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SU(Nf) cubic-anomaly check for proposed SQCD dual data

Physical scope

Algebraic anomaly-coefficient check for the proposed four-dimensional N=1 electric/magnetic SQCD field data in the conformal window; it is not evidence that the two continuum theories are equivalent.

Example

Let integers $N_{c} \geq 2$ and $N_{f}$ satisfy $\frac{3}{2} N_{c} < N_{f} < 3 N_{c}$ , and put ${\tilde{N}}_{c} = N_{f} - N_{c}$ . Consider the following proposed four-dimensional $N = 1$ SQCD field data, in natural units. In the electric $S U (N_{c})$ theory, there are $N_{f}$ quark chiral multiplets $Q$ in the fundamental of the flavor group $S U (N_{f})_{L}$ , with $N_{c}$ color copies; the antiquark multiplets are singlets under $S U (N_{f})_{L}$ . In the proposed magnetic $S U ({\tilde{N}}_{c})$ theory, the quark multiplets $q$ are in the antifundamental of $S U (N_{f})_{L}$ , with ${\tilde{N}}_{c}$ color copies, and the meson multiplet $M$ transforms in $(N_{f}, \overline{N_{f}})$ under $S U (N_{f})_{L} \times S U (N_{f})_{R}$ , giving $N_{f}$ copies of the fundamental of $S U (N_{f})_{L}$ . The magnetic antiquarks and both gauginos are $S U (N_{f})_{L}$ singlets. Normalize the cubic-anomaly coefficient of a left-handed Weyl fermion to $+ 1$ in $N_{f}$ and $- 1$ in $\overline{N_{f}}$ , and add coefficients over independent copies. Then the electric and proposed magnetic $S U (N_{f})_{L}^{3}$ coefficients are

$A_{e l e c t r i c} = N_{c}, A_{m a g n e t i c} = - (N_{f} - N_{c}) + N_{f} = N_{c} .$

This equality is one anomaly-matching consistency check of the proposed field data; it is not a proof of Seiberg duality or of equality of the infrared continuum theories.

Example references

Seiberg, §3, equations (3.1)–(3.4).

Tong, §6.5.1, printed pp. 198–200.

Facts & Assumptions

Given: The electric and proposed magnetic field representations, ${\tilde{N}}_{c} = N_{f} - N_{c}$ , and the anomaly normalization stated in the Example.

[A1]

The magnetic field data are a proposed infrared-dual description; this example assumes the field representations for the coefficient calculation and does not assume or prove equivalence of the electric and magnetic theories.

[A2]

For a left-handed Weyl fermion, the chosen cubic-anomaly coefficient is $+ 1$ in the fundamental of $S U (N_{f})_{L}$ , $- 1$ in its antifundamental, and zero in a singlet; independent copies contribute additively.

Verification

Given: [A1]–[A2] and the explicit electric/magnetic matter multiplicities in the Example.

1.1A1A2givenalgebra

In the electric theory, each of the $N_{c}$ color components of $Q$ is a left-handed Weyl fermion in $N_{f}$ of $S U (N_{f})_{L}$ , so these fields contribute $N_{c}$ . The antiquarks and gaugino are $S U (N_{f})_{L}$ singlets and contribute zero. Hence $A_{e l e c t r i c} = N_{c}$ .

1.2A1A2givenalgebra

In the magnetic theory, the ${\tilde{N}}_{c}$ color copies of $q$ are antifundamentals and contribute $- {\tilde{N}}_{c}$ . For each of the $N_{f}$ right-flavor components of $M$ , its left-flavor index is fundamental, so the meson fermions contribute $+ N_{f}$ . The magnetic antiquarks and gaugino are left-flavor singlets. Therefore $A_{m a g n e t i c} = - {\tilde{N}}_{c} + N_{f}$ .

2.1step 1.1step 1.2givenalgebra

Substituting ${\tilde{N}}_{c} = N_{f} - N_{c}$ gives $A_{m a g n e t i c} = - (N_{f} - N_{c}) + N_{f} = N_{c} = A_{e l e c t r i c}$ . This verifies the stated cubic-anomaly match.

3.1A1step 2.1∎

The calculation checks only this specified global-symmetry coefficient under the proposed field assignments. It supplies no map of all observables, no equivalence of renormalization-group flows, and no continuum construction; therefore it does not prove the proposed duality.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.