Local Operators from the Space of Vacua of Four Dimensional SUSY Gauge Theories

Richard Eager ^{1}^{1}1eager@mathi.uni-heidelberg.de

Mathematical Institute, Heidelberg University, Heidelberg, Germany

[.1cm]

Abstract

We construct local operators in short representations of supersymmetry algebras from polyvector fields on the quantum moduli space of vacua of supersymmetric gauge theories. These operators form a super Lie algebra under a natural bracket operation with structure constants determined by terms in the operator product expansion of the corresponding operators. We propose a formula for the superconformal index in terms of an index over polyvector fields on the moduli space of vacua.

Along the way, we construct several models with moduli space of vacua corresponding to affine cones over smooth bases using the classical geometry of Severi varieties and the Landsberg-Manivel projective geometries corresponding to the Freudenthal magic square of exceptional Lie algebras. Curiously, we relate the Landsberg-Manivel projective geometries to the exceptional enhanced symmetry surprises of Dimofte and Gaiotto. Finally, we determine Beasley-Witten higher-derivative F-terms in new examples arising from Severi varieties and remark on their origin in classical projective duality.

###### Contents

- 1 Introduction
- 2 Overview
- 3 Smooth models
- 4 Geometry of orbit closures
- 5 SQCD review
- 6 Local operators from polyvector fields
- 7 Superconformal index from the moduli space
- 8 Polvectorfields on moduli spaces of vacua
- 9 The surprise revisited
- 10 Projective duality and Beasley-Witten F-terms
- 11 Schouten-Nijenhuis bracket and the OPE
- 12 Product structure on cyclic homology
- 13 Conclusions and outlook
- A Polyvectorfields from Borel-Weil-Bott
- B Tables of polyvector fields

## 1 Introduction

Supersymmetric quantum field theories have furnished a rich testing ground for many ideas in quantum field theory. The theories often have classical flat directions, and as a result, the theories often have a space of inequivalent vacua called the classical moduli space. Remarkably, the strong constraints of supersymmetry often allow the quantum corrected moduli space of vacua to be exactly determined [1]. Supersymmetric quantum field theories are often connected by a rich web of electric-magnetic Seiberg dualities that relate strongly coupled gauge dynamics in one theory to weakly coupled gauge dynamics in the dual theory [2]. Traditionally, these dualities have been studied by ‘t Hooft anomaly matching, comparing deformations, matching local operators, and matching the quantum moduli space of vacua [2].

In this paper, we attempt to derive the spectrum of BPS local operators directly from the quantum moduli space of vacua. In the rare instances when the moduli space of vacua is an affine complex cone over a smooth Kähler base, we are surprisingly successful. We determine the operators that are BPS with respect to a fixed supercharge. As a result, we can recover the superconformal index directly from the quantum moduli space of vacua. Our basic strategy is to view the low energy effective theory as an supersymmetric nonlinear sigma model from Minkowski space or to the quantum moduli space of vacua. The local operators arise as cohomology classes of polyvector fields on , which is familiar as the ring of local observables in the topological B-model on These operators had largely been ignored in supersymmetric QCD (SQCD) until Beasley and Witten’s study of multi-fermion F-terms [3]. We systematically compute these operators using an extension of the Borel-Weil-Bott theorem.

Using the spectrum of BPS local operators, we find a candidate expression for the superconformal index in terms of the moduli space of vacua as

where is an alternating sum of Euler characteristics of polyvector fields on the moduli space of vacua that will be defined more precisely and is the inverse of the plethystic exponential. The formula is typically not exact due to extra degrees of freedom arising from the singularity at the origin of the moduli space [1]. However, in the examples we consider, the formula agrees remarkably well with the superconformal index.

A similar expression for the superconformal index was found for large-N gauge theories dual to type IIB string theory on with Sasaki-Einstein in [4]. There, the local operators were studied using cyclic homology. The local operators in the large-N quiver gauge theory are related to polyvector fields on the Calabi-Yau cone over via the Hochschild-Kostant-Rosenberg theorem [5]. The moduli space of vacua in these theories is roughly the large limit of , which is -th symmetric product of . A similar relation was also found in [6, 7].

