---|---|---|---|---|---|---|---|---|---|---
MW Disk
Barker78 | 32 | 3400-7400 | 0.106 |  $\cdots$ | 3.63E-05 | 2.63E-04 | 5.02E-05 | 2.43E-06 |  $\cdots$ |  $\cdots$
Aller83+87 | 72 | 3400-7800 | 0.107 | 7.28E-04 | 1.11E-04 | 4.07E-04 | 8.32E-05 | 8.91E-06 | 1.51E-07 | 2.29E-06
Kingsburgh94 | 58 | 3150-7400; | 0.112 | 3.98E-04 | 1.74E-04 | 4.58E-04 | 1.10E-04 | 7.38E-06 |  $\cdots$ | 1.85E-06
Perinotto04ccThe abundances in this study were computed by the authors but were based either partially or entirely upon spectrophotometric data from other researchers. | 114 | Optical+UV | 0.106 |  $\cdots$ | 1.34E-04 | 4.27E-04 | 1.01E-04 | 4.96E-06 |  $\cdots$ | 1.59E-06
Stanghellini06ccThe abundances in this study were computed by the authors but were based either partially or entirely upon spectrophotometric data from other researchers. | 78 | Optical | 0.114 |  $\cdots$ | 1.50E-04 | 3.25E-04 | 8.20E-05 |  $\cdots$ |  $\cdots$ | 8.25E-07
Girard07 | 37 | 3700-7500 | 0.106 |  $\cdots$ | 1.32E-04 | 4.90E-04 | 9.34E-05 | 8.13E-06 | 1.03E-07 | 2.32E-06
Pottasch10ccThe abundances in this study were computed by the authors but were based either partially or entirely upon spectrophotometric data from other researchers. | 32 | Op+IR+UV | 0.107 | 4.03E-04 | 1.50E-04 | 4.15E-04 | 1.31E-04 | 7.20E-06 |  $\cdots$ | 2.70E-06
Maciel17 | 230 | 3600-7900 | 0.112 | 5.37E-04 | 1.51E-04 | 4.47E-04 | 8.91E-05 | 7.41E-06 |  $\cdots$ | 2.51E-06
KH20 | 152 | Optical+UV | 0.121 | 7.71E-04 | 1.67E-04 | 3.95E-04 | 1.02E-04 | 4.59E-06 | 8.91E-08 | 2.16E-06
MW Bulge
Aller87 | 15 | 3400-7400 | 0.107 |  $\cdots$ | 2.69E-04 | 4.27E-04 | 7.94E-05 | 7.93E-06 | 4.02E-07 | 3.16E-06
Webster88 | 33 | 3700-7350 | 0.120 |  $\cdots$ | 1.30E-04 | 5.45E-04 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Ratag97 | 20 | 3500-7600 | 0.106 |  $\cdots$ | 1.32E-04 | 4.42E-04 | 6.76E-05 | 6.31E-06 | 2.30E-06 | 3.13E-07
Stasińska98ccThe abundances in this study were computed by the authors but were based either partially or entirely upon spectrophotometric data from other researchers. | 85 | Optical | 0.100 |  $\cdots$ | 1.60E-04 | 4.00E-04 | 9.60E-05 |  $\cdots$ |  $\cdots$ |  $\cdots$
Exter04 | 32 | 3600-7400 | 0.112 |  $\cdots$ | 2.47E-04 | 4.43E-04 | 8.41E-05 | 7.06E-06 |  $\cdots$ | 3.43E-06
Wang07 | 20 | Op+IR+UV | 0.115 | 2.99E-04 | 1.78E-04 | 5.25E-4 | 1.24E-4 | 1.18E-05 | 2.14E-07 | 1.82E-06
Chiappini09 | 167 | 3500-7400 | 0.122 |  $\cdots$ | 1.22E-04 | 3.37E-04 | 7.78E-05 | 6.28E-06 |  $\cdots$ | 2.00E-06
Cavichia10+17 | 37 | 3600-7900 | 0.110 |  $\cdots$ | 1.05E-04 | 2.29E-04 | 5.96E-05 | 3.80E-06 |  $\cdots$ | 1.48E-06
Solar |  $\cdots$ |  $\cdots$ | 0.085 | 2.69E-04 | 6.76E-05 | 4.90E-04 | 8.51E-05 | 1.32E-05 | 3.16E-07 | 2.51E-06

