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ADE Singularities and Coset Models

We consider the compactification of the IIA string to (1+1) dimensions on non-compact 4-folds that are ALE fibrations. Supersymmetry requires that the compactification include 4-form fluxes, and a particular class of these models has been argued by Gukov, Vafa and Witten to give rise to a set of perturbed superconformal coset models that also have a Landau-Ginzburg description. We examine all these ADE models in detail, including the exceptional cosets. We identify which perturbations are induced by the deformation of the singularity, and compute the Landau-Ginzburg potentials exactly. We also show how the the Landau-Ginzburg fields and their superpotentials arise from the geometric data of the singularity, and we find that this is most naturally described in terms of non-compact, holomorphic 4-cycles.


Introduction
The link between the classification of singularity types and quantum effective actions of field theories now has a fairly long and interesting history. The key to making this work is for the field theory to have enough supersymmetry so that a non-renormalization "theorem" protects the sector of the field theory that is determined by the classical singularity type. For theories with a mass gap there are generically BPS states whose spectrum can be computed semi-classically, and which can be used to identify the field theory. This was first used with considerable success in N = 2 superconformal field theories, in which there is a Landau-Ginzburg superpotential [1]. Most particularly the ADE classification of complex singularities with no moduli was shown to correspond to precisely the modular invariant N = 2 superconformal field theories with central charge c < 3. If one topologically twists these N = 2 theories the result is topological matter [2], and the coupling to topological gravity could be naturally incorporated into the singularity theory [3,4].
The advent of string dualities gave rise to many new constructions of string and field theories, and in particular, low energy effective actions. An early offshoot of this general program was to realize that the complete Seiberg-Witten quantum effective action of N = 2 super-Yang-Mills theories in four dimensions [5,6] could be obtained from the period integrals of K3-fibrations that were developing an ADE singularity [7]. In this construction, the singularity type corresponds to the gauge group of the Yang-Mills theory.
This subject has now, in a sense, come full circle. In [8] it was shown how Landau-Ginzburg models in (1 + 1)-dimensions could be constructed using compactifications of type IIA strings on Calabi-Yau 4-folds. If the 4-fold in this construction is a K3-fibration, and if the fiber develops an ADE singularity, then the (1 + 1)-dimensional field theory has been argued to be a massive perturbation of a coset model based upon the corresponding ADE group. The mass scale is set by the deformation of the singularity, and a conformal field theory emerges when the fiber is singular.
Extracting field theories from singular limits of compactifications is often a subtle process, and there are sometimes multiple limits being taken at the same time. For example, to extract the Seiberg-Witten effective actions one needs to decouple gravity and yet maintain a cut-off whose dimensions are inherited from the string tension. This is done by scaling the size of the base of the fibration while scaling the K3 so as to isolate the singular fiber. This results in a non-trivial role for the fibration even in the field theory limit. A conformal field theory can occur in this context if the moduli of the singularity are fine-tuned to an Argyres-Douglas point, but the generic theory has a scale. Things are considerably simpler for the Landau-Ginzburg models that arise from singular K3 fibers in 4-folds: there is no cut-off, and the field theory is conformal when the K3 fiber is singular.
All the essential physics comes from the isolated ADE singularity in the fiber, and the base of the fibration only plays a trivial role.
While the singularity type determines the numerator of the coset conformal field the- have a mass gap, and the associated coset model was argued in [8] to be (a perturbation of) the N = 2, Kazama-Suzuki coset model based upon the level one, A, D or E hermitian symmetric space. This was checked in detail in [8] for the A-type models, and here we will verify that the construction works for the D-type and the E-type singularities. It is not immediately clear what the corresponding models are for larger (non-minimal) fluxes, but it is tempting to try to identify the magnitude of the flux with the level of the coset model.
The identification of these models, even at level one, is a non-trivial problem. If one uses a non-compact Calabi-Yau 4-fold with an ADE singularity as outlined above, it turns out that all the dynamics are frozen because the kinetic terms of the the model are not normalizable. On the other hand, in [8] a Landau-Ginzburg superpotential, W , was conjectured for compact Calabi-Yau 4-folds, and for non-compact Calabi-Yau manifolds this yields expressions for the topological charges of solitons in terms of period integrals on the 4-fold. The problem is that for non-compact Calabi-Yau manifolds, there does not appear to be an obvious geometric characterization of the Landau-Ginzburg fields themselves: one does not, a priori, know how many fields there are, let alone their U (1) charges. Thus the singularity only yields topological data about the model, that is, one knows only the ground states and the topological charges of the solitons. On the other hand, knowing the coset conformal field theory determines the chiral primary fields and the 1 A minimal flux is one that corresponds to a miniscule weight of G.
Landau-Ginzburg potential (if there is one). Such a theory also generically has many more deformations of the superpotential than there are deformations of the singularity of the 4-fold. Thus there is a special, "canonically" deformed Landau-Ginzburg superpotential whose deformations are precisely generated from the deformations of the 4-fold.
One of our purposes here is to characterize and compute all the canonically deformed superpotentials associated with the ADE singularities with minimal fluxes. We first do this rather abstractly by developing an algorithm for computing such superpotentials. In so doing, we find that there are obvious choices in the procedure, and that these choices correspond to determining the coset denominator, and hence the correct set of Landau-Ginzburg variables. We then look at the period integrals on the 4-fold, and find how to determine the superpotentials from such calculations. A direct computation appears to lead to only one superpotential for each A, D, or E singularity. However, we also show that there are, once again, ambiguities in the calculation of the periods, and that different choices lead to the full set of canonically deformed superpotentials. We will show that the resolution of the ambiguity amounts to selecting holomorphic representatives of the non-compact homology of the singularity. These representatives are characterized by weights of G, and each choice is equivalent to selecting a Weyl orbit of fluxes. In doing the calculations of the period integrals, we also find the vestigial remnants of the Landau-Ginzburg variables, and this leads to some natural conjectures as to the role of the Landau-Ginzburg variables in the geometric picture on 4-folds.
