Five- and Six-Dimensional Supersymmetric Fixed Points, Branches, and Status

Above four dimensions, a Yang–Mills or tensor-branch Lagrangian is generally an infrared description, not a definition of an ultraviolet fixed point. A credible fixed-point claim must combine dimension-appropriate field-theory constraints with independent evidence and an explicit regime of validity. This page applies that standard to three rank-one records: the $E_1$ and $\widetilde E_1$ endpoints reached from pure five-dimensional $SU(2)$ with its two discrete theta choices, and the six-dimensional $A_1$ $(2,0)$ theory.

Required background. Dimensions and reality conditions fixes higher-dimensional spinors. Superconformal algebras identifies the allowed higher-dimensional algebras. Ultraviolet and infrared fixed points supplies the renormalization-group criterion. Scale separation and locality supplies the effective-field-theory cutoff. Anomaly polynomials and inflow supplies the characteristic-class conventions.

Helpful background. Extended supersymmetry gauge dynamics gives the four-dimensional comparison. BPS particles and central charges, gauge instantons, and Bogomolny bounds supply the protected objects used as probes. Controlled interacting higher-dimensional CFTs gives independent conformal diagnostics, while non-invertible topological defects and fusion supplies the generalized-symmetry refinement relevant after compactification.

Five-dimensional rank-one branch data

Five-dimensional $\mathcal N=1$ supersymmetry has eight real supercharges. An interacting superconformal endpoint must realize the exceptional superalgebra $F(4)$, whose bosonic subalgebra is $SO(5,2)\times SU(2)_R$. On a gauge-theory branch, a vector multiplet contains a real scalar $\phi$. The two-derivative Coulomb-branch action is controlled by a prepotential that is at most cubic within each Weyl chamber. For a gauge algebra with roots $\alpha$ and hypermultiplet weights $w$,

$$\mathcal F(\phi) =\frac{1}{2g_0^2}h_{ij}\phi^i\phi^j +\frac{\kappa}{6}d_{ijk}\phi^i\phi^j\phi^k +\frac1{12} \left( \sum_{\alpha}\lvert\alpha\cdot\phi\rvert^3 -\sum_{f,w}\lvert w\cdot\phi+m_f\rvert^3 \right).$$

Here $h_{ij}$ is an invariant metric on the Cartan subalgebra, $d_{ijk}$ is the symmetric cubic invariant when the algebra admits one, and the quantized coefficient $\kappa$ is the five-dimensional Chern–Simons level. The first sum runs over roots $\alpha$; the second runs over every hypermultiplet flavor $f$ and every weight $w$ of its gauge representation, with real mass $m_f$.

The classical coefficient $\kappa$ is quantized whenever the gauge algebra admits the corresponding cubic invariant. The absolute-value cubic terms are one-loop exact: further perturbative corrections would violate the supersymmetric restriction on $\mathcal F$. This makes the Hessian a strong branch-level constraint, not an all-observable definition of the endpoint. Intriligator, Morrison, and Seiberg derive the prepotential and use its positivity to obtain necessary fixed-point conditions in Intriligator, Morrison, and Seiberg 1997, §§2–3.

For pure $SU(2)$ choose the Weyl chamber $\phi\geq0$, normalize the roots as $\alpha=\pm2$, and take $g_0^2$ to be the five-dimensional Yang–Mills coupling in this prepotential convention. Dimensional analysis gives

$$[\phi]=1, \qquad [g_0^{-2}]=1, \qquad [\mathcal F]=3.$$

There is no continuous classical $SU(2)$ cubic invariant. With no hypermultiplets,

$$\mathcal F(\phi) =\frac{1}{2g_0^2}\phi^2+\frac1{12} \left(\lvert2\phi\rvert^3+\lvert-2\phi\rvert^3\right) =\frac{1}{2g_0^2}\phi^2+\frac43\phi^3.$$

The Abelian kinetic coefficient is

$$g_{\rm eff}^{-2}(\phi) =\mathcal F''(\phi) =g_0^{-2}+8\phi.$$

For $g_0^{-2}\geq0$ and $\phi\geq0$, this coefficient is nonnegative and is strictly positive except at the joint limit $g_0^{-2}=\phi=0$. The zero there signals that the Abelian kinetic description has reached the strongly coupled conformal origin; it is not a regular free-field metric. The $W$-boson threshold is $m_W=2\phi$ in this normalization, so the Abelian branch action requires energies well below $m_W$ and every other charged threshold. At finite $g_0^2$, the non-Abelian five-dimensional Yang–Mills description is itself an effective theory with a strong-coupling scale of order $g_0^{-2}$, up to convention-dependent factors. Sending $g_0^{-2}\to0$ removes the explicit dimensionful deformation and makes the origin strongly coupled; it does not extend the weakly coupled Lagrangian through the origin. Absence of a negative branch metric and removal of the explicit scale are necessary tests for a candidate ultraviolet endpoint, not an existence proof.

