Applications of gauged linear sigma models
Zhuo Chen
Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Physics
Eric Sharpe, Chair Lara Anderson James Gray Uwe T¨auber
March 26, 2019 Blacksburg, Virginia
Keywords: Heterotic compactification, Mirror symmetry, Gauged linear sigma model, Topological field theory, Superconformal field theory Copyright 2019, Zhuo Chen Applications of gauged linear sigma models
Zhuo Chen
(ABSTRACT)
This thesis is devoted to a study of applications of gauged linear sigma models. First, by constructing (0,2) analogues of Hori-Vafa mirrors, we have given and checked proposals for (0,2) mirrors to projective spaces, toric del Pezzo and Hirzebruch surfaces with tangent bundle deformations, checking not only correlation functions but also e.g. that mirrors to del Pezzos are related by blowdowns in the fashion one would expect. Also, we applied the recent proposal for mirrors of non-Abelian (2,2) supersymmetric two-dimensional gauge theories to examples of two-dimensional A-twisted gauge theories with exceptional gauge
groups G2 and E8. We explicitly computed the proposed mirror Landau-Ginzburg orbifold and derived the Coulomb ring relations (the analogue of quantum cohomology ring relations). We also studied pure gauge theories, and provided evidence (at the level of these topological- field-theory-type computations) that each pure gauge theory (with simply-connected gauge group) flows in the IR to a free theory of as many twisted chiral multiplets as the rank of the gauge group. Last, we have constructed hybrid Landau-Ginzburg models that RG flow to a new family of non-compact Calabi-Yau threefolds, constructed as fiber products of genus g curves and noncompact K¨ahlerthreefolds. We only considered curves given as branched double covers of P1. Our construction utilizes nonperturbative constructions of the genus g curves, and so provides a new set of exotic UV theories that should RG flow to sigma models on Calabi-Yau manifolds, in which the Calabi-Yau is not realized simply as the critical locus of a superpotential. Applications of gauged linear sigma models
Zhuo Chen
(GENERAL AUDIENCE ABSTRACT)
This thesis is devoted to a study of vacua of supersymmetric string theory (superstring theory) by gauged linear sigma models. String theory is best known as the candidate to unify Einstein’s general relativity and quantum field theory. We are interested in theories with a symmetry exchanging bosons and fermions, known as supersymmetry. The study of superstring vacua makes it possible to connect string theory to the real world, and describe the Standard model as a low energy effective theory. Gauged linear sigma models are one of the most successful models to study superstring vacua by, for example, providing insights into the global structure of their moduli spaces. We will use gauged linear sigma models to study mirror symmetry and its heterotic generalization “(0, 2) mirror symmetry.” They are both world-sheet dualities relating different interpretations of the same (internal) superstring vacua. Mirror symmetry is a very powerful duality which exchanges classical and quantum effects. By studying mirror symmetry and (0, 2) mirror symmetry, we gain more knowledge of the properties of superstring vacua. Acknowledgments
I would like to express my sincere gratitude to my advisor professor Eric Sharpe for his endless help and patient guidance. From you, I not only learned a great deal about string theory, but also the way to conduct research. I am very fortunate to have had the opportunity to work with you and I am really grateful for all that you have done for me. I would also like to acknowledge two other professors in the string theory group at Virginia Tech, professor Lara Anderson and professor James Gray, for all their help and valuable conversations. In addition, I appreciate all useful discussions with other graduate students and postdocs of string theory group at Virginia Tech.
Special thanks to my wife and my parents. Without your consistent supports, I wouldn’t be able to accomplish my Ph.D. study.
iv Contents
1 Introduction 1
1.1 The N = 2 superconformal algebra ...... 3
1.2 Nonlinear sigma models ...... 5
1.3 Gauged linear sigma models ...... 13
1.4 Mirror symmetry ...... 19
2 Toda-like (0,2) mirrors 26
2.1 Introduction ...... 26
2.1.1 Review of (2,2) Toda dual theories ...... 29
2.1.2 Review of (0,2) Landau-Ginzburg models ...... 31
2.2 Pn × Pm ...... 34
2.2.1 The A/2-twisted nonlinear sigma model ...... 34
2.2.2 The Toda-like mirror theory ...... 35
2.3 Example: Toda-like duals to P1 × P1 ...... 39
2.3.1 The (0,2) NLSM ...... 39
2.3.2 The Toda-like mirror theory ...... 41
2.3.3 Moduli ...... 45
2.3.4 Redundancies and equivalent descriptions ...... 45
v 2.4 Del Pezzo surfaces ...... 50
2.4.1 The first del Pezzo surface, dP1 ...... 50
2.4.2 The second del Pezzo surface, dP2 ...... 58
2.4.3 The third del Pezzo surface, dP3 ...... 70
2.5 Hirzebruch surfaces ...... 78
2.5.1 Review of the (2,2) mirror ...... 78
2.5.2 (0,2) deformations and proposed (0,2) mirrors ...... 80
2.6 Correlation functions in some examples ...... 84
2.6.1 A/2 correlation functions on P1 × P1 ...... 84
2.6.2 Toda-like dual to P1 × P1 ...... 85
2.6.3 A/2 correlation functions on P1 × P2 ...... 86
2.7 Tangent bundle moduli ...... 89
2.8 Quantum cohomology of dP1 ...... 93
3 Two-dimensional supersymmetric gauge theories with exceptional gauge groups 95
3.1 Introduction ...... 95
3.2 G2 ...... 98
3.2.1 Mirror Landau-Ginzburg orbifold ...... 98
3.2.2 Weyl group ...... 101
3.2.3 Coulomb ring relations ...... 104
3.2.4 Vacua ...... 108
3.2.5 Pure gauge theory ...... 111
3.2.6 Comparison with A model results ...... 114
3.2.7 Comparison to other bases for weight lattice ...... 116
vi 3.3 E8 ...... 117
3.3.1 Mirror Landau-Ginzburg orbifold ...... 117
3.3.2 Superpotential ...... 119
3.3.3 Coulomb ring relations ...... 128
3.3.4 Pure gauge theory ...... 144
4 Landau-Ginzburg models for certain fiber products with curves 146
4.1 Introduction ...... 146
4.2 GLSM for P2g+1[2, 2] and curves of genus g ...... 148
4.3 Hybrid Landau-Ginzburg models for fiber products ...... 149
4.3.1 Fiber products with vector bundles on P1 ...... 149
4.3.2 Fiber products with hypersurfaces in vector bundles ...... 153
4.4 Fiber products with twistor spaces ...... 154
4.4.1 Fiber product with twistor space of R4 ...... 155
2 4.4.2 Fiber product with twistor space of C /Zk ...... 156
4.4.3 Fiber product with twistor space of S1 × R3 ...... 157
4.5 Review of pertinent mathematics ...... 158
5 Conclusion 161
References 164
vii List of Figures
1.1 Mirror symmetry for Calabi-Yau manifolds ...... 22
1.2 Mirror symmetry for Fano spaces ...... 22
2.1 A toric fan of P2 can be obtained by removing the edge (0, −1) from the toric
fan of dP1...... 56
2.2 A toric fan for dP1 can be obtained by removing the edge (−1, 0) from the
toric fan for dP2...... 64
2.3 A toric fan for P1 × P1 can be obtained by removing the edge (−1, −1) from
the toric fan for dP2...... 69
2.4 A toric fan for dP2 can be obtained by removing the edge (1, 1) from the toric
fan for dP3...... 76
3.1 Roots of G2...... 99
3.2 Weights of 7 of G2...... 100
viii List of Tables
3.1 Roots and weights for G2 and associated fields...... 98
3.2 Weyl group actions on the vacua of the case n = 4 ...... 110
3.3 Weyl group actions on the vacua of five fundamental matter multiplets ... 112
3.4 Roots and weights for G2 and associated fields...... 116
3.5 First set of roots of E8 and associated fields...... 118
3.6 Second set of roots of E8 and associated fields...... 119
3.7 Third set of roots of E8 and associated fields...... 120
ix Chapter 1
Introduction
Ever since the beginning of superstring theory, the study of superstring vacua has been one of the most interesting problems in string theory. String theory predicts that the uni- verse has ten dimensions, nine space and one time. To get the four-dimensional world we observe, string theorists typically make the ansatz that the universe is a product of our four-dimensional universe and a compact six-dimensional space, the ‘internal space.’ Such an ansatz is known as a compactification of string theory, and the data defining the compact- ification correspond to a vacuum of string theory. Furthermore, to make close contact with the real world, one typically works with a compactification of a heterotic string, which de- scribes a non-Abelian gauge field in ten dimensions. Heterotic string theory was introduced by Gross, Harvey, Martinec and Rohm [5–7], based on the fact that the left-moving and right-moving modes are decoupled in closed string theories. Therefore, one can construct a closed string theory as a hybrid of two different kinds of modes which contains the features of two different theories. The right-moving modes are taken from superstrings, and exhibit supersymmetry. Then, the theory contains fermions and is free of tachyons. The left-moving modes are taken from bosonic strings, and exhibit an SO(32) or E8 × E8 current algebra, which realizes target space gauge degrees of freedom. Internal consistency of this hybrid construction at the quantum level is ensured by the Green-Schwarz anomaly cancellation condition,
ch2(E) = ch2(TX), where TX is the tangent bundle of the internal space X and E is a vector bundle on X corresponding to the gauge group. In particular, heterotic string theory flows to a D = 10,
1 2
N = 1 supergravity theory at low energies, coupling to supersymmetric Yang-Mills theory
with gauge group SO(32) or E8 × E8.
To compactify a heterotic string, one specifies both a (compact, six-dimensional) space, as well as a bundle on that space, satisfying certain conditions (including the Green-Schwarz anomaly cancellation condition above). In addition, for phenomenological reasons, one also typically begins by requiring that the low-energy four-dimensional theory have N = 1 su- persymmetry, which constrains the spaces and bundles. The simplest vacua are determined by two equations [8],
Dmη = 0, (1.1) ij a Γ Fijη = 0, (1.2)
where η is the supersymmetry parameter we are looking for, m = 0, ··· , 9 are the ten- dimensional spacetime coordinates, i, j are four-dimensional spacetime coordinates, and a is a gauge index. The first equation (1.1) implies that the internal manifold has SU(3) holon- omy. Together with the K¨ahlerconditions, it leads to Calabi-Yau manifolds. The second equation is a condition on the vector bundles, restricting them to be poly-stable and slope zero holomorphic vector bundles. On the other hand, N = 1 spacetime supersymmetry in four dimensions implies the existence of right-moving worldsheet N = 2 superconformal symmetry [9,10]. Therefore, perturbatively consistent superstring vacua are described by, at least, two-dimensional (0, 2) superconformal field theory. Then, in typical string compactifi- cations, we are interested in studying a product of two superconformal field theories. In fact, string theorists often generalize the compactification ansatz given above: instead of speci- fying a space and bundle, one specifies superconformal field theories. This is slightly more general, in that although every space (with bundle) defines a superconformal field theory, there exist superconformal field theories which do not correspond to spaces, and yet can also be used in string compactifications. It is also known that the total central charge should be twelve in order to have a consistent spectrum. The conformal field theory on Minkowski space is trivial with central charge c = 3 in light cone gauge. The nontrivial and interesting part is the two-dimensional superconformal field theory with either (0, 2) or (2, 2) supersym- metry for the internal Calabi-Yau manifold of central charge c = 9. Generally speaking, there are three ways to construct N = 2 superconformal field theories that we will study in this thesis. The first is nonlinear sigma models which give a geometric realization of N = 2 superconformal field theories. The second is Landau-Ginzburg models. In both types of 3
models, one obtains superconformal field theories at the endpoint of renormalization group flow. Another type consists of Kazama-Suzuki coset models which are gauged Wess-Zumino- Witten models describing strings propagating on group manifolds [12,13]. In the rest of this chapter, we will review different aspects of N = 2 superconformal field theories as well as mirror symmetry which is one of the most important tools to study properties of superstring vacua. We will focus on (2, 2) theories only which are comparably well-understood.
1.1 The N = 2 superconformal algebra
To begin with, let us first review the two-dimensional N = 2 superconformal algebra which is generated by three types of currents: the energy momentum tensor T (z) of conformal weight h = 2; the supercurrent G(z) of conformal weight 3/2; the U(1) current J(z) under which the supercurrent G(z) has charge 1. The algebra is determined by operator product expansion of those currents:
c 2T (w) ∂ T (w) T (z)T (w) ∼ 2 + + w , (z − w)4 (z − w)2 z − w 3 G(w) ∂ G(w) T (z)G(w) ∼ 2 + w , (z − w)2 z − w J(w) ∂ J(w) T (z)J(w) ∼ + w , (z − w)2 z − w (1.3) 2c 2J(w) 2T (w) + ∂ J(w) G+(z)G−(w) ∼ 3 + + w , (z − w)3 (z − w)2 z − w G(w) J(z)G(w) ∼ , z − w c J(z)J(w) ∼ 3 , (z − w)2
where c is the central charge of the theory. In terms of the Fourier mode expansions of the currents, ∑ ∑ ∑ − − − − 3 − − n 2 (n a) 2 n 1 T (z) = Lnz ,G (z) = Gnaz ,J(z) = Jnz , n n n 4
the N = 2 superconformal algebra is given by, c ( ) [L ,L ] = (m − n)L + m m2 − 1 δ , m n m+n 12 m+n,0 c [J ,J ] = mδ , m n 3 m+n,0 [L ,J ] = −mJ , n m ( m+n ) [ ] (1.4) n − Ln,Gma = (m a) Gm+na, [ ] 2 J ,G = G , n ma n+ma ( ) { } c 1 G+ ,G− = 2L + (n − m + 2a)J + (n + a)2 − δ . n+a m−a m+n n+m 3 4 m+n,0
Notice that the superconformal algebras are parametrized by a continuous parameter a ∈ [0, 1). The parameter a determines the boundary conditions of the fermions. When a = 0, the algebra corresponds to Ramond boundary conditions. When a = 1/2, the algebra corresponds to Neveu-Schwarz boundary conditions. All the algebras are isomorphic and related by spectral flow [11].
N = 2 superconformal theories have a very rich structure. Most significantly, a finite subsector of the operators, the chiral primary operators, form a ring [11]. This ring structure plays a central role in N = 2 superconformal theories. So, let’s give a brief review here. Given an algebra, one can build the unitary irreducible representations from highest weight states by acting by creation and annihilation operators. Here, the creation operators can be identified with the negative modes as they raise the L0 eigenvalue of a state and similarly annihilation operators with positive modes as they lower the L0 eigenvalue of a state. Therefore, the highest weight state satisfies,
| ⟩ | ⟩ | ⟩ Ln ϕ = 0,Gr ϕ = 0,Jm ϕ = 0, n, r, m > 0.
As usual, the states are labeled by the eigenvalues of L0 and J0,
L0|ϕ⟩ = hϕ|ϕ⟩,J0|ϕ⟩ = qϕ|ϕ⟩.
On the other hand, the highest weight state |ϕ⟩ is created by (superconformal) primary field ϕ by ϕ|0⟩ = |ϕ⟩. 5
But, what is interesting is a subset of the primary fields defined by
+ | ⟩ G−1/2 ϕ = 0, which is known as the set of chiral primary fields. Similarly, there are antichiral primary fields given by − | ⟩ G−1/2 ϕ = 0. Together with the anti-holomorphic operator G−1/2, this leads to the notion of (c, c), (a, c), (a, a) and (c, a) primary fields. For example, the (a, c) primary fields are anti-chiral in the holomorphic sector and chiral in the antiholomorphic sector. The (a, a) and (c, a) primary fields are complex conjugates of the (c, c) and (a, c) primary fields. What’s important is that under the operator product expansion the chiral primary fields are closed and form a finite closed non-singular ring, k ϕiϕj = Cijϕk, which is called the chiral ring. There are four types of rings: the (c, c), (a, c) rings and their complex conjugations. As we will see, there is a deep connection between chiral rings and cohomology rings of Calabi-Yau manifolds. Besides, studying those two different chiral rings led to the idea of mirror symmetry of Calabi-Yau manifolds. Moreover, the physical states of topological field theories (A/B model) obtained by the topological twists (A/B twists) turn out to be the same as the chiral primary states of (a, c) ring and (c, c) ring.
1.2 Nonlinear sigma models
Another important tool in string theory is the nonlinear sigma model (NLSM). A nonlinear sigma model is a two-dimensional quantum field theory of a map ϕ from the worldsheet Σ to the target space X. String propagation in the space-time is described by the map ϕ which is exactly the embedding of the two-dimensional surface Σ formed by the motion of the string in the spacetime X. We would be most interested in the case where the target space X is K¨ahlerdue to N = 2 worldsheet supersymmetry. The action for a nonlinear sigma models 6
is ∫ 2 i ¯¯ȷ¯ i ¯¯ȷ¯ S =2t d z(giȷ¯∂ϕ ∂ϕ + iBiȷ¯∂ϕ ∂ϕ Σ ȷ¯ i ȷ¯ i i ȷ¯ k ¯l + igiȷ¯ψ−Dzψ− + igiȷ¯ψ+Dz¯ψ+ + Riȷk¯ ¯lψ+ψ+ψ−ψ−), (1.5)
where t is the coupling constant; giȷ¯ is the K¨ahlermetric of the target space pulled back
to the worldsheet; Biȷ¯ is anti-symmetric closed two-form on the target space pulled back to
the worldsheet; Riȷk¯ ¯l is the Riemann curvature tensor on the target space pulled back to the i,ȷ¯ worldsheet. The fermions ψ have two indices. The lower indices indicate their worldsheet chirality and the upper indices i,ȷ ¯ are target space indices due to the supersymmetry. In i,ȷ¯ all, the fermions ψ are Grassmannian-valued sections of the following bundles, ( ) ( ( ) ) i 1/2 ∗ 1,0 i 1/2 ∗ 0,1 ∨ ∈ ∞ ⊗ ∈ ∞ ⊗ ψ+ ΓC KΣ ϕ T X , ψ− ΓC KΣ ϕ T X , ( ( ) ) ( ) ı 1/2 ∗ 1,0 ∨ ¯ı 1/2 ∗ 0,1 ∈ ∞ ⊗ ∈ ∞ ⊗ ψ+ ΓC KΣ ϕ T X , ψ− ΓC KΣ ϕ T X ,
with K and K the holomorphic and anti-holomorphic canonical bundles of the worldsheet
Σ. As a result, the covariant derivatives Dz, Dz¯ are given by
1 D ψi = ∂ψi + ∂ϕjΓi ψm + ω ψi , z − − jm − 2 z − 1 D ψ¯ı = ∂ψ¯ ¯ı + ∂¯ϕ¯ȷ¯Γ¯ı ψm¯ + ω ψ¯ı , z¯ + + ȷ¯m¯ + 2 z¯ + where Γ is the Christoffel connection on the target space and ωz,z¯ are spin connections asso- ciate to K, K. For completeness, let’s also write down the supersymmetry transformations
i i i δϕ = iα−ψ+ + iα+ψ−, ¯i ¯ı ¯ı δϕ = iα˜−ψ+ + iα˜+ψ−, i − i − j i m δψ+ = α˜−∂ϕ iα+ψ−Γjmψ+ , ¯ı − ¯ı − ȷ¯ ¯ı m¯ δψ+ = α−∂ϕ iα˜+ψ−Γȷ¯m¯ ψ+ , i − ¯ i − j i m δψ− = α˜+∂ϕ iα−ψ+Γjmψ− , ¯ı − ¯ ¯ı − ȷ¯ ¯ı m¯ δψ− = α+∂ϕ iα˜−ψ+Γȷ¯m¯ ψ− .
The supersymmetric transformation parameters α,α ˜ are Grassmannian-valued smooth −1/2 sections of bundle K−1/2, K . There is a more concise way to write down the action with 7 explicit supersymmetry. In terms of chiral superfields Φ (the definition will be reviewed later (1.19)), the above action (1.5) can be rewritten as, ∫ 1 S = d2zd4θK(Φ , Φ ). α′ i i
This action is determined by one single real function on the superspace, the K¨ahlerpotential K(Φ, Φ). In addition, the new action is invariant under superfield K¨ahlertransformations:
K(Φ, Φ†) → K(Φ, Φ†) + F (Φ) + F †(Φ†), which is the origin of K¨ahlergeometry. The metric is simply given by
∂ ∂ ¯ giȷ¯ = ¯ K(ϕ, ϕ). ∂ϕi ∂ϕȷ¯
As stated in the last section, N = 2 superconformal field theories are particularly important as superstring vacua. The nonlinear sigma models give geometric representations of some superconformal field theories. First, notice that classically the action (1.5) is invariant under the scaling of the worldsheet coordinates,
z → λz, with a rescaling of fermions, ψ → λ−1/2ψ.
However, the scale invariance may be spoiled by the quantum corrections. To be a conformal field theory at the quantum level, the beta function must vanish. In the current case, the beta function is proportional to the Ricci tensor of target space X,
βµν ∝ Rµν.
The vanishing of the beta function implies the vanishing of the Ricci tensor. Combined with the existence of a K¨ahlerstructure, this implies that the target space X is a Calabi-Yau manifold. In other words, a nonlinear sigma model on a Calabi-Yau manifold will flow to a superconformal field theory in the IR, a fixed point of RG flow. The central charge of the corresponding superconformal field theory is determined by the complex dimension d of the 8
target Calabi-Yau space. The d complex bosonic field contribute 2d to the central charge and their superpartners, the d fermionic fields, contribute d. Together, we have
c = 3d.
The nonlinear sigma model gives a geometric realization of this N = 2 superconformal field theory. Thus, one should be able to find geometric counterparts for basic ingredients of N = 2 superconformal field theory in nonlinear sigma models. First, the currents in the algebra (1.3) can be realized as,
1 1 T = −g ∂ ϕi∂ ϕȷ¯ + g ψi∂ ψȷ¯ + g ψȷ¯∂ ψi, iȷ¯ z z 2 iȷ¯ z 2 iȷ¯ z + 1 i ȷ¯ G = giȷ¯ψ ∂zϕ , 2 (1.6) 1 G− = g ψȷ¯∂ ϕi, 2 iȷ¯ z 1 J = g ψiψȷ¯. 4 iȷ¯
In addition, the chiral rings are associated with the cohomology rings of the target Calabi-Yau manifold in a nonlinear sigma model. More precisely, the chiral primary operator in (a, c) ring 0,s ∧r ∨ with U(1) charge (r, s) correpsond to elements of H∂¯ (X, TX ) and the chiral primary 0,s ∧r operator in (c, c) ring with U(1) charge (r, s) correpsond to elements of H∂¯ (X, TX). Suppose the target space is a Calabi-Yau threefold, then the (0, s) forms valued in ∧rTX∨ are in one-to-one correspondence with (3−r, s) forms on the Calabi-Yau threefold by contracting with the holomorphic (3, 0)-form.
There are also two important subsectors of nonlinear sigma models, corresponding to the A and B model topological field theories. They are obtained by a procedure called topological twisting and the resulting theories turn out to be topological field theories [14, 15]. At the level of superconformal field theory, the two twistings correpond to restricting to operators in the (c, c) and (a, c) rings, respectively. From the definitions of the chiral rings, it is natural to consider − + QA = G−1/2 + G−1/2, whose cohomology contains physical states corresponding to the (a, c) ring, and
+ + QB = G−1/2 + G−1/2 9
whose cohomology contains physical states corresponding to the (c, c) ring. The fact that those operators are fermionic and are nilpotent
2 2 QA = QB = 0,
implies the theories of each subsector will be topological field theory. However, the energy momentum tensor is not exact with respect to either operator and the BRST operators are not globally defined. Therefore, the topological twists are introduced and the energy mo- mentum tensors are modified by the U(1) current J. For A twist, the new energy momentum tensor is 1 1 T = T + ∂J, T = T − ∂J, A 2 A 2 which correspond to (a, c) ring. For B twist, the new energy momentum tensor is given similarly by 1 1 T = T − ∂J, T = T − ∂J, B 2 B 2 which correspond to (c, c) ring. The geometric realization of the above structure gives the A-model if restricts to the (a, c) ring and B-model if restricts to the (c, c) ring.
A model
On a nonlinear sigma model with target space a Calabi-Yau manifold, both topological twists exist and modify the worldsheet fermions. For the A-twist, the worldsheet fermions are sections of, ( ( )) ( ( )) i i ∗ 1,0 ¯ı ¯ı ∗ 0,1 ψ ≡ χ ∈ Γ ϕ T X ψ− ≡ χ ∈ Γ ϕ T X , + ( ( )) ( ( )) (1.7) ¯ı ≡ ¯ı ∈ ∗ ⊗ 0,1 i ≡ i ∈ ∗ ⊗ 1,0 ψ+ ψz Γ ϕ K T X ψ− ψz¯ Γ ϕ K T X .
The fermions χi and χ¯ı combine into a worldsheet scalar and the other two fermions are worldsheet holomorphic and antiholomorphic one-forms. The resulting theory is an example of a topological field theory known as the A model, and the action is given by, ∫ ( ( ) ) 2 i ¯ı i ¯ı ¯ı i i χ − i ¯ı j ȷ¯ S = 2t d z gi¯ı ∂zϕ ∂z¯ϕ + ∂z¯ϕ ∂zϕ + iψz∂z¯χ gii + iψz¯∂zχ g¯ıi Ri¯ıjȷ¯ψz¯ψzχ χ . (1.8) Σ 10
By setting α+ = αe− = 0 and α = α− =α ˜+, the supersymmetry transformations are simplified,
δϕi = iαχi, δϕ¯ı = iαχ¯ı, δχi = δχ¯ı = 0, (1.9) ¯ı − ¯ı − ȷ¯ ¯ı m¯ δψz = α∂zϕ iαχ Γȷ¯m¯ ψz , i − i − j i m δψz¯ = α∂z¯ϕ iαχ Γjmψz¯ .
Notice that the supersymmetry generator is a BRST operator,
QA = Q+ + Q−, and as mentioned before, it is a scalar operator on the worldsheet. The observables of interest are BRST-invariant operators which can be built from products of χ. Actually, there is a one-to-one correspondence between BRST-closed operators and closed differential forms on the target space, ψi ↔ dzi, χ¯ı ↔ dz¯¯ı, with
QA ↔ d = ∂ + ∂.
Moreover, the BRST cohomology is isomorphic to the de Rham cohomology of the target space. The action can be rewritten as, ∫ ∫ S = it d2z{Q, V } + t ϕ∗(K), (1.10) Σ Σ
where ( ) i j i j V = gij ψz∂zϕ + ∂zϕ ψz ,
and ∫ ∫ ( ) ∗ 2 i j − i j Φ (K) = d z ∂zϕ ∂zϕ gij ∂zϕ ∂zϕ gij . Σ Σ This simple rewriting reveals some crucial properties of the A model. First, notice that K is the K¨ahlerform of the target space. Also, it can be shown that all information about the complex structure of the target space is buried in the term V . Therefore, if the theory is only deformed by a change of complex structure, the correlation functions only pick up an extra 11
Q-exact term. As a result, all correlation functions remain unchanged. Thus, the A model is independent of the choice of complex structure on the target space and it only depends on the choice of K¨ahlerstructure. On the other hand, path integrals for all correlation functions and also partition function localize onto the zero modes of the action which are described by holomorphic maps ∂ϕ¯ = 0.
In general, the moduli space of holomorphic maps will be a disjoint union
⨿dMd of spaces correponding to holomorphic maps of different degrees. For example, when the degree d = 0, M0 is given by all degree zero maps which are just constant maps and M0
is the same as the target space. The correlation functions of BRST-closed operators Oi are
given by integrals of wedge products of corresponding differential forms ωi on the moduli i ¯ı space Md for each degree. Then, for cases that there are only χ and χ zero modes, the correlation functions are given by ∑ ∫ ⟨O1 ···On⟩ = ωq ∧ · · · ∧ ωn, M d d
which is non-vanishing only if the corresponding wedge product is a top form. More generally, ¯ı i one can cancel out the ψz and ψz¯ zero modes in the path integral by utilizing the four-fermi interaction term i ȷ¯ k ¯l Riȷk¯ ¯lψ+ψ+ψ−ψ−, and the correlation functions are ∑ ∫ ⟨O1 ···On⟩ = ω1 ∧ ... ∧ ωn ∧ e(n), (1.11) d Md
where e(n) is an Euler class, representing zero modes of the worldsheet 1-form fields. The sectors of d > 0 are called worldsheet instanton sectors which are quantum corrections to the correlation functions. The geometric realization of the (a, c) chiral ring should also take the effects of instanton sectors into account. Thus, the classical chiral ring is modified and becomes the quantum cohomology ring. For example, consider the A model on Pn. All BRST-cohomology classes of local operators are generated by a single operator ϕ, corre- 12 sponding to a degree-two cohomology class on Pn, with correlation functions of the form
⟨ϕn⟩ = 1, ⟨ϕ2n+1⟩ = q, (1.12) ⟨ϕn+d(n+1)⟩ = qd, and the quantum cohomology ring relation is ϕn+1 = q.
B model
The B model is similarly defined. The worldsheet fermions, due to the topological twist, become sections of, ( ( )) ¯ı ∗ 0,1 ψ ∈ Γ ϕ T X , ( ( )) ψi ∈ Γ K ⊗ ϕ∗ T 1,0X , (1.13) + ( ( )) i ∗ 1,0 ψ− ∈ Γ K ⊗ ϕ T X .
The action can be written with a convenient field redefinitions,
i i i i ψ+ = ρz, ψ− = ρz¯, ¯ı ¯ı ¯ı ¯ı − ¯ı η = ψ+ + ψ−, θi = gi¯ı(ψ+ ψ−), as follows: ∫ ( ( ) ( ) 2 i ¯ı i ¯ı ¯ı i i S =t d z gi¯ı ∂zϕ ∂z¯ϕ + ∂z¯ϕ ∂zϕ + iη Dzρz¯ + Dz¯ρz gi¯ı+ Σ ( ) ) (1.14) i − i i j ¯ı kȷ¯ + iθi Dz¯ρz Dzρz¯ + Riijȷ¯ρzρz¯η θkg .
By setting α+ = α− = 0 andα ˜+ =α ˜− = α, the supersymmetry transformations are
δϕi = 0, δϕ¯ı = iαη¯ı, (1.15) ¯ı δη = δθi = 0, δρi = −αdϕi. 13
Similar to the A model, there is a one-to-one correspondence
∂ η¯ı ↔ d¯z¯ı, θ ↔ ,Q ↔ ∂, i ∂zi B so the BRST cohomology can be identified with the sheaf cohomology group ( ) Hp X, ΛqT 1,0X .
The action can be rewritten as ∫ S = it {Q, V } + tW, where ( ) i ȷ¯ i ȷ¯ V = giȷ¯ ρz∂z¯ϕ + ρz¯∂zϕ , and ∫ ( ) i i i j ¯ı kȷ¯ W = −θiDρ − Ri¯ıjȷ¯ρ ∧ ρ η θkg . Σ 2 This implies that the B model is independent of K¨ahlerdeformations and only depend on the complex deformations of target space. As in the A model, the correlation functions of BRST-closed operators will be of the form ∑ ∫ ⟨O1 ···On⟩ = ω1 ∧ · · · ∧ ωn, M d d
p q where the corresponding differential forms ωi ∈ H (Md, Λ T Md). However, since B model is K¨ahler-deformations-independent, one can take the large radius limit of the target space. The B model reduces to its weakly coupling limit and becomes purely classical. As a result, only the d = 0 sector contributes to correlation functions, ∫
⟨O1 ···On⟩ = ω1 ∧ · · · ∧ ωn. X
1.3 Gauged linear sigma models
Gauged linear sigma models, ever since they were initially constructed by Witten [18], have been extraordinarily successful models for the study of superstring vacua. They not only pro- 14
vide a global description of nonlinear sigma models which allow us to study non-perturbative aspects of sigma models, but also enable and simplify numerous computations. In nonlinear sigma models, the kinetic terms are of the form
i µ j gij(ϕ)∂µϕ ∂ ϕ .
The non-linearity is reflected in the target space metric gij(ϕ) which depends on the map ϕ. A perturbative expansion of the kinetic term above will lead to infinite interaction terms proportional to powers of 1/r. Therefore, perturbation theory is only valid when the radius r of the target space is large so that all the interaction terms are suppressed, the theory is weakly coupled and close to being a free field theory. If r is small, then the theory is strongly coupled and the renormalization flow is difficult to analyze. The basic idea of GLSMs is to describe nonlinear sigma models as the low energy limit of a linear theory with non-trivial potential. The two theories are equivalent in the sense that they have the same IR fixed point which means they flow to the same superconformal field theory in the IR limit. The advantages of gauged linear sigma models are that they do not contain nonlinear interactions, and moreover they give information about the global structure of the moduli space.
Now, let’s review the construction of gauged linear sigma models and set notation. Note that here we will only consider gauged linear sigma models with Abelian gauge groups. We take x0,1 to be worldsheet coordinates. In the light-cone coordinates, we have
1 0 1 1 x = (x x ), ∂ = (∂ ∂ ). (1.16) 2 2 0 1
There are also four superspace coordinates θ+, θ−, θ¯+, θ¯− which are related by complex ¯ † conjugation θ = (θ ) . The spin indices raise and lower by the antisymmetric tensor ϵij
with ϵ+− = 1/2 and ϵ−+ = −1/2. There are four supercharges (supersymmetry generators)
for N = (2, 2) theories Q and Q,
∂ − ∂ − Q = + iθ ∂, Q = iθ ∂. (1.17) ∂θ ∂θ
The only non-vanishing anti-commutator of supercharges is
{Q, Q} = −2i∂. 15
The supercovariant derivative D, D are derivatives on the superspace,
∂ − − ∂ D = iθ ∂, D = + iθ ∂. (1.18) ∂θ ∂θ
They anti-commute with supercharges { } {D,Q} = 0, D, Q = 0, and have one non-vanishing anti-commutator, { } D, D = 2i∂.
To construct the action, we need three different superfields.
Superfields:
Chiral superfields: D+Φ = D−Φ = 0,
m ¯ m + m − m + − m Φ(x , θ , θ ) = ϕ(y ) + θ ψ+(y ) + θ ψ−(y ) + θ θ F (y ), (1.19) with y = x − iθθ .
Vector superfields in Wess-Zumino gauge:
− ¯− + ¯+ − ¯+ + ¯− V =θ θ (v0 − v1) + (−)θ θ (v0 + v1) − θ θ σ − θ θ σ¯ (1.20) − + ¯−¯ ¯+¯ ¯+ ¯− − + − + ¯+ ¯− + iθ θ (θ λ− + θ λ+) + iθ θ (θ λ− + θ λ+) + θ θ θ θ D.
In the expression above, σ is a complex scarlar, v0,1 are gauge fields, λ−,+ are fermion fields (gauginos) introduced to perserve the supersymmetry, D is a auxiliary real scalar.
Twisted chiral superfields: D+Σ = D−Σ = 0. In general, the twisted chiral superfields can be written similar to chiral superfilds with θ− and θ¯− exchanged. In the introduction, 16
we will only consider gauged linear sigma models with Abelian gauge groups.
Σ =D+D−V + ¯− + ¯− − ¯− + ¯+ =σ + iθ λ+ − iθ λ− + θ θ (D − iv01) + iθ θ ∂−σ − iθ θ ∂+σ (1.21) − + ¯− + ¯− ¯+ + ¯− − ¯+ − θ θ θ ∂−λ+ + θ θ θ ∂+λ− + θ θ θ θ ∂−∂+σ,
which is twisted chiral and also gauge invariant.
The action contains four parts
S = Skin + SW + Sgauge + SF I,θ. (1.22)
The first term determines the kinetic energy of chiral superfields, ∫ ( ) ∑ ∑ 2 4 Qi,aVa Skin = d xd θ Φie a Φi , (1.23) i
where Qi,a is the charge of the i-th chiral superfield under the a-th U(1) gauge group. The second term is the superpotential term, ∫ 2 2 SW = d xd θW (Φ) + c.c., (1.24)
where W (Φ) is a gauge invariant holomorphic quasi-homogeneous polynomial in Φi. The third term determines the kinetic energy of gauge fields, ∫ ∑ 1 S = − d2xd4θΣ Σ , (1.25) gauge 4e2 a a a a
where ea are gauge coupling constants. The last term of the action contains the Fayet-
Iliopoulos parameters ra and the theta angles θa, ∫ ∑ − 2 + SF I,θ = ita d xdθ dθ Σa + c.c., (1.26) a where θ t = ir + a . a a 2π 17
Expanding the superfields, one can find the scalar potential is, ( ) 2 ∑ ∑ 2 ∑ ∑ 2 2 2 ea 2 ∂W U (ϕi, σ) = |Qi,aσa| |ϕi| + Qi,a|ϕ| − ra + . (1.27) 2 ∂ϕ i,a a i i i
The scalar potential determines the geometry of the target space of the nonlinear sigma model. With σ = 0, the vanishing of D terms, ∑ 2 Qi,a|ϕ| − ra = 0, i
determines the ambient space and vanishing of F terms,
∂W = 0, ∂ϕi
describe the embedding of the target space of the nonlinear sigma model into the ambient space. Here we only consider Abelian gauge groups, so at low energies a gauged linear sigma model without superpotential reduces to a nonlinear sigma model on a toric variety. With a superpotential, the target space will be a submanifold of the toric variety. This is how one can describe a complete intersection in a toric variety.
The prototype for gauged linear sigma models is the supersymmetric CPN model. This is
a U(1) gauged linear sigma model containing N + 1 chiral superfields Φ1,..., ΦN+1 which all have charge one. The classical supersymmetric lie on the locus where potential energy vanishes, ( )2 N∑+1 e2 N∑+1 U = |σ|2|ϕ |2 + |ϕ |2 − r = 0. i 2 i i=1 i=1 On the Higgs branch (σ = 0), if r >> 0, the vacua are described by
N∑+1 2 |ϕi| − r = 0, i=1 which is just S2N+2. Quotienting the U(1) gauge symmetry, the vacuum manifold becomes the desired CPN . Following the Higgs mechanism, we expand the Lagrangian around points N N on the vacuum manifold CP . For bosons ϕi, the modes√ tangent to the CP are massless and the modes perpendicular to the CPN have mass e 2r. Considering low-energy fluctuations 18
√ with energy lower than e 2r, the massive modes of bosons and fermions are frozen and the massless modes form a nonlinear sigma model with target space CPN . So far, we have seen that the gauged linear sigma model reduces to a nonlinear sigma model on CPN at the classical level. Under renormalization group flow, only the parameter r receives quantum corrections which are one-loop exact and proportional to the sum of the U(1) charges of all the superfields, ∑ δr ∝ Qi. (1.28) i In general, if there are multiple U(1) gauge groups, the quantum corrections take the similar form ∑ δra ∝ Qi,a, (1.29) i n where ra are the FI-parameters for the corresponding U(1) gauge group. Thus, for the CP model, the FI parameter r is not a true parameter but flows under renormalization. A quick way to determine the low energy physics’s geometry is to take the limit e → ∞. If one take the gauge coupling constants to infinity, gauge fields will become non-dynamical and act as Lagrange multipliers. In the current case, one found the metric is proportional to the FI parameter r, ds2 = rgFS, (1.30)
where gFS is the standard Fubini-Study metric. Then, the renormalization group flow of r reflects the fact that the size of the CPn shrinks under RG flow. The nonlinear sigma model flows to a trivial superconformal field theory corresponding to a sigma model on a set of points. Following the same ideas, any toric variety can be realized by GLSMs with gauge group U(1)n for some n [19].
One may wonder what happens when the sum of charges equal to zero. The sum of charges equal to zero implies that FI parameters are true parameters and do not receive quantum
corrections. One can choose ra freely. In different regions of the space parametrized by FI
parameters ra, the corresponding low energy theories will be different. To be specific, let us consider the example of a hypersurface in CPn. To describe a hypersurface of degree n, we add another chiral superfield P with charge −n − 1 together with a superpotential W (ϕ),
W = PG(Φ),
which is gauge invariant if G(Φ) is a degree n + 1 holomorphic polynomial in Φi. When 19
r >> 0, the GLSM reduces to a nonlinear sigma model with target space describe by the hypersurface G(ϕ) = 0. It is easy to check that the hypersurface is a Calabi-Yau surface by calculating the first Chern class. In the opposite limit r << 0, the low energy physics is de- scribed by a Landau-Ginzburg orbifold on Cn+1 with superpotential W = G(ϕ). In this case,
the U(1) gauge group is broken to a Zn+1 subgroup. This example demonstrates the precise relation between nonlinear sigma models and Laudau-Ginzburg models for hypersurfaces.