Table 2: MW Halo Planetary Nebula Abundance Studies PN: Common Name (PNG) | SourceaaReferences: Adams84=Adams et al. (1984); Asplund09=Asplund et al. (2009); Barker80=Barker (1980); Clegg87=Clegg, Peimbert, & Torres-Peimbert (1987); Dinerstein03=Dinerstein et al. (2003); Garnett9=Garnett & Dinerstein (1989); Hawley78=Hawley & Miller (1989); Howard97=Howard, Henry, & McCartney (1997); Jacoby97=Jacoby et al. (1997); Jacoby06=Jacoby et al. (2006); Kholtygin98=Kholtygin (1998); Kniazev08=Kniazev et al. (2008); KH20=Kwitter & Henry (2020); Milanova09=Milanova & Kholtygin (2009); Otsuka09=Otsuka et al. (2009); Otsuka10=Otsuka et al. (2010); Otsuka13=Otsuka & Tajitsu (2013); Otsuka14=Otsuka et al. (2014); Otsuka15=Otsuka, Hyung, & Tajitsu (2015); Pagomeno18=Pagomenos et al. (2018); Peña89=Peña et al. (1989); Peña90=Peña et al. (1990); Peña91=Peña, Torres-Peimbert, & Ruíz (1991); Péquignot05=Péquignot & Tsamis (2005); Pereira07=Pereira & Miranda (2007); Sandin10=Sandin et al. (2010); Stasińska10=Stasińska et al. (2010); TP79=Torres-Peimbert & Peimbert (1979); TP81=Torres-Peimbert, Rayo, & Peimbert (2081); TP90=Torres-Peimbert, Peimbert, & Peña (1990) | He/H | C/HbbAn asterisk ($\ast$) indicates that the C abundance was derived from optical line strengths; unmarked abundances were derived from UV line strengths. | N/H | O/H | Ne/H | S/H | Cl/H | Ar/H
---|---|---|---|---|---|---|---|---|---
BoBn1 (108.4-76.1)ccHalo PN classification is uncertain, based upon one or more relatively low radial velocity observations (see §3.1.3). | Hawley78 | 0.111 |  $\cdots$ | 3.30E-04 | 7.70E-05 | 5.20E-05 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Barker80 | 0.115 |  $\cdots$ | 2.19E-04 | 7.59E-05 | 5.25E-05 |  $\cdots$ |  $\cdots$ | 3.89E-08
| TP81 | 0.095 | 1.20E-03$\ast$ | 2.19E-04 | 7.94E-05 | 1.00E-04 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Peña91 | 0.105 | 1.45E-03$\ast$ | 8.71E-05 | 4.79E-05 | 5.75E-05 | 6.31E-07 |  $\cdots$ | 5.50E-08
| Kniazev08 | 0.100 | 1.58E-03$\ast$ | 4.37E-05 | 6.46E-05 | 8.13E-05 | 1.45E-07 | 1.38E-09 | 3.72E-08
| Milanova09 | 0.129 |  $\cdots$ | 3.02E-05 | 3.09E-04 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| Otsuka10 | 0.118 | 1.05E-03 | 1.07E-04 | 5.51E-05 | 9.04E-05 | 2.07E-07 | 2.47E-09 | 2.13E-08
| KH20 | 0.088 | 8.75E-04$\ast$ | 4.81E-05 | 6.69E-05 | 1.14E-04 | 7.34E-08 |  $\cdots$ | 1.49E-08
DdDm1 (061.9+41.3) | Clegg87 | 0.100 | 1.40E-05 | 2.50E-05 | 1.40E-04 | 2.00E-05 | 2.90E-06 |  $\cdots$ |  $\cdots$
| Dinerstein03 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.40E-05 | 2.10E-06 |  $\cdots$ |  $\cdots$
| Milanova09 | 0.110 | 1.51E-05 | 9.77E-05 | 1.17E-04 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| Otsuka09 | 0.102 | 1.05E-05 | 2.31E-05 | 1.18E-04 | 2.84E-05 | 2.23E-06 | 4.07E-08 | 6.10E-07
| KH20 | 0.093 | 1.12E-05 | 2.51E-05 | 1.15E-04 | 2.09E-05 | 2.04E-06 | 2.63E-08 | 6.46E-07
H4-1 (049.3+88.1) | Hawley78 | 0.106 |  $\cdots$ | 6.30E-05 | 2.20E-04 | 5.20E-06 | 4.00E-06 |  $\cdots$ |  $\cdots$
| TP79 | 0.098 | 2.45E-03$\ast$ | 7.41E-05 | 3.16E-04 | 6.31E-06 | 7.94E-07 |  $\cdots$ |  $<$1.58E-07
| Barker80 | 0.107 |  $\cdots$ | 6.31E-05 | 2.19E-04 | 5.25E-06 |  $\cdots$ |  $\cdots$ | 4.90E-08
| Dinerstein03 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\leq$8.00E-06 |  $\leq$5.00E-07 |  $\cdots$ |  $\cdots$
| Milanova09 | 0.120 | 1.55E-03 | 3.55E-05 | 2.57E-04 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| Otsuka13 | 0.108 | 1.04E-03 | 3.85E-05 | 1.50E-04 | 2.67E-06 | 1.36E-07 | 7.57E-09 | 3.63E-08
| KH20 | 0.122 | 1.68E-03$\ast$ | 5.16E-05 | 1.46E-04 | 2.95E-06 | 1.32E-07 |  $\cdots$ | 2.71E-08
K648 (009.8-07.5) | Hawley78 | 0.100 |  | 1.30E-05 | 4.60E-05 | 2.50E-06 |  $\cdots$ |  $\cdots$ |  $\cdots$
| TP79 | 0.098 |  $\cdots$ | 2.46E-06 | 6.61E-05 | 6.17E-06 |  $<$1.66E-06 |  $\cdots$ |  $<$3.31E-07
| Barker80 | 0.100 |  $\cdots$ | 1.17E-06 | 4.47E-05 | 2.51E-06 |  $\cdots$ |  $\cdots$ | 1.82E-08
| Adams84 | 0.104 | 5.40E-04 | 3.00E-06 | 4.70E-05 | 5.00E-06 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Milanova09 | 0.100 | 1.35E-03 | 7.41E-05 | 8.32E-05 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| Otsuka15 | 0.104 | 9.41E-04 | 2.28E-06 | 5.39E-05 | 2.75E-05 | 2.53E-07 | 3.76E-09 | 4.00E-08
| KH20 | 0.094 | 1.14E-03$\ast$ | 2.89E-06 | 4.89E-05 | 6.63E-06 | 1.85E-07 |  $\cdots$ | 6.06E-08
NGC 2242 (170.3+15.8)ccHalo PN classification is uncertain, based upon one or more relatively low radial velocity observations (see §3.1.3). | Garnett89 | 0.100 | 9.77E-05 | 1.26E-05 | 5.89E-05 | 3.63E-05 |  $\cdots$ |  $\cdots$ | 1.51E-06
| TP90 | 0.100 | 2.45E-04$\ast$ | 5.25E-05 | 1.07E-04 | 6.31E-05 |  $\cdots$ |  $\cdots$ | 7.76E-07
| KH20 | 0.125 |  $\cdots$ |  $\cdots$ | 1.08E-05 | 1.36E-06 |  $\cdots$ |  $\cdots$ | 1.12E-06
SaSt 2-3 (232.0+05.7) | Pereira07 |  $\cdots$ |  $\cdots$ | 8.40E-05 | 3.60E-04 |  $\cdots$ | 3.10E-06 |  $\cdots$ |  $\cdots$
| Otsuka14 |  $\cdots$ | 5.24E-04 | 3.09E-05 | 1.70E-04 | 4.79E-05 | 1.48E-06 |  $\cdots$ | 8.51E-07
| Pagomeno18 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 2.60E-05 | 7.00E-07 |  $\cdots$ | 1.66E-06
SBS 1150+599A (135.9+55.9) | Péquignot05 | 0.082 | 9.00E-05$\ast$ |  | 3.00E-05 | 4.50E-06 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Jacoby06 | 0.076 | 3.09E-05 | 8.71E-06 | 1.95E-05 | 4.17E-06 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Sandin10 | 0.076 | 7.94E-05 | 2.95E-05 | 5.50E-06 | 9.12E-06 |  $\cdots$ |  $\cdots$ |  $\cdots$
| Stasińska10 | 0.089 | 6.92E-05 | 1.41E-05 | 6.61E-06 | 6.76E-06 |  $<$3.16e-7 |  $\cdots$ |  $<$3.16e-8
NGC 4361 (294.1+43.6)ccHalo PN classification is uncertain, based upon one or more relatively low radial velocity observations (see §3.1.3). | TP90 | 0.107 | 9.12E-05 | 2.45E-05 | 6.46E-05 | 3.24E-05 |  $\cdots$ |  $\cdots$ | 1.23E-06
| Howard97 | 0.100 | 7.76E-05 | 2.24E-05 | 1.41E-04 | 1.78E-05 | 2.00E-06 |  $\cdots$ | 2.51E-07
| Kholtygin98 |  $\cdots$ | 2.90E-05 |  $\cdots$ | 5.10E-05 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
PRTM 1 (243.8-37.1)ccHalo PN classification is uncertain, based upon one or more relatively low radial velocity observations (see §3.1.3). | Peña90 | 0.107 |  $<$3.98E-05 |  $<$1.00E-04 | 2.24E-04 | 7.94E-05 |  $\cdots$ |  $\cdots$ | 2.00E-06
| Howard97 | 0.100 | 3.98E-06$\ast$ | 3.16E-05 | 1.26E-04 | 2.29E-05 |  $<$1.00E-05 |  $\cdots$ | 3.39E-07
PRMG 1 (006.0-41.9)ccHalo PN classification is uncertain, based upon one or more relatively low radial velocity observations (see §3.1.3). | Pena89 | 0.091 |  $\cdots$ |  $\cdots$ | 1.26E-04 | 3.16E-05 |  $\cdots$ |  $\cdots$ | 6.31E-07
| Howard97 | 0.091 | 1.00E-04$\ast$ | 1.91E-04 | 1.15E-04 | 1.70E-05 | 1.00E-05 |  $\cdots$ | 3.16E-07
JaFu1 (002.1+01.7) | Jacoby97 | 0.141 |  $\cdots$ | 9.33E-05 | 3.09E-04 |  $<$1.00E-04 | 4.17E-06 |  |
JaFu2 (353.5-05.0) | Jacoby97 | 0.115 |  $<$3.16E-05 |  $<$9.12E-05 | 5.37E-05 | 6.17E-06 |  $<$5.89E-06 |  $\cdots$ |  $<$2.88E-07
M2-29 (004.0-03.0) | Howard97 | 0.093 | 3.55E-05$\ast$ | 1.82E-05 | 5.13E-05 | 8.71E-06 | 8.51E-07 |  $\cdots$ | 4.57E-07
| Peña91 | 0.129 |  $\cdots$ | 9.55E-06 | 2.04E-05 | 5.25E-06 | 8.13E-07 |  $\cdots$ | 1.82E-07
Solar | Asplund09 | 0.085 | 2.69E-04 | 6.76E-05 | 4.90E-04 | 8.51E-05 | 1.32E-05 | 3.16E-07 | 2.51E-06