The two discrete-theta endpoints

Pure $SU(2)$ also has a discrete theta angle

$$\theta_5\in\{0,\pi\},$$

associated with $\pi_4(SU(2))=\mathbb Z_2$. The local prepotential cannot see it, so the identical Hessian above does not identify the two quantum theories. At their proposed endpoints,

$$\begin{array}{c|c|c} \theta_5 & \text{endpoint} & \text{flavor symmetry}\\ \hline 0 & E_1 & SU(2)\\ \pi & \widetilde E_1 & U(1)_I \end{array}$$

where $U(1)_I$ is the instanton-number symmetry visible in the infrared gauge theory. Morrison and Seiberg distinguish $E_1$ from the $U(1)$-flavored $\widetilde E_1$ endpoint in Morrison and Seiberg 1997, §§2–4.

The original string-theory construction of the exceptional sequence, including the $E_1$ fixed point with $SU(2)$ flavor symmetry, appears in Seiberg 1996, pp. 754–759. This construction is independent evidence for the endpoint; it is not a derivation from the low-energy Hessian.

Fix Hermitian generators by $\operatorname{Tr}_{\mathbf 2}(T_aT_b)=\delta_{ab}/2$ and write

$$\mathcal L_{\rm YM} =-\frac{1}{2g_5^2}\operatorname{Tr}_{\mathbf 2} F_{\mu\nu}F^{\mu\nu}, \qquad Q=\frac{1}{8\pi^2}\int_{\mathbb R^4} \operatorname{Tr}_{\mathbf 2}(F\wedge F)\in\mathbb Z.$$

In this normalization an instanton particle in the unbroken non-Abelian description has classical mass

$$M_{\rm inst}=\frac{8\pi^2\lvert Q\rvert}{g_5^2}.$$

In the prepotential convention above this mass is proportional to $g_0^{-2}$. Its becoming light as $g_0^{-2}\to0$ is an intrinsic field-theory signal. More sharply, the $\theta_5=0$ one-instanton sector supplies the additional conserved-current operators needed to enhance $U(1)_I$ to $SU(2)$, while the discrete-theta projection removes that enhancement for $\theta_5=\pi$. Tachikawa derives this current-multiplet test from instanton zero modes in Tachikawa 2015, §3.

There is an important logical qualifier: a one-instanton calculation alone cannot rule out a new current first appearing at higher instanton number. The standard $U(1)_I$ assignment for $\widetilde E_1$ therefore rests on the combined classification and construction evidence above, not on that projection test alone.

Protected indices, instanton-current multiplets, Calabi–Yau degenerations, and five-brane webs are complementary evidence channels. Bergman, Rodríguez-Gómez, and Zafrir compute the discrete-theta-dependent superconformal index and show how its protected spectrum distinguishes the $E_1$ and $\widetilde E_1$ candidates in Bergman, Rodríguez-Gómez, and Zafrir 2014, §§2–4. The geometric and brane constructions are external and require their gravity, string, and decoupled sectors to be controlled. The prepotential alone neither distinguishes $E_1$ from $\widetilde E_1$ nor constructs either conformal theory.

Six-dimensional A₁ tensor-branch data

The six-dimensional $(2,0)$ superconformal algebra is $OSp(8^*|4)$, with bosonic subalgebra $Spin(6,2)\times USp(4)_R$ and $USp(4)_R\simeq Spin(5)_R$. It has sixteen Poincaré supercharges. This page does not use a conventional local, Lorentz-covariant non-Abelian Lagrangian as a definition of the interacting theory. On a generic tensor branch, the available low-energy description instead consists of Abelian tensor multiplets containing a two-form with self-dual three-form field strength and five real scalars.