In addition, gauged linear sigma models gave the first examples of topology change, demon- strating that stringy quantum mechanics could interpolate between different geometries, just as ordinary QM can jump between classically isolated vacua. Consider a U(1) GLSM of four chiral superfields with charges +1, +1, −1, −1. The two phases of the low energy theory correspond to two nonlinear sigma models with different geometries. This is an example of a birational transformation called a flop. The theory has a singularity at r = 0, midway between the two phases r >> 0 and r << 0. However, the theta angle should also be taken into account and the parameter really is
iθ t = r + . 2π
In the case of on FI parameter, the phase diagram is an infinitely long cylinder instead of a straight line, such that one can choose a continuous path without crossing the singularity to continuously connect the two different phases. Essentially, the GLSM phase transition realizes the birational equivalence implicitly.
The above review only covers Abelian gauged linear sigma models. Non-Abelian cases are also discussed in the original paper [18]. More recently, physicists have also successfully constructed GLSMs for more general spaces than complete intersections. Toric stacks can be similarly constructed by a non-perturbative trick with non-minimal charges, Pfaffian varieties can be realized both by perturbative and nonperturbative methods, intersections of Pfaffian varieties with hypersurfaces and so on [20–28].
1.4 Mirror symmetry
Early days of mirror symmetry: The original idea of mirror symmetry can be traced back to the mid-1980s, where physicists found that there is a nontrivial equivalence between 20
string theory compactified on a circle with radius R and string theory compactified on a circle with radius 1/R. This is the famous T-duality which exchanges winding numbers of one theory with quantized momenta of the other theory [29, 30] (for a modern review of T-duality see [31]). As mentioned before, Calabi-Yau threefolds plays a significant role in superstring compactification. Naturally, physicists wondered whether the same T-duality story would happen for Calabi-Yau spaces and speculated on the existence of Calabi-Yau mirror pairs that correspond to the same superconformal field theory [11, 17]. The Greene- Plesser construction provided the first non-trivial example of a mirror relationship [32]. There was also a computational survey of the Calabi-Yau manifolds embedded in weighted projective spaces satisfying the mirror condition which provided further evidence for the existence of mirror symmetry [33]. With the celebrated paper [34], mirror symmetry really began to catch physicists and mathematicians’ eyes. In [34], the authors provide a new way, through mirror symmetry, to calculate Gromov-Witten invariants and solve long-standing problems in enumerative geometry.
Mirror symmetry is a duality between two interpretations of the same N = (2, 2) supercon- formal field theory. One can see directly from the superconformal algebra that there is an automorphism given by flipping the left-moving U(1) current J 7→ −J. The moduli space of N = 2 superconformal field theory is described by two types of deformations generated by two different truly marginal operators. Those two types of truely marginal operators can be constructed from the (c, c) ring and the (a, c) ring [16,17]. Corresponding to an operator ϕ in the (c, c) ring of conformal weight h = h¯ = 1/2 and U(1) charge Q = Q = 11, the truly marginal operator is given by { [ ]} − − Φ1,1(w, w) = G (¯z), G (z), ϕ(w, w) , I (I ) − = dz G (z) dz G−(z)ϕ(w, w) .
¯ ¯ ′ Φ1,1 has conformal weight h = h = 1 and zero charge Q = Q = 0. For an operator ϕ in the (a, c) ring with h = h¯ = 1/2 and Q = −Q = 1, the truly marginal operator is given similarly
1Note, h and Q denote the conformal weight and the U(1) R-charge associate to the left-moving super- conformal algebra and h¯, Q denote the right-moving weight and charge. 21 by [ ] − + Φ−1,1 = G (z), {G (¯z), ϕ(w, w¯)} , I (I ) − = dzG+(z) d¯zG (z)ϕ(w, w) .
¯ ¯ Φ−1,1 has the same conformal weight h = h = 1 and the same charges Q = Q = 0 as Φ1,1. The moduli space of N = (2, 2) superconformal field theories can be divided into two subspaces generated by the Φ1,1 marginal operators and the Φ−1,1 marginal operators, respectively. At least locally, the moduli space can be written as a product of those two subspaces. Then, mirror symmetry is a symmetry exchanging the interpretation of the moduli space in terms of these two types of deformations. Notice that the two types of marginal operators Φ1,1 and
Φ−1,1 only differ by the sign of the U(1) charge at the level of superconformal field theory which reflects the fact that mirror symmetry is a superconformal field theory automorphism flipping the sign of the left-moving U(1) current. Even though mirror symmetry is only a sign change of the U(1) charge, it have significant effects on the geometric counterparts. Suppose the Calabi-Yau target manifold X has complex dimension d. The Φ−1,1 marginal operators 1,1 are associated with the cohomology group H (X) and the Φ1,1 marginal operators are associated with the cohomology group Hd−1,1(X). The two cohomology groups have entirely different geometrical meanings. Elements of the cohomology group H1,1(X) parametrize K¨ahlermoduli of Calabi-Yau manifolds. On the other hand, elements of the cohomology group Hd−1,1(X) parametrize complex moduli of Calabi-Yau manifolds. Mirror symmetry relates different Calabi-Yau manifolds and states that two Calabi-Yau manifolds (X,Y ) are equivalent at the quantum level if the K¨ahlerstructure of one Calabi-Yau manifold is exchanged with the complex structure of the other. The pair of Calabi-Yau manifolds (X,Y ) are called a mirror pair. Note that one direct consequence of mirror symmetry is that,
h1,1(X) = hd−1,1(Y ), (1.31) hd−1,1(X) = h1,1(Y ), and more generally the Hodge diamonds of Calabi-Yau mirror pairs are exchanged across the diagonal axis. There is also a version of mirror symmetry for topological field theories. Mirror symmetry relates the A model on a Calabi-Yau manifold X, which only depends upon the K¨ahlermoduli of X, to the B model on Y , which only depends upon the complex structure moduli of Y . Recall that A model correlation functions receive worldsheet instanton 22 corrections and B model correlation functions are purely classical. Immediately, we see the power of mirror symmetry, in that it exchanges classical and quantum effects. Difficult quantum amplitude computations in the A model can be done in the B model by classical calculations.
Hori-Vafa construction: Based on T-duality, Kentaro Hori and Cumrun Vafa gave a physics proof of mirror symmetry in 2000 [109]. The Hori-Vafa construction also generalizes mirror symmetry to more spaces than Calabi-Yau manifolds. They constructed Landau- Ginzburg model from any given Abelian gauged linear sigma model and established an equivalence between the two theories, in which the A twisted sector of the gauged linear sigma model along with all topological data is equivalent to the B twisted sector of the Landau-Ginzburg model. As we have seen, in some cases, the gauged linear sigma model flows to a nonlinear sigma model at low energy. On the other hand, certain Landau-Ginzburg models are related to nonlinear sigma models on Calabi-Yau manifolds via phase transitions. Then, when the gauged linear sigma model flows to nonlinear sigma model on Calabi-Yau manifolds, the original mirror symmetry is recovered.
GLSM Hori-Vafa mirror LG
RG flow phase transition
mirror symmetry NLSM on CY NLSM on CY
Figure 1.1: Mirror symmetry for Calabi-Yau manifolds
GLSM Hori-Vafa mirror LG
RG flow generalized mirror symmetry
NLSM on Fano
Figure 1.2: Mirror symmetry for Fano spaces
The construction of the mirror Landau-Ginzburg model is relatively simple. Consider a k U(1) gauged linear sigma model with chiral superfields Φi of charge Qi,a under the a-th 23
U(1) gauge group. There are k D terms ∑ 2 Da = Qi,a|ϕi| − ra. i
For each chiral superfield Φi, the mirror Landau-Ginzburg model will have a corresponding periodic fundamental field Yi = Yi + 2πi. The mirror Landau-Ginzburg model is defined by the superpotential
W = exp (−Y1) + ··· + exp (−YN ) , (1.32)
together with constraints coming from D terms, ∑ Qi,aYi − ra = 0. (1.33) i
Also, the constraints can be embedded into the superpotential by Lagrange multipliers Σa, ( ) ∑ ∑ ∑ W = exp (−Yi) + Σa Qi,aYi − ra . (1.34) i a i
Let’s check the construction in an example, the CPn model. The CPn model is given by a U(1) gauged linear sigma model with n + 1 chiral superfields of charge one. The D-term is given by, ∑ 2 D = |ϕi| − r. i Following the general procedure, the mirror Landau-Ginzburg model will have n + 1 funda- mental fields Yi with the superpotential
W = exp (−Y1) + ··· + exp (−Yn+1) , (1.35)
and the constraint ∑ Yi = r. (1.36) i The superpotential could also be written as, ( ) ∑ ∑ W = exp (−Yi) + Σ Yi − r . (1.37) i i 24
Using the constraint, we can eliminate one fundamental field,
Yn+1 = r − Y1 − · · · − Yn. (1.38)
Therefore, effectively the mirror Landau-Ginzburg model is described by n fundamental fields with the superpotential
W = exp (−Y1) + ··· + exp (−Yn) + exp(−r) exp (Y1 + ··· + Yn) . (1.39)
For this case, the correlation functions for the twisted sectors are easy to calculate. We have seen the correlation functions of the A-twisted CPn model (1.12). Correlation functions of B-twisted Landau-Ginzburg models are also known [35]. For theories with discrete vacua, we have ∑ f . . . f ⟨f . . . f ⟩ = 1 m , (1.40) 1 m H dW =0
where H = det(∂i∂jW ) is the Hessian and fi are functions of physical operators. In the current case, there is only one physical operator x and the correlation functions are
⟨xn⟩ = 1, ⟨x2n+1⟩ = q, (1.41) ⟨xn+d(n+1)⟩ = qd,
where q = exp(−r). One can easily see that the correlation functions (1.12) and (1.41) match after making the identification ϕ ≡ x.
For hypersurfaces in toric varieties, there is another way to construct mirror manifolds by exchanging Newton polytopes of toric varieties [45]. Also, there is another approach to understand mirror symmetry, known as the SYZ conjecture, proposed by Strominger, Yau and Zaslow [44].
Mathematicians take the topological field theory version of mirror symmetry as the definition of mirror symmetry since topological field theories has a rigorous definition in mathematics [36]. One generalization of mirror symmetry studied by mathematicians is homological mirror symmetry. It was first proposed in [37] and proved later in [38–43]. 25
In this thesis, we will be interested in two directions in mirror symmetry. First, we will discuss progress on a heterotic generalization of mirror symmetry, known as (0,2) mirror symmetry. It is not as well-understood as ordinary mirror symmetry. In chapter 2, we will construct some concrete examples. Second, we will also discuss work on an extension of the Hori-Vafa construction of mirrors to abelian gauged linear sigma models, to nonabelian gauged linear sigma models. We will discuss this in chapter 3. Finally, in chapter 4 of this thesis, we will discuss other applications of gauged linear sigma models. Chapter 2
Toda-like (0,2) mirrors
The contents of this chapter were adapted with minor modifications, with permission from the Journal of High Energy Physics, from our publications [1,2]. Part of the contents in this chapter has also appeared in the thesis of my collaborator Ruoxu Wu.
2.1 Introduction
In heterotic string compactifications, there is a natural generalization of mirror symmetry, known as (0, 2) mirror symmetry, see e.g. [73–77]. For mathematics, (0, 2) mirror symmetry yields quantum sheaf cohomology, a generalization of quantum cohomology in (2, 2) theories, see e.g. [77–90].
A perturbative heterotic compactification is defined by a worldsheet theory with (0,2) su- persymmetry. A (0, 2) nonlinear sigma model is defined by a pair (X, E), with X a K¨ahler manifold and E → X a holomorphic vector bundle, satisfying Green-Schwarz anomaly can- cellation
ch2(E) = ch2(TX).
In cases in which X is Calabi-Yau, so that the nonlinear sigma model above flows to a SCFT, (0, 2) mirror symmetry states that there is a dual pair (X′, E ′) which gives rise to the same SCFT. If X is Fano, then the (0, 2) mirror will be a (0, 2) Landau-Ginzburg model.
Although (0, 2) mirror symmetry has not been developed to nearly the same extent as ordi-
26 27
nary mirror symmetry, a number of crucial results do exist. One of the first accomplishments was a numerical scan through anomaly-free examples demonstrating the existence of pairs of (0, 2) theories with matching spectrum computations [73], giving strong evidence for the existence of (0, 2) mirrors. Other work includes a version [74] of the old Greene-Plesser orb- ifold construction [46], work on GLSM-based dualities [47], and most recently, a proposal for a generalization of Batyrev’s construction involving reflexively plain polytopes [48]. In addition, there has been considerable work on quantum sheaf cohomology [49–69], the (0, 2) analogue of ordinary quantum cohomology. Other recent work on two-dimensional (0,2) theories from different directions includes e.g. [94–103].
All that said, many basic gaps remain. For example, there is not yet a systematic description of (0, 2) Landau-Ginzburg mirrors to (0, 2) nonlinear sigma models on Fano spaces, aside from a special case discussed in [47] (see e.g. [91, 92] for a few early references on ordinary mirror symmetry for Fano spaces). The work in this chapter is a first pass at filling that gap.
In this chapter, we will focus on topological twists of these theories. In (0, 2) theories, broadly speaking, two topological twists exist, now known as the A/2 and B/2 twists. In the case of the nonlinear sigma models above, the A/2 twist will exist when
∗ ∼ det E = KX , and the B/2 twist will exist when ∼ det E = KX .
Clearly, both the Green-Schwarz condition and the conditions for the twists will be satisfied when one takes E = TX, in which case, the A/2 theory becomes the ordinary A model topological field theory, and the B/2 theory becomes the ordinary B model topological field theory.
Quantum sheaf cohomology emerges as the OPE algebra of the A/2-twisted theory, forming a precise (0, 2) analogue of ordinary quantum cohomology. To be specific, recall that the ordinary quantum cohomology ring is a ring of local operators defined in the A twist of a nonlinear sigma model on X as BRST-closed states of the form
i1 ··· ip ¯ı1 ··· ¯ıq bi1···ip¯ı1···¯ıq χ χ χ χ . 28
These BRST-closed states can be identified with closed differential forms on the target space X, elements of Hq(X, Ωp) = Hp,q(X). Similarly, the quantum sheaf cohomology ring is a ring of local operators defined in the A/2 twist of a nonlinear sigma model on X as right- BRST-closed states of the form
a ı a1 ··· p ı1 ··· q ba1···apı1···ıq λ− λ− ψ+ ψ+ .
These right-BRST-closed states can be identified with ∂-closed bundle-valued differential forms, elements of Hq(X, ∧pE ∗).
Ordinary mirror symmetry exchanges A twists with B twists, which means an A twisted nonlinear sigma model is equivalent to a B twisted nonlinear sigma model on the mirror Calabi-Yau manifold [15]. Similarly, (0, 2) mirror symmetry exchanges A/2 twists with B/2 twists, meaning the A/2 twisted nonlinear sigma model on (X, E) is equivalent to the B/2 twisted nonlinear sigma model on the mirror (X′, E ′).
In this chapter, we explore (0, 2) Toda-like mirrors to A/2-twisted theories on Pn × Pm, toric del Pezzo surfaces and Hirzebruch1 surfaces, as part of an on-going program to understand (0,2) mirror symmetry. These ansatzes will be tested in several different ways:
• First, each case reduces to an ordinary (2,2) mirror along the (2,2) locus.
• We check that the fields in the Landau-Ginzburg vacua obey the quantum sheaf coho- mology relations of their A/2-model partners.
• We check in each case that all genus zero correlation functions of the proposed B/2- twisted Landau-Ginzburg mirror match those of the original A/2-twisted (0,2) theory.
• Amongst the toric del Pezzo mirrors, we check that our proposed mirrors are related by blowdowns as dictated by2 geometry.
• As an implicit check, we also give a proposal for (0,2) mirrors of Hirzebruch surfaces3 of arbitrary degree, which not only correctly captures the genus zero correlation functions,
1More precisely, as we will explain later, gauged linear sigma models (GLSMs) [18] for Hirzebruch surfaces. Most Hirzebruch surfaces are not Fano, and so the UV limits of their GLSMs are not Hirzebruch surfaces but rather different geometries, so in principle we are actually describing mirrors to those different surfaces. 2As by definition the del Pezzos are Fano, the UV GLSM phases correspond to the naive geometries. 3For degree greater than one, Hirzebruch surfaces are not Fano; nevertheless, one expects their sigma models to have isolated vacua in the IR, hence a Toda-type mirror is expected. 29
but also includes as special cases our previous proposed mirror for P1 × P1 [1] and for 1 1 the del Pezzo dP1 above, thereby demonstrating that the P × P and dP1 mirrors are indeed elements of a sequence of mirrors, as one would expect.
There is another subtlety we shall encounter in the form of the J functions defining the (0,2) superpotential. Specifically, they will sometimes have poles away from the origin. Now, ordinary (2,2) mirrors to projective spaces and Fano varieties will often have superpotential terms proportional to 1/Xn for n > 0, but it is understood that those Landau-Ginzburg models are defined over algebraic tori of the form (C×)k, so that the target space does not include places where X = 0, and hence the theory never encounters a divergent superpoten- tial. By contrast, in this chapter we will encounter some examples which have poles at points which are not disallowed. As a result, we interpret these theories in a low-energy effective theory sense – so long as no vacua are located at those poles, we can understand the theory in a neighborhood of the vacua, which excludes the poles. (Similar remarks have been ap- plied to understand GLSMs for generalized Calabi-Yau complete intersections [104–106].) Of course, this also means that these theories are not UV-complete, but we will leave searches for UV-complete descriptions for other work.
2.1.1 Review of (2,2) Toda dual theories
Consider a (2,2) supersymmetric abelian GLSM, with gauge group U(1)k and n chiral su- a perfields Φi. Let Qi denote the charge of the ith chiral superfield under the ath factor in the gauge group. Following [93], the mirror4 of an A-twisted theory of this form is a Landau-Ginzburg model with a superpotential of the form ( ) ∑k ∑n ∑n a − W = Σa Qi Yi ra + exp(Yi), (2.1) a=1 i=1 i=1 where the Yi are twisted chiral superfields in one-to-one correspondence with chiral superfields in the original theory. We integrate out the Σa’s to recover the usual form.
It will also be useful to track R-symmetries, as a consistency test on our proposals. Recall that a two-dimensional N = (2, 2) theory has classical left-moving U(1)L and a right-moving
4If the toric variety is Fano, this will yield the mirror of the Fano phase. Otherwise, it may yield the mirror of a different phase. 30
U(1)R R-symmetries,
+ −iκ + U(1)R : θ 7→ e θ , − −iκ − U(1)L : θ 7→ e θ .
Denoting the generators of the R-symmetry U(1)L × U(1)R as JL and JR respectively, then one can combine them to get the vector R-symmetry U(1)V and axial R-symmetry U(1)A with generators 1 1 J = (J + J ),J = (J − J ). V 2 R L A 2 R L Chiral superfields transform under the R-symmetries as follows [93][equ’ns (2.11)-(2.12)],
− i − ¯ iαqV iα iα ¯ RV Φi(x, θ , θ ) = e Φi(x, e θ , e θ ), − i ∓ ¯ iβqA iβ iβ ¯ RAΦi(x, θ , θ ) = e Φi(x, e θ , e θ ),
i where qV,A denote the vector and axial R-charges of Φi, chosen so that the superpotential has vector charge 2 and axial charge 0. (A twisted superpotential, a function of twisted chiral superfields, has vector charge 0 and axial charge 2.) In components,
iαqV iα(qV +1) RV : x 7→ e x, ψ 7→ e ψ,
iβqA iβ(qA1) RA : x 7→ e x, ψ 7→ e ψ.
In the quantum theory, the axial R symmetry is typically anomalous.
Now we turn to the mirror theory. We assume the original theory has no superpotential (as we are taking the mirror of a toric variety), so the vector and axial R-charges of the original 5 chiral superfields both vanish. The twisted chiral superfields Yi transform as [93][equ’ns (3.29)-(3.30)]
¯ −iα iα ¯ RV Yi(x, θ , θ ) = Yi(x, e θ , e θ ), (2.2) ¯ ∓iβ iβ ¯ RAYi(x, θ , θ ) = Yi(x, e θ , e θ ) − 2iβ. (2.3)
It is straightforward to see that in the mirror Landau-Ginzburg model defined by (2.1), the vector R-symmetry is unbroken, but the axial R-symmetry is broken classically by the
5We follow the conventions of [93] in using the same ‘axial,’ ‘vector’ terminology to describe both the original symmetry and its mirror, to assist in tracking the symmetries. 31
superpotential, corresponding to the fact that in the original theory, the axial R-symmetry is anomalous.
For example, consider the mirror to Pn, which (after integrating out Σ) is a Landau-Ginzburg theory defined by the (twisted) superpotential ∫ ∫ ( ) ∑n q dθ+ dθ¯−Wf + c.c. = dθ+ dθ¯− X + ∏ + c.c., i n X i=1 i=1 i ∏ − −1 where Xi = exp Yi and q = exp( r). Here, the last term, q Xi , classically breaks the axial R symmetry unless exp(2iβ(n + 1)) = 1,
corresponding to the anomaly of the original theory, breaking the original U(1) symmetry
to a Z2(n+1) subgroup.
2.1.2 Review of (0,2) Landau-Ginzburg models
In this section, we’d like to review the basics of (0,2) Landau-Ginzburg models which will play an important role in our mirror constructions in this chapter. For (0,2) theories, there is only a right-moving N = 2 supersymmetry. Therefore, there are only two supercharges
Q+, Q+ and two (0,2) super-derivatives D+, D+ which are defined the same as in the (2,2) cases (1.17) and (1.18). Also, there are only two superspace coordinates θ+ and θ¯+. To build actions for (0,2) Landau-Ginzburg models, we will need two sets of superfields: “(0,2) chiral superfields” Φi and “(0,2) Fermi superfields” Γa.
(0,2) chiral superfields are superfields annihilated by the operator D+,
i D+Φ = 0.
In general, they can be expanded as
i i + i − + + i Φ = ϕ + θ ψ+ iθ θ ∂+ψ .
As in the (2,2) theories, the bosonic fields ϕi can be interpreted as coordinates on the target 32
i ¯ı space X and the fermion fields ψ+, ψ+ are sections of the following bundles, ( ) ( ( ) ) i 1/2 ∗ 1,0 ¯ı 1/2 ∗ 1,0 ∨ ∈ ∞ ⊗ ∈ ∞ ⊗ ψ+ ΓC KΣ ϕ T X , ψ+ ΓC KΣ ϕ T X .
(0,2) Fermi superfields satisfy √ a a D+Γ = 2E ,
where Ea are (0,2) chiral superfields. Expanding in θ, we have √ √ a a + a ¯+ a + + a Γ = γ− + 2θ G − 2θ E − iθ θ ∂+γ−,
a a where G are auxiliary fields and γ− are left-moving fermions. Unlike the (2, 2) theories, we a can couple γ− to the pullback of a holomorphic vector bundle E over X, ( ) ( ) ( )∨ a 1/2 ∗ a 1/2 ∗ ∈ ∞ ⊗ E ∈ ∞ ⊗ E γ− ΓC KΣ ϕ , γ− ΓC KΣ ϕ .
The action contains two parts,
S = Skin + SW .
The kinetic part is given by ∫ ( ) ¯ 2 2 i − ¯ı a b Skin = 2t d xd θ Ki∂−Φ Kı∂−Φ + ha¯bΓ Γ , (2.4)
where the one-form Ki determines the metric on the target space and ha¯b is a Hermitian fiber metric on the vector bundle E. The other part of the action is the (0,2) analogue of superpotential which is called the (0, 2) superpotential. Introducing a holomorphic function of (0,2) chiral superfields J(Φ), the contribution to the action from the (0, 2) superpotential is given by, ∫ 2 + a SW = 2t d xdθ Γ Ja(Φ) + c.c.
Expanding the action in component fields, we have ∫ ( 2 i ¯¯ȷ¯ i ȷ¯ i ¯b a i ȷ a b S =2t d z giȷ¯∂ϕ ∂ϕ + giȷ¯ψ+Dz¯ψ+ + iha¯bγ−Dzγ− + Fiȷa¯ ¯bψ+ψ+λ−λ− Σ 2 a¯b i a ¯ı b + 2h J J¯ + ψ λ D J + ψ λ D J (2.5) a b + − i a + − ¯ı b ( ) ) ¯ ( ) ¯ a b i a¯ b ¯ı a b + 2ha¯bE E + ψ+λ− DiE hab¯ + ψ+λ− D¯ıE ha¯b , 33
E where Fiȷa¯ ¯b is the curvature tensor of the Hermitian vector bundle with the connections defined by a ac¯ a¯ ab¯ Aib = h ∂ihcb¯ ,A¯ı¯b = h ∂¯ıhbc¯. The covariant derivatives acting on the left-moving fermions are given by
α α µ α β Dzγ− = ∂γ− + (∂ϕ ) Aµβγ−.
Because of supersymmetry, there is a constraint on E and F , ∑ a E (ϕ)Fa(ϕ) = 0. a
The anomaly cancellation condition restricts the vector bundle to have
c2(E) = c2(TX).
For completeness, we also give the supersymmetry transformations here,
i i δϕ = iα−ψ+, ¯ı ¯ı δϕ = iα˜−ψ+, i i δψ = −α˜−∂ϕ , + (2.6) ¯ı − ¯ı δψ+ = α−∂ϕ , a − j a c a¯b a δγ− = iα−ψ+Ajcγ− + iα−h J¯b + iα˜−E , a¯ − ȷ¯ a¯ c¯ ab¯ a¯ δγ− = iα˜−ψ+Aȷ¯c¯γ− + iα˜−h Jb + iα−E .
The (2, 2) Landau-Ginzburg models can be recoverd by taking the holomorphic vector bundle E to be the tangent bundle TX, taking all Ei = 0, and taking the (0,2) superpotential to be i Γ ∂iW with the corresponding (2,2) superpotential W . 34
2.2 Pn × Pm
2.2.1 The A/2-twisted nonlinear sigma model
Let us begin by briefly reviewing pertinent properties of the (0,2) nonlinear sigma model on Pn × Pm, whose dual we shall describe. First, the gauge bundle in this (0,2) theory is a deformation E of the tangent bundle of Pn × Pm, which can be described as a cokernel
0 −→ O2 −→E O(1, 0)n+1 ⊕ O(0, 1)m+1 −→ E −→ 0,
where [ ] AB E = , CD in which A, B are (n + 1) × (n + 1) matrices and C, D are (m + 1) × (m + 1) matrices. The quantum sheaf cohomology ring of an A/2-twisted nonlinear sigma model on Pn × Pm with the bundle above takes the form [51,52,64,67]
˜ ˜ det(Aψ + Bψ) = q1, det(Cψ + Dψ) = q2,
and for later use, we expand the determinants as follows:
∑n ˜ n+1 ˜n+1 i ˜n+1−i det(Aψ + Bψ) = aψ + bψ + µiψ ψ , (2.7) i=1 ∑m ˜ m+1 ˜m+1 k ˜m+1−k det(Cψ + Dψ) = cψ + dψ + νkψ ψ , (2.8) k=1
where a = det A, b = det B, c = det C, d = det D,
µi is a sum of determinants of matrices, each of which is formed by taking i rows of A and
n + 1 − i rows of B, and νi is formed similarly from C, D. 35
2.2.2 The Toda-like mirror theory
We claim the (0,2) superpotential of the (0,2) Landau-Ginzburg Toda-like mirror to Pn ×Pm is ∑n ∑m ˜ ˜ W = FiJi + FkJk, (2.9) i=1 k=1 where ( ) ˜ n+1 ∑n ˜ i (1−n)/n q1 X1 X1 J = a aX − + b + µ − , (2.10) i i X ··· X Xn n+1 i Xi−1 ( 1 n 1 i=1 1 ) q Xm+1 ∑m Xk J˜ = d(1−m)/m dX˜ − 2 + c 1 + ν 1 , (2.11) k k ˜ ··· ˜ ˜ m k ˜ k−1 X1 Xm X1 k=1 X1 which clearly generalizes the dual to P1 × P1 discussed in section 2.3.2.
First, note that if the parameters a, b, c, d, and the µi, νk are related to the matrices A, B, ˜ C, D of the A/2 model as above, then the vacua of this theory, defined by Ji = 0 = Jk, are the solutions of
˜ ˜ ˜ ˜ X1 = X2 = ··· = Xn ≡ X, X1 = X2 = ··· = Xm ≡ X,
˜ ˜ det(AX + BX) = q1, det(CX + DX) = q2, identical to the solutions of the quantum sheaf cohomology relations, as one would expect for a sensible Toda-like dual.
One can show that the correlation functions of this B/2-twisted Landau-Ginzburg model computed by equation (2.28) equal the correlation functions of A/2-twisted model on Pn ×Pm [67]: [ ] ∑ ∏ −1 ⟨ ··· ⟩ ··· J σa1 σa = σa1 σa det a,b det M(α) (2.12) l l a,b σ|J =0 α with [ ] ∏ Qa J −1 (α) a = ln qa det M(α) . (2.13) α 36
n m In the present case, for P × P , there are only two σ’s, which we label σ1, σ2, and [ ] J = ln q−1 det(Aσ + Bσ ) , 1 [ 1 1 2 ] J −1 2 = ln q2 det(Cσ1 + Dσ2) .
To show that the two expressions for correlation functions match, it suffices to show that ∏ det |Ji,j| = det |Ja,b| det M(α), (2.14) a,b α
˜ by identifying Xi with σ1 and Xk with σ2 on the space of vacua, since [ ] ∏ ∂σ1 det(Aσ1 + Bσ2) ∂σ2 det(Aσ1 + Bσ2) det |Ja,b| det M(α) = det (2.15) a,b α ∂σ1 det(Cσ1 + Dσ2) ∂σ2 det(Cσ1 + Dσ2)
on the classical vacua Ja(σ) = 0.
In order to show (2.14), we will need a minor linear algebra result. For an (n + m) × (n + m) matrix of the form ··· ··· a11 a12 a13 a1n β 0 0 −α α 0 ··· 0 0 0 ··· 0 −α 0 α ··· 0 0 0 n ...... ...... −α 0 ··· 0 α 0 ··· 0 , (2.16) ρ 0 ··· 0 d d d ··· d 11 12 13 1m 0 0 ··· 0 −δ δ 0 ··· 0 m 0 0 −δ 0 δ ··· 0 ...... ...... . . . . 0 ··· 0 −δ 0 ··· 0 δ its determinant has the form
(det ζ)(det η) − βραn−1δm−1, (2.17) where det ζ is the determinant of the upper-left n×n submatrix and det η is the determinant 37 of the lower-right m × m submatrix, given by
∑n n−1 det ζ = α a1i, (2.18) i=1 ∑m m−1 det η = δ d1k. (2.19) k=1
Next, we need to compute
∂Y J1 ··· ∂Y J1 ∂ ˜ J1 ··· ∂ ˜ J1 1 n Y1 Ym ...... ...... ∂Y Jn ··· ∂Y Jn ∂ ˜ Jn ··· ∂ ˜ Jn det |J | = det 1 n Y1 Ym , i,j ˜ ˜ ˜ ˜ ∂Y J1 ··· ∂Y J1 ∂ ˜ J1 ··· ∂ ˜ J1 1 n Y1 Ym ...... ...... ∂ J˜ ··· ∂ J˜ ∂ J˜ ··· ∂ J˜ Y1 m Yn m Y˜1 m Y˜m m
˜ ˜ where Xi = exp(Yi) and Xi = exp(Yi). By taking suitable linear combinations, one can rewrite the matrix above in the form of the matrix (2.16), with the following identifications: ( ) ˜ n+1 ∑n ˜ i (1−n)/n q1 X1 X1 a = a aX + − nb + (1 − i)µ − , 11 1 X ··· X Xn n+1 i Xi−1 ( 1 n 1 i=1 ) 1 ˜ n+1 ∑n ˜ i (1−n)/n X X = a 2aX + (1 − n)b + (2 − i)µ − , Xn n+1 i Xi−1 ( i=1 ) q ··· (1−n)/n 1 a12 = a13 = = a1n = a + ··· , ( X1 Xn ) ˜ n+1 ∑n ˜ i (1−n)/n X X = a aX + b + µ − , Xn n+1 i Xi−1 i=1 38
(1−n)/n α = a (aX1), = a1/nX, ( ) ˜ n+1 ∑n ˜i (1−n)/n X1 X1 β = a (n + 1)b + iµ − , Xn n+1 i Xi−1 ( 1 i=1 1 ) ˜ n+1 ∑n ˜ i (1−n)/n X X = a (n + 1)b + iµ − , Xn n+1 i Xi−1 i=1
( ) m+1 ∑m k (1−m)/m ˜ q2 X1 X1 d11 = d dX1 + − cm + (1 − k)νk , X˜ ··· X˜ X˜ m X˜ k−1 ( 1 m 1 k=1 ) 1 m+1 ∑m k (1−m)/m ˜ X X = d 2dX + (1 − m)c + (2 − k)νk , X˜ m X˜ k−1 ( k=1 ) q d = d = ··· = d = d(1−m)/m + 2 , 12 13 1n ˜ ··· ˜ ( X1 X)m m+1 ∑m k (1−m)/m ˜ X X = d dX + c + νk , ˜ m ˜ k−1 X k=1 X
(1−m)/m ˜ δ = d (dX1), = d1/mX,˜ ( ) m+1 ∑m k (1−m)/m X1 X1 ρ = d (m + 1)c + kνk , X˜ m X˜ k−1 ( 1 k=1 1 ) m+1 ∑m k (1−m)/m X X = d (m + 1) + kνk . ˜ m ˜ k−1 X k=1 X
(In the expressions above, the second line is obtained by evaluation on vacua.)
Putting this together, we can write [ ] det ζ βαn−1 det |Ji,j| = det ρδm−1 det η [ ] αn−1(a + (n − 1)a ) αn−1β = det 11 12 m−1 m−1 δ ρ δ (d11 + (n − 1)d12) 39
which is easily checked to be the determinant of ∑ n n i n−i (n + 1)aX + (n + 1 − i)µ − X˜ X i=1 n+1 i ∑ ˜ n+1 −1 n ˜ i n−i (n + 1)bX X + i=1 iµn+1−iX X ∑ . (m − 1)cXm+1X˜ −1 + m kν XkX˜ m−k k=1 k ∑ − ˜ m m − k ˜ m−k (m 1)dX + k=1(m + 1 k)νkX X
˜ By identifying Xi with σ1 and Xk with σ2, we see that the determinant above matches (2.15).
Thus, all genus-zero correlation functions in our proposed Toda dual match those of the (0,2) theory on Pn × Pm with a deformation of the tangent bundle. In addition to constructing a general argument that correlation functions should match, we have also compared correlation functions in special cases, as we shall outline next.
2.3 Example: Toda-like duals to P1 × P1
In this section, we’d like to study a special case P1 × P1 in detail.
2.3.1 The (0,2) NLSM
In the case of X = P1 × P1, one can describe a general deformation E of the tangent bundle as the cokernel of the following sequence:
0 −→ O ⊗ O −→OE (1, 0)2 ⊕ O(0, 1)2 −→ E −→ 0, where [ ] Ax Bx E = , Cx˜ Dx˜ with A, B, C, D 2 × 2 matrices and [ ] [ ] x x˜ x = 1 , x˜ = 1 x2 x˜2 are homogeneous coordinates on the two P1 factors. The tangent bundle corresponds to A = D = I, and B = C = 0. For more general A, B, C, D, the vector bundle is (generically) 40
a deformation of the tangent bundle. In this model, it has been argued in [51,52,64,67] that the OPE ring relations in the A/2 twist (defining the quantum sheaf cohomology ring) are given by
˜ det(Aψ + Bψ) = q1, (2.20) ˜ det(Cψ + Dψ) = q2. (2.21)
Correlation functions in A/2 twisted theories on P1 × P1 with a deformation of the tangent bundle can be computed in several ways. One method is to use direct Cech techniques to compute sheaf cohomology products on P1 × P1, as has been discussed in e.g. [50, 54, 68]. Another method is to use GLSM-based Coulomb branch results, as described in [67]. A third, more recent, method is to use residue formulas obtrained via localization, as in [64]. In this last approach, correlation functions in the A/2 twisted theory on P1 × P1 are of the form6
⟨ ˜ ⟩ f(ψ, ψ) ( ) ∑ 1 1 k1 k2 − ˜ = q1 q2 JKG Res f(ψ, ψ) . det(Aψ + Bψ˜)k1+1 det(Cψ + Dψ˜)k2+1 k1,k2
However one computes the correlation functions, the results have the following form, in terms of the matrices A, B, C, D above. Let
a = det(A), b = det(B), c = det(C), d = det(D),
e = det(A + B), f = det(C + D).
Define µ = e − a − b, ν = f − c − d,
6As a matter of principle, there is a phase ambiguity in expressions of this form, due geometrically to ∗ ∼ possible phases of the isomorphism det E −→ KX , and physically to chiral left and right global U(1) actions on the worldsheet, that play a role closely analogous to that of the Bagger-Witten line bundle. The expression given here implicitly determines such phases. 41
ϕ1 = νb − µd = ad + bf − de − bc,
ϕ2 = ad − bc,
ϕ3 = µc − νa = ad + ce − af − bc,
2 − ∆ = ϕ2 ϕ1ϕ3, = (c − d)(bc − ad)e + cde2 + (a − b)(ad − bc)f − (bc + ad)ef + abf 2.
Then the two-point correlation functions, for example, can be expressed as:
ϕ ϕ ϕ ⟨ψψ⟩ = 1 , ⟨ψψ˜⟩ = 2 , ⟨ψ˜ψ˜⟩ = 3 . (2.22) ∆ ∆ ∆
Higher-point correlation functions have a similar form. We list four-point functions in this A/2-twisted theory in section 2.6.1. More general correlation functions at genus zero are straightforward to compute with residue techniques, but the resulting expressions are rather unwieldy, so we do not include them in this chapter.
2.3.2 The Toda-like mirror theory
We claim the mirror theory to the A/2 twisted theory just described, is a (0,2) Landau- Ginzburg model, defined by a (0,2) superpotential of the form
W = FJ + F˜J,˜ (2.23) where F and F˜ are Fermi superfields, and
X˜ 2 q J = X−1(det(AX + BX˜) − q ) = aX + b + µX˜ − 1 , (2.24) 1 X X 2 ˜ ˜ −1 ˜ ˜ X q2 J = X (det(CX + DX) − q2) = dX + c + νX − , (2.25) X˜ X˜
for X = exp(Y ), X˜ = exp(Y˜ ), where Y , Y˜ are the fundamental fields, and
a = det A, b = det B, c = det C, d = det D, 42
µ = det(A + B) − det A − det B, ν = det(C + D) − det C − det D, for A, B, C, D the matrices defining the tangent bundle deformation of the A/2 theory.
In passing, the form written here does not manifestly match the expression in [47] for the special case they considered. In section 2.3.4, we will study various field redefinitions yielding non-obviously-equivalent expressions, and discover the expression in [47] arising as a special case.
We will check the ansatz above by comparing correlation functions between the original A/2 theory and the B/2 twist of the Landau-Ginzburg theory above, but first, let us make a few quick observations.
As one consistency check, note that for
A = D = I,B = C = 0,
then the vector bundle E is the tangent bundle, and the theory has (2,2) supersymmetry. This also can be seen from the (0,2) superpotential ( ) ( ) q q W = F X − 1 + F˜ X˜ − 2 , X X˜
which matches the (2,2) superpotential in this case.
As another check, note that the space of classical vacua of this theory (J = J˜ = 0) matches the space of solutions to the quantum sheaf coholomogy ring relations:
˜ det(AX + BX) = q1, (2.26) ˜ det(CX + DX) = q2. (2.27)
Now, let us compute and compare genus zero correlation functions. Given a B/2-twisted Landau-Ginzburg model with superpotential W over a vector space or a product of C×’s, 43 correlation functions at genus zero are given by7 [66] ∑ i1 ik i1 ik −1 ⟨ϕ (x1) ··· ϕ (xk)⟩ = ϕ (x1) ··· ϕ (xk)[det Ji,j] (2.28) i,j Ji(ϕ)=0 where the sum is over classical vacua.
