SOLUTIONS TO EXERCISES (BY SIYE WU) General Remarks

The classical moduli space

We recall that the quark superfield Q in an N = 1 supersymmetric gauge theory is valued in a representation R of the gauge group G. The bosonic potential is V = |D|^{2}, where D: R → 𝔤^{∗} ≅ 𝔤 is the moment map of the Hamiltonian G-action on R. Therefore the space M_{c} of classical ground states (modulo gauge transformations), called the moduli space of classical vacua, is the symplectic quotient D^{−1}(0)/G.

The space M_{c} = D^{−1}(0)/G is also the categorical quotient of R by G^{ℂ}.^{1} That is to say, M_{c} is a complex space whose coordinate ring is isomorphic to ℂ[R]^{G}. In fact, for any Q ∈ R, the orbit G^{ℂ} · Q is closed if and only if G^{ℂ} · Q ∩ D^{−1}(0) ≠ ∅. If Q ∈ D^{−1}(0), then G^{ℂ} · Q ∩ D^{−1}(0) = G · Q. Let the semi-stable part of R be R^{ss} = {Q ∈ R | G^{ℂ} · Q ∩ D^{−1}(0) ≠ ∅}. Let π: R^{ss} → R^{ss}/G^{ℂ} be the quotient map. We have

1) the the inclusion D^{−1}(0) → R^{ss} induces a homeomorphism D^{−1}(0)/G → R^{ss}/G^{ℂ};

2) for any point O ∈ R^{ss}/G^{ℂ}, π^{−1}(O) contains exactly one closed G^{ℂ}-orbit in R;

3) for any open set U ⊂ R^{ss}/G^{ℂ}, we have an isomorphism O(U) ≅ O(π^{−1}(U))^{G}. This justifies the identification of the structure sheaf O_{D^{−1}(0)/G} with (O_{R})^{G}. (However the invariants in ℂ[R]^{G} do not distinguish the G^{ℂ}-orbits in R.)

In what follows we will consider the case R = Hom(F, C). Here C, called the space of colors, is an irreducible unitary representation of G. F, the space of flavors, is an Hermitian vector space on which G acts trivially. At the classical level, the global symmetry group is U(F) × U(1)_{X}, where U(1)_{X} is the rotation on the fermionic coordinates in the N = 1 superspace. We write U(F) = SU(F) ×_{Z(SU(F))} U(1)_{A}, where Z(SU(F)) is the center of SU(F) and U(1)_{A} is the scalar multiplication of U(1) on F.

Axial anomaly and the global symmetry

40 N. SEIBERG, DYNAMICS OF N = 1 SUSY THEORIES

The classical action depends on the gauge group G, the matter representation R, the coupling constant g and the θ-term. In the quantum theory, the coupling constant g depends on the renormalization scale μ: g(μ) runs according to μ\frac{dg}{dμ} = β(g). Here β(g) is the β-function; at one-loop it is given by

1G. Kempf, L. Ness, in: Lecture Notes of Math. 732, (1979) pp. 233-244; G. W. Schwarz, in:

Progress in Math. 80, (1989) pp. 135-151; P. Heinzner, F. Loose, Geom. Funct. Anal. 4 (1994)

288-297

β(g) = −\frac{b_{0}}{(4π)^{2}} g^{3}, (4.1)

where b_{0} = \frac{3}{2}c_{2}(G) − \frac{1}{2}c_{2}(R). The quantum theory thus has a parameter of mass dimension 1: the (complex) scale Λ defined by

Λ^{b_{0}} = μ^{b_{0}}e^{− \frac{8π^{2}}{g^{2}(μ)}} e^{iθ} = μ^{b_{0}}e^{2πiτ(μ)}, (4.2)

where θ is the theta angle and τ(μ) = \frac{θ}{2π} + \frac{4πi}{g^{2}(μ)}.

Recall that if R = Hom(F, C), then the classical theory has a global symmetry U(F) × U(1)_{X}. Upon quantization, both U(1)_{A} and U(1)_{X} shift the theta angle θ, and hence Λ^{b_{0}}, whereas SU(F) does not. Consequently neither of these U(1) groups is a true symmetry of a fixed quantum theory; this is called the axial anomaly. Rather, the group U(1)_{A} × U(1)_{X} acts on a family of quantum theories. The U(1)_{A} and U(1)_{X} charges of Q, its fermionic component ψ_{Q}, the gluino λ and the scale Λ are summarized as follows:

\begin{aligned}& U(1)_{A} & × & U(1)_{X} & U(1)_{R} \\ Q & −1 & & 0 & \frac{c_{2}(R)−c_{2}(G)}{c_{2}(R)} \\ ψ_{Q} & −1 & & −1 & −\frac{c_{2}(G)}{c_{2}(R)} \\ λ & 0 & & 1 & 1 \\ Λ^{b_{0}} & c_{2}(R) & & c_{2}(R) − c_{2}(G) & 0 \end{aligned} (4.3)

Here and below, representations of U(1) are labeled by their weights.

The anomaly-free subgroup of U(1)_{A} × U(1)_{X} is a product of a U(1) subgroup, denoted by U(1)_{R}, and a finite group containing the center of SU(F). We normalize the generator of the Lie algebra 𝔲(1)_{R} so that it projects to that of 𝔲(1)_{X}. The weights of U(1)_{R}, which are also shown in the above table, are related to those of U(1)_{A} and U(1)_{X} by

R = X + \frac{c_{2}(G) − c_{2}(R)}{c_{2}(R)}A. (4.4)

The true quantum symmetry is SU(F) × U(1)_{R} up to a discrete factor.

Global gauge anomalies

Consider a principal G-bundle P over the (Euclidean) space-time. Let A be the space of connections and G, the group of gauge transformations. Let D̸_{R}^{+} be the Dirac operator acting on Weyl spinors twisted by a representation R. A gauge theory with Weyl fermions has gauge anomalies if the determinant line bundle Det D̸_{R}^{+} over A/G is not trivial. When the gauge group G is semi-simple, the curvature F of Det D̸_{R}^{+} is proportional to a multi-linear form d_{R} on the Lie algebra 𝔤 defined by

d_{R}(x, y, z) = tr_{R}x{y, z} (x, y, z ∈ 𝔤). (4.5)

Local gauge anomalies come from the non-vanishing of F. For G = Sp(n) or G = SO(n), we have d_{R} = 0 for any representation R of G. Consequently the bundle Det D̸_{R}^{+} is flat and there are no local gauge anomalies.

However if there is a non-contractible loop in A/G, the holonomy along it may be non-trivial even if F = 0. This is called global gauge anomaly. Since the space A is contractible, π_{1}(A/G) ≅ π_{0}(G). When the (Euclidean) space-time is S^{4}, we have π_{0}(G) ≅ π_{4}(G). When G is a simple Lie group, π_{4}(G) ≠ 1 only if only if the universal covering group is Sp(n) for some n ≥ 1, in which case π_{4}(G) ≅ ℤ_{2}. Thus the holonomy of Det D̸_{R}^{+} for any R is ±1, and consequently, the bundle (Det D̸_{R}^{+})^{⊗2} is always trivial. In particular, consider R = Hom(F, C), where G acts on F trivially. When G = Sp(n) and C ≅ ℂ^{2n} is its defining representation, the bundle Det D̸_{R}^{+} = (Det D̸_{C}^{+})^{⊗ dim F} is trivial if dim F is even. This condition will guarantee the absence of global gauge anomalies. That the bundle Det D̸_{C}^{+} itself is non-trivial is a deeper fact related to the Bott periodicity.

For G = SO(n), the curvature F of Det D̸_{R}^{+} vanishes for any representation R. If n ≥ 6, since π_{4}(G) = 1, the bundle Det D̸_{R}^{+} is trivial for any representation R (including the defining representation). If G is simple and the representation R is non-trivial and of real type, then up to a finite subgroup of the center, G can be embedded into SO(d), where d = dim R. The G-bundle P induces a principal SO(d)-bundle P_{0}. Let A_{0} be the space of connections on P_{0}, and G_{0}, the group of gauge transformations. Then there exist maps A → A_{0}, G → G_{0} and A/G → A_{0}/G_{0}. The determinant line bundle Det D̸_{R}^{+} over A/G is the pull-back of corresponding bundle over A_{0}/G_{0}. If d ≥ 6, the latter is trivial. If 2 ≤ d ≤ 5, the case can be reduced to d ≥ 6 since Det D̸_{R⊕R⊕R}^{+} = (Det D̸_{R}^{+})^{⊗3} ≅ Det D̸_{R}^{+}. Therefore Det D̸_{R}^{+} is trivial for any real representation R of any compact Lie group G.

We list a few consequences of this result. First, R and R′ are both real or quarternionic representations of G, then the representation R ⊗ R′ is real. So the bundle Det D̸_{R⊗R′}^{+} is trivial. Second, if G = SO(n) for any n ≥ 3, and C ≅ ℂ^{n}, the defining representation, then the bundle Det D̸_{C}^{+} is trivial. Finally, since the adjoint representation is always of real type, the bundle Det D̸_{adj}^{+} is trivial. Thus the gluino λ does not contribute to the global gauge anomaly.

SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 1–2

Solution to Problem 1

Let C ≅ ℂ^{2N_{c}} be an Hermitian vector space with a compatible symplectic form ω_{c}. C is the defining representation of the compact group G = Sp(C) ≅ Sp(N_{c}). We write the complexification G^{ℂ} = Sp(C, ℂ). The representation on C is quarternionic. We have c_{2}(C) = 1. The dual Coxeter number of G is h = N_{c} + 1. Therefore c_{2}(G) = 2(N_{c} + 1). Let the flavor space F be of dimension 2N_{f}. (It is even to ensure the absence of the global gauge anomaly.) Then c_{2}(R) = 2N_{f} and b_{0} = 3(N_{c} + 1) − N_{f}.

Classical theory

The moduli space of classical vacua is M_{c} ≅ Hom(F, C)^{ss}/Sp(C, ℂ). We define gauge invariant quantities

M = Q^{∗}ω_{c} ∈ ∧^{2}F^{∗}. (5.1)

Clearly, M satisfies the constraint

∧^{N_{c}+1}M = 0. (5.2)

It is a classical result of Weyl that the ring of invariants ℂ[Hom(F, C)]^{G} is generated by M ∈ ∧^{2}F^{∗} with the (only) relations (5.2).^{1} So the moduli space M_{c} is parametrized by M subject to the constraint (5.2).

Recall that C has a quarternionic structure. We have an orthogonal decomposition of the Lie algebra

(5.3) 𝔲(C) = 𝔲(C, ℍ) ⊕ i{ℍ-self-adjoint operators on C}.

Since 𝔲(C, ℍ) = 𝔰𝔭(C), the moment map is

(5.4) D(Q) = orthogonal projection of iQQ^{†} onto 𝔲(C, ℍ).

1H. Weyl, Classical groups, §VI.1

So D(Q) = 0 if and only if QQ^{†} is ℍ-self-adjoint on C. In this case, there is an ℍ-unitary basis of C such that QQ^{†} has the form

QQ^{†} ∼ (\begin{matrix}a_{1}^{2} \\ & . & . & . \\ & & & & a_{r}^{2} \end{matrix} ) ⊗ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (5.5)

where a_{1}, …, a_{r} > 0, r = \frac{1}{2} rank Q. Hence there is a unitary basis in F such that Q is of the form

Q ∼ (\begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \end{matrix} ) ⊗ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) . (5.6)

In the complex language, any Q ∈ Hom(F, C)^{ss} can be put into the standard form (5.6) by a complexified gauge transformation and a global symmetry. The stabilizer of Q in Sp(C, ℂ) is isomorphic to Sp(N_{c} − r, ℂ) and that in Sp(C) is Sp(N_{c} − r). At a generic point in M_{c}, the gauge group is broken to Sp(N_{c} − N_{f}) if N_{c} > N_{f}; it is completely broken if N_{c} ≤ N_{f}.