Alternatively, we can determine the local operators from gauge theory using the cohomology of a nilpotent supercharge The classical Q-cohomology can be reformulated in terms of a generalization of Lie algebra cohomology. This classical space of states coincides with the local operators in the holomorphic twist of the theory [8, 9]. However, the differential of the quantum corrected Q-cohomology is different due to the Konishi anomaly. After taking into account the Konishi anomaly [10], we find that the leading contributions to Q-cohomology match the space of local operators computed from polyvector fields on the quantum moduli space of vacua in accordance with [11].

The quantum behaviour of SQCD was studied long ago using instanton techniques. For , instantons generate a superpotential and deform the classical moduli space of vacua. The Konishi anomaly provides a consistency check on these calculations and can partially simplify them. It is therefore reassuring that the Konishi anomaly also corrects the gauge theory Q-cohomology in order to match the polyvector fields on the quantum moduli space of vacua. Similarly, the multi-fermion operators studied by Beasley and Witten arise from instanton effects in SQCD.

Finally, we verify that our counting of operators is consistent with the superconformal index. The superconformal index [12, 13, 14] has been used to match protected operators in dual quantum field theories. In a superconformal theory, the index counts protected operators satisfying a BPS condition that cannot be combined to form long multiplets. The equality of the index for a theory and its electric-magnetic dual often lead to very interesting integral identities between products of elliptic gamma functions [15, 16]. Conversely, recently discovered integral identities have led to the derivation of new dualities between quantum field theories. One of the great features of the superconformal index is that it can be easily computed, at least in a perturbative expansion. Part of its simplicity follows from its construction, which is insensitive to the precise form of the superpotential. However, the Q-cohomology does depend on the explicit form of the superpotential.

One of the main motivations of this study is to develop a strategy to prove that the -cohomology groups of local operators in two Seiberg dual theories are isomorphic. This can be viewed as a categorification of the equality of superconformal indices and in particular Spiridonov’s elliptic beta integral [17]. Since the superconformal index is the partition function for the holomorphically twisted theory [7], it is natural to conjecture that even more is true. Namely, not just the operators, but the correlation functions should be equivalent. In the language of holomorphic factorization algebras developed by Costello and Gwilliam [18], our conjecture is that

###### Conjecture.

The holomorphic factorization algebras associated to the holomorphic twists of two Seiberg dual theories are (quasi-)isomorphic.

Since two Seiberg dual theories have equivalent quantum moduli spaces of vacua, it is natural to prove the equality of the -cohomology groups by relating both cohomologies to the local operators that can be described directly on the quantum moduli space of vacua. In two-dimensions, Ando and Sharpe [19] showed that the superconformal index (elliptic genus) of a Landau-Ginzburg model is equal to that of its low-energy sigma model using a Thom class computation. It is natural to expect that a similar result holds in four dimensions. The full superconformal index can be derived using the holomorphically twisted sigma model appearing in [20]. The calculations in this paper represent the first steps toward evaluating these indices. We find many indications that there might be simplifications in the full formula, perhaps arising from cohomology vanishing theorems.

Identifying local operators in terms of polyvector fields has the added virtue that we can adapt several classical results on polyvector fields to gauge theory. In particular, polyvector fields have a Schouten-Nijenhuis bracket operation that generalizes the ordinary Lie bracket of vector fields. On a complex manifold, the Schouten-Nijenhuis bracket on global sections of the sheaf of polyvector fields extends to a Gerstenhaber algebra on the cohomology of the sheaf of polyvector fields. Following a suggestion of Costello, we propose that the Shouten-Nijenhuis bracket computes certain protected OPE coefficients. These can be viewed as the leading OPE coefficients in the holomorphically twisted theory [8, 21]. The bracket is an example of a secondary products in supersymmetric field theory [22].