Using the formula above for B/2-twisted Landau-Ginzburg correlation functions, one finds that the two-point correlation functions in this model are given by
⟨XX⟩ = ∆−1(bν − dµ), ⟨XX˜⟩ = ∆−1(ad − bc), ⟨X˜X˜⟩ = ∆−1(cµ − aν), where ∆ = b2c2 − 2abcd + a2d2 + cdµ2 − (bc + ad)µν + abν2.
These match the A/2 correlation functions in equation (2.22), if we identify X with ψ and X˜ with ψ˜.
We also checked that all four-point functions for general A, B, C, D (as listed in section 2.6.2) match the results from the A/2 model. For the special case in which det B = det C = 0, we have checked that all correlation functions up to ten-point correlation functions and one twelve-point correlation function ⟨X6X˜ 6⟩ match the results from the A/2 model.
Beyond special cases, there is also a general argument that all correlation functions must match. We will utilize a formula for the A/2 model correlation functions given in [67][section 3.4], which is similar in form to the formula above for B/2 Landau-Ginzburg model correla- tion functions, and argue that after some algebra, the formula for A/2 correlation functions in [67] matches the formula for B/2 correlation functions above. As a result, all correlation functions in our B/2-twisted Landau-Ginzburg model will necessarily match those of the
7Correlation functions for more general B/2-twisted Landau-Ginzburg models are discussed in [63]. In passing, we should comment on the absence of worldsheet instanton corrections to the formulas above. On the (2,2) locus, the Toda duals to A model topological field theories are B-twisted, and correlation functions in the B model do not have worldsheet instanton corrections. In the present case, our Toda-like mirrors to A/2 model pseudo-topological field theories are B/2 twisted. Unlike the (2,2) case, however, in general B/2 twisted models can and will receive worldsheet instanton corrections. However, our Toda-like theories are defined by superpotentials over algebraic tori, i.e. (C×)n, and there are no non-constant holomorphic maps from P1 (or any projective variety) to an algebraic torus. All holomorphic maps are constant maps, hence there are no worldsheet instanton corrections in these theories [70]. Thus, we need only compute classically in the B/2 model, just as in ordinary Toda mirrors. 44
A/2 nonlinear sigma model.
Let us describe this argument for general matching correlation functions. From [67][section 3.4], all correlation functions in an A/2-twisted (0,2) nonlinear sigma model on P1 × P1, at genus zero, take the form
⟨f(ψ, ψ˜)⟩ [( ) ] ∑ ( ) ( ) −1 ˜ ˜ ˜ = f(ψ, ψ) det Ja,b det Aψ + Bψ det Cψ + Dψ , a,b ψ,ψ˜|J =0 a [ ] −1 ∑ ∂ det(Aψ + Bψ˜) ∂ det(Aψ + Bψ˜) = f(ψ, ψ˜) det ψ ψ˜ ˜ ˜ ∂ψ det(Cψ + Dψ) ∂ψ˜ det(Cψ + Dψ) ψ,ψ˜|Ja=0
˜ where the Ja (not to be confused with the J, J we used in our dual theory earlier) are defined by ( ( )) −1 ˜ J1 = ln q det Aψ + Bψ , ( 1 ( )) J −1 ˜ 2 = ln q2 det Cψ + Dψ .
To compare the correlation functions above with the B/2 correlation functions in our dual ˜ theory, which take a similar form, first note that the constraint Ja = 0 implies
˜ ˜ det(Aψ + Bψ) = q1, det(Cψ + Dψ) = q2, the quantum sheaf cohomology relations and also the relations defining the vacua of the B/2 Landau-Ginzburg model. Then, matching follows as a consequence of ( ) ( ) ( ) ˜ ˜ det Ji,j = det Ja,b det Aψ + Bψ det Cψ + Dψ , i,j a,b or more explicitly ( ) ( ) ˜ 2 ˜ − ˜ 2 ˜ − ∂Y aX + bX /X + µX q1/X ∂Y˜ aX + bX /X + µX q1/X det ( ) ( ) ˜ 2 ˜ ˜ ˜ 2 ˜ ˜ ∂Y dX + cX /X + νX − q2/X ∂ ˜ dX + cX /X + νX − q2/X [ Y ] ∂ det(Aψ + Bψ˜) ∂ det(Aψ + Bψ˜) = det ψ ψ˜ , ˜ ˜ ∂ψ det(Cψ + Dψ) ∂ψ˜ det(Cψ + Dψ) 45
where X = exp(Y ), X˜ = exp(Y˜ ), after identifying X with ψ and X˜ with ψ˜, which is straightforward to verify. Thus, all genus zero correlation functions of our B/2 Landau- Ginzburg model, the proposed dual to P1 × P1, do indeed match the correlation functions of the (0,2) theory on P1 × P1.
2.3.3 Moduli
On the face of it, the correlation functions above are determined by six numbers:
det A, det B, det C, det D, det(A + B), det(C + D),
(in addition, of course, to q1, q2). Not all of the individual elements of each of the four matrices A, B, C, D are pertinent, essentially because this theory admits global GL(2) actions rotating those matrices. In addition, in principle field redefinitions could be used to also eliminate some of the parameters above.
Mathematically, the tangent bundle of P1 ×P1 also has six moduli (as counted in section 2.7), matching the count above. However, if one deforms to a finite distance away from the tangent bundle, the number of mathematical bundle moduli (counted by H1(P1 × P1, End E) for bundle E) may drop, as we discuss in section 2.7. Furthermore, not all of those moduli need necessarily be expressible monadically, as polynomial deformations of the GLSM, so the true number of parameters that the GLSM can access may be significantly smaller (reflecting e.g. the symmetries and field redefinitions mentioned above). We will see this in an example in section 2.3.4, where we will take models with matrices B such that det B ≠ 0, and construct equivalent theories with det B = 0.
2.3.4 Redundancies and equivalent descriptions
As the moduli counts in the last section suggest, our description of the theories in terms of four matrices A, B, C, D has a great deal of redundancy. This can be expressed in the fact that there are three GL(2) actions8 on these matrices. Specifically, three matrices P , Q, R
8We would like to thank R. Donagi for making this observation originally. See also a related discussion in [69]. 46
each in GL(2) act on the matrices A, B, C, D as follows: [ ] [ ] AB PAPB 7→ R CD QC QD
at the same time that [ ] [ ] ψ ψ 7→ R , ψ˜ ψ˜ and
q1 7→ (det P )q1, q2 7→ (det Q)q2.
Of course, these three GL(2) actions are not completely independent, but in broad brush- strokes, they are the reason that there are no more than six independent moduli yet sixteen naive parameters (the elements of the four 2 × 2 matrices).
To understand how correlation functions behave, let us consider a residue expression for correlation functions from [64]:
⟨f(ψ, ψ˜)⟩ ∑ f(ψ, ψ˜) k1 k2 − = q1 q2 JKG Res . (det(Aψ + Bψ˜))k1+1 det(Cψ + Dψ˜))k2+1 k1,k2
Formally, if we rotate ψ, ψ˜ by the matrix R at the same time that A, B, C, D are also rotated by R, the new resulting expression is equivalent to the original one, after a linear field redefinition. In other words,
1 ⟨f(R(ψ, ψ˜))⟩ = ⟨f(ψ, ψ˜)⟩ . R(A,B,C,D) | det R| A,B,C,D
That said, the expressions for correlation functions we utilize in this chapter assume that A and D are both invertible, and a general R-rotation could change that. In such cases, the pole prescription implicit in the definition of the JKG residue in [64] would yield different results, so one should be careful in applying the formal statement above.
An example of such equivalences is as follows. Define β to be a solution of
(det A)β2 + µβ + (det B) = 0, 47 i.e. ( ) 1 √ β = − µ µ2 − 4ab , 2a (where A is assumed invertible,) and let us assume C = 0. Take [ ] 1 β R = . 0 1
This matrix R rotates B to a noninvertible matrix. Specifically, under the action of R,
B 7→ B′ = B + βA, and the other matrices are invariant. It is straightforward to check that det B′ = 0. In principle, correlation functions in the original theory should match correlation functions with these parameters so long as ψ, ψ˜ are suitably rotated:
˜ ˜ ˜ ⟨f(ψ, ψ)⟩original = ⟨f(ψ + βψ, ψ + γψ)⟩new.
Now, having constructed an equivalent model for which det B′ = 0, we can construct the dual (0,2) Landau-Ginzburg theory. This is defined by the (0,2) superpotential with
q J ′ = aX + µ′X˜ − 1 , X q J˜′ = dX˜ + ν′X − 2 , X˜ where
µ′ = det(A + B′) − det A − det B′, ν′ = det(C + D) − det C − det D = ν.
This is just the specialization of our previous proposed dual to case with B′ instead of B and with C = 0, so that the X˜ 2/X and X2/X˜ terms vanish.
Given the rotation on the original ψ, ψ˜, we see that in principle the original correlation functions should match the correlation functions in the final Landau-Ginzburg model above as ˜ ˜ ˜ ⟨f(ψ, ψ)⟩original = ⟨f(X + βX, X)⟩final. 48
Now, let us turn to a particular special case, appearing in [47]. This special case is the sole previous example of a (0,2) Landau-Ginzburg mirror to a A/2-twisted theory that had previously appeared in the literature. More to the point, this sole example in the literature does not fit the pattern we have discussed in previous sections, and instead is related to them via a field redefinition of the form discussed in this section.
Specifically, let us consider the case [ ] ϵ 0 A = D = I,C = 0,B = 1 . 0 ϵ2
Following the methods we have discussed prior to this section, the dual Landau-Ginzburg theory has the parameters
a = 1, b = ϵ1ϵ2, c = 0, d = 1, µ = ϵ1 + ϵ2, ν = 0, and superpotential ( ) ( ) ˜ 2 X ˜ q1 ˜ ˜ q2 W = F X + ϵ1ϵ2 + (ϵ1 + ϵ2)X − + F X − . X X X˜
The two-point correlation functions in this theory, for example, are
⟨XX⟩ = −(ϵ1 + ϵ2), ⟨XX˜⟩ = 1, ⟨X˜X˜⟩ = 0.
Unfortunately, although this does correctly capture the A/2 correlation functions, neither the superpotential nor the correlation functions above match those given in [47] as the dual.
To find the presentation of the dual given in [47], one must instead perform a R-rotation of the sort described above, rotating B and C to noninvertible matrices. One then computes
1 β = (−ϵ − ϵ + (ϵ − ϵ )) = −ϵ , 2 1 2 1 2 2 γ = 0, 49
(taking the positive square root in β). After transforming by [ ] [ ] 1 β 1 −ϵ R = = 2 , γ 1 0 1 one has the new dual defined by parameters [ ] ϵ − ϵ 0 a′ = 1, b′ = det 1 2 = 0, c′ = 0, d′ = 1, 0 0
′ ′ ′ ′ ′ µ = det(A + B ) − det A − det B = ϵ1 − ϵ2,
ν′ = det(C′ + D′) − det C′ − det D′ = 0, hence the superpotential ( ) ( ) ˜ q1 ˜ ˜ q2 W = F X + (ϵ1 − ϵ2)X − + F X − . X X˜
From the results in section 2.6.2, the two-point functions in this Landau-Ginzburg model are given by ˜ ˜ ˜ ⟨XX⟩ = ϵ2 − ϵ1, ⟨XX⟩ = 1, ⟨XX⟩ = 0, matching the results of [47].
Also note that, in this same theory,
⟨ − ˜ 2⟩ ⟨ 2⟩ − ⟨ ˜⟩ 2⟨ ˜ 2⟩ − − (X ϵ2X) = X 2ϵ2 XX + ϵ2 X = ϵ1 ϵ2, ˜ ˜ ˜ ˜ 2 ⟨(X − ϵ2X)X⟩ = ⟨XX⟩ − ϵ2⟨X ⟩ = 1, ⟨X˜X˜⟩ = 0, matching the correlation functions of the original A/2-twisted theory, as expected. In [54], the change of variables above was given to correlate A/2 correlation functions with those of the proposed dual theory, and here we see that this is a special case of a much more general redundancy in the description. 50
2.4 Del Pezzo surfaces
In this section, we will discuss mirrors to toric del Pezzo surfaces. We will use the notation 2 dPk to indicate P blown up at k points.
2.4.1 The first del Pezzo surface, dP1
2 The first del Pezzo surface we will consider, dP1, corresponding to a single blowup of P , is isomorphic to the first Hirzebruch surface F1. As mirrors to higher del Pezzo surfaces will be constructed on the ‘foundation’ of dP1, let us begin by describing its (2,2) and (0,2) mirrors.
(Section 2.8 reviews some standard results on quantum cohomology of dP1, standard in the math community but perhaps less well-known in the physics community, that are pertinent for the mirror.)
(2,2) and proposed (0,2) mirrors
The del Pezzo surface dP1 can be described as a toric variety by a fan with edges (1, 0), (0, 1),
(−1, −1), (0, −1). A corresponding GLSM is defined by four chiral superfields ϕi, i = 1 ... 4 charged under the gauge group U(1) × U(1) as follows:
(1,0) (-1,-1) (0,1) (0,-1) 1 1 1 0 0 0 1 1
The quantum cohomology relations9 are
2 ˜ ψ (ψ + ψ) = q1, ˜ ˜ (ψ + ψ)ψ = q2.
As reviewed in section 2.1.1, the (2,2) mirror to a sigma model on dP1 = F1 is [93] a Landau-
9Note that example 7.3 in [107] gives the same quantum cohomology ring relations after identifying ψ ∼ f, ˜ ∼ ∼ −1 ∼ ψ e, q1 r, q1q2 q. 51
Ginzburg theory with superpotential
W = exp(Y1) + exp(Y2) + exp(Y3) + exp(Y4), where the fields obey the constraints
Y1 + Y2 + Y3 = r1,Y3 + Y4 = r2.
We will describe ansatzes for (0,2) mirrors based on two different solutions of the constraints above.
Our first description of the (2,2) B-twisted mirror to the A-twisted theory is written in terms of Y1 and Y3. Define X1 = exp(Y1) and X3 = exp(Y3), then the mirror can be described as a Landau-Ginzburg model over (C×)2 with superpotential
q2 q1 W = X1 + X3 + + . (2.29) X3 X1X3
(This matches the mirror given in [93][equ’n (5.19)].)
An alternative description of the mirror to the same theory is written in terms of Y1 and Y4.
Define X1 = exp(Y1) and X4 = exp(Y4), then on the (2, 2) locus, the mirror superpotential is q1 X4 q2 W = X1 + X4 + + . (2.30) q2 X1 X4 On the (2,2) locus, this can be related to the previous expression via the field redefinition
q2 X4 = . X3
(Analogous field redefinitions can be computed to relate the (0,2) mirrors we discuss next, but their expressions for general parameters are both extremely unwieldy and unhelpful, so we omit them from this chapter.)
The (0, 2) deformations of dP1 are defined by a pair of 2 × 2 matrices A, B, and complex numbers γ1, γ2, α1, α2, that define a deformation E of the tangent bundle
0 −→ O⊕2 −→OE (1, 0)⊕2 ⊕ O(1, 1) ⊕ O(0, 1) −→ E −→ 0, 52 where E is Ax Bx E = γ1s γ2s ,
α1t α2t with [ ] u x = . v The (2, 2) locus is given by the special case
A = I,B = 0, γ1 = 1, γ2 = 1, α1 = 0, α2 = 1.
If we define
˜ ˜ ˜ Q(k) = det(ψA + ψB),Q(s) = ψγ1 + ψγ2,Q(t) = ψα1 + ψα2, then the quantum sheaf cohomology ring relations are given by [77]
Q(k)Q(s) = q1,Q(s)Q(t) = q2. (2.31)
Next, we shall give an ansatz for a (B/2-twisted) (0,2) Landau-Ginzburg theory which is mirror to the A/2 model on dP1 with deformed tangent bundle as above. For readers not familiar with (0,2) Landau-Ginzburg models, the analogue of the superpotential interactions are described in superspace in the form ∑ ∫ i dθΛ Ji(Φ), i
α where the Jα are a set of holomorphic functions and Λ are Fermi superfields (forming half of a (2,2) chiral superfield). This reduces to a (2,2) superpotential in the special case that
Ji = ∂iW for some holomorphic function W .
Our proposal for the (0, 2) Toda-like mirror of the A/2 model on dP1 = F1 with a deformation 53
of the tangent bundle is defined by
2 (X3 − X1) q1 J1 = aX1 + µAB(X3 − X1) + b − , (2.32) X1 X1(γ1X1 + γ2(X3 − X1)) (X − X )2 q J = aX + µ (X − X ) + b 3 1 − 1 2 1 AB 3 1 X X (γ X + γ (X − X )) (( 1 )( 1 1 1 2 3 )) 1 q −1 − − − 2 + X3 γ1X1 + γ2(X3 X1) α1X1 + α2(X3 X1) . (2.33) X3
(Because the J’s have poles away from origins, we interpret the resulting action in a low- energy effective field theory sense, as discussed in the introduction.)
We have chosen the labels on the J’s to match q’s, but that also means they are slightly
inconsistent with bosons on the (2,2) locus. Here, for example, J2 on the (2,2) locus corre-
sponds to the Y3 derivative of W .
It is straightforward to check that the J’s above have the correct (2,2) locus, and that they
are invariant under the U(1)V but the U(1)A symmetry is classically broken in the fashion expected.
Previously we gave two forms for the B-twisted Landau-Ginzburg mirror to dP1, on the (2,2) locus. So far, we have given the mirror that reduces on the (2,2) locus to the first form. An expression for a (0,2) mirror that reduces on the (2,2) locus to the second form is
2 X4 q1 α1X1 + α2X4 J1 = aX1 + µABX4 + b − , (2.34) X1 q2 X1 2 X1 q1 (α1X1 + α2X4)(γ1α2 + γ2α1) − q2 J2 = α2γ2X4 + α1γ1 + 2 −1 . (2.35) X4 q2 aX1 + µABX4 + bX4 X1 X4
As above, we have chosen subscripts on the J’s to match q’s, which means that J2 on the
(2,2) locus corresponds to the Y4 derivative of W .
As above, it is straightforward to check that the J’s above have the correct (2,2) locus, and that they are invariant under the U(1)V but the U(1)A symmetry is classically broken in the fashion expected.
We will check our proposal by arguing that all genus zero A/2 model correlation functions will match those of the B/2-twisted mirror Landau-Ginzburg theory given above, using a variation of an argument in [1] which can be adapted to apply to potential (0,2) Landau- 54
Ginzburg model mirrors to any toric variety realized as a GLSM.
Given a B/2 Landau-Ginzburg model with a superpotential Ji, the genus zero correlation functions are given by [85] [ ] ∑ −1 i1 ik i1 ik ⟨ϕ (x1) . . . ϕ (xk)⟩ = ϕ (x1) . . . ϕ (xk) det Ji,j , (2.36) i,j Ji(ϕ)=0 where the sum is taken over the classical vacua.
From [86], the one-loop effective theory is described by the following J functions in general: [ ] ∏ qa J −1 (α) a = ln qa Q(α) , α where Q(α) encodes the tangent bundle deformations (as opposed to gauge charges). In the present case of dP1, the superpotential is given by [ ] J = ln q−1 det(Aψ + Bψ˜)(ψγ + ψγ˜ ) , (2.37) 1 [ 1 1 ]2 J −1 ˜ ˜ 2 = ln q2 (ψγ1 + ψγ2)(ψα1 + ψα2) , (2.38) and the correlation functions are given by [ ] ∑ ∏ −1 ˜ ˜ ⟨f(ψ, ψ)⟩ = f(ψ, ψ) det Ja,b Q(α) . a,b J =0 α
Comparing to the formula calculating the correlation functions of Toda dual Landau-Ginzburg model (2.36), in order to claim the correlation functions match, we only need to verify ∏ det |Ji,j| = det |Ja,b| Q(α) (2.39) a,b α
˜ on the space of vacua after identifying X1 with ψ and X2 with ψ. Expanding the right side of above formula, we get [ ] 1 1 Qn ∂ψ(Q(k)Q(s)) Qn ∂ψ˜(Q(k)Q(s)) det (s) (s) . ∂ψ(Q(s)Q(t)) ∂ψ˜(Q(s)Q(t)) 55
One can then easily verify equation (2.39) holds on the space of vacua with X1 ∼ ψ and ˜ X3 ∼ nψ + ψ for both of the presentations of (0,2) mirrors we have given here.
We will use analogous arguments throughout this chapter to compare genus zero correlation functions in proposed (0,2) mirrors to the original A/2 theories, but for brevity in later sections will only mention the result, not walk through the details of the computation.
So far we have checked that the genus zero correlation functions in this proposed (0,2) mirror
to dP1 match those of the original A/2-twisted theory. In the next section, we will check that there is an analogue of a blowdown in the mirror. In later sections we will describe proposals for (0,2) mirrors to higher del Pezzo surfaces that blow down to this proposal, and we will also describe a family of proposals for (0,2) mirrors to Hirzebruch surfaces that include the 1 1 proposal of this section for dP1 = F1 as well as our earlier proposal for P × P as special cases.
Consistency check: mirrors of blowdowns
2 Geometrically, dP1 can be blown down to P , which is visible in the toric fan in figure 2.1 by removing the edge (0, −1). In the GLSM, although in general K¨ahlermoduli of non-Calabi- Yau manifolds need not correspond to operators in the physical theory, it is nevertheless straightforward to see that there is an analogous limit10 in which one recovers P2. The (2,2)
mirror of this blowdown is manifest that we only need to take the limit q2 → 0 in (2.29), which reduces to the Toda dual superpotential of P2.
In this section we will show that the blowdown limit of the (0,2) mirror of dP1 with a tangent bundle deformations is also equivalent (as a UV theory) to the mirror of P2. This will provide a consistency test of our proposed (0,2) mirror.
To that end, it will be helpful to first revisit the (2,2) case, albeit in (0,2) language. Recall
1 2 W = Λ J1 + Λ J2,
10The same statement will be true of the other blowdown examples considered in this chapter – all involving blowups of Fano spaces at smooth points. 56
(0, 1)
(1, 0)
(−1, −1)
Figure 2.1: A toric fan of P2 can be obtained by removing the edge (0, −1) from the toric fan of dP1.
where Λi, i = 1, 2 are Fermi superfields, and
q1 J1 = X1 − , X1X3 q1 J2 = X3 − . X1X3
We can rewrite the (0,2) superpotential as follows,
˜ 1 ˜ ˜ 2 ˜ W = Λ J1 + Λ J2,
where Λ˜ 1 = Λ1 + Λ2, Λ˜ 2 = Λ2,
and
˜ q1 J1 = J1 = X1 − , X1X3 ˜ J2 = J2 − J1 = X3 − X1.
Then, one can integrate out the Fermi superfield Λ˜ 2 and obtain a constraint,
X1 = X3. 57
Plugging the constraint back in, we get ( ) q ˜ 1 ˜ ˜ 1 − 1 W = Λ J1 = Λ X1 2 . (2.40) X1
Now let us analyze the (0,2) superpotential of dP1 in the blowdown limit q2 → 0,
1 2 W = Λ J1 + Λ J2, where Λi, i = 1, 2 are Fermi superfields, and
2 (X3 − X1) q1 J1 = aX1 + µAB(X3 − X1) + b − , X1 X1(γ1X1 + γ2(X3 − X1)) (X − X )2 q J = aX + µ (X − X ) + b 3 1 − 1 2 1 AB 3 1 X X (γ X + γ (X − X )) (( 1 )( 1 1 1 2 3 )) 1 −1 − − + X3 γ1X1 + γ2(X3 X1) α1X1 + α2(X3 X1) .
We can rewrite it as ˜ 1 ˜ ˜ 2 ˜ W = Λ J1 + Λ J2, where Λ˜ 1 = Λ1 + Λ2, Λ˜ 2 = Λ2, and
− 2 ˜ (X3 X1) q1 J1 = J1 = aX1 + µAB(X3 − X1) + b − , X1 X1(γ1X1 + γ2(X3 − X1)) ˜ − −1 − − J2 = J2 J1 = +X3 ((γ1X1 + γ2(X3 X1)) (α1X1 + α2(X3 X1))) .
˜ 2 Then, we integrate out Λ and obtain the following constraint on X1,X3:
˜ −1 − − J2 = X3 ((γ1X1 + γ2(X3 X1)) (α1X1 + α2(X3 X1))) = 0.
Notice that γ1X1 + γ2(X3 − X1) ≠ 0 since it is in the denominator of J1. Solving the constraint, one obtain the relation
α2 − α1 X3 = X1, α2 58
where for simplicity we have assumed α2 ≠ 0. ˜ Lastly, plugging the above relation back into J1, we find ( ) ( ) q W = Λ˜ 1J˜ = Λ˜ 1 a − µ α α−1 + bα2α−2 X − ( 1 ) . 1 AB 1 2 1 2 1 − −1 2 γ1 γ2α1α2 X1
(We assume for simplicity that γ1 ≠ γ2α1/α2.) One can easily see the above superpotential is equivalent to (2.40) for the mirror to P2, after suitable field redefinitions. Thus, as expected, mirrors and blowdowns commute with one another.
2.4.2 The second del Pezzo surface, dP2
Review of the (2,2) mirror
2 The next del Pezzo surface, dP2, is P blown up at two points, which can be described as a toric variety by a fan with edges (1, 0), (0, 1), (−1, −1), (0, −1), (−1, 0). The gauged linear
sigma model has five chiral superfields ϕi, i = 1 ... 5 which are charged under the gauge group U(1)3 as follows:
(1,0) (-1,-1) (0,1) (0,-1) (-1,0) 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1
The quantum cohomology relations of the A-twisted theory are
(ψ1 + ψ3)ψ1(ψ1 + ψ2) = q1,
(ψ1 + ψ2)ψ2 = q2,
(ψ1 + ψ3)ψ3 = q3.
As reviewed in section 2.1.1, the superpotential of the (2,2) mirror theory is
∑5 W = exp(+Yi) i=1 59
where the Yi obey constraints
Y1 + Y2 + Y3 = r1,Y3 + Y4 = r2,Y1 + Y5 = r3.
One solution is to solve the constraints for Y4 and Y5. Defining X4 = exp(Y4) and X5 =
exp(Y5), the superpotential is then
q3 q1 q2 W = X4 + X5 + + X4X5 + . (2.41) X5 q2q3 X4
(The mirror map relates ψ2 ∼ X4 and X5 ∼ ψ3.)
However, we will not use the form of the Toda dual above when building (0,2) deformations. Instead, we will use an alternative form of the Toda dual, which is obtained by retaining an explicit Lagrange multiplier Z, so that one of the constraints naturally embeds into the superpotential, ( ) q1 q2 q3 W = X1 + X3 + X5 + + + Z 1 − , (2.42) X1X3 X3 X1X5
for Xi = exp(Yi). (If we solve the constraint by taking X1 = q3/X5, then this form can
be related to the previous expression by the holomorphic coordinate transformation X4 =
q2/X3.) On the space of vacua which is given by,
q1 q3 X1∂1W = X1 − + Z = 0, X1X3 X1X5 q1 q2 X3∂3W = X3 − − = 0, X1X3 X3 q3 X5∂5W = X5 + Z = 0, X1X5 q3 ∂Z W = 1 − = 0, X1X5
where ∂ ∂i = . ∂Xi The quantum cohomology relations are satisfied with the identifications
X1 ∼ ψ1 + ψ3,X3 ∼ ψ1 + ψ2,X5 ∼ ψ3,
It is straightforward to check that the vector R symmetry is preserved, but the axial R 60 symmetry is broken, as expected. Furthermore, it is easy to check that all correlation 11 functions of the alternative description match those of A-twisted theory for dP2. This alternative description turns out to be convenient for constructing ansatzes for (0,2) mirrors.
The choice of constraint embedding in the superpotential should be arbitrary. For example, we could also take the mirror superpotential to be ( ) q2 q3 q1 W = X1 + X2 + X3 + + + Z 1 − , (2.43) X3 X1 X1X2X3 with vacua,
q3 q1 X1∂1W = X1 − + Z = 0, X1 X1X2X3 q1 X2∂2W = X2 + Z = 0, X1X2X3 q2 q1 X3∂3W = X3 − + Z = 0, X3 X1X2X3 q1 ∂Z W = 1 − = 0. X1X2X3
This can be related to the previous form using the holomorphic coordinate transformation
X2 = (q1/q3)(X5/X3). We will also describe (0,2) deformations of this presentation.
(0,2) deformations and proposed (0,2) mirrors
The (0, 2) deformation of dP2 is defined by fifteen complex numbers αi, βj, γk, δm, ϵn, with i, j, k, m, n = 1, 2, 3, which define a deformation E of the tangent bundle as follows:
0 −→ O3 −→OE (1, 0, 1) ⊕ O(1, 0, 0) ⊕ O(1, 1, 0) ⊕ O(0, 1, 0) ⊕ O(0, 0, 1) −→ E −→ 0,
11Technically, keeping an explicit Lagrange multiplier turns out to introduce a sign in correlation functions, which can easily be accounted for. 61 where α s α s α s 1 1 2 1 3 1 β1s2 β2s2 β3s2 E = γ1s3 γ2s3 γ3s3 . δ1s4 δ2s4 δ3s4
ϵ1s5 ϵ2s5 ϵ3s5 E reduces to the tangent bundle when
α1 = 1, α2 = 0, α3 = 1,
β1 = 1, β2 = β3 = 0,
γ1 = γ2 = 1, γ3 = 0,
δ1 = 0, δ2 = 1, δ3 = 0,
ϵ1 = ϵ2 = 0, ϵ3 = 1.
The quantum sheaf cohomology relations are
Q(1)Q(2)Q(3) = q1, (2.44)
Q(3)Q(4) = q2, (2.45)
Q(1)Q(5) = q3, (2.46) where
∑3 ∑3 ∑3 Q(1) = αiψi,Q(2) = βiψi,Q(3) = γiψi, i=1 i=1 i=1 ∑3 ∑3 Q(4) = δiψi,Q(5) = ϵiψi. i=1 i=1
We will propose below two (0,2) Toda-like mirrors based on the (2,2) mirrors (2.42) and (2.43) which have a Lagrange multiplier (labelled Z). 62
Our first (0,2) mirror proposal for dP2 is defined by the following four holomorphic functions
q1 q3 (α · X)(ϵ · X) (α · X)(β · X) J1 = − + Z + + , (2.47) X1(γ · X) X1(ϵ · X) X1 X1
q2 q1 (α · X)(β · X) (γ · X)(δ · X) J3 = − − + + , (2.48) X3 X1(γ · X) X1 X3 q J = (ϵ · X) + Z 3 , (2.49) 5 (α · X)(ϵ · X)
(ϵ · X) q3 JZ = − , (2.50) X5 X5(α · X) where on the (2,2) locus, it can be shown that
∂W ∂W ∂W ∂W J1 = ,J3 = ,J5 = ,JZ = , ∂Y1 ∂Y3 ∂Y5 ∂Z for W defined by (2.42), and
(α · X) = α1(X1 − X5) + α2(X3 − X1 + X5) + α3X5, . .
(ϵ · X) = ϵ1(X1 − X5) + ϵ2(X3 − X1 + X5) + ϵ3X5.
Because the J’s have poles away from origins, we interpret the resulting action in a low- energy effective field theory sense, as discussed in the introduction. It is straightforward to check that this proposal has the correct (2,2) locus.
Our second (0,2) mirror ansatz for dP2 is defined by the following data:
q3 (α · X)(ϵ · X) q1 J1 = − + + (β · X) + Z , (2.51) X1 X1 (α · X)(β · X)(γ · X) q J = (β · X) + Z 1 , (2.52) 2 (α · X)(β · X)(γ · X)
q2 (γ · X)(δ · X) q1 J3 = − + + (β · X) + Z , (2.53) X3 X3 (α · X)(β · X)(γ · X)
(β · X) q1 JZ = + − , (2.54) X2 X2(α · X)(γ · X) 63
where
(α · X) = α1X2 + α2(X3 − X2) + α3(X1 − X2), . .
(ϵ · X) = ϵ1X2 + ϵ2(X3 − X2) + ϵ3(X1 − X2).
On the (2, 2) locus, the above data reduces to (2.43) (in the sense that each Ji becomes a suitable derivative of W ).
With the identifications,
X1 ∼ ψ1 + ψ3,X2 ∼ ψ1,X3 ∼ ψ1 + ψ2,X5 ∼ ψ3,
both proposals pass our standard consistency checks: the quantum sheaf cohomology re-
lations are satisfied on the vacua, the U(1)V symmetry is unbroken but the U(1)A broken classically, and all correlation functions match those of A/2 twisted theory as before.
Consistency check: mirrors of blowdowns to dP1
The del Pezzo surface dP2 can be blown down to dP1, which one can see from the toric fan by removing the edge (−1, 0) in figure 2.2. (Moreover, essentially because we are discussing blowups of smooth points on Fano varieties, the UV phases of the GLSMs are the geometries described here, so in the cases described here there are no subtleties involving the GLSM giving results for unexpected geometries.)
On the (2,2) locus, the mirror of the blowdown from dP2 to dP1 is described in the Toda
dual theory (2.43) by taking the limit q3 → 0 after integrating out the Lagrange multiplier Z. Next, we will analyze both of the proposed (0,2) mirror theories in section (2.4.2) under the same blowdown limit.
First, to illustrate the method, let us explain how to explicitly follow the blowdown in the (2,2) Toda dual (2.43). That (2,2) superpotential can be rewritten in the (0, 2) language as follows, ∫ ∑ ∫ i dθ W (Φ) = dθ Λ Ji(Φ), i i where the Λ are Fermi superfields and the Ji are derivatives of W , which in the current case 64
(0, 1)
(1, 0)
(−1, −1) (0, −1)
Figure 2.2: A toric fan for dP1 can be obtained by removing the edge (−1, 0) from the toric fan for dP2. are given by
q3 q1 J1 = X1 − + Z , X1 X1X2X3 q1 J2 = X2 + Z , X1X2X3 q2 q1 J3 = X3 − + Z , X3 X1X2X3 q1 JZ = 1 − . X1X2X3
To integrate out the Lagrange multiplier Z, one integrates out the Fermi field ΛZ corre- sponding to JZ = ∂Z W , which implies
q1 JZ = 1 − = 0, X1X2X3 or q1 X2 = . X1X3 As one might expect, the above constraint is the same constraint arising from integrating out the Lagrange multiplier Z in (2.43). Imposing this constraint, the remaining J functions 65 become
q3 J1 = X1 − + Z, X1 q1 J2 = + Z, X1X3 q2 J3 = X3 + Z − . X3
Since we have removed explicit X2 dependence from the Ji above, we should also integrate 2 − out the Fermi field Λ corresponding to J2 = X2∂X2 W , which implies q Z = − 1 . X1X3
Applying the constraint above, one reaches the form
q3 q1 J1 = X1 − − , X1 X1X3 q1 q2 J3 = X3 − − . X1X3 X3
Finally, taking the limit q3 → 0, we see that the J functions above precisely coincide with those for the (2,2) Toda dual of dP1 presented in (2.29).
Now that we have illustrated the method, let us analyze the mirror of the (0, 2) theory
(2.47)-(2.50) in the blowdown limit q3 → 0. After integrating out the Lagrange multiplier, one obtains the constraints
q J = (ϵ · X) + Z 3 = 0, 5 (α · X)(ϵ · X)
(ϵ · X) q3 JZ = − = 0, X5 X5(α · X) where
(α · X) = α1(X1 − X5) + α2(X3 − X1 + X5) + α3X5, . .
(ϵ · X) = ϵ1(X1 − X5) + ϵ2(X3 − X1 + X5) + ϵ3X5. 66
In the limit q3 → 0, the constraint JZ = 0 implies
ϵ1X1 + ϵ2(X3 − X1) X5 = . ϵ1 − ϵ2 − ϵ3
(For simplicity, we assume ϵ1 − ϵ2 − ϵ3 ≠ 0.) The mirror blowdown is then given by,
q1 J1 = − X1(Γ1X1 + Γ2(X3 − X1)) (A X + A (X − X ))(B X + B (X − X )) + 1 1 2 3 1 1 1 2 3 1 , X1 q1 J3 = − X1(Γ1X1 + Γ2(X3 − X1)) (A X + A (X − X ))(B X + B (X − X )) + 1 1 2 3 1 1 1 2 3 1 X1 (Γ X + Γ (X − X ))(∆ X + ∆ (X − X )) q + 1 1 2 3 1 1 1 2 3 1 − 2 , X3 X3
where,
ϵ1(α2 + α3) − α1(ϵ2 + ϵ3) ϵ2(α3 − α1) + α2(ϵ1 − ϵ3) A1 = ,A2 = , ϵ1 − ϵ2 − ϵ3 ϵ1 − ϵ2 − ϵ3 ϵ1(β2 + β3) − β1(ϵ2 + ϵ3) ϵ2(β3 − β1) + β2(ϵ1 − ϵ3) B1 = ,B2 = , ϵ1 − ϵ2 − ϵ3 ϵ1 − ϵ2 − ϵ3 ϵ1(γ2 + γ3) − γ1(ϵ2 + ϵ3) ϵ2(γ3 − γ1) + γ2(ϵ1 − ϵ3) Γ1 = , Γ2 = , ϵ1 − ϵ2 − ϵ3 ϵ1 − ϵ2 − ϵ3 ϵ1(δ2 + δ3) − δ1(ϵ2 + ϵ3) ϵ2(δ3 − δ1) + δ2(ϵ1 − ϵ3) ∆1 = , ∆2 = . ϵ1 − ϵ2 − ϵ3 ϵ1 − ϵ2 − ϵ3
One can see that the resulting superpotential is the same as the superpotential (2.32), (2.33) after adjusting the parameters as follows,
a = A1B1, b = A2B2, µAB = A1B2 + A2B1,
γ1 = Γ1, γ2 = Γ2,
α1 = ∆1, α2 = ∆2.
Thus, as expected, the (0,2) mirror to the blowdown, is the blowdown limit of the mirror. This provides a consistency check on the form of the proposed mirror.
Next, we repeat the analysis for the second form of the (0, 2) mirror (2.51)-(2.54). We first 67 obtain the constraints
q J = (β · X) + Z 1 = 0, 2 (α · X)(β · X)(γ · X)
(β · X) q1 JZ = − = 0. X2 X2(α · X)(γ · X)
In principle, one can use these constraints to eliminate the dependence on X2 and Z in the remaining J functions. Then, taking the limit q3 → 0 one should recover the J functions of the (0, 2) mirror of dP1. However, JZ is effectively a cubic polynomial in X2, so directly solving for X2 in arbitrary (0,2) deformations is rather complex. For simplicity, we will only consider the blowdown in the second form of the (0,2) mirror for a special family of deformations, of the form
α1 = 1, α2 = 0, α3 = 1, γ1 = 1, γ2 = 1, γ3 = 0, leaving other deformation parameters arbitrary.
Now, for this family of deformations, the constraints become ( ) −1 q1 X2 = (β1 − β2 − β3) − β3X1 − β2X3 , X1X3 q Z = − 1 = −β · X. X1X3
Plugging back into the other J functions, we find
′ J1 = EJ1,
q1 β1ϵ3 − β2ϵ3 − β3ϵ1 + β3ϵ2 β1ϵ2 − β3ϵ2 − β2ϵ1 + β2ϵ3 = − − X1 − X3, X1X3 ϵ1 − ϵ2 − ϵ3 ϵ1 − ϵ2 − ϵ3 ′ J3 = ∆ J3, ′ − − q2 q1 β1δ3 β2δ3 β3δ1 + β3δ2 = − − − X1 X3 X1X3 δ1 − δ2 − δ3 β1δ2 − β3δ2 − β2δ1 + β2δ3 − X3, δ1 − δ2 − δ3 68
where
β − β − β E = − 1 2 3 , ϵ1 − ϵ2 − ϵ3 β − β − β ∆ = − 1 2 3 , δ1 − δ2 − δ3 − − ′ −β1 β2 β3 q2 = q2. δ1 − δ2 − δ3
We assume that
δ1 − δ2 − δ3 ≠ 0, ϵ1 − ϵ2 − ϵ3 ≠ 0.