The next two sections of this paper contain a detailed review of the N = 2 superconformal coset models of Kazama and Suzuki, and the construction of associated Landau-Ginzburg superpotentials. In particular, in section 2 we will review that standard construction of the chiral rings and Landau-Ginzburg potentials of the simply laced, level one, Hermitian symmetric space (SLOHSS) models. We then generalize this to obtain the "canonically" deformed superpotentials. In section 3 we will construct some specific examples of these deformed superpotentials, and evolve an algorithm for generating all the superpotentials associated with the numerator Lie algebra, G, from any one such superpotential. In section 4 we first review the part of [8] that is relevant to the SLOHSS models, and then use the formula of [8] to calculate topological charges of solitons in terms of period integrals of the singularity. We then use this calculation, in combination with the algorithm developed in section 3, to reconstruct the Landau-Ginzburg superpotentials from the geometry of the singularity. Our methods of computing the period integrals can be more naturally cast in terms excising holomorphic surfaces, and computing intersection numbers. We also discuss this in section 4, and show how the selection of the holomorphic surfaces is equivalent to choosing a flux at infinity. We also find that the Landau-Ginzburg variables appear very naturally in the parameterization of these surfaces. Finally, in section 5 we make some remarks about fermion numbers of solitons, and about generalizations of the results presented here.
They are based upon the coset construction using: where the SO(dim(G/H)) is a level one current algebra, and represents the bosonized fermions in a supersymmetrization of the coset model based upon G/H. In (2.1), H is embedded diagonally into G and SO(dim(G/H)), the embedding into G is an index one embedding, and the embedding into SO(dim(G/H)) is a conformal embedding [10] of level g −h where g and h are the dual Coxeter numbers of G and H respectively. If G has a level k, then H has a level k + g − h. If G/H is Kähler, then H = H 0 × U (1), with U (1) inducing the complex structure, and the corresponding coset model has an N = 2 superconformal algebra.
The simply laced, level one, hermitian symmetric space (SLOHSS) models have been studied even more extensively. In these models, G is represented by a level one current algebra, and G/H is a hermitian symmetric space. These cosets are: .
It was shown in [11] that the chiral ring of these coset models is isomorphic to the de Rham cohomology ring H * (S, IR). It has also been argued that these models have a Landau-Ginzburg formulation [11]. Most of the Landau-Ginzburg potentials have been determined.
The most direct way to compute the chiral ring and potential is to use the isomorphism with the cohomology ring of S. The latter can be generated by the Chern classes of H-bundles on S, and these can be generated from the irreducible H-representations. One should also observe that elements of the coset W (G)/W (H), and hence the ground states, are in one-to-one correspondence with the weights of a "miniscule" representation of G. A miniscule representation, R, is defined as one in which all the weights lie in a single Weyl orbit. The subgroup H is then defined to be the one whose semi-simple part (not the U (1)) fixes the highest weight of R. Since all other weights lie in the Weyl orbit of the highest weight, it follows that the representation is indeed in one-to-one correspondence with W (G)/W (H). In this manner one can generate the list (2.2). We thus see that ground states of the SLOHSS models are naturally labeled by the weights of a miniscule representation of G. This will be important throughout this paper.
The "reduction of Casimirs" procedure was used in [11,12] to construct the chiral rings. The remarkable, and rather unexpected fact is that the vanishing relations appear to be integrable to produce a single superpotential for each SLOHSS model 2 . In particular, the Landau-Ginzburg potentials for E 6 and E 7 were computed in [12].
The basic procedure is therefore as follows: Let m i and m i , i = 1, . . . , r, be the exponents of G and H respectively. To get the Landau-Ginzburg potential one wants to integrate the vanishing relations to obtain a potential W (x i ) such that the set of equations V i = 0 are equivalent to the set of equations ∂W ∂x i = 0. The only complexity in this task is that ∂W ∂x j will generically be some constant multiple of V j plus V k 's of lower degree multiplied by polynomials in the x i 's.
One has thus to find the proper combinations of V k 's before integration is possible, and as we indicated earlier, it seems remarkable that such integration is possible.

The canonically deformed Landau-Ginzburg potential
The conformal Landau-Ginzburg potential is, of course, quasi-homogeneous, and therefore multi-critical. The index of the singularity, µ = W (G) W (H) , is the degeneracy of the Ramond ground states. We now wish to deform this potential in such a manner as to make a mass gap, and yield the corresponding theories associated with the ADE singularities in [8]. The key to seeing how this must happen is to observe that the versal deformation of the singularity preserves the W (G) symmetry of the singularity, and therefore so must the corresponding deformation of the coset model. In particular, the ground states of the deformed model must be a W (G)-invariant family. This leads to a unique versal deformation procedure for the coset model. Instead of setting the G-Casimirs, V i (x j ), to zero one can set them to constant values: V i = v i for some v i . This is manifestly W (G) invariant, and represents a set of equations that define the ground states of the canonically deformed model.