Many of the techniques developed here can be applied to theories in various dimensions and varying amounts of supersymmetry. In particular in two dimensions, a gauge theory that has a Calabi Yau manifold as its moduli space of vacua in the IR, has the same elliptic genus as the sigma model with target The elliptic genus is the two dimensional analog of the superconformal index, and it is explicitly expressible in terms of vector bundles on the Calabi Yau target [23, 24].

## 2 Overview

Typically, supersymmetric theories have moduli spaces of vacua with several branches and high-dimensional singular loci. To simplify the analysis of the physics at the singular loci, we first consider models where the moduli space of vacua is an affine complex cone over a smooth Kähler base. In this case, the only singularity is at the origin of the cone. These models are not very common. We consider two types of families of theories with moduli space of vacua that are affine cones of a smooth Kähler base. The first family is SQCD with gauge group and flavors. Its moduli space of vacua is an affine cone over the Grassmannians The second family consists of affine cones over the four Severi varieties and their hyperplane sections. They can be described by generalized Wess-Zumino models with cubic superpotentials. Surprisingly all of these models arise as projective geometries associated to Freudenthal’s magic square [25]. The most complicated member of the family is the Cayley plane which was recently considered in [26].

Finally, we will consider models with singular loci. Our main tool will be to utilize the geometry of orbit closures. While seemingly esoteric, the geometry of orbit closures elegantly recovers the structure of the moduli space of SQCD. For exceptional Lie algebras, the geometry of orbit closures will be used to describe the various smooth and singular strata of moduli spaces given by superpotentials of degree four or more.

When the quantum moduli space of vacua is an affine cone over a smooth projective Kähler base we review how the polyvector fields on can be pulled back to polyvector fields on following [3]. We will construct local operators from cohomology classes where is the line bundle obtained from pulling back to .

We find that the alternating sum of the Euler characters of polyvector fields on have a particularly simple expression when is a cone over a Severi variety. This includes both the Grassmannian and the Cayley plane . The simple expression arises from a small correction due to removing the identity operator and taking the plethystic logarithm. From this index, we can determine the superconformal index and conjecture a general form of the index in terms of polyvector fields. We then test this conjecture for SQCD with gauge group and four flavors. We find an surprise where the plethystic logarithm is almost entirely expressible of characters of the exceptional Lie group [27]. While the symmetry has a natural explanation in terms of a five dimensional gauge theory or coupling two four dimensional gauge theories together [27], it would be desirable to have an explanation entirely in terms of the geometry of the Grassmannian

## 3 Smooth models

### Freudenthal’s magic square

From a pair of normed real division algebras Freudenthal and Tits define a Lie algebra

where is the space of imaginary elements of , is the Jordan algebra of -Hermitian matrices, and is the subspace of traceless matrices. The corresponding Lie algebras are shown in table 1.

To each Lie algebra in Freudenthal’s magic square, Landsberg and Manivel [25] associate a projective geometry. These geometries are listed in table 2. The Lie algebra of the projective geometry’s isometry group is the corresponding Lie algebra in the magic square.

Family Name | |||||
---|---|---|---|---|---|

hyperplane section of Severi | |||||

Severi variety | |||||

lines through a points of | |||||

All of the “exceptionally simple exceptional models” studied by Razamat and Zafrir [26] can be naturally associated to one of the Manivel-Landsberg projective geometries. The first two families of Severi varieties and their hyperplane sections are described by the critical locus of a cubic superpotential. The third family, also known as the “subexceptional series,” describes varieties that are the critical locus of a quartic superpotential. The fourth and final row corresponds to the Deligne-Cvitanović exceptional series and appears in the classification of rank one SCFTs [28].

The third family of Freudenthal Lie groups appears in another interesting way. Dimofte and Gaiotto found an “surprise” in the theory of 28 five-dimensional free hypermultiplets coupled to four-dimensional SQCD with four flavors (eight chiral multiplets). The flavor symmetry group enhanced from to Similarly, they found enhanced symmetry with 16 four-dimensional free hypermultiplets coupled to three-dimensional SQCD with three flavors. These two enhanced symmetry groups appear in the third row of the Freudenthal magic square. It is natural to conjecture that there is an enhanced flavor symmetry group for 10 three dimensional matter multiplets coupled to two dimensional SQCD with two flavors extending the pattern.