For the $A_1$ theory, the interacting relative sector has rank one. Its BPS self-dual strings carry the root lattice

$$\Lambda_{\rm string}=A_1, \qquad \Omega=(2).$$

The string tension is proportional to the tensor-branch scalar expectation value and vanishes at the conformal origin. Since $[T_{\rm str}]=2$, the Abelian tensor description requires

$$E\ll \sqrt{T_{\rm str}}$$

and energies below any other massive threshold; saying merely “below the tension” is dimensionally wrong. The discriminant group $\Lambda_{\rm string}^{*}/\Lambda_{\rm string}\simeq\mathbb Z_2$ means that the theory naturally supplies a vector of partition functions until a polarization, or equivalent global completion, is chosen. Tachikawa derives the partition-vector and maximal-isotropic-lattice description in Tachikawa 2014, §§2–4. Compactification must retain this information.

Let $T$ and $N$ denote the tangent and $Spin(5)_R$ bundles, with Pontryagin classes in integral normalization. The anomaly polynomial of one free $(2,0)$ tensor multiplet is

$$I_8^{\rm tens} =\frac1{48} \left[ p_2(N)-p_2(T) +\frac14\bigl(p_1(T)-p_1(N)\bigr)^2 \right].$$

For a simply-laced type $G$ theory,

$$I_8[G] =r_G I_8^{\rm tens} +\frac{d_Gh_G^\vee}{24}p_2(N).$$

Using $r_{A_1}=1$, $d_{A_1}=3$, and $h_{A_1}^\vee=2$ gives

$$I_8[A_1] =I_8^{\rm tens}+\frac14p_2(N).$$

The derivation from tensor-branch anomaly matching and the identification of the string charge pairing with the Cartan matrix are given in Ohmori et al. 2014, §2.3.

On the tensor branch, $Spin(5)_R$ is broken to $Spin(4)_R$. The infrared relative sector is one free tensor multiplet, so its mismatch with the interacting $A_1$ anomaly is

$$\Delta I_8 =I_8[A_1]-I_8^{\rm tens} =\frac14p_2(N).$$

For the restricted $Spin(4)_R$ bundle, $p_2(N)=\chi_4(N)^2$. Choosing the branch orientation so that

$$I_4=\frac12\chi_4(N), \qquad \Omega=(2),$$

gives the explicit Green–Schwarz/Wess–Zumino factorization

$$\Delta I_8 =\frac12\,\Omega I_4^2 =\frac14\chi_4(N)^2 =\frac14p_2(N).$$

Here $dH=I_4$; reversing the branch orientation changes the sign of $I_4$ but not the square. The formula uses the displayed integral characteristic-class convention. Ohmori and collaborators derive the ADE string pairing, the $Spin(4)_R$ Euler-class relation, and the anomaly polynomial in Ohmori et al. 2014, §2.3. Factorization is a stringent consistency condition: the infrared tensor and its Green–Schwarz coupling reproduce the ultraviolet ’t Hooft anomaly. It does not construct the operator algebra at the origin.

Two coincident M5-branes furnish an external realization containing the interacting $A_1$ relative sector plus a decoupled free center-of-mass tensor. Their anomaly bookkeeping makes the subtraction explicit:

$$I_8[2\,\mathrm{M5}]-I_8^{\rm tens} =I_8[A_1] =I_8^{\rm tens}+\frac14p_2(N).$$

Thus the free tensor must be subtracted exactly once before comparing intrinsic $A_1$ anomalies Ohmori et al. 2014, eqs. (2.15)–(2.16). Seiberg and Witten’s analysis of six-dimensional string dynamics provides the early tensionless-string and tensor-branch picture Seiberg and Witten 1996, §§2–3.

Compare the three intrinsic inputs

The following table puts unlike theories into common evidential columns without pretending that their branch descriptions are interchangeable. In particular, the shared five-dimensional Hessian cannot see $\theta_5$, while the six-dimensional record has no Yang–Mills prepotential at all. The seven columns are intentionally kept distinct; on a narrow screen, scroll horizontally rather than reading a construction or an open question as if it were an intrinsic check.