Table 3: s-Process Emission Line Observations AuthoraaReferences: Aleman19=Aleman et al. (2019), Asplund09=Asplund et al. (2009), Dinerstein01=Dinerstein (2001), García-Rojas15=García-Rojas et al. (2015), Geballe91=Geballe, Burton, & Isaacman (1991), Hora99=Hora, Latter, & Deutsch (1999), Likkel06=Likkel et al. (2006), Luhman96=Luhman & Rieke (1996), Lumsden01=Lumsden, Puxley, & Hoare (2001), Madonna17=Madonna et al. (2017), Madonna18=Madonna et al. (2018), Otsuka20=Otsuka & Hyung (2020), Péquignot94=Péquignot & Baluteau (1994), Rudy01=Rudy et al. (2001), Sharpee07=Sharpee, Zhang, & Williams (2007), Sterling15=Sterling et al. (2015), Sterling16=Sterling et al. (2016), Sterling17=Sterling et al. (2017). | PNe Observed |  $\lambda$ Range | Ge (32) | Se (34) | Br (35) | Kr (36) | Rb (37) | Cd (48) | Te (52) | Xe (54) | Ba (56) | Pb (82)
---|---|---|---|---|---|---|---|---|---|---|---|---
Geballe91 | 13 MW PNe | 2.04-2.45 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Pequignot94 | NGC 7027 | 4220-10300 Å |  $\cdots$ | III | III | III, IV, V | IV |  $\cdots$ |  $\cdots$ | III, IV, VI | II, IV |  $\cdots$
Luhman96 | Hubble 12 | 1.6,2.2 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Hora99 | 41 MW PNe | 1-2.5 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Dinerstein01 | IC 5117, N7027 | 2.199, 2.287 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Lumsden01 | 26 MW PNe | H+K bands |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Ruby01 | IC 5117 | 0.8-2.5 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Likkel06 | 16 MW PNe | K band |  $\cdots$ |  $\cdots$ |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Sharpee07 | 5 MW PNe | 3280-7580 Å |  $\cdots$ |  $\cdots$ | III, IV | III, IV, V | IV, V |  $\cdots$ | III | III, IV, V, VI | II, IV | II
Sterling15 | 103 MW PNe | 2.14-2.30 $\mu$m |  $\cdots$ | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
García-Rojas15 | NGC 3918 | 3100-10420 Å |  $\cdots$ | III |  $\cdots$ | III, IV, V | IV, V |  $\cdots$ |  $\cdots$ | III, IV, VI |  $\cdots$ |  $\cdots$
Sterling16 | NGC 7027, IC 5117 | 1.45-2.45 $\mu$m | VI |  $\cdots$ |  $\cdots$ |  $\cdots$ | IV | IV |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Madonna17 | NGC 5315 | 0.31-2.50 $\mu$m |  $\cdots$ | III, IV |  $\cdots$ | III, IV |  $\cdots$ |  $\cdots$ |  $\cdots$ | IV |  $\cdots$ |  $\cdots$
Sterling17 | 2 MW/2 LMC PNe | 0.83-2.45 $\mu$m |  $\cdots$ | III, IV |  $\cdots$ | VI |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Madonna18 | NGC 7027, IC 418 | 1.17-2.45 $\mu$m |  $\cdots$ | IV | V | III, VI | IV |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$
Aleman19 | Tc 1 | 3200-10100 Å |  $\cdots$ |  $\cdots$ |  $\cdots$ | III |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Otsuka20 | J900 | UV, OP, NIR |  $\cdots$ |  $\cdots$ |  $\cdots$ | III, IV | IV |  $\cdots$ |  $\cdots$ | III, IV |  $\cdots$ |  $\cdots$

Table 4: s-Process Abundances PN | AuthoraaReferences: Aleman19=Aleman et al. (2019), Asplund09=Asplund et al. (2009), García-Rojas15=García-Rojas et al. (2015), Madonna17=Madonna et al. (2017), Madonna18=Madonna et al. (2018), Otsuka20=Otsuka & Hyung (2020), Péquignot94=Péquignot & Baluteau (1994), Sharpee07=Sharpee, Zhang, & Williams (2007), Sterling15=Sterling et al. (2015), Sterling16=Sterling et al. (2016), Sterling17=Sterling et al. (2017). | Ge/H (32) | Se/H (34) | Br/H (35) | Kr/H (36) | Rb/H (37) | Cd/H (48) | Te/H (52) | Xe/H (54)
---|---|---|---|---|---|---|---|---|---
IC418 | Sharpee07 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.70E-09 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4.66E-10
| Madonna18 |  $\cdots$ | 2.51E-10 |  $\cdots$ | 5.89E-09 |  $\cdots$ |  $\cdots$ | 5.13E-10 |  $\cdots$
IC2501 | Sharpee07 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 9.62E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 9.40E-11
IC4191 | Sharpee07 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.25E-09 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.81E-10
IC5117 | Sterling16 |  $\cdots$ | 4.14E-09 |  $\cdots$ | 7.13E-09 | 9.52E-10 | 1.71E-10 |  $\cdots$ |  $\cdots$
J900 | Otsuka20 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 7.11E-09 | 7.28E-10 |  $\cdots$ |  $\cdots$ | 2.61E-09
NGC2440 | Sharpee07 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 9.59E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4.70E-11
NGC3918 | García-Rojas15 |  $\cdots$ | 3.31E-09 |  $\cdots$ | 6.31E-09 | 3.24E-10 |  $\cdots$ |  $\cdots$ | 3.09E-10
| Sterling17 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 8.91E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC5315 | Madonna17 |  $\cdots$ | 3.98E-09 | 3.39E-09 | 3.98E-09 |  $<$7.41E-10 |  $\cdots$ |  $\cdots$ | 2.69E-09
| Sterling17 |  $\cdots$ | 1.90E-08 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC7027 | Péquignot94 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 2.74E-08 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.36E-09
| Sharpee07 |  $\cdots$ |  $\cdots$ | 7.34E-10 | 9.69E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 9.92E-11
| Sterling16 | 1.81E-09 | 4.72E-09 |  $\cdots$ | 1.16E-08 | 1.14E-09 | 1.08E-10 |  $\cdots$ |  $\cdots$
| Sterling17 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 2.63E-09 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| Madonna18 |  $\cdots$ | 3.80E-09 | 4.37E-10 | 1.35E-08 | 1.07E-09 |  $\cdots$ | 5.62E-10 |  $\cdots$
TC1 | Aleman19 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.26E-08 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
SMP47 | Sterling17 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 6.61E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
SMP99 | Sterling17 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.89E-09 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Multi-Object Survey
120 PNebbTop row: Median Se/H for 68 objects and median Kr/H abundances for 39 objects. Bottom row: Abundance maximum, minimum. | Sterling15 |  $\cdots$ | 2.43E-09 |  $\cdots$ | 6.91E-09 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
|  |  $\cdots$ | 2.29E-08, 2.69E-10 |  $\cdots$ | 2.82E-08, 7.59E-10 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Solar Abundances
SOLAR | Asplund09 | 4.47E-09 | 2.19E-09 | 3.47E-10 | 1.78E-09 | 3.31E-10 | 5.13E-11 | 1.51E-10 | 1.74E-10