Note that we rescaled J1 and J3: the rescaling parameters E and ∆ can always be absorbed in the corresponding Fermi fields. We also rescaled q2 to match the form of the J functions of dP1. As a result, one can see that the J functions reduce to those of dP1 in equations (2.32)- (2.33), with the parameters related as follows:
γ1 = γ2 = 1, b = 0, β ϵ − β ϵ + β ϵ − β ϵ a = − 1 2 2 1 1 3 3 1 , ϵ1 − ϵ2 − ϵ3 β1ϵ2 − β3ϵ2 − β2ϵ1 + β2ϵ3 µAB = − , ϵ1 − ϵ2 − ϵ3 β1ϵ3 − β3ϵ1 + β1ϵ2 − β2ϵ1 β1δ2 − β3δ1 − β2δ1 + β1δ3 α1 = − , ϵ1 − ϵ2 − ϵ3 δ1 − δ2 − δ3 β1ϵ2 − β3ϵ2 − β2ϵ1 + β2ϵ3 β1δ2 − β3δ2 − β2δ1 + β2δ3 α2 = − . ϵ1 − ϵ2 − ϵ3 δ1 − δ2 − δ3
Consistency check: mirrors of blowdowns to P1 × P1
1 1 We can also blowdown dP2 to P × P , which can be represented in the toric fan we have used previously by removing the edge (−1, −1), as shown in figure 2.3. (As before, since we are discussing Fano varieties, the geometries described all correspond to UV phases of the GLSMs.)
1 1 On the (2, 2) locus, the mirror of the blowdown from dP2 to P × P is described in the
mirror theory (2.43) by taking the limit q1 → 0 after integrating out the Lagrange multiplier Z. Off the (2, 2) locus, we can follow the same procedure as before, integrating out the
Lagrange multiplier in (2.51)-(2.54) and taking the limit q1 → 0 to blow down the (0, 2) 69
(0, 1)
(−1, 0) (1, 0)
(0, −1)
Figure 2.3: A toric fan for P1 × P1 can be obtained by removing the edge (−1, −1) from the toric fan for dP2.
1 1 mirror dual J functions of dP2 to P × P . Integrating out the Lagrange multiplier, we obtain the constraints
q J = (β · X) + Z 1 = 0, 2 (α · X)(β · X)(γ · X)
(β · X) q1 JZ = − = 0, X2 X2(α · X)(γ · X) where
(α · X) = α1X2 + α2(X3 − X2) + α3(X1 − X2), . .
(ϵ · X) = ϵ1X2 + ϵ2(X3 − X2) + ϵ3(X1 − X2).
In the limit q1 → 0, the only solution of {JZ = 0} for X2 is
β2X3 + β3X1 X2 = − . β1 − β2 − β3
(For simplicity we assume β1 − β2 − β3 ≠ 0.) Then, in this limit, the resulting J functions 70 are given by
q3 (A1X1 + A2X3)(E1X1 + E2X3) J1 = − + , X1 X1 q2 (Γ1X1 + Γ2X3)(∆1X1 + ∆2X3) J3 = − + , X3 X3
where
β3(α2 − α1) + α3(β1 − β2) α2(β1 − β3) + β2(α3 − α1) A1 = ,A2 = , β1 − β2 − β3 β1 − β2 − β3 β3(ϵ2 − ϵ1) + ϵ3(β1 − β2) ϵ2(β1 − β3) + β2(ϵ3 − ϵ1) E1 = ,E2 = , β1 − β2 − β3 β1 − β2 − β3 β3(γ2 − γ1) + γ3(β1 − β2) γ2(β1 − β3) + β2(γ3 − γ1) Γ1 = , Γ2 = , β1 − β2 − β3 β1 − β2 − β3 β3(δ2 − δ1) + δ3(β1 − β2) δ2(β1 − β3) + β2(δ3 − δ1) ∆1 = , ∆2 = . β1 − β2 − β3 β1 − β2 − β3
The J’s above are equivalent to (2.24), (2.25) in the mirror to the A/2 model on P1 × P1, if we take [ ] [ ] A 0 A 0 A = 1 ,B = 2 , 0 E1 0 E2 [ ] [ ] Γ 0 Γ 0 C = 1 ,D = 2 . 0 ∆1 0 ∆2
1 1 In principle one could also similarly analyze the mirror of the blowdown dP2 → P × P in the same limit q1 → 0 in terms of the J functions (2.47)-(2.50), but we will not do so here.
2.4.3 The third del Pezzo surface, dP3
Review of the (2, 2) mirror
In this section, we will consider the last toric del Pezzo dP3, which can be described by a fan with edges (1, 0), (0, 1), (−1, −1), (1, 1), (−1, 0), (0, −1). The corresponding GLSM has six 71
4 chiral superfields ϕi, i = 1,..., 6 which are charged under the gauge group U(1) as follows:
(1,0) (-1,-1) (0,1) (0,-1) (-1,0) (1, 1) 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1
Following section 2.1.1, the superpotential of the (2,2) mirror is given by,
∑6 ∑6 W = exp(Yi) = Xi i=1 i=1
for Xi = exp(Yi), with constraints:
Y1 + Y2 + Y3 = r1,Y3 + Y4 = r2,Y1 + Y5 = r3,Y2 + Y6 = r4.
Eliminating X2 and X4 via two of the constraints above, and introducing two Lagrange
multipliers Z1 and Z2 to implement the remaining constraints, the superpotential can be written as ( ) ( ) q1 q2 q3 q4 X1X3 W = X1 + X3 + X5 + X6 + + + Z1 1 − + Z2 1 − . (2.55) X1X3 X3 X1X5 q1 X6
The vacua solve the following algebraic equations:
q1 q3 q4 X1X3 X1∂1W = X1 − + Z1 − Z2 = 0, X1X3 X1X5 q1 X6 q2 q1 q4 X1X3 X3∂3W = X3 − − − Z2 = 0, X3 X1X3 q1 X6 q3 X5∂5W = X5 + Z1 = 0, X1X5 q4 X1X3 X6∂6W = X6 + Z2 = 0, q1 X6 − q3 ∂Z1 W = 1 = 0, X1X5 − q4 X1X3 ∂Z2 W = 1 = 0. q1 X6 72
The quantum cohomology relations are
(ψ1 + ψ3)(ψ1 + ψ4)(ψ1 + ψ2) = q1,
(ψ1 + ψ2)ψ2 = q2,
(ψ1 + ψ3)ψ3 = q3,
(ψ1 + ψ4)ψ4 = q4.
One can check that these quantum cohomology ring relations are satisfied on the space of vacua of the Toda theory after identifying
X1 ∼ ψ1 + ψ3,X3 ∼ ψ1 + ψ2,X5 ∼ ψ3,X6 ∼ ψ4.
One can also check that all the correlation functions match those of the A-twisted theory on dP3.
(0, 2) deformations and proposed (0,2) mirrors
To describe the (0, 2) deformation of dP3, we will need 24 complex parameters αi, βj, γk,
δl, ϵm, ζn, i, j, k, l, m, n = 1 ... 4. Those parameters define a deformation E of the tangent bundle as follows,
0 −→ O3 −→OE (1, 0, 1, 0)⊕O(1, 0, 0, 1) ⊕ O(1, 1, 0, 0) ⊕ O(0, 1, 0, 0) ⊕ O(0, 0, 1, 0) ⊕ O(0, 0, 0, 1) −→ E −→ 0, where E is defined by: α s α s α s α s 1 1 2 2 3 3 4 4 β1s1 β2s2 β3s3 β4s4 γ s γ s γ s γ s 1 1 2 2 3 3 4 4 E = , δ1s1 δ2s2 δ3s3 δ4s4 ϵ1s1 ϵ2s2 ϵ3s3 ϵ4s4
ζ1s2 ζ2s2 ζ3s3 ζ4s4 73
for si the chiral superfields of the GLSM. The (2,2) locus is given by the special case
α1 = 1, α2 = 0, α3 = 1, α4 = 0,
β1 = 1, β2 = β3 = 0, β4 = 1,
γ1 = γ2 = 1, γ3 = γ4 = 0,
δ1 = 0, δ2 = 1, δ3 = δ4 = 0,
ϵ1 = ϵ2 = 0, ϵ3 = 1, ϵ4 = 0,
ζ1 = ζ2 = ζ3 = 0, ζ4 = 1.
If we define:
∑4 ∑4 ∑4 Q(1) = αiψi,Q(2) = βiψi,Q(3) = γiψi, i=1 i=1 i=1 ∑4 ∑4 ∑4 Q(4) = δiψi,Q(5) = ϵiψi,Q(6) = ζiψi, i=1 i=1 i=1 then the quantum sheaf cohomology ring relations are
Q(1)Q(2)Q(3) = q1, (2.56)
Q(3)Q(4) = q2, (2.57)
Q(1)Q(5) = q3, (2.58)
Q(2)Q(6) = q4, (2.59) which reduce to the ordinary quantum cohomology ring relations on the (2, 2) locus.
Our proposal for the (0,2) mirror of the A/2-twisted theory on dP3 with a deformation of 74
the tangent bundle is defined by the following six J functions:
q1 q3 q4 (α · X)(γ · X) (α · X)(ϵ · X) J1 = − + Z1 − Z2 + X1(γ · X) X1(ϵ · X) q1 (ζ · X) X1 (α · X)(β · X) − (ζ · X) + , X1 q2 q1 q4 (α · X)(γ · X) (α · X)(β · X) J3 = − − − Z2 + − (ζ · X) X3 X1(γ · X) q1 (ζ · X) X1 (γ · X)(δ · X) + , X3 q J = (ϵ · X) + Z 3 , 5 1 (α · X)(ϵ · X)
q4 (α · X)(γ · X) J6 = (ζ · X) + Z2 , q1 (ζ · X)
(ϵ · X) q3 JZ1 = − , X5 X5(α · X)
(ζ · X) q4 (α · X)(γ · X) JZ2 = − , X6 q1 X6 where
(α · X) = α1(X1 − X5) + α2(−X1 + X3 + X5) + α3X5 + α4X6, . .
(ζ · X) = ζ1(X1 − X5) + ζ2(−X1 + X3 + X5) + ζ3X5 + ζ4X6.
(Because the J’s have poles away from origins, we interpret the resulting action in a low- energy effective field theory sense, as discussed in the introduction.)
On the (2,2) locus, the above J functions reduce to derivatives of the (2,2) superpotential
(2.55) as expected. It is also easy to check that this theory respects the U(1)V symmetry,
but U(1)A is broken classically. One can also show that on the space of vacua all quantum sheaf cohomology ring relations (2.56)-(2.59) are satisfied after identifying
X1 = ψ1 + ψ3,X3 = ψ1 + ψ2,X5 = ψ3,X6 = ψ4.
In all other examples in this chapter, we have checked that all of the genus zero correlation functions of the proposed B/2-twisted Landau-Ginzburg mirror match those of the original 75
A/2 theory, for all deformations. However, for dP3, we have only checked that the genus zero correlation functions match in several families of deformation parameters, described below:
1. families parametrized by αi, βi, γi, δi, ϵi, ζi, for fixed i ∈ {1, ··· , 4}, and other param- eters set to their (2,2) locus values,
2.
α2 = α4 = 0, β2 = β3 = 0,
γ3 = γ4 = 0, δ1 = δ3 = δ4 = 0,
ϵ1 = ϵ2 = ϵ4 = 0, ζ1 = ζ2 = ζ3 = 0,
for a family parametrized by α1,3, β1,4, γ1,2, δ2, ϵ3, ζ4,
3.
α1 = 1, α2 = 0, α3 = 1, α4 = 0,
β1 = 1, β2 = β3 = 0, β4 = 1,
for a family parametrized by γ1−4, δ1−4, ϵ1−4, ζ1−4,
4.
γ1 = γ2 = 1, γ3 = γ4 = 0,
ϵ1 = ϵ2 = 0, ϵ3 = 1, ϵ4 = 0,
for a family parametrized by α1−4, β1−4, δ1−4, ζ1−4.
For each of the families of deformation parameters above, we have checked that all of the genus zero correlation functions of the proposed B/2-twisted Landau-Ginzburg model match those of the original A/2-twisted theory.
Consistency check: mirrors of blowdowns to dP2
In this section we will describe the mirror of the blowdown dP3 → dP2, verifying that the blowdown of the mirror is the mirror of the blowdown. This can be represented torically 76
by removing the edge (1, 1) from the toric fan previously discussed for dP3, as shown in figure 2.4. (As before, since all of the varieties in question are Fano, the UV phases of the GLSMs correspond to the geometries described here.)
(0, 1)
(−1, 0) (1, 0)
(−1, −1) (0, −1)
Figure 2.4: A toric fan for dP2 can be obtained by removing the edge (1, 1) from the toric fan for dP3.
The (2,2) mirror of this blowdown is given by applying the limit q4 → 0 to the superpotential
(2.55) after integrating out one of the Lagrange multiplier Z2. Following the same procedure
for (0,2) mirror dual, one first integrate out the Lagrange multiplier Z2 and obtain two constraints:
q4 (α · X)(γ · X) J6 = (ζ · X) + Z2 = 0, q1 (ζ · X)
(ζ · X) q4 (α · X)(γ · X) JZ2 = − = 0. X6 q1 X6
In the limit q4 → 0, the constraints above imply ζ · X = 0 and Z2 = 0, or for ζ4 ≠ 0 (which we assume for simplicity),
− −1 − − X6 = ζ4 (ζ1(X1 X5) + ζ2( X1 + X3 + X5) + ζ3X5) ,Z2 = 0. 77
Plugging those constraints into the Js, one gets
q1 q3 (α · X)(ϵ · X) (α · X)(β · X) J1 = − + Z1 + + , X1(γ · X) X1(ϵ · X) X1 X1
q2 q1 (α · X)(β · X) (γ · X)(δ · X) J3 = − − + + , X3 X1(γ · X) X1 X3 q J = (ϵ · X) + Z 3 , 5 1 (α · X)(ϵ · X)
(ϵ · X) q3 JZ1 = − , X5 X5(α · X) where
(α · X) = A1(X1 − X5) + A2(−X1 + X3 + X5) + A3X5,
(β · X) = B1(X1 − X5) + B2(−X1 + X3 + X5) + B3X5,
(γ · X) = G1(X1 − X5) + G2(−X1 + X3 + X5) + G3X5,
(δ · X) = D1(X1 − X5) + D2(−X1 + X3 + X5) + D3X5,
(ϵ · X) = E1(X1 − X5) + E2(−X1 + X3 + X5) + E3X5,
with
− −1 − −1 − −1 A1 = α1 α4ζ1ζ4 ,A2 = α2 α4ζ2ζ4 ,A3 = α3 α4ζ3ζ4 , − −1 − −1 − −1 B1 = β1 β4ζ1ζ4 ,B2 = β2 β4ζ2ζ4 ,B3 = β3 β4ζ3ζ4 , − −1 − −1 − −1 G1 = γ1 γ4ζ1ζ4 ,G2 = γ2 γ4ζ2ζ4 ,G3 = γ3 γ4ζ3ζ4 , − −1 − −1 − −1 D1 = δ1 δ4ζ1ζ4 ,D2 = δ2 δ4ζ2ζ4 ,D3 = δ3 δ4ζ3ζ4 , − −1 − −1 − −1 E1 = ϵ1 ϵ4ζ1ζ4 ,E2 = ϵ2 ϵ4ζ2ζ4 ,E3 = ϵ3 ϵ4ζ3ζ4 . 78
The resulting J functions are the same as those of dP2 in (2.47)-(2.50) with parameters
α1 = A1, α2 = A2, α3 = A3,
β1 = B1, β2 = B2, β3 = B3,
γ1 = B1, γ2 = B2, γ3 = B3,
δ1 = D1, δ2 = D2, δ3 = D3,
ϵ1 = E1, ϵ2 = E2, ϵ3 = E3.
2.5 Hirzebruch surfaces
2.5.1 Review of the (2,2) mirror
We will first review the construction of Hirzebruch surfaces Fn and their (2,2) Toda duals along with the ordinary quantum cohomology relations.
Recall a Hirzebruch surface Fn is a toric variety which can be described by the fan with edges (1, 0), (0, 1), (0, −1), (−1, −n). The corresponding gauged linear sigma model has four chiral 2 superfields ϕ1, ϕ2, ϕ3, ϕ4 which are charged under the gauge group U(1) as follows
u v s t 1 1 n 0 0 0 1 1
Now, for n > 1, Hirzebruch surfaces Fn are not Fano. This fact manifests itself in the RG flow: en route to the IR, the GLSM for a Hirzebruch surface will enter a different phase, and describe a different geometry. Nevertheless, we expect them to flow in the IR to a discrete set of isolated vacua, and so one can reasonably expect a Toda-type mirror, despite the fact that they are not Fano. On the other hand, for the same reasons, for n > 2 the mirrors we describe should not be interpreted as mirrors to Hirzebruch surfaces per se but rather 2 to different phases of the same GLSM, specifically to geometries P[1,1,n] which appear as the UV phases of the same GLSMs. In any event, for simplicity, we will speak loosely of ‘mirrors to Hirzebruch surfaces’ with the understanding that we are speaking of mirrors to GLSMs, and the actual geometries being mirrored are the UV phases, which for n > 2 will not in fact be Hirzebruch surfaces. 79
Following the methods of [93] (with the non-Fano caveat above), the mirror is a Landau- Ginzburg model with superpotential
W = exp(Y1) + exp(Y2) + exp(Y3) + exp(Y4), with constraints
Y1 + Y2 + nY3 = r1,Y3 + Y4 = r2.
We can then use those two constraints to write the result in terms of only Y1, Y3, yielding [93][equ’n (5.19)] q2 q1 W = X1 + X3 + + n , X3 X1X3 where we defined X1 = exp(Y1), X3 = exp(Y3), q1 = exp(r1) and q2 = exp(r2).
As discussed in e.g. [19], the quantum cohomology ring relations are given by
2 ˜ n ˜ ˜ ψ (nψ + ψ) = q1, ψ(nψ + ψ) = q2.
To translate to the present case, we use the dictionary
˜ ψ ∼ X1, nψ + ψ ∼ X3, (2.60)
so on the space of vacua, we expect that the fields X1,X3 of the mirror should obey
2 n − X1 X3 = q1, (X3 nX1)X3 = q2. (2.61)
Vacua are computed by taking derivatives of the superpotential with respect to ln X1, ln X3. Doing so one finds that the vacua are defined by
− −1 −n X1 q1X1 X3 = 0, (2.62) − −1 − −1 −n X3 q2X3 nq1X1 X3 = 0. (2.63)
After some algebra one can show that these conditions for vacua imply conditions (2.61), as expected.
For geometries previously described, we have given alternative mirrors, and this is no ex- ception. Solving the original constraints for Y1 and Y3, and defining X1 = exp(Y1) and
X3 = exp(Y3), we get an alternative expression for the superpotential defining the (2,2) 80 mirror: −n −1 n −1 W = X1 + q1q2 X1 X4 + X4 + q2X4 . This expression is related to the one above by the field redefinition
q2 X4 = . X3
2.5.2 (0,2) deformations and proposed (0,2) mirrors
In this section, we will give a proposal for the mirror (0, 2) Landau-Ginzburg model to an A/2-twisted nonlinear sigma model on a Hirzebruch surface with a deformation of the tangent bundle, which we will check by matching correlation functions.
The (0, 2) deformations of a Hirzebruch surface Fn are defined by a pair of 2 × 2 matrices
A, B, and complex numbers γ1, γ2, α1, α2, that define a deformation E of the tangent bundle
0 −→ O⊕2 −→OE (1, 0)⊕2 ⊕ O(n, 1) ⊕ O(0, 1) −→ E −→ 0, where E is Ax Bx E = γ1s γ2s ,
α1t α2t with [ ] u x = . v The (2, 2) locus is given by the special case
A = I,B = 0, γ1 = n, γ2 = 1, α1 = 0, α2 = 1.
If we define
˜ ˜ ˜ Q(k) = det(ψA + ψB),Q(s) = ψγ1 + ψγ2,Q(t) = ψα1 + ψα2, then the quantum sheaf cohomology ring relations are given by [77]
n Q(k)Q(s) = q1,Q(s)Q(t) = q2. 81
Our proposal for the (0, 2) Toda mirror of the A/2 model on Fn with a deformation of the tangent bundle is defined by
2 (X3 − nX1) J1 = aX1 + µAB(X3 − nX1) + b X1 − − q X−1 (γ X + γ (X − nX )) n , (2.64) ( 1 1 1 1 )2 3 1 2 (X3 − nX1) J2 = n aX1 + µAB(X3 − nX1) + b X1 nq1 q2 − n − X1 (γ1X1 + γ2(X3 − nX1)) X3 (γ X + γ (X − nX )) (α X + α (X − nX )) + 1 1 2 3 1 1 1 2 3 1 . (2.65) X3
(Because the J’s have poles away from origins, we interpret the resulting action in a low- energy effective field theory sense, as discussed in the introduction.) In the expression above, we used the same notation as in our description of the A/2 theory, namely
a = det A, b = det B, µAB = det(A + B) − det A − det B.
We will check our proposal by arguing that A/2 model correlation functions will match those of the B/2-twisted mirror Landau-Ginzburg theory given above. Before doing so, let us first make a few elementary observations. As one might expect, our proposal reduces to the (2, 2)
Toda dual when E = TX, corresponding to A = I,B = 0, γ1 = n, γ2 = 1, α1 = 0, α2 = 1,
as in this case each Ji becomes the derivative of the (2,2) superpotential with respect to
Y1 = ln X1 and Y3 = ln X3. As another consistency check, one can show X1,X3 satisfy the quantum sheaf cohomology relations on the space of vacua. Specifically, the vacua are
defined by J1 = 0, J2 = 0, which imply
n det (AX1 + B(X3 − nX1)) (γ1X1 + γ2(X3 − nX1)) = q1,
(α1X1 + α2(X3 − nX1)) (γ1X1 + γ2(X3 − nX1)) = q2.
With the correspondence (2.60), it is straightforward to show that the quantum sheaf coho- mology relations are satisfied on the vacua.
As another consistency check, we observe that this naturally specializes to results obtained in [1] and reviewed in section 2.3.2 for the mirror to the A/2 model on P1 × P1 with a 82
deformation of the tangent bundle. If we take n = 0, then the resulting Hirzebruch surface with tangent bundle deformation corresponds to P1 × P1 with [ ] [ ] γ 0 γ 0 C = 1 ,D = 2 , 0 α1 0 α2
so that, after simplification,
2 X2 q1 J1 = aX1 + µABX2 + b − , (2.66) X1 X1 2 X1 q2 J2 = α1γ1 + (α1γ2 + α2γ1)X1 + α2γ2X2 − . (2.67) X2 X2
One can easily observe that the J functions above (2.66), (2.67) are the same as (2.64), (2.65) after setting n 7→ 0.
Similarly, for the case n = 1, the mirror here matches the mirror to dP1 = F1 described previously in section 2.4.1.
We have checked that all genus zero correlation functions in this proposed (0,2) mirror match those of the original A/2-twisted theory, following the arguments outlined in section 2.4.1.
So far we have presented a (0,2) mirror proposal that reduces on the (2,2) locus to the first expression for a (2,2) mirror. As we have done for other geometries, we next present a (0,2) mirror proposal that reduces on the (2,2) locus to the second expression for a (2,2) mirror.
Specifically, a proposal for a (0, 2) mirror of the A/2 model on Fn (with a deformation of the tangent bundle) that reduces on the (2,2) locus to the model above is given by ( ) 2 n X4 − q1 (α1X1 + α2X4) J1 = aX1 + µABX4 + b n , (2.68) ( X1 q2 X1 ) 2 n X1 q1 (α1X1 + α2X4) (γ1α2 + γ2α1) − q2 J2 = α2γ2X4 + γ1α1 + n 2 −1 . (2.69) X4 q2 aX1 + µABX4 + bX4 X1 X4
In passing, we should mention that an alternative expression which also has matching cor-
relation functions and the correct (2,2) locus can be written which has the same J1 but a 83
different J2 given by ( ) 2 n − X4 − q1 (α1X1 + α2X4) J2 = n aX1 + µABX4 + b n ( X1 q2 X1) 2 X1 q2 + α2γ2X4 + γ1α1 + (γ1α2 + γ2α1)X1 − . (2.70) X4 X4
Of course, by taking suitable field redefinitions, we can simplify the second J2 above to write it in the form ( ) 2 X1 q2 J2 = α2γ2X4 + γ1α1 + (γ1α2 + γ2α1)X1 − . (2.71) X4 X4
This third model, with the altered J2 above, does not reduce to the (2,2) locus expression given previously, but we felt important to point out its existence.
One can check that these alternative proposals also reduce to the (2, 2) mirror, and X1, X4 satisfy the quantum sheaf cohomology relations on the space of vacua with identification ˜ X1 ∼ ψ, X4 ∼ ψ. More importantly, all (genus zero) A/2 model correlation functions again match those of the B/2 Landau-Ginzburg theory given above. Thus, this is another expression for the mirror.
Setting n = 0, we also have a new expression for the B/2 mirror Landau-Ginzburg theory to P1 × P1,
2 X4 q1 J1 = aX1 + µABX4 + b − , X1 X1 2 X1 q1(γ1α2 + γ2α1) − q2 J2 = α2γ2X4 + γ1α1 + 2 −1 . X4 aX1 + µABX4 + bX4 X1 X4
On the (2, 2) locus the J functions above reduce to those of the (2, 2) mirror of P1 × P1 written in (0, 2) language,
q1 J1 = X1 − , X1 q2 J1 = X4 − . X4
All correlation functions of the new mirror theory given above are the same as those given in section (2.5.2). In both case, all correlation functions match the correlation functions of the same one-loop effective action of the A/2 theory on P1 × P1. 84
2.6 Correlation functions in some examples
2.6.1 A/2 correlation functions on P1 × P1
In this section we list the two- and four-point correlation functions for A/2 twisted nonlinear sigma models on P1 × P1 with a deformation E of the tangent bundle, defined by
0 −→ O ⊗ O −→OE (1, 0)2 ⊕ O(0, 1)2 −→ E −→ 0,
defined as in section 2.3.1 by four matrices A, B, C, D.
In writing the correlation functions, we use the following notation:
a = det(A), b = det(B), c = det(C), d = det(D),
e = det(A + B), f = det(C + D),
µ = e − a − b, ν = f − c − d,
ϕ1 = νb − µd = ad + bf − de − bc,
ϕ2 = ad − bc,
ϕ3 = µc − νa = ad + ce − af − bc,
2 − ∆ = ϕ2 ϕ1ϕ3, = (c − d)(bc − ad)e + cde2 + (a − b)(ad − bc)f − (bc + ad)ef + abf 2.
The two-point correlation functions are given by
ϕ ϕ ϕ ⟨ψψ⟩ = 1 , ⟨ψψ˜⟩ = 2 , ⟨ψ˜ψ˜⟩ = 3 . (2.72) ∆ ∆ ∆ 85
The four-point correlation functions are given by
ϕ 1 ⟨ψψψψ⟩ = 1 (νϕ + 2ϕ d) = (ϕ ((f − c)ϕ + ad2 + d2e − bcd − bdf)), 10 ∆2 1 2 ∆2 1 1 1 ( ) ⟨ψψψψ˜⟩ = −ϕ2c + ϕ2d , 10 ∆2 1 2 ϕ ⟨ψψψ˜ψ˜⟩ = 2 (ϕ d − ϕ c) , 10 ∆2 3 1 1 ( ) ⟨ψψ˜ψ˜ψ˜⟩ = ϕ2d − ϕ2c , 10 ∆2 3 2 ϕ ⟨ψ˜ψ˜ψ˜ψ˜⟩ = 3 (νϕ + 2ϕ c) , 10 ∆2 3 2 1 = (ϕ (ce(c + d − f) + bc(c − d + f) + a((d − f)2 − c(d + f)))), ∆2 3 ϕ ⟨ψψψψ⟩ = − 1 (µϕ + 2ϕ b) , 01 ∆2 1 2 1 = (−ϕ (2b(ad − bc) − d(a + b − e)2 + b(a + b − e)(c + d − f))), ∆2 1 1 ( ) ⟨ψψψψ˜⟩ = ϕ2a − ϕ2b , 01 ∆2 1 2 ϕ ⟨ψψψ˜ψ˜⟩ = 2 (−ϕ b + ϕ a) , 01 ∆2 3 1 1 ( ) ⟨ψψ˜ψ˜ψ˜⟩ = −ϕ2b + ϕ2a , 01 ∆2 3 2 ϕ 1 ⟨ψ˜ψ˜ψ˜ψ˜⟩ = − 3 (µϕ − 2ϕ a) = (ϕ ((e − b)ϕ + a2d + a2f − abc − acd)), 01 ∆2 3 2 ∆2 3 3 where the subscripts 10 and 01 denote contributions from the degree one sector on either P1 factor:
⟨O1O2O3O4⟩ = q1⟨O1O2O3O4⟩10 + q2⟨O1O2O3O4⟩01.
2.6.2 Toda-like dual to P1 × P1
In this section we list the two-point and four-point correlation functions for our proposed Toda-like dual (0,2) Landau-Ginzburg model, with superpotential of the form
X˜ 2 q J = aX + b + µX˜ − 1 , X X X2 q J˜ = dX˜ + c + νX − 2 . X˜ X˜ 86
The two-point correlation functions in this (0,2) Landau-Ginzburg model can be shown to be ⟨XX⟩ = γ−1(bν − dµ), ⟨XX˜⟩ = γ−1(ad − bc), ⟨X˜X˜⟩ = γ−1(cµ − aν), where γ = b2c2 − 2abcd + a2d2 + cdµ2 − (bc + ad)µν + abν2.
The four-point correlation functions in this (0,2) Landau-Ginzburg model can be shown to be −2 2 ⟨XXXX⟩10 = γ (−(dµ − bν)(d(2ad − µν) + b(−2cd + ν ))), −2 2 2 ⟨XXXX⟩01 = γ (−dµ + bν)(2b c + dµ − b(2ad + µν)), ˜ −2 2 2 2 2 ⟨XXXX⟩10 = γ (d((bc − ad) − cdµ ) + 2bcdµν − b cν ), ˜ −2 3 2 2 2 2 2 ⟨XXXX⟩01 = γ (−b c + ad µ − adb(ad + 2µν) + ab (2cd + ν )), ˜ ˜ −2 ⟨XXXX⟩10 = γ (bc − ad)(−2cdµ + bcν + adν), ˜ ˜ −2 ⟨XXXX⟩01 = γ (bc − ad)(bcµ + adµ − 2abν), ˜ ˜ ˜ −2 2 2 2 2 ⟨XXXX⟩10 = γ (c(−(bc − ad) + cdµ ) − 2acdµν + a dν ), ˜ ˜ ˜ −2 3 2 2 2 2 2 ⟨XXXX⟩01 = γ (a d − bc µ + abc(bc + 2µν) − a b(2cd + ν )), ˜ ˜ ˜ ˜ −2 2 2 ⟨XXXX⟩10 = γ (cµ − aν)(2bc − cµν + a(−2cd + ν )), ˜ ˜ ˜ ˜ −2 2 2 ⟨XXXX⟩01 = γ (cµ − aν)(2a d + cµ − a(2bc + µν)),
where the 10 and 01 subscripts indicate the coefficients of q1, q2, as in the previous subsection.
As remarked in section 2.3.2, if we identify the parameters above with matrix determinants as a = det A, b = det B, c = det C, d = det D, µ = det(A + B) − det A − det B, ν = det(C + D) − det C − det D, for A, B, C, D the matrices appearing in the A/2-twisted (0,2) model on P1 × P1, the correlation functions in the Landau-Ginzburg model above match those of the A/2 model.
2.6.3 A/2 correlation functions on P1 × P2
In this section we list the three-point, five-point and six-point correlation functions for A/2 twisted nonlinear sigma models on P1 × P2 with a deformation E of the tangent bundle, defined by 0 −→ O2 −→E O(1, 0)2 ⊕ O(0, 1)3 −→ E −→ 0, 87
with A, B 2 × 2 matrices and C, D 3 × 3 matrices.
Correlation functions in this theory can be computed in a variety of methods, such as e.g. residues [64]. In writing the correlation functions, we use the following notation:
a = det A, b = det B, c = det C, d = det D,
µ = det(A + B) − det A − det B, g is a sum of determinants of three matrices, each formed from two rows of C and one row of D, and f is similarly a sum of three determinants, each having two rows of D and one row of C.
Three-point functions in the A/2 theory are given by
⟨ψ3⟩ = ∆−1(−abd + b2g − bfµ + dµ2), ⟨ψ2ψ˜⟩ = ∆−1(−b2c + abf − adµ), ⟨ψψ˜2⟩ = ∆−1(a2d − abg + bcµ), ⟨ψ˜3⟩ = ∆−1(abc − a2f + agµ − cµ2), where ( ) ∆ = a3d2 + b (bc − af)2 − 2a2dg + abg2 + (bcf + adg)µ2 − cdµ3 −(ad(−3bc + af) + b(bc + af)g)µ. 88
Five-point correlation functions in the A/2 theory are given by ( 5 −2 4 2 3 2 2 2 2 ⟨ψ ⟩ = q1∆ b (c d − 2cfg + g ) + d µ (3a d − 2afµ + gµ ) +2b3(ag(f 2 − 2dg) + (cf 2 + cdg − fg2)µ) +b2(a2d(−f 2 + 5dg) − 2af(f 2 − dg)µ + (−4cdf + g(f 2 + 2dg))µ2) ) −2bd(a3d2 + a2dfµ − 2a(f 2 − dg)µ2 + (−cd + fg)µ3) ,
( 4 ˜ −2 3 2 2 2 ⟨ψ ψ⟩ = q1∆ 2a d (bf − dµ) + 2ab c(−bf + 2bdg + dfµ) +a2(b2(−3cd2 + f 3 − 2dfg) + 2bd(−f 2 + dg)µ + d2fµ2) +c(b4(cf − g2) − b2(f 2 + 2dg)µ2 + 2bdfµ3 − d2µ4 ) +b3(−2cdµ + 2fgµ)) ,
( 3 ˜2 −2 4 3 3 2 2 2 2 2 ⟨ψ ψ ⟩ = q1∆ a d − 2a bd g + b c (b g − 2bfµ + 3dµ ) +a2(b2(2cdf − f 2g + dg2) + 2bdfgµ − d2gµ2) ) −2ac(b3cd + 2bdfµ2 − d2µ3 + b2(−f 2µ + dgµ)) ,
( 2 ˜3 −2 4 3 3 2 2 2 2 2 ⟨ψ ψ ⟩ = q1∆ −b c + 2ab c f − a d (a f − 2agµ + 3cµ ) +b2(a2(fg2 − c(f 2 + 2dg)) − 2acfgµ + c2fµ2) ) +2bd(a3cd + a2(cf − g2)µ + 2acgµ2 − c2µ3) ,
( ˜4 −2 4 2 3 2 ⟨ψψ ⟩ = q1∆ a d(f − dg) + 2a d(−2bcf + bg + cdµ − fgµ) +a2(b2(3c2d + 2cfg − g3) − 2bcdgµ + d(2cf + g2)µ2) ) +c2µ(2b3c − b2gµ + dµ3) − 2ac(b3cg + b2(cf − g2)µ + dgµ3) ,
( ˜5 −2 4 2 3 2 2 2 2 ⟨ψ ⟩ = q1∆ −a (cd + f − 2dfg) − c µ (3b c − 2bgµ + fµ ) +2a3(bf(2cf − g2) − (cdf − f 2g + dg2)µ) +a2(b2c(−5cf + g2) + 2bg(−cf + g2)µ − (2cf 2 − 4cdg + fg2)µ2) ) +2ac(b3c2 + b2cgµ + 2b(cf − g2)µ2 + (−cd + fg)µ3) . 89
Six-point correlation functions in the A/2 theory are given by ( ⟨ψ6⟩ = q ∆−2 (abd − b2g + bfµ − dµ2)(−2b3c + dµ3 − bµ(3ad + fµ) 2 ) +b2(2af + gµ)) ,
( 5 ˜ −2 5 2 4 2 2 4 2 ⟨ψ ψ⟩ = q2∆ −b c + ab (2cf + g ) + ad µ − abdµ (3ad + 2fµ) −ab3(a(f 2 + 2dg) + 2(cd + fg)µ) + ab2(a2d2 + 4adfµ ) +(f 2 + 2dg)µ2) ,
( ) 4 ˜2 −2 2 2 2 2 2 ⟨ψ ψ ⟩ = q2∆ (b c − abf + adµ)(2a bd + b cµ + a(−2b g + bfµ − dµ )) ,
( ⟨ψ3ψ˜3⟩ = q ∆−2 −a4bd2 − a2b3(2cf + g2) − b3c2µ2 + ab3c(bc + 2gµ) 2 ) +a3(b2(f 2 + 2dg) − 2bdfµ + d2µ2) ,
( 2 ˜4 −2 2 2 2 ⟨ψ ψ ⟩ = q2∆ −(a d − abg + bcµ)(−bcµ + a (−2bf + dµ) +ab(2bc + gµ))) ,
( ˜5 −2 5 2 4 2 2 4 2 ⟨ψψ ⟩ = q2∆ a d − a b(f + 2dg) − bc µ + abcµ (3bc + 2gµ) +a3b(b(2cf + g2) + 2(cd + fg)µ) ) −a2b(b2c2 + 4bcgµ + (2cf + g2)µ2) ,
( ˜6 −2 2 2 3 3 2 ⟨ψ ⟩ = q2∆ −(a f + cµ − a(bc + gµ))(2a d − cµ − a (2bg + fµ) +aµ(3bc + gµ))) .
2.7 Tangent bundle moduli
In this section we compute12 the dimension of the tangent space to the moduli space of tangent bundle deformations, at the tangent bundle and ‘near’ the tangent bundle. We will
12These computations were originally worked out in collaboration with R. Donagi and J. Guffin for another project. 90
see that the rank of the tangent space to the moduli space of tangent bundle deformations can change as one moves away from the (2,2) locus.
Define W = V ⊗ O(1, 0) + V˜ ⊗ O(0, 1),
where ∼ n+1 ∼ m+1 V = C , V˜ = C ,
so that we can write the definition of the tangent bundle deformation E as
0 −→ O2 −→ W −→ E −→ 0.
First, if we dualize the definition above and take the associated long exact sequence, then from the fact that Hq(W ∗) = for all q,
(from the Bott formula, [71][section 1.1]), we have that
Hq(E ∗) = Hq−1(O2)
and so vanishes unless q = 1.
Then, applying Hom(E, −) to the definition of E and taking the associated long exact se- quence, one finds
0 → H0(E ∗ ⊗ W ) → H0(E ∗ ⊗ E) → C4 → H1(E ∗ ⊗ W ) → H1(E ∗ ⊗ E) → 0.
From this expression we find ( ) h1(E ∗ ⊗ E) = h0(E ∗ ⊗ E) − h0(E ∗ ⊗ W ) − h1(E ∗ ⊗ W ) − 4. (2.73)
Next, we will derive a relation between h0(E ∗ ⊗ W ) and h1(E ∗ ⊗ W ). Apply Hom(−,W ) to the definition of E to get, from the associated long exact sequence,
0 → H0(E ∗ ⊗ W ) → H0(W ∗ ⊗ W ) → H0(O2 ⊗ W ) → H1(E ∗ ⊗ W ) → 0, 91 where we have used the fact that
Hq(W ∗ ⊗ W ) = 0 for q > 0, as none of O, O(1, −1), O(−1, 1) have any cohomology in degree greater than zero. From the sequence above, we have that
h0(E ∗ ⊗ W ) − h1(E ∗ ⊗ W ) = h0(W ∗ ⊗ W ) − h0(O2 ⊗ W ).
To simplify further, we use the fact that
H0(W ∗ ⊗ W ) = V ⊗ V ∗ + V˜ ⊗ V˜ ∗, and so has dimension (n + 1)2 + (m + 1)2
Similarly, from Bott-Borel-Weil,
H0(W ) = V ⊗ V ∗ + V˜ ⊗ V˜ ∗, and so has the same dimension. Thus,
h0(E ∗ ⊗ W ) − h1(E ∗ ⊗ W ) = −(n + 1)2 − (m + 1)2.
Plugging into equation (2.73), we find
h1(End E) = h0(End E) + (n + 1)2 + (m + 1)2 − 4. (2.74)
From the relation above, we immediately see that
h1(End E) ≥ (n + 1)2 + (m + 1)2 − 4 = n(n + 2) + m(m + 2) − 2.