To understand the interrelationship between all the Landau-Ginzburg potentials for cosets G/H with the same numerator G, but with different denominators H, it is important to understand how these ground states, and hence the canonical deformation, is realized in terms of the CSA variables, ξ j . In terms of the Cartan subalgebra, X , setting V i = v i uniquely defines a general point, ξ j = ξ (0) j , in X up to the action of W (G). That is, the v j define a general point, ξ Remarkably enough, we find that the deformed "vanishing relations:" V i = v i are still integrable to a Landau-Ginzburg potential W (x j ; v i ). This will be proven in the next section. Based upon the foregoing observation we develop an algorithm for computing the desired Landau-Ginzburg potential for any SLOHSS model, G/H, given one such superpotential for any SLOHSS model with the same numerator, G. We then compute a superpotential for each choice of G, including the exceptional cosets, and we also give several examples of the application of the algorithm.
Finally, we note that the deformed potential is quasi-homogeneous: where N = m r is the dual Coxeter number (i.e. the degree of the highest Casimir) of G.
Once again, we stress that because we have preserved the Weyl symmetry of G, it is these deformed potentials that must be related to those of the versal deformations of the ADE singularities.

Grassmannians
Here one has G = SU (m + n) and H = SU (m) × SU (n) × U (1), but it is simpler to think of this coset as having G = U (m + n) and H = U (m) × U (n). We will take m ≤ n.
The CSA of U (m + n) can be parameterized by (ξ 1 , . . . , ξ m+n ), and the Casimirs of G are the permutation invariants: ξ k ℓ , k = 1, . . . , m + n while those of H are: It is convenient to introduce an equivalent set of Casimirs: along with a corresponding family of new variables: Since x k +x k = V k = v k , we can use this linear equation to eliminate all thex k in terms of x k and v k . Note that for k > m one must write x k as a polynomial in the x j (j ≤ m), and that for k > n one must writex k in terms of a polynomial in thex j (j ≤ n).
For m = 1 one can set x = x 1 = ξ 1 , and one has v n+1 = x n+1 +x n+1 . One then writesx n+1 as a polynomial inx j , j ≤ n and then eliminates thex The result is a superpotential of the form: The n + 1 critical points of this superpotential are then given by the components of the weight, ξ ℓ . In particular, the lowest dimension chiral primary, For the general Grassmannian model we need the m variables, x k , or equivalently and more conveniently, the z k of (3.2). These characterize a subset, ξ 1 , . . . , ξ m , of the m + n roots of dW 1,n dx = 0. Moreover, from (3.2) it follows that the individual roots, ξ 1 , . . . , ξ m , are related to the z k as the m roots of the auxiliary equation: Thus, given a critical point of W m,n , one can use this formula to extract the set of m critical , that characterize a single ground state of the Grassmannian model One now reverses this procedure: the multi-variable critical points are precisely characterized by adjusting the z k (or x k ) so that all m roots of (3.5) are critical points of (3.3).
The job is thus to properly recast (3.3) in terms of the z k using the fact that we now wish to sum (3.3) over subsets of m roots of One can do this by using (3.5) to eliminate all powers, x ℓ , then this will vanish for a (generic) set of m roots of dW 1,n dx = 0 if and only if all of the B ℓ (z j ) vanish. These must therefore be the deformed "vanishing relations" that characterize the chiral ring of the multi-variable model. It is these that must be integrated to give the multi-variable superpotential.
There is, however, a simpler way to get the Landau-Ginzburg superpotential: Perform the same elimination procedure, using (3.5), on the superpotential (3.3). The result is once again a function, W (z j ; x), that is a polynomial of degree at most m − 1 in x. Now replace x j in this function by 1 m x j , and rewrite everything in terms of either z j or x j . We claim that this results in the requisite multi-variable potential, W m,n+1−m (x j ; v j ). To see why this is so, observe that this prescription for replacing x j is the same as summing 1 m W (z j ; x) over the m roots of (3.5). Thus W m,n+1−m (x j ; v j ) represents an average of the values of (3.3) over a set of roots of = 0 on all m of these roots. Also recall that the values of the superpotential encodes the topological charge of solitons, and this averaging procedure reproduces the proper topological charge in the multi-variable case.
To illustrate this procedure we consider the potentials for which m = 2. Equation (3.5) implies: One now substitutes this into (3.3) and sums over both roots, which amounts to dropping all square-roots from the result. For n = 4 and 5 one obtains: In the first of the superpotentials we have made the change of variables: Note, in particular that the value of the modulus (the coefficient of yz 4 , with y and z suitably normalized) in W 2,3 is consistent with the results of [12,13,14].

The SO(2n) superpotentials
There are two infinite series of coset models with this numerator, and we begin with , leaves an SO(2n) pair of vectors invariant, and so ground states of this model are characterized by the weights of the vector representation of SO(2n). This model may also be thought of as the D 2n minimal model, but we will see that the canonical deformation leads to a subset of the full set of deformations of the standard D 2n potential.
The Casimirs of SO(2n) are defined by where the a i are the skew eigenvalues of an SO(2n) matrix in the vector representation.
with v 2j being some constant and eliminate x 2 , . . . , x 2n−4 from the . We are then left with and V n −ṽ n = 0. Introduce the superpotential so that then the desired D 2n potential is obtained as where we have set x 1 = x andx n−1 = y.
Note that our procedure yields only an n-dimensional subset of the 2n possible relevant deformations of the conformal D 2n superpotential: The foregoing canonical deformation is Any other deformations would destroy the W (D n ) symmetry of the ground states.
We now turn to the SO(2n)/SU (n) × U (1) coset model, whose central charge is In view of the decomposition 2n = n 2 +n Here [ n 2 ] stands for the integral part of n 2 .
For n = 2, 3, 4 the coset models are identified with the A 2 , A 4 and D 8 minimal models, respectively. It is easily checked that the present method produces the corresponding Landau-Ginzburg superpotentials with deformations 4 .