Quantum theory

Let Λ be the (complex) scale of the quantum theory. Since Λ^{b_{0}} has U(1)_{A}-charge c_{2}(R) = 2N_{f}, it is appropriate to write

Λ^{3(N_{c}+1)−N_{f}} ∈ ∧^{2N_{f}} F^{∗}. (5.7)

The anomaly-free subgroup in U(1)_{A} ×U(1)_{X} is U(1)_{R} ×ℤ_{2N_{f}}, where the R-charge can be computed by, following (4.4),

R = X + \frac{N_{c} + 1 − N_{f}}{N_{f}} A. (5.8)

Since ℤ_{2N_{f}} = Z(SU(F)), the global symmetry of the quantum theory is SU(F) × U(1)_{R}. The superfields Q and M transform as

\begin{aligned}& SU(F) & × & U(1)_{R} \\ Q & F^{∗} & & 1 − \frac{N_{c}+1}{N_{f}} \\ M & ∧^{2}F^{∗} & & 2(1 − \frac{N_{c}+1}{N_{f}}) \end{aligned} (5.9)

The vacuum degeneracy in the classical moduli space M_{c} may be lifted by a dynamically generated superpotential W for the light fields M. W must be invariant under the gauge and global symmetries. The fields in M are already gauge invariant; the only combination invariant under SU(F) is the Pfaffian Pf M. Also, the superpotential is holomorphic on M_{c} hence in M. These restrictions imply that W is of the form

W ∼ (Λ^{b_{0}})^{x}(Pf M)^{y}. (5.10)

The exponents x, y can be fixed as follows:

1) Since Λ^{b_{0}}, Pf M ∈ ∧^{2N_{f}} F^{∗} and W is a number, we have x + y = 0.

2) As a superpotential, W has U(1)_{X} charge 2(N_{c} + 1 − N_{f})x = 2.

3) W has mass dimension b_{0}x + 2N_{f}y = 3.

Notice that 1) and 2) together imply that W has U(1)_{R}-charge 2. The above three equations for two unknowns x and y are consistent and we get x = −y = \frac{1}{N_{c}+1−N_{f}}. Hence the superpotential is

W = A (\frac{Λ^{3(N_{c}+1)−N_{f}}}{Pf M} )^{\frac{1}{Nc+1−N_{f}}} , (5.11)

where A is a constant that depends only on N_{c}, N_{f} and the renormalization scheme. Notice that if N_{f} ≥ N_{c} + 1, then the superpotential W in (5.11) does not exist because Pf M = 0. In this case, the classical vacuum degeneracy is not lifted by quantum effects. When N_{f} ≤ N_{c}, the decay of W at M → ∞ is consistent with asymptotic freedom, which holds when N_{f} < 3(N_{c} + 1).

Solution to Problem 2

Let C ≅ ℂ^{N_{c}} be a complex vector space with a real structure, i.e., C is the complexification of C_{ℝ} ≅ ℝ^{N_{c}}. Suppose C_{ℝ} is equipped with a metric g_{c} and a compatible volume form v_{c}, which provides a trivialization of det C_{ℝ}^{∗}. (In the symplectic case, v_{c} is fixed by the symplectic form ω_{c}.) C is the defining representation of the compact Lie group G = SO(C) ≅ SO(N_{c}). (The complexification is denoted by G^{ℂ} = SO(C, ℂ).) Let the flavor space F have dimension N_{f}. For N_{c} ≥ 5, the dual Coxeter number h = N_{c} − 2 and c_{2}(C) = 2. So c_{2}(G) = 2(N_{c} − 2), c_{2}(R) = 2N_{f} and b_{0} = 3(N_{c} −2)−N_{f}. For N_{c} = 4, G = (G_{+} ×G_{−})/ℤ_{2}, where G_{±} ≅ SU(2). For each factor G_{±}, we have c_{2}(G_{±}) = 4, c_{2}(C) = 2, c_{2}(R) = 2N_{f}. So b_{0} = 6 − N_{f} for the coupling constant of each G_{±}. These are formally the continuation of the cases N_{c} ≥ 5. For N_{c} = 3, G ≅ SU(2) up to a ℤ_{2} factor. We have c_{2}(G) = c_{2}(C) = 4, c_{2}(R) = 4N_{f} and b_{0} = 6 − 2N_{f}.

Classical theory

The moduli space of classical vacua is M_{c} ≅ Hom(F, C)^{ss}/SO(C, ℂ). The gauge invariants are the mesons M = Q^{∗}g_{c} ∈ S^{2}F^{∗} and, if rank Q = N_{c}, the baryons B = Q^{∗}v_{c} ∈ ∧^{N_{c}}F^{∗}. They satisfy the constraints

rank M ≤ N_{c}, B ⊗ B = ∧^{N_{c}}M. (5.12)

Here ∧^{N_{c}}M ∈ S^{2}(∧^{N_{c}}F^{∗}) corresponds to the product of non-zero eigenvalues of M. The ring of invariants ℂ[Hom(F, C)]^{G} is generated by M ∈ S^{2}F^{∗} and, if N_{f} ≥ N_{c}, B ∈ ∧^{N_{c}}F^{∗} with the relations (5.12).^{2} So M and B subject to the constraints in (5.12) parametrize the moduli space M_{c}.

Consider the orthogonal decomposition

(5.13) 𝔲(C) = 𝔰𝔬(C) ⊕ i{ℝ-self-adjoint operators on C_{ℝ}}.

Note that 𝔰𝔬(C) is the set of skew-self-adjoint operators on C_{ℝ}. The moment map of the G-action on Hom(F, C) is

(5.14) D(Q) = orthogonal projection of iQQ^{†} onto 𝔰𝔬(C).

So D(Q) = 0 if and only if QQ^{†} is ℝ-self-adjoint on C_{ℝ}. In this case, there is an orthogonal basis of C (i.e., up to a gauge transformation) such that QQ^{†} has the form

2H. Weyl, Classical groups, §II.9

QQ^{†} ∼ (\begin{matrix}a_{1}^{2} \\ & . & . & . \\ & & & & a_{r}^{2} \end{matrix} ) , (5.15)

where a_{1}, …, a_{r} > 0 if r = rank Q < N_{c} and a_{1}, …, a_{N_{c}−1} > 0, a_{N_{c}} ≠ 0 if r = N_{c}. In the latter case, B^{1,...,N_{c}} = a_{1} … a_{N_{c}} is the only non-zero component of B. Hence up to gauge and global symmetries Q is of the form

Q ∼ (\begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \end{matrix} ) . (5.16)

The stabilizer of Q in SO(C, ℂ) is isomorphic to SO(N_{c}−r, ℂ), and that in SO(C_{ℝ}), SO(N_{c} − r). At a generic point in M_{c}, the gauge group is broken to SO(N_{c} − N_{f}) if N_{c} ≥ N_{f} + 2; it is totally broken if N_{c} < N_{f} + 2.

Quantum theory

Let Λ be the (complex) scale of the quantum theory. We have

Λ^{3(N_{c}−2)−N_{f}} ∈ (∧^{N_{f}} F^{∗})^{⊗2} (5.17)

if N_{c} ≥ 4. (For N_{c} = 4, this holds for the scale of each SU(2) theory.) If N_{c} = 3, then

Λ^{6−2N_{f}} ∈ (∧^{N_{f}} F^{∗})^{⊗4}. (5.18)

The anomaly-free subgroup of U(1)_{A} × U(1)_{X} is U(1)_{R} × ℤ_{2N_{f}} if N_{c} ≥ 4 and U(1)_{R} × ℤ_{4N_{f}} if N_{c} = 3. Here the R-charges are computed by

R = X + \frac{N_{c} − 2 − N_{f}}{N_{f}} A. (5.19)

The ℤ_{N_{f}} subgroup in ℤ_{2N_{f}} or ℤ_{4N_{f}} is the center of SU(F) ≅ SU(N_{f}). Therefore the global symmetry of the quantum theory is (SU(F)⋊ℤ_{2})×U(1)_{R} if N_{c} ≥ 4 and (SU(F) ⋊ ℤ_{4}) × U(1)_{R} if N_{c} = 3. Here SU(F) ⋊ ℤ_{n} = {U ∈ U(F) | U^{n} ∈ SU(F)}. For all N_{c} ≥ 3, the superfields Q and M transform as

\begin{aligned}& SU(F) & × & U(1)_{R} \\ Q & F^{∗} & & 1 − \frac{N_{c}−2}{N_{f}} \\ M & S^{2}F^{∗} & & 2(1 − \frac{N_{c}−2}{N_{f}}) \end{aligned} (5.20)

Invariance under the gauge and global symmetries and holomorphicity imply that the superpotential, if it exists, is of the form

W ∼ (Λ^{b_{0}})^{x}(det M)^{y} (5.21)

for some exponents x, y. When N_{c} ≥ 4, considerations exactly the same as in the symplectic case yield three equations x + y = 0, 2 = 2(N_{c} − N_{f} − 2)x and b_{0}x + 2N_{f}y = 3 for the two unknowns x and y. Again, they have a solution x = −y = \frac{1}{N_{c}−2−N_{f}}. Hence

W = A (\frac{Λ^{3(N_{c}−2)−N_{f}}}{det M} )^{\frac{1}{Nc−2−N_{f}}} , (5.22)

where A is a constant that depends only on N_{c}, N_{f} and the renormalization scheme. Clearly W does not exist if N_{f} = N_{c} − 2. If N_{f} > N_{c}, then rank M < N_{f} and det M = 0. So W does not exist either. If N_{c} − 2 < N_{f} ≤ N_{c}, W → ∞ as Q → ∞; this is in contradiction with asymptotic freedom (b_{0} > 0 in this range). So the superpotential is not generated when N_{f} ≥ N_{c} − 2. When N_{c} = 4, (5.22) is the superpotential generated by each of the SU(2) factors. We limit the analysis to the case of equal scales for the two SU(2) factors. When N_{c} = 3, the three equations 2x + y = 0, 2 = (4 − 4N_{f})x and b_{0}x + 2N_{f}y = 3 are also consistent and have a solution 2x = −y = \frac{1}{1−N_{f}}. So,

W = A (\frac{Λ^{3−N_{f}}}{det M} )^{\frac{1}{1−N_{f}}} . (5.23)

Similarly, W does not exist if N_{f} = 1 or N_{f} > 3. When N_{f} = 2, the form of W is not consistent with asymptotic freedom (b_{0} = 2 > 0 in this case). When N_{f} = 3, we have b_{0} = 0, but the two-loop β-function is positive. The theory is infrared free and does not have a superpotential. In summary, no superpotential can be generated when N_{c} = 3.

Solution to Problem 3

N_{f} = N_{c}: instanton generated superpotential

Recall that when N_{f} ≤ N_{c}, it is possible to have a dynamically generated superpotential (5.11). This superpotential is generated by instantons when N_{f} = N_{c}, because the following three requirements are independently met:

1) The result of any calculation in instanton perturbation theory is proportional to e^{2πiτ(μ)} ∼ Λ^{b_{0}}. The superpotential W in (5.11) has the same power of Λ when N_{f} = N_{c}.

2) An instanton vertex absorbs 2(N_{c} + 1) of the λ and 2N_{f} of the ψ_{Q}. The usual vertex is given by the term λψ_{Q}Q. When N_{f} = N_{c}, it is possible to form a diagram with 2 external lines of ψ_{Q}.

3) The Sp(N_{c}) gauge symmetry is completely broken when the renormalization scale μ is small compared to ⟨Q⟩. This means the coupling constant stays small for all μ. Hence the instanton calculation is trustworthy. Thus we expect A ≠ 0 when N_{f} = N_{c}.

We study how the constant A depends on N_{c} and N_{f} using deformations on the moduli space. Suppose Q acquires an expectation value ⟨Q⟩ ∈ Hom(F, C) with \frac{1}{2} rank ⟨Q⟩ = r. Then C = Ĉ ⊕ Ĉ^{⊥}, F = F̂ ⊕ F̂^{⊥}, where Ĉ = (im⟨Q⟩)^{⊥}, F̂ = ker⟨Q⟩ have dimensions 2(N_{c} − r), 2(N_{f} − r), respectively. The expectation value ⟨Q⟩ breaks the gauge group to Sp(Ĉ). Let ⟨M^{⊥}⟩ ∈ ∧^{2}F̂^{⊥∗} be the restriction of ⟨M⟩ to F̂^{⊥}; ⟨M^{⊥}⟩ is non-degenerate. Because of the superpotential (5.11), the excitations of M in the F̂^{⊥} directions become massive. In the low energy theory, M is of the form

M = ( M̂ ; ⟨M^{⊥}⟩ ) , (6.1)

where ⟨M^{⊥}⟩ is fixed and M̂ ∈ ∧^{2}F̂^{2} contains the light modes. The space of light flavors is F̂.