Table 5: Iron Abundances and Depletion Valuesaa$[X_{gas}/H]\equiv\log(X/H)_{gas}-\log(X/H)_{\odot}$, where X represents the abundance of the element in question. | Shields78 | Garstang78 | Perinotto99
---|---|---|---
PN | Fe/H | Depletion | Fe/H | Depletion | Fe/H | Depletion
Cn 3-1 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 6.0E-07 | -1.72
Hu 2-1 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 9.0E-07 | -1.54
IC 2165 | 1.00E-06 | -1.51 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 1565 | 6.31E-06 | -0.70 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 2022 | 1.58E-06 | -1.30 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 6543 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.0E-06 | -1.04
NGC 6720 |  $\cdots$ |  $\cdots$ | 7.00E-06 | -0.70 |  $\cdots$ |  $\cdots$
NGC 6741 | 2.51E-06 | -1.11 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 6886 | 2.51E-07 | -2.11 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 7009 |  $\cdots$ |  $\cdots$ | 2.40E-05 | -0.11 |  $\cdots$ |  $\cdots$
NGC 7027 | 1.00E-06 | -1.51 |  $\cdots$ |  $\cdots$ | 4.0E-07 | -1.86
NGC 7662 |  $\cdots$ |  $\cdots$ | 4.20E-07 | -1.88 |  $\cdots$ |  $\cdots$
Table 6A: Extragalactic Planetary Nebula Abundance StudiesaaGalaxy morphological types were taken from NASA/IPAC Extragalactic Database https://ned.ipac.caltech.edu/, while the T-number for each galaxy was obtained from the RC3 catalog (de Vaucouleurs et al., 1991). Author.yearbbReferences: Aller83=Aller & Czyzak (1983); Balick13=Balick et al. (2013); Bernard-Salas08=Bernard-Salas et al. (2008); Chiappini09=Chiappini et al. (2009); Corradi15b=Corradi et al. (2015b); Danziger78=Danziger et al. (1978); Delgado-Inglada11=Delgado-Inglada et al. (2011); Dopita97=Dopita et al. (1997); Dufour77=Dufour & Killen (1977); Fang 13,15,18=Fang et al. (2013, 2015, 2018); Jacoby86=Jacoby & Ford (1986); Jacoby99=Jacoby & Ciardullo (1999); Jenner79=Jenner, Ford, & Jacoby (1979); Kwitter12=Kwitter et al. (2012); Leisy96=Leisy & Dennefeld (1996); Leisy06=Leisy & Dennefeld (2006); Magrini09a=Magrini & Gonçalves (2009a); Maran82=Maran et al. (1982); Osmer76=Osmer (1976); Richer99=Richer, Stasińska, & McCall (1999); Richer08=Richer & McCall (2008); Sanders12=Sanders et al. (2012); Stanghellini05=Stanghellini, Shaw, & Gilmore (2005) | Number of Objects | Elements | Spectral Region | O/H*ccO/H* values for studies of three or more objects are medians. For studies of one or two PNe, values correspond to the single value or the pair average, respectively. | Comments
---|---|---|---|---|---
|  | FORNAX [dE4, -5] |  |  |
Danziger78 | 1 | He,N,O,Ar | OPT | 3.21E-04 |
|  | IC 10 [IBm, 10] |  |  |
Magrini09a | 2 | He,N,O,Ne,S,Ar | OPT | 1.30E-04 |
|  | LARGE MAGELLANIC CLOUD [SB(s)m, 9] |  |  |
Osmer76 | 3 | He,N,O | OPT | 1.58E-04 |
Dufour77 | 2 | He,N,O,Ne,S,Ar | OPT | 2.88E-04 |
Maran82 | 1 | He,C,N,O,Ne,S,Ar | UV, OPT | 2.14E-04 |
Aller83 | 8 | He,C,N,O,Ne,S,Ar | OPT | 1.79E-04 |
Leisy96 | 16 | He,C,N,O,Ne,S,Ar | UV, OPT | 2.76E-04 |
Dopita97 | 10 | He,C,N,O,Ne,Mg,Si,S,Ar | UV, OPT | 2.46E-04 |
Stanghellini05 | 24 | C | UV |  $\cdots$ |
Leisy06 | 120 | He,N,O,Ne,S,Ar | OPT | 2.04E-04 |
Bernard-Salas08 | 18 | Ne,S | IR |  $\cdots$ |
Chiappini09 | 110 | He,N,O,Ne,S,Ar |  | 1.70E-04 |
Delgado-Inglada11 | 3 | O,Fe | OPT ,NIR |  $\cdots$ | Fe/O only
|  | M31 [SA(s)b, 3] |  |  |
Jacoby86 | 1 | He,N,O,Ne | OPT | 4.67E-04 | disk
Jacoby86 | 2 | He,N,O,Ne | OPT | 2.35E-04 | halo
Jacoby99 | 12 | He,C,N,O,Ne,S,Ar | OPT | 2.19E-04 | bulge
Jacoby99 | 3 | He,C,N,O,Ne,S,Ar | OPT | 2.88E-04 | disk
Richer99 | 14 | O | OPT | 3.80E-04 | bulge
Kwitter12 | 16 | He,N,O,Ne,S,Cl,Ar | OPT | 4.17E-04 | outer disk
Sanders12 | 51 | O | OPT | 2.82E-04 | disk
Sanders12 | 17 | O | OPT | 3.16E-04 | halo
Balick13 | 2 | He,C,N,O,Ne,S,Cl,Ar | OPT | 3.29E-04 | outer disk
Corradi15 | 9 | He,N,O,Ne,S,Cl,Ar | OPT |  $\cdots$ |
Fang13,15,18 | 9 | He,C,N,O,Ne,S,Ar | OPT | 3.06E-04 | northern spur
Fang13,15,18 | 7 | He,C,N,O,Ne,S,Ar | OPT | 3.74E-04 | giant stream
|  | M32 [cE2, -6] |  |  |
Jenner79 | 1 | He | OPT |  $\cdots$ |
Richer99 | 4 | O | OPT | 2.49E-04 |
Richer08 | 11 | He,N,O,Ne | OPT | 3.09E-04 |

Table 6B: Extragalactic Planetary Nebula Abundance StudiesaaGalaxy morphological types were taken from NASA/IPAC Extragalactic Database https://ned.ipac.caltech.edu/, while the T-number for each galaxy was obtained from the RC3 catalog (de Vaucouleurs et al., 1991). Author.yearbbReferences: Aller81=Aller et al. (1981); Annibali17=Annibali, Tosi, & Romano (2017); Bernard-Salas08=Bernard-Salas et al. (2008); Bresolin10=Bresolin et al. (2010); Dufour77=Dufour & Killen (1977); Dufour80=Dufour & Talent (1980); Flores-Dirán17=Flores-Duŕan, Peña, & Ruiz (2017); Ford83=Ford (1983); García-Rojas16=García-Rojas et al. (2016); Gonçalves07=Gonçalves et al. (2007); Gonçalves14=Gonçalves et al. (2014); Hernández-Martínez09=Hernández-Martínez et al. (2009); Leisy96=Leisy & Dennefeld (1996); Leisy06=Leisy & Dennefeld (2006); Magrini05=Magrini et al. (2005); Magrini09b=Magrini, Stanghellini, & Villaver (2009b); Maran82=Maran et al. (1982); Osmer76=Osmer (1976); Richer08=Richer & McCall (2008); Shaw10=Shaw et al. (2010); Stanghellini09=Stanghellini et al. (2009); Stanghellini10=Stanghellini et al. (2010b); Stasińska13=Stasińska et al. (2013); Walsh99=Walsh et al. (1999); Walsh12=Walsh et al. (2012); Zijlstra06=Zijlstra et al. (2006) | Number of Objects | Elements | Spectral Region | O/H*ccO/H* values for studies of three or more objects are medians. For studies of one or two PNe, values correspond to the single value or the pair average, respectively. | Comments
---|---|---|---|---|---
|  | M33 [SA(s)cd, 6] |  |  |
Magrini09b | 91 | He,N,O,Ne,S,Ar | OPT | 1.64E-04 | gradient
Bresolin10 | 16 | He,N,O,Ne,Ar | OPT | 2.75E-04 | gradient
|  | M81 [SA(s)ab, 2] |  |  |
Stanghellini10 | 19 | He,N,O,Ne,S,Ar | OPT | 1.86E-04 | gradient
|  | NGC 147 [E5 pec, -5] |  |  |
Gonçalves07 | 6 | He,N,O,Ne,S,Ar | OPT | 1.29E-04 |
|  | NGC 185 [E3 pec, -5] |  |  |
Ford83 | 1 | He,N,O,Ne,Ar | OPT | 7.59E-05 |
Richer08 | 5 | He,N,=O,Ne | OPT | 1.35E-04 |
|  | NGC 205 [E5 pec, -5] |  |  |
Richer08 | 13 | He,N,O,Ne | OPT | 1.29E-04 |
Gonçalves14 | 14 | He,N,O,Ne,S,Ar | OPT | 1.15E-04 |
|  | NGC 300 [SA(s)d, 7] |  |  |
Stasińska13 | 25 | He,N,O,Ne,S,Ar | OPT | 1.95E-04 | gradient
|  | NGC 3109 [SB(s)m edge-on, 9] |  |  |
Peña07 | 7 | He,N,O,Ne,S,Ar | OPT | 1.26E-04 |
Flores-Durán17 | 7 | He,N,O,Ne,S,Ar | OPT | 1.32E-04 |
|  | NGC 4449 [IBm, 10] |  |  |
Annibali17 | 4 | He,N,O,Ne,S,Ar | OPT | 2.00E-04 |
|  | NGC 5128 [S0 pec, -2] |  |  |
Walsh99 | 2 | N,O | OPT | 2.37E-04 |
Walsh12 | 40 | He,N,O,Ne,S,Ar | OPT | 2.88E-04 |
|  | NGC 6822 [IB(s)m, 10] |  |  |
Dufour80 | 1 | He,N,O,Ne | OPT | 1.26E-04 |
Hernández-Martínez09 | 11 | He,N,O,Ne,S,Ar |  | 8.91E-05 |
Garcia-Rojas16 | 18 | He,N,O,Ne,S,Ar | OPT | 7.11E-05 |
|  | SEXTANS A [IBm, 10] |  |  |
Magrini05 | 1 | He,N,O,Ne,S,Ar | OPT | 1.3E-04 |
|  | SEXTANS B [IB(s)m, 10] |  |  |
Magrini05 | 5 | He,N,O,Ne,S,Ar | OPT | 9.6E-05 |
|  | SAGITTARIUS [dSp(t),xxx] |  |  |
Zijlstra06 | 4ddZijlstra et al. (2006) include BoBn 1, usually assumed to be located in the MW halo (see Table 2), in their abundance survey of PNe in the Sagittarius dwarf spheroidal galaxy. | He,C,N,O,Ne,Mg,S,Cl,Ar,K | OPT | 2.21E-04 |
|  | SMALL MAGELLANIC CLOUD [SB(s) pec, 9] |  |  |
Osmer76 | 3 | He,N,O | OPT | 1.00E-04 |
Dufour77 | 1 | He,N,O | OPT | 3.80E-05 |
Aller81 | 7 | He,N,O,Ne | OPT | 1.38E-04 |
Maran82 | 2 | He,C,N,O,Ne,S,Ar | UV, OPT | 2.15E-04 |
Leisy96 | 15 | He,C,N,O,Ne,S,Ar | UV, OPT | 1.29E-04 |
Leisy06 | 37 | He,C,N,O,Ne,S,Ar | OPT | 1.00E-04 |
Bernard-Salas08 | 7 | Ne,S | IR |  $\cdots$ |
Stanghellini09 | 11 | C | UV |  $\cdots$ | no O/H
Shaw10 | 14 | He,C,N,O,Ne,S,Ar | IR, OPT | 1.10E-04 |