### Severi varieties

Affine cones over the four Severi varieties arise as the moduli space of vacua of simple generalized Wess-Zumino models consisting of chiral multiplets with a specific cubic superpotential. The four Severi varieties correspond to the four real division algebras and The cubic polynomial arises as the determinant of symmetric three-by-three matrix over the corresponding division algebra. It is therefore a cubic polynomial in real variables, which is the number of chiral multiplets. Since the superpotential has -charge 2, the chiral multiplets have -charge 2/3. The models have an extended flavor symmetry group and the chiral multiplets are in a representation of . The corresponding groups are listed in Table 3. The algebras and correspond to the Veronese embedding of in and the Segre embedding of in respectively. The algebra corresponds to the Grassmannian . Finally the algebra corresponds to the Cayley plane.

# chirals | Matter Rep | |||
---|---|---|---|---|

6 | 2 | |||

9 | 4 | |||

15 | 8 | |||

27 | 16 |

The four Severi varieties arise as the projective duals of cubic hypersurfaces

in dimensional projective space [29]. The cubic polynomial defining the hypersurface is the cubic superpotential. The space of cubics polynomials has dimension

This is equal to the dimension of the conformal manifold of the corresponding Wess-Zumino theory [30].

#### Grassmannian

The most familiar member of the Severi varieties is the Grassmannian . It arises as the moduli space of vacua of QCD with three flavors. QCD with three flavors has a magnetic dual consisting of 15 chiral multiplets and cubic superpotential

The F-term relations are the Plücker relations for the Grassmannian . The moduli space of vacua is

This moduli space is an affine cone over the Grassmannian . Finally, there is one syzygy that arises since all of the relations are obtained from a superpotential. In general, the Plücker embedding for has Plücker variables, Plücker relations, and syzygies. Only for do the number of Plücker variables and Plücker relations agree as required for a theory with only chiral multiplets. It is then an extra condition that there is only one syzygy and no higher syzygies.

#### Cayley plane

A similar model to the Grassmannian was considered in [31] and more recently in [26]. It consists of 27 chiral multiplets with an invariant superpotential The cubic polynomial was first written down by Cartan in his thesis [32]. Another more recent description is in [33]. The moduli space of vacua is an affine cone over the Cayley plane . This implies that the Cayley plane is also described by 27 variables, 27 relations, and one syzygy. The relations are described in [34] and arise from the appearance of the in .

### Hyperplane sections of Severi varieties

The hyperplane sections of Severi varieties also occur as the critical locus of a cubic superpotential. Their geometry is described in [35] Section 6.3. The flavor symmetry groups are obtained by “folding” the corresponding Dynkin diagram in the series of Severi varieties. The number of chiral multiplets, the dimension of the moduli space of vacua, the corresponding flavor symmetry group, and matter representation of the flavor symmetry group are displayed in Table 4.

# chirals | Matter Rep | |||
---|---|---|---|---|

5 | 1 | |||

8 | 3 | |||

15 | 7 | |||

26 | 15 |

## 4 Geometry of orbit closures

Given a group and a representation the action of the group provides a decomposition of into orbits. If view as a global symmetry group and as a matter representation, then the orbits correspond to various symmetry breaking patterns. In general, the geometry of these orbits is quite complicated. Classical invariant theory provides invariants that can distinguish between the various orbits. The invariants can be thought of as Landau-Ginzburg order parameters. The irreducible representations of (reductive) groups with finitely many orbits were classified by Kac in [36]. Quite remarkably, almost all of these representations arise from gradings on Lie algebras. One common way the gradings arise is from a Dynkin diagram with a distinguished node.

Our interest is that orbit closures often arise as moduli spaces of vacua in supersymmetric gauge theory. When the largest orbit is a hypersurface defined by a polynomial the moduli space of vacua for the theory of free chiral multiplets with superpotential is a smaller orbit closure. The results of [37, 38] describe the geometry of the moduli space of vacua and allow us to determine its Hilbert series.