When the renormalization scale μ >> (Pf (⟨M^{⊥}⟩/2))^{1/2r}, the coupling constant g(μ) runs according to

(Λ/μ)^{3(N_{c}+1)−N_{f}} = e^{− \frac{8π^{2}}{g^{2}(μ)}} . (6.2)

When μ << (Pf (⟨M^{⊥}⟩/2))^{1/2r}, the low energy theory has N_{c} −r colors and N_{f} −r flavors. Let Λ̂ be its scale. Then

(Λ̂/μ)^{3(N_{c}−r+1)−(N_{f} −r)} = e^{− \frac{8π^{2}}{g^{2}(μ)}} . (6.3)

We use the \overline{DR} subtraction scheme. Matching the coupling constant g(μ) in (6.2) and (6.3) at μ = (Pf (⟨M^{⊥}⟩/2))^{1/2r}, we get

Λ^{3(N_{c}+1)−N_{f}} = Pf (⟨M^{⊥}⟩/2) Λ̂^{3(N_{c}−r+1)−(N_{f} −r)}. (6.4)

This relation is valid for any N_{c}, N_{f} and r ≤ N_{c}, N_{f}. Using (6.4), the superpotential of the low energy theory has the same form (5.11) but with a different coefficient Â related to A by

A = 2^{\frac{r}{Nc+1−N_{f}}} Â. (6.5)

When N_{f} = N_{c}, if we choose r = N_{c} − 1, then \frac{1}{2} dim Ĉ = \frac{1}{2} dim F̂ = 1. Due to the accidental isomorphism Sp(1) ≅ SU(2) and their defining representations, the low energy theory is identical to the SU(2) supersymmetric gauge theory with a single flavor of quark. We know from the SU(2) theory that Â = 1 under the \overline{DR} scheme. So A = 2^{N_{c}−1} when N_{f} = N_{c}.

N_{f} < N_{c} and the pure gauge theory

To derive the superpotential for smaller values of N_{f} but with a fixed N_{c}, we need to apply a gauge invariant mass deformation. We perturb the theory by adding at the tree level a superpotential W_{tree} = ⟨M, m⟩, where m ∈ ∧^{2}F. Let r = \frac{1}{2} rank m. We write F = F̂ ⊕ F̂^{⊥} such that \frac{1}{2} dim F̂ = N_{f} − r and the projection m^{⊥} ∈ ∧^{2}F̂^{⊥} of m is non-degenerate. The gauge group of the low energy theory is still Sp(C), but the flavor space is F̂. Let Λ̂ be the scale. When μ >> (Pf m^{⊥})^{1/r}, g(μ) runs according to (6.2). However when μ << (Pf m^{⊥})^{1/r}, there are N_{f} − r flavors of light quarks, and we get

(Λ̂/μ)^{3(N_{c}+1)−(N_{f} −r)} = e^{− \frac{8π^{2}}{g^{2}(μ)}} . (6.6)

Matching the coupling constant g(μ) in (6.2) and (6.6) at μ = (Pf m^{⊥})^{1/r}, we get

Λ̂^{3(N_{c}+1)−(N_{f} −r)} = Pf m^{⊥}Λ^{3(N_{c}+1)−N_{f}} . (6.7)

This relation is valid for any N_{c}, N_{f} and r ≤ N_{f}.

If N_{f} ≤ N_{c} in the ultraviolet theory, the total superpotential after the mass deformation is

W = A (\frac{Λ^{3(N_{c}+1)−N_{f}}}{Pf M} )^{\frac{1}{Nc+1−N_{f}}} + ⟨M, m⟩. (6.8)

We eliminate the heavy fields in M by minimizing bosonic potential. At the extremum of the superpotential (6.8), M is of the block-diagonal form (6.1), where

⟨M^{⊥}⟩ = (\frac{A}{N_{c} + 1 − N_{f}} )^{\frac{Nc+1−N_{f}}{Nc+1−(N_{f} −r)}} (\frac{Pf m^{⊥}Λ^{3(N_{c}+1)−N_{f}}}{Pf M̂} )^{\frac{1}{Nc+1−(N_{f} −r)}} (m^{⊥})^{−1}. (6.9)

Here (m^{⊥})^{−1} ∈ ∧^{2}F̂^{⊥∗} is induced by m^{⊥} ∈ ∧^{2}F̂^{⊥}. We compute the superpotential of the light mesons M̂ by substitute in (6.8) the solution of ⟨M^{⊥}⟩. Using ⟨(m^{⊥})^{−1}, m^{⊥}⟩ = r, d Pf M = Pf M⟨dM, M^{−1}⟩ and (6.7), we deduce that the superpotential has precisely the same form as (5.11), with the coefficient

Â = (N_{c} + 1 − (N_{f} − r)) (\frac{A}{N_{c} + 1 − N_{f}} )^{\frac{Nc+1−N_{f}}{Nc+1−(N_{f} −r)}} . (6.10)

If we have N_{f} = N_{c} to begin with, then A = 2^{N_{c}−1} and Â = (N_{c} + 1 − (N_{f} −

r))2^{\frac{Nc−1}{Nc+1−(N_{f} −r)}} . Equivalently, for a theory with N_{f} ≤ N_{c}, the coefficient A in (5.11) is given by

A = (N_{c} + 1 − N_{f})2^{\frac{Nc−1}{Nc+1−N_{f}}} . (6.11)

In all cases of N_{f} ≤ N_{c}, the superpotential is non-zero. When 1 ≤ N_{f} ≤ N_{c}, the classical vacuum degeneracy is lifted by W and the quantum theory has no vacuum.

In the pure Sp(C) ≅ Sp(N_{c}) gauge theory, N_{f} = 0, and the superpotential

W = (N_{c} + 1)2^{\frac{Nc−1}{Nc+1}} Λ^{3} = (N_{c} + 1)μ^{3}2^{\frac{Nc−1}{Nc+1}} e^{\frac{2πi}{Nc+1} τ} . (6.12)

This is (N_{c} + 1)-valued, corresponding to the fact that there are N_{c} + 1 vacua in quantum theory. In particular, when N_{c} = 1, (6.12) agrees with the result for SU(2) gauge theory. Notice also that the superpotential W is a constant: although non-zero, it does not contribute to the effective Lagrangian. Nevertheless, the result (6.12) implies gaugino condensation. We consider an extended theory in which the parameter τ becomes a superfield and assume that supersymmetry is unbroken after quantization. The partition function

Z(τ) = ∫ DA Dλ e^{iS(A,λ,τ)} = e^{i ∫ d^{4}xd^{2}θ W (τ)} (6.13)

is holomorphic in τ and W (τ) has the same form as (6.12). Since the Lagrangian density of the pure gauge theory is

L = \frac{1}{4πi} ∫ d^{2}θ τ trW_{α}W^{α} + c.c., (6.14)

where W_{α} = λ_{α} + ⋯ is the superfield strength, we get

\begin{aligned}⟨λλ⟩ & = 4π \frac{δ}{δF_{τ}} log Z(τ) = 4π \frac{∂}{∂F_{τ}} ∫ d^{2}θ W (τ) \\ & = 4π \frac{∂}{∂τ} W (τ) = 2^{4− \frac{2}{Nc+1}} π^{2}iΛ^{3} = 2^{4− \frac{2}{Nc+1}} π^{2}iμ^{3}e^{\frac{2πi}{Nc+1} τ} . \end{aligned} (6.15)

Again, this is (N_{c} + 1)-valued.

N_{f} = N_{c} + 1

In the above mass deformation of theories with N_{f} ≤ N_{c}, if r = N_{f}, then F̂^{⊥} = F, m^{⊥} = m and M^{⊥} = M. The extremum of the total superpotential (6.8) is reached at

⟨M⟩ = 2^{\frac{Nc−1}{Nc+1}} (Λ^{3(N_{c}+1)−N_{f}} Pf m)^{\frac{1}{Nc+1}} m^{−1}. (6.16)

Just as in the supersymmetric SU(N_{c}) gauge theories, the above result is valid in the range N_{f} ≥ N_{c} + 1. For N_{f} = N_{c} + 1, the solution (6.16) satisfies

Pf M = 2^{N_{c}−1}Λ^{2(N_{c}+1)}. (6.17)

This is independent of m, and shall remain valid in the absence of the mass term. We claim that in the low energies the quantum theory is described by M subject to the constraint (6.17). To check that the theory reduces to N_{f} = N_{c} via mass deformation, we add a mass term ⟨M, m⟩ with \frac{1}{2} rank m = 1. Then \frac{1}{2} dim F̂ = N_{c}. Because of the constraint (6.17), we have

⟨M, m⟩ = \frac{2^{N_{c}−1}m^{⊥}Λ^{2(N_{c}+1)}}{Pf M̂} = 2^{N_{c}−1} \begin{matrix}Λ̂ & 2N_{c}+3 \\ Pf & M̂ \end{matrix} , (6.18)

where the last equality follows from (6.7) when N_{f} = N_{c} + 1, r = 1. This is the superpotential (5.11) with the correct coefficient (6.11) when N_{f} = N_{c}.

The quantum moduli space M_{q} is smooth. There are no additional massless fields other than M subject to the constraint (6.17). The absence of massless gluons is known as the confinement of color charges. We check the consistency of these postulates using ’t Hooft’s anomaly matching conditions. Each point M_{0} ∈ ∧^{2}F^{∗} on M_{q} is a symplectic form on F. Let Sp(F, ℂ) be the (complex) symplectic group preserving M_{0}. The most symmetric points on M_{q} are those compatible with the Hermitian structure on F. There the global symmetry SU(F) × U(1)_{R} is broken to Sp(F) × U(1)_{R}, where the (compact) symplectic group Sp(F) = Sp(F, ℂ) ∩ U(F) depends on the choice of M_{0}. Notice that when N_{f} = N_{c} + 1, U(1)_{X} = U(1)_{R} is anomaly free. In the ultraviolet theory, the massless fermions are ψ_{Q} and λ. In the infrared theory, they are the fermionic partners ψ_{M} of M in T_{M_{0}}M_{q}. Under the unbroken global symmetry, these light fermions transform according to

Sp(F) ×  U(1)_{R}
ultraviolet ψ_{Q}  F^{∗}  −\frac{N_{c}+1}{N_{f}} = −1
λ  1  1
infrared ψ_{M} ∧^{2}F^{∗} ⊖ 1  1 − 2\frac{N_{c}+1}{N_{f}} = −1

(6.19)

Here and below, 1 denotes the trivial representation. The non-zero combinations to ’t Hooft’s condition are given by

ultraviolet  infrared
𝔰𝔭(F)^{2}𝔲(1)_{R} 2N_{c}c_{2}(F) · (−1)  c_{2}(∧^{2}F ⊖ 1) · (−1)
𝔲(1)_{R}  4N_{c}N_{f} (−1) + N_{c}(2N_{c} + 1) 1  (N_{f}(2N_{f} − 1) − 1) (−1)
𝔲(1)_{R}^{3}  4N_{c}N_{f} (−1)^{3} + N_{c}(2N_{c} + 1) 1^{3}  (N_{f}(2N_{f} − 1) − 1) (−1)^{3}

(6.20)

To calculate c_{2}(∧^{2}F ⊖ 1), we note that the root system of 𝔰𝔭(N_{f}) is {ϵ_{i} ± ϵ_{j}, 2ϵ_{i} | 1 ≤ i ≠ j ≤ N_{f}} and the positive roots are ϵ_{i} ± ϵ_{j} (1 ≤ i ≤ j ≤ N_{f}). The highest weights of the representations of F and ∧^{2}F ⊖ 1 are λ = ϵ_{1} and λ_{2} = ϵ_{1} + ϵ_{2}, respectively. Therefore c_{2}(∧^{2}F ⊖ 1) = \frac{(λ_{2}+2ρ,λ_{2}) dim(∧^{2}F ⊖1)}{(λ_{1}+2ρ,λ_{1}) dim F}c_{2}(F) =

2(N_{f} − 1)c_{2}(F), where ρ = ∑_{i=1}^{N_{f}}(N_{f} + 1 − i)ϵ_{i} is half of the sum of positive roots. The two columns match when N_{f} = N_{c} + 1.