Table 7: Median Gas Phase AbundancesaaNumber abundance ratios for He/H and Ar/H are given in the format 12+log(X/H), while for N/O, the values are log(N/O). These values are taken from García-Hernández & Górny (2014). for the Galactic Bulge and Disk Dust Type | He/H | N/O | Ar/H
---|---|---|---
Bulge |  |  |
Featureless (F) |  $\cdots$ |  $\cdots$ |  $\cdots$
C-Rich (CC) |  $\cdots$ |  $\cdots$ |  $\cdots$
O-Rich (OC) | 11.04 | -0.80 | 6.29
Dual-Dust (DC) | 11.09 | -0.23 | 6.51
Disk |  |  |
Featureless (F) | 11.07 | -0.65 | 6.21
C-Rich (CC) | 11.03 | -0.70 | 6.07
O-Rich (OC) | 11.02 | -0.81 | 6.04
Dual-Dust (DC) | 11.10 | -0.27 | 6.56
SolarbbAsplund et al. (2009) | 10.93 | -0.85 | 6.40
Table 8: Disk Metallicity GradientsaaGradients are given in units of dex/R25 From Planetary Nebulae Element | Milky WaybbPeña & Flores-Durán (2019) | M31bbPeña & Flores-Durán (2019) | M33bbPeña & Flores-Durán (2019) | NGC 300bbPeña & Flores-Durán (2019) | M81ccStanghellini et al. (2010b)
---|---|---|---|---|---
O/H | -0.276$\pm{.035}$ | -0.021$\pm{.021}$ | -0.342$\pm{.144}$ | -0.159$\pm{.058}$ | -0.803$\pm{.292}$
Ne/H | -0.242$\pm{.058}$ | -0.041$\pm{.021}$ | -0.324$\pm{.270}$ | -0.154$\pm{.069}$ | -0.686$\pm{.438}$
Ar/H | -0.207$\pm{.173}$ | -0.041$\pm{.021}$ | -0.279$\pm{.126}$ | -0.270$\pm{.074}$ |  $\cdots$
S/H |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | -1.01$\pm{.146}$
Table 9: Least Squares Fit Parameters For Alpha Elements Relation | y-Intercept | Slope | Correlation Coefficient | Objects
---|---|---|---|---
Planetary NebulaeaaLeast-squares fits shown with solid lines to the PN data of
KH20 displayed here.
Ne/H vs. O/H | -0.056$\pm$.08 | +0.94$\pm$.083 | +0.72 | 116
S/H vs. O/H | +0.024$\pm$1.17 | +0.77$\pm$.14 | +0.47 | 117
Cl/H vs. O/H | -2.26$\pm$1.03 | +0.83$\pm$.12 | +0.56 | 107
Ar/H vs. O/H | -1.45$\pm$.83 | +0.91$\pm$.097 | +0.66 | 117
H II Regions and Blue Compact GalaxiesbbLeast-squares fits shown with dashed
lines to a sample of H II region and blue compact galaxy data computed by
Milingo et al. (2010) in the cases of Ne, S, and Ar. For Cl, we used the MW
disk abundance gradients reported by Esteban, García-Rojas, & Pérez-Mesa
(2015, §3.3) to derive the fit given here.
Ne/H vs. O/H | -0.84$\pm$.27 | +1.02$\pm$.03 | +0.96 | 85
S/H vs. O/H | -1.63$\pm$.22 | +1.01$\pm$.03 | \+ 0.97 | 73
Cl/H vs. O/H | -3.44$\pm$.11 | +1.00$\pm$.01 |  $\cdots$ | 9
Ar/H vs. O/H | -3.03$\pm$.75 | +1.10$\pm$.09 | +0.80 | 82
Table 10: Properties of AGB Models SourceaaEach source name refers to one or more investigators and their associated publications, all of whom make use of a specific modelling code. The source names are explained in the text here as well as in Henry et al. (2018). | Metallicity | Progenitor Mass (M⊙) | HBB Onset (M⊙)bbThese values were estimated by plotting log(C/O) versus progenitor mass and determining the mass maximum where log(C/O) begins to descend as mass continues to increase.
---|---|---|---
MONASH | 0.007 | 1.0-7.5 | 3.00
MONASH | 0.014 | 1.0-8.0 | 3.25
MONASH | 0.030 | 1.0-8.0 | 3.50
LPCODE | 0.001 | 1.0-3.0 | 2.50
LPCODE | 0.010 | 1.0-3.0 | 2.50
LPCODE | 0.020 | 1.0-4.0 | 3.00
ATON | 0.004 | 0.0-8.0 | 2.50
ATON | 0.008 | 1.0-8.0 | 2.50
ATON | 0.014 | 1.0-8.0 | 3.00
FRUITY | 0.003 | 1.3-6.0 | 2.50
FRUITY | 0.008 | 1.3-6.0 | 2.50
FRUITY | 0.014 | 1.3-6.0 | 2.50

Figure 1: Montage of 22 PNe showing relative sizes. The 5 LY scale bar is
equivalent to $\sim$1.5 pc. Credit: Ivan Bojic̆ić, Quentin Parker, and David
Frew, Laboratory for Space Research, HKU. Original images: ESA/Hubble & NASA.
Figure 2: Aitoff projection showing the Galactic distribution of all 3540 PNe
currently in HASH. True, likely, and possible PNe are the red, green, and
black symbols, respectively. Credit: Fig. 1 in Frew (2016) ©Cambridge
University Press. Reproduced with permission of the Licensor through PLSclear.
Figure 3: Hertzsprung-Russell diagram of a complete evolutionary track for a 2
M⊙solar metallicity star from the main sequence to the white dwarf phase. In
the cooler section of the post-AGB phase, wiggles in the track are caused by
numerical convergence difficulties. The blue track shows a born-again
evolution model [triggered by a very late thermal pulse – see §4.2 in Herwig
(2005)] of the same mass, however, shifted by approximately $\Delta
log~{}T_{eff}=-0.2$ and $\Delta log~{}L/L_{\odot}=-0.5$ for clarity. The red
and green stars mark the position of the central stars of planetary nebulae
for which spectra are shown in Herwig (2005, Fig. 8). The number labels for
each evolutionary phase indicates the log of the approximate duration for a 2
M⊙ case. Larger or smaller mass cases would have smaller or larger
evolutionary timescales, respectively. This figure and its caption are
republished with permission of Annual Reviews, from Annual Reviews of
Astronomy and Astrophysics, Falk Herwig, 43, 435, 2005; permission conveyed
through Copyright Clearance Center, Inc. Figure 4: Crossing time vs. main
sequence mass for solar metallicity. Crossing time is the interval between the
star having an effective temperature $\sim$104K and when it reaches its
maximum effective temperature. Based on data in Table 3 from Miller Bertolami
(2016). Figure 5: Median abundance values for disk PNe in Table 1 are plotted
as $12+\log{X/H}$ for each element displayed along the horizontal axis for the
surveys listed in the legend, using colored filled circles. Surveys are linked
to their related references in a Table 1 footnote. Survey symbols are plotted
left to right, early to late, in the order of survey date. Solar values from
Asplund et al. (2009) are shown as black stars and are connected by a solid
line. Figure 6: Median abundance values for bulge PNe in Table 1 are plotted
as $12+\log{X/H}$ for each element displayed along the horizontal axis for the
surveys listed in the legend using colored filled circles. Surveys are linked
to their related references in a Table 1 footnote. Survey symbols are plotted
left to right, early to late, in the order of survey date. Solar values from
Asplund et al. (2009) are shown as black stars and are connected by a solid
line. Figure 7: Median abundance values for halo PNe in Table 2 are plotted as
$12+\log{X/H}$. For each element ratio on the horizontal axis, each symbol in
the plot represents the median abundance of all studies that pertain to an
individual halo PN. Objects are identified by colored filled circles as shown
in the legend. Solar values from Asplund et al. (2009) are shown with black
filled circles and are connected by a solid black line to aid comparisons with
PN values.

