Casimirs. It drops to one when its determinant is equal to zero. This happens when that has rank 1. As a result, according to (3.36), we find that the total number of zero modes is: where we recall that d R is the dimension of the representation, r is the rank of (4.29) and r is the rank of A . The zero-modes have charge +1 with respect to the U (1) generator. Analogously, we find 4 localized zero-modes with charge −1 in the (2, 2, 2) −1 representation.
• When the c i 's fulfill (4.34), the rank of B d drops to 1. This produces a change in the zero-mode counting. We can parametrize a solution of (4.34) in terms of two parameters q 1 , q 4 as: where ε can take the values ±1. Now we have When q 2 1 +εq 1 q 4 +q 2 4 = 0, the 2×2 matrix is diagonalizable with the non-zero eigenvalue equal to 4 3 (q 2 1 + εq 1 q 4 + q 2 4 ). The corresponding B d is This matrix has rank less than two only when one of the c i 's vanishes (and consequently the other two are equal to each other), that we excluded.
When q 2 1 + εq 1 q 4 + q 2 4 = 0 (i.e. all the eigenvalues vanish) the 2 × 2 matrix has still rank 1 and the corresponding B d is also forced to have rank 2 (for non-vanishing c i 's).
We can finally count the localized zero-modes using formula (3.36), finding: (4.40) Notice that, with respect to the case (4.36) in which the Casimirs were totally generic, we have found an enhancement in the number of modes on a specific locus in the space of the partial Casimirs. This is the same locus where one conifold singularity that was at w = 0 collides onto the origin.
Let us summarize our findings for the modes localized at w = 0 for the simple flop of length 2 and partial Casimirs given by i (w) = w c i , with c i constants.   Non-constant c i 's. For simplicity, we have analyzed cases when the partial Casimirs i are just a constant c i multiplied by w. Of course, one can also let c i depend on w and rerun the algorithm.
One can in particular find the dependence of the c i (w)'s such that the threefold X has only one isolated singularity at the origin. An easy solution is when  [16] in the form that appeared in [3] (that has the expected GV invariants).

Simple flop with length 3
In this section we engineer a threefold X with a simple flop of length three. Analogously to the previous sections, we are going to define a suitable Higgs field, valued in the E 6 Lie algebra, that generates a family of deformed E 6 surfaces with an E 6 singularity at w = 0. The resolution of the isolated singularity in the threefold X will blow-up only the trivalent node of the E 6 Dynkin diagram (see Figure 2). To achieve this result, we pick the following Levi subalgebra where the factors A (i,j) 2 are associated, as subalgebras, to the roots α i , α j of the E 6 Dynkin diagram (we follow the labels in Figure 4) and A (6) 1 is the algebra associated to the root α 6 . Again, we pick X + ≡ Φ| w=0 to be an element of the principal nilpotent orbit of each simple factor of L. The partial Casimirs relative to L are the total Casimirs of each simple factor of (4.44), plus the coefficient along the Cartan element α * 3 . I.e. the generic Φ will be such that 20 2 e −α 6 and Φ| α * 3 = We now explicitly construct a threefold, by making the choice with c 3 , c 6 , c 12 , c 45 constant numbers. By plugging this choice into the Higgs field vev Φ, and following the procedure described in Section 3.1, one obtains the threefold as an hypersurface of (x, y, z, w) ∈ C 4 .
As an example, if we pick c 3 = 0, c 6 = −3, c 12 = 1, c 45 = −1, one gets the following threefold, which is singular at the origin (as well as at other three points with non-zero w): Via a change of coordinates, this exactly coincides with the length 3 threefold explicitly presented by [5].
Zero modes. We now proceed (with the same procedure of the previous sections) to the mode counting. The branching of the adjoint representation 78 of E 6 w.r.t L in 20 To match the conventions of Section 2, one takes where the subscripts denote the charges under α * 3 . For the E-cases the explicit computations done for length one and two become convoluted. We present here only the results. We have worked out a Mathematica routine, presented in Appendix D, that implements the algorithm described in Section 3.3 and that can be used to check the results. Running this code for a generic choice of the parameters c 6 , c 3 , c 12 , c 45 , we obtained, for each irreducible representation appearing in (4.48), the 5d modes shown in Table 1. In the table, we also write how many elements of the given representation support a mode localized in C[w]/(w k ), for each k; we find that k ≤ 2. We get a total of 20 5d modes:  • one hyper with charge three, inside (1, 1, 2) 3 ⊕ (1, 1, 2) −3 ; • three hypers with charge two inside (3, 3, 1) 2 ⊕ (3, 3, 1) −2 ; 21 It can be better understood starting from the one of the maximal subalgebra A