Finally, we propose a global anomaly matching condition and show that it is satisfied in the above situation. Consider two theories with the same (anomaly free) global symmetry group H but with different contents of light fermionic fields, which form representations R and R′ of H, respectively. If the two theories describe the same low energy physics, then if H is hypothetically gauged, the determinant line bundles Det D̸_{R}^{+} and Det D̸_{R′}^{+} of the Dirac operators twisted by R and R′ shall be isomorphic. ’t Hooft’s anomaly matching condition checks whether the curvatures of the two bundles are the same. This is a stringent condition for the two theories to describe the same low energy physics. However the two bundles can still differ by a flat bundle, which can be non-trivial if π_{4}(H) ≠ 1. If the representation R^{∗} ⊕ R′ of H is real, then the determinant line bundle Det D̸_{R^{∗}⊕R′}^{+} ≅ (Det D̸_{R}^{+})^{−1} ⊗ Det D̸_{R′}^{+} is trivial. Therefore the reality of the representation R^{∗} ⊕R′ is a sufficient condition of global anomaly matching. In the above problem, suppose only the Sp(F) factor of the global symmetry is gauged. In the ultraviolet theory, the representation is an even number (2N_{c}) of copies of F^{∗} and the trivial representations. In the infrared theory, the representation ∧^{2}F^{∗} ⊖ 1 ⊂ F^{∗} ⊗ F^{∗} is real because F^{∗} is quarternionic. So the global anomalies match.

N_{f} = N_{c} + 2

When N_{f} = N_{c} + 2, we obtain from (6.16) that ∧^{N_{c}+1}M → 0 as m → 0. So there is no quantum correction to the classical moduli space M_{c}. We claim that in the low energy theory, all components of M become dynamical and that there is a superpotential

W = −\frac{Pf M}{2^{N_{c}−1}Λ^{2N_{c}+1}} . (6.21)

Indeed, varying W with respect to M, we get the constraint (5.2). If we add a mass term W_{tree} = ⟨M, m⟩ with \frac{1}{2} rank m = 1, then \frac{1}{2} dim F̂ = N_{c} + 1. At the extremum of W + W_{tree}, M is of the block-diagonal form (6.1). From varying M^{⊥}, we get

Pf M̂ = 2^{N_{c}−1}m^{⊥}Λ^{2N_{c}+1}. (6.22)

By (6.7), this is exactly the constraint (6.17).

The most symmetric point on M_{c} is M = 0, where the full global symmetry SU(F)×U(1)_{R} is unbroken. This is also the most singular point. The singularity is the result of additional massless fields. In fact all the light fields in M are massless at the origin. We check this using ’t Hooft’s anomaly matching condition. Under the global symmetry, the light fermions in the ultraviolet and the infrared theories transform as

SU(F) × U(1)_{R}
ultraviolet ψ_{Q}  F^{∗}  −\frac{N_{c}+1}{N_{f}}
λ  1  1
infrared ψ_{M} ∧^{2}F^{∗}  1 − 2\frac{N_{c}+1}{N_{f}}

(6.23)

The non-zero combinations to be matched are

ultraviolet  infrared
𝔰𝔲(F)^{3}  2N_{c} (−d_{F})  −d_{∧^{2}F}
𝔰𝔲(F)^{2}𝔲(1)_{R} 2N_{c}c_{2}(F) · (−\frac{N_{c}+1}{N_{f}})  c_{2}(∧^{2}F) · (1 − 2\frac{N_{c}+1}{N_{f}})
𝔲(1)_{R}  4N_{c}N_{f} (−\frac{N_{c}+1}{N_{f}}) + N_{c}(2N_{c} + 1) 1 N_{f}(2N_{f} − 1) (1 − 2\frac{N_{c}+1}{N_{f}})
𝔲(1)_{R}^{3}  4N_{c}N_{f} (−\frac{N_{c}+1}{N_{f}})^{3} + N_{c}(2N_{c} + 1) 1^{3} N_{f}(2N_{f} − 1) (1 − 2\frac{N_{c}+1}{N_{f}})^{3}

(6.24)

We claim that c_{2}(∧^{2}F) = 2(N_{f} − 1)c_{2}(F) and d_{∧^{2}F} = 2(N_{f} − 2)d_{F}. Indeed, let ϵ_{1}, …, ϵ_{2N_{f}} be the weights of the fundamental representation of SU(F). Since ∑_{i=1}^{2N_{f}} ϵ_{i} = 0, we have ∑_{i<j}(ϵ_{i} + ϵ_{j})^{2} = 2(N_{f} − 1) ∑_{i=1}^{2N_{f}} ϵ_{i}^{2} and ∑_{i<j}(ϵ_{i} + ϵ_{j})^{3} =

2(N_{f} − 2) ∑_{i=1}^{2N_{f}} ϵ_{i}^{3}. Therefore the two columns match when N_{f} = N_{c} + 2.

Solution to Problem 4

N_{f} ≤ N_{c} − 5 and pure gauge theory

When N_{f} ≤ N_{c} −5, it is possible to have a dynamically generated superpotential of the form (5.22). Since most of the analysis is parallel to the Sp(N_{c}) gauge theories, we only give an outline here.

Suppose M acquires a large expectation value ⟨M⟩ of rank r. Let F̂ = ker⟨M⟩. Then F = F̂ ⊕ F̂^{⊥} and the restriction ⟨M^{⊥}⟩ ∈ S^{2}F̂^{⊥∗} of ⟨M⟩ to F̂^{⊥} is nondegenerate. The low energy theory has N_{c}−r colors and dim F̂ = N_{f} −r flavors. So b_{0} = 3(N_{c}−r−2)−(N_{f} −r). Matching the coupling constant at μ = (det⟨M^{⊥}⟩)^{1/2r}, we obtain

Λ^{3(N_{c}−2)−N_{f}} = det⟨M^{⊥}⟩Λ̂^{3(N_{c}−r−2)−(N_{f} −r)}. (6.25)

In fact, this relation remains valid for all values of N_{c}, N_{f} as long as N_{c} − r ≥ 5. For future needs, we also consider here the cases N_{c} − r ≤ 4. For N_{c} − r = 4, in the low energy SO(4) theory, both scales Λ̂_{±} of the SU(2)_{±} factors are given by (6.25). If N_{c} − r = 3, we derive the relation in two stages: first from N_{c} ≥ 5 to N_{c} = 4, then from N_{c} = 4 to N_{c} = 3. In the second stage, N_{c} = 4, r = 1. The low energy theory has 3 colors and N_{f} −1 flavors. Above the scale Λ̂, the gauge group is SU(2)_{+} ×SU(2)_{−}; the coupling constants g_{±}(μ), or τ_{±}(μ) run according (4.2) with b_{0} = 6 − N_{f}. Since SU(2) is diagonally embedded, its coupling constant is given by τ(μ) = τ_{+}(μ)+τ_{−}(μ). Therefore the scale is (Λ̂_{+}^{6−N_{f}} Λ̂_{−}^{6−N_{f}})^{1/2(6−N_{f})} = (Λ̂_{+}Λ̂_{−})^{1/2}. At low energies, the SU(2) gauge theory has b_{0} = 6 − 2(N_{f} − 1). Matching the coupling constant g(μ) at μ = \frac{1}{2}⟨M^{⊥}⟩, we get

4Λ̂_{+}^{6−N_{f}} Λ̂_{−}^{6−N_{f}} = ⟨M^{⊥}⟩^{2}Λ̂^{6−2(N_{f} −1)}. (6.26)

Combining (6.25) and (6.26), we get, for any N_{c} ≥ 4, N_{f} and r = N_{c} − 3,

4(Λ^{3(N_{c}−2)−N_{f}})^{2} = (det⟨M^{⊥}⟩)^{2}Λ̂^{6−2(N_{f} −N_{c}+3)}. (6.27)

When N_{f} ≤ N_{c} − 5 and N_{c} − r ≥ 5, a comparison of the superpotential (5.22) above and below μ = (det⟨M^{⊥}⟩)^{1/2r} yields A = Â. If N_{f} = N_{c} − 5 and r = N_{f}, the low energy theory is a pure SO(5) ≅ Sp(2)/ℤ_{2} gauge theory. Using (6.11), we get Â = 3 · 2^{\frac{1}{3}} . So A = 3 · 2^{\frac{1}{3}} when N_{f} = N_{c} − 5.

Now we add a mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = r. Again F = F̂ ⊕ F̂^{⊥}. Here dim F̂ = N_{c} − r and the projection m^{⊥} ∈ S^{2}F̂^{⊥} of m is nondegenerate. The low energy theory has the same number of colors and N_{f} −r flavors. So b_{0} = 3(N_{c} − 2) − (N_{f} − r). Matching the coupling constant at μ = (det m^{⊥})^{1/r}, we obtain

Λ̂^{3(N_{c}−2)−(N_{f} −r)} = det m^{⊥}Λ^{3(N_{c}−2)−N_{f}} . (6.28)

This is valid for N_{c} ≥ 5 and all N_{f}. When N_{c} = 4, the scales Λ̂_{±} of the SU(2)_{±} factors are related to the scales Λ_{±} of the ultraviolet theory by the same formula.

When N_{c} = 3, (6.28) should be replaced by

Λ̂^{6−2(N_{f} −r)} = (det m^{⊥})^{2}Λ^{6−2N_{f}} . (6.29)

When N_{f} ≤ N_{c} − 5, the superpotential W before mass deformation is given by (5.22). Varying the heavy modes in M, the extremum of the total superpotential W + W_{tree} is reached when M is of the diagonal form (6.1) with

⟨M^{⊥}⟩ = (\frac{2A}{N_{c} − 2 − N_{f}} )^{\frac{Nc−2−N_{f}}{Nc−2−(N_{f} −r)}} (\frac{det m^{⊥}Λ^{3(N_{c}−2)−N_{f}}}{det M̂} )^{\frac{1}{Nc−2−(N_{f} −r)}} (m^{⊥})^{−1}. (6.30)

In terms of the of the light fields M̂, this extremum has the same form as (5.22) with

Â = \frac{N_{c} − 2 − (N_{f} − r)}{2} (\frac{2A}{N_{c} − 2 − N_{f}} )^{\frac{Nc−2−N_{f}}{Nc−2−(N_{f} −r)}} . (6.31)

If we start from N_{f} = N_{c} − 5, then A = 3 · 2^{\frac{1}{3}} and Â = \frac{1}{2}(N_{c} − 2 − (N_{f} −

r))2^{\frac{4}{Nc−2−(N_{f} −r)}} . Equivalently, for a theory with N_{f} ≤ N_{c} − 5, the coefficient A in (5.22) is given by

A = \frac{1}{2}(N_{c} − 2 − N_{f})2^{\frac{4}{Nc−2−N_{f}}} . (6.32)

For pure gauge theories, N_{f} = 0 and A = \frac{1}{2}(N_{c} − 2)2^{\frac{4}{Nc−2}} . As in (6.33), for the pure SO(N_{c}) gauge theory with N_{c} ≥ 5,

⟨λλ⟩ = 2^{2+ \frac{4}{Nc−2}} π^{2}iΛ^{3} = 2^{2+ \frac{4}{Nc−2}} π^{2}iμ^{3}e^{\frac{2πi}{Nc−2} τ}; (6.33)

this is (N_{c} − 2)-valued, corresponding to the N_{c} − 2 vacua. Finally, the result is consistent with another accidental isomorphism SO(6) ≅ SU(4)/ℤ_{2}. Indeed, when N_{c} = 6, A = 4; this agrees with the coefficient of the pure SU(4) gauge theory.

N_{f} = N_{c} − 4

A generic flat deformation on the moduli space breaks the gauge group to SO(4) ≅ (SU(2)_{+}×SU(2)_{−})/ℤ_{2}. From (6.25), the scales Λ̂_{±} of the two SU(2) gauge theories (without matter) are related to the scale Λ of the ultraviolet theory by

Λ̂_{±}^{6} = \frac{Λ^{2(N_{c}−1)}}{det M} . (6.34)

In the infrared, the superpotential of the two SU(2)_{±} gauge theories are W_{±} = 2Λ̂_{±}^{3}, respectively. But both of them are two-valued, corresponding to the two vacua of each SU(2) theory. We fix a square root. Then the superpotential of the original theory is

W = 2(ϵ_{+} + ϵ_{−}) (\frac{Λ^{2(N_{c}−1)}}{det M} )^{1/2} , (6.35)

where ϵ_{+}, ϵ_{−} = ±1. There are two physically inequivalent branches ϵ_{+} = ϵ_{−} and ϵ_{+} = −ϵ_{−}.

