$\mathcal{N}=4$ supersymmetry on a space-time lattice

Maximally supersymmetric Yang–Mills theory in four dimensions can be formulated on a space-time lattice while exactly preserving a single supersymmetry. Here we explore in detail this lattice theory, paying particular attention to its strongly coupled regime. Targeting a theory with gauge group $\mathrm{SU}(N)$, the lattice formulation is naturally described in terms of gauge group $\mathrm{U}(N)$. Although the U(1) degrees of freedom decouple in the continuum limit we show that these degrees of freedom lead to unwanted lattice artifacts at strong coupling. We demonstrate that these lattice artifacts can be removed, leaving behind a lattice formulation based on the $\mathrm{SU}(N)$ gauge group with the expected apparently conformal behavior at both weak and strong coupling.

Figure 1

The absolute value of the Polyakov loop plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$ for fixed $\mu =1$ indicates that the lattice theory defined by Eqs. (14) and (15) transitions into a confining phase at strong coupling, characterized by $⟨|PL|⟩\ll 1$.

Figure 2

The real part of the plaquette determinant is a gauge-invariant quantity associated with the U(1) sector. As a function of the ’t Hooft coupling ${\lambda }_{\text{lat}}$ for fixed $\mu =1$, it shows the same confinement transition at strong coupling as the Polyakov loop in Fig. 1, suggesting that this transition is related to the U(1) sector.

Figure 3

The density of monopole world lines in the U(1) sector plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$ for fixed $\mu =1$ indicates that the strong-coupling confinement transition observed in Figs. 1 and 2 is caused by the compact U(1) degrees of freedom in the $\mathrm{U}(N)$ gauge group.

Figure 4

The real part of the plaquette determinant vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$ on $6^4$ lattices with a variety of $\kappa$ in Eq. (32) and fixed $\mu =1$. As $\kappa$ increases the confinement transition associated with the U(1) sector moves to larger coupling, and disappears entirely for $\kappa \ge 0.5$. Lines connect points with the same $\kappa$ to guide the eye.

Figure 5

The absolute value of the Polyakov loop with nonzero $\kappa >0.5$ on $8^3\times 24$ lattices. The six data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$. Despite the long temporal extent of lattice, $N_t=24$, the results remain $\mathcal{O}(1)$ for all investigated couplings, as expected for a deconfined system. No significant dependence on $\mu$ or $\kappa$ is visible. At stronger couplings ${\lambda }_{\text{lat}}\gtrsim 3$ larger fluctuations in the nonunitarized link fields cause $⟨|PL|⟩$ to increase, and require that we use larger values of $\mu$ to maintain stability. Lines connect points with fixed $(\mu ,\kappa )$ to guide the eye.

Figure 6

The real part of the plaquette determinant with nonzero $\kappa >0.5$ on $8^3\times 24$ lattices. The six data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$. The results remain $\mathcal{O}(1)$ for all investigated couplings, with no visible dependence on $\mu$. As expected, larger $\kappa$ keeps $⟨\mathrm{Re}\det \mathcal{P}⟩$ closer to unity, especially at stronger couplings. Lines connect points with fixed $(\mu ,\kappa )$ to guide the eye.

Figure 7

Deviations of the bosonic action $⟨s_B⟩$ from its exact supersymmetric value $9N^2/2$ serve as a measure of $\mathcal{Q}$ supersymmetry breaking due to non-zero $\mu$ and $\kappa$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$. The results indicate mild $\mathcal{O}(10\%)$ supersymmetry breaking in these $8^3\times 24$ lattice ensembles that are the focus of our current investigations. As expected, larger $\mu$ and $\kappa$ each increase supersymmetry breaking, though $s_B$ is more sensitive to $\kappa$ than to $\mu$. The deviations also grow with the coupling, and approach zero in the free-field limit ${\lambda }_{\text{lat}}\to 0$. Lines connect points with fixed $(\mu ,\kappa )$ to guide the eye.

Figure 8

Violations of the Ward identity $⟨\mathcal{Q}\mathcal{O}⟩=0$ as defined by Eq. (36) serve as a measure of $\mathcal{Q}$ supersymmetry breaking due to nonzero $\mu$ and $\kappa$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$. The results indicate mild $\mathcal{O}(10\%)$ supersymmetry breaking in these $8^3\times 24$ lattice ensembles that are the focus of our current investigations. As expected, larger $\mu$ and $\kappa$ each increase supersymmetry breaking, though $⟨\mathcal{Q}\mathcal{O}⟩$ is more sensitive to $\kappa$ than to $\mu$. The Ward identity violations also grow with the coupling, and approach zero in the free-field limit ${\lambda }_{\text{lat}}\to 0$. Lines connect points with fixed $(\mu ,\kappa )$ to guide the eye.

