# Elliptic Integrable Systems and Modularity

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## The Semi-Classical Configurations and the Classification Problem

In this section, we discuss how the classification of vacuum expectation values for the adjoint scalars in the chiral multiplets of N = 1∗ reduces to the problem of the classification of nilpotent orbits of the complexified Lie algebra g of the gauge group. The idea of using nilpotent orbits in the context of classifying vacua in N = 1∗ theory was mentioned in [90]. See also [91] for an application of nilpotent orbit theory to a supersymmetric index calculation.

### Semi-classical Configurations and sl(2) algebras

Our starting point is the N = 1∗ super Yang-Mills theory with compact gauge group G on R4. As we explained in section 2.3, the vacua are classified by solving the F-term equations for constant fields, and dividing the solution space by the complexified gauge group GC. The equations dictate that the (rescaled) adjoint scalar fields ˜Φi (where i ∈ {1, 2, 3}) form an sl(2) algebra, see equation (2.13). Thus, the scalars provide us with a map from an sl(2) algebra into the complexified Lie algebra g of the gauge group. To find the supersymmetric vacua, we are to classify all sl(2) triples inside the Lie algebra g, up to gauge equivalence. Configurations are gauge equivalent if they are mapped to each other by the adjoint action of the complexified gauge group GC on the Lie algebra g. Thus, our first step is to review what is known about the classification of inequivalent sl(2) triples embedded in the adjoint representation.

#### The Gauge Group, Triples and Nilpotent Orbits

From now on, we will denote the complexified gauge group as GC ≡ G. We need to make a distinction between various groups that have the same Lie algebra. One canonical group associated to the Lie algebra g is the adjoint group Gad = Aut(g)o, namely the identity component of the group of automorphisms of the Lie algebra g. The adjoint group Gad is alternatively characterized by the fact that it is the group with algebra g and trivial center. We can now lay the groundwork for the first classification problem. Note that amongst our complex adjoint fields ˜Φi, we can identify a linear combination ˜Φ+ which is nilpotent by the equations of motion. Indeed, we can consider the complex combinations ˜Φ ± = ±˜Φ1 + i˜Φ2 and ˜Φ 0 = 2i˜Φ3. Then the non-vanishing commutation relations amongst these fields are h ˜Φ 0,˜Φ+ i = 2˜Φ h + ˜Φ 0,˜Φ − i = −2˜Φ h − ˜Φ +,˜Φ − i = ˜Φ0.

The Supersymmetric Index for the Classical Groups

We have determined the set of inequivalent semi-classical configurations for the adjoint scalar fields ˜Φi, as well as the subgroup of the gauge group that is left unbroken by the vacuum expectation values. To compute the semi-classical number of massive vacua, we compute the Witten indices of the pure N = 1 theories that arise upon fixing a given semi-classical configuration for the fields ˜Φi, and add them up for all possible inequivalent semi-classical configurations. The global properties of the unbroken gauge group come into play at this stage – we have already stressed that generically, the unbroken gauge group will not be connected (even if we started out with a connected gauge group). We start with a demonstration of how to take into account this complication in the elementary case of the groups O(4) and SO(4). With this example in mind, we can generalize this third step and compute the supersymmetric index for all N = 1∗
theories with classical gauge groups.

The Counting for the Exceptional Groups

In this section, we count the number of massive vacua for mass-deformed N = 4 super Yang-Mills theory with exceptional gauge group. For determining the centralizer subgroups, we assume that our gauge group is the adjoint group G = Gad. The supersymmetric indices for other choices of centers are identical.

The Orbits and the Centralizers

The first two steps in our program consist of listing the nilpotent orbits, in bijection with the sl(2) triples, and the centralizers, the subgroup of the gauge group left unbroken by the adjoint scalar field vacuum expectation values. While there is a handy list of nilpotent orbits (see e.g. [93]) of exceptional Lie algebras available, we need to delve slightly deeper into the mathematics to understand the centralizers of the associated triples.
Let n be a nilpotent element of a Lie algebra g and trip the span of an sl(2) triple corresponding to n. The centralizer of the triple is reductive (i.e. semi-simple plus abelian factors) and is a factor in the centralizer of the nilpotent element CG(n) = CG(trip) ⋉ U where U is the unipotent radical of CG(n). From chapter 13 of [93] we can read off both the type and the component group of CG(trip) (called C there), which is almost all we need to compute the contribution to the Witten index for a given orbit. However, we still have to know precisely how the component group of CG(trip) acts on its connected components. These are final gauge equivalence identifications that we will need to perform on the vacua of effective pure N = 1 supersymmetric Yang-Mills theories. These actions can be deduced from the detailed reference [95].