The foregoing procedure is, in general, very hard to integrate directly to obtain the general Landau-Ginzburg superpotential, and so we illustrate it in the first non-trivial example: n = 5. This model has c = 10 3 > 3 and turns out to be instructive since its shares interesting properties with the Grassmannian as well as the E 6 model. To decompose the Once again, setting V i to constant values, v i (ṽ 5 for V 5 ), we solve the first three equations of (3.10) for z 2 , z 4 and z 5 . We then substitute these values of z j , j = 2, 4, 5 into equations for V 6 and V 8 . With the notation z 1 = x/5, z 3 = y, the equation for V 6 reads Similarly the equation for V 8 becomes In order to obtain the Landau-Ginzburg superpotential we set (3.11) The second term of ∂W ∂x is introduced so that the integrability condition is satisfied ∂ ∂x ∂W ∂y = ∂ ∂y ∂W ∂x .
By integrating (3.11) we obtain the deformed Landau-Ginzburg superpotential for the This procedure does not expose any obviously generalizable structure, and so it seems rather hard to use it to extract the superpotential for general n. We start by eliminating, or integrating out, the variable y in (3.9) using ∂W ∂y = 0 to get: The 2n roots of this equation are, by construction, the 2n vector weights, x = ±a j . The critical points of the SO(2n)/(SU (n) × U (1)) model are determined by the 2 n−1 spinorial combinations: ± 1 2 a 1 ± . . . ± 1 2 a n , with an even (or odd) number of − signs. The simplest auxiliary equation has roots x = a j , j = 1, . . . , n and so determines the vector weights up to a sign. One now rewrites this polynomial equation in terms of the z j : That is, we define the polynomial When calculating the coefficients c k (z j ) we utilize the deformed relations V 2i (z j ) = v 2i to eliminate z 2i (i = 1, 2, . . .) in favor of z 2j−1 with j = 1, 2, . . . , [ n 2 ]. Then we have whereas the relation V n (z j ) =ṽ n yields For example, for n = 5 we find: The Landau-Ginzburg superpotential for the SO(2n)/(SU (n) × U (1)) coset model is then computed by summing W (x) over all the roots of R(x) = 0: where the sums of powers of the a j that appear on the right-hand side are evaluated by making use of the auxiliary equation R(x) = 0. In particular, the sum of the last term of where (3.15) and (3.14) have been used.
We have confirmed that this method correctly recovers the deformed superpotentials obtained by the procedure of linear elimination for the models with n = 2, 3, 4 and 5.

The canonically deformed E 6 superpotential
There is only one SLOHSS model involving each of E 6 and E 7 , and so there are no short-cuts: we need to use the Casimir decompositions. Here we present the computation for E 6 . The details for E 7 are similar, but more complicated, and so they have been included in an appendix.
The denominator of the E 6 model involves SO (10), which fixes a weight in the 27 of E 6 , and there are thus 27 ground states. The first step to obtaining the superpotential is to decompose the six Casimirs, V j , of E 6 into the Casimirs, x j , of SO(10) ×U (1). This was done explicitly in [12], but we reproduce the result here since there was a typographical error in the printed version.
We then solve the equations V i = v i , where v i are constants, for i = 2, 5, 6, 8. These are the equations that are linear in x i , and so we use them to directly eliminate x 2 , x 5 , x 6 and x 8 . This leaves the equations: V 9 = v 9 and V 12 = v 12 . These are integrable, and former is proportional to ∂W ∂x 4 , and the latter must be proportional to ∂W For reasons that will become apparent in sections 3.4 and 4.2, we replace the v j with some other constants, w α according to: 1 , v 9 = −28 (w 5 + 2 w 2 1 w 2 ) , v 12 = 5 w 6 + 26 3 w 2 1 w 4 + 1 3 w 2 3 + 4 w 3 1 w 3 + 19 w 1 w 2 2 + w 6 1 . (3.17) Finally, setting x 1 = 1 4 x and x 4 = 60 13 z + 117 16 x 4 + 5w 1 x 2 + 2w 2 1 one arrives at the superpotential: This potential then has the property that:

Single variable potentials
One can always partially eliminate variables from the Landau-Ginzburg potential by using some of the equations, ∂W ∂x j = 0. Indeed we have already done this for all the quadratic variables in W : this is what is meant by using the linear vanishing relations.
One can go further and eliminate all variables except the one of lowest degree, the Casimir, In general one will not be able to solve (3.20) explicitly, and so the one variable potential is generically implicit. On the other hand, for several important examples, the equations can be explicitly solved and one obtains a polynomial or irrational potential.
There is also another more direct way to get the single variable potential, and this approach is more directly related to the applications considered in [15,4]. In this approach one takes the characteristic polynomial, P (x; v j ), of a general CSA matrix in the representation of G that corresponds to the ground state of the SLOHSS model of interest. One then shifts the Casimir of highest degree according to v r → v r + τ , and solves P (x; v 1 , . . . , v v−1 , v r + τ ) = 0 for τ (x; v j ). The roots of the equation τ (x; v j ) = 0 are precisely the ground states of the SLOHSS model, and so the single variable potential in terms of the U (1) Casimir, x, is given by: We thus refer to τ (x; v j ) as the pre-potential. This function is discussed further in the Appendix.
Given an irrational or implicit potential in a single variable, one can reconstruct the multi-variable, polynomial potentials in a relatively straightforward manner: One replaces every algebraically independent irrational or implicit form by a new chiral primary field.