The superpotential (6.35) is compatible with mass deformation. In the branch ϵ_{+} = ϵ_{−}, it is the continuation to N_{f} = N_{c} − 4 of (5.22). As before, adding a mass term \frac{1}{2}⟨M, m⟩ of rank m = r reduces the theory to N_{f} = N_{c} − 4 − r with the same superpotential (5.22) in the low energies. In the other branch ϵ_{+} = −ϵ_{−}, the total superpotential is W_{tree} = \frac{1}{2}⟨M, m⟩. The theory has no vacuum and hence does not contribute to the low energy theory with a smaller N_{f}.

For ϵ_{+} = ϵ_{−}, the superpotential (6.35) lifts the classical vacuum degeneracy and the quantum theory has no vacuum. In the branch ϵ_{+} = −ϵ_{−}, the vacuum degeneracy is not lifted by quantum effects. Classically there is a singularity at M = 0, where the full gauge symmetry is restored and all the gluons become massless. We claim that in quantum theory, the singularity is smoothed out and the color charges are confined. That is, there are no massless gluons and all the massless fermions are contained in M, even at the origin of M. We check this claim by the ’t Hooft and the global anomaly matching conditions. At M = 0, the identity component of the unbroken global symmetry is SU(F) × U(1)_{R}. The light fermions in high and low energy theories transform according to

SU(F) × U(1)_{R}
ultraviolet ψ_{Q}  F^{∗}  −\frac{N_{c}−2}{N_{f}}
λ  1  1
infrared ψ_{M} S^{2}F^{∗}  1 − 2\frac{N_{c}−2}{N_{f}}

(6.36)

The non-zero combinations to be matched are given by

ultraviolet  infrared
𝔰𝔲(F)^{3}  −N_{c}d_{F}  −d_{S^{2}F}
𝔰𝔲(F)^{2}𝔲(1)_{R} N_{c}c_{2}(F) · (−\frac{N_{c}−2}{N_{f}})  c_{2}(S^{2}F) · (1 − 2\frac{N_{c}−2}{N_{f}})
𝔲(1)_{R}  N_{c}N_{f} (−\frac{N_{c}−2}{N_{f}}) + \frac{1}{2}N_{c}(N_{c} − 1) 1  \frac{1}{2}N_{f}(N_{f} + 1) (1 − 2\frac{N_{c}−2}{N_{f}})
𝔲(1)_{R}^{3}  N_{c}N_{f} (−\frac{N_{c}−2}{N_{f}})^{3} + \frac{1}{2}N_{c}(N_{c} − 1) 1^{3} \frac{1}{2}N_{f}(N_{f} + 1) (1 − 2\frac{N_{c}−2}{N_{f}})^{3}

(6.37)

Let ϵ_{i} (i = 1, …, N_{f}) be the weights in the defining representation F. Using ∑_{i=1}^{N_{f}} ϵ_{i} = 0, it is easy to check ∑_{i≤j}(ϵ_{i} + ϵ_{j})^{2} = (N_{f} + 2) ∑_{i=1}^{N_{f}} ϵ_{i}^{2} and ∑_{i≤j}(ϵ_{i} +

ϵ_{j})^{3} = (N_{f} + 4) ∑_{i=1}^{N_{f}} ϵ_{i}^{3}. So c_{2}(S^{2}F) = (N_{f} + 2)c_{2}(F) and d_{S^{2}F} = (N_{f} + 4)d_{F}. It is now straightforward to check that the two columns of (6.37) match when N_{f} = N_{c} − 4, fulfilling ’t Hooft’s criterion. For N_{f} = 2, SU(F) ≅ Sp(1). In the ultraviolet theory, the number of copies of F^{∗} is even (N_{c} = 6). In the infrared, the representation S^{2}F^{∗} ⊂ F^{∗} ⊗ F^{∗} is real. So the global anomalies also match.

N_{f} = N_{c} − 3

We first find the superpotential by flat deformations on the moduli space. A generic expectation value of Q breaks the gauge group to SO(3). Let F̂ ⊂ F a subspace of dimension 1. We consider the limit where a large expectation value ⟨M^{⊥}⟩ ∈ S^{2}F̂^{⊥∗} with rank N_{f} − 1 breaks the theory to SO(4) ≅ (SU(2)_{+} × SU(2)_{−})/ℤ_{2} with one quark Q̂ ∈ Hom(F̂, C), which further break the gauge group to a diagonal subgroup SU(2)/ℤ_{2}.^{1} In the first stage of deformation, from (6.25), the scales Λ̂_{±} of the SU(2)_{+} × SU(2)_{−} theory are given by

1The decomposition of SO(4) into SU(2)_{+} × SU(2)_{−} is canonical. The isotropy group of Q̂ ∈ Hom(F̂, C) (in SO(4)) and its isomorphisms onto SU(2)_{±} depend on Q̂.

Λ̂_{±}^{5} = \frac{Λ^{2N_{c}−3}}{det⟨M^{⊥}⟩}. (6.38)

The effective SU(2)_{±} theories have instanton generated superpotentials

Ŵ_{±} = 2\begin{matrix}Λ̂ & 5 ± \\ det & M̂ \end{matrix} = 2\frac{Λ^{2N_{c}−3}}{det M} . (6.39)

In the second stage, we have the low energy pure SU(2) gauge theory whose scale Λ̂ is given by (6.27). It has a superpotential

Ŵ = 2Λ̂^{3} = 4ϵ\frac{Λ^{2N_{c}−3}}{det M} , (6.40)

where ϵ = ±1 comes from taking the square root of (6.27) and labels the two vacua of the pure SU(2) theory. Combining (6.39) and (6.40), we find that the superpotential of the N_{f} = N_{c} − 3 theory is

W = Ŵ_{+} + Ŵ_{−} + Ŵ = 4(1 + ϵ)\frac{Λ^{2N_{c}−3}}{det M} . (6.41)

The first term in (6.41) is due to the instanton effect whereas the second term comes from gaugino condensation in the unbroken SU(2) theory. As in SU(N_{c}) and Sp(N_{c}) gauge theories, the superpotential (6.41) is proportional to the exponential of the instanton action, and it is possible to form a diagram with two external fermion lines using the instanton vertex. The novelty here is that the unbroken gauge group SU(2) is non-Abelian. However, when the gauge group is broken to the diagonal SU(2), an instanton in one SU(2)_{±} cannot be rotated into the unbroken group. Therefore the instanton computation is still reliable because there is no infrared divergence.

We check that the result (6.41) is consistent with mass deformation. The theory has two branches: ϵ = 1 and ϵ = −1. After adding a mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = 1, the two branches should correspond to those of N_{f} = N_{c} − 4. In fact, when ϵ = 1, the total superpotential (6.41) is the continuation of (5.22), just as (6.35) when ϵ_{+} = ϵ_{−}. So the extremum of W + W_{tree} matches (6.35) in the branch ϵ_{+} = ϵ_{−}. In particular, from (6.30), the extremum is reached when

⟨M^{⊥}⟩ = 4 (\frac{Λ^{2N_{c}−3}}{m^{⊥} det M̂} )^{1/2} . (6.42)

The ambiguity in the sign of the square root corresponds to the two possibilities ϵ_{+} = ϵ_{−} = ±1. In the other branch ϵ = −1, we claim that the theory has additional light fields q ∈ F ⊗ ∧^{N_{f}} F coupling to M and that the superpotential near M = 0 behaves as

W ∼ \frac{1}{2μ}M(q, q), (6.43)

where μ ∈ (∧^{N_{f}} F)^{⊗2} is a parameter of mass dimension 1. The R-charges of M, q are −\frac{2}{N_{f}}, 1 + \frac{1}{N_{f}}, respectively. This is consistent with requirement that W has R-

charge 2. Upon mass deformation, we write F = F̂ ⊕ F̂^{⊥} as before. The extremum of W + W_{tree} is reached at q^{⊥} = ±(−mμ)^{1/2}, q̂ = 0, M(q, ·) = 0. The light fields are in M = M̂ ∈ S^{2}F̂. This matches the low energy behavior of N_{f} = N_{c} − 4. The signs of q^{⊥} correspond to the choices of ϵ_{+}, ϵ_{−} in ϵ_{+} = −ϵ_{−} = ±1.

We check the postulate on the existence of q using ’t Hooft’s and global anomaly matching conditions. In the low energy theory, the global symmetry SU(F)×U(1)_{R} is unbroken at M = 0. Also at M = 0, the fields q are massless. The R-charge of ψ_{q} is \frac{1}{N_{f}}. So their contribution to the left column of (6.37) is

infrared: additional contribution from q

𝔰𝔲(F)^{3}  d_{F}
𝔰𝔲(F)^{2}𝔲(1)_{R}  c_{2}(F) · \frac{1}{N_{f}}
𝔲(1)_{R}  N_{f}(\frac{1}{N_{f}})
𝔲(1)_{R}^{3}  N_{f}(\frac{1}{N_{f}})^{3}

(6.44)

It is straightforward to check that for N_{f} = N_{c} − 3, each term in the left column of (6.37) is equal to the sum of the corresponding one in the right column and that in (6.44). When N_{f} = 2, the number of copies of F^{∗} in the ultraviolet theory is odd (N_{c} = 5). In the infrared theory, the representation S^{2}F^{∗} is real, but there is an additional quark q in the representation F. Therefore the global anomalies match.

N_{f} = N_{c} − 2

Classically, a generic expectation value ⟨M⟩ breaks the gauge group to the Abelian group SO(2) ≅ U(1). At the singularities, where ⟨M⟩ is of a lower rank, the unbroken gauge group is larger and hence there are more massless gauge bosons. At the quantum level, since no superpotential is generated, the vacuum degeneracy is not lifted. In fact when N_{f} = N_{c} − 2, U(1)_{X} = U(1)_{R} is anomaly free and M is neutral, whereas a superpotential should have R-charge 2. We expect the classical picture to be valid when ⟨M⟩ is large. There the theory is in the Higgs phase. Modification is needed near the singularities due to strong coupling effects. In particular, the quantum moduli space may have different singularities.

Over a smooth point M in the moduli space, we have a U(1) gauge theory whose scale is set by M. By N = 1 supersymmetry, the effective gauge coupling τ_{eff}(M) is holomorphic in M. However, due to the periodicity in the (effective) θ-angle and the electric-magnetic duality, τ_{eff} needs not be single-valued in M; it is defined up to fractional linear transformations of SL(2, ℤ). At the singularities, τ_{eff} is not defined because there are extra massless fields. Consequently, there are possible non-trivial SL(2, ℤ) monodromies around the singularities. These information can be summarized by a family of elliptic curves over the smooth part M_{smooth} of the moduli space whose fibre over M has modular parameter τ_{eff}(M). The jinvariant of the family is a well-defined holomorphic function on M_{smooth}. In our situation, the j-invariant is a function of U = det M because of the (anomaly free) SU(F) symmetry. We construct a family of elliptic curves over the smooth part of U = (∧^{N_{f}} F^{∗})^{⊗2}; the family over M_{smooth} is then the pull back by the determinant map.