Figure 9

Deviations of the bosonic action $⟨s_B⟩$ from its exact supersymmetric value $9N^2/2$, as in Fig. 7. Nine $4^4$ data sets with $0.2\le \mu \le 1$ are plotted vs. $\kappa$ for fixed ${\lambda }_{\text{lat}}=1$. While the results again show little dependence on $\mu$, the deviations steadily grow as $\mu$ increases, as expected. The lines are cubic $\kappa \to 0$ extrapolations. A subsequent quadratic $\mu \to 0$ extrapolation of their intercepts produces $\underset{(\mu ,\kappa )\to (0,0)}{\lim }⟨s_B⟩=17.956(37)$, consistent with the restoration of the $\mathcal{Q}$ supersymmetry up to minor effects attributable to the small lattice volume.

Figure 10

Violations of the Ward identity $⟨\mathcal{Q}\mathcal{O}⟩=0$ as defined by Eq. (36), as in Fig. 8. Nine $4^4$ data sets with $0.2\le \mu \le 1$ are plotted vs. $\kappa$ for fixed ${\lambda }_{\text{lat}}=1$. Although the results are somewhat noisy, the Ward identity violations grow as $\mu$ increases, as expected. The lines are cubic $\kappa \to 0$ extrapolations. A subsequent quadratic $\mu \to 0$ extrapolation of their intercepts produces $\underset{(\mu ,\kappa )\to (0,0)}{\lim }⟨\mathcal{Q}\mathcal{O}⟩=-0.011(10)$, consistent with the restoration of the $\mathcal{Q}$ supersymmetry up to minor effects attributable to the small lattice volume.

Figure 11

Violations of $R_a$ symmetry in the plaquette, based on Eq. (40). Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the ’t Hooft coupling ${\lambda }_{\text{lat}}$. The results indicate mild $\mathcal{O}(10\%)$ R symmetry breaking in these $8^3\times 24$ lattice ensembles that are the focus of our current investigations. No dependence on $\mu$ is visible, while larger $\kappa$ reduces R symmetry breaking, which may hint at a potential connection to the U(1) sector. The $R_a$ violations also grow with the coupling, and approach zero in the free-field limit ${\lambda }_{\text{lat}}\to 0$. Lines connect points with fixed $(\mu ,\kappa )$ to guide the eye.

Figure 12

Semi-log plot of $1-⟨\cos \alpha ⟩$ vs. lattice volume, where $\alpha$ is the phase of the Pfaffian. All points are for the U(2) gauge group with fixed $({\lambda }_{\text{lat}},\mu ,\kappa )=(1,1,1)$. The inset zooms in on the four largest-volume results, for $3^3\times 8$, $4^4$, $4^3\times 5$ and $4^3\times 6$. We find that the phase of the Pfaffian is small, $1-⟨\cos \alpha ⟩\lesssim 0.003$, and does not grow on the larger volumes with $L>3$.

Figure 13

Deviations of the bosonic action $⟨s_B⟩$ from its exact supersymmetric value $9N^2/2$, as in Fig. 7. Results from $4^4$ ensembles with $({\lambda }_{\text{lat}},\mu ,\kappa )=(1,1,1)$ are plotted vs. $1/N^2$ for gauge groups U(2), U(3) and U(4). The deviations clearly scale $\propto 1/N^2$. The red line is a linear fit constrained to vanish in the large-$N$ limit.

Figure 14

Violations of the Ward identity $⟨\mathcal{Q}\mathcal{O}⟩=0$ as defined by Eq. (36), as in Fig. 8. Results from $4^4$ ensembles with $({\lambda }_{\text{lat}},\mu ,\kappa )=(1,1,1)$ are plotted vs. $1/N^2$ for gauge groups U(2), U(3) and U(4). The Ward identity violations clearly scale $\propto 1/N^2$. The red line is a linear fit constrained to vanish in the large-$N$ limit.

Figure 15

To extract the static potential $V(r)$ we must determine the appropriate range $t_{\min }\le t<N_t/2$ over which to fit the Coulomb gauge fixed Wilson loops to $W(r,t)=\exp (-V(r)t)$. We do this by searching for a plateau in each $V(r)$ plotted vs. $t_{\min }$. These representative results for the $8^3\times 24$ lattice ensemble with $({\lambda }_{\text{lat}},\mu ,\kappa )=(0.5,0.2,0.6)$ show plateaus appearing to begin at $t_{\min }\approx 5$ for the four smallest $r=\sqrt{3/4}$, 1, $\sqrt{2}$ and $\sqrt{11/4}$ on the $A_4^{*}$ lattice. Other data sets are similar.