Algebraic vanishing relations are then introduced so that their solution yields the relationships between the new variables and the original irrational or implicit forms. One then integrates these vanishing relations (where possible) to obtain a Landau-Ginzburg potential. For all the SLOHSS potentials that we have studied in detail, we have found that the irrational, single variable potentials are equivalent to the full algebraic, multi-variable potentials.
There are several reasons why the single variable potentials are useful. First, we will find that the results of the period integrals in the 4-fold will most directly reduce to the single variable potential. Knowing the single variable potentials is thus valuable data in the reconstruction process. More generally, one finds such single variable potentials also naturally arise in the construction of Seiberg-Witten Riemann surfaces from period integrals on Calabi-Yau 3-folds [15,7,16].
There is a further application of the single variable potentials in the coupling of topological matter to topological gravity [3,4]. In this context, the physical operators and their correlators are most directly expressed in terms of a single variable potential and residue formulae.
We will therefore illustrate both approaches to computing the single variable superpotentials. We will use the first method on the E 6 superpotential, and arrive at results closely related to those of [16,4]. We will then use the second approach to rederive the result from section 3.2 for SO(2n)/U (n).
We now start from the other end for the SO(10)/(SU (5) × U (1)) model: The ground states are classified by the 16 of SO(10), and the relevant characteristic polynomial is therefore given by: x − (± 1 2 a 1 ± 1 2 a 2 · · · ± 1 2 a 5 ) with an even number of "−" signs. Expanding this we have where we have put the Casimirs to V 2i = v 2i and V 5 =ṽ 5 . Here q 0 , q 1 and q 2 are polynomials Note that dq 2 dx − 16 q 1 = 0 . (3.28) Then we obtain the pre-potential, τ , by solving P 16 SO(10) (x; v 8 + τ ) = 0: This expression can then be integrated with the use of (3.28) to give: To get to a multi-variable, rational potential one simply has to replace √ q 2 by a new Landau-Ginzburg variable. It is convenient to mix this with other degree 3 terms in x and v j to arrive at the definition: One now replaces all occurences of √ q 2 in W(x) using (3.29), and the result is precisely W (x, y) of (3.12). Moreover, the equation (3.29) is exactly the solution of ∂W ∂y = 0.

Review
The construction of [8] starts by identifying the vacuum states of the IIA theory that correspond to the Landau-Ginzburg vacua. These IIA vacua are obtained by first choosing a Calabi-Yau 4-fold, Y , and then choosing a set of fluxes for the 4-form field strength, G, on Y . There is a quantization condition on the fluxes which requires that the quantity: be an integer. This means that the characteristic class ξ ≡ [ G 2π ] satisfies a shifted Dirac quantization condition in that the difference between any two fluxes G and G ′ must satisfy ξ −ξ ′ ∈ H 4 (Y ; Z Z). The integer N is then the number of "world-filling" fundamental strings in the vacuum state of the theory, and these fundamental strings are to be located at some points, P i , chosen in Y . For the vacuum to have a mass gap one must thus have N = 0.
Domain walls, or solitons are represented by kinks that interpolate between spatial regions in which the G-fluxes are different. In particular, for any element S of the integral homology H 4 (Y, Z Z), we may choose a D4-brane that wraps this cycle and appears as a Having chosen a G-flux, the complex and Kähler structures on Y are required to satisfy constraints [18,8] in order to preserve the requisite supersymmetry with a zero cosmological constant. Specifically, if G p,q are the (p, q) parts of G one requires that G 0,4 = G 1,3 = 0, and if K is the complexified Kähler form, one requires that G ∧ K = 0. These imply that G must be a self-dual (2, 2)-form. In [8] a superpotential was proposed for the complex structure and Kähler moduli. We will only consider the former here, and it is given by: The variation of Hodge structure means that W and dW vanish in the vacuum states, and the result is the Landau-Ginzburg theory at a conformal point. More generally, one can seek vacua of massive Landau-Ginzburg theories, and then one only impose the condition that dW = 0 for a vacuum state.
A Landau-Ginzburg soliton is a BPS state whose mass is equal to its topological charge, and the latter is given by the change in the value of the superpotential along the soliton. Since the solitons are represented by D4-branes wrapping integral 4-cycles, one thus concludes that: For non-compact Calabi-Yau manifolds, the quantization condition, (4.1), is replaced by a boundary condition at infinity. That is, there is a new conserved quantity, the "flux at infinity:" 4) and the value of Φ must be given in order to specify the model. Going to non-compact Calabi-Yau manifolds also affects the dynamics in other non-trivial ways. In particular, some of the complex structure moduli will give rise to scalar fields whose kinetic terms are non-normalizable. These moduli will thus have to have zero kinetic energy, and thus their dynamics is frozen for the non-compact manifold. These complex structure moduli thus become true moduli, or coupling constants of the Landau-Ginzburg model.
In this paper we will follow [8] and consider only non-compact Calabi-Yau manifolds that are fibrations of ALE singularities over some complex 2-dimensional base. Specifically, we consider non-compact 4-folds defined by the equation P (z 1 , . . . , z 5 ) = 0 where P (z 1 , . . . , z 5 ) = H(z 1 , z 2 ) − z 2 3 − z 2 4 − z 2 5 and: We now wish to examine how the Landau-Ginzburg potentials of the SLOHSS models emerge from the period integrals of the surfaces defined by (4.5). From the foregoing discussion there is an obvious problem that will arise: all the dynamics is frozen. The singularity, and its period integrals encode only topological data about the Landau-Ginzburg theory, and do not contain dynamical fields. It turns out that we will still be able to extract the Landau-Ginzburg potentials from this topological data, and thus implicitly find some Landau-Ginzburg fields. While we cannot give definitive geometric characterizations of these Landau-Ginzburg fields, we find that they emerge in the calculation in a very interesting and natural manner. We will also find that the evaluation of the period integrals has implicit ambiguities whose resolution corresponds to selecting the miniscule weight, or flux at infinity, and thus determines the Landau-Ginzburg variables and chiral ring. We will remark further upon this in sections 4.4, and we will begin by direct evaluation of periods.