We consider the range where M is of the form (6.1): the large expectation value ⟨M^{⊥}⟩ of rank N_{c}−3 breaks the gauge group to SO(3) ≅ SU(2)/ℤ_{2}, which is further broken to U(1) by M̂. The SO(3) theory in the intermediate stage has a single flavor. It is the N = 2 theory which gives rise to Donaldson and Seiberg-Witten invariants. Using (6.27), the scale Λ̂ is given by

Λ̂^{4} = 4 (\frac{Λ^{2N_{c}−4}}{det⟨M^{⊥}⟩})^{2} . (6.45)

The family of elliptic curve is

y^{2} = (x − Λ̂^{2})(x + Λ̂^{2})(x − u), (6.46)

where the parameter u is related to our theory by u = M̂/4. After making a change of variables x ↦ x/4 det⟨M^{⊥}⟩, y ↦ y/(4 det⟨M^{⊥}⟩)^{3/2}, (6.46) becomes

y^{2} = (x − 8Λ^{2N_{c}−4})(x + 8Λ^{2N_{c}−4})(x − U). (6.47)

This is a good approximation for large U. When U goes around ∞, the monodromy is P T^{−2}. For large U, we have

e^{2πiτ_{eff} (M)} = (\frac{Λ}{U^{1/2(N_{c}−2)}} )^{4N_{c}−8} . (6.48)

The exact family of curves which agrees with (6.48) for large U must be of the form

y^{2} = (x − 8Λ^{2N_{c}−4})(x + 8Λ^{2N_{c}−4})(x − U + αΛ^{2N_{c}−4}). (6.49)

The curve is singular at U = (α ± 8)Λ^{2N_{c}−4}. To determine α, we add a mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = 1. Then the vacuum degeneracy is lifted except at two singularities, which shall correspond to the two branches ϵ = ±1 of the N_{f} = N_{c} − 3 theory. The ϵ = 1 branch has a superpotential

\frac{1}{2}M^{⊥}m^{⊥} = 8\begin{matrix}Λ̂ & 2N_{c}−3 \\ det & M̂ \end{matrix} . (6.50)

So U = 16Λ̂^{2N_{c}−3}/m^{⊥} = 16Λ^{2N_{c}−4}. The ϵ = −1 branch has det M̂ = 0, hence U = 0. Therefore we have a unique solution α = 8. After making another change of variable x ↦ x − 8Λ^{2N_{c}−4}, we get the family of elliptic curves of our theory

y^{2} = x(x − 16Λ^{2N_{c}−4})(x − U). (6.51)

The monodromies when U goes around 16Λ^{2N_{c}−4} and 0 are (T^{2}S)T (T^{2}S)^{−1} and ST^{2}S^{−1}, and the fields that become massless at U = 16Λ^{2N_{c}−4} and U = 0 are dyons and magnectic monopoles, respectively. Unlike the N = 2 gauge theory, the theory here does not have the ℤ_{2} symmetry that relates the two singular components in the modular space. We will see that the physics in these branches are inequivalent.

Taking M around the surface det M = 16Λ^{2N_{c}−4} in M, U goes around 16Λ^{2N_{c}−4} once, so the monodromy is (T^{2}S)T (T^{2}S)^{−1}. From the one-loop βfunction, we conclude that a pair of dyons E^{±} of charges ±1 become massless at U = 16Λ^{2N_{c}−4}. The effective action when U is near 16Λ^{2N_{c}−4} is of the form

W ∼ (U − 16Λ^{2N_{c}−4}) E^{+}· E^{−}, (6.52)

where · is the pairing between fields of opposite U(1) charges. After adding the mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = 1, the M^{⊥} equation of motion implies that E^{+} · E^{−} = −m^{⊥}/2 det M̂. The light fields are in M̂ and the superpotential (6.50) agrees with the ϵ = 1 branch of N_{f} = N_{c} − 3. The U(1) gauge bosons becomes heavy due to the expectation values of E^{±}. But since E^{±} are dyons, this is called oblique confinement.

We check ’t Hooft’s anomaly condition at a generic point M_{0} on the moduli space. The light fermions in the infrared theory are the mesons M, the photino λ_{0} and, if det M = 16Λ^{2N_{c}−4}, the dyons E^{±}. When M_{0} is non-degenerate and is compatible with the Hermitian form on F, the global symmetry is broken to SO(F) × U(1)_{R}. The light fermions in the high and low energy theories transform as

SO(F) ×  U(1)_{R}
ultraviolet ψ_{Q}  F^{∗}  −\frac{N_{c}−2}{N_{f}} = −1
λ  1  1
infrared ψ_{M} S^{2}F^{∗}  1 − 2\frac{N_{c}−2}{N_{f}} = −1
λ_{0}  1  1
(E^{±})  1  0

(6.53)

With or without E^{±}, the non-zero combinations to be matched are

ultraviolet  infrared
𝔰𝔬(F)^{2}𝔲(1)_{R} N_{c}c_{2}(F) · (−1)  c_{2}(S^{2}F) · (−1)
𝔲(1)_{R}  N_{c}N_{f} (−1) + \frac{1}{2}N_{c}(N_{c} − 1) 1  \frac{1}{2}N_{f}(N_{f} + 1) (−1) + 1
𝔲(1)_{R}^{3}  N_{c}N_{f} (−1)^{3} + \frac{1}{2}N_{c}(N_{c} − 1) 1^{3}  \frac{1}{2}N_{f}(N_{f} + 1) (−1)^{3} + 1^{3}

(6.54)

It is easy to check that the two columns match when N_{f} = N_{c} − 2. The global anomalies also match because all the representations of SO(F) are of real type here. We compare this and the Sp(N_{c}) theory with N_{f} = N_{c} + 1. There, the meason M is constrained on the surface (6.17) and there is no massless photon. In the low energies, the fermion ψ_{M} satisfies a traceless conditon. In the present theory, M can be deformed away from the singular branch and hence there is no traceless condition on ψ_{M}.

At the other singularity U = 0, a non-trivial loop around a point M_{0} with rank M_{0} = r is mapped one that goes N_{f} − r times around 0. So the monodromy is ST^{2(N_{f} −r)}S^{−1}. Therefore the number of pairs of monopoles becoming massless at M_{0} is N_{f} − r. This suggest a superpotential

W ∼ \frac{1}{2μ}⟨M, Q̃^{+}· Q̃^{−}⟩, (6.55)

where μ is of mass dimension 1 and Q̃^{±} represent magnetic monopoles that become massless at the singularity. After adding a mass term W = \frac{1}{2}⟨M, m⟩ with rank m = 1, the fermionic content reduces to the ϵ = −1 branch of N_{f} = N_{c} − 3. Again, the expectation values of Q̃^{⊥} lift the U(1) gauge boson. Consequently we see confinement in the original description. The light superfield q in the N_{f} = N_{c} − 3 theory can be constructed by

q = \frac{1}{2\sqrt{m^{⊥}μ}}( ˆQ̃^{+}· (Q̃^{−})^{⊥} − (Q̃^{+})^{⊥}· ˆ Q̃^{−}). (6.56)

We check ’t Hooft’s anomaly matching condition at M = 0. In addition to M, the fermionic component ψ_{Q̃_{±}} of Q̃^{±} and the photino λ_{0} transform as

\begin{aligned}& SU(F) & × & U(1)_{R} \\ ψ_{Q̃_{±}} & F & & 0 \\ λ_{0} & 1 & & 1 \end{aligned} (6.57)

Their contribution to anomaly is

infrared: additional contribution from λ_{0}, Q̃^{±}

𝔰𝔲(F)^{3}  2d_{F}
𝔰𝔲(F)^{2}𝔲(1)_{R}  0
𝔲(1)_{R}  1
𝔲(1)_{R}^{3}  1^{3}

(6.58)

It is easy to check that the left column of (6.37) is the sum of the right column and (6.58). The global anomalies also match.

Solution to Problem 5

The supersymmetric gauge theory with gauge group G = Sp(C) and quarks Q ∈ Hom(F, C) is called the electric theory. When N_{f} ≥ N_{c}+3, we define a dual theory, called the magnetic theory, with gauge group G̃ = Sp(C̃). The color space C̃ is an Hermitian vector space of dimension 2Ñ_{c} = 2(N_{f} − N_{c} − 2) equipped with a compatible symplectic form ω̃_{c}. The flavor space is

F̃ = F^{∗} ⊗ (∧^{2N_{f}} F)^{\frac{1}{2(Ñc+1)}} . (7.1)

In addition to the dual quarks Q̃ ∈ Hom(F̃, C̃), the magnetic theory has elementary mesons M ∈ ∧^{2}F^{∗}. The fields Q̃, M, their fermionic components ψ_{Q̃}, ψ_{M} and the gluino λ̃ of the magnetic theory transform under the group SU(F) × U(1)_{R} according to

\begin{aligned}& SU(F) & × & U(1)_{R} \\ Q̃ & F & & \frac{N_{c}+1}{N_{f}} \\ ψ_{Q̃} & F & & \frac{N_{c}+1}{N_{f}} − 1 \\ M & ∧^{2}F^{∗} & & 2(1 − \frac{N_{c}+1}{N_{f}}) \\ ψ_{M} & ∧^{2}F^{∗} & & 1 − 2\frac{N_{c}+1}{N_{f}} \\ λ̃ & 1 & & 1 \end{aligned} (7.2)

The global symmetry SU(F) × U(1)_{R} is anomaly free in the dual theory; this can be checked by a simple calculation (\frac{N_{c}+1}{N_{f}} − 1) · 2N_{f}c_{2}(C̃) + c_{2}(G̃) = 0. The fields

Q̃ and M couple through a superpotential

W = \frac{1}{2μ}⟨M, Q̃^{∗}ω̃_{c}⟩, (7.3)

where

μ ∈ (∧^{2N_{f}} F^{∗})^{\frac{1}{Ñc+1}} (7.4)

has mass dimension 1. The superpotential has R-charge 2 and is invariant under SU(F). Therefore the dual theory has the same global symmetry SU(F) × U(1)_{R}.

The β-functions of the electric and magnetic theories are given by b_{0} = 3(N_{c} + 1) − N_{f} and b̃_{0} = 3(Ñ_{c} + 1) − N_{f} = 2N_{f} − 3(N_{c} + 1), respectively. The scale Λ̃ of the dual theory should be related to Λ and μ. Because Λ, Λ̃ and μ all have mass dimension 1 and because of (7.4), (5.7) and its magnetic counterpart, the only possible relation among them is

Λ^{3(N_{c}+1)−N_{f}} Λ̃^{3(Ñ_{c}+1)−N_{f}} = C(−1)^{Ñ_{c}+1}μ^{N_{f}} , (7.5)

where C is a constant to be determined later.

We check that duality is an involution. In the double dual theory, the number of colors is N_{f} −Ñ_{c} −2 = N_{c}, and the flavor space is F̃^{∗} ⊗(∧^{2N_{F}} F̃)^{\frac{1}{2(Nc+1)}} ≅ F. There are new mesons M̃ ∈ ∧^{2}F̃^{∗} and quarks Q ∈ Hom(F, C). The total superpotential is

W = \frac{1}{2μ}⟨M, M̃⟩ + \frac{1}{2μ̃}⟨M̃, Q^{∗}ω_{c}⟩, (7.6)

where μ̃ ∈ (∧^{2N_{f}} F̃^{∗})^{\frac{1}{Nc+1}} ≅ (∧^{2N_{f}} F^{∗})^{\frac{1}{Ñc+1}} . We propose that μ̃ = −μ. Then the double dual theory has the same scale Λ as the electric theory, and the equation of motion of M̃ is M = Q^{∗}ω_{c}. Therefore we recover the electric theory.

The electric theory is infrared free when b_{0} ≤ 0, i.e., when N_{f} ≥ 3(N_{c} + 1). (When b_{0} = 0, the two-loop β-function is positive.) It is ultraviolet free when the dual theory is infrared free, i.e., when b̃_{0} ≤ 0 or N_{f} ≤ \frac{3}{2}(N_{c} + 1). We claim that for \frac{3}{2}(N_{c} + 1) < N_{f} < 3(N_{c} + 1), the theory flows to an interacting superconformal field theory in the infrared. The infrared behavior of the Sp(N_{c}) theories with N_{f} ≥ N_{c} + 3 and their duals is summarized in the following table.

electric theory  magnetic theory
N_{c} ≥ 1, N_{f} ≥ 3(N_{c} + 1)  infrared free  strongly coupled

(Ñ_{c} + 3 ≤ N_{f} ≤ \frac{3}{2}(Ñ_{c} + 1))

\frac{3}{2}(N_{c} + 1) < N_{f} < 3(N_{c} + 1)  non-trivial  non-trivial
(\frac{3}{2}(Ñ_{c} + 1) < N_{f} < 3(Ñ_{c} + 1)) infrared fixed point infrared fixed point
N_{c} + 3 ≤ N_{f} ≤ \frac{3}{2}(N_{c} + 1)  strongly coupled  infrared free

(Ñ_{c} ≥ 1, N_{f} ≥ 3(Ñ_{c} + 1))

That the electric and the magnetic theories describe the same low energy physics will be established through the following highly nontrivial consistency tests.