0 Présentation
0.1 Difficultés en Théorie Quantique des Champs
0.2 Théories de Yang-Mills en quatre dimensions
0.2.1 Dualité électromagnétique classique
0.2.2 Physique non-perturbative
0.2.3 Phases des théories de Yang-Mills
0.2.4 Structure globale du groupe de jauge
0.3 Théories supersymétriques et modularité
0.3.1 La théorie maximalement supersymétrique
0.3.2 De la S-dualité à l’invariance modulaire
0.3.3 La structure des vides
0.3.4 Compactification sur le cylindre
0.4 Les vides des théories N = 1∗
1 Presentation
1.1 Yang-Mills Gauge Theories in Four Dimensions
1.1.1 Classical Electric-Magnetic Duality
1.1.2 Non-perturbative Physics
1.1.3 Phases of Yang-Mills Theories
1.1.4 Line Operators
1.1.5 The Global Structure of the Gauge Group
1.2 Supersymmetric Gauge Theories and Modularity
1.2.1 The maximally supersymmetric theory
1.2.2 From S-duality to modular invariance
1.2.3 The vacuum structure
1.2.4 Argyres-Douglas points
1.2.5 Compactification on a cylinder
1.3 Vacua of the N = 1∗ Theory: a Summary
2 Massive Vacua of N = 1∗ Theory on R4
2.1 Introduction
2.2 The N = 1∗ Theory on R4
2.2.1 The N = 4 Gauge Theory
2.2.2 The Mass Deformation
2.2.3 Field Theoretic Properties of N = 1∗
2.3 The Vacuum Structure
2.3.1 Classical Vacua
2.3.2 The Witten Index
2.3.3 Quantum Vacua
2.3.4 Modularity of the Theory
2.4 The Semi-Classical Configurations and the Classification Problem
2.4.1 Semi-classical Configurations and sl(2) algebras
2.4.2 The Gauge Group, Triples and Nilpotent Orbits
2.4.3 Nilpotent Orbit Theory
2.4.4 Number of Partitions
2.4.5 The Centralizer and the Index
2.5 The Counting for the Classical Groups
2.5.1 The Nilpotent Orbits
2.5.2 The Centralizers
2.5.3 The Supersymmetric Index for the Classical Groups
2.5.4 The Generating Functions
2.5.5 Illustrative Examples
2.6 The Counting for the Exceptional Groups
2.6.1 The Orbits and the Centralizers
2.6.2 The Supersymmetric Index for the Exceptional Groups
3 Elliptic Integrable Systems and Modularity
3.1 Introduction
3.2 The Calogero-Moser system
3.2.1 General Definitions
3.2.2 Why Calogero-Moser Systems ?
3.2.3 Classical and Quantum Integrability
3.2.4 Twisted and Untwisted Elliptic Calogero-Moser Models
3.2.5 Complexified Models
3.2.6 The Symmetries of the Potential
3.3 Elliptic Integrable Systems and Modularity
3.3.1 Langlands Duality
3.3.2 Langlands Duality at Rank Two
3.4 Semi-Classical Limits of Elliptic Integrable Systems
3.4.1 Calogero-Moser systems and Toda systems
3.4.2 The Dual Affine Algebra and Non-Perturbative Contributions
3.4.3 Semi-Classical Limits
3.4.4 The Trigonometric, Affine Toda and Intermediate Limits
4 Isolated Extrema of the Twisted Elliptic Calogero-Moser System
4.1 The Case AN−1 = su(N)
4.2 The B,C,D Models
4.3 The Case C2 = sp(4) = so(5) and Vector Valued Modular Forms
4.3.1 The Positions of the Extrema
4.3.2 Series Expansions of the Extrema
4.3.3 Modular Forms of the Hecke Group and the Γ0(4) Subgroup
4.3.4 A Remark on a Manifold of Extrema
4.4 The Case D4 = so(8) and the Point of Monodromy
4.4.1 The Singlet
4.4.2 The Triplet
4.4.4 The Duodecuplet and a Point of Monodromy