Simple flop with length 4
In the following section we are going to engineer, by means of a Higgs field Φ valued in the E 7 Lie algebra, a flop of length four. By looking at the dual Coxeter labels of the E 7 Dynkin diagram in Figure 2, we see that the simultaneous resolution should involve the trivalent node. Analogously to the previous examples, this means that we have to pick the Higgs field in the Levi subalgebra where the superscripts refer to the roots of the E 7 Dynkin diagram numbered as in the Figure 5, and α 3 is the trivalent root of E 7 . Analogously to the E 6 case, we choose the Higgs field as follows: The corresponding threefold is a hypersurface in C 4 , that is a family of deformed E 7 singularities over C w . To make the equation of the threefold more readable, we set the parameters to specific values, picking c 3 = 0, c 7 = 3, c 12 = 1 2 , c 456 = − 1 2 , obtaining where we neglected terms of high degree, irrelevant for the singularity at w = 0.
Zero modes. We now proceed with the modes counting. We will again perform the gauge-fixing separately in each irreducible representation of the branching of the adjoint representation 133 of E 7 under the subalgebra L: 22 (4.53) Running the Mathematica routine described in Appendix D, we find the results displayed in table 2. As in the E 6 case, there are no five-dimensional modes localized 22 The first entry of each summand is a representation of A (4,5,6) 3 , the second one is a representation of A (1,2) 2 , and the third on a representation of A  • one hyper with charge four, inside (1, 3, 1) −4 ⊕ (1, 3, 1) 4 ; • two hypers with charge three inside (4, 1, 2) −3 ⊕ (4, 1, 2) 3 ; • five hypers with charge two inside (6, 3, 1) −2 ⊕ (6, 3, 1) 2 .
In terms of the GV invariants, one then reads Finally, we find (as in the E 6 case) that no particular choice of the constants c i can enhance the number of zero modes at w = 0 (without generating non-isolated singularities).

Simple flop with length 5
A flop with length 5 is obtained from an E 8 family over C w . The node that should be simultaneously resolved at w = 0 is depicted in Figure 6.
In terms of the GV invariants, one then reads Again, we notice that we can not enhance the number of zero-modes at w = 0 without generating a non-isolated singularity.

Simple flop with length 6
In this section we conclude our analysis of simple flops by dealing with the highest length case, i.e. a flop of length 6 arising from a E 8 singularity deformed over the plane C w . We choose the Higgs Φ ∈ E 8 in such a way to resolve only the central node of the E 8 Dynkin diagram as depicted in Figure 7. According to the principles outlined in previous sections, the Higgs field resolving the central node must lie in the Levi subalgebra defined by: where as usual the upper indices label the simple roots. Again we choose Φ of the 41 following form: To make the equation more readily understandable, we set the parameters to a specific value c 3 = 0, c 8 = 1, c 12 = −1, c 4567 = 1. In this way we obtain the threefold 61) where we neglected terms of high degree, irrelevant for the singularity at w = 0.
In terms of the GV invariants, one then reads

43
Finally, analyzing the rank of the matrix B d , we find that no enhancement in the number of localized modes at w = 0 is feasible without generating a non-isolated singularity.