Chiral operators and the unitarity bound

Though their field contents seem rather different, the electric and the magnetic theories have the same gauge invariant operators. We construct the mapping of gauge invariant chiral operators. The composite meson Q^{∗}ω_{c} of the electric theory corresponds to the elementary meson M of the magnetic theory, whereas the composite meson M̃ = Q̃^{∗}ω̃_{c} of the magnetic theory vanishes by the equation of motion of M.

At the non-trivial infrared fixed-points, the dimensions of the chiral fields can be calculated by their R-charges. In fact,

D(M) = \frac{3}{2}R(M) = 3 (1 − \frac{N_{c} + 1}{N_{f}}) . (7.7)

The unitarity bound D(M) ≥ 1 is satisfied if N_{f} ≥ \frac{3}{2}(N_{c} + 1).

’t Hooft anomaly matching condition

Under SU(F) × U(1)_{R}, the fermions ψ_{Q̃}, ψ_{M}, λ̃ transform as (7.2). At M = 0, all matter fields are massless and the whole global symmetry group is unbroken. The non-zero contributions to the ’t Hooft condition are

magnetic

𝔰𝔲(F)^{3}  2Ñ_{c} d_{F} −d_{∧^{2}F}
𝔰𝔲(F)^{2}𝔲(1)_{R}  2Ñ_{c} c_{2}(F) · (\frac{N_{c}+1}{N_{f}} −1) + c_{2}(∧^{2}F) · (1−2\frac{N_{c}+1}{N_{f}})
𝔲(1)_{R}  4Ñ_{c}N_{f} (\frac{N_{c}+1}{N_{f}} −1) + N_{f}(2N_{f} −1) (1−2\frac{N_{c}+1}{N_{f}}) + Ñ_{c}(2Ñ_{c} + 1) 1
𝔲(1)_{R}^{3}  4Ñ_{c}N_{f} (\frac{N_{c}+1}{N_{f}} −1)^{3} + N_{f}(2N_{f} −1) (1−2\frac{N_{c}+1}{N_{f}})^{3} + Ñ_{c}(2Ñ_{c} + 1) 1^{3}

(7.8)

Using d_{∧^{2}F} = 2(N_{f} − 2)d_{F} and c_{2}(∧^{2}F) = 2(N_{f} − 1), the above matches the left column of (6.24).

Deformation along the flat directions

The magnetic theory has the same space of ground states as the electric theory. In the electric theory, the classical moduli space M_{c} consists of M satisfying (5.2) as a classical constraint. Since the superpotential is absent, the quantum moduli M_{q} space is the same as M_{c}; we denote both by M. In the magnetic theory, the equations of motion of M set Q̃ = 0. The same constraint (5.2) are enforced by non-perturbative quantum effects. In fact, if \frac{1}{2} rank M > N_{c}, then the number of flavors in the low energy magnetic theory is N_{f} − \frac{1}{2} rank M < Ñ_{c} + 2. When N_{f} − \frac{1}{2} rank M ≤ Ñ_{c}, a superpotential of the form (5.11) is generated, and consequently there is no vacuum. When N_{f} − \frac{1}{2} rank M = Ñ_{c} + 1, the light components of M̃ = Q̃^{∗}ω̃_{c} are constrained by an equation like (6.17) which implies Q̃ ≠ 0. Therefore the superpotential (7.6) does not have an extremum on the constraint surface and there is no vacuum in this case either. Thus \frac{1}{2} rank M ≤ N_{c} in the low energy magnetic theory.

We show that the procedure of deforming M in M commutes with taking the dual. Suppose in the electric theory, M has an expectation value ⟨M⟩ with \frac{1}{2} rank ⟨M⟩ = r. At low energies, the gauge group is broken to Sp(N_{c} − r), and there are N_{f} − r light flavors. Using the previous notations, the mass scale Λ̂ of the effective theory is given by (6.4). The original magnetic theory has Ñ_{c} colors and N_{f} flavors, and the scale Λ̃ is related to Λ by (7.5). Now the expectation value ⟨M⟩ gives mass to r components of the dual quark Q̃. The low energy magnetic theory has Ñ_{c} colors and N_{f} − r flavors. Its scale ˆ Λ̃ is given by

ˆΛ̃^{3(Ñ_{c}+1)−(N_{f} −r)} = Pf (⟨M^{⊥}⟩/2μ)Λ̃^{3(Ñ_{c}+1)−N_{f}} . (7.9)

It follows from (6.4) and (7.9) that the relation (7.5) is preserved at low energies. Thus the low energy magnetic theory is dual to the low energy electric theory.

Mass deformation

We now check that mass deformation commutes with taking the dual. Suppose a mass term \frac{1}{2}⟨M, m⟩ with \frac{1}{2} rank m = r is added to the electric theory. We follow the previous notations. The low energy theory has quarks Q̂ ∈ Hom(F̂, C). Its scale Λ̂ is given by (6.7). When r ≤ N_{f} − N_{c} − 2, it has a dual description as well. In the

magnetic theory dual to the original electric theory, let ˆ F̃ = F̂ ⊗ (∧^{2N_{f}} F)^{\frac{1}{2(Ñc+1)}} . The equations of motion of the heavy components of M imply that

Q̃^{∗}ω̃_{c} = ( ˆ M̃ ; −2μm^{⊥} ) , (7.10)

where ˆ M̃ ∈ S^{2} ˆF̃ contains the light components. This breaks the Sp(Ñ_{c}) gauge symmetry to Sp(Ñ_{c} − r). Integrating out the heavy components in M, Q̃, we obtain a low energy effective superpotential

Ŵ = \frac{1}{2μ}⟨M̂, ˆ M̃⟩. (7.11)

This is exactly a magnetic theory of Ñ_{c} − r flavors and N_{f} − r quarks. Its mass

scale ˆ Λ̃ is given by the magnetic counterpart of (6.4), i.e.,

Λ̃^{3(Ñ_{c}+1)−N_{f}} = (−μ)^{r}Pf m^{⊥} ˆΛ̃^{3(Ñ_{c}−r+1)−(N_{f} −r)}. (7.12)

From (6.7) and (7.12), we see that the relation (7.5) is preserved at low energies. So the low energy theory of the magnetic theory after mass deformation is dual to that of the electric theory.

If r = N_{f} −N_{c} −2, then in the low energy electric theory, the number of flavors is N_{f} − r = N_{c} + 2. In the low energy magnetic theory, the gauge symmetry is completely broken since Ñ_{c} − r = 0. Therefore, the instanton contribution should be included in the magnetic theory just as in the case N_{f} = N_{c} for the electric theory. Now the number of zero modes of ψ_{Q̃} is 2N_{f} > 2Ñ_{c}; these 2(N_{c} + 2) extra ones are absorbed by the interaction (7.3), generating a term proportional to the (N_{c} + 2)-th power of (M/2μ)_{light} in the effective superpotential. So the instanton contribution is

(7.13) 2^{Ñ_{c}−1} \frac{Λ̃^{3(Ñ_{c}+1)−N_{f}} Pf (M/2μ)_{light}}{Pf (M̃)_{heavy}} = \frac{(−1)^{r}Λ̃^{3(Ñ_{c}+1)−N_{f}} Pf M̂}{2^{N_{c}+3}μ^{N_{f}} Pf m^{⊥}}

= −\frac{C Pf M̂}{2^{N_{c}+3}Pf m^{⊥}Λ^{3(N_{c}+1)−N_{f}}} = −\frac{C Pf M̂}{2^{N_{c}+3}Λ̂^{2N_{c}+1}} ,

where (7.5), (6.7) have been used. This is exactly (6.21) if we choose C = 16.

Solution to Problem 6

Recall that the electric theory has a gauge group G = SO(C) and quarks Q ∈ Hom(F, C). The cases with N_{c} ≥ 4 and N_{f} ≤ N_{c} − 2 have been treated earlier. When N_{f} ≥ N_{c} −1, we define a magnetic theory whose gauge group is G̃ = SO(C̃). Here the space of dual colors C̃ is a complex vector space of dimension Ñ_{c} = N_{f} − N_{c} + 4. It has a linear metric g̃_{c} and a compatible volume form ṽ_{c}. The space of flavors is

F̃ = F^{∗} ⊗ (∧^{N_{f}} F)^{\frac{1}{Ñc−2}} . (7.14)

The matter fields of the magnetic theory are quarks Q̃ ∈ Hom(F̃, C̃) and (elementary) mesons M ∈ ∧^{2}F^{∗}.

We first treat the cases N_{c} ≥ 4, N_{f} ≥ N_{c}. We concentrate on the identity component SU(F)×U(1)_{R} of the global symmetry. The fields Q̃, M, their fermionic components ψ_{Q̃}, ψ_{M} and the gluino λ̃ transform as

\begin{aligned}& SU(F) & × & U(1)_{R} \\ Q̃ & F & & \frac{N_{c}−2}{N_{f}} \\ ψ_{Q̃} & F & & \frac{N_{c}−2}{N_{f}} − 1 \\ M & S^{2}F^{∗} & & 2(1 − \frac{N_{c}−2}{N_{f}}) \\ ψ_{M} & S^{2}F^{∗} & & 1 − 2\frac{N_{c}−2}{N_{f}} \\ λ̃ & 1 & & 1 \end{aligned} (7.15)

It is easy to check that SU(F) × U(1)_{R} is anomaly free in the dual theory. The fields Q̃ and M couple through a superpotential

W = \frac{1}{2μ}⟨M, Q̃^{∗}g̃_{c}⟩, (7.16)

where

μ ∈ (∧^{N_{f}} F^{∗})^{\frac{2}{Ñc−2}} (7.17)

has mass dimension 1. The superpotential has R-charge 2 and is invariant under SU(F).

When N_{f} ≥ N_{c}, the β-functions of the electric and magnetic theories are given by b_{0} = 3(N_{c} − 2) − N_{f} and b̃_{0} = 3(Ñ_{c} − 2) − N_{f} = 2N_{f} − 3(N_{c} − 2), respectively. The scale Λ̃ of the magnetic theory is given by

Λ^{3(N_{c}−2)−N_{f}} Λ̃^{3(Ñ_{c}−2)−N_{f}} = C(−1)Ñ_{c}μ^{N_{f}} , (7.18)

where C is a constant to be determined. It is easy to check that duality is an involution just as the symplectic case. The electric theory is infrared free when N_{f} ≥ 3(N_{c} − 2). It is ultraviolet free when the dual theory is infrared free, i.e., when N_{f} ≤ \frac{3}{2}(N_{c} −2). We claim that when \frac{3}{2}(N_{c} −2) < N_{f} < 3(N_{c} −2), the theory flows to an interacting superconformal field theory in the infrared. The infrared behavior of the SO(N_{c}) theories with N_{f} ≥ N_{c} + 3 and their duals is summarized in the following table.

electric theory  magnetic theory
N_{c} ≥ 4, N_{f} ≥ 3(N_{c} − 2)  infrared free  strongly coupled
(Ñ_{c} ≤ N_{f} ≤ \frac{3}{2}(Ñ_{c} − 2))
\frac{3}{2}(N_{c} − 2) < N_{f} < 3(N_{c} − 2)  non-trivial  non-trivial
(\frac{3}{2}(Ñ_{c} − 2) < N_{f} < 3(Ñ_{c} − 2)) infrared fixed point infrared fixed point
N_{c} + 3 ≤ N_{f} ≤ \frac{3}{2}(N_{c} − 2)  strongly coupled  infrared free
(Ñ_{c} ≥ 4, N_{f} ≥ 3(Ñ_{c} − 2))

Chiral operators and the unitarity bound

We construct the mapping of gauge invariant chiral operators. Obviously, the composite mesons Q^{∗}ω_{c} of the electric theory correspond to the elementary meson M of the magnetic theory. Recall the baryons B = Q^{∗}v_{c} ∈ ∧^{N_{c}}F^{∗} in the electric theory. Since the (fermionic) superfield strength W_{α} is valued in the Lie algebra 𝔰𝔬(C) ≅ ∧^{2}C, we can define additional gauge invariant fields b = Q^{∗}v_{c}(W_{α}, W^{α}, ·) ∈ ∧^{N_{c}−4}F^{∗} and w_{α} = Q^{∗}v_{c}(W_{α}, ·) ∈ ∧^{N_{c}−2}F^{∗}. The field q in the N_{f} = N_{c} − 3 theory is proportional to b.