Let us compute h0(End E) on the (2,2) locus, where E is the tangent bundle of Pn × Pm. From the Bott formula [71][section 1.1], one has
H0(Pn, Ω1) = 0 92
and from applying Hom(T Pn, −) to the Euler sequence, one can similarly derive
h0(End T Pn) = h1(Ω1) = 1,
from which one quickly derives that
h0(End E) = 2.
Thus, on the (2,2) locus, we find
h1(End E) = n(n + 2) + m(m + 2).
For P1 × P1, the predicted number of infinitesimal deformations of the tangent bundle is 3 + 3 = 6, matching the number of parameters on which (0,2) computations depend, namely
a, b, c, d, µ, ν,
as described in section 2.3.3.
Away from the tangent bundle itself, the computations above suggest that the correct number of moduli is smaller, which can be confirmed from other computations. For example, if we
twist the tangent bundle by O(0, −1), we get a rank two vector bundle of c2 = 2, and from [72][chapter 6, theorem 20], the moduli space of such vector bundles has dimension13
4c2 − 3 = 5. As twisting by line bundles does not affect Mumford stability, the space of tangent bundle deformations should have the same dimension, so we see that the tangent bundle of P1 × P1 represents an unstable point on the moduli space.
For higher-dimensional products, not all of the deformations can be realized in the Euler sequence, or as E moduli in the GLSM [69]. For example, the predicted number of infinitesi- mal moduli of the tangent bundle of P1 ×P2 is 3+2(4) = 11. However, only seven parameters appear in the (0,2) GLSMs:
a, b, c, d, µ, ν1, ν2. 13We would like to thank Z. Lu for pointing out this computation to us. 93
2.8 Quantum cohomology of dP1
In this section we briefly outline standard results on the quantum cohomology of dP1, fol- lowing [108][section II.5].
2 For any β = dH − αE ∈ H2(dP1, Z), where H is the pullback of the hyperplane class of P 2 and E is the class of the exceptional divisor (viewing dP1 as the blowup of P at one point), define the Gromov-Witten invariant
nd,α Nd,α = I0,nd,α,β((pt) ) with the expected dimension nd,α = 3d − 1 − α. Using results from [108][section II.5], one can compute these Nd,α recursively, and thus determine all the Gromov-Witten invariants.
For example, N0,−1 = 1,N0,−2 = 0,N1,2 = 0,N1,1 = 1, from which we see
I0,3,0(E,E,X) = E · E = −1,
I0,3,0,−1(E,E,E) = −1,
I0,3,(1,1)(E, E, pt) = N1,1 = 1, and all other three point Gromov-Witten invariants containing two E’s vanish. Thus ∑ [ E ∗ E = I0,3,β(E,E,X)pt + I0,3,β(E,E,E)(−E) + I0,3,β(E,E,H)H β ] β + I0,3,β(E, E, pt)X q , − −1 = pt + Eq1 + Xq0q1 ,
E H F where q1 = q , q0 = q . If we define F = H − E to be the class of the fiber and q2 = q , then the relation can be written as
E ∗ E = −pt + Eq1 + Xq2. 94
Similarly one can verify all the relations
E ∗ F = pt − Eq1,
F ∗ F = Eq1,
E ∗ pt = F q2,
F ∗ pt = Xq1q2,
pt ∗ pt = Hq1q2. Chapter 3
Two-dimensional supersymmetric gauge theories with exceptional gauge groups
The contents of this chapter were adapted with minor modifications, from our publication [4].
3.1 Introduction
The Hori-Vafa construction [93] only computes Landau-Ginzburg mirrors to Abelian gauged linear sigma models. The problem of constructing analogous mirrors to non-Abelian gauged linear sigma models was only solved very recently, in [110]. The proposal of [110] can be summarized as follows. For an A-twisted two-dimensional (2,2) supersymmetric gauge theory with connected gauge group G, the mirror is a B-twisted Landau-Ginzburg orbifold, defined by (twisted) chiral multiplets
• Yi, corresponding to the N matter fields of the original gauge theory,
• Xµ˜, corresponding to nonzero rootsµ ˜ of the Lie algebra g of G, of dimension n,
• σa = D+D−Va, as many as the rank r of G, corresponding to a choice of Cartan subalgebra of g, the Lie algebra of G,
95 96
with superpotential ( ) ∑r ∑N ∑n−r a − a − W = σa ρi Yi αµ˜ ln Xµ˜ ta a=1 i=1 µ˜=1 ∑N ∑n−r ∑ + exp (−Yi) + Xµ˜ − m˜ iYi. (3.1) i=1 µ˜=1 i
a In the expression above, the ρi are components of weight vectors for the matter representa- a tions appearing in the original gauge theory, and αµ˜ are components of nonzero roots (here viewed as defining a sublattice of the weight lattice). The ta are constants, corresponding to Fayet-Iliopoulos parameters of the original gauge theory, and them ˜ i are twisted masses in the original gauge theory. One then orbifolds by the Weyl group, which acts naturally on all the fields above, and leaves the superpotential invariant. The expression above was written for A-twisted gauge theories without a superpotential, but can be generalized to mirrors of gauge theories with superpotentials by assigning suitable R-charges and changing the fundamental fields accordingly, as explained in [110].
In the analysis of this theory, it was argued that some loci are dynamically excluded – specifi-
cally, loci where any Xµ˜ vanishes. These loci turn out to reproduce excluded loci on Coulomb branches of the original gauge theories. Furthermore, critical loci of the superpotential above obey relations which correspond to relations in the OPE ring of the original A-twisted gauge theory. For gauge theories with U(1) factors in G, one has continuous Fayet-Iliopoulos pa- rameters, so one can speak of weak coupling limits, and those OPE relations are known as quantum cohomology relations. In cases in which G has no U(1) factors, so that there are no continuous Fayet-Iliopoulos parameters, there is no weak coupling limit, and so referring to such relations as ‘quantum cohomology’ relations is somewhat misleading. In such cases, we refer to the relations as defining the Coulomb ring or Coulomb branch ring.
The work [110] checked the predictions of this proposal for excluded loci and Coulomb branch and quantum cohomology relations against known results for two-dimensional gauge theories in e.g. [20, 21, 26, 114], and gave general arguments for why correlation functions in this B- twisted theory should match correlation functions in corresponding A-twisted gauge theories, such as in e.g. [115–117]. It also studied mirrors to pure gauge theories, to test and refine predictions for IR behavior described in [111]. In this chapter, we will apply this mirror construction to make predictions for two-dimensional (2,2) supersymmetric gauge theories 97
with exceptional gauge groups. To make all of these comparisons, the paper [110] utilized the following operator mirror map:
∑r − − a exp( Yi) = m˜ i + σaρi , (3.2) a=1 ∑r a Xµ˜ = σaαµ˜, (3.3) a=1
which we shall also use in this chapter.
The σs encode theta angles in the Cartan subalgebra of the original gauge theory, and have periodicities reflecting the weight lattice, or at least the sublattice generated by the matter representations. However, the weight lattice need not be normalized in the same way as a charge lattice. It is always possible to find a basis for the weight lattice (in terms of fundamental weights) so that the coefficients in the σ terms are all integers, reflecting 2π theta angle periodicities and standard charge lattice conventions, but one can also consistently
work in other bases as well. For the case of G2 gauge theories, we will use a naive basis which results in nonstandard theta angle periodicities and charge lattices.
In both case, we shall also study the mirror to the pure gauge theory, to follow up observations in [111]. In particular, [111] argued that two-dimensional pure (2,2) supersymmetric SU(k) gauge theories flow in the IR to a free theory of k − 1 twisted chiral multiplets, which [110] checked at the level of topological field theory computations and extended to SO(n) theories with discrete theta angles and to Sp(k) gauge theories. In each case, for one discrete theta angle, evidence in TFT computations was given that the theory flowed to a pure gauge theory of as many twisted chiral multiplets as the rank of the gauge group. We shall check
the analogous claim for pure gauge theories with exceptional gauge groups G2 and E8 (the
other pure gauge theories with exceptional gauge groups F4, E6, E7 are also discussed in [4]) in this chapter, at the level of topological field theory computations, and will find evidence for the same result – that the pure gauge theories (for simply-connected gauge groups) flow in the IR to a theory of as many twisted chiral multiplets as the rank of the gauge group.
Combining the results of this chapter with those in [110], a simple conjecture emerges: a pure two-dimensional (2,2) supersymmetric gauge theory with connected and simply-connected semisimple gauge group flows in the IR to a free theory of as many twisted chiral superfields as the rank of the gauge group. A check of this conjecture for Spin gauge theories can be 98
derived from the results for SO gauge theories in [110]. Now, SO groups are not simply- connected; however, we can apply two-dimensional decomposition [112, 113] and the results for SO theories with various discrete theta angles to argue that a pure Spin gauge theory flows in the IR to a free theory of as many twisted chiral superfields as the rank. Combined
with the results in this chapter for pure two-dimensional supersymmetric G2 and E6 gauge theories, we have the conjecture above.
3.2 G2
In this section we will consider the mirror Landau-Ginzburg orbifold of G2 gauge theory with matter fields in copies of the 7 representation, and then we compute quantum cohomology ring.
3.2.1 Mirror Landau-Ginzburg orbifold
The mirror Landau-Ginzburg model has fields
• Yiβ, i ∈ {1, ··· , n}, β ∈ {0, ··· , 6}, corresponding to the matter fields in n copies of
the 7 of G2, ˜ • Xm, Xm, m ∈ {1, ··· , 6}, corresponding to the short, respectively long roots of G2,
• σa, a ∈ {1, 2}.
We associate the roots and weights to fields as listed in table 3.1 and figures 3.1, 3.2.
Field Short root Field Long√ root Field Weight ˜ − X1 (1, 0) X1 ( 3/2, √3/2) Yi1 (1, 0) − ˜ − − X2 ( 1√, 0) X2 (3/2, √ 3/2) Yi2 ( 1√, 0) ˜ X3 (1/2, √3/2) X3 (3/2, √3/2) Yi3 (1/2, √3/2) − − ˜ − − − − X4 ( 1/2, √ 3/2) X4 ( 3/2, √ 3/2) Yi4 ( 1/2, √ 3/2) − ˜ − X5 ( 1/2, √3/2) X5 (0, √3) Yi5 ( 1/2, √3/2) ˜ X6 (1/2, − 3/2) X6 (0, − 3) Yi6 (1/2, − 3/2) Yi0 (0, 0)
Table 3.1: Roots and weights for G2 and associated fields. 99
˜ X5
˜ ˜ X1 X5 X3 X3
X2 X1
˜ ˜ X4 X4 X6 X2
˜ X6
Figure 3.1: Roots of G2.
The mirror superpotential takes the form ( ∑ W = σ1 (Yi1 − Yi2 + (1/2)Yi3 − (1/2)Yi4 − (1/2)Yi5 + (1/2)Yi6) i − − − + (Z1 Z2 + (1/2)Z3 (1/2)Z4 (1/2)Z5 + (1)/2)Z6) ( ) ˜ ˜ ˜ ˜ + −(3/2)Z1 + (3/2)Z2 + (3/2)Z3 − (3/2)Z4 ( √ ∑ √ + σ2 ( 3/2) (Yi3 − Yi4 + Yi5 − Yi6) + ( 3/2) (Z3 − Z4 + Z5 − Z6) i ) √ ( ) ˜ ˜ ˜ ˜ ˜ ˜ + ( 3/2) Z1 − Z2 + Z3 − Z4 + 2Z5 − 2Z6
∑ ∑6 ∑6 ∑6 ∑ ˜ + exp (−Yiα) + Xm + Xm − m˜ iYiα, (3.4) i α=0 m=1 m=1 i,α
˜ ˜ ˜ where Xm = exp(−Zm), Xm = exp(−Zm), with Xm, Xm the fundamental fields andm ˜ i are the twisted masses. ˜ The logic of the assignments above is that Xodd, Xodd correspond to positive roots, Xeven, 100
Yi5 Yi3
Yi2 Yi0 Yi1
Yi4 Yi6
Figure 3.2: Weights of 7 of G2.
˜ Xeven correspond to their opposites, and the weight vectors are associated to matter fields similarly. We follow the conventions of [118][chapter 22]: short roots are given by √ √ (1, 0), (+1/2, 3/2), (−1/2, 3/2),
long roots are given by √ √ √ (−3/2, 3/2), (+3/2, 3/2), (0, 3),
and the weights of the 7 are given by √ √ (1, 0), (1/2, 3/2), (−1/2, 3/2), (0, 0).
Before moving on, there is an important subtlety in the expression for the mirror super- potential above, involving the theta angle periodicities. As described in [110], the factors multiplied by σs are not single-valued, reflecting the fact that the σ terms encode theta angles in the abelian subgroup determined by the choice of Cartan subgroup of the original gauge group. The periodicities1 of these theta angles are determined by 2π times the weight lattice, or at least the sublattice generated by the matter representations. However, the
1On a noncompact worldsheet, the theta angles generate electric fields with periodicities determined by the matter representations – as theta increases, the electric field density eventually becomes strong enough to allow pair creation of matter fields. 101
weight lattice need not be normalized in the same way as a charge lattice. For example,
in our conventions for the weight lattice of G2 above, the σ1 terms determine a theta angle periodicity√ of 2π/√2 = π rather than 2π, and the σ2 terms determine a theta angle periodicity of ( 3/2)(2π) = 3π rather than 2π.
Now, on the one hand, the normalization of the charge lattice is ultimately a convention, and so long as one is consistent, one can work with alternative conventions. On the other hand, it is also often helpful to work with standard conventions.
For the case of G2, we shall use the normalization above, hence a nonstandard charge lattice normalization. However, it is always possible to rotate to a conventional charge lattice normalization by writing the weights in a basis of fundamental weights, for which any other weight is an integer linear combination. In terms of that mathematical basis, the theta angle periodicities determined by σs are all 2π, reflecting a standard charge lattice normalization.
We will discuss this alternative basis in more detail for F4, and in fact will use that alternative basis (and standard charge normalization) to study all the other gauge theories in this chapter, after G2. We study G2 in nonstandard conventions for illustrative purposes.
3.2.2 Weyl group
Now, let us explicitly describe the action of the Weyl group on the fields of this theory and outline explicitly why the superpotential is invariant in this case. (General arguments appeared in [110], but as the Weyl group action is more complicated here than in the examples in that paper, a more detailed verification seems in order.)
For any root α, recall that the Weyl group reflection generated by α acts on a weight µ as follows: 2(α · µ) µ 7→ µ − α. (3.5) α2 For example, for the Weyl reflection generated by α = (1, 0), it is straightforward to compute that the group action on fields corresponding to roots is given by
X1 ↔ X2,X3 ↔ X5,X4 ↔ X6, (3.6)
˜ ˜ ˜ ˜ X1 ↔ X3, X2 ↔ X4, (3.7) 102
˜ and X5,6 are invariant. The action on matter fields is
Yi1 ↔ Yi2,Yi3 ↔ Yi5,Yi4 ↔ Yi6, (3.8)
with Yi7 invariant. This is just a reflection about the y axis, which multiplies the first coordinate by −1 but leaves the second invariant. It is straightforward to check that the superpotential will be invariant under this reflection so long as
σ1 ↔ −σ1, (3.9)
and σ2 is invariant. √ For another example, for Weyl reflections generated by α = (3/2, 3/2), it is straightforward to compute that the group action on fields corresponding to roots is given by
X1 ↔ X4,X2 ↔ X3, (3.10)
with X5,6 invariant, and ˜ ˜ ˜ ˜ ˜ ˜ X1 ↔ X5, X2 ↔ X6, X3 ↔ X4. (3.11)
The action on matter fields is the same as on the mirrors to the short roots:
Yi1 ↔ Yi4,Yi2 ↔ Yi3, (3.12)
with Yi5, Yi6 invariant. The σa fields are similarly rotated: √ 1 3 σ1 7→ − σ1 − σ2, √2 2 3 1 σ 7→ − σ + σ . 2 2 1 2 2
(Note that if we describe the action above as mapping ⃗σ 7→ A⃗σ for a 2×2 matrix A, then for the choice of A implicit above, it is straightforward to check A = A−1.) It is straightforward 103
to check that the superpotential is invariant under the action above. For example, the terms
σ (Z − Z + (1/2)Z − (1/2)Z − (1/2)Z + (1/2)Z ) 1 1 2 √ 3 4 5 6 + σ ( 3/2) (Z − Z + Z − Z ) ( 2 √3 4 ) 5 6 7→ −(1/2)σ1 − ( 3/2)σ2 (Z4 − Z3 + (1/2)Z2 − (1/2)Z1 − (1/2)Z5 + (1/2)Z6) ( √ ) √ + −( 3/2)σ1 + (1/2)σ2 ( 3/2) (Z2 − Z1 + Z5 − Z6) , which is easily checked to be the same as the starting point, √ σ1 (Z1 − Z2 + (1/2)Z3 − (1/2)Z4 − (1/2)Z5 + (1/2)Z6) + σ2( 3/2) (Z3 − Z4 + Z5 − Z6) .
Similar statements are true of other terms, and so the superpotential is preserved.
To be thorough, we will consider one more example of a√ Weyl group action, this time a reflection defined by a short root, specifically α = (1/2, 3/2). It is straightforward to compute that the group action on fields corresponding to roots is given by
X1 ↔ X6,X2 ↔ X5,X3 ↔ X4, (3.13)
and ˜ ˜ ˜ ˜ X3 ↔ X6, X4 ↔ X5, (3.14) ˜ with X1,2 invariant. The action on the matter fields is the same as on the mirrors to the short roots:
Yi1 ↔ Yi6,Yi2 ↔ Yi5,Yi3 ↔ Yi4. (3.15)
This is another reflection about the axis pass through X5 and X6. The σa fields are similarly rotated: √ 1 3 σ1 7→ σ1 − σ2, 2 √ 2 3 1 σ 7→ − σ − σ . 2 2 1 2 2
It is straightforward to check that the superpotential is invariant. 104
3.2.3 Coulomb ring relations
Integrating out the sigma fields in the superpotential (3.4), we obtain two constraints: ∑ (2Yi1 − 2Yi2 + Yi3 − Yi4 − Yi5 + Yi6) + (2Z1 − 2Z2 + Z3 − Z4 − Z5 + Z6) + i ( ) ˜ ˜ ˜ ˜ + −3Z1 + 3Z2 + 3Z3 − 3Z4 = 0, ∑ (Yi3 − Yi4 + Yi5 − Yi6) + (Z3 − Z4 + Z5 − Z6) + i ( ) ˜ ˜ ˜ ˜ ˜ ˜ + Z1 − Z2 + Z3 − Z4 + 2Z5 − 2Z6 = 0.
With the two constraints above, we are free to eliminant two fundamental fields, which we will take to be Yn3 and Yn6:
∑n ∑n−1 −Yn3 = (Yi1 − Yi2 − Yi4) + Yi3 + (Z1 − Z2 + Z3 − Z4) i=1 i=1 ( ) ˜ ˜ ˜ ˜ ˜ ˜ + −Z1 + Z2 + 2Z3 − 2Z4 + Z5 − Z6 , ∑n ∑n−1 −Yn3 = (Yi1 − Yi2 − Yi5) + Yi6 + (Z1 − Z2 − Z5 + Z6) i=1 i=1 ( ) ˜ ˜ ˜ ˜ ˜ ˜ + −2Z1 + 2Z2 + Z3 − Z4 − Z5 + Z6 .
For convenience, let’s define:
Π3 ≡ exp (−Yn3) , − ∏n n∏1 X X X˜ X˜ 2X˜ = exp (Y − Y − Y ) exp (Y ) 2 4 1 4 6 , (3.16) i1 i2 i4 i3 X X ˜ ˜ 2 ˜ i=1 i=1 1 3 X2X3 X5
Π6 ≡ exp (−Yn6) , − ∏n n∏1 X X X˜ 2X˜ X˜ = exp (Y − Y − Y ) exp (Y ) 2 5 1 4 5 . (3.17) i1 i2 i5 i6 X X ˜ 2 ˜ ˜ i=1 i=1 1 6 X2 X3X6 105
Then, the superpotential (3.4) reduces to
∑n [ ] W = exp (−Yi0) + exp (−Yi1) + exp (−Yi2) + exp (−Yi4) + exp (−Yi5) i=1 ∑n−1 [ ] ∑6 ( ) ˜ + exp (−Yi3) + exp (−Yi6) + Π3 + Π6 + Xm + Xm i=1 m=1 ∑n ∑n−1 − m˜ i (Yi0 + Yi1 + Yi2 + Yi4 + Yi5) − m˜ i (Yi3 + Yi6) +m ˜ n(ln Π3 + ln Π6). i=1 i=1
˜ ˜ Notice that the superpotential has poles at X1 ≠ 0, X3 ≠ 0, X6 ≠ 0, X2 ≠ 0, X3 ≠ 0, ˜ ˜ X5 ≠ 0 and X6 ≠ 0. With the mirror maps, ∑ ∑ ∑ − − a a ˜ a exp( Yiβ) = m˜ i + σaρiβ,Xm = σaαm, Xm = σaα˜m, (3.18) a=1,2 a=1,2 a=1,2 one can get the excluded loci:
2 2 2 2 σ1σ2(σ − 3σ )(3σ − σ ) ≠ 0, (3.19) 1 2 1 2 ( √ )( √ ) ∏ 1 3 1 3 (−m + σ )(−m − σ ) −m + σ + σ −m − σ − σ · i 1 i 1 i 2 1 2 2 i 2 1 2 2 i ( √ )( √ ) 1 3 1 3 · −m − σ + σ −m + σ − σ ≠ 0. (3.20) i 2 1 2 2 i 2 1 2 2 106
The critical locus is given by
∂W : exp (−Yi0) = −m˜ i, for i = 1, ··· , n, ∂Yi0 ∂W : exp (−Yi1) = Π3 + Π6 − m˜ i + 2m ˜ n, for i = 1, ··· , n, ∂Yi1 ∂W : exp (−Yi2) = −Π3 − Π6 − m˜ i − 2m ˜ n, for i = 1, ··· , n, ∂Yi2 ∂W : exp (−Yi3) = Π3 − m˜ i +m ˜ n, for i = 1, ··· , n − 1, ∂Yi3 ∂W : exp (−Yi4) = −Π3 − m˜ i − m˜ n, for i = 1, ··· , n, ∂Yi4 ∂W : exp (−Yi5) = −Π6 − m˜ i − m˜ n, for i = 1, ··· , n, ∂Yi5 ∂W : exp (−Yi6) = Π6 − m˜ i +m ˜ n, for i = 1, ··· , n − 1, ∂Yi6
∂W : X1 = Π3 + Π6 + 2m ˜ n, ∂X1 ∂W : X2 = −Π3 − Π6 − 2m ˜ n, ∂X2 ∂W : X3 = Π3 +m ˜ n, ∂X3 ∂W : X4 = −Π3 − m˜ n, ∂X4 ∂W : X5 = −Π6 − m˜ n, ∂X5 ∂W : X6 = Π6 +m ˜ n, ∂X6 107
∂W : X˜ = −Π − 2Π − 3m ˜ , ˜ 1 3 6 n ∂X1 ∂W : X˜ = Π + 2Π + 3m ˜ , ˜ 2 3 6 n ∂X2 ∂W : X˜ = 2Π + Π + 3m ˜ , ˜ 3 3 6 n ∂X3 ∂W : X˜ = −2Π − Π − 3m ˜ , ˜ 4 3 6 n ∂X4 ∂W : X˜ = Π − Π , ˜ 5 3 6 ∂X5 ∂W : X˜ = −Π + Π , ˜ 6 3 6 ∂X6
Plug the above equations back to (3.16), (3.17), one obtains the Coulomb branch relations:
n∏−1 −1 Π3 = (Π3 − m˜ i +m ˜ n) · i=1 ∏n −1 · (Π3 + Π6 − m˜ i + 2m ˜ n) (−Π3 − Π6 − m˜ i − 2m ˜ n)(−Π3 − m˜ i − m˜ n), (3.21) i=1 n∏−1 −1 Π6 = (Π6 − m˜ i +m ˜ n) · i=1 ∏n −1 · (Π3 + Π6 − m˜ i + 2m ˜ n) (−Π3 − Π6 − m˜ i − 2m ˜ n)(−Π6 − m˜ i − m˜ n). (3.22) i=1
With the mirror map (3.18), on the critical locus relations, one finds √ √ 1 3 1 3 Π = σ + σ − m˜ , Π = σ − σ − m˜ . 3 2 1 2 2 n 6 2 1 2 2 n
Plugging them back in, one obtains the Coulomb (quantum cohomology) ring relations for 108
G2, ( √ ) ( √ ) ∏n 1 3 ∏n 1 3 (−σ − m˜ ) − σ − σ − m˜ = (σ − m˜ ) σ + σ − m˜ , (3.23) 1 i 2 1 2 2 i 1 i 2 1 2 2 i i=1 ( √ ) i=1 ( √ ) ∏n 1 3 ∏n 1 3 (−σ − m˜ ) − σ + σ − m˜ = (σ − m˜ ) σ − σ − m˜ . (3.24) 1 i 2 1 2 2 i 1 i 2 1 2 2 i i=1 i=1
Combining the above two relations, one gets ( √ )( √ ) ∏n 1 3 1 3 (−σ − m˜ )2 − σ − σ − m˜ − σ + σ − m˜ 1 i 2 1 2 2 i 2 1 2 2 i i=1 ( √ )( √ ) ∏n 1 3 1 3 = (σ − m˜ )2 σ + σ − m˜ σ − σ − m˜ , (3.25) 1 i 2 1 2 2 i 2 1 2 2 i ( √ i)(=1 √ ) ∏n 1 3 1 3 σ + σ − m˜ − σ + σ − m˜ 2 1 2 2 i 2 1 2 2 i i=1 ( √ )( √ ) ∏n 1 3 1 3 = − σ − σ − m˜ σ − σ − m˜ . (3.26) 2 1 2 2 i 2 1 2 2 i i=1
3.2.4 Vacua
In this section, we will count the number of vacua in cases with small numbers n of funda- mental fields. To solve the Coulomb branch (quantum cohomology) relations (3.23), (3.24) in general is not easy. However, since the superpotential is invariant under the Weyl group, the Coulomb ring relations (3.23), (3.24) will be covariant under the Weyl group action, which we check explicitly.
The Weyl group of G2 is the dihedral group D12 of degree 6 and order 12, which can be described as [119][section 7] { } i j 6 2 −1 D12 = a x | a = 1 = x , xax = a .
(See e.g. [119][section 47] for a discussion of representations of the dihedral groups.) Among the twelve elements of the Weyl group, there are six reflections, and below we list group 109
elements and the field corresponding to the root about which the reflection takes place:
3 5 X1 ↔ a x, X3 ↔ a x, X5 ↔ ax, ˜ 2 ˜ 4 ˜ X1 ↔ a x, X3 ↔ a x, X5 ↔ x.
Notice that the reflections are also generated by the Weyl group reflection (3.5) and we denote the reflection matrices by the fields correspond to the positive simple roots. There
are also five nontrivial rotations, corresponding to ⟨a⟩ ⊂ D12.
Now we can start to solve for the vacua (solutions of the Coulomb ring relations (3.23), (3.24)) begining with the case of small number of fundamental matter fields.
• n = 1, the only solution is σ1 = σ2 = 0 and it is excluded by the constraints (3.19), (3.20),
• n = 2, there are seven solutions but all of them are excluded by the constraints (3.19), (3.20),
• n = 3, there are ninteen solutions but all of them are excluded by the constraints (3.19), (3.20).
Starting with the case n = 4, we begin to obtain non-trivial solutions. First, let us analyze
the case of n = 4 in detail. For simplicity, from now on, we will take mi = mj = m, ∀i ≠ j
and will rescale the σi fields to σi = σi/m. There are thirty-seven solutions in total and twelve of them are true vacua (meaning, not on the excluded locus): { } √ √ i = 1, ··· , 4, si = σ1 = i 5, σ2 = i 3 , √ { √ √ } 7 3 5 3 √ i = 5, ··· , 8, si = σ1 = i − , σ2 = i (3 + 5) , √2 2 2 { √ √ } 7 3 5 3 √ i = 9, ··· , 12, s = σ = i + , σ = i (3 − 5) . i 1 2 2 2 2
Signs are assigned in each group of four solutions in the order {−, −}, {−, +}, {+, −}, 110
˜ ˜ ˜ e X1 X3 X5 X1 X3 X5 a5 a4 a3 a2 a1 1 1 3 8 9 5 12 2 7 11 4 6 10 2 2 4 10 7 11 6 1 9 5 3 12 8 3 3 1 11 6 10 7 4 12 8 2 9 5 4 4 2 5 12 8 9 3 6 10 1 7 11 5 5 7 4 10 1 11 6 3 12 8 2 9 6 6 8 9 3 12 2 5 10 1 7 11 4 7 7 5 12 2 9 3 8 11 4 6 10 1 8 8 6 1 11 4 10 7 2 9 5 3 12 9 9 11 6 1 7 4 10 5 3 12 8 2 10 10 12 2 5 3 8 9 1 7 11 4 6 11 11 9 3 8 2 5 12 4 6 10 1 7 12 12 10 7 4 6 1 11 8 2 9 5 3
Table 3.2: Weyl group actions on the vacua of the case n = 4
{+, +}. For example, √ √ √ √ X = {σ = −i 5, σ = −i 3},X = {σ = −i 5, σ = +i 3}, 1 1 √ 2 √ 2 1 √ 2 √ X3 = {σ1 = +i 5, σ2 = −i 3},X4 = {σ1 = +i 5, σ2 = +i 3}.
Under the Weyl group actions, the solutions transform as in table 3.2.2 One can see that the twelve vacua are covariant and form one Weyl orbit under the Weyl group action.
When there are five fundamental matter multiplets, there are sixty-one solutions and twenty- four of them are non-trivial. Following the same conventions, those non-trivial vacua are { √ √ } 6 6 i = 1, ··· , 4, si = σ1 = i 5 − √ , σ2 = i 3 − √ , 5 5 { √ √ } 6 6 i = 5, ··· , 8, si = σ1 = i 5 + √ , σ2 = i 3 + √ , 5 5
2Note that the Weyl group acts on σs by the inverse of the group elements. In the table, we denote the action by the original group elements instead of the inverse of the elements. Table 3.3 adopts the same notation. 111 { √ √ √ 1/2 i = 9, ··· , 12, si = σ1 = i (1/10)(35 + 12 5 + 3(185 + 80 5) , } √ √ √ 1/2 σ2 = i (3/10)(15 + 4 5 − (185 + 80 5) , { √ √ √ 1/2 i = 13, ··· , 16, si = σ1 = i (1/10)(35 + 12 5 − 3(185 + 80 5) , } √ √ √ 1/2 σ2 = i (3/10)(15 + 4 5 + (185 + 80 5) , { √ √ √ 1/2 i = 17, ··· , 20, si = σ1 = i (1/10)(35 − 12 5 + 3(185 + 80 5) , } √ √ √ 1/2 σ2 = i (3/10)(15 − 4 5 − (185 + 80 5) , { √ √ √ 1/2 i = 21, ··· , 24, si = σ1 = i (1/10)(−35 + 12 5 + 3(185 + 80 5) , } √ √ √ 1/2 σ2 = i (3/10)(15 − 4 5 + (185 + 80 5) .
The vacua form two Weyl orbits, each of which contains twelve elements. The first orbit consists of the first through fourth solutions, the seventeenth through twentieth solutions, and the twenty-first through twenty-fourth solutions. The rest of the solutions form the second Weyl orbit. We summarize the results for the Weyl group actions in table 3.3.
We checked one more case, n = 6. In this case, there are ninety-one solutions in total, of which forty-eight solutions are not on the excluded locus. As expected, these vacua form four Weyl orbits under the Weyl group action and each orbit contain twelve vacua. The solutions in this case are much more complicated, and so we do not list them explicitly.
3.2.5 Pure gauge theory
In this section, we will check (at the level of topological field theory computations) that the pure G2 theory flows in the IR to a free theory of two chiral multiplets. The superpotential 112
˜ ˜ ˜ e X1 X3 X5 X1 X3 X5 a5 a4 a3 a2 a1 1 1 3 22 17 23 20 2 21 19 4 24 18 2 2 4 18 21 19 24 1 17 23 3 20 22 3 3 1 19 24 18 21 4 20 22 2 17 23 4 4 2 23 20 22 17 3 24 18 1 21 19 17 17 19 24 1 21 4 18 23 3 20 22 2 18 18 20 2 23 3 22 17 1 21 19 4 24 19 19 17 3 22 2 23 20 4 24 18 1 21 20 20 18 21 4 24 1 19 22 2 17 23 3 21 21 23 20 2 17 3 22 19 4 24 18 1 22 22 24 1 19 4 18 21 2 17 23 3 20 23 23 21 4 18 1 19 24 3 20 22 2 17 24 24 22 17 3 20 2 23 18 1 21 19 4 5 5 7 16 9 13 12 6 15 11 8 14 10 6 6 8 10 15 11 14 5 9 13 7 12 16 7 7 5 11 14 10 15 8 12 16 6 9 13 8 8 6 13 12 16 9 7 14 10 5 15 11 9 9 11 14 5 15 8 10 13 7 12 16 6 10 10 12 6 13 7 16 9 5 15 11 8 14 11 11 9 7 16 6 13 12 8 14 10 5 15 12 12 10 15 8 14 5 11 16 6 9 13 7 13 13 15 8 10 5 11 14 7 12 16 6 9 14 14 16 9 7 12 6 13 10 5 15 11 8 15 15 13 12 6 9 7 16 11 8 14 10 5 16 16 14 5 11 8 10 15 6 9 13 7 12
Table 3.3: Weyl group actions on the vacua of five fundamental matter multiplets 113
of the pure gauge theory is
∑2 ( ) ∑ a a ˜ ˜ W = σa αmZm +α ˜mZm + (Xm + Xm), a=1[( m )
=σ1 Z1 − Z2 + (1/2)Z3 − (1/2)Z4 − (1/2)Z5 + (1/2)Z6 ( )] ˜ ˜ ˜ ˜ + − (3/2)Z1 + (3/2)Z2 + (3/2)Z3 − (3/2)Z4 ( ) √ ˜ ˜ ˜ ˜ ˜ ˜ + σ2( 3/2) Z3 − Z4 + Z5 − Z6 + Z1 − Z2 + Z3 − Z4 + 2Z5 − 2Z6
∑6 ˜ + (Xm + Xm). m=1
˜ Integrating out Xm and Xm, one obtains the constraints, ∑ a Xm = σaαm, ∑a ˜ a Xm = σα˜m. a
˜ The point is that all the Xm fields and Xm fields correspond to the nonzero roots of G2 which come in pairs, positive roots and their negatives. As a result, pluging the constraints above back into the superpoential, one gets W = 0. Therefore, the pure gauge theory indeed flows to a free theory of two twisted chiral multiplies in the IR limit.
On the other hand, integrating out σ1 and σ2, one obtains the constraints,
− ln X1 + ln X2 − (1/2) ln X3 + (1/2) ln X4 + (1/2) ln X5 − (1/2) ln X6 + (3/2) ln X˜ − (3/2) ln X˜ − (3/2)X˜ + (3/2)X˜ = 0, √ 1 2 √4 4 3 3 − (ln X − ln X + ln X − ln X ) − (X˜ − X˜ + X˜ − X˜ + 2X˜ − 2X˜ ) = 0 2 3 4 5 6 2 1 2 3 4 5 6
With those two constraints, one can eliminate two fields in the superpotential,
X X X˜ X˜ 2X˜ X X X˜ 2X˜ X˜ X = a 1 3 2 3 5 ,X = b 1 6 2 3 6 , 4 ˜ ˜ 2 ˜ 5 ˜ 2 ˜ ˜ X2X1X4 X6 X2X1 X4X5 114 with a = 1 and b = 1. Plugging this back into the superpotential, we get
˜ ˜ ˜ ˜ ˜ ˜ W =X1 + X2 + X3 + X6 + X1 + X2 + X3 + X4 + X5 + X6 X X X˜ X˜ 2X˜ X X X˜ 2X˜ X˜ + a 1 3 2 3 5 + b 1 6 2 3 6 . (3.27) ˜ ˜ 2 ˜ ˜ 2 ˜ ˜ X2X1X4 X6 X2X1 X4X5
The critical loci are given by
a = b = 1,
X1 = −X2 = X3 + X6, ˜ ˜ X1 = −X2 = −X3 − 2X6, ˜ ˜ X3 = −X4 = 2X3 + X6, ˜ ˜ X5 = −X6 = X3 − X6.
One can easily see that, on the critical locus, the above superpotential (3.27) vanishes with two free fields X3 and X6. Therefore, the pure gauge theory again flows to free theories of two chiral multiplies in the IR.
3.2.6 Comparison with A model results
In this section, we will discuss the A-twisted gauge theory with gauge group G2 and n chiral superfields in the 7, and compare it to results from our proposed mirror, as a check of our methods. In principle, this should necessarily work, for reasons discussed in [110][section 3]; however, we will check for the special case of G2 that indeed everything works as it should, which will also give us the opportunity to discuss the role of theta angle periodicities and charge lattice normalizations. ˜ The one-loop effective twisted superpotential Weff of the A-twisted gauge theory is given 115
by [115][equ’ns (2.17), (2.19)], [116][equ’ns (3.16), (3.17)] ∑ ( ) ˜ − ′1 ′2 − ′1 ′2 − − Weff = (σ1ρi,α + σ2ρi,α m˜ i) ln(σ1ρi,α + σ2ρi,α m˜ i) 1 i,α∑ ( ) ′1 ′2 ′1 ′2 − + (σ1αm + σ2αm) ln(σ1αm + σ2αm) 1 ∑m ( ) ′1 ′2 ′1 ′2 − + (σ1α˜m + σ2α˜m) ln(σ1α˜m + σ2α˜m) 1 . m
Since the logarithm branch cuts in the expressions above are supposed to reflect (standard) theta angle periodicities of 2π, we have rescaled ρ and α to ρ′ and α′. Specifically, we have rescaled√ all the charges under σ1 by a factor of 2 and all the charges under σ2 by a factor of 2/ 3.
Since the roots and weights of G2 come in positive/negative pairs, we can further simplify the effective superpotential: ∑ ∑ ˜ − ′1 ′2 − ′1 ′2 − − Weff = (σ1ρi,α + σ2ρi,α m˜ i) ln(σ1ρi,α + σ2ρi,α m˜ i) 7 m˜ i + 6πi(σ1 + σ2). i,α i
The vacua are given by
∏ ′ ′ ′ 1 1 2 − ρi,α (σ1ρi,α + σ2ρi,α m˜ i) = 1, i,α ∏ ′ ′ ′ 2 1 2 − ρi,α (σ1ρi,α + σ2ρi,α m˜ i) = 1. i,α
Plugging in the charges, we get ∏ 2 (2σ1 − m˜ i) (σ1 + σ2 − m˜ i)(σ1 − σ2 − m˜ i) i ∏ 2 = (−2σ1 − m˜ i) (−σ1 − σ2 − m˜ i)(−σ1 + σ2 − m˜ i), ∏ i (σ1 + σ2 − m˜ i)(−σ1 + σ2 − m˜ i) i ∏ = (−σ1 − σ2 − m˜ i)(σ1 − σ2 − m˜ i). i
The relations above are the same as the Coulomb ring relations (3.25), (3.26) we derived 116
from the B model with a suitable rescaling of the σ fields, √ 1 3 σ → σ , σ → σ . 1 2 1 2 2 2
Thus, we see that A model results match those of the B model mirror, as expected, after correctly taking into account subtleties in theta angle periodicities.
3.2.7 Comparison to other bases for weight lattice
So far in this section, we have used a particular basis for the weight lattice for G2. In principle, other bases are related by field redefinitions. To make this more explicit, in this section we will outline corresponding results in a different basis for the weight lattice, specifically a basis of fundamental weights. We will describe this basis in greater detail in the section on
F4, as it will be used for the rest of the exceptional gauge groups in this chapter, but for the moment, will content ourselves to briefly outline results.
In terms of that basis of fundamental weights, it can be shown that the roots and pertinent
weights of G2 are expanded as given in table 3.4.