If we gauge , then many of the theories we study have played an important role in the dynamics of supersymmetric gauge theory. Many of these connections are described in [39]. We first illustrate how orbit closures describe the strata of the moduli space of vacua in SQCD and then describe more exotic orbit closures.

### Sqcd

Let be a vector space of dimension and let be the space of bivectors. The group acts on the projective space . The nontrivial orbits range in dimension from 2 to The k-th Pfaffian variety is the orbit closure of bivectors with rank The Pfaffian variety is the Grassmannian of -planes in The orbits are nested as

In SQCD with flavors, the moduli space of vacua is the Pfaffian variety [40].

For with , the moduli space of vacua is the Pfaffian variety However, this variety occurs as the singular locus of Since is described by the vanishing of a single Pfaffian, the singular locus is given by the simultaneous vanishing of the partial derivatives of a single quartic polynomial. This is precisely the “magnetic dual” description.

Despite the fact that the theory is not superconformal, we can formally consider the superconformal index given by 28 free chiral multiplets with -charge Then the index is

The leading terms come from the Hilbert series of the Pfaffian variety [41]

and the coefficient of the term is , where is the flavor symmetry group.

### Subexceptional series

There is a family of quartic superpotentials corresponding to the subexceptional series of Lie groups.

# chirals | Matter Rep | |||
---|---|---|---|---|

14 | 6 | |||

20 | 9 | |||

32 | 16 | |||

56 | 28 |

Analogous to SQCD, these theories are sigma models on an orbit closure. For the theory with 20 chiral multiplets, the orbit structure is given by the following inclusion of strata [37]

(4.1) |

The orbit closure is a quartic hypersurface in Letting the quartic polynomial be the superpotential, the moduli space of vacua is the orbit closure . For the theory with 56 chiral multiplets, the orbit closures are [42]

(4.2) |

The orbit closure is a quartic hypersurface in Again using this quartic polynomial as a superpotential for 56 chiral multiplets with -charge 1/2, the moduli space of vacua is the orbit closure . The orbit closure is the Freudenthal variety The moduli space of vacua was incorrectly identified as the Freudenthal variety in [31]. Using the free resolution in [42], the Hilbert series of is straightforward to compute. Combining this with the isometry group of the orbit closure, we can match the leading terms of the index.

## 5 SQCD review

Supersymmetric QCD with gauge group and massless flavors of quarks has a vector multiplet and chiral multiplets transforming in the fundamental representation of and chiral multiplets transforming in the anti-fundamental representation of The theory has a global symmetry. When the theory is in the conformal window, the IR charge of the chiral multiplets is . When the gauge group is the fundamental and anti-fundamental representation are isomorphic, so it is equivalent to think of a theory with doublets of The number of doublets is necessarily even because of a global anomaly [43]. When the gauge group is the global symmetry enhances to

The quarks are chiral superfields transforming in the fundamental representation of and is the color index for The gauge invariant mesons are chiral superfields that are color singlets obtained from

The mesons transform in the anti-symmetric representation of The classical moduli space of vacua is parametrized by the space of possible expectation values of the mesons subject to the constraint

or equivalently

These equations imply that has rank at most two. Viewing the mesons as coordinates on projective space , the above equations are the Plücker relations that describe the embedding of the Grassmannian as an algebraic subvariety of However, the space of expectation values of the meson fields is not projectivized. Therefore, the classical moduli space of vacua is the affine cone over the Grassmannian

## 6 Local operators from polyvector fields

We view the low energy effective theory of a four-dimensional theory as an supersymmetric nonlinear sigma model from Minkowski space or to the quantum moduli space of vacua. Similar to the B-model in two dimensions, local BPS operators arise from polyvector fields on Recall that the B-model with target space has

as its space of local observables [44]. We will consider the case when the moduli space can be described as an affine cone over a base . Then the local operators arise from the polyvector fields

where is a line bundle over In the examples we consider of algebraic varieties embedded in projective space, is the pull-back of on the ambient projective space. This picture of local operators is described in more detail by Beasley and Witten [3].