Intrinsic system	Branch calculation	Quantized data	Intrinsic checks	External construction and decoupled sectors	Branch regime	Open field and evidence ceiling
Pure 5D $SU(2)$, $\theta_5=0$	$\mathcal F=\phi^2/(2g_0^2)+4\phi^3/3$ and $\mathcal F''=g_0^{-2}+8\phi\geq0$ for $g_0^{-2},\phi\geq0$, with equality only at their joint zero	$\theta_5=0$ in $\pi_4(SU(2))=\mathbb Z_2$; no continuous cubic $SU(2)$ Chern–Simons parameter	The instanton-current multiplet and protected indices support the $E_1$ endpoint and $U(1)_I\to SU(2)$ enhancement.	Calabi–Yau degeneration, type I′ engineering, or a five-brane web supplies independent support only after its decoupling limit is controlled; the prepotential alone removes no extra sector.	The Abelian branch requires $E\ll m_W=2\phi$ and energies below instanton and other charged thresholds; 5D Yang–Mills is effective below a scale of order $g_0^{-2}$.	A regulator-independent intrinsic construction and general unprotected correlators are not supplied here; the mutually reinforcing evidence is strong, not a theorem of existence.
Pure 5D $SU(2)$, $\theta_5=\pi$	The same local $\mathcal F$ and nonnegative Hessian as the $\theta_5=0$ theory, with the same zero at the joint endpoint	$\theta_5=\pi$ changes the instanton-sector projection while leaving the local prepotential unchanged.	Protected spectra distinguish the $\widetilde E_1$ endpoint, whose flavor symmetry remains $U(1)_I$.	The same controlled geometric or brane constructions provide independent support; their decoupling limits must again be checked explicitly.	The same Abelian and five-dimensional Yang–Mills cutoffs apply as for $E_1$.	The missing one-instanton multiplet alone cannot exclude a current first appearing at higher instanton number; a complete intrinsic construction and unprotected data are not supplied here.
Relative 6D $A_1$ $(2,0)$ theory	One free relative tensor multiplet away from the origin; a primitive self-dual string has $T_{\rm str}\to0$ at the origin.	$A_1$ string lattice with $\Omega=2$ and discriminant group $\mathbb Z_2$; compactification requires a polarization.	$I_8[A_1]=I_8^{\rm tens}+p_2(N)/4$ and $\Delta I_8=\Omega I_4^2/2$ with $I_4=\chi_4(N)/2$ test anomaly matching.	Two coincident M5-branes supply independent support only after subtracting the decoupled free center-of-mass tensor.	The tensor-branch EFT requires $E\ll\sqrt{T_{\rm str}}$ and energies below every other massive threshold.	Lattice integrality, anomaly matching, and M-theory give strong evidence, but no complete construction of every intrinsic or unprotected observable is supplied here.

The machine-readable numerical, convention, source, and provenance data (JSON) reproduce this comparison and its stated limits.

The six-dimensional (1,0) generalization

Six-dimensional $(1,0)$ theories have eight supercharges and can contain tensor, vector, and hypermultiplets. On a tensor branch, a scalar may set an inverse gauge coupling. A consistent branch description organizes the anomaly polynomial as

$$I_8^{\rm total} =I_8^{\rm one-loop}+I_8^{\rm GS}, \qquad I_8^{\rm GS}=\frac12\Omega_{ij}I_4^i I_4^j.$$

Irreducible gauge-anomaly terms must vanish, and the reducible gauge-dependent part of $I_8^{\rm one-loop}$ must be canceled by the Green–Schwarz term in the chosen sign convention. Surviving terms involving only background tangent, R-symmetry, and flavor bundles are global ’t Hooft anomalies to be matched, not canceled. The lattice $\Omega$ must be compatible with integral self-dual-string charges. One should additionally record branch dimensions, BPS strings, flavor symmetries, decoupled free sectors, and whether the construction is intrinsic, geometric, brane-based, or conjectural.

These tests can rule out a candidate. Passing them does not prove that a unitary interacting fixed point exists. Anomaly matching is necessary because anomalies cannot change along the flow; it is not sufficient because positivity, locality at the fixed point, the spectrum, and operator products are not encoded completely in $I_8$.

What compactification may infer

In standard physics usage, $E_1$, $\widetilde E_1$, and the $A_1$ $(2,0)$ theory are well-established interacting SCFTs supported by several mutually reinforcing constructions and protected checks. That is stronger than a lone necessary-condition test, but it is not a theorem-level, regulator-independent construction of every observable.