Field Short root Field Long root Field Weight ˜ X1 (1, 0) X1 (0, 1) Yi1 (1, 0) ˜ X2 ( − 1, 1) X2 (3, −1) Yi2 ( − 1, 1) ˜ X3 (2, −1) X3 ( − 3, 2) Yi3 (2, −1) ˜ X4 ( − 1, 0) X4 (0, −1) Yi4 ( − 1, 0) ˜ X5 (1, −1) X5 ( − 3, 1) Yi5 (1, −1) ˜ X6 ( − 2, 1) X6 (3, −2) Yi6 ( − 2, 1) Yi0 (0, 0)
Table 3.4: Roots and weights for G2 and associated fields.
Repeating the same mirror analysis as described earlier in this section, we derive the Coulomb branch relations
∏n ∏n ′ − ′ − ′ − − ′ − − ′ ′ − (σ1 m˜ i)(2σ1 σ2 m˜ i) = ( σ1 m˜ i)( 2σ1 + σ2 m˜ i), (3.28) i=1 i=1 ∏n ∏n ′ − ′ − ′ − ′ − − ′ ′ − − ′ ′ − (σ1 σ2 m˜ i)(2σ1 σ2 m˜ i) = ( σ1 + σ2 m˜ i)( 2σ1 + σ2 m˜ i), (3.29) i=1 i=1 117 and excluded loci
′ ′ ′ − ′ ′ − ′ ′ − ′ ′ − ′ ̸ σ1σ2(σ1 σ2)(2σ1 σ2)(3σ1 σ2)(3σ1 2σ2) = 0, (3.30)
∏n ′ − − ′ ′ − ′ − ′ − (σ1 m˜ i)( σ1 + σ2 m˜ i)(2σ1 σ2 m˜ i) i=1 · − ′ − ′ − ′ − − ′ ′ − ̸ ( σ1 m˜ i)(σ1 σ2 m˜ i)( 2σ1 + σ2 m˜ i) = 0. (3.31)
Comparing with earlier reuslts for the critical locus (3.23), (3.24) and excluded loci (3.19), (3.20), computed in the earlier basis, we find that the results above are related by the following linear field redefinitions √ ′ 1 3 σ1 = σ1 + σ2, (3.32) √2 2 ′ σ2 = 3σ2, (3.33) or equivalently
′ − ′ σ1 = 2σ1 σ2, (3.34) 1 ′ σ2 = √ σ . (3.35) 3 2
3.3 E8
In this section, we will discuss the mirror theory to a two-dimensional 8 gauge theory. The group E8 and its algebra are the largest and most complicated exceptional groups, we shall only list results.
3.3.1 Mirror Landau-Ginzburg orbifold
We will consider an E8 gauge theory with n matter fields in the 248, the lowest-dimensional representation, which also happens to be the adjoint representation. The mirror Landau- Ginzburg model has fields 118
• Yiα, i ∈ {1, ··· , n}, α ∈ {1, ··· , 248}
• Xm, m ∈ {1, 2, ··· , 120}, correponding to positive roots, and X120+m, associated with
the negative roots of those associated to Xm,
• σa, a ∈ {1, 2, ··· , 8}.
As before, we work with an integer-lattice-basis for the roots and weights, corresponding to standard theta angle periodicities. We associate the roots and weights to fields as listed in the tables 3.5, 3.6, and 3.7.
Field Positive root/weight Field Positive root/weight X1, Yi,1, (0, 0, 0, 0, 0, 0, 1, 0) X2, Yi,2 (0, 0, 0, 0, 0, 1, −1, 0) X3, Yi,3 (0, 0, 0, 0, 1, −1, 0, 0) X4, Yi,4 (0, 0, 0, 1, −1, 0, 0, 0) X5, Yi,5 (0, 0, 1, −1, 0, 0, 0, 0) X6, Yi,6 (0, 1, −1, 0, 0, 0, 0, 1) X7, Yi,7 (0, 1, 0, 0, 0, 0, 0, −1) X8, Yi,8 (1, −1, 0, 0, 0, 0, 0, 1) X9, Yi,9 (−1, 0, 0, 0, 0, 0, 0, 1) X10, Yi,10 (1, −1, 1, 0, 0, 0, 0, −1) X11, Yi,11 (−1, 0, 1, 0, 0, 0, 0, −1) X12, Yi,12 (1, 0, −1, 1, 0, 0, 0, 0) X13, Yi,13 (−1, 1, −1, 1, 0, 0, 0, 0) X14, Yi,14 (1, 0, 0, −1, 1, 0, 0, 0) X15, Yi,15 (−1, 1, 0, −1, 1, 0, 0, 0) X16, Yi,16 (0, −1, 0, 1, 0, 0, 0, 0) X17, Yi,17 (1, 0, 0, 0, −1, 1, 0, 0) X18, Yi,18 (−1, 1, 0, 0, −1, 1, 0, 0), X19, Yi,19 (0, −1, 1, −1, 1, 0, 0, 0) X20, Yi,20 (1, 0, 0, 0, 0, −1, 1, 0) X21, Yi,21 (−1, 1, 0, 0, 0, −1, 1, 0) X22, Yi,22 (0, −1, 1, 0, −1, 1, 0, 0) X23, Yi,23 (0, 0, −1, 0, 1, 0, 0, 1) X24, Yi,24 (1, 0, 0, 0, 0, 0, −1, 0) X25, Yi,25 (−1, 1, 0, 0, 0, 0, −1, 0) X26, Yi,26 (0, −1, 1, 0, 0, −1, 1, 0) X27, Yi,27 (0, 0, −1, 1, −1, 1, 0, 1) X28, Yi,28 (0, 0, 0, 0, 1, 0, 0, −1) X29, Yi,29 (0, −1, 1, 0, 0, 0, −1, 0) X30, Yi,30 (0, 0, −1, 1, 0, −1, 1, 1) X31, Yi,31 (0, 0, 0, −1, 0, 1, 0, 1) X32, Yi,32 (0, 0, 0, 1, −1, 1, 0, −1) X33, Yi,33 (0, 0, −1, 1, 0, 0, −1, 1) X34, Yi,34 (0, 0, 0, −1, 1, −1, 1, 1) X35, Yi,35 (0, 0, 0, 1, 0, −1, 1, −1) X36, Yi,36 (0, 0, 1, −1, 0, 1, 0, −1) X37, Yi,37 (0, 0, 0, −1, 1, 0, −1, 1) X38, Yi,38 (0, 0, 0, 0, −1, 0, 1, 1) X39, Yi,39 (0, 0, 0, 1, 0, 0, −1, −1) X40, Yi,40 (0, 0, 1, −1, 1, −1, 1, −1) X41, Yi,41 (0, 1, −1, 0, 0, 1, 0, 0) X42, Yi,42 (0, 0, 0, 0, −1, 1, −1, 1) X43, Yi,43 (0, 0, 1, −1, 1, 0, −1, −1) X44, Yi,44 (0, 0, 1, 0, −1, 0, 1, −1) X45, Yi,45 (0, 1, −1, 0, 1, −1, 1, 0) X46, Yi,46 (1, −1, 0, 0, 0, 1, 0, 0) X47, Yi,47 (−1, 0, 0, 0, 0, 1, 0, 0) X48, Yi,48 (0, 0, 0, 0, 0, −1, 0, 1)
Table 3.5: First set of roots of E8 and associated fields. 119
Field Positive root/weight Field Positive root/weight X49, Yi,49 (0, 0, 1, 0, −1, 1, −1, −1) X50, Yi,50 (0, 1, −1, 0, 1, 0, −1, 0) X51, Yi,51 (0, 1, −1, 1, −1, 0, 1, 0) X52, Yi,52 (1, −1, 0, 0, 1, −1, 1, 0) X53, Yi,53 (−1, 0, 0, 0, 1, −1, 1, 0) X54, Yi,54 (0, 0, 1, 0, 0, −1, 0, −1) X55, Yi,55 (0, 1, −1, 1, −1, 1, −1, 0) X56, Yi,56 (0, 1, 0, −1, 0, 0, 1, 0) X57, Yi,57 (1, −1, 0, 0, 1, 0, −1, 0) X58, Yi,58 (1, −1, 0, 1, −1, 0, 1, 0) X59, Yi,59 (−1, 0, 0, 0, 1, 0, −1, 0) X60, Yi,60 (−1, 0, 0, 1, −1, 0, 1, 0) X61, Yi,61 (0, 1, −1, 1, 0, −1, 0, 0) X62, Yi,62 (0, 1, 0, −1, 0, 1, −1, 0) X63, Yi,63 (1, −1, 0, 1, −1, 1, −1, 0) X64, Yi,64 (1, −1, 1, −1, 0, 0, 1, 0) X65, Yi,65 (−1, 0, 0, 1, −1, 1, −1, 0) X66, Yi,66 (−1, 0, 1, −1, 0, 0, 1, 0) X67, Yi,67 (0, 1, 0, −1, 1, −1, 0, 0) X68, Yi,68 (1, −1, 0, 1, 0, −1, 0, 0) X69, Yi,69 (1, −1, 1, −1, 0, 1, −1, 0) X70, Yi,70 (1, 0, −1, 0, 0, 0, 1, 1) X71, Yi,71 (−1, 0, 0, 1, 0, −1, 0, 0) X72, Yi,72 (−1, 0, 1, −1, 0, 1, −1, 0) X73, Yi,73 (−1, 1, −1, 0, 0, 0, 1, 1) X74, Yi,74 (0, 1, 0, 0, −1, 0, 0, 0) X75, Yi,75 (1, −1, 1, −1, 1, −1, 0, 0) X76, Yi,76 (1, 0, −1, 0, 0, 1, −1, 1) X77, Yi,77 (1, 0, 0, 0, 0, 0, 1, −1) X78, Yi,78 (−1, 0, 1, −1, 1, −1, 0, 0) X79, Yi,79 (−1, 1, −1, 0, 0, 1, −1, 1) X80, Yi,80 (−1, 1, 0, 0, 0, 0, 1, −1) X81, Yi,81 (0, −1, 0, 0, 0, 0, 1, 1) X82, Yi,82 (1, −1, 1, 0, −1, 0, 0, 0) X83, Yi,83 (1, 0, −1, 0, 1, −1, 0, 1) X84, Yi,84 (1, 0, 0, 0, 0, 1, −1, −1) X85, Yi,85 (−1, 0, 1, 0, −1, 0, 0, 0) X86, Yi,86 (−1, 1, −1, 0, 1, −1, 0, 1) X87, Yi,87 (−1, 1, 0, 0, 0, 1, −1, −1) X88, Yi,88 (0, −1, 0, 0, 0, 1, −1, 1) X89, Yi,89 (0, −1, 1, 0, 0, 0, 1, −1) X90, Yi,90 (1, 0, −1, 1, −1, 0, 0, 1) X91, Yi,91 (1, 0, 0, 0, 1, −1, 0, −1) X92, Yi,92 (−1, 1, −1, 1, −1, 0, 0, 1) X93, Yi,93 (−1, 1, 0, 0, 1, −1, 0, −1) X94, Yi,94 (0, −1, 0, 0, 1, −1, 0, 1)
Table 3.6: Second set of roots of E8 and associated fields.
For the rest of the fields, the roots and weights are given by
Xa+120 = −Xa, a = 1, ··· , 120,
Yi,a+120 = −Yi,120, a = 1, ··· , 120.
3.3.2 Superpotential
In this section, we give the superpotential for the Landau-Ginzburg orbifold mirror to the theory above.
∑8 ∑n ∑248 ∑n ∑248 ∑240 a W = σaC − m˜ i Yi,α + exp(−Yi,α) + Xm, (3.36) a=1 i=1 α=1 i=1 α=1 m=1 120
Field Positive root/weight Field Positive root/weight X95, Yi,95 (0, −1, 1, 0, 0, 1, −1, −1) X96, Yi,96 (0, 0, −1, 1, 0, 0, 1, 0) X97, Yi,97 (1, 0, 0, −1, 0, 0, 0, 1) X98, Yi,98 (1, 0, 0, 1, −1, 0, 0, −1) X99, Yi,99 (−1, 1, 0, −1, 0, 0, 0, 1) X100, Yi,100 (−1, 1, 0, 1, −1, 0, 0, −1) X101, Yi,101 (0, −1, 0, 1, −1, 0, 0, 1) X102, Yi,102 (0, −1, 1, 0, 1, −1, 0, −1) X103, Yi,103 (0, 0, −1, 1, 0, 1, −1, 0) X104, Yi,104 (0, 0, 0, −1, 1, 0, 1, 0) X105, Yi,105 (1, 0, 1, −1, 0, 0, 0, −1) X106, Yi,106 (−1, 1, 1, −1, 0, 0, 0, −1) X107, Yi,107 (0, −1, 1, −1, 0, 0, 0, 1) X108, Yi,108 (0, −1, 1, 1, −1, 0, 0, −1) X109, Yi,109 (0, 0, −1, 1, 1, −1, 0, 0) X110, Yi,110 (0, 0, 0, −1, 1, 1, −1, 0) X111, Yi,111 (0, 0, 0, 0, −1, 1, 1, 0) X112, Yi,112 (1, 1, −1, 0, 0, 0, 0, 0) X113, Yi,113 (−1, 2, −1, 0, 0, 0, 0, 0) X114, Yi,114 (0, −1, 2, −1, 0, 0, 0, −1) X115, Yi,115 (0, 0, −1, 0, 0, 0, 0, 2) X116, Yi,116 (0, 0, −1, 2, −1, 0, 0, 0) X117, Yi,117 (0, 0, 0, −1, 2, −1, 0, 0) X118, Yi,118 (0, 0, 0, 0, −1, 2, −1, 0) X119, Yi,119 (0, 0, 0, 0, 0, −1, 2, 0) X120, Yi,120 (2, −1, 0, 0, 0, 0, 0, 0) Yi,241,Yi,242 (0, 0, 0, 0, 0, 0, 0, 0) Yi,243,Yi,244 (0, 0, 0, 0, 0, 0, 0, 0) Yi,245,Yi,246 (0, 0, 0, 0, 0, 0, 0, 0) Yi,247,Yi,248 (0, 0, 0, 0, 0, 0, 0, 0)
Table 3.7: Third set of roots of E8 and associated fields. where the Ca are:
∑n ( 1 C = Yi,8 − Yi,9 + Yi,10 − Yi,11 + Yi,12 − Yi,13 + Yi,14 − Yi,15 + Yi,17 − Yi,18 + Yi,20 − Yi,21 i=1
+ Yi,24 − Yi,25 + Yi,46 − Yi,47 + Yi,52 − Yi,53 + Yi,57 + Yi,58 − Yi,59 − Yi,60 + Yi,63 + Yi,64
− Yi,65 − Yi,66 + Yi,68 + Yi,69 + Yi,70 − Yi,71 − Yi,72 − Yi,73 + Yi,75 + Yi,76 + Yi,77 − Yi,78
− Yi,79 − Yi,80 + Yi,82 + Yi,83 + Yi,84 − Yi,85 − Yi,86 − Yi,87 + Yi,90 + Yi,91 − Yi,92 − Yi,93
+ Yi,97 + Yi,98 − Yi,99 − Yi,100 + Yi,105 − Yi,106 + Yi,112 − Yi,113 + 2Yi,120 − Yi,128 + Yi,129
− Yi,130 + Yi,131 − Yi,132 + Yi,133 − Yi,134 + Yi,135 − Yi,137 + Yi,138 − Yi,140 + Yi,141
− Yi,144 + Yi,145 − Yi,166 + Yi,167 − Yi,172 + Yi,173 − Yi,177 − Yi,178 + Y179 + Yi,180 − Yi,183
− Yi,184 + Yi,185 + Yi,186 − Yi,188 − Yi,189 − Yi,190 + Yi,191 + Yi,192 + Yi,193 − Yi,195
− Yi,196 − Yi,197 + Yi,198 + Yi,199 + Yi,200 − Yi,202 − Yi,203 − Yi,204 + Yi,205 + Yi,206 + Y − Y − Y + Y + Y − Y − Y + Y + Y − Y + Y i,207 i,210 i,211 ) i,212 i,213 i,217 i,218 i,219 i,220 225 i,226 − Yi,232 + Yi,233 − 2Yi,240 121
+ Z8 − Z9 + Z10 − Z11 + Z12 − Z13 + Z14 − Z15 + Z17 − Z18 + Z20 − Z21 + Z24 − Z25
+ Z46 − Z47 + Z52 − Z53 + Z57 + Z58 − Z59 − Z60 + Z63 + Z64 − Z65 − Z66 + Z68
+ Z69 + Z70 − Z71 − Z72 − Z73 + Z75 + Z76 + Z77 − Z78 − Z79 − Z80 + Z82 + Z83
+ Z84 − Z85 − Z86 − Z87 + Z90 + Z91 − Z92 − Z93 + Z97 + Z98 − Z99 − Z100 + Z105
− Z106 + Z112 − Z113 + 2Z120 − Z128 + Z129 − Z130 + Z131 − Z132 + Z133 − Z134 + Z135
− Z137 + Z138 − Z140 + Z141 − Z144 + Z145 − Z166 + Z167 − Z172 + Z173 − Z177 − Z178
+ Z179 + Z180 − Z183 − Z184 + Z185 + Z186 − Z188 − Z189 − Z190 + Z191 + Z192 + Z193
− Z195 − Z196 − Z197 + Z198 + Z199 + Z200 − Z202 − Z203 − Z204 + Z205 + Z206 + Z207
− Z210 − Z211 + Z212 + Z213 − Z217 − Z218 + Z219 + Z220 − Z225 + Z226 − Z232 + Z233
− 2Z240,
∑n ( 2 C = Yi,6 + Yi,7 − Yi,8 − Yi,10 + Yi,13 + Yi,15 − Yi,16 + Yi,18 − Yi,19 + Yi,21 − Yi,22 + Yi,25 i=1
− Yi,26 − Yi,29 + Yi,41 + Yi,45 − Yi,46 + Yi,50 + Yi,51 − Yi,52 + Yi,55 + Yi,56 − Yi,57 − Yi,58
+ Yi,61 + Yi,62 − Yi,63 − Yi,64 + Yi,67 − Yi,68 − Yi,69 + Yi,73 + Yi,74 − Yi,75 + Yi,79 + Yi,80
− Yi,81 − Yi,82 + Yi,86 + Yi,87 − Yi,88 − Yi,89 + Yi,92 + Yi,93 − Yi,94 − Yi,95 + Yi,99 + Yi,100
− Yi,101 − Yi,102 + Yi,106 − Yi,107 − Yi,108 + Yi,112 + 2Yi,113 − Yi,114 − Yi,120 − Yi,126
− Yi,127 + Yi,128 + Yi,130 − Yi,133 − Yi,135 + Yi,136 − Yi,138 + Yi,139 − Yi,141 + Yi,142
− Yi,145 + Yi,146 + Yi,149 − Yi,161 − Yi,165 + Yi,166 − Yi,170 − Y171 + Yi,172 − Yi,175
− Yi,176 + Yi,177 + Yi,178 − Yi,181 − Yi,182 + Yi,183 + Yi,184 − Yi,187 + Yi,188 + Yi,189
− Yi,193 − Yi,194 + Yi,195 − Yi,199 − Yi,200 + Yi,201 + Yi,202 − Yi,206 − Yi,207 + Yi,208 + Y − Y − Y + Y + Y − Y − Y + Y + Y − Y i,209 i,212 i,213 i,214 i,215 i,219 ) i,220 i,221 i,222 i,226 + Yi,227 + Yi,228 − Yi,232 − 2Yi,233 + Yi,234 + Yi,240 122
+ Z6 + Z7 − Z8 − Z10 + Z13 + Z15 − Z16 + Z18 − Z19 + Z21 − Z22 + Z25 − Z26 − Z29
+ Z41 + Z45 − Z46 + Z50 + Z51 − Z52 + Z55 + Z56 − Z57 − Z58 + Z61 + Z62 − Z63 − Z64
+ Z67 − Z68 − Z69 + Z73 + Z74 − Z75 + Z79 + Z80 − Z81 − Z82 + Z86 + Z87 − Z88 − Z89
+ Z92 + Z93 − Z94 − Z95 + Z99 + Z100 − Z101 − Z102 + Z106 − Z107 − Z108 + Z112 + 2Z113
− Z114 − Z120 − Z126 − Z127 + Z128 + Z130 − Z133 − Z135 + Z136 − Z138 + Z139 − Z141
+ Z142 − Z145 + Z146 + Z149 − Z161 − Z165 + Z166 − Z170 − Z171 + Z172 − Z175 − Z176
+ Z177 + Z178 − Z181 − Z182 + Z183 + Z184 − Z187 + Z188 + Z189 − Z193 − Z194 + Z195
− Z199 − Z200 + Z201 + Z202 − Z206 − Z207 + Z208 + Z209 − Z212 − Z213 + Z214 + Z215
− Z219 − Z220 + Z221 + Z222 − Z226 + Z227 + Z228 − Z232 − 2Z233 + Z234 + Z240,
∑n ( 3 C = Yi,5 − Yi,6 + Yi,10 + Yi,11 − Yi,12 − Yi,13 + Yi,19 + Yi,22 − Yi,23 + Yi,26 − Yi,27 i=1
+ Yi,29 − Yi,30 − Yi,33 + Yi,36 + Yi,40 − Yi,41 + Yi,43 + Yi,44 − Yi,45 + Yi,49 − Yi,50
− Yi,51 + Yi,54 − Yi,55 − Yi,61 + Yi,64 + Yi,66 + Yi,69 − Yi,70 + Yi,72 − Yi,73 + Yi,75
− Yi,76 + Yi,78 − Yi,79 + Yi,82 − Yi,83 + Yi,85 − Yi,86 + Yi,89 − Yi,90 − Yi,92 + Yi,95
− Yi,96 + Yi,102 − Yi,103 + Yi,105 + Yi,106 + Yi,107 + Yi,108 − Yi,109 − Yi,112 − Yi,113
+ 2Yi,114 − Yi,115 − Yi,116 − Yi,125 + Yi,126 − Yi,130 − Yi,131 + Yi,132 + Yi,133 − Yi,139
− Yi,142 + Yi,143 − Yi,146 + Yi,147 − Yi,149 + Yi,150 + Yi,153 − Yi,156 − Yi,160 + Yi,161
− Yi,163 − Yi,164 + Yi,165 − Yi,169 + Yi,170 + Yi,171 − Yi,174 + Yi,175 + Yi,181 − Yi,184
− Yi,186 − Yi,189 + Yi,190 − Yi,192 + Yi,193 − Yi,195 + Yi,196 − Yi,198 + Yi,199 − Yi,202 + Y − Y + Y − Y + Y + Y − Y + Y − Y + Y i,203 i,205 i,206 i,209 i,210 i,212 i,215 i,216 i,222 i,223 ) − Yi,225 − Yi,226 − Yi,227 − Yi,228 + Yi,229 + Yi,232 + Yi,233 − 2Yi,234 + Yi,235 + Yi,236 123
+ Z5 − Z6 + Z10 + Z11 − Z12 − Z13 + Z19 + Z22 − Z23 + Z26 − Z27 + Z29 − Z30
− Z33 + Z36 + Z40 − Z41 + Z43 + Z44 − Z45 + Z49 − Z50 − Z51 + Z54 − Z55 − Z61
+ Z64 + Z66 + Z69 − Z70 + Z72 − Z73 + Z75 − Z76 + Z78 − Z79 + Z82 − Z83 + Z85
− Z86 + Z89 − Z90 − Z92 + Z95 − Z96 + Z102 − Z103 + Z105 + Z106 + Z107 + Z108
− Z109 − Z112 − Z113 + 2Z114 − Z115 − Z116 − Z125 + Z126 − Z130 − Z131 + Z132
+ Z133 − Z139 − Z142 + Z143 − Z146 + Z147 − Z149 + Z150 + Z153 − Z156 − Z160
+ Z161 − Z163 − Z164 + Z165 − Z169 + Z170 + Z171 − Z174 + Z175 + Z181 − Z184
− Z186 − Z189 + Z190 − Z192 + Z193 − Z195 + Z196 − Z198 + Z199 − Z202 + Z203
− Z205 + Z206 − Z209 + Z210 + Z212 − Z215 + Z216 − Z222 + Z223 − Z225 − Z226
− Z227 − Z228 + Z229 + Z232 + Z233 − 2Z234 + Z235 + Z236,
∑n ( 4 C = Yi,4 − Yi,5 + Yi,12 + Yi,13 − Yi,14 − Yi,15 + Yi,16 − Yi,19 + Yi,27 + Yi,30 − Yi,31 i=1
+ Yi,32 + Yi,33 − Yi,34 + Yi,35 − Yi,36 − Yi,37 + Yi,39 − Yi,40 − Yi,43 + Yi,51 + Yi,55
− Yi,56 + Yi,58 + Yi,60 + Yi,61 − Yi,62 + Yi,63 − Yi,64 + Yi,65 − Yi,66 − Yi,67 + Yi,68
− Yi,69 + Yi,71 − Yi,72 − Yi,75 − Yi,78 + Yi,90 + Yi,92 + Yi,96 − Yi,97 + Yi,98 − Yi,99
+ Yi,100 + Yi,101 + Yi,103 − Yi,104 − Yi,105 − Y106 − Yi,107 + Yi,108 + Yi,109 − Yi,110
− Yi,114 + 2Yi,116 − Yi,117 − Yi,124 + Yi,125 − Yi,132 − Yi,133 + Yi,134 + Yi,135 − Yi,136
+ Yi,139 − Yi,147 − Yi,150 + Yi,151 − Yi,152 − Yi,153 + Yi,154 − Yi,155 + Yi,156 + Yi,157
− Yi,159 + Yi,160 + Y163 − Yi,171 − Yi,175 + Yi,176 − Yi,178 − Yi,180 − Yi,181 + Yi,182
− Yi,183 + Yi,184 − Yi,185 + Yi,186 + Yi,187 − Yi,188 + Yi,189 − Yi,191 + Yi,192 + Yi,195 + Y − Y − Y − Y + Y − Y + Y − Y − Y − Y i,198 i,210 i,212 i,216 i,217 i,218 i,219 i,220 i,221 i,223) + Yi,224 + Yi,225 + Yi,226 + Yi,227 − Y228 − Yi,229 + Yi,230 + Yi,234 − 2Yi,236 + Yi,237 124
+ Z4 − Z5 + Z12 + Z13 − Z14 − Z15 + Z16 − Z19 + Z27 + Z30 − Z31 + Z32 + Z33
− Z34 + Z35 − Z36 − Z37 + Z39 − Z40 − Z43 + Z51 + Z55 − Z56 + Z58 + Z60
+ Z61 − Z62 + Z63 − Z64 + Z65 − Z66 − Z67 + Z68 − Z69 + Z71 − Z72 − Z75
− Z78 + Z90 + Z92 + Z96 − Z97 + Z98 − Z99 + Z100 + Z101 + Z103 − Z104
− Z105 − Z106 − Z107 + Z108 + Z109 − Z110 − Z114 + 2Z116 − Z117 − Z124
+ Z125 − Z132 − Z133 + Z134 + Z135 − Z136 + Z139 − Z147 − Z150 + Z151
− Z152 − Z153 + Z154 − Z155 + Z156 + Z157 − Z159 + Z160 + Z163 − Z171
− Z175 + Z176 − Z178 − Z180 − Z181 + Z182 − Z183 + Z184 − Z185 + Z186
+ Z187 − Z188 + Z189 − Z191 + Z192 + Z195 + Z198 − Z210 − Z212 − Z216
+ Z217 − Z218 + Z219 − Z220 − Z221 − Z223 + Z224 + Z225 + Z226 + Z227
− Z228 − Z229 + Z230 + Z234 − 2Z236 + Z237,
∑n ( 5 C = Yi,3 − Yi,4 + Yi,14 + Yi,15 − Yi,17 − Yi,18 + Yi,19 − Yi,22 + Yi,23 − Yi,27 + Yi,28 i=1
− Yi,32 + Yi,34 + Yi,37 − Yi,38 + Yi,40 − Yi,42 + Yi,43 − Yi,44 + Yi,45 − Yi,49 + Yi,50
− Yi,51 + Yi,52 + Yi,53 − Yi,55 + Yi,57 − Yi,58 + Yi,59 − Yi,60 − Yi,63 − Yi,65 + Yi,67
− Yi,74 + Yi,75 + Yi,78 − Yi,82 + Yi,83 − Yi,85 + Yi,86 − Yi,90 + Yi,91 − Yi,92 + Yi,93
+ Yi,94 − Yi,98 − Yi,100 − Yi,101 + Yi,102 + Yi,104 − Yi,108 + Yi,109 + Yi,110 − Yi,111
− Yi,116 + 2Yi,117 − Yi,118 − Yi,123 + Yi,124 − Yi,134 − Yi,135 + Yi,137 + Yi,138 − Yi,139
+ Yi,142 − Yi,143 + Yi,147 − Yi,148 + Yi,152 − Yi,154 − Yi,157 + Yi,158 − Yi,160 + Yi,162
− Yi,163 + Yi,164 − Y165 + Yi,169 − Yi,170 + Yi,171 − Yi,172 − Yi,173 + Yi,175 − Yi,177
+ Yi,178 − Yi,179 + Yi,180 + Yi,183 + Yi,185 − Yi,187 + Yi,194 − Yi,195 − Yi,198 + Yi,202 − Y + Y − Y + Y − Y + Y − Y − Y + Y + Y i,203 i,205 i,206 i,210 i,211 i,212 i,213 i,214 i,218 i,220 ) + Yi,221 − Yi,222 − Yi,224 + Yi,228 − Y229 − Yi,230 + Yi,231 + Yi,236 − 2Yi,237 + Yi,238 125
+ Z3 − Z4 + Z14 + Z15 − Z17 − Z18 + Z19 − Z22 + Z23 − Z27 + Z28 − Z32 + Z34
+ Z37 − Z38 + Z40 − Z42 + Z43 − Z44 + Z45 − Z49 + Z50 − Z51 + Z52 + Z53
− Z55 + Z57 − Z58 + Z59 − Z60 − Z63 − Z65 + Z67 − Z74 + Z75 + Z78 − Z82
+ Z83 − Z85 + Z86 − Z90 + Z91 − Z92 + Z93 + Z94 − Z98 − Z100 − Z101 + Z102
+ Z104 − Z108 + Z109 + Z110 − Z111 − Z116 + 2Z117 − Z118 − Z123 + Z124 − Z134
− Z135 + Z137 + Z138 − Z139 + Z142 − Z143 + Z147 − Z148 + Z152 − Z154 − Z157
+ Z158 − Z160 + Z162 − Z163 + Z164 − Z165 + Z169 − Z170 + Z171 − Z172 − Z173
+ Z175 − Z177 + Z178 − Z179 + Z180 + Z183 + Z185 − Z187 + Z194 − Z195 − Z198
+ Z202 − Z203 + Z205 − Z206 + Z210 − Z211 + Z212 − Z213 − Z214 + Z218 + Z220
+ Z221 − Z222 − Z224 + Z228 − Z229 − Z230 + Z231 + Z236 − 2Z237 + Z238,
∑n ( 6 C = Yi,2 − Yi,3 + Yi,17 + Yi,18 − Yi,20 − Yi,21 + Yi,22 − Yi,26 + Yi,27 − Yi,30 + Yi,31 i=1
+ Yi,32 − Yi,34 − Yi,35 + Yi,36 − Yi,40 + Yi,41 + Yi,42 − Yi,45 + Yi,46 + Yi,47 − Yi,48
+ Yi,49 − Yi,52 − Yi,53 − Yi,54 + Yi,55 − Yi,61 + Yi,62 + Yi,63 + Yi,65 − Yi,67 − Yi,68
+ Yi,69 − Yi,71 + Yi,72 − Yi,75 + Yi,76 − Yi,78 + Yi,79 − Yi,83 + Yi,84 − Yi,86 + Yi,87
+ Yi,88 − Yi,91 − Yi,93 − Yi,94 + Yi,95 − Yi,102 + Yi,103 − Yi,109 + Yi,110 + Yi,111
− Yi,117 + 2Yi,118 − Yi,119 − Yi,122 + Yi,123 − Yi,137 − Yi,138 + Yi,140 + Yi,141 − Yi,142
+ Yi,146 − Yi,147 + Yi,150 − Yi,151 − Yi,152 + Yi,154 + Yi,155 − Yi,156 + Yi,160 − Yi,161
− Yi,162 + Yi,165 − Y166 − Yi,167 + Yi,168 − Yi,169 + Yi,172 + Yi,173 + Yi,174 − Yi,175
+ Yi,181 − Yi,182 − Yi,183 − Yi,185 + Yi,187 + Yi,188 − Yi,189 + Yi,191 − Yi,192 + Yi,195 − Y + Y − Y + Y − Y + Y − Y − Y + Y + Y i,196 i,198 i,199 i,203 i,204 i,206 i,207 i,208 i,211 i,213 ) + Yi,214 − Yi,215 + Yi,222 − Yi,223 + Y229 − Yi,230 − Yi,231 + Yi,237 − 2Yi,238 + Yi,239 126
+ Z2 − Z3 + Z17 + Z18 − Z20 − Z21 + Z22 − Z26 + Z27 − Z30 + Z31 + Z32 − Z34
− Z35 + Z36 − Z40 + Z41 + Z42 − Z45 + Z46 + Z47 − Z48 + Z49 − Z52 − Z53 − Z54
+ Z55 − Z61 + Z62 + Z63 + Z65 − Z67 − Z68 + Z69 − Z71 + Z72 − Z75 + Z76 − Z78
+ Z79 − Z83 + Z84 − Z86 + Z87 + Z88 − Z91 − Z93 − Z94 + Z95 − Z102 + Z103 − Z109
+ Z110 + Z111 − Z117 + 2Z118 − Z119 − Z122 + Z123 − Z137 − Z138 + Z140 + Z141
− Z142 + Z146 − Z147 + Z150 − Z151 − Z152 + Z154 + Z155 − Z156 + Z160 − Z161
− Z162 + Z165 − Z166 − Z167 + Z168 − Z169 + Z172 + Z173 + Z174 − Z175 + Z181
− Z182 − Z183 − Z185 + Z187 + Z188 − Z189 + Z191 − Z192 + Z195 − Z196 + Z198
− Z199 + Z203 − Z204 + Z206 − Z207 − Z208 + Z211 + Z213 + Z214 − Z215 + Z222
− Z223 + Z229 − Z230 − Z231 + Z237 − 2Z238 + Z239,
∑n ( 7 C = Yi,1 − Yi,2 + Yi,20 + Yi,21 − Yi,24 − Yi,25 + Yi,26 − Yi,29 + Yi,30 − Yi,33 + Yi,34 i=1
+ Yi,35 − Yi,37 + Yi,38 − Yi,39 + Yi,40 − Yi,42 − Yi,43 + Yi,44 + Yi,45 − Yi,49 − Yi,50
+ Yi,51 + Yi,52 + Yi,53 − Yi,55 + Yi,56 − Yi,57 + Yi,58 − Yi,59 + Yi,60 − Yi,62 − Yi,63
+ Yi,64 − Yi,65 + Yi,66 − Yi,69 + Yi,70 − Yi,72 + Yi,73 − Yi,76 + Yi,77 − Yi,79 + Yi,80
+ Yi,81 − Yi,84 − Yi,87 − Yi,88 + Yi,89 − Yi,95 + Yi,96 − Yi,103 + Yi,104 − Yi,110 + Yi,111
− Yi,118 + 2Yi,119 − Yi,121 + Yi,122 − Yi,140 − Yi,141 + Yi,144 + Yi,145 − Yi,146 + Yi,149
− Yi,150 + Yi,153 − Yi,154 − Yi,155 + Yi,157 − Yi,158 + Yi,159 − Yi,160 + Yi,162 + Yi,163
− Yi,164 − Yi,165 + Yi,169 + Yi,170 − Yi,171 − Yi,172 − Yi,173 + Yi,175 − Yi,176 + Yi,177
− Yi,178 + Yi,179 − Yi,180 + Yi,182 + Yi,183 − Yi,184 + Yi,185 − Yi,186 + Yi,189 − Yi,190 + Y − Y + Y − Y + Y − Y − Y + Y + Y + Y i,192 i,193 i,196 i,197 i,199 i,200 i,201 i,204 i,207 ) i,208 − Yi,209 + Yi,215 − Yi,216 + Yi,223 − Yi,224 + Yi,230 − Yi,231 + Yi,238 − 2Yi,239 127
+ Z1 − Z2 + Z20 + Z21 − Z24 − Z25 + Z26 − Z29 + Z30 − Z33 + Z34 + Z35 − Z37
+ Z38 − Z39 + Z40 − Z42 − Z43 + Z44 + Z45 − Z49 − Z50 + Z51 + Z52 + Z53
− Z55 + Z56 − Z57 + Z58 − Z59 + Z60 − Z62 − Z63 + Z64 − Z65 + Z66 − Z69
+ Z70 − Z72 + Z73 − Z76 + Z77 − Z79 + Z80 + Z81 − Z84 − Z87 − Z88 + Z89
− Z95 + Z96 − Z103 + Z104 − Z110 + Z111 − Z118 + 2Z119 − Z121 + Z122 − Z140
− Z141 + Z144 + Z145 − Z146 + Z149 − Z150 + Z153 − Z154 − Z155 + Z157 − Z158
+ Z159 − Z160 + Z162 + Z163 − Z164 − Z165 + Z169 + Z170 − Z171 − Z172 − Z173
+ Z175 − Z176 + Z177 − Z178 + Z179 − Z180 + Z182 + Z183 − Z184 + Z185 − Z186
+ Z189 − Z190 + Z192 − Z193 + Z196 − Z197 + Z199 − Z200 − Z201 + Z204 + Z207
+ Z208 − Z209 + Z215 − Z216 + Z223 − Z224 + Z230 − Z231 + Z238 − 2Z239,
∑n ( 8 C = Yi,6 − Yi,7 + Yi,8 + Yi,9 − Yi,10 − Yi,11 + Yi,23 + Yi,27 − Yi,28 + Yi,30 + Yi,31 i=1
− Yi,32 + Yi,33 + Yi,34 − Yi,35 − Yi,36 + Yi,37 + Yi,38 − Yi,39 − Yi,40 + Yi,42 − Yi,43
− Yi,44 + Yi,48 − Yi,49 − Yi,54 + Yi,70 + Yi,73 + Yi,76 − Yi,77 + Yi,79 − Yi,80 + Yi,81
+ Yi,83 − Yi,84 + Yi,86 − Yi,87 + Yi,88 − Yi,89 + Yi,90 − Yi,91 + Yi,92 − Yi,93 + Yi,94
− Yi,95 + Yi,97 − Yi,98 + Yi,99 − Yi,100 + Yi,101 − Yi,102 − Yi,105 − Yi,106 + Yi,107
− Yi,108 − Yi,114 + 2Yi,115 − Yi,126 + Yi,127 − Yi,128 − Yi,129 + Yi,130 + Yi,131 − Yi,143
− Yi,147 + Yi,148 − Yi,150 − Yi,151 + Yi,152 − Yi,153 − Yi,154 + Yi,155 + Yi,156 − Yi,157
− Yi,158 + Yi,159 + Yi,160 − Yi,162 + Yi,163 + Yi,164 − Yi,168 + Yi,169 + Yi,174 − Yi,190
− Yi,193 − Yi,196 + Yi,197 − Yi,199 + Yi,200 − Yi,201 − Yi,203 + Yi,204 − Yi,206 + Yi,207 − Y + Y − Y + Y − Y + Y − Y + Y − Y + Y i,208 i,209 i,210 i,211 i,212 i,213 i,214 i,215 i,217 i,218 ) − Yi,219 + Yi,220 − Yi,221 + Yi,222 + Yi,225 + Yi,226 − Yi,227 + Yi,228 + Yi,234 − 2Yi,235 128
+ Z6 − Z7 + Z8 + Z9 − Z10 − Z11 + Z23 + Z27 − Z28 + Z30 + Z31 − Z32 + Z33 + Z34
− Z35 − Z36 + Z37 + Z38 − Z39 − Z40 + Z42 − Z43 − Z44 + Z48 − Z49 − Z54 + Z70
+ Z73 + Z76 − Z77 + Z79 − Z80 + Z81 + Z83 − Z84 + Z86 − Z87 + Z88 − Z89 + Z90
− Z91 + Z92 − Z93 + Z94 − Z95 + Z97 − Z98 + Z99 − Z100 + Z101 − Z102 − Z105
− Z106 + Z107 − Z108 − Z114 + 2Z115 − Z126 + Z127 − Z128 − Z129 + Z130 + Z131
− Z143 − Z147 + Z148 − Z150 − Z151 + Z152 − Z153 − Z154 + Z155 + Z156 − Z157
− Z158 + Z159 + Z160 − Z162 + Z163 + Z164 − Z168 + Z169 + Z174 − Z190 − Z193
− Z196 + Z197 − Z199 + Z200 − Z201 − Z203 + Z204 − Z206 + Z207 − Z208 + Z209
− Z210 + Z211 − Z212 + Z213 − Z214 + Z215 − Z217 + Z218 − Z219 + Z220 − Z221
+ Z222 + Z225 + Z226 − Z227 + Z228 + Z234 − 2Z235.