Integrating over 4-cycles
The integral of the 4-form, Ω, over a cycle, S, of the non-compact surfaces defined by (4.5) can be written: where the 4-cycle is defined as follows: z 3 runs around branch-cut of the square-root; z 4 runs between z 4 = ± H(z 1 , z 2 ), i.e. points at which the branch cut shrinks to a point; the variables z 1 , z 2 are then integrated over a 2-cycle, S 2 , of the singularity H(z 1 , z 2 )−z 2 5 = 0. The first two integrals are elementary, and reduce to the following (up to an overall normalization): For the A n singularity, one has H(z 1 , z 2 ) = P (z 1 ; a j ) + z 2 2 , where P (z 1 ; a j ) is a polynomial of degree n + 1. As was discussed in [8], the integral (4.7) then reduces to: evaluated between the zeroes of P (z 1 ; a j ). That is, we have recovered the superpotential, W, of (3.3), with the integral (i.e. topological charge of the soliton) expressed in terms of the values of W at its critical points.
It is almost as elementary to perform the integrals for the D n singularity. In this instance one has: H(z 1 , z 2 ) = P (z 1 ; a j ) + z 1 z 2 2 + a 0 z 2 , where P (z 1 ; a j ) = z n−1 1 + n−2 j=0 a n−1−j z j 1 . One makes the change of variable z 1 = x 2 , and one performs the elementary integral over z 2 to obtain: which is to be evaluated between the zeroes of the integrand.
This is the single variable potential (3.13) for the SO(2n) SO(2n−2)×U(1) coset model. As we remarked earlier, it is obtained from (3.9) by using ∂W ∂y = 0 to eliminate y. The superpotential, (3.9), can be recovered by removing the singular term by introducing the variable y = a 0 2x . Also observe that (4.9) is an odd function and contains only half of the general versal deformations of the D n singularity.
The calculation for the E 6 singularity is essentially the same as that performed in [16], and so we will only sketch the details here.
One starts with the singularity: where we have made a convenient choice for the deformation parameters, w j . One reparameterizes the singularity by setting 11) and the result is a quartic in y. One now sets where p 2 is given by (3.22). The functions β and α are chosen so that the y 3 and y 1 terms in the quartic vanish respectively. The integral (4.7) now takes the form: for some functions, A(x) and B(x). By taking a contour at large y this integral reduces to: and thus we regenerate the one variable superpotential (3.23) with all the correct moduli.
As we remarked earlier, one can reconstruct (3.18) by replacing the singular irrational part using:

Other superpotentials
Our integration procedure appears to have led us directly to a single superpotential for the A n and D n models. The reason for this is that we must have implicitly chosen a flux, or boundary condition at infinity. Indeed, the sleight-of-hand occurred when we passed from the definite integral to the indefinite integral with a single end-point to the integration.
We know from the algorithms of section 3 that we can get the multi-variable potentials for any of the Grassmannian models and for the SO(2n)/U (n) series by summing the potentials (3.3) and (3.9) over carefully selected sets of their critical points. Exactly the same choice emerges in the period integrals. For example, in the Grassmannian we could take the end-points of the integration to be a subset of m solutions of dW dx = 0, with this subset defined by (3.5). The result would be an indefinite integral that depends upon z 1 , . . . , z m , and reproduces the multi-variable potential for the Grassmannian.
We therefore see that the selection of the G-flux in H 4 (Y, Z Z), or the choice of the flux at infinity, amounts to selecting a set of critical points of the single variable potentential.
In the next subsection we will demonstrate how this works more explicitly by showing that the "boundaries" of the integration procedure, and in particular these critical points, correspond to special holomorphic divisors of the surfaces defined by (4.5).
It is also interesting to note that if we now consider the period integral defined by two critical points of the Grassmannian potential, W m,n (z 1 , . . . , z m ), then it will decompose into a difference of sums of critical values of the superpotential W 1,m+n−1 (x). Generically this will represent the sum of topological charges of a multi-soliton (and thus non-BPS) state. Thus not all pairs of ground states represent boundary conditions for BPS solitons.
This fact was first discovered by considering the integrable Landau-Ginzburg models, where it was seen that one could not make a factorizable S-matrix involving only solitons for certain classes of integrable model. Some boundary conditions could only give rise to multi-soliton states [19,12]. Indeed, based on the conserved quantities of the E 6 model it was shown in [12] that for the E 6 integrable model, the correct soliton spectrum was exactly given by the prescription that two ground states labeled by weights λ 1 and λ 2 would give rise to a single, fundamental BPS soliton if and only if λ 1 − λ 2 was exactly a root. It is now very satisfying to have a simpler, and far more general string theoretic explanation of this result: A wrapped of D4-branes on the ADE singularity is a fundamental BPS soliton if and only if it wraps a cycle that is represented by a root of the Lie algebra.

Intersection forms and holomorphic 4-cycles
We have just seen how the Landau-Ginzburg potentials emerge from some kind of "semi-periods" of the singularity: That is, we get the potentials by making the indefinite integral of the single variable potential, and then parameterizing families of its critical points. This suggests that one should try to characterize these families of critical points and the "semi-periods" more geometrically. As we will see, this is indeed possible since they represent non-compact homology cycles of the singularity. Ideally we would like to integrate Ω over these non-compact cycles, but such an integral will diverge. This can be fixed by compactifying the singularity. Our approach is based upon the techniques used in section 17 of [6], and we will therefore only summarize the key steps here.