Compactification may transport only the stated theory type, charge or defect lattice, anomaly convention, branch scale, discrete theta datum, decoupled sectors, and the strength and limits of their supporting evidence. A lower-dimensional duality match cannot be fed back as an independent proof that the higher-dimensional input exists. String and brane constructions belong to the M2/M5 interface in Volume 15; theorem-level existence questions belong to the dated mathematical-QFT handoff in Volume 16.

Exercises

1. Differentiate the prepotential. Starting from the pure $SU(2)$ expression, compute $\mathcal F'$ and $\mathcal F''$.

Solution

$\mathcal F'=g_0^{-2}\phi+4\phi^2$ and $\mathcal F''=g_0^{-2}+8\phi$. The latter is nonnegative for $\phi\geq0$ and $g_0^{-2}\geq0$, and it vanishes only when both are zero. Everywhere else in that closed chamber it is strictly positive.

2. Check the six-dimensional cutoff. A self-dual string has tension $T_{\rm str}$. What energy scale can be formed from it, and what happens at the conformal origin?

Solution

In six dimensions a string tension has mass dimension two, so the associated mass scale is $\sqrt{T_{\rm str}}$. The Abelian tensor-branch description requires $E\ll\sqrt{T_{\rm str}}$. At the conformal origin $T_{\rm str}\to0$, this window closes rather than defining the interacting theory.

3. Reproduce the $A_1$ mismatch. Use $\Omega=2$ and $I_4=\chi_4(N)/2$ to compute the Green–Schwarz contribution.

Solution

Direct substitution gives

$$\frac12\Omega I_4^2 =\frac12(2)\left(\frac{\chi_4(N)}2\right)^2 =\frac14\chi_4(N)^2 =\frac14p_2(N),$$

which is exactly $I_8[A_1]-I_8^{\rm tens}$ in the stated convention.

4. Distinguish the two five-dimensional endpoints. Why does equality of $\mathcal F''$ for $\theta_5=0$ and $\theta_5=\pi$ not imply that $E_1=\widetilde E_1$?

Solution

The Hessian is local branch data and cannot detect the $\mathbb Z_2$ theta angle. The two theories differ in their instanton-sector projection and flavor-current multiplets: $E_1$ has $SU(2)$ enhancement, whereas $\widetilde E_1$ retains only $U(1)_I$. Complete theory data, not one effective coupling, determine the endpoint.

References

• Bergman, Oren, Diego Rodríguez-Gómez, and Gabi Zafrir. “Discrete θ and the 5d Superconformal Index.” Journal of High Energy Physics 01 (2014): 079. doi:10.1007/JHEP01(2014)079.
• Intriligator, Kenneth, David R. Morrison, and Nathan Seiberg. “Five-Dimensional Supersymmetric Gauge Theories and Degenerations of Calabi–Yau Spaces.” Nuclear Physics B 497 (1997): 56–100. doi:10.1016/S0550-3213(97)00279-4.
• Morrison, David R., and Nathan Seiberg. “Extremal Transitions and Five-Dimensional Supersymmetric Field Theories.” Nuclear Physics B 483 (1997): 229–247. doi:10.1016/S0550-3213(96)00592-5.
• Ohmori, Kantaro, Hiroyuki Shimizu, Yuji Tachikawa, and Kazuya Yonekura. “Anomaly Polynomial of General 6D SCFTs.” Progress of Theoretical and Experimental Physics 2014 (2014): 103B07. doi:10.1093/ptep/ptu140.
• Seiberg, Nathan. “Five-Dimensional SUSY Field Theories, Non-Trivial Fixed Points and String Dynamics.” Physics Letters B 388 (1996): 753–760. doi:10.1016/S0370-2693(96)01215-4.
• Seiberg, Nathan, and Edward Witten. “Comments on String Dynamics in Six Dimensions.” Nuclear Physics B 471 (1996): 121–134. doi:10.1016/0550-3213(96)00189-7.
• Tachikawa, Yuji. “On the 6d Origin of Discrete Additional Data of 4d Gauge Theories.” Journal of High Energy Physics 05 (2014): 020. doi:10.1007/JHEP05(2014)020.
• Tachikawa, Yuji. “Instanton Operators and Symmetry Enhancement in 5d Supersymmetric Gauge Theories.” Progress of Theoretical and Experimental Physics 2015 (2015): 043B06. doi:10.1093/ptep/ptv040.