3.3.3 Coulomb ring relations
For the group E8, we obtain eight Coulomb ring relations, using the same methods as before. The results for these ring relations are listed below. 129
The first ring relation is
∏n 2 (m ˜ i + 2σ1 − σ2) (−m˜ i − σ1 + 2σ2 − σ3)(−m˜ i − σ1 − σ2 + σ3)(−m˜ i − σ1 + σ3 − σ4) i=1
· (−m˜ i − σ1 + σ2 − σ3 + σ4)(−m˜ i − σ1 + σ3 − σ5)(−m˜ i − σ1 + σ4 − σ5)
· (−m˜ i − σ1 + σ2 − σ3 + σ5)(−m˜ i − σ1 + σ2 − σ4 + σ5)(−m˜ i − σ1 + σ2 − σ6)
· (−m˜ i − σ1 + σ4 − σ6)(−m˜ i − σ1 + σ5 − σ6)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ6)
· (−m˜ i − σ1 + σ6)(−m˜ i − σ1 + σ2 − σ4 + σ6)(−m˜ i − σ1 + σ2 − σ5 + σ6)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)(−m˜ i − σ1 + σ2 − σ7)
· (−m˜ i − σ1 + σ5 − σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ7)(−m˜ i − σ1 + σ6 − σ7)
· (−m˜ i − σ1 + σ3 − σ4 + σ6 − σ7)(−m˜ i − σ1 + σ2 − σ5 + σ6 − σ7)
· (−m˜ i − σ1 + σ7)(−m˜ i − σ1 + σ3 − σ4 + σ7)(−m˜ i − σ1 + σ2 − σ5 + σ7)·
· (−m˜ i − σ1 + σ4 − σ5 + σ7)(−m˜ i − σ1 + σ2 − σ6 + σ7)
· (−m˜ i − σ1 + σ5 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)·
· (−m˜ i − σ1 + σ3 − σ8)(−m˜ i − σ1 + σ2 + σ3 − σ4 − σ8)(−m˜ i − σ1 + σ4 − σ8)
· (−m˜ i − σ1 + σ2 + σ4 − σ5 − σ8)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i − σ1 + σ3 − σ5 + σ6 − σ8)(−m˜ i − σ1 + σ3 − σ7 − σ8)
· (−m˜ i − σ1 + σ2 + σ7 − σ8)(−m˜ i − σ1 + σ3 − σ6 + σ7 − σ8)(−m˜ i − σ1 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ8)(−m˜ i − σ1 + σ2 − σ4 + σ8)(−m˜ i − σ1 − σ3 + σ4 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)(−m˜ i − σ1 − σ4 + σ5 + σ8)·
· (−m˜ i − σ1 − σ5 + σ6 + σ8)(−m˜ i − σ1 − σ7 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ7 + σ8)(−m˜ i − σ1 − σ6 + σ7 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ7)(−m˜ i − σ1 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)(−m˜ i − σ1 + σ2 + σ6 − σ7 − σ8)
· (−m˜ i − σ1 + σ2 + σ5 − σ6 − σ8)(−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8) 130
∏n 2 = (m ˜ i − 2σ1 + σ2) (−m˜ i + σ1 + σ2 − σ3)(−m˜ i + σ1 − 2σ2 + σ3) i=1
· (−m˜ i + σ1 − σ2 + σ3 − σ4)(−m˜ i + σ1 − σ3 + σ4)(−m˜ i + σ1 − σ2 + σ3 − σ5)
· (−m˜ i + σ1 − σ2 + σ4 − σ5)(−m˜ i + σ1 − σ3 + σ5)(−m˜ i + σ1 − σ4 + σ5)
· (−m˜ i + σ1 − σ6)(−m˜ i + σ1 − σ2 + σ4 − σ6)(−m˜ i + σ1 − σ2 + σ5 − σ6)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ6)(−m˜ i + σ1 − σ4 + σ6)
· (−m˜ i + σ1 − σ5 + σ6)(−m˜ i + σ1 − σ3 + σ4 − σ5 + σ6)(−m˜ i + σ1 − σ7)
· (−m˜ i + σ1 − σ3 + σ4 − σ7)(−m˜ i + σ1 − σ2 + σ5 − σ7)(−m˜ i + σ1 − σ4 + σ5 − σ7)
· (−m˜ i + σ1 − σ2 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ7)
· (−m˜ i + σ1 − σ5 + σ7)(−m˜ i + σ1 − σ2 + σ4 − σ5 + σ7)(−m˜ i + σ1 − σ6 + σ7)
· (−m˜ i + σ1 − σ3 + σ4 − σ6 + σ7)(−m˜ i + σ1 − σ2 + σ5 − σ6 + σ7)
· (−m˜ i + σ1 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)
· (−m˜ i + σ1 − σ2 + σ4 − σ8)(−m˜ i + σ1 + σ4 − σ5 − σ8)
· (−m˜ i + σ1 + σ5 − σ6 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)
· (−m˜ i + σ1 + σ6 − σ7 − σ8)(−m˜ i + σ1 + σ7 − σ8)(−m˜ i + σ1 + σ3 − σ4 − σ8)
· (−m˜ i + σ1 − σ2 + σ8)(−m˜ i + σ1 − σ3 + σ8)(−m˜ i + σ1 − σ4 + σ8)
· (−m˜ i + σ1 − σ2 − σ3 + σ4 + σ8)(−m˜ i + σ1 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i + σ1 − σ3 + σ5 − σ6 + σ8)(−m˜ i + σ1 − σ2 − σ5 + σ6 + σ8)
· (−m˜ i + σ1 − σ3 + σ6 − σ7 + σ8)(−m˜ i + σ1 − σ3 + σ7 + σ8)
· (−m˜ i + σ1 − σ2 − σ7 + σ8)(−m˜ i + σ1 − σ2 − σ6 + σ7 + σ8)
· (−m˜ i + σ1 − σ2 − σ4 + σ5 + σ8)(−m˜ i + σ1 − σ2 + σ3 − σ7 − σ8)
· (−m˜ i + σ1 − σ4 + σ5 − σ6 + σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ7)(−m˜ i + σ1 − σ5 + σ6 − σ7). 131
The second ring relation is
∏n 2 (−m˜ i + 2σ1 − σ2)(−m˜ i + σ1 − 2σ2 + σ3) (−m˜ i − σ1 − σ2 + σ3) i=1
· (−m˜ i + σ1 − σ2 + σ3 − σ4)(−m˜ i − σ2 + σ4)(−m˜ i + σ1 − σ2 + σ3 − σ5)
· (−m˜ i + σ1 − σ2 + σ4 − σ5)(−m˜ i − σ2 + σ5)(−m˜ i − σ2 + σ3 − σ4 + σ5)
· (−m˜ i − σ2 + σ3 − σ6)(−m˜ i + σ1 − σ2 + σ4 − σ6)(−m˜ i + σ1 − σ2 + σ5 − σ6)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ6)
· (−m˜ i − σ2 + σ3 − σ5 + σ6)(−m˜ i − σ2 + σ4 − σ5 + σ6)(−m˜ i − σ2 + σ3 − σ7)
· (−m˜ i − σ2 + σ4 − σ7)(−m˜ i + σ1 − σ2 + σ5 − σ7)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ7)
· (−m˜ i + σ1 − σ2 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ7)(−m˜ i + σ1 − σ2 + σ4 − σ5 + σ7)
· (−m˜ i − σ2 + σ4 − σ6 + σ7)(−m˜ i + σ1 − σ2 + σ5 − σ6 + σ7)
· (−m˜ i − σ2 + σ3 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ8)(−m˜ i − σ2 + 2σ3 − σ4 − σ8)
· (−m˜ i + σ1 − σ2 + σ4 − σ8)(−m˜ i − σ2 + σ3 + σ4 − σ5 − σ8)
· (−m˜ i − σ2 + σ3 + σ5 − σ6 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ7 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ2 + σ3 − σ5 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ7)(−m˜ i − σ2 + σ3 − σ4 + σ6)
· (−m˜ i − σ2 + σ3 + σ6 − σ7 − σ8)(−m˜ i − σ2 + σ3 + σ7 − σ8)
· (−m˜ i − σ2 + σ8)(−m˜ i + σ1 − σ2 + σ8)(−m˜ i − σ2 + σ3 − σ4 + σ8)
· (−m˜ i + σ1 − σ2 − σ3 + σ4 + σ8)(−m˜ i − σ2 + σ4 − σ5 + σ8)
· (−m˜ i − σ2 + σ5 − σ6 + σ8)(−m˜ i + σ1 − σ2 − σ5 + σ6 + σ8)
· (−m˜ i − σ2 + σ6 − σ7 + σ8)(−m˜ i − σ2 + σ7 + σ8)(−m˜ i − σ2 + σ3 − σ6 + σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)(−m˜ i + σ1 − σ2 − σ4 + σ5 + σ8)
· (−m˜ i + σ1 − σ2 − σ7 + σ8)(−m˜ i + σ1 − σ2 − σ6 + σ7 + σ8) 132
∏n 2 = (−m˜ i − 2σ1 + σ2)(−m˜ i + σ1 + σ2 − σ3)(m ˜ i + σ1 − 2σ2 + σ3) i=1
· (−m˜ i + σ2 − σ4)(−m˜ i − σ1 + σ2 − σ3 + σ4)(−m˜ i + σ2 − σ5)
· (−m˜ i + σ2 − σ3 + σ4 − σ5)(−m˜ i − σ1 + σ2 − σ3 + σ5)(−m˜ i − σ1 + σ2 − σ4 + σ5)
· (−m˜ i − σ1 + σ2 − σ6)(−m˜ i + σ2 − σ3 + σ4 − σ6)(−m˜ i + σ2 − σ3 + σ5 − σ6)
· (−m˜ i + σ2 − σ4 + σ5 − σ6)(−m˜ i + σ2 − σ3 + σ6)(−m˜ i − σ1 + σ2 − σ4 + σ6)
· (−m˜ i − σ1 + σ2 − σ5 + σ6)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ7)(−m˜ i + σ2 − σ3 + σ5 − σ7)
· (−m˜ i + σ2 − σ3 + σ6 − σ7)(−m˜ i + σ2 − σ4 + σ6 − σ7)
· (−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ7)
· (−m˜ i − σ1 + σ2 − σ5 + σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ7)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)(−m˜ i + σ2 − σ3 + σ5 − σ6 + σ7)
· (−m˜ i + σ2 − σ8)(−m˜ i − σ1 + σ2 − σ8)(−m˜ i − σ1 + σ2 + σ3 − σ4 − σ8)
· (−m˜ i + σ2 − σ3 + σ4 − σ8)(−m˜ i − σ1 + σ2 + σ4 − σ5 − σ8)
· (−m˜ i − σ1 + σ2 + σ5 − σ6 − σ8)(−m˜ i + σ2 − σ5 + σ6 − σ8)(−m˜ i + σ2 − σ7 − σ8)
· (−m˜ i − σ1 + σ2 + σ6 − σ7 − σ8)(−m˜ i + σ2 − σ6 + σ7 − σ8)
· (−m˜ i + σ2 − σ3 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ8)(−m˜ i − σ1 + σ2 − σ4 + σ8)
· (−m˜ i + σ2 − 2σ3 + σ4 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)(−m˜ i + σ2 − σ3 − σ5 + σ6 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ7 + σ8)
· (−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8)(−m˜ i + σ2 − σ3 − σ7 + σ8)
· (−m˜ i + σ2 − σ3 − σ4 + σ5 + σ8)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ4 + σ5 − σ7)(−m˜ i − σ1 + σ2 − σ5 + σ6 − σ7)
· (−m˜ i − σ1 + σ2 − σ6 + σ7)(−m˜ i + σ2 − σ4 + σ5 − σ8)
· (−m˜ i − σ1 + σ2 − σ7)(−m˜ i + σ2 − σ4 + σ7)(−m˜ i − σ1 + σ2 + σ7 − σ8). 133
The third ring relation is
∏n (−m˜ i + σ1 + σ2 − σ3)(−m˜ i − σ1 + 2σ2 − σ3)(−m˜ i − σ3 + σ4) i=1
· (−m˜ i + σ1 − σ3 + σ4)(−m˜ i − σ1 + σ2 − σ3 + σ4)(−m˜ i + σ2 − σ3 + σ4 − σ5)
· (−m˜ i − σ3 + 2σ4 − σ5)(−m˜ i + σ1 − σ3 + σ5)(−m˜ i − σ1 + σ2 − σ3 + σ5)
· (−m˜ i + σ2 − σ3 + σ4 − σ6)(−m˜ i + σ2 − σ3 + σ5 − σ6)(−m˜ i − σ3 + σ4 + σ5 − σ6)
· (−m˜ i + σ2 − σ3 + σ6)(−m˜ i + σ1 − σ3 + σ4 − σ5 + σ6)
· (−m˜ i + σ1 − σ3 + σ4 − σ7)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ7)(−m˜ i + σ2 − σ3 + σ5 − σ7)
· (−m˜ i + σ2 − σ3 + σ6 − σ7)(−m˜ i − σ3 + σ4 + σ6 − σ7)
· (−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ7)
· (−m˜ i − σ3 + σ4 + σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ7)(−m˜ i + σ1 − σ3 + σ4 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)(−m˜ i + σ2 − σ3 + σ5 − σ6 + σ7)
· (−m˜ i + σ1 − σ3 + σ8)(−m˜ i + σ2 − σ3 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ8) 2 · (−m˜ i + σ2 − 2σ3 + σ4 + σ8) (−m˜ i − σ1 − σ3 + σ4 + σ8)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i + σ2 − σ3 − σ4 + σ5 + σ8)(−m˜ i − σ3 + σ4 − σ6 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)(−m˜ i − σ3 + σ6 + σ8)
· (−m˜ i − σ3 + σ4 − σ5 + σ6 + σ8)(−m˜ i + σ2 − σ3 − σ7 + σ8)(−m˜ i − σ3 + σ4 − σ7 + σ8)
· (−m˜ i − σ3 + σ5 − σ7 + σ8)(−m˜ i + σ1 − σ3 + σ6 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ7 + σ8)
· (−m˜ i + σ1 − σ3 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ6 + σ7 + σ8)(−m˜ i − σ3 + σ5 − σ6 + σ7 + σ8)(−m˜ i − σ3 + 2σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)
· (−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8)(−m˜ i + σ2 − σ3 − σ5 + σ6 + σ8)
· (−m˜ i + σ1 − σ3 + σ5 − σ6 + σ8)(−m˜ i + σ1 − σ2 − σ3 + σ4 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)(−m˜ i + σ2 − σ3 + σ4 − σ8)
· (−m˜ i − σ3 + σ5 + σ8)(−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8) 134
∏n = (−m˜ i + σ1 − 2σ2 + σ3)(−m˜ i − σ1 − σ2 + σ3)(−m˜ i + σ3 − σ4) i=1
· (−m˜ i − σ1 + σ3 − σ4)(−m˜ i + σ1 − σ2 + σ3 − σ4)(−m˜ i − σ1 + σ3 − σ5)
· (−m˜ i + σ1 − σ2 + σ3 − σ5)(−m˜ i + σ3 − 2σ4 + σ5)(−m˜ i − σ2 + σ3 − σ4 + σ5)
· (−m˜ i − σ2 + σ3 − σ6)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ6)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)(−m˜ i − σ2 + σ3 − σ4 + σ6)
· (−m˜ i − σ2 + σ3 − σ5 + σ6)(−m˜ i + σ3 − σ4 − σ5 + σ6)
· (−m˜ i − σ2 + σ3 − σ7)(−m˜ i + σ3 − σ4 − σ7)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ7)
· (−m˜ i − σ1 + σ3 − σ4 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i − σ1 + σ3 − σ4 + σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ7)
· (−m˜ i − σ2 + σ3 − σ6 + σ7)(−m˜ i + σ3 − σ4 − σ6 + σ7)
· (−m˜ i + σ3 − 2σ8)(−m˜ i − σ1 + σ3 − σ8)(−m˜ i − σ2 + σ3 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ8)(−m˜ i + σ1 + σ3 − σ4 − σ8)(−m˜ i − σ1 + σ2 + σ3 − σ4 − σ8)
· (−m˜ i + σ3 − σ5 − σ8)(−m˜ i − σ2 + σ3 + σ4 − σ5 − σ8)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)(−m˜ i + σ3 − σ6 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 − σ8)(−m˜ i + σ3 − σ4 + σ6 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)(−m˜ i − σ1 + σ3 − σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ7 − σ8)(−m˜ i − σ2 + σ3 + σ6 − σ7 − σ8)
· (−m˜ i + σ3 − σ5 + σ6 − σ7 − σ8)(−m˜ i − σ2 + σ3 + σ7 − σ8)(−m˜ i + σ3 − σ4 + σ7 − σ8)
· (−m˜ i + σ3 − σ5 + σ7 − σ8)(−m˜ i − σ1 + σ3 − σ6 + σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)(−m˜ i − σ2 + σ3 − σ4 + σ8) 2 · (−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)(m ˜ i + σ2 − 2σ3 + σ4 + σ8)
· (−m˜ i + σ3 − σ4 + σ6 − σ7 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ7 − σ8)
· (−m˜ i − σ2 + σ3 + σ5 − σ6 − σ8)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ7)(−m˜ i − σ2 + σ3 − σ5 + σ6 − σ7)
· (−m˜ i − σ1 + σ3 − σ5 + σ6 − σ8). 135
The fourth ring relation is
∏n (−m˜ i + σ2 − σ4)(−m˜ i + σ3 − σ4)(−m˜ i − σ1 + σ3 − σ4) i=1 2 · (−m˜ i + σ1 − σ2 + σ3 − σ4)(−m˜ i + σ3 − 2σ4 + σ5)
· (−m˜ i − σ4 + σ5)(−m˜ i + σ1 − σ4 + σ5)(−m˜ i − σ1 + σ2 − σ4 + σ5)
· (−m˜ i − σ2 + σ3 − σ4 + σ5)(−m˜ i + σ2 − σ4 + σ5 − σ6)
· (−m˜ i − σ1 + σ3 − σ4 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)
· (−m˜ i − σ4 + 2σ5 − σ6)(−m˜ i + σ1 − σ4 + σ6)(−m˜ i − σ1 + σ2 − σ4 + σ6)
· (−m˜ i − σ2 + σ3 − σ4 + σ6)(−m˜ i + σ3 − σ4 − σ5 + σ6)(−m˜ i + σ3 − σ4 − σ7)
· (−m˜ i + σ1 − σ4 + σ5 − σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ7)
· (−m˜ i − σ2 + σ3 − σ4 + σ5 − σ7)(−m˜ i + σ2 − σ4 + σ6 − σ7)
· (−m˜ i − σ1 + σ3 − σ4 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i − σ4 + σ5 + σ6 − σ7)(−m˜ i + σ2 − σ4 + σ7)(−m˜ i − σ1 + σ3 − σ4 + σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ7)(−m˜ i − σ4 + σ5 + σ7)(−m˜ i + σ3 − σ4 − σ6 + σ7)
· (−m˜ i + σ1 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)(−m˜ i + σ1 + σ3 − σ4 − σ8)
· (−m˜ i − σ1 + σ2 + σ3 − σ4 − σ8)(−m˜ i − σ2 + 2σ3 − σ4 − σ8)
· (−m˜ i + σ2 − σ4 + σ5 − σ8)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ6 − σ8)
· (−m˜ i + σ3 − σ4 + σ6 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ6 − σ7 − σ8)(−m˜ i + σ3 − σ4 + σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)(−m˜ i + σ1 − σ4 + σ8)
· (−m˜ i − σ1 + σ2 − σ4 + σ8)(−m˜ i − σ2 + σ3 − σ4 + σ8)(−m˜ i − σ1 − σ4 + σ5 + σ8)
· (−m˜ i + σ1 − σ2 − σ4 + σ5 + σ8)(−m˜ i + σ2 − σ3 − σ4 + σ5 + σ8)
· (−m˜ i − σ4 + σ5 − σ6 + σ8)(−m˜ i − σ4 + σ6 + σ8)(−m˜ i − σ4 + σ5 − σ7 + σ8)
· (−m˜ i − σ4 + σ6 − σ7 + σ8)(−m˜ i − σ4 + σ7 + σ8)(−m˜ i − σ4 + σ5 − σ6 + σ7 + σ8) 136
∏n = (−m˜ i − σ2 + σ4)(−m˜ i − σ3 + σ4)(−m˜ i + σ1 − σ3 + σ4) i=1
· (−m˜ i − σ1 + σ2 − σ3 + σ4)(−m˜ i + σ4 − σ5)(−m˜ i − σ1 + σ4 − σ5) 2 · (−m˜ i + σ1 − σ2 + σ4 − σ5)(−m˜ i + σ2 − σ3 + σ4 − σ5)(m ˜ i + σ3 − 2σ4 + σ5)
· (−m˜ i − σ1 + σ4 − σ6)(−m˜ i + σ1 − σ2 + σ4 − σ6)(−m˜ i + σ2 − σ3 + σ4 − σ6)
· (−m˜ i − σ3 + σ4 + σ5 − σ6)(−m˜ i + σ4 − 2σ5 + σ6)(−m˜ i − σ2 + σ4 − σ5 + σ6)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ6)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)
· (−m˜ i − σ2 + σ4 − σ7)(−m˜ i + σ1 − σ3 + σ4 − σ7)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ7)
· (−m˜ i + σ4 − σ5 − σ7)(−m˜ i − σ3 + σ4 + σ6 − σ7)(−m˜ i − σ1 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i − σ3 + σ4 + σ7)(−m˜ i − σ1 + σ4 − σ5 + σ7)(−m˜ i + σ1 − σ2 + σ4 − σ5 + σ7)
· (−m˜ i + σ2 − σ3 + σ4 − σ5 + σ7)(−m˜ i − σ2 + σ4 − σ6 + σ7)
· (−m˜ i + σ1 − σ3 + σ4 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)
· (−m˜ i + σ4 − σ5 − σ6 + σ7)(−m˜ i − σ1 + σ4 − σ8)(−m˜ i + σ1 − σ2 + σ4 − σ8)
· (−m˜ i + σ2 − σ3 + σ4 − σ8)(−m˜ i + σ1 + σ4 − σ5 − σ8)
· (−m˜ i − σ1 + σ2 + σ4 − σ5 − σ8)(−m˜ i − σ2 + σ3 + σ4 − σ5 − σ8)
· (−m˜ i + σ4 − σ6 − σ8)(−m˜ i + σ4 − σ5 + σ6 − σ8)(−m˜ i + σ4 − σ7 − σ8)
· (−m˜ i + σ4 − σ5 + σ6 − σ7 − σ8)(−m˜ i + σ4 − σ5 + σ7 − σ8)
· (−m˜ i + σ4 − σ6 + σ7 − σ8)(−m˜ i + σ2 − 2σ3 + σ4 + σ8)(−m˜ i − σ1 − σ3 + σ4 + σ8)
· (−m˜ i + σ1 − σ2 − σ3 + σ4 + σ8)(−m˜ i − σ2 + σ4 − σ5 + σ8)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i − σ3 + σ4 − σ6 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 + σ8)
· (−m˜ i − σ3 + σ4 − σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ5 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ6 + σ7 + σ8). 137
The fifth ring relation is
∏n (−m˜ i + σ2 − σ5)(−m˜ i − σ1 + σ3 − σ5)(−m˜ i + σ1 − σ2 + σ3 − σ5) i=1
· (−m˜ i + σ4 − σ5)(−m˜ i − σ1 + σ4 − σ5)(−m˜ i + σ1 − σ2 + σ4 − σ5) 2 · (−m˜ i + σ2 − σ3 + σ4 − σ5)(−m˜ i − σ3 + 2σ4 − σ5)(−m˜ i + σ4 − 2σ5 + σ6)
· (−m˜ i − σ5 + σ6)(−m˜ i + σ1 − σ5 + σ6)(−m˜ i − σ1 + σ2 − σ5 + σ6)
· (−m˜ i − σ2 + σ3 − σ5 + σ6)(−m˜ i + σ3 − σ4 − σ5 + σ6)
· (−m˜ i − σ2 + σ4 − σ5 + σ6)(−m˜ i + σ1 − σ3 + σ4 − σ5 + σ6)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)(−m˜ i + σ4 − σ5 − σ7)
· (−m˜ i + σ1 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ2 − σ5 + σ6 − σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i − σ5 + 2σ6 − σ7)(−m˜ i + σ1 − σ5 + σ7)(−m˜ i − σ1 + σ2 − σ5 + σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ7)(−m˜ i − σ1 + σ4 − σ5 + σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ7)
· (−m˜ i + σ4 − σ5 − σ6 + σ7)(−m˜ i − σ5 + σ6 + σ7)(−m˜ i + σ3 − σ5 − σ8)
· (−m˜ i + σ1 + σ4 − σ5 − σ8)(−m˜ i − σ1 + σ2 + σ4 − σ5 − σ8)
· (−m˜ i − σ2 + σ3 + σ4 − σ5 − σ8)(−m˜ i + σ2 − σ5 + σ6 − σ8)
· (−m˜ i − σ1 + σ3 − σ5 + σ6 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)
· (−m˜ i + σ4 − σ5 + σ6 − σ8)(−m˜ i + σ3 − σ5 + σ6 − σ7 − σ8)
· (−m˜ i + σ4 − σ5 + σ6 − σ7 − σ8)(−m˜ i + σ3 − σ5 + σ7 − σ8)
· (−m˜ i + σ4 − σ5 + σ7 − σ8)(−m˜ i − σ5 + σ8)(−m˜ i − σ2 + σ4 − σ5 + σ8)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i − σ1 − σ5 + σ6 + σ8)(−m˜ i + σ1 − σ2 − σ5 + σ6 + σ8)
· (−m˜ i + σ2 − σ3 − σ5 + σ6 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 + σ8)
· (−m˜ i − σ5 + σ6 − σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8)
· (−m˜ i − σ5 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ7 + σ8) 138
∏n = (−m˜ i − σ2 + σ5)(−m˜ i + σ1 − σ3 + σ5)(−m˜ i − σ1 + σ2 − σ3 + σ5) i=1
· (−m˜ i + σ3 − 2σ4 + σ5)(−m˜ i − σ4 + σ5)(−m˜ i + σ1 − σ4 + σ5)
· (−m˜ i − σ1 + σ2 − σ4 + σ5)(−m˜ i − σ2 + σ3 − σ4 + σ5)(−m˜ i + σ5 − σ6)
· (−m˜ i − σ1 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ5 − σ6)(−m˜ i + σ2 − σ3 + σ5 − σ6)
· (−m˜ i + σ2 − σ4 + σ5 − σ6)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ6)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)(−m˜ i − σ3 + σ4 + σ5 − σ6) 2 · (m ˜ i + σ4 − 2σ5 + σ6) (−m˜ i − σ1 + σ5 − σ7)(−m˜ i + σ1 − σ2 + σ5 − σ7)
· (−m˜ i + σ2 − σ3 + σ5 − σ7)(−m˜ i + σ1 − σ4 + σ5 − σ7)
· (−m˜ i − σ1 + σ2 − σ4 + σ5 − σ7)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ7)
· (−m˜ i + σ5 − σ6 − σ7)(−m˜ i − σ4 + σ5 + σ6 − σ7)(−m˜ i − σ4 + σ5 + σ7)
· (−m˜ i + σ5 − 2σ6 + σ7)(−m˜ i − σ1 + σ5 − σ6 + σ7)(−m˜ i + σ1 − σ2 + σ5 − σ6 + σ7)
· (−m˜ i + σ2 − σ3 + σ5 − σ6 + σ7)(−m˜ i + σ1 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i + σ5 − σ8)(−m˜ i + σ2 − σ4 + σ5 − σ8)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)(−m˜ i + σ1 + σ5 − σ6 − σ8)
· (−m˜ i − σ1 + σ2 + σ5 − σ6 − σ8)(−m˜ i − σ2 + σ3 + σ5 − σ6 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 − σ8)(−m˜ i + σ5 − σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ7 − σ8)(−m˜ i + σ5 − σ6 + σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)(−m˜ i − σ3 + σ5 + σ8)
· (−m˜ i − σ1 − σ4 + σ5 + σ8)(−m˜ i + σ1 − σ2 − σ4 + σ5 + σ8)
· (−m˜ i + σ2 − σ3 − σ4 + σ5 + σ8)(−m˜ i − σ2 + σ5 − σ6 + σ8)
· (−m˜ i + σ1 − σ3 + σ5 − σ6 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)
· (−m˜ i − σ4 + σ5 − σ6 + σ8)(−m˜ i − σ3 + σ5 − σ7 + σ8)(−m˜ i − σ4 + σ5 − σ7 + σ8)
· (−m˜ i − σ3 + σ5 − σ6 + σ7 + σ8)(−m˜ i − σ4 + σ5 − σ6 + σ7 + σ8). 139
The sixth ring relation is
∏n (−m˜ i + σ1 − σ6)(−m˜ i − σ1 + σ2 − σ6)(−m˜ i − σ2 + σ3 − σ6) i=1
· (−m˜ i − σ1 + σ4 − σ6)(−m˜ i + σ1 − σ2 + σ4 − σ6)(−m˜ i + σ2 − σ3 + σ4 − σ6)
· (−m˜ i + σ5 − σ6)(−m˜ i − σ1 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ5 − σ6)
· (−m˜ i + σ2 − σ3 + σ5 − σ6)(−m˜ i + σ2 − σ4 + σ5 − σ6)
· (−m˜ i − σ1 + σ3 − σ4 + σ5 − σ6)(−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ6)
· (−m˜ i − σ3 + σ4 + σ5 − σ6)(−m˜ i − σ4 + 2σ5 − σ6)(−m˜ i + σ5 − σ6 − σ7) 2 · (−m˜ i + σ5 − 2σ6 + σ7) (−m˜ i − σ6 + σ7)(−m˜ i + σ1 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ6 + σ7)(−m˜ i − σ2 + σ3 − σ6 + σ7)
· (−m˜ i + σ3 − σ4 − σ6 + σ7)(−m˜ i − σ2 + σ4 − σ6 + σ7)
· (−m˜ i + σ1 − σ3 + σ4 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)
· (−m˜ i + σ4 − σ5 − σ6 + σ7)(−m˜ i − σ1 + σ5 − σ6 + σ7)
· (−m˜ i + σ1 − σ2 + σ5 − σ6 + σ7)(−m˜ i + σ2 − σ3 + σ5 − σ6 + σ7)
· (−m˜ i + σ1 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ6 + 2σ7)(−m˜ i + σ3 − σ6 − σ8)
· (−m˜ i + σ4 − σ6 − σ8)(−m˜ i + σ1 + σ5 − σ6 − σ8)
· (−m˜ i − σ1 + σ2 + σ5 − σ6 − σ8)(−m˜ i − σ2 + σ3 + σ5 − σ6 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 − σ8)(−m˜ i + σ2 − σ6 + σ7 − σ8)
· (−m˜ i − σ1 + σ3 − σ6 + σ7 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)
· (−m˜ i + σ4 − σ6 + σ7 − σ8)(−m˜ i + σ5 − σ6 + σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)(−m˜ i − σ6 + σ8)
· (−m˜ i − σ3 + σ4 − σ6 + σ8)(−m˜ i − σ2 + σ5 − σ6 + σ8)
· (−m˜ i + σ1 − σ3 + σ5 − σ6 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)
· (−m˜ i − σ4 + σ5 − σ6 + σ8)(−m˜ i − σ1 − σ6 + σ7 + σ8)
· (−m˜ i + σ1 − σ2 − σ6 + σ7 + σ8)(−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ6 + σ7 + σ8)(−m˜ i − σ3 + σ5 − σ6 + σ7 + σ8)
· (−m˜ i − σ4 + σ5 − σ6 + σ7 + σ8) 140
∏n = (−m˜ i − σ1 + σ6)(−m˜ i + σ1 − σ2 + σ6)(−m˜ i + σ2 − σ3 + σ6) i=1
· (−m˜ i + σ1 − σ4 + σ6)(−m˜ i − σ1 + σ2 − σ4 + σ6)(−m˜ i − σ2 + σ3 − σ4 + σ6)
· (−m˜ i + σ4 − 2σ5 + σ6)(−m˜ i − σ5 + σ6)(−m˜ i + σ1 − σ5 + σ6)
· (−m˜ i − σ1 + σ2 − σ5 + σ6)(−m˜ i − σ2 + σ3 − σ5 + σ6)
· (−m˜ i + σ3 − σ4 − σ5 + σ6)(−m˜ i − σ2 + σ4 − σ5 + σ6)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ6)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ6)
· (−m˜ i + σ6 − 2σ7)(−m˜ i + σ6 − σ7)(−m˜ i − σ1 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ6 − σ7)
· (−m˜ i + σ2 − σ4 + σ6 − σ7)(−m˜ i − σ1 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)(−m˜ i − σ3 + σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ2 − σ5 + σ6 − σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7) 2 · (−m˜ i − σ4 + σ5 + σ6 − σ7)(m ˜ i + σ5 − 2σ6 + σ7) (−m˜ i − σ5 + σ6 + σ7)
· (−m˜ i + σ6 − σ8)(−m˜ i + σ3 − σ4 + σ6 − σ8)(−m˜ i + σ2 − σ5 + σ6 − σ8)
· (−m˜ i − σ1 + σ3 − σ5 + σ6 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)
· (−m˜ i + σ4 − σ5 + σ6 − σ8)(−m˜ i + σ1 + σ6 − σ7 − σ8)
· (−m˜ i − σ1 + σ2 + σ6 − σ7 − σ8)(−m˜ i − σ2 + σ3 + σ6 − σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ6 − σ7 − σ8)(−m˜ i + σ3 − σ5 + σ6 − σ7 − σ8)
· (−m˜ i + σ4 − σ5 + σ6 − σ7 − σ8)(−m˜ i − σ3 + σ6 + σ8)(−m˜ i − σ4 + σ6 + σ8)
· (−m˜ i − σ1 − σ5 + σ6 + σ8)(−m˜ i + σ1 − σ2 − σ5 + σ6 + σ8)
· (−m˜ i + σ2 − σ3 − σ5 + σ6 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 + σ8)
· (−m˜ i − σ2 + σ6 − σ7 + σ8)(−m˜ i + σ1 − σ3 + σ6 − σ7 + σ8)
· (−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)(−m˜ i − σ4 + σ6 − σ7 + σ8)
· (−m˜ i − σ5 + σ6 − σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8). 141
The seventh ring relation is
∏n (−m˜ i − σ7)(−m˜ i + σ1 − σ7)(−m˜ i − σ1 + σ2 − σ7)(−m˜ i − σ2 + σ3 − σ7) i=1
· (−m˜ i + σ3 − σ4 − σ7)(−m˜ i − σ2 + σ4 − σ7)(−m˜ i + σ1 − σ3 + σ4 − σ7)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ7)(−m˜ i + σ4 − σ5 − σ7)(−m˜ i − σ1 + σ5 − σ7)
· (−m˜ i + σ1 − σ2 + σ5 − σ7)(−m˜ i + σ2 − σ3 + σ5 − σ7)
· (−m˜ i + σ1 − σ4 + σ5 − σ7)(−m˜ i − σ1 + σ2 − σ4 + σ5 − σ7)
· (−m˜ i − σ2 + σ3 − σ4 + σ5 − σ7)(−m˜ i + σ5 − σ6 − σ7)(−m˜ i + σ6 − σ7)
· (−m˜ i − σ1 + σ6 − σ7)(−m˜ i + σ1 − σ2 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ6 − σ7)
· (−m˜ i + σ2 − σ4 + σ6 − σ7)(−m˜ i − σ1 + σ3 − σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ6 − σ7)(−m˜ i − σ3 + σ4 + σ6 − σ7)
· (−m˜ i + σ1 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ2 − σ5 + σ6 − σ7)
· (−m˜ i − σ2 + σ3 − σ5 + σ6 − σ7)(−m˜ i − σ1 + σ4 − σ5 + σ6 − σ7)
· (−m˜ i + σ1 − σ2 + σ4 − σ5 + σ6 − σ7)(−m˜ i + σ2 − σ3 + σ4 − σ5 + σ6 − σ7) 2 · (−m˜ i − σ4 + σ5 + σ6 − σ7)(−m˜ i − σ5 + 2σ6 − σ7)(m ˜ i − σ6 + 2σ7)
· (−m˜ i + σ2 − σ7 − σ8)(−m˜ i − σ1 + σ3 − σ7 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ7 − σ8)
· (−m˜ i + σ4 − σ7 − σ8)(−m˜ i + σ5 − σ7 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ7 − σ8)
· (−m˜ i + σ1 + σ6 − σ7 − σ8)(−m˜ i − σ1 + σ2 + σ6 − σ7 − σ8)
· (−m˜ i − σ2 + σ3 + σ6 − σ7 − σ8)(−m˜ i + σ3 − σ4 + σ6 − σ7 − σ8)
· (−m˜ i + σ3 − σ5 + σ6 − σ7 − σ8)(−m˜ i + σ4 − σ5 + σ6 − σ7 − σ8)
· (−m˜ i − σ1 − σ7 + σ8)(−m˜ i + σ1 − σ2 − σ7 + σ8)(−m˜ i + σ2 − σ3 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ7 + σ8)(−m˜ i − σ3 + σ5 − σ7 + σ8)
· (−m˜ i − σ4 + σ5 − σ7 + σ8)(−m˜ i − σ2 + σ6 − σ7 + σ8)
· (−m˜ i + σ1 − σ3 + σ6 − σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)
· (−m˜ i − σ4 + σ6 − σ7 + σ8)(−m˜ i − σ5 + σ6 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8) 142
∏n 2 = (m ˜ i + σ6 − 2σ7) (σ7 − m˜ i)(−m˜ i − σ1 + σ7)(−m˜ i + σ1 − σ2 + σ7) i=1
· (−m˜ i + σ2 − σ3 + σ7)(−m˜ i + σ2 − σ4 + σ7)(−m˜ i − σ1 + σ3 − σ4 + σ7)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ7)(−m˜ i − σ3 + σ4 + σ7)(−m˜ i + σ1 − σ5 + σ7)
· (−m˜ i − σ1 + σ2 − σ5 + σ7)(−m˜ i − σ2 + σ3 − σ5 + σ7)
· (−m˜ i − σ1 + σ4 − σ5 + σ7)(−m˜ i + σ1 − σ2 + σ4 − σ5 + σ7)
· (−m˜ i + σ2 − σ3 + σ4 − σ5 + σ7)(−m˜ i − σ4 + σ5 + σ7)(−m˜ i + σ5 − 2σ6 + σ7)
· (−m˜ i − σ6 + σ7)(−m˜ i + σ1 − σ6 + σ7)(−m˜ i − σ1 + σ2 − σ6 + σ7)
· (−m˜ i − σ2 + σ3 − σ6 + σ7)(−m˜ i + σ3 − σ4 − σ6 + σ7)
· (−m˜ i − σ2 + σ4 − σ6 + σ7)(−m˜ i + σ1 − σ3 + σ4 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ3 + σ4 − σ6 + σ7)(−m˜ i + σ4 − σ5 − σ6 + σ7)
· (−m˜ i − σ1 + σ5 − σ6 + σ7)(−m˜ i + σ1 − σ2 + σ5 − σ6 + σ7)
· (−m˜ i + σ2 − σ3 + σ5 − σ6 + σ7)(−m˜ i + σ1 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ1 + σ2 − σ4 + σ5 − σ6 + σ7)(−m˜ i − σ2 + σ3 − σ4 + σ5 − σ6 + σ7)
· (−m˜ i − σ5 + σ6 + σ7)(−m˜ i + σ1 + σ7 − σ8)(−m˜ i − σ1 + σ2 + σ7 − σ8)
· (−m˜ i − σ2 + σ3 + σ7 − σ8)(−m˜ i + σ3 − σ4 + σ7 − σ8)
· (−m˜ i + σ3 − σ5 + σ7 − σ8)(−m˜ i + σ4 − σ5 + σ7 − σ8)
· (−m˜ i + σ2 − σ6 + σ7 − σ8)(−m˜ i − σ1 + σ3 − σ6 + σ7 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)(−m˜ i + σ4 − σ6 + σ7 − σ8)
· (−m˜ i + σ5 − σ6 + σ7 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)
· (−m˜ i − σ2 + σ7 + σ8)(−m˜ i + σ1 − σ3 + σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ7 + σ8)
· (−m˜ i − σ4 + σ7 + σ8)(−m˜ i − σ5 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ7 + σ8)
· (−m˜ i − σ1 − σ6 + σ7 + σ8)(−m˜ i + σ1 − σ2 − σ6 + σ7 + σ8)
· (−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ6 + σ7 + σ8)
· (−m˜ i − σ3 + σ5 − σ6 + σ7 + σ8)(−m˜ i − σ4 + σ5 − σ6 + σ7 + σ8). 143
The eighth ring relation is
∏n (−m˜ i + σ1 − σ8)(−m˜ i + σ2 − σ8)(−m˜ i − σ1 + σ2 − σ8) i=1
· (−m˜ i − σ1 + σ3 − σ8)(−m˜ i − σ2 + σ3 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ8)
· (−m˜ i + σ1 + σ3 − σ4 − σ8)(−m˜ i − σ1 + σ2 + σ3 − σ4 − σ8)
· (−m˜ i − σ2 + 2σ3 − σ4 − σ8)(−m˜ i − σ1 + σ4 − σ8)(−m˜ i + σ1 − σ2 + σ4 − σ8)
· (−m˜ i + σ2 − σ3 + σ4 − σ8)(−m˜ i + σ3 − σ5 − σ8)(−m˜ i + σ1 + σ4 − σ5 − σ8)
· (−m˜ i − σ1 + σ2 + σ4 − σ5 − σ8)(−m˜ i − σ2 + σ3 + σ4 − σ5 − σ8)
· (−m˜ i + σ5 − σ8)(−m˜ i + σ2 − σ4 + σ5 − σ8)(−m˜ i − σ1 + σ3 − σ4 + σ5 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ4 + σ5 − σ8)(−m˜ i + σ3 − σ6 − σ8)
· (−m˜ i + σ4 − σ6 − σ8)(−m˜ i + σ1 + σ5 − σ6 − σ8)
· (−m˜ i − σ1 + σ2 + σ5 − σ6 − σ8)(−m˜ i − σ2 + σ3 + σ5 − σ6 − σ8)
· (−m˜ i + σ3 − σ4 + σ5 − σ6 − σ8)(−m˜ i + σ6 − σ8)(−m˜ i + σ3 − σ4 + σ6 − σ8)
· (−m˜ i + σ2 − σ5 + σ6 − σ8)(−m˜ i − σ1 + σ3 − σ5 + σ6 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ5 + σ6 − σ8)(−m˜ i + σ4 − σ5 + σ6 − σ8)
· (−m˜ i + σ2 − σ7 − σ8)(−m˜ i − σ1 + σ3 − σ7 − σ8)(−m˜ i + σ1 − σ2 + σ3 − σ7 − σ8)
· (−m˜ i + σ4 − σ7 − σ8)(−m˜ i + σ5 − σ7 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ7 − σ8)
· (−m˜ i + σ1 + σ6 − σ7 − σ8)(−m˜ i − σ1 + σ2 + σ6 − σ7 − σ8)
· (−m˜ i − σ2 + σ3 + σ6 − σ7 − σ8)(−m˜ i + σ3 − σ4 + σ6 − σ7 − σ8)
· (−m˜ i + σ3 − σ5 + σ6 − σ7 − σ8)(−m˜ i + σ4 − σ5 + σ6 − σ7 − σ8)
· (−m˜ i + σ1 + σ7 − σ8)(−m˜ i − σ1 + σ2 + σ7 − σ8)(−m˜ i − σ2 + σ3 + σ7 − σ8)
· (−m˜ i + σ3 − σ4 + σ7 − σ8)(−m˜ i + σ3 − σ5 + σ7 − σ8)(−m˜ i + σ4 − σ5 + σ7 − σ8)
· (−m˜ i + σ2 − σ6 + σ7 − σ8)(−m˜ i − σ1 + σ3 − σ6 + σ7 − σ8)
· (−m˜ i + σ1 − σ2 + σ3 − σ6 + σ7 − σ8)(−m˜ i + σ4 − σ6 + σ7 − σ8) 2 · (−m˜ i + σ5 − σ6 + σ7 − σ8)(−m˜ i + σ3 − σ4 + σ5 − σ6 + σ7 − σ8)(m ˜ i − σ3 + 2σ8) 144
∏n 2 = (m ˜ i + σ3 − 2σ8) (−m˜ i − σ1 + σ8)(−m˜ i − σ2 + σ8)(−m˜ i + σ1 − σ2 + σ8) i=1
· (−m˜ i + σ1 − σ3 + σ8)(−m˜ i + σ2 − σ3 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ8)
· (−m˜ i + σ1 − σ4 + σ8)(−m˜ i − σ1 + σ2 − σ4 + σ8)(−m˜ i − σ2 + σ3 − σ4 + σ8)
· (−m˜ i + σ2 − 2σ3 + σ4 + σ8)(−m˜ i − σ1 − σ3 + σ4 + σ8)
· (−m˜ i + σ1 − σ2 − σ3 + σ4 + σ8)(−m˜ i − σ5 + σ8)(−m˜ i − σ2 + σ4 − σ5 + σ8)
· (−m˜ i + σ1 − σ3 + σ4 − σ5 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ4 − σ5 + σ8)
· (−m˜ i − σ3 + σ5 + σ8)(−m˜ i − σ1 − σ4 + σ5 + σ8)
· (−m˜ i + σ1 − σ2 − σ4 + σ5 + σ8)(−m˜ i + σ2 − σ3 − σ4 + σ5 + σ8)
· (−m˜ i − σ6 + σ8)(−m˜ i − σ3 + σ4 − σ6 + σ8)(−m˜ i − σ2 + σ5 − σ6 + σ8)
· (−m˜ i + σ1 − σ3 + σ5 − σ6 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ5 − σ6 + σ8)
· (−m˜ i − σ4 + σ5 − σ6 + σ8)(−m˜ i − σ3 + σ6 + σ8)(−m˜ i − σ4 + σ6 + σ8)
· (−m˜ i − σ1 − σ5 + σ6 + σ8)(−m˜ i + σ1 − σ2 − σ5 + σ6 + σ8)
· (−m˜ i + σ2 − σ3 − σ5 + σ6 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ6 + σ8)
· (−m˜ i − σ1 − σ7 + σ8)(−m˜ i + σ1 − σ2 − σ7 + σ8)(−m˜ i + σ2 − σ3 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ7 + σ8)(−m˜ i − σ3 + σ5 − σ7 + σ8)
· (−m˜ i − σ4 + σ5 − σ7 + σ8)(−m˜ i − σ2 + σ6 − σ7 + σ8)
· (−m˜ i + σ1 − σ3 + σ6 − σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ6 − σ7 + σ8)
· (−m˜ i − σ4 + σ6 − σ7 + σ8)(−m˜ i − σ5 + σ6 − σ7 + σ8)
· (−m˜ i − σ3 + σ4 − σ5 + σ6 − σ7 + σ8)(−m˜ i − σ2 + σ7 + σ8)
· (−m˜ i + σ1 − σ3 + σ7 + σ8)(−m˜ i − σ1 + σ2 − σ3 + σ7 + σ8)
· (−m˜ i − σ4 + σ7 + σ8)(−m˜ i − σ5 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ5 + σ7 + σ8)
· (−m˜ i − σ1 − σ6 + σ7 + σ8)(−m˜ i + σ1 − σ2 − σ6 + σ7 + σ8)
· (−m˜ i + σ2 − σ3 − σ6 + σ7 + σ8)(−m˜ i − σ3 + σ4 − σ6 + σ7 + σ8)
· (−m˜ i − σ3 + σ5 − σ6 + σ7 + σ8)(−m˜ i − σ4 + σ5 − σ6 + σ7 + σ8).