Figure 8: Values of 12+log(O/H∗) versus morphological T-type for 19 external
galaxies and the MW listed in Tables 6A,B. Each circular symbol represents the
measured median value(s) of PNe within each galaxy as a function of its
morphological T-Type. Symbols for individual galaxies are color-coded, and
multiple symbols for that color represent measurements by different authors,
as detailed in Tables 6A and 6B.

.

Figure 9: The relative percentages of dust types in the Galactic bulge, disk,
and Magellanic Clouds. Galactic data are from García-Hernández & Górny (2014,
Table 5), while the MC data are provided in Stanghellini et al. (2012, Table
6). Sample size for each region is indicated in parentheses on the horizontal
axis. Figure 10: Starting with the upper left panel and proceeding in
clockwise fashion, Ne/H, S/H, Ar/H and Cl/H, are plotted versus O/H, where
values are expressed in the form of 12+log(X/H). Observational data (open
circles) are from KH20 error bars represent median uncertainties and are shown
in the lower right of each panel. We also show least-squares fits in each
panel, where the solid line represents the fit to the PN data. For Ne, S, and
Ar, the dashed line shows the fit to abundances for H II regions and blue
compact galaxies compiled by Milingo et al. (2010, Table 6) for comparison
purposes. The analogous fit for Cl was inferred from a study of Galactic disk
H II regions by Esteban, García-Rojas, & Pérez-Mesa (2015, §3.3). Table 9
displays the regression parameters and uncertainties. Figure 11: A comparison
of observations with model predictions of final AGB surface abundances. Upper
panels: log(C/O) vs. He/H for Galactic disk PNe, with observational data taken
from KH20. Observed C abundances forming the C/O ratios in the upper three
panels were determined strictly from UV lines (C III] 1909Å and C IV 1549Å;
filled circles) or optical lines (C II 4267Å; open circles). Models produced
by the MONASH (left), LPCODE (center), and ATOM (right) codes are shown, with
model metallicity indicated by the color-coding defined in the legends. Each
model within a metallicity set is represented by a cross (X) and has a unique
progenitor mass, where the mass range for each set is given in Table 10. Lower
panels: Same as the upper panels but for log(N/O) vs. He/H. Error bars for the
observations represent median values of uncertainties and are shown in the
lower right corner of the two left panels. The black dashed lines in the lower
three panels define the upper-right quadrant containing Type I PNe. Figure 12:
Like Fig. 11 but for log(C/O) vs. log(N/O), showing AGB surface abundance
predictions from the MONASH (ul), LPCODE (ur), ATON (ll) and FRUITY (lr) codes
along with observational data of disk PNe from KB20. Figure 13: Upper:
Predicted values of final AGB surface abundances of He/H vs. progenitor mass
in solar units from Karakas & Lugaro (2016); Karakas et al. (2018) for the
three metallicity values indicated in the legend. Lower: Same as the upper
panel but for log(N/O) versus progenitor mass. Figure 14: Top: Comparison with
proto-solar abundances of the contribution of LIMS plus rotating massive stars
(red filled symbols), LIMS plus non-rotating massive stars (green crosses,
only up to A=96 to avoid confusion at higher A), and LIMS only (red open
symbols). Abundance uncertainties from Lodders, Palme, & Gail (2009) for each
isotope are indicated. The dotted horizontal lines show deviations from the
proto-solar values of 10% and 50%. Middle: Contribution of LIMS to the total
production of s-only isotopes. Bottom: Production factors normalized to 150Sm.
(Note: This plot is a copy of Fig. 11 in the paper titled Chemical evolution
with rotating massive star yields I. The solar neighbourhood and the s-process
elements, Prantzos et al. 2018, Monthly Notices of the Royal Astronomical
Society, 476, 3432. We are grateful to Oxford University Press for permitting
us to display it.) Figure 15: PNLF for PNe in M31 out to $\sim$30 kpc. The
characteristic sharp cutoff at the bright end occurs at m${}_{5007}\sim$20.17.
Credit: R. Ciardullo. Figure 16: The PNLF for five galaxies in the Leo I group
(d$\sim$10 Mpc). Note the remarkably consistent bright-end cutoff across the
different Hubble types. Credit: Based on a figure by R. Ciardullo. Figure 17:
Diffraction-limited V-band ($\lambda$=554 nm) and NR-band ($\lambda$=645.9 nm)
composite of L2 Puppis showing the dust ring and bipolar outflows in primarily
scattered light. Note the stellar components (separated by $\sim$33 mas)
visible as white blobs above the narrow waist. The image is 38.4 AU on a side.
Credit: ESO/P. Kervella. Figure 18: Abell 39: Example of a round PN. The
nebular diameter is roughly 1.2 pc. Image taken in [O III]. Credit:
WIYN/NOIRLab/NSF. Figure 19: IC 4634: Example of a point-symmetric PN. The
major axis from outermost arc to outermost arc is about 0.23 pc. Blue:
H$\beta$, [O III]; Cyan: V; Red: H$\alpha$, [N II]. Credit: ESA/Hubble and
NASA. Figure 20: Mz 3: Example of a bipolar/bilobed PN. The major axis of the
bulbous main nebula is very roughly 0.15 pc. Purple: [O III]; Blue: H$\alpha$;
Green: [N II]; Red: [S II]. Credit: R. Sahai (JPL) et al., Hubble Heritage
Team, ESA, NASA. Figure 21: NGC 6543 main nebula. The major axis is roughly
0.25 pc. Blue: X-ray; Orange and Purple: optical. Note that X-rays are
confined in the interior of the nebula. FLIERs are visible along the major
axis. Concentric rings surround the main nebula. Credit: X-ray: NASA/CXC/SAO;
Optical: NASA/STScI. Figure 22: NGC 6543 extended halo. This halo has a
diameter of about 1.7 pc. Blue: [O III]; Red: [N II]. Credit: Nordic Optical
Telescope and Romano Corradi (Isaac Newton Group of Telescopes, Spain). Figure
23: Toy model of a wind-blown bubble. From the star outward: stellar wind
(gray); inner shock; shocked stellar wind (blue); ionized swept-up gas (the
visible PN; red); ionization front; neutral swept-up gas (tan); outer shock;
unaffected AGB wind (white). Figure 24: NGC 7027. The main nebula is about
0.08 pc along the major axis. Blue: F343N ([Ne V]); Green: 502N ([O III]); and
Red: F164N ([Fe II]). Note the concentric dust rings. Credit NASA, ESA, and J.
Kastner (RIT). Figure 25: NGC 6302. The blue-green “butterfly” nebula is about
0.9 pc in diameter. Blue: F343N ([Ne V]); Green: F128N (Pa$\beta$); Red: F164N
([Fe II]). The point-symmetic arcs emitting [Fe II] are suggestive of fast
shocks. Credit: NASA, ESA, and J. Kastner (RIT). Figure 26: NGC 7293 knots
seen in H2 at 2.12 $\mu$m. Image size: 5.10’ x 3.45’. The central star is
outside of this image toward the bottom. Reprinted from Matsuura et al. (2009)
Figure 2. Figure 27: Two PNe with prominent FLIERs. The major axis of each
nebula is about 0.2 pc. Left: NGC 6826. Blue: [O III]; Green: [O I]; Red: [N
II] and [S II]. Credit: Bruce Balick (University of Washington), Jason
Alexander (University of Washington), Arsen Hajian (U.S. Naval Observatory),
Yervant Terzian (Cornell University), Mario Perinotto (University of Florence,
Italy), Patrizio Patriarchi (Arcetri Observatory, Italy) and NASA/ESA. Right:
NGC 7009. Violet: H$\beta$; Blue: [O II], [O III]; Green: He II, He I; Red: [O
I], H$\alpha$, [N II]. Credit: ESA/J. Walsh. Figure 28: Fg 1: The S-shaped
jets appear in point symmetry across the nebula. The main nebula is about 1.5
pc in diameter. Blue: [O II]; Green: [O III]; Red: H$\alpha$, [N II]. Credit:
ESO/H. Boffin. Figure 29: Sh2-188. Lines indicate the central star. This is
one of the largest known PNe, with a diameter of about 2.8 pc. Original image
credit: T.A. Rector/University of Alaska Anchorage, H. Schweiker/WIYN and
NOIRLab/NSF/AURA. Figure 30: Abell 21. Orange: H$\alpha$; Blue: [O III]. Lines
indicate the central star. This is a large PN, with a diameter of roughly 1.6
pc. Original image credit: H. Schwelker/NOIRLab/NSF/AURA and T.A.
Rector/University of Alaska Anchorage, and NOIRLab/NSF/AURA.


along the $x$-$y$ plane and acts as a symmetry operator. Our strategy, as
shown in Fig. 9, is to put two parallel subsystem KW operators with a
separation of $\epsilon$ and compute the partition function in the region
between two operators. As we take the limit $\epsilon\to 0$, we get the fusion
of two operators. It is equivalent to performing 1-gauging on a co-dimension
one surface [81].