The first step is to introduce a new coordinate and make the equation of the singularity homogeneous (see [6] for details). The resulting compact manifold will not be a Calabi-Yau manifold and so Ω will have some kind of problem at infinity: If one wants to preserve holomorphicity, then it will be singular. Since we wish to consider period integrals, we wish to keep Ω regular, but the cost is the loss of holomorphy in the patch at infinity. This will not affect the computation of topological charges since they only involve integrals of Ω over compact cycles of the original singularity. This means that such a compactification will only affect the Landau-Ginzburg superpotential by some additive constant.
The basic idea now is to try to write Ω = d λ for some 3-form, λ. This is not possible precisely because Ω is a non-trivial element of cohomology. However, if one excises all the non-trivial homology from the compactified singularity then it is possible to write Ω = d λ.
Let C a be a basis of the non-trivial 4-cycles. One can then write where [C a ] denotes the Poincaré dual of the cycle C a , and is a 4-form with delta-function support on C a . The parameters, m a are determined from the integral of Ω over C a : Specifically, if M ab = I(C a , C b ) is the intersection form of the basis of cycles, then: Given any other cycle, S, one then has: where I(S, C a ) denotes the intersection numbers of the cycle S with C a .
The issue now is to find a good basis, C a ; and there is a particularly nice way to do this using holomorphic lines in the ADE singularity in two complex dimensions, or using holomorphic planes in four complex dimensions. The important point is that Ω = dz 1 dz 2 dz 3 dz 4 z 5 is odd under any reflection z j → −z j , and so any holomorphically defined dual homology cycle must similarly be odd.
In practice, given the singularity type: one starts by looking for complex "2-planes" of the form where ζ is a complex variable, and a, b, c and d are constants. For generic a, b, c, d (4.14) and (4.15) define an irreducible rational surface that is even under z 5 → −z 5 . At special values of a, b, c, d one finds that H(a ζ +b, c ζ +d) is a perfect square and the rational surface degenerates to two intersecting planes: The odd 4-cycles that we seek are the differences of such pairs of 2-planes 6 . More generally, one can seek holomorphic 2-surfaces of higher degree (see, for example, [20]). Such holomorphic surfaces form Weyl orbits whose order depends upon the degree of the surface. Here we are focussing on "planes," or surfaces of lowest degree so as to recover the miniscule orbits. One could also go to surfaces of higher degree, and these are presumably relevant to the physics of more general fluxes at infinity.
Going back to the earlier parts of this section, one sees that the planes described above played a crucial role in the explicit evaluation of the integrals. The cycles were defined by families of contour integrals and the planes defined where these contours collapse to points, that is, the planes defined the limits of the integration. In simple terms, a 4-cycle is essentially an S 4 and its intersection points with 2-planes define the extremities, or "north" and "south" poles of the S 4 . Thus the difference of two lines, C a − C b defines the compact cycle over which we integrate, and the fact that integration is the difference in values of the superpotential at the two extremities simply reflects the fact that the integral is given For the A n singularity the zeroes of P (z 1 ; a j ) are precisely the points where the singularity contains the planes: For the D n singularity the solutions of P (x 2 ; a j ) = a 2 0 4x 2 define precisely the points where the singularity contains the planes: Indeed, more generally, the D n superpotential is given by W (x, y) ≡ P (x 2 ; a j )dx−x y 2 + a 0 y, and reduces to (4.9) upon eliminating y via: ∂W ∂y = 0. The critical points of the full superpotential W (x, y) precisely define the planes: lying in the D-type singularity.
The story is similar for the E 6 and E 7 singularities. The details for the E 7 singularity may be found in Appendix A. For the E 6 , the surfaces are quadratic in z 5 , but linear in the other z j . Once again take z 3 = ± i z 4 and introduce ζ ∈ C with The parameters of these surfaces are x and z. When the superpotential W (x, z) of (3.18) has a critical point then the foregoing defines a "plane" in the E 6 singularity (4.10) with: Indeed, the 27 critical points of W (x, z) corresponding to the weights of the fundamental of E 6 define the celebrated 27 lines in a cubic hypersurface in IP 3 .
We therefore see that the Landau-Ginzburg variables naturally emerge in the parameterization of representatives of the non-compact homology cycles. In retrospect this is rather natural: we know that the Landau-Ginzburg variables must parameterize a family of fluxes, and have non-normalizable kinetic terms in the non-compact singularity. Dual to such fluxes are non-compact homology 4-cycles, and so the Landau-Ginzburg dynamics can be converted into a description of these 4-cycles: they intersect the compact cycles of the singularity in a manner determined by the inner products of the corresponding weight and root vectors, and the dynamics is such that these supersymmetric ground states correspond holomorphic surfaces. It is therefore tempting to think of the ground-state flux as being created by a non-compact D4-brane that threads the singularity along one of these non-compact cycles. The solitons then intersect these non-compact branes and then combine with them to yield another non-compact D4-brane that threads the compact part of the singularity with different set of intersection numbers.

Conclusions
The N = 2 superconformal models that arise from the non-compact ADE singularities are determined by a flux at infinity. For the minimal, or miniscule, fluxes the versal deformations of the singularity lead to a theory with a mass gap. It was argued in [8] that these particular models should be perturbations of the SLOHSS models. Here we observed that the perturbations of the singularity respect the underlying Weyl symmetry, and thus the corresponding perturbations of the Landau-Ginzburg superpotential must do the same.
