Supersymmetric SU(10) Chiral Gauge Theory

• Supersymmetric SU(10) Chiral Gauge Theory is a four-dimensional N=1 theory featuring chiral matter with an antisymmetric tensor and multiple fundamental flavors.
• The theory exhibits infinite dual descriptions and transitions from superconformal fixed points to confinement and supersymmetry-breaking as the flavor number F is varied.
• Key insights include intricate anomaly cancellation, a-maximization determining R-charges, and novel nonchiral dual limits resembling vector-like regimes.

Supersymmetric SU(10) chiral gauge theory is a strongly-coupled four-dimensional $\mathcal{N}=1$ gauge theory characterized by a gauge group $SU(10)$, chiral matter content including an antisymmetric tensor and multiple fundamental flavors, and a cubic superpotential. The system displays a rich phase structure—controlled by the number $F$ of flavors—and provides a setting in which new dualities, confining dynamics, and emergent fixed points are manifest. Key features include the existence of an infinite family of dual descriptions, intricate anomaly cancellation structure, interacting superconformal phases, and transitions to confinement and supersymmetry-breaking as $F$ is varied. All these features are rigorously analyzed in detail and specialized to $SU(10)$ in (Craig et al., 2011).

1. Field Content, Symmetries, and Anomaly Cancellation

The gauge group is $SU(10)$. The chiral superfield content comprises:

• $Q_\alpha^i$, with $\alpha=1,\dots,10$ transforming as the fundamental of $SU(10)$ and $i=1,\dots,F$ in the fundamental of a global $SU(10)$0 flavor symmetry.
• $SU(10)$1, with $SU(10)$2 as another fundamental of $SU(10)$3 and $SU(10)$4 in the fundamental of a distinct global $SU(10)$5 flavor symmetry.
• $SU(10)$6, the antisymmetric tensor of $SU(10)$7, singlet under all flavor groups.

The model admits a non-R $SU(10)$8 global symmetry and an $SU(10)$9 R-symmetry:

• $F$0, $F$1.
• The superpotential is invariant under $F$2 by construction.

Anomaly cancellation for $F$3 requires addition of fundamentals. This can be realized either by introducing two distinct fundamentals $F$4 and $F$5, each transforming under different non-Abelian flavor symmetries ($F$6 and $F$7 for $F$8), or by adding additional fields ($F$9) to satisfy $F$0. In all constructions, both pure $F$1 and mixed gauge-global anomalies vanish once the spectrum is properly supplemented (Craig et al., 2011).

2. Superpotential, Classical Constraints, and Chiral Ring

The fundamental superpotential is

$F$2

where $F$3 are $F$4 indices and $F$5 are flavor indices (any two $F$6's can be coupled to $F$7).

The F-term constraint,

$F$8

eliminates those composite mesons involving the $F$9-coupled $SU(10)$0 flavors from the chiral ring.

3. Conformal Window and a-Maximization

The NSVZ $SU(10)$1-function for $SU(10)$2 with chiral matter and the imposed superpotential is: $SU(10)$3 where $SU(10)$4, $SU(10)$5.

At a conformal fixed point, setting $SU(10)$6 gives a constraint for the $SU(10)$7-charges: $SU(10)$8 with $SU(10)$9 from the cubic superpotential.

For $SU(10)$0, solutions exist only for $SU(10)$1, defining the conformal window; outside this range, the theory runs away in the infrared.

a-maximization yields—for $SU(10)$2:

• $SU(10)$3
• $SU(10)$4
• $SU(10)$5

No gauge-invariant operator in the chiral ring has $SU(10)$6, so the spectrum is interacting and non-trivial through the conformal window (Craig et al., 2011).

4. Infinite Families of Magnetic Duals

For $SU(10)$7 and $SU(10)$8 (even), there exists an infinite family of magnetic duals, labeled by integer $SU(10)$9 (same parity as $Q_\alpha^i$0), with gauge group

$Q_\alpha^i$1

and global symmetry $Q_\alpha^i$2. Their field content includes fundamentals $Q_\alpha^i$3 (in $Q_\alpha^i$4), $Q_\alpha^i$5 (in $Q_\alpha^i$6), $Q_\alpha^i$7 (in $Q_\alpha^i$8), an antisymmetric $Q_\alpha^i$9 of $\alpha=1,\dots,10$0, and various gauge singlets ($\alpha=1,\dots,10$1, $\alpha=1,\dots,10$2, $\alpha=1,\dots,10$3, $\alpha=1,\dots,10$4). The magnetic superpotential, in schematic form, is: $\alpha=1,\dots,10$5 Non-perturbative effects remove $\alpha=1,\dots,10$6 and $\alpha=1,\dots,10$7 from the chiral ring—any vev triggers an Affleck-Dine-Seiberg (ADS) runaway superpotential—ensuring that $\alpha=1,\dots,10$8 does not act on gauge-invariant operators. All these magnetic duals flow to the same infrared fixed point as the electric theory; this duality is established using product-group interpolation and deconfinement techniques (Craig et al., 2011).

5. Phase Structure for Varying Flavor Number

The phase behavior is highly sensitive to $\alpha=1,\dots,10$9:

• $SU(10)$0: Superconformal fixed point as above, with infinite duals.
• $SU(10)$1: Two distinct IR behaviors arise depending on which flavor is integrated out. If a $SU(10)$2-type flavor is given mass and decouples, the theory $SU(10)$3-confines, with mesons and singlet composites and a cubic superpotential. If a $SU(10)$4--$SU(10)$5 pair is integrated out, the theory flows to a new interacting superconformal field theory (SCFT); for $SU(10)$6 even, this is self-dual and admits again an infinite set of duals.
• $SU(10)$7: Confinement with chiral symmetry breaking occurs. The magnetic dual is a collection of singlets subject to a quantum-modified constraint, analogous to $SU(10)$8 in SQCD. The symmetry breaking pattern is $SU(10)$9.
• $i=1,\dots,F$0: No supersymmetric vacuum exists; an ADS-type runaway (nonperturbative instability) is generated and the theory does not possess stable vacua below this flavor number.

6. Nonchiral Duals and Vector-Like Limits

Special choices of dual magnetic gauge group parameters $i=1,\dots,F$1 allow for scenarios where the antisymmetric becomes equivalent to a fundamental, yielding a vector-like dual. For instance, with $i=1,\dots,F$2, the magnetic theory reduces to an $i=1,\dots,F$3 gauge theory with only fundamentals and singlets, entirely devoid of two-index representations. This logic can be extended to other values of $i=1,\dots,F$4 by selecting $i=1,\dots,F$5 so that $i=1,\dots,F$6 is minimized, thereby arriving at duals without antisymmetric representations (Craig et al., 2011).

7. Significance and Context

Supersymmetric $i=1,\dots,F$7 chiral gauge theory with an antisymmetric and fundamental matter exemplifies nontrivial dynamics of chiral gauge theories with $i=1,\dots,F$8 supersymmetry, including rich conformal and confining phases, infinite dual descriptions, and new renormalization group phenomena. The structure revealed in (Craig et al., 2011) generalizes to higher and lower $i=1,\dots,F$9, with implications for the study of dualities, anomaly cancellation, and the construction of consistent chiral models in supersymmetric field theory. The existence of such duals and the nonperturbative phase structure deepen understanding of chiral dynamics and strengthen links between chiral and nonchiral theories in four dimensions.