$z=0$$z=\epsilon$$\mathcal{N}$$\mathcal{N}$ Figure 9: Fusion of two subsystem
KW operators along $z$ direction.

Consider the symmetry operator $\mathcal{N}^{\textrm{sub}}$ located at $z=0$
as an example. The theory at $z\geq 0$ is defined to be,

$Z[M_{3}^{\geq 0},\hat{A}]=\frac{1}{|H^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})|}\sum_{A\in H^{1}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})}Z[M_{3}^{\geq 0},A](-1)^{\int_{M_{3}^{\geq
0}}A\cdot\hat{A}},$ (C.14)

where $H^{1}_{\textrm{sub}}(M_{3}^{\geq 0},M_{2|0},\mathbb{Z}_{2})$ is the
relative cohomology such that $A^{xy}=0$ at the boundary $M_{2|0}$ and we use
$M_{2|0}$ to emphasize $M_{2}$ is located at $z=0$. To compute the fusion
$\mathcal{N}^{\textrm{sub}}\times\mathcal{N}^{\textrm{sub}}$, we insert
another $\mathcal{N}^{\textrm{sub}}$ at $z=\epsilon$ such that the theory
living on $M_{3}^{\geq 0}$ is given by,

$\frac{1}{|H^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||H^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{\begin{subarray}{c}A\in
H^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},M_{2|0},\mathbb{Z}_{2})\\\
\widetilde{A}\in
H^{1}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}+\int_{M_{3}^{\geq\epsilon}}(A-\hat{A})\cdot\widetilde{A}}.$
(C.15)

Using the relations $|H^{1}|=|Z^{1}|/|B^{1}|$, we can write the sum over
cohomologies into a sum over cocycles as $\sum_{A\in
H^{1}}=\frac{1}{|B^{1}|}\sum_{A\in Z^{1}}$. Moreover, notice that
$|B^{1}|=|C^{0}|/|Z^{0}|$ and $|B^{0}|=1$, we can rewrite the integral as,

$\displaystyle\frac{1}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{\begin{subarray}{c}A\in
Z^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},M_{2|0},\mathbb{Z}_{2})\\\
\widetilde{A}\in
Z^{1}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}+\int_{M_{3}^{\geq\epsilon}}(A-\hat{A})\cdot\widetilde{A}}.$
(C.16)

The cocycle condition can further be relaxed by introducing Lagrange
multiplier $\phi\in C^{0}(M_{3}^{\geq 0},\mathbb{Z}_{2})$ and
$\widetilde{\phi}\in C^{0}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})$,

$\displaystyle\frac{1}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}$
$\displaystyle\times\sum_{\begin{subarray}{c}A\in
C^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2}),\widetilde{A}\in
C^{1}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\\\ \phi\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2}),\widetilde{\phi}\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}+\int_{M_{3}^{\geq\epsilon}}(A-\hat{A})\cdot\widetilde{A}+\int_{M_{3}^{>0}}\phi\delta
A-\int_{M_{2|0}}\phi
A^{xy}+\int_{M_{3}^{>\epsilon}}\widetilde{\phi}\delta\widetilde{A}-\int_{M_{2|\epsilon}}\widetilde{\phi}\widetilde{A}^{xy}}.$
(C.17)

Summing over $\phi$ in the bulk $M_{3}^{\geq 0}$ enforces $A$ to be a cocycle
due to the coupling $\int_{M_{3}^{>0}}\phi\delta A$, and summing over $\phi$
on the boundary $M_{2|0}$ enforces $A^{xy}=0$, which makes the cocycle
relative to $M_{2|0}$. Same to $\hat{A}$.

We can firstly perform the sum over $\widetilde{A}$ and one has,

$(-1)^{\int_{M_{3}^{>\epsilon}}\widetilde{\phi}\delta\widetilde{A}-\int_{M_{2|\epsilon}}\widetilde{\phi}\widetilde{A}^{xy}}=(-1)^{\int_{M_{3}^{\geq\epsilon}}\delta\widetilde{\phi}\cdot\widetilde{A}},$
(C.18)

where we use integration by part and
$\delta\widetilde{\phi}\cdot\widetilde{A}=\partial_{z}\widetilde{\phi}\widetilde{A}^{xy}+\partial_{x}\partial_{y}\widetilde{\phi}\widetilde{A}^{z}$.
Sum over $\widetilde{A}$ will produce a factor
$|C^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|$ and enforce
$A-\hat{A}-\delta\widetilde{\phi}=0$,

$\displaystyle\frac{|C^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}$
$\displaystyle\times\sum_{\begin{subarray}{c}A\in
C^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2})\\\ \phi\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2}),\widetilde{\phi}\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}+\int_{M_{3}^{>0}}\phi\delta
A-\int_{M_{2|0}}\phi
A^{xy}}\delta(A-\hat{A}-\delta\widetilde{\phi})|_{M_{3}^{\geq\epsilon}}.$
(C.19)

We then integrate out $\phi$, which produces a factor
$|C^{0}_{\textrm{sub}}(M_{3}^{\geq 0},\mathbb{Z}_{2})|$ and enforces $A\in
Z^{1}_{\textrm{sub}}(M_{3}^{\geq 0},M_{2|0},\mathbb{Z}_{2})$,

$\displaystyle\frac{|C^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}$
$\displaystyle\times\sum_{\begin{subarray}{c}A\in
Z^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},M_{2|0},\mathbb{Z}_{2})\\\
\widetilde{\phi}\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}}\delta(A-\hat{A}-\delta\widetilde{\phi})|_{M_{3}^{\geq\epsilon}}.$
(C.20)

The summand is independent of $\widetilde{\phi}$ and we can set
$\widetilde{\phi}$ to zero in the delta function and add a normalization
factor $|C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|$. The
delta function then fixes $A=\hat{A}$ in $M_{3}^{\geq\epsilon}$ and make $A$
an element of $Z^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})$,

$\displaystyle\frac{|C^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{A\in
Z^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})}Z[M_{3}^{\geq
0},A+\hat{A}_{M_{3}^{\geq\epsilon}}](-1)^{\int_{M_{3}^{[0,\epsilon]}}A\cdot\hat{A}},$
(C.21)

where $\hat{A}|_{M_{3}^{\geq\epsilon}}$ is equal to $\hat{A}$ if we are on
$M_{3}^{\geq\epsilon}$ and vanishes elsewhere.

Let’s introduce the Euler factor $\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]$
as,

$\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]\equiv\frac{|H^{2}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})||H^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|H^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}=\frac{|C^{2}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|C^{1}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}.$
(C.22)

where in the second expression we use the fact $H^{0}=Z^{0}$, $Z^{2}=C^{2}$
since $C^{2}$ is the top one and $|B^{n+1}|=|C^{n}|/|Z^{n}|$. The
normalization factor in (C.21) can be written as,

$\frac{|C^{2}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})|}\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]^{-1}.$
(C.23)

The first factor can be further simplified by using
$|C^{2}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|=|C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})|$
since the elements in
$C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})$ and
the elements in $C^{2}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})$
are Fourier partners under integration over $M_{3}$. Finally, use the fact
that,

$|C^{n}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})|=|C^{n}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})||C^{n}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})|,$ (C.24)

which is a decomposition of cochains on $M_{3}^{\geq 0}$ into the sum of
cochains on $M_{3}^{[0,\epsilon]}$ with fixed boundary condition at
$M_{2|\epsilon}$ and cochains on $M_{3}^{\geq\epsilon}$ with free boundary
conditions. Substituting the simplified normalization into (C.21) we have,

$\frac{\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]^{-1}}{|C^{0}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{A\in
Z^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})}Z[M_{3}^{\geq
0},A+\hat{A}_{M_{3}^{\geq\epsilon}}](-1)^{\int_{M_{3}^{[0,\epsilon]}}A\cdot\hat{A}}.$
(C.25)

Write the summation of $Z^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})$ back to
$H^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})$ as $\sum_{A\in Z^{1}}=|B^{1}|\sum_{A\in
H^{1}}$, we obtain the finial result,

$\frac{\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]^{-1}}{|H^{0}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{A\in
H^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})}Z[M_{3}^{\geq
0},A+\hat{A}_{M_{3}^{\geq\epsilon}}](-1)^{\int_{M_{3}^{[0,\epsilon]}}A\cdot\hat{A}},$
(C.26)

where we use the relations $|B^{1}||Z^{0}|=|C^{0}|$ and $|B^{0}|=1$ again.