We then used this to either explicitly construct, or provide a precise computational algorithm for the construction of the Landau-Ginzburg superpotentials of the SLOHSS models with the most general perturbations consistent with the Weyl symmetry. We then showed that exactly the same superpotentials and algorithms emerged from the topological data and period integrals of corresponding ADE singularities. This provides some extremely detailed checks on the results of [8]. There are several interesting by-products of our work which may find application in the study of Landau-Ginzburg models and in topological matter coupled to topological gravity.
There are also quite a number of interesting open questions. Our work suggests an interpretation of the Landau-Ginzburg variables in terms of moduli of non-compact cycles that thread the singularity. It would be interesting to verify these ideas more explicitly by considering nearly singular, compact Calabi-Yau manifolds in more detail.
Here we have considered only the theories with a mass gap. It would be interesting to find out exactly what perturbed conformal theories emerge for non-miniscule fluxes.
It is tempting to conjecture that the flux at infinity, Φ, encodes the level of the numerator current algebra of the coset model. Since versal deformation of the singularity is not supposed to yield a mass gap, the corresponding perturbed Landau-Ginzburg models will have to remain multi-critical. This is certainly possible, but there is a potentially interesting conundrum here: We know from the results in [21] that simple perturbative attempts at adding marginally relevant operators can create non-trivial higher order corrections, and indeed result in a theory with a mass gap. For example, the N = 2 minimal models perturbed by the least relevant chiral primary results in a theory with a Chebyshev superpotential. Indeed, the such a perturbation would be a natural candidate for the A 1 singularity with higher flux. It would thus be interesting to see if the resulting theory really does have massless modes, or if, somehow the theory does indeed develop a Chebyshev superpotential and hence have a mass gap. It is also quite possible that the conformal field theories associated with the singularities with non-miniscule fluxes are not the various coset models at higher levels and with different denominators.
One way to approach this problem that might be particularly interesting would be to find a method of computing the elliptic genus of the conformal field theory using only the topological data of the singularity and its flux at infinity. For Landau-Ginzburg theories this is equivalent to finding the fundamental Landau-Ginzburg fields and determining their U (1) charges. For more general theories, the elliptic genus gives a lot of valuable information about the partition function.
Finally, there are some interesting properties of Landau-Ginzburg solitons that may have interesting consequences for D4-branes more generally. Most notable, but probably least relevant is the fact that some very special perturbations lead to quantum integrable models, with factorizable S-matrices. This sort of property is not at all robust, and so it likely to be only a property of the field theory in the near singular limit. Integrability will very likely be spoiled in the full string theory.
More interesting is the fact that solitons have fractional fermion number [22,23]. In particular, for a Landau-Ginzburg soliton running between two vacua, the fermion number is given by [23]: where ∆ indicates the change in the value of the quantity between the two critical points.
Note that this quantity is, in principle, a very complicated function of the moduli and can presumably assume arbitrary (even irrational) values. However, for singularities with discrete geometric symmetries (like the Coxeter resolution), this fermion number will take simple fractional values. One might naturally wonder whether this fractional fermion number might also be an artifact of the field theory that emerges near the singularity, and that this fermion number would disappear in the full string theory. There are no global conserved currents in string theory, and so one might expect that fermion number would not persist in string theory. However, the formula (5.1) only defines the fermion number mod 1, and indeed fermion number mod 2 is a well defined concept in string theory. The fact that (5.1) follows from an index calculation gives some further hope that fractional fermion number might persist in string theory. Finally, it seems particularly likely that fractional fermion number will persist if the fractional fermion number is fixed by discrete geometric symmetries of the singularity.
If some version of fractional fermion number persists in string theory then it would be very interesting to find the geometric meaning of (5.1). As regards physical consequences: there are the obvious phases that would arise in scattering, but perhaps more interesting would be the consequences for quantization conditions under compactification, and thus for partition functions.
Before moving on to the multi-variable potential, we wish to describe how this singlevariable potential is related to the characteristic polynomial of the 56 of E 7 . Generally the characteristic polynomial of a representation R of the Lie algebra G is defined by which is of degree dim R in x. Here a is an r-dimensional vector in the Cartan subspace spanned by H and the V i (i = 1, . . . , r = rank G) are the Casimirs built out of a among which V r is the top Casimir whose degree equals the Coxeter number h of G. Let λ be the weight of R, then we have For the 56 of E 7 we obtain where the Casimirs have been set to V i = v i . Note that this has the degree 3 in the top Casimir v 18 , which is analogous to (A.12). In fact, one can show explicitly that Hence, (A.12) for τ E 7 may be expressed as This also makes it clear that each of the lines in the E 7 singularity we have constructed is in correspondence with a weight of the 56 of E 7 .
More generally, for any ADE group, the single-variable version of the Landau-Ginzburg potential can be obtained from τ (x)dx , where τ (x) is obtained by solving P R G (x; v 1 , . . . , v r + τ (x)) = 0 .
Thus the pre-potential, τ , depends upon the representation, R, and takes the form 7 : where F R is some irrational, or implicit function of x whose expansion at x = ∞ starts with the x h term and does not carry the pole term 1 x as can be seen by degree counting.
A.2. The Landau-Ginzburg potential for the deformed E 7 /(E 6 × U (1)) coset model The calculation is also parallel to that for the E 6 /(SO(10) × U (1)) model. We start with the decomposition of the seven Casimirs, V j , of E 7 into the Casimirs x j of E 6 × U (1).
Performing the integral over y along a contour at large |y| we obtain where τ E 7 (x) is given by (A.11). The remaining integral has already been evaluated. The