3.3.4 Pure gauge theory
In this part we will consider the mirror to the pure E8 gauge theory. For brevity, we will not rewrite the superpotential here, explicitly omitting Y fields, but instead merely refer to the 145
expression (3.36) given earlier, leaving the reader to omit Y fields.
Now, let us consider the critical locus of the superpotential above. For each root µ, the
fields Xµ and X−µ appear paired with opoosite signs coupling to each σ. Therefore, one impliciation of the derivatives ∂W = 0 ∂Xµ is that, on the critical locus,
Xµ = −X−µ. (3.37)
(Furthermore, on the critical locus, each Xµ is determined by σs.) Next, each derivative
∂W
∂σa
is a product of ratios of the form X µ = −1. X−µ
It is straightforward to check that in the superpotential above that each σa is multiplied by an even number of such ratios (i.e. the number of Z’s is a multiple of four). Specifically, for each σ, the sum of the absolute values of the coefficients of the Z’s multiplying it is 116 = 4 · 29. Thus, the constraint implied by the σ’s is automatically satisfied.
As a result, following the same analysis as in [110] and previous sections, we see that the critical locus is nonempty, and in fact is determined by eight σs. In other words, at the level of these topological field theory computations, we have evidence that the pure supersymmetric
E8 gauge theory in two dimensions flows in the IR to a theory of eight free twisted chiral superfields. Chapter 4
Landau-Ginzburg models for certain fiber products with curves
The contents of this chapter were adapted with minor modifications, with permission from the Journal of Geometry and Physics, from our publication [3].
4.1 Introduction
Over the years, there has been much work on non-K¨ahlersolutions of heterotic compacti- fications with H flux, known as the Strominger system [120]. The work in this chapter is inspired by the recent work [121] in which a new family of compact non-K¨ahleranalogues of Calabi-Yau threefolds, a new set of potential solutions to the Strominger system, was constructed. In this chapter, we do not construct physical theories for non-K¨ahlertargets, but instead apply recent tricks in GLSMs to build physical theories for K¨ahleranalogues of the fiber products discussed in [121].
Specifically, suppose M is a compact hyperK¨ahlermanifold with real dimension four and Σ is a compact Riemann surface of genus g ≥ 3. Then, a manifold X with holomorphically-trivial
146 147 canonical bundle can be constructed as the pullback
∗ 1 X = φ Z = Z ×P1 Σ Z = M × P
π
φ Σ P1 where Z is the twistor space of M together with the natural holomorphic projection π and φ is a nonconstant holomorphic map. It was argued in [121] that the threefold X has trivial canonical bundle as long as ∗ ∼ φ O(2) = KΣ.
In this paper, the curve Σ will be constructed as a branched double cover of P1, for which case the condition above for the fiber product to have trivial canonical bundle reduces to simply g = 3, independent of the details of M.
Motivated by the mathematical construction above, we will give a physical realization of threefolds of trivial canonical bundle constructed as fiber products of genus g curves with noncompact K¨ahlerthreefolds, including as special cases certain noncompact K¨ahler1 twistor spaces. We will take the Riemann surface of genus g and the holomorphic map φ to be a branched double cover over P1, realized physically via nonperturbative tricks as in [24]. Because we describe fiber products with K¨ahlerthreefolds, including K¨ahlertwistor spaces of certain hyperK¨ahlerfour-manifolds, the Calabi-Yau threefolds we realize are non-compact and K¨ahler,as opposed to non-K¨ahlerspaces of trivial canonical bundle which were the focus of [121].
We will construct higher-energy theories that realize these geometries as (2,2) supersym- metric hybrid Landau-Ginzburg models. These hybrid models do not seem to have a UV description as GLSMs, though some GLSMs do come close, as we shall explain later.
We begin in section 4.2 with a review of GLSMs for genus g curves, constructed via non- perturbative methods as branched double covers. In section 4.3 we construct (2,2) super- symmetric hybrid Landau-Ginzburg models for the fiber products above, of curves with a few noncompact K¨ahlerthreefolds. In section 4.4 we specialize to fiber products of curves and twistor spaces, which arise as special cases. In section 4.5 we review some pertinent mathematics. 1Most twistor spaces are not K¨ahler. 148
Although we are not able to give physical realizations of any non-K¨ahlergeometries in this paper, it is our hope that the ideas we present here will later be extended to non-K¨ahler fiber product constructions.
Finally, before starting, we should add a caution. We discuss non-compact K¨ahlermanifolds with trivial canonical bundle. However, Yau’s theorem does not apply to non-compact cases, so it is possible2 that some might not have Ricci-flat metrics. (Nevertheless, we will sometimes call these noncompact K¨ahlerspaces of trivial canonical bundle, “Calabi-Yau’s,” though this terminology is inaccurate.) This is an issue for both the spaces themselves as well as for Landau-Ginzburg models on such spaces that do not have known UV completions as GLSMs. In the case of Landau-Ginzburg models, if the metric is not Ricci-flat, not even asymptotically, then, RG flow would be more complicated, and our analysis likely too naive. Our proposed hybrid Landau-Ginzburg models are constructed on the assumption that they have Ricci-flat metrics, at least asymptotically, so that the renormalization group flow works as expected.
It is not entirely out of the realm of possibility that complications in RG flow, alluded to above, might actually generate nonzero H flux in a low-energy theory, especially in (0,2) supersymmetric versions of this construction where one has less control over RG flow. We will leave this possibility to future work.
4.2 GLSM for P2g+1[2, 2] and curves of genus g
One essential piece of our construction will be a trick from [24], in which GLSMs describe geometries nonperturbatively, rather than as the critical locus of a superpotential. As it plays a critical role in this paper, we review the highlights in this section.
Section 4.1 of [24] discusses a gauged linear sigma model for P2g+1[2, 2] (with g ≥ 1) which
2One set of examples is discussed in [122]. A second set of examples arises by removing an anticanonical divisor with deep singularities from a projective manifold, then the complement will not have a complete Ricci-flat metric, though it may still have a non-complete Ricci-flat metric. A third set of examples arises from the cotangent bundle of a compact K¨ahlermanifold X. On a tubular neighborhood of the zero section, one can find a unique Ricci-flat metric which restricts to the given K¨ahlermetric on the zero section and is circle invariant. This metric extends to a Ricci-flat metric on the whole cotangent bundle if and only if X is a homogeneous Fano or semi-abelian variety. For example, if X is a Riemann surface of genus at least two, there is a Ricci-flat metric on a disk bundle of fixed radius but it cannot be extended beyond that bundle to the entire cotangent bundle. 149
realizes a genus g Riemann surface, via nonperturbative tricks, in its r ≪ 0 phase. That model will play an essential role in this paper, so we shall quickly review it here.
The GLSM in question is a U(1) gauge theory with (2,2) supersymmetry and 2g + 2 chiral
superfields ϕi of charge 1 and two chiral superfields p1, p2 of charge −2, with superpotential ∑ ij W = p1Q1(ϕ) + p2Q2(ϕ) = ϕiA (p)ϕj, (4.1) ij
ij where Qi are quadratic functions of ϕ’s, and A (p) is a (2g + 2) × (2g + 2) symmetric matrix
whose entries are linear in the pa.
For r ≫ 0 (geometric phase), one can do the usual analysis of the critical loci to argue that the GLSM flows to a sigma model on a complete intersection of two quadrics in P2g+1.
The r ≪ 0 phase is more interesting. D terms imply that p1 and p2 can not simultaneously vanish, and the superpotential generically gives a mass to the ϕ’s. On that open set where all the ϕ’s are massive, since the p’s have nonminimal charges, physics sees a double cover of the P1 mapped out by p’s [24, 25, 112]. On the locus where any ϕ becomes massless, specifically the degree 2g + 2 locus {det A = 0} where the mass matrix develops at least one zero eigenvalue, the double cover collapses to a single cover.
Put simply, {det Aij = 0} defines the branch locus on the double cover of P1. (Monodromies about the branch locus correspond to Berry phases and are described in [24].) The resulting geometry, a double cover of P1 branched over a degree 2g + 2 loci, is a compact Riemann surface of genus g.
4.3 Hybrid Landau-Ginzburg models for fiber prod- ucts
4.3.1 Fiber products with vector bundles on P1
In this section we will consider a general set of fiber products, between curves and vector bundles on P1. Specifically, let V be the total space of the rank-two vector bundle O(a) + O(b) → P1. 150
Mathematically, we are considering the fiber product3
∗ X = φ V = V ×P1 Σ V π
φ Σ P1
The fiber product X will have trivial canonical bundle if
∗ KΣ = φ (det V ). (4.2)
We will consider the special case of genus g curves Σ constructed as branched double covers, for which the condition above reduces to
a + b = g − 1. (4.3)
Now, in general, we will want to describes cases in which a or b are positive, and the total space of such V is challenging to describe with a GLSM. Recall that the total space of O(−1) ⊕ O(−1) → P1 can be described by a GLSM with a single U(1) gauge field and four chiral superfields:
• two chiral superfields pa of charge +1 corresponding to homogeneous coordinates on the base P1,
• two chiral superfields ya of charge −1 corresponding to the two line bundles O(−1).
Naively, one could try a similar GLSM with the charges of the chiral superfields ya flipped to +1 describing two O(+1) line bundles. However, D terms in the resulting GLSM make it clear that that GLSM will describe the space P3. To describe the total space of the bundle above, one would need to remove a different exceptional locus than the one canonically dictated by the D terms for a quotient of flat space.
Setting aside the issue above, the fiber product would formally be described by the GLSM with gauge grop U(1) and matter
• 2g + 2 chiral superfields ϕi of charge −1, 3As an aside, if π :Σ → P1 is the projection from the genus g curve to P1, then the fiber product X is ∗ the total space of La ⊕ LB → Σ, where Ln = π O(n). 151
• 2 chiral superfields pa of charge +2,
• 2 chiral superfields ya, yb of charges 2a, 2b,
and superpotential ∑ ij W = ϕiϕjA (p), ij where Aij is a symmetric (2g + 2) × (2g + 2) matrix with entries linear in the p’s.
We have taken the matrix Aij to be independent of the y’s, to preserve translation invariance along the fibers as well as a global SU(2) rotation symmetry between the y’s. In models discussed later we will make similar restrictions so as to reproduce the desired geometries.
In passing, if we were to add terms to the superpotential to realize the most general case compatible with gauge invariance, i.e. adding terms involving ya, yb, and sufficient ϕ factors, we would get the GLSM for P2g+1[2, 2, 2a, 2b] (with an overall sign flip on the charges, inverting the r ≫ 0 and r ≪ 0 phases). (This includes the GLSM for P7[2, 2, 2, 2], studied in [24] because of the geometric realization of its r ≪ 0 phase.) Note that the Calabi-Yau condition for that complete intersection also reduces to (4.3).
1 Assuming that p1 and p2 are homogeneous coordinates on P , one can easily see, modulo the issue with D terms, that the mass matrix in the F term imply the fiber product geometry
in the phase r ≫ 0. First, following the same argument in section (4.2), the fields ϕi and 1 pa describe the genus g Riemann surface as the branched double cover of P . On the other
hand, the fields ya and yb correspond to the fiber coordinates on O(a) ⊕ O(b) since we 1 assumed that p1 and p2 are homogeneous coordinates on P . The fiber product structure is achieved by identifying two different P1 in the holomorphic map and the total space of V . The identification is manifest in our theory.
Finally, let us consider the Calabi-Yau condition. The sum of the charges in this theory is precisely −2g + 2 + 2a + 2b,
and so vanishes precisely when the mathematical condition (4.3) for trivial canonical bundle is satisfied.
As mentioned above, the putative GLSM above does not quite work, because the D terms will not describe the correct exceptional locus in general. To evade the issue of positive- 152
degree line bundles on P1 in GLSMs, we construct a hybrid Landau-Ginzburg model, an ungauged sigma model on the total space of
O(−1/2)2g+2 ⊕ O(a) ⊕ O(b) → P1, (4.4)
with superpotential ∑ ij W = ϕiϕjA (p), (4.5) ij where the mass matrix Aij(p) should now be interpreted as a generic symmetric (2g + 2) × (2g + 2) matrix of sections of O(+1) → P1.
1 1 Notice that the P in the target space (4.4) is actually a Z2 gerbe on P indicated by the line bundle denoted O(−1/2). The bundle O(−1/2) is a special kind of line bundle that only exist for gerbes. On the other hand, it is also a fiber bundle of P1 whose fibers are
the orbifolds [C/Z2]. More details of such line bundles on gerbes over projective spaces are discussed in appendix B of [123].
1 Generically on the P , the ϕi are massive, away from the locus {det A = 0}, and so the Z2 gerbe implies a branched double cover, as usual, and hence the fiber product of the genus g Riemann surface and the vector bundle V over P1.
The condition for the canonical class to be trivial in this hybrid model is that the first Chern class of the vector bundle (4.4) match the first Chern class of the canonical bundle, meaning specifically that (2g + 2)(−1/2) + a + b = −2,
which again reduces to the mathematical condition (4.3).
There is a possible technical issue with this construction, due to the fact that there is no non-compact version of Yau’s theorem, as described in the introduction. We describe above a hybrid Landau-Ginzburg model over a K¨ahlerspace with holomorphically trivial canonical bundle (in the case g = 3); however, that does not guarantee that a Ricci-flat metric exists in the noncompact case. If the metric is not Ricci-flat, at least asymptotically, then the RG flow may be more complicated than we have naively supposed. If the model arose from a GLSM, we could appeal to RG flow from the GLSM, but as we have not been able to write down a UV GLSM, we cannot guarantee that an asymptotically Ricci-flat metric exists. Analogous potential issues arise in every hybrid Landau-Ginzburg model described in this 153
paper.
4.3.2 Fiber products with hypersurfaces in vector bundles
Let V denote the rank-three vector bundle
O(a) ⊕ O(b) ⊕ O(c) → P1,
and consider the hypersurface f(x, y, z) = 0, where x, y, z are coordinates along the fibers of V , and f is of degree4 d. Mathematically, for the fiber product of this hypersurface with the curve Σ of genus g to have trivial canonical bundle, we must require that the degree d match the degree of φ∗V , for φ :Σ → P1, which means, for Σ realized as a branched double cover of P1, d = a + b + c + 1 − g. (4.6)
Modulo the same issue with D terms and positive-degree line bundles discussed in the last section, we can construct a ‘fake’ GLSM for the fiber product with the hypersurface in V as a U(1) gauge theory with matter
• 2g + 2 chiral superfields ϕi of charge −1,
• 2 chiral superfields pa of charge +2,
• 3 chiral superfields x, y, z of charges 2a, 2b, 2c, respectively,
• 1 chiral superfield q of charge −2d,
and superpotential ∑ ij W = ϕiϕjA (p) + qf(x, y, z), ij where Aij is a symmetric (2g + 2) × (2g + 2) matrix with entries linear in the p’s. (As before, we do not consider more general possible terms, in order to preserve pertinent symmetries.) The sum of the charges in this theory vanishes when
g + d = a + b + c + 1, (4.7)
4In the sense of weighted projective spaces, so that the monomials x, y, z have weights a, b, c, respectively. 154 matching the mathematical condition given above for the canonical bundle of the fiber prod- uct to be trivial.
As before, there is a problem involving D terms in the putative GLSM above. To evade this issue, we can construct a hybrid Landau-Ginzburg model which describes the geometry. Specifically, this will be an ungauged sigma model on the total space of
O(−1/2)2g+2 ⊕ O(a) ⊕ O(b) ⊕ O(c)O(−d) → P1, (4.8)
1 where we interpret the bundle in terms of a Z2 gerbe on the P , and with superpotential ∑ ij W = ϕiϕjA (p) + qf(x, y, z), ij where the mass matrix Aij(p) is a generic symmetric (2g + 2) × (2g + 2) matrix of sections O(+1) → P1.
This hybrid Landau-Ginzburg model realizes the same fiber product structure as the fake
GLSM above. The superpotential contains a mass matrix for the ϕi, i = 1,..., 2g + 2, that gives them a mass away from the locus {det A = 0}. As a result, at generic points on the P1, the remaining massless fields are invariant under the gerbe Z2, which physics sees [24, 112] as a double cover of P1, branched over the locus {det A = 0}. Consequently, one obtains a fiber product of the genus g curve and the hypersurface.
The Calabi-Yau condition for the hybrid Landau-Ginzburg model is the condition that c1 of 1 the bundle (4.8) match c1 of the canonical bundle of P , which in this case implies
(2g + 2)(−1/2) + a + b + c − d = −2.
It is straightforward to check that this matches the condition (4.7) given earlier.
4.4 Fiber products with twistor spaces
In this section, we will construct (2,2) supersymmetric hybrid Landau-Ginzburg theories which should RG flow to sigma models on the non-compact K¨ahlerCalabi-Yau threefolds constructed as fiber products of genus three curves and twistor spaces, as explained in 155
the introduction. The (K¨ahler)twistor spaces we consider here are the twistor spaces5 of 4 2 1 3 R , C /Zk and S × R . These will all correspond to special cases of the constructions in section 4.3, so we will strive to be brief. In each case, since the curve is realized as a branched double cover of P1, the Calabi-Yau condition is that the curve be of genus 3 – details of the hyperK¨ahlermanifold are otherwise irrelevant. In each of our models, we will recover the genus three condition as a consistency check.
4.4.1 Fiber product with twistor space of R4
From [124], the twistor space of R4 can be described as the total space of O(+1) ⊕ O(+1) → P1. Our construction for this case is a special case of the construction in section 4.3.1.
As discussed in section 4.3.1, there is a technical question of how to realize positive-degree line bundles in GLSMs, so we instead construct a lower energy theory, a hybrid Landau- Ginzburg model. Specifically, this will be an ungauged sigma model on the total space of O(−1/2)2g+2 ⊕ O(+1) ⊕ O(+1) → P1, (4.9)
with superpotential ∑ ij W = ϕiϕjA (p), ij where the mass matrix Aij(p) should now be interpreted as a generic symmetric (2g + 2) × (2g + 2) matrix of sections of O(+1) → P1.
The superpotential contains a mass matrix for the ϕi, i = 1,..., 2g + 2, that gives them a mass away from the locus {det A = 0}. Therefore, at generic points on the P1, the remaining massless fields are all non-minimally charged. The Riemann surface of genus g is given by 1 a double cover of P branched over a degree 2g + 2 locus as before. Also, the fields y1 and 1 y2 are the coordinates on the fibers of O(+1) ⊕ O(+1) of the same P . Consequently, one obtains a fiber product of the genus g Riemann surface and twistor space of R4 over P1.
The Calabi-Yau condition for the total space of a vector bundle over P1 is that the first Chern class of the vector bundle should be the same as the first Chern class of the canonical 5Sometimes, blowdowns of the twistor spaces. 156
bundle of P1. In this case, one gets
(2g + 2)(−1/2) + 1 + 1 = −2.
It implies that the genus of the Riemann surface is three, as expected.
2 4.4.2 Fiber product with twistor space of C /Zk
Next, we will give a physical theory describing the fiber product with (the blowdown of) a 2 different twistor space, namely the the twistor space of C /Zk [124,125]. The group Γ = Zk acts on C2 as follows: 2πin/k −2πin/k (z1, z2) → (e z1, e z2).
k k Notice that the monomials x = z1 , y = z2 , z = z1z2 are invariant under the group Γ. Therefore the singular surface C2/Γ can be described by a hypersurface embedding in C3,
{xy = zk} ⊂ C3 = Spec C[x, y, z].
One can turn on a universal family of complex structure deformations which is given by adding lower order terms in z. As a result, the hypersurface defining equation becomes
∏k k k−1 xy = z + a1z + ··· + ak = (z − fi), i=1 where ai and fi are constant parameters. The twistor space is a resolution of the hypersuface
∏k 1 {xy = (z − fi(p))} ⊂ Tot(O(+k) ⊕ O(+k) ⊕ O(+2) → P ), i=1
where x, y are fiber coordinates on the bundle O(+k), z on O(+2), and fi are sections of 1 1 2 O(+2) → P . In particular, for each point of P , the fiber is a deformation of C /Zk. In the special case k = 2, the fiber space is also known as an Eguchi-Hanson space.
We can realize the hypersurface above, a blowdown of the twistor space, using the same ideas as in section 4.3.2. Specifically, we propose a hybrid Landau-Ginzburg model, an ungauged 157
sigma model whose target space is the total space of
O(−1/2)2g+2 ⊕ O(+k)2 ⊕ O(+2) ⊕ O(−2k) → P1,
2g+2 2 with fiber coordinates ϕi on O(−1/2) , x, y on O(+k) , z on O(+2) and q on O(−2k). The superpotential is
∑ ∏k ij W = ϕiϕjA (p) + q(xy − (z − fi(p))), (4.10) ij i=1
1 ij where fi are sections of O(2) → P and A (p) is symmetric (2g + 2) × (2g + 2) matrix with elements which are sections of O(1) → P1.
Going through the same analysis as in section 4.3.2, one obtains the desired fiber product geometry. Note that the Calabi-Yau condition in this case is
(1/2)(2g + 2) + k + k + 2 + (2k) = −2
hence g = 3, as expected.
4.4.3 Fiber product with twistor space of S1 × R3
The last case we will discuss here is the fiber product with (the blowdown of) the twistor space of S1 × R3 [124]. Since the analyses is similar to previous sections, we will present our proposition briefly here. The space S1 × R3 can be defined as C2/Γ where Γ is given by
(z1, z2) → (z1 + 1, z2).
Following the same process as in section 4.4.2, the twistor space is defined by a resolution of the hypersurface
2 2 2 2 2 2 {y + zx =z + (f1(p) + f2(p) ) + 2f1(p) f2(p) x} ⊂ Tot (O ⊕ O(+2) ⊕ O(+4) → P1),
where x is a fiber coordinate on the line bundle O, y on O(+2), z on O(+4) and f1, f2 are two sections of O(+2) → P1. 158
We can construct a hybrid Landau-Ginzburg model realizing this geometry as a special case of the construction in section 4.3.2. This hybrid Landau-Ginzburg model is defined on the total space of
O(−1/2)2g+2 ⊕ O(0) ⊕ O(+2) ⊕ O(+4) ⊕ O(−4) → P1,
with superpotential ∑ ij 2 2 2 2 2 2 W = ϕiϕjA (p) + q(y + zx − z − (f1(p) + f2(p) ) − 2f1(p) f2(p) x), (4.11) ij
where Aij is a symmetric (2g+2)×(2g+2) matrix with entries that are sections of O(1) → P1.
The Calabi-Yau condition
(1/2)(2g + 2) + 3(+2) + (4) = 2
implies that g = 3, as expected.
4.5 Review of pertinent mathematics
According to proposition 2.2 of [121], the fiber product of a twistor space X and a genus g curve Σ with some map φ :Σ → P1 has trivial canonical bundle if and only if
∗ ∼ φ O(2) = KΣ. (4.12)
We can see this as follows. Let π : X → P1 be the twistor space for any hyperK¨ahler ∗ surface, then the relative symplectic form is a nowhere-zero section of KX/P1 ⊗ π O(2), ∼ ∗ hence KX/P1 = π O(−2). Furthermore, by definition,
⊗ ∗ −1 KX/P1 = KX π KP1 , ∗ = KX ⊗ π O(2).
This gives ∗ KX = π O(−4). 159
Next, for the fiber product Z = X ×P1 Σ,
∗ KZ/Σ = pX KX/P1 , ∗ ∗O − ∗ O − = pX π ( 2) = πZ ( 2),
1 using πZ = π ◦ pX = φ ◦ pΣ, and where pΣ : Z → Σ, πZ : Z → P are projections. Hence,
∗ ⊗ KZ = pΣKΣ KZ/Σ, ∗ ⊗ ∗ O − = pΣKΣ πZ ( 2).
To double-check, we can also compute
∗ ⊗ KZ = pX KX KZ/X , ∗ ⊗ ∗ = pX KX pΣKΣ/P1 , ∗ ∗O − ⊗ ∗ ⊗ ∗O = pX π ( 4) pΣ (KΣ φ (2)) , ∗ O − ⊗ ∗ ⊗ ∗ O = πZ ( 4) pΣKΣ πZ (2), ∗ ⊗ ∗ O − = pΣKΣ πZ ( 2), matching the result above. In any event, using the fact that πZ = π ◦ pX = φ ◦ pΣ, we see that KZ is trivial if and only if ∗ KΣ = φ O(2).
The fact that this condition does not depend upon X follows from the fact that KX is always a pullback of O(−4).
Now, under what circumstances is (4.12) satisfied?
Let us consider the case that Σ is a spectral cover of P1, the case of relevance for this paper. Let d be the degree of the projection map φ :Σ → P1, then 2d = 2g − 2 for g the genus of Σ, ∗ hence g = d + 1. From Hurwitz, KΣ = φ O(−2) ⊗ O(R), where R ⊂ Σ is the ramification divisor of φ :Σ → P1. Thus, to satisfy (4.12), we must satisfy O(R) = φ∗O(4).
For a hyperelliptic double cover, the degree of the ramification divisor is 2g + 2 and O(R) = f ∗O(g + 1). Thus, in such a case, one must require g = 3.
More generally spectral covers have the property that their ramification divisor is a pullback. If φ :Σ → P1 is a spectral cover of degree d emebedded in the total space of O(k), then 160
O(R) = φ∗O((d − 1)k). To satisfy (4.12), one must require (d − 1)k = 4, which gives three options (for the case that Σ is a spectral cover of P1):
• d − 1 = 1, k = 4, which gives a hyperelliptic curve of genus 3 (the case that arises in this paper),
• d − 1 = 2, k = 2, which gives a 3-sheeted spectral cover of P1 of genus 4,
• d − 1 = 4, k = 1, which gives a 5-sheeted spectral cover of P1 of genus 6.
Note this condition is not satisfied for Σ a genus one curve.
If we drop the spectral curve constraint on Σ, then there are solutions in any genus ≥ 3. We can see this as follows. First, the condition (4.12) for the fiber product to have trivial canonical class can be rephrased as the statement that φ is given by a basepoint-free pencil of sections in some spin structure L on Σ. From a theorem of Harris [126], if g ≥ 3, the moduli space of pairs (Σ,L) such that Σ is smooth of genus g and L is a spin structure which has a pencil of sections has dimension 3g − 4.
For another example, there is a theorem of Farkas [127] which says that for any g ≥ 10, the moduli space of pairs (Σ,L) such that Σ is a smooth curve of genus g and L is a spin structure which gives an embedding of Σ in P3 is of dimension 3g − 9. Given such a pair and 3 compose the embedding ϕL :Σ → P with a generic projection from some point not in the image, one will get a morphism Σ → P1 which has the desired property. Thus, for g ≥ 10, there is a 3g − 9-dimensional space of pairs with the desired property.
So far we have discussed conditions for the fiber product Z to have holomorphically trivial canonical bundle. Next, let us turn to the question of when Z is K¨ahler.Since Z is a finite cover of X, Z is K¨ahlerif and only if X is K¨ahler. Chapter 5
Conclusion
Gauged linear sigma models are one of the most successful constructions utilized in string theory. In particular, gauged linear sigma models provide a way to study superstring vacua providing numerous insights into both specific theories as well as the global structure of the moduli space. In this thesis, we first reviewed certain general aspects of N = 2 superconfor- mal field theory with a short introduction to mirror symmetry. One of the open problems in understanding (0,2) mirror symmetry, an extension of ordinary mirror symmetry, concerned the construction of Toda-like Landau-Ginzburg mirrors to (0,2) theories on Fano spaces. In chapter 2, we began to fill this gap by making an ansatz for (0,2) Toda-like theories mirror to (0,2) supersymmetric nonlinear sigma models on products of certain spaces, with the most general deformations of the tangent bundle. Specifically, we proposed (0,2) mirrors to GLSMs on projective spaces, toric del Pezzo surfaces and Hirzebruch surfaces with defor- mations of the tangent bundle. We checked the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0,2) Toda-like mirrors. In chapter 3, we applied the recent proposal for mirrors to non-Abelian (2,2) supersymmetric two-dimensional gauge theories to make predictions for two-dimensional su-
persymmetric gauge theories with exceptional gauge groups G2 and E8. We computed the mirror Landau-Ginzburg models and predicted excluded Coulomb loci and Coulomb branch relations (quantum cohomology). We also discussed the behavior of pure gauge theories with exceptional gauge groups under RG flow, and described evidence that any pure super- symmetric two-dimensional gauge theory with connected and simply-connected semisimple gauge group flows in the IR to a free theory of as many twisted chiral superfields as the rank of the gauge group, extending previous results for SU, SO, and Sp gauge theories. In
161 162 chapter 4, we described a physical realization of a family of non-compact K¨ahlerthreefolds with trivial canonical bundle in hybrid Landau-Ginzburg models, motivated by some recent non-K¨ahlersolutions of Strominger systems, and utilizing some recent ideas from GLSMs. We considered threefolds given as fiber products of compact genus g Riemann surfaces and noncompact threefolds. Each genus g Riemann surface is constructed using recent GLSM tricks, as a double cover of P1 branched over a degree 2g + 2 locus, realized via nonpertur- bative effects rather than as the critical locus of a superpotential. We focus in particular on special cases corresponding to a set of K¨ahlertwistor spaces of certain hyperK¨ahlerfour- 4 2 1 3 manifolds, specifically the twistor spaces of R , C /Zk , and S ×R . We checked in all cases that the condition for trivial canonical bundle arising physically matches the mathematical constraint.
In the future, I would like to continue to explore two dimensional supersymmetric field theories. First, I’d like to formulate a systematic method to generate (0, 2) B/2-twisted mirrors to A/2 models on toric Fano spaces with arbitrary deformations of tangent bundles, based on our current results on (0, 2) mirrors and related work [129]. Ideally, one would like to understand (0, 2) mirrors for Calabi-Yau manifolds. After construct mirrors to toric ambient spaces, I plan to extend the conjectures to (0, 2) mirrors to compact Calabi-Yau hypersurfaces by standard tricks. It will provide a solid starting point for extending our understanding of (0, 2) mirror symmetry.
For the non-Abelian cases, there is enough information to realize (0, 2) B/2-twisted mirrors to A/2 models on Grassmannian manifolds and even flag manifolds. As mentioned above, we now have a solid understanding of the mirrors to non-Abelian (2, 2) supersymmetric two- dimensional gauge theories [4,128]. Moreover, quantum sheaf cohomology has been computed for Grassmannians and flag manifolds with deformations of tangent bundles [88, 130]. I also plan to apply our method of construction of (0, 2) mirrors to Grassmannians and flag manifolds, and eventually understand (0, 2) B/2-twisted mirrors for non-Abelian gauged linear sigma models.
So far, we only have a basic understanding of (0, 2) mirror symmetry near the (2, 2) locus, since we are only able to consider deformations of tangent bundles. But, more general (0, 2) theories without a (2, 2) locus do exist and there is evidence for (0, 2) mirror symmetry with vector bundles that are not deformations of tangent bundles [131,132]. Hopefully, ideas from developing (0, 2) theories with a (2, 2) locus will pave the way to a more general structure. On the other hand, for now we are only able to check the genus zero correlation functions 163
in our ansatz, it would be useful and interesting to check at higher genera in the future.
It would be very interesting to realize the new family of potential solutions to the Strominger system that were reviewed in section 4.1 in gauged linear sigma models. As mentioned in section 4.2, GLSM realizations of Riemann surfaces of genus g and the holomorphic map ϕ are known. In addition, Witten constructed two dimensional (0, 4) theories on HyperK¨ahler manifolds [133]. In unpublished work, we considered an analogous construction of sigma models on the fiber products above by considering a field-dependent (0, 2) subset of Wit- ten’s (0, 4) theories. Based on the (0, 2) Lagrangian, we generated a new Lagrangian that is invariant under a P1-dependent (0, 2) supersymmetry. In this construction, the supersymme- try parameter is a function of the chiral superfields, which can be thought of as coordinates on the P1. Thus, we were able to realize the two ingredients of the fiber product structure separately. On the other hand, although GLSMs for some non-K¨ahlerspaces are known in the literature [134–136], this class is significantly different, and new ideas are required in order to describe these new models in GLSMs. Bibliography
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