Now let’s evaluate the integral $\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}$
with Dirichlet boundary condition $A^{xy}|_{z=0}=A^{xy}|_{z=\epsilon}=0$. At
this stage, we need to regularize the spacetime and write,

$\displaystyle\int_{M_{2}\times[0,\epsilon]}(A^{xy}\hat{A}^{z}+A^{z}\hat{A}^{xy})=$
$\displaystyle\sum_{i=1}^{L_{x}}\left(w_{y,i+\frac{1}{2}}\left[\sum_{k=1}^{L_{\epsilon}}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}\right]^{y}\right)+\sum_{j=1}^{L_{y}}\left(w_{x,j+\frac{1}{2}}\left[\sum_{k=1}^{L_{\epsilon}}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}\right]^{x}\right)$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{L_{x}}\left(w_{z,y;i}\sum_{j=1}^{L_{y}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right)+\sum_{j=1}^{L_{y}}\left(w_{z,x;j}\sum_{i=1}^{L_{x}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right),$
(C.27)

where $L_{\epsilon}$ is the number of sites between $[0,\epsilon)$ and
$\omega$ are the holonomies of $A$.
$\sum_{j=1}^{L_{y}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}},\sum_{i=1}^{L_{x}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}$
are strip operators of the dual field $\hat{A}$ and they generate the
subsystem symmetry as mentioned in (3.30),

$U_{i}^{y}=\exp\left(i\pi\sum_{j=1}^{L_{y}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right),\quad
U_{j}^{x}=\left(i\pi\sum_{i=1}^{L_{x}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right).$
(C.28)

The line operator
$\sum_{k=1}^{\epsilon}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}$ of the dual
field $\hat{A}^{z}$ are subsystem symmetry defects. Recall that the line
operator can be decomposed into two line operators separately movable along
$x$ and $y$ directions and we use the labels $[\cdots]^{x}$ and $[\cdots]^{y}$
to represent them.

We will take the limit $\epsilon\rightarrow 0$ while fixing the holonomies
$w$. The first line vanishes in the limit. Another point of view is, since
$A^{xy}|_{z=0}=A^{xy}|_{z=\epsilon}=0$ the holonomies of $A^{xy}$ vanishes and
$w_{y,i+\frac{1}{2}}=w_{x,j+\frac{1}{2}}=0$ and first line is trivial.
Therefore we only need to consider the second line which implies the fusion
rule,

$\mathcal{N}^{\text{sub}}{}^{\dagger}\times\mathcal{N}^{\text{sub}}=\sum_{w_{z,y;i},w_{z,x;j}/\sim}(U_{i}^{y})^{w_{z,y;i}}(U_{j}^{x})^{w_{z,x;j}}=\frac{1}{2}\prod_{i=1}^{L_{x}}\left(1+U^{y}_{i}\right)\prod_{j=1}^{L_{y}}\left(1+U^{x}_{j}\right),$
(C.29)

where we have used $|H^{0}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})|=1$ ******The elements $f$ in
$H^{0}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})$ should satisfies
$\partial_{x}\partial_{y}f=\partial_{z}f=0$ and $f=0$ at the boundary. They
fix $f$ to be trivial. and $\mathcal{N}^{\text{sub}}{}^{\dagger}$ is
normalized as $\mathcal{N}^{\text{sub}}{}^{\dagger}=\chi[M_{3}^{\geq
0},\mathbb{Z}_{2}]\mathcal{N}^{\text{sub}}$. In the sum, we mod out the gauge
redundancy $\sim$ of the holonomies.

Let’s then consider the case where the defect $\mathcal{N}^{\textrm{sub}}$ is
along the $x$-$z$ plane and acts as a symmetry defect. The derivation of the
fusion rule is similar and we have,

$\displaystyle\frac{1}{|C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},M_{2|0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},M_{2|\epsilon},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq
0},\mathbb{Z}_{2})||C^{0}_{\textrm{sub}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})|}$
$\displaystyle\times$ $\displaystyle\sum_{\begin{subarray}{c}A\in
C^{1}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2}),\widetilde{A}\in
C^{1}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\\\ \phi\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq 0},\mathbb{Z}_{2}),\widetilde{\phi}\in
C^{0}_{{\textrm{sub}}}(M_{3}^{\geq\epsilon},\mathbb{Z}_{2})\end{subarray}}Z[M_{3}^{\geq
0},A](-1)^{\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}+\int_{M_{3}^{\geq\epsilon}}(A-\hat{A})\cdot\widetilde{A}+\int_{M_{3}^{>0}}\phi\delta
A+\int_{M_{2|0}}\phi\partial_{x}A^{z}+\int_{M_{3}^{>\epsilon}}\widetilde{\phi}\delta\widetilde{A}+\int_{M_{2|\epsilon}}\widetilde{\phi}\partial_{x}\widetilde{A}^{z}}.$
(C.30)

where the difference is that $M_{2|0}$ and $M_{2|\epsilon}$ are the $x$-$z$
plane located at $y=0$ and $y=\epsilon$, and summing over the Lagrangian
multiplier $\phi$ at $M_{2}$ enforces $\partial_{x}A^{z}=0$ as discussed at
the beginning of this section. It has the advantage that,

$(-1)^{\int_{M_{3}^{>\epsilon}}\widetilde{\phi}\delta\widetilde{A}+\int_{M_{2|\epsilon}}\widetilde{\phi}\partial_{x}\widetilde{A}^{z}}=(-1)^{\int_{M_{3}^{\geq\epsilon}}\delta\widetilde{\phi}\cdot\widetilde{A}},$
(C.31)

which is the same as before. The remaining derivations are exactly the same
and we get,

$\frac{\chi[M_{3}^{\geq\epsilon},\mathbb{Z}_{2}]^{-1}}{|H^{0}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})|}\sum_{A\in
H^{1}_{\textrm{sub}}(M_{3}^{[0,\epsilon]},M_{2|0}\cup
M_{2|\epsilon},\mathbb{Z}_{2})}Z[M_{3}^{\geq
0},A+\hat{A}_{M_{3}^{\geq\epsilon}}](-1)^{\int_{M_{3}^{[0,\epsilon]}}A\cdot\hat{A}}.$
(C.32)

We then regularize the integral $\int_{M_{3}^{[0,\epsilon)}}A\cdot\hat{A}$ in
the same way,

$\displaystyle\int_{M_{2}\times[0,\epsilon]}(A^{xy}\hat{A}^{z}+A^{z}\hat{A}^{xy})=$
$\displaystyle\sum_{i=1}^{L_{x}}\left(w_{y,i+\frac{1}{2}}\left[\sum_{k=1}^{L_{z}}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}\right]^{y}\right)+\sum_{j=1}^{L_{\epsilon}}\left(w_{x,j+\frac{1}{2}}\left[\sum_{k=1}^{L_{z}}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}\right]^{x}\right)$
$\displaystyle+\sum_{i=1}^{L_{x}}\left(w_{z,y;i}\sum_{j=1}^{L_{\epsilon}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right)+\sum_{j=1}^{L_{\epsilon}}\left(w_{z,x;j}\sum_{i=1}^{L_{x}}\hat{A}^{xy}_{i,j,k+\frac{1}{2}}\right).$
(C.33)

If we take $\epsilon\rightarrow 0$ while fixing the holonomies $w$, we only
need to keep the first term. Recall that from (3.30) we have,

$\exp\left(i\pi\left[\sum_{k=1}^{L_{z}}\hat{A}^{z}_{i+\frac{1}{2},j+\frac{1}{2},k}\right]^{y}\right)\equiv\hat{W}_{z,y}(x_{i+\frac{1}{2}})\leftrightarrow\prod_{i^{\prime}\leq
i}U^{yz}_{0,i}.$ (C.34)

The fusion rule is then,

$\mathcal{N}^{\textrm{sub}}{}^{\dagger}\times\mathcal{N}^{\textrm{sub}}=\sum_{w_{y,i+\frac{1}{2}}}\prod_{i=1}^{L_{x}}\left(\prod_{i^{\prime}\leq
i}U^{yz}_{0,i}\right)^{w_{y,i+\frac{1}{2}}}.$ (C.35)

We can recover (2.22) using $w_{y,i+\frac{1}{2}}=t_{i}^{y}+t_{i+1}^{y}$.

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