Conclusions
In the present work, we constructed and studied the GV invariants of examples of threefold simple flops of any length. Building on our previous results based on the Mtheory/Type IIA duality [2,3], we introduced techniques to deal with intrinsically nonperturbative cases, where the Type IIA limit is not available, involving one-parameter families of deformed E 6 , E 7 , E 8 singularities. By means of a group-theoretic computational algorithm (included in an ancillary Mathematica file), the determination of the GV invariants for flops of any length has been reduced to a linear algebra problem. Our method is explicit, and also enables us to detect special cases of threefold simple flops exhibiting enhanced GV invariants.
Furthermore, from a physical perspective, our work allows to study the dynamics of M-theory on threefold simple flops, which are notably non-toric, characterizing the Higgs Branches of the resulting 5d SCFTs as complex algebraic varieties. This is possible thanks to the correspondence between the GV invariants of threefold simple flops and BPS states arising from M2-branes wrapped on the curve inflated by the resolution of the simple flops. Indeed, M2-brane states descend to hypermultiplets in 5d, whose charges under the flavor group are determined by the degrees of the GV invariants, which we computed explicitly in flops of all lengths. In particular, we are able to fully characterize the action of the flavor symmetries on the Higgs Branch.
These results pave the way for a range of engaging perspectives. For example, it would be interesting to apply our methods to the so-called (A j , E k ) (with k = 6, 7, 8) singularities, which require an intrinsically non-perturbative description. Furthermore, achieving a more thorough understanding of the construction of general Higgs backgrounds would prove hugely profitable for the study of more general one-parameter fibrations of exceptional singularities. One can start from the family over t (2.14), that is the expression of the versal deformation of the ADE singularity in which the deformation parameters are functions of the t i 's that encode the volumes of the 2-cycles. To write down the equation of the fibration over t/W , one first needs to write the W invariant coordinates i 's in terms of the t i 's; inverting these relations and plugging the result in (2.14) one obtains the hypersurface equation of the partial simultaneous resolution [1].
Let us immediately clarify this statement with the example of the deformed A 3 singularity we have introduced before. According to (2.14) the defining equation reads: Now suppose again that we wish to produce a simple flop in which only the central node is resolved. The original Weyl group W = S 4 , acts as the permutation group on the four t i 's. Here, the desired simultaneous resolution defines for us the subgroup W = Z 2 × Z 2 , which acts by exchange of t 1 ↔ t 2 and t 3 ↔ t 4 , respectively. It is easy to see that the W -invariant coordinates are: and comparing them to the expression (A.1) in terms of the t i it is a matter of mechanical computation 24 to extract the fibration over t/W : x 2 + y 2 + (z 2 −s 2 z +s 1 )(z 2 +s 2 z +s 3 ) = 0. which gives the hypersurface x 2 + y 2 + z 4 − w 2 . In principle, this procedure can be carried out for any deformed ADE singularity and any resolution pattern, although for the exceptional cases it can get particularly time-consuming.

B Casimirs for the equation of E r families
Let us consider the deformed E n singularities in equation (2.14). In each one, we have n deformation parameters: exceptional algebras 25 and a basis of the Cartan subalgebras. We used, to produce these matrices, the results of [21][22][23]. These text files should be saved in one of the folders of the variable $Paths of Mathematica. In the folder, one can find also the notebook file "CodeHiggsBranchData.nb" which is divided into two sections. The first section "Main Code" contains the routines that we used to produce the Higgs branch data having, as input, the Higgs field describing the type IIA dual of M-theory on the considered threefold. The second sections "Examples" contains the application of the functions defined in the section "Main Code" to the simple flops we analyzed in this paper. We now describe the main routines of the ancillary Mathematica code.
The arguments of the function are • ADE: is a Symbol to be picked among "A, DD, E6, E7, E8" and specifies the type of ADE algebra associated to the threefold.
• rank: is a positive Integer that specifies the rank of the ADE algebra associated to the threefold.
• listhiggs: is a List of Lists. Each sublist represents a root such that the Higgs field has a non-zero coefficient along the corresponding root-vector in g. The root vectors considered here do not lie in the Cartan subalgebra: the elements in the Cartan subalgebra will be separately input with the variables cartanhiggs and coeffcartan. To order the roots, we followed the labelling of Figures 4, 5, 6 and 7 for the exceptional algebras. We labelled roots from the left to the right in the A r Dynkin diagram. For the D r case, the first r − 3 integers are associated to the A r−3 subalgebra associated to the longest tail of the D r Dynkin diagram (again, labelling roots from left to right), the third last integer number appearing in the sublists of listhiggs is associated to the trivalent root, the last two integers are associated to the two A 1 short tails of the D r diagram.
For example, if we again consider the Higgs field of section 4.5, we have to insert, as input in (4.56).
The root system, in our convention, can be printed on screen calling the function PrintRootSystem[ADE,rank] (the first argument being again the ADE type of g, and the second argument its rank).
• cartanhiggs: is a List of positive Integers n i , with n i = 1, ..., rank, describing the elements of the Cartan subalgebra C of g along which the Higgs field has a non-zero coefficient. The generators of the Cartan subalgebra has been chosen as the dual elements α * j of the simple roots. For example, let's consider again the E 8 algebra. We know by construction that the Higgs field has to preserve the Cartan element dual to α 4 in Figure 6. This Cartan element is defined as the one that commutes with all the simple root vectors different from e α 4 , and [α * 4 , e α 4 ] = 1.
In order to pick a Higgs field with a non-zero component along α 4 we then input cartanhiggs = {4} .