In the dual theory, the baryons B̃ ∈ ∧Ñ_{c} F̃^{∗} ≅ ∧^{N_{f} −N_{c}+4}F ⊗ (∧^{N_{f}} F^{∗}) ⊗ (∧^{N_{f}} F^{∗})^{\frac{2}{Ñc−2}} and b̃ ∈ ∧^{Ñ_{c}−4}F̃^{∗} ≅ ∧^{N_{f} −N_{c}}F ⊗ (∧^{N_{f}} F^{∗}) ⊗ (∧^{N_{f}} F^{∗})^{− \frac{2}{Ñc−2}} , w̃_{α} ∈ ∧^{Ñ_{c}−2}F̃^{∗} ≅ ∧^{N_{f} −N_{c}+2}F ⊗ (∧^{N_{f}} F^{∗}). It is easy to check that R(B) = R(b̃) = \frac{N_{c}(N_{f} −N_{c}+2)}{N_{f}}, R(b) = R(B̃) = \frac{(N_{c}−2)(N_{f} −N_{c}+4)}{N_{f}} and R(w_{α}) = R(w̃_{α}) = 1 +

\frac{(N_{c}−2)(N_{f} −N_{c}+2)}{N_{f}}. So B and b̃, b and B̃, w_{α} and w̃_{α} have the same quantum numbers, respectively. The mapping of chiral operators in the electric and magnetic theories are given by B ↦ b̃, b ↦ B̃ and w_{α} ↦ w̃_{α}, upon choosing a trivialization of ∧^{N_{f}} F.

At non-trivial infrared fixed-points, the dimension of any chiral field is given by Φ is D(Φ) = \frac{3}{2}R(Φ). Since R(Q) = 1 − \frac{N_{c}−2}{N_{f}} and R(W_{α}) = 1, we have D(B), D(b), D(w_{α}) ≥ 1 and D(M) = 3(1 − \frac{N_{c}−2}{N_{f}}) ≥ 1 if N_{f} ≥ \frac{3}{2}(N_{c} − 2). The unitarity bounds are satisfied.

’t Hooft anomaly matching condition

Under the global symmetry, the light fermions ψ_{Q̃}, ψ_{M}, λ̃ transform according to (7.15). At M = 0 in M_{q}, the whole group is unbroken. The non-zero combinations in the ’t Hooft condition are

magnetic

𝔰𝔲(F)^{3}  Ñ_{c} d_{F} − d_{S^{2}F}
𝔰𝔲(F)^{2}𝔲(1)_{R}  Ñ_{c} c_{2}(F) · (\frac{N_{c}−2}{N_{f}} − 1)+c_{2}(S^{2}F) · (1 − 2\frac{N_{c}−2}{N_{f}})
𝔲(1)_{R}  Ñ_{c}N_{f} (\frac{N_{c}−2}{N_{f}} − 1)+ \frac{1}{2}N_{f}(N_{f} +1) (1 − 2\frac{N_{c}−2}{N_{f}})+ \frac{1}{2} Ñ_{c}(Ñ_{c} − 1) 1
𝔲(1)_{R}^{3}  Ñ_{c}N_{f} (\frac{N_{c}−2}{N_{f}} − 1)^{3}+ \frac{1}{2}N_{f}(N_{f} +1) (1 − 2\frac{N_{c}−2}{N_{f}})^{3}+ \frac{1}{2} Ñ_{c}(Ñ_{c} − 1) 1^{3}

(7.19)

Using d_{S^{2}F} = (N_{f} + 4)d_{F} and c_{2}(S^{2}F) = (N_{f} + 2)c_{2}(F), the above matches the left column of (6.37).

Deformation along the flat directions

First, the dual theory has the same moduli space of vacua. In the electric theory, the condition rank M ≤ N_{c} is a classical constraint. If rank M > N_{c} in the magnetic theory, then the number of light flavors is N_{f} − rank M < Ñ_{c} − 4. So a superpotential of the form (5.22) is generated and consequently the quantum theory has no vacuum. If rank M = N_{c}, then according to (5.12), there are two choices of sign of B = ±(∧^{N_{c}}M)^{1/2} in the electric theory. In the dual theory, the number of light flavors is N_{f} − rank M = Ñ_{c} − 4. So there are two branches. One branch has a superpotential (6.35) with ϵ_{+} = ϵ_{−} and hence does not have a vacuum. The other branch has two vacua, labeled by ϵ_{+} = −ϵ_{−} = ±1. Therefore the magnetic theory also has two vacua for each M with rank M = N_{c}.

We check that flat deformations on the moduli space commutes with taking the dual. If M acquires an expectation value ⟨M⟩ with rank ⟨M⟩ = r, then the low energy electric theory has N_{c} colors and N_{f} − r flavors. In the magnetic theory, the expectation value ⟨M⟩ gives mass to some of the dual quarks. The low energy magnetic theory has Ñ_{c} −r colors and N_{f} −r flavors. This is dual to the low energy electric theory. Further, using (6.25) and the magnetic analog of (6.28), one can show that the relation (7.18) is preserved in the low energy theories.

Mass deformation

If we add a mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = r in the electric theory, then the low energy theory has N_{c} colors and N_{f} − r flavors. Its scale Λ̂ is given by (6.28). When r ≤ N_{f} −N_{c} +1, this low energy theory has a dual theory. The latter

has Ñ_{c} − r colors and N_{f} − r flavors. Its scale ˆ Λ̃ is given by the magnetic analog of (6.25). It is easy to show that the relation (7.18) is preserved in low energy theories and so is the form of the superpotential (7.16) after integrating out the heavy fields.

When r = N_{f} − N_{c} + 1, the number of colors in the low energy magnetic theory is Ñ_{c} − r = 3. Just as the electric theory when N_{f} = N_{c} − 3, we consider an intermediate stage in which the gauge group is broken to SO(4) ≅ (SU(2)_{+} × SU(2)_{−})/ℤ_{2}. When the groups SU(2)_{±} are further broken, there are instanton generated superpotentials. Now the number of zero modes of ψ_{Q̃} is 2N_{f} > 2(Ñ_{c}−3). The extra 2(N_{c} − 1) ones are absorbed by the interaction (7.16), generating a term proportional to the (N_{c} − 1)-th power of (μ^{−1}M)_{light}. Thus the superpotential contains an extra term

(7.20)  4\frac{Λ̃^{3(Ñ_{c}−2)−N_{f}} det(μ^{−1}M)_{light}}{(det M̃)_{heavy}} = \frac{4(−1)^{r}Λ̃^{3(Ñ_{c}−2)−N_{f}} det M̂}{μ^{N_{f}} det m^{⊥}}

= −\frac{4C det M̂}{det m^{⊥}Λ^{3(N_{c}−2)−N_{f}}} = −\frac{4C det M̂}{Λ̂^{2N_{c}−5}} .

Here we have used (7.18) and (6.28). So the total superpotential in the low energy magnetic theory is

ˆW̃ = \frac{1}{2μ}⟨M̂, ˆ Q̃^{∗}g̃_{c}⟩ − \frac{4C det M̂}{Λ̂^{2N_{c}−5}} . (7.21)

Finally, we derive the relation between the scales of the low energy theories. The scale Λ̂ of the low energy electric theory is given by (6.28). The scale ˆ Λ̃ of the low energy magnetic theory is given by the magnetic analog of (6.27), i.e.,

4(Λ̃^{3(Ñ_{c}−2)−N_{f}})^{2} = det(−μm^{⊥})^{2} ˆΛ̃^{6−2(N_{f} −Ñ_{c}+3)}. (7.22)

Using (6.28) and (7.22), the relation (7.18) reduces at low energies to

(Λ̂^{2N_{c}−5})^{2} ˆΛ̃^{6−2(N_{c}−1)} = 4C^{2}μ^{2N_{c}−2}. (7.23)

N_{f} = N_{c} − 1

We consider the theory with N_{f} = N_{c} − 1 and scale Λ. The electric theory is strongly coupled at low energies. From (7.21), the superpotential of the magnetic theory is

W = \frac{1}{2μ}⟨M, Q̃^{∗}g̃_{c}⟩ − \frac{4C det M}{Λ^{2N_{c}−5}} . (7.24)

From (7.23), the scale of the dual theory is given by

(Λ^{2N_{c}−5})^{2}Λ̃^{6−2(N_{c}−1)} = 4C^{2}μ^{2N_{c}−2}. (7.25)

In fact, up to a constant, (7.25) is the only possible relation consistent with (7.17), (5.17), the magnetic analog of (5.18), and the fact that Λ, Λ̃, μ all have mass dimension 1. Under a flat deformation, (7.25) is satisfied by the low energy electric and magnetic theories.

We show that adding the mass term W_{tree} = \frac{1}{2}⟨M, m⟩ with rank m = 1 reduces the theory to N_{f} = N_{c} − 2. The total superpotential is W + W_{tree}. Following

previous notations, we have F̃ = ˆ F̃ ⊕ ˆ F̃^{⊥} with dim ˆ F̃^{⊥} = 1. The dual quarks have the decomposition Q̃ = ( ˆQ̃, Q̃^{⊥}) ∈ Hom( ˆF̃, C̃) ⊕ Hom( ˆF̃^{⊥}, C̃). The M^{⊥} equations of motion imply

(Q̃^{⊥})^{∗}g̃_{c} = \frac{8Cμ det M̂}{Λ^{2N_{c}−5}} − μm^{⊥}. (7.26)

In particular, im( ˆQ̃^{⊥}) is not isotropic in C̃. We write C̃ = ˆ C̃ ⊕ ˆ C̃^{⊥}, where ˆ C̃^{⊥} = im( ˆQ̃^{⊥}). The value of Q̃^{⊥} breaks the dual gauge group SO(C̃) to SO( ˆC̃) ≅ SO(2). The equations of motion imply that M is block-diagonal and that g̃_{c}( ˆQ̃, ˆ Q̃^{⊥}) = 0, i.e., ˆ Q̃ ∈ Hom( ˆF̃, ˆ C̃). Let ˆ C̃ = ˆ C̃^{+} ⊕ ˆ C̃^{−}, where the weights of SO( ˆC̃) in ˆ C̃^{±} are ±1, respectively. We write ˆ Q̃ = ( ˆQ̃^{+}, ˆ Q̃^{−}), where ˆ Q̃^{±} ∈ Hom( ˆF̃, ˆ C̃^{±}). Then the superpotential W + W_{tree} reduces in the low energies to

Ŵ ∼ \frac{1}{2μ}⟨M̂, ˆ Q̃^{+}· ˆ Q̃^{−}⟩, (7.27)

where · is the pairing between ˆ C̃^{+} and ˆ C̃^{−}. This matches (6.55). Due to the instanton effects in symmetry breaking, it is only an approximate result near M̂ = 0. When det M̂ is large, the quarks ˆ Q̃ become heavy. After integrating out the heavy fields, the effective superpotential is

W = \frac{1}{2μ}M^{⊥} (4u − \frac{8Cμ det M̂}{Λ^{2N_{c}−5}} + μm^{⊥}) , (7.28)

where u = g̃_{c}(Q̃^{⊥}, Q̃^{⊥})/4. The effective theory, which is the SO(C̃) ≅ SO(3) gauge theory with a single flavor Q̃^{⊥}, has N = 2 supersymmetry. Its scale Λ̃′ is given by

Λ̃′^{4} = det(μ^{−1}M̂)^{2}Λ̃^{6−2(N_{c}−1)} = (\frac{2Cμ det M̂}{Λ^{2N_{c}−5}} )^{2} . (7.29)

The superpotential (7.28) lifts the vacuum degeneracy of the u-plane except at the two singularities u = ±Λ̃′^{2} = ±8Cμ det M̂/Λ^{2N_{c}−5}. There is still no vacuum with the plus sign. With the minus sign, there is a vacuum if M̂ satisfies

det M̂ = \frac{m^{⊥}Λ^{2N_{c}−5}}{16C} = \frac{Λ̂^{2N_{c}−4}}{16C} . (7.30)

This matches the non-zero singular branch of the N_{f} = N_{c}−2 theory when C = 2^{−8}.

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