In the case that is a homogeneous space, we can evaluate the relevant cohomology groups using a generalization of the Borel-Weil-Bott theorem. The first step is to express the tangent bundle of as a homogeneous bundle. Although not strictly necessary for the logical development, we make a brief detour to explain a physical description of the tangent bundle to the Grassmannian in SQCD.

### Homogeneous bundles and Sqcd

The Grassmannian can be viewed as the space of complex -planes in The Grassmannian has a natural vector bundle called the universal subbundle which is the subbundle of whose fiber at each point is the subspace . The universal quotient bundle is the quotient bundle . The subbundle and the quotient bundle fit into the tautological exact sequence

on the Grassmannian The tangent bundle of the moduli space arising from fluctuations about a generic point on recovers the algebraic geometry description of the tangent bundle as

Alternatively, can be described by the fluctuations about a fixed supersymmetric vacuum. Up to gauge and global symmetry transformations, the classical moduli space of supersymmetric vacua has the form

where is an arbitrary complex number. Unlike the gauge invariant description in terms of the mesons this description explicitly depends on a choice of gauge. When is non-zero, the expectation values of break the global symmetry group from to the subgroup The gauge group is completely Higgsed. We recover that the moduli space of vacua is the quotient space

The massless fluctuations of the quarks about the vacuum transform in representations of the unbroken gauge group and are listed in table 6.

The field represents a tangent vector to the moduli space Using the Kähler metric, it can be converted to a holomorphic one-form on It transforms in the bundle

### Index of polyvector fields

The description of the tangent bundle to the moduli space as the homogeneous bundle is precisely what is needed in computation of the space of polyvector fields using a generalization of the Borel-Weil-Bott theorem. This theorem describes the polyvector fields as representations of the global symmetry group. We typically list only the dimensions of the representations, but stress that the actual representations can easily be determined.

Let by the line bundle on which is the restriction of of the ambient projective space. Then for any coherent sheaf on we denote . The Euler character is a polynomial in called the Hilbert polynomial. For these polynomials precisely count the contributions of local operators. However, for small some of the cohomology classes are not actually realized as local operators. The unitarity bound excludes these cohomology elements from representing local operators. However, we will see later that they can correspond to Beasley-Witten F-terms.

### Polyvector fields on Severi varieties

We define the alternating sum of Euler characters of polyvector fields on the Severi varieties to be

The restriction of implements the unitarity bound.

For Severi varieties, this alternating sum of Euler characteristics is

(6.1) |

where is the number of chiral multiplets. In terms of the plethystic logarithm

## 7 Superconformal index from the moduli space

In this section, we will explain how the local operators constructed from the moduli space of vacua can be related to the superconformal index. First, we review the definition of the superconformal index following [15]. Then we will propose a formula for the superconformal index in terms of the index of polyvector fields on the moduli space of vacua.

### Superconformal index review

The superconformal index is defined as

where the trace is taken over the Hilbert space of the theory quantized on By the operator-state correspondence, the index can also be viewed as a trace over the Hilbert space of local operators. In the formula is the fermion number, are the left and right spins, and is the operator scaling dimension. If the theory has flavor symmetries, then we can introduce an equivariant index with flavor fugacities for the flavor symmetry charges For a theory with a UV Lagrangian description with gauge group and chiral multiplets in the representation of the index is given by the following integral

over with respect to the Haar measure . The integrand is the plethystic exponential of the single letter index

(7.1) | ||||

(7.2) |

where and are the charges of the chiral multiplets. The first contribution is from the vector multiplet accounts and is the adjoint character of the gauge group. Similarly, are the characters of the representations of the gauge group and are the characters of the representations of the chiral multiplets in the flavor symmetry group. The plethystic exponential is defined by

and can similarly be extended to several variables.