Figure 16

Representative results for the static potential $V(r)$ vs. $r$ from the usual Wilson loops, for the $({\lambda }_{\text{lat}},\mu ,\kappa )=(0.5,0.2,0.6)$ ensemble. The curves are fits of $V(r)$ to the Coulombic form of Eq. (49), for different $t_{\min }=5$ and 7 in the earlier fits to the Coulomb gauge fixed $W(r,t)$.

Figure 17

Representative results for the static potential $V_D(r)$ vs. $r$ from the determinant-divided Wilson loops, for the $({\lambda }_{\text{lat}},\mu ,\kappa )=(0.5,0.2,0.6)$ ensemble. The curves are fits of $V_D(r)$ to the Coulombic form of Eq. (49), for different $t_{\min }=5$ and 7 in the earlier fits to the Coulomb gauge fixed $W_D(r,t)$.

Figure 18

Representative results for the static potential $V_{\text{pol}}(r)$ vs. $r$ from the polar-projected Wilson loops, for the $({\lambda }_{\text{lat}},\mu ,\kappa )=(0.5,0.2,0.6)$ ensemble. The curves are fits of $V_{\text{pol}}(r)$ to the Coulombic form of Eq. (49), for different $t_{\min }=5$ and 7 in the earlier fits to the Coulomb gauge fixed $W_{\text{pol}}(r,t)$.

Figure 19

Representative results for the static potential Coulomb coefficients $C$ vs. the $t_{\min }$ used in the fits to Coulomb gauge fixed Wilson loop data, for the $({\lambda }_{\text{lat}},\mu ,\kappa )=(0.5,0.2,0.6)$ ensemble. The three $C$, $C_D$ and $C_{\text{pol}}$ correspond to the usual Wilson loops, determinant-divided loops and polar-projected loops, respectively. Plateaus in the Coulomb coefficients extend over a wide range of $t_{\min }$, and the other data sets behave similarly.

Figure 20

Static potential Coulomb coefficients $C$ for the usual Wilson loops, based on fits with $t_{\min }=6$ and $r\le 2.6$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the continuum ’t Hooft coupling ${\lambda }_{\text{lat}}/\sqrt{5}$. No significant dependence on $\mu$ or $\kappa$ is visible. The line is the leading-order perturbative prediction from Eq. (51), not a fit. The results agree quite well with this perturbative prediction, and show no sign of the $C\propto \sqrt{\lambda }$ scaling predicted at strong coupling in the large-$N$ limit.

Figure 21

Static potential Coulomb coefficients $C_D$ for the determinant-divided Wilson loops, based on fits with $t_{\min }=6$ and $r\le 2.6$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the continuum ’t Hooft coupling ${\lambda }_{\text{lat}}/\sqrt{5}$. No significant dependence on $\mu$ or $\kappa$ is visible. The line is the leading-order perturbative prediction from Eq. (51), not a fit. The results agree quite well with this perturbative prediction, and show no sign of the $C_D\propto \sqrt{\lambda }$ scaling predicted at strong coupling in the large-$N$ limit. The axes cover the same range as those in Fig. 20.

Figure 22

Static potential Coulomb coefficients $C_{\text{pol}}$ for the polar-projected Wilson loops, based on fits with $t_{\min }=6$ and $r\le 2.6$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the continuum ’t Hooft coupling ${\lambda }_{\text{lat}}/\sqrt{5}$. No significant dependence on $\mu$ or $\kappa$ is visible. The line is the leading-order perturbative prediction from Eq. (51), not a fit. The results agree quite well with this perturbative prediction, and show no sign of the $C_{\text{pol}}\propto \sqrt{\lambda }$ scaling predicted at strong coupling in the large-$N$ limit. The axes cover the same range as those in Fig. 20.

Figure 23

Ratios $C_D/C$ of static potential Coulomb coefficients for the determinant-divided Wilson loops relative to those for the usual loops, based on fits with $t_{\min }=6$ and $r\le 2.6$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the continuum ’t Hooft coupling ${\lambda }_{\text{lat}}/\sqrt{5}$. No significant dependence on $\mu$ or $\kappa$ is visible. The results, although noisy, are consistent with the expected ratio of $3/4$ from Eq. (51).

Figure 24

Ratios $C_{\text{pol}}/C$ of static potential Coulomb coefficients for the polar-projected Wilson loops relative to those for the usual loops, based on fits with $t_{\min }=6$ and $r\le 2.6$. Six $8^3\times 24$ data sets with $(\mu ,\kappa )=(0.2,0.6)$, (0.2, 0.8), (0.4, 0.6), (0.4, 0.8), (0.8, 0.6) and (0.8, 0.8) are plotted vs. the continuum ’t Hooft coupling ${\lambda }_{\text{lat}}/\sqrt{5}$. No significant dependence on $\mu$ or $\kappa$ is visible. The results, although noisy, are consistent with the expected ratio of $1/2$ from Eq. (51).