]>NUPHB14605S0550-3213(19)30079-310.1016/j.nuclphysb.2019.03.017The Author(s)High Energy Physics – TheoryTable 13d N=2 SU(2) with Image 1.Table 1SU(2)SU(6)U(1)U(1)R

QImage 2Image 21r

MQQ:=QQ1Image 322r

Y11−64 − 6r

Table 23d N=2 SU(2) with Image 4.Table 2SU(2)SU(4)SU(2)U(1)U(1)U(1)R

QImage 2Image 2110r

Q˜Image 51Image 201r

M:=QQ˜1Image 2Image 2112r

B:=Q21Image 31202r

B¯:=Q˜2111022r

Y111−4−24 − 6r

Table 3USp(2) dual theory.Table 3SU(2)SU(6)U(1)U(1)R

qImage 2Image 2−11 − r

MQQ1Image 322r

Y11−64 − 6r

Y˜116−2 + 6r

Table 4USp(2) dual theory in an SU(4)×SU(2) basis.Table 4SU(2)SU(4)SU(2)U(1)U(1)U(1)R

qImage 2Image 61−101 − r

q˜Image 21Image 20−11 − r

M1Image 2Image 2112r

B1Image 31202r

B¯111022r

Y111−4−24 − 6r

Y˜11142−2 + 6r

Table 5“Chiral” SU(2) dual theory.Table 5SU(2)SU(4)SU(2)U(1)U(1)U(1)R

qImage 2Image 6110r

q˜Image 61Image 2−2−12 − 3r

M1Image 2Image 2112r

B:=q21Image 31202r

Y∼q˜2111−4−24 − 6r

B¯∼YSU(2)111022r

Table 6Third SU(2) dual theory.Table 6SU(2)SU(4)SU(2)U(1)U(1)U(1)R

qImage 2Image 21−101 − r

q˜Image 21Image 221−1 + 3r

B1Image 31202r

B¯111022r

Y111−4−24 − 6r

M∼qq˜1Image 2Image 2112r

Y˜SU(2)1110−22 − 2r

Table 73d N=2 SU(4) with Image 9.Table 7SU(4)SU(2)SU(3)SU(3)U(1)U(1)U(1)U(1)R

AImage 3Image 211100rA

QImage 21Image 21010r

Q˜Image 611Image 2001r¯

M0:=QQ˜11Image 2Image 2011r+r¯

M2:=QA2Q˜11Image 2Image 22112rA+r+r¯

T:=A21Image 10112002rA

B1:=AQ21Image 2Image 61120rA + 2r

B¯1:=AQ˜21Image 21Image 6102rA+2r¯

Z:=Y1Y2Y31111−4−3−34−4rA−3r−3r¯

Y:=Y1Y22Y31111−2−3−34−2rA−3r−3r¯

Table 8Dual of SU(4) with Image 11.Table 8SU(4)SU(2)SU(3)SU(3)U(1)U(1)U(1)U(1)R

aImage 3Image 211100rA

qImage 21Image 61−1−101 − rA − r

q˜Image 611Image 6−10−11−rA−r¯

M011Image 2Image 2011r+r¯

M211Image 2Image 22112rA+r+r¯

B11Image 2Image 61120rA + 2r

B¯11Image 21Image 6102rA+2r¯

Z1111−4−3−34−4rA−3r−3r¯

Y1111−2−3−34−2rA−3r−3r¯

T ∼ a21Image 10112002rA

Z˜:=Y˜1Y˜2