### Superconformal index from the moduli space

All of the short multiplets contributing to the superconformal limits can be viewed as special cases of the multiplets with possibly unphysical spins and -charges. A multiplet satisfies the shortening condition [45]. Conserved currents reside in multiplets and have Defining , the contribution of a multiplet to the index is

For a fixed set of quantum numbers , the net degeneracy is defined to be the number of short multiplets with integer spin minus the number of short multiplets with half integer spin. The net degeneracy can be determined from the superconformal index using the relation [46]

For quantum numbers , the net degeneracy is

which gives the dimension of the conformal manifold [30]. In general we expect contributions not just from the tangent bundle but from all exterior powers However, these operators are no longer sufficient to account for all of the contributions to the net degeneracy. From experience with holographic duality, it is natural to expect that the operators arising from polyvector fields generate a Fock space of operators that contribute to the index.

### Superconformal index from polyvectorfields

Given the index of polyvector fields we propose that the superconformal index can be expressed as

where the correction terms arise from massless matter at the singularity.

For the cones over the Severi varieties using from Equation (6.1)

We therefore recover the superconformal index

In general, there is not an exact match. For SQCD with four flavors,

while the index constructed from polyvector fields is

The two indices are remarkably close, but already differ by differ at their term. The extra massless degrees of freedom at the origin of the moduli space are responsible for the failure of equality of the two indices. By taking the difference of the two indices, we can isolate the new extra massless degrees of freedom at the origin of moduli space. Alternatively, it is an intriguing problem to take into account polyvector fields localized at the origin to give an exact formula for the superconformal index. In the next section we will explain the computation of these indices in more detail.

## 8 Polvectorfields on moduli spaces of vacua

### Polyvector fields on and

The polyvector fields on the Veronese embedding of in are listed in table 7. They polyvector fields that are sections of are in the representation of The sections of are in representation of in the notation of LiE. The sections of transform in the representations of

dim | ||||
---|---|---|---|---|

0 | 1 | 0 | 0 | 1 |

1 | 6 | 0 | 0 | 6 |

2 | 15 | 0 | 0 | 15 |

3 | 28 | -8 | 0 | 20 |

4 | 45 | -24 | 0 | 21 |

5 | 66 | -48 | 3 | 21 |

6 | 91 | -80 | 10 | 21 |

For the Segre embedding of the polyvector fields transform in representations of and are listed in table 8.

dim | ||||||
---|---|---|---|---|---|---|

0 | 1 | 0 | 0 | 0 | 0 | 1 |

1 | 9 | 0 | 0 | 0 | 0 | 9 |

2 | 36 | 0 | 0 | 0 | 0 | 36 |

3 | 100 | -16 | 0 | 0 | 0 | 84 |

4 | 225 | -90 | 0 | 0 | 0 | 135 |

5 | 441 | -288 | 9 | 0 | 0 | 162 |

6 | 784 | -700 | 84 | 0 | 0 | 168 |

7 | 1296 | -1440 | 315 | 0 | 0 | 171 |

8 | 2025 | -2646 | 828 | -36 | 0 | 171 |

9 | 3025 | -4480 | 1785 | -160 | 1 | 171 |

10 | 4356 | -7128 | 3384 | -450 | 9 | 171 |

### Polyvector fields on

### Polyvector fields on the Cayley plane

dim | |||||||

0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

1 | 27 | 0 | 0 | 0 | 0 | 0 | 27 |

2 | 351 | 0 | 0 | 0 | 0 | 0 | 351 |

3 | 3003 | -78 | 0 | 0 | 0 | 0 | 2925 |

4 | 19305 | -1728 | 0 | 0 | 0 | 0 | 17577 |

5 | 100386 | -19305 | 0 | 0 | 0 | 0 | 81081 |

6 | 442442 | -146432 | 2925 | 0 | 0 | 0 | 298935 |

7 | 1706562 | -853281 | 51975 | 351 | 0 | 0 | 905607 |

8 | 5895396 | -4088448 | 494208 | 0 | 0 | 0 | 2301156 |

9 | 18559580 | -16812796 | 3309696 | -70070 | -650 | 0 | 4985760 |

The Cayley plane where is the parabolic subgroup corresponding to the root of The Dynkin diagram and labelling of roots for is shown in Figure 1. The root is dual to the weight The first three tensors powers of tangent bundle of are [47]

The Euler characteristics of polyvector fields on the Cayley plane are listed in table 9.