4.4.5 The List of Extrema for so(8)
4.5 The Dual Cases B3 = so(7) and C3 = sp(6)
4.5.1 Exact Multiplets
4.5.2 The Duodecuplet, the Quattuordecuplet and the Points of Monodromy .
4.5.3 The List of Extrema for so(7) and sp(6)
4.6 Partial Results for Other Lie Algebras
4.7 Conclusions
5 The N = 1∗ Gauge Theories on R3 × S1
5.1 Supersymmetric gauge theories and integrable systems
5.1.1 Seiberg-Witten curves for N = 2 theories
5.1.2 Integrable Systems and N = 2 theories
5.1.3 The Calogero-Moser poteneial and N = 2∗
5.1.4 Breaking N = 2 to N = 1
5.2 The non-perturbative superpotential
5.2.1 A Strategy : Compactification
5.2.2 The gauge field on the cylinder
5.2.3 The Non-perturbative Superpotential for N = 1 theories
5.2.4 Electric-Magnetic Duality
5.3 The Massive Vacua of N = 1∗ gauge theories
5.3.1 Comparison with Vacua on R4
5.3.2 Tensionless Domain Walls, Colliding Quantum Vacua and Masslessness .
6 The SU(N) theory
6.1 Introduction
6.2 Semi-Classical Preliminaries
6.3 The Gauge Algebra su(3), the Massless Branch and the Singularity
6.3.1 Semi-classical analysis
6.3.2 The Massless Branch and the Singularity
6.3.3 The Massless Branch in the Toroidal Variables
6.3.4 The points λ6 6= −1
6.3.5 The Z2 symmetric points
6.3.6 Dualities at the Z2 symmetric points
6.3.7 Summary Remarks
6.4 The Gauge Algebra su(4)
6.4.1 The Partition 2 + 1 + 1
6.4.2 The Partition 3 + 1
6.4.3 The Duality Diagram
6.4.4 Summary Remarks
6.5 A Word on the su(N) Theory
7 Topological properties of Groups and Lines
7.1 Introduction
7.2 The N = 1∗ Theory with Gauge Algebra so(5)
7.2.1 The Semi-Classical Analysis and Nilpotent Orbit Theory
7.2.2 The Elliptic Integrable System
7.2.3 Global Properties of the Gauge Group and Line Operators
7.2.4 Summary and Motivation
7.2.5 Representations of the Vacua for B2 Theories
7.3 More on Nilpotent Orbit Theory
7.3.1 The Nilpotent Orbit Theory of Bala-Carter and Sommers
7.3.2 The Bridge between Gauge Theory and Integrable System
7.4 Discrete Gauge Groups and Wilson Lines
7.4.1 Discrete Gauge Groups and Wilson Lines
7.4.2 The Semi-Classical Vacua for G2
7.4.3 The Elliptic Integrable System and the Semi-classical Limits
7.4.4 Results Based on Numerics
7.4.5 Langlands Duality and the Duality Diagram
7.5 The so(5) Massless Branch
7.5.1 The Local Description of the Massless Branch
7.5.2 Duality and the Massless Branch
7.5.3 The Moduli Space of Vacua for the Different Gauge Theories
A Supersymmetric gauge theories
A.1 Supersymmetry in various dimensions
A.2 Supersymmetric gauge theories in four dimensions
A.2.1 The non supersymmetric gauge theories
A.2.2 The N = 1 gauge theory
A.2.3 The N = 2 gauge theory
A.2.4 The N = 4 gauge theory
B Lie Algebra
B.1 Basic definitions
B.2 Lattices
B.3 Lie algebra data
C Modular Forms and Elliptic Functions
C.1 Modular and Automorphic forms
C.1.1 Definitions
C.1.2 Eisenstein Series
C.2 Elliptic functions
C.2.1 The Weierstrass function
C.2.2 The twisted Weierstrass functions
C.3 Theta and Eta Functions
C.4 Modular Forms and Sublattices

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