### Hilbert series and free resolutions

A striking feature of the table of local operators, is that there is a diagonal of zeros. This motivates the idea of slicing the table diagonally instead of horizontally or vertically. The first non-trivial diagonal is closely related to a free resolution of the structure sheaf. For the Veronese embedding of in we have the (graded) free resolution

of [48]. From the free resolution of , we can determine the Hilbert series of the Veronese embedded

For Segre embedded the free resolution is constructed in [49] using the methods of [50]. The free resolution of the Grassmannian is described in [51, 52]. For the complex Cayley plane the free resolution is given in Lemma 7.2 of [53].

The Hilbert series of the Severi varieties is also

where is the weight of the defining representation of the corresponding Wess-Zumino model. Then is the representation with weight This representation geometrically is the space of sections of the line bundle

The Hilbert series of the Severi varieties are listed in Table 10 using results from [54]. The Hilbert series can also be expressed in the form

For the first three Severi varieties, the terms in the numerator determines its minimal free resolution. It is natural to conjecture that this also holds for The numerators of and are listed in Table 11. The corresponding betti tables of the minimal free resolutions are

for and

for .

Intriguingly the dimension of the cohomology groups along one of the diagonals precisely matches the number of simplices in the conjectured minimal triangulation of the projective plane [55]^{2}^{2}2The author would like to thank Sergey Galkin for this observation..

## 9 The surprise revisited

Dimofte and Gaiotto argued that four dimensional SQCD with eight chiral multiplets defines an superconformal invariant boundary condition for a five dimensional theory of 28 free hypermultiplets [27]. One strong hint for the symmetry is that there are 72 dual descriptions of the SQCD theory corresponding to elements of the quotient of Weyl groups [16]. We will find that the index of polyvector fields on the moduli space of vacua of SQCD with eight chiral multiplets can be expressed in terms of the Hilbert series of the Freudenthal variety . As a consequence, our conjectured form of the index in terms of polyvector fields also predicts the enhancement of the superconformal index.

Similarly three dimensional SQCD with six chiral multiplets defines an superconformal invariant boundary condition for 16 four dimensional free hypermultiplets [27]. Again we will see that the index of polyvector fields can be expressed in terms of the Hilbert series of the spinor variety

Quite remarkably the enhanced symmetry groups and occur in the third row of the Freudenthal magic square. Their corresponding Landsberg-Manivel projective geometries are the spinor variety and the Freudenthal variety.

### Polyvector fields on

dim | |||||||

0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

1 | 28 | 0 | 0 | 0 | 0 | 0 | 28 |

2 | 336 | -63 | 0 | 0 | 0 | 0 | 273 |

3 | 2520 | -1280 | 36 | 0 | 0 | 0 | 1276 |

4 | 13860 | -12474 | 1890 | 0 | 0 | 0 | 3276 |

5 | 60984 | -80640 | 26180 | -1280 | 36 | 0 | 5280 |

6 | 226512 | -396396 | 205920 | -30030 | 336 | -63 | 6279 |

The moduli space of vacua of four dimensional SQCD with four flavors is an affine cone over the Grassmannian The theory does not have a dual magnetic description as a Wess-Zumino model. Instead it has a rich duality web. Nevertheless, we will see that we can still determine a rich set of operators from its moduli space of vacua. The calculation of cohomologies of local operators is practically identical to the case of again using the Borel-Weil-Bott theorem. The first few cohomology groups are shown in table 12 and more cohomology groups are listed in table 16 in the appendix. The first few terms in the betti table of the minimal free resolution are

and as representations of the betti table is

where we have listed only the first few terms.

The r-charges of the chiral multiplets are so the alternating sum of polyvector fields takes the form

The alternating sum of Euler characters of polyvector fields on can be written as

where

The polynomial is the numerator of the Hilbert series

of the Freudenthal variety [54] where is the parabolic subgroup corresponding to the root of

Taking the plethystic logarithm of the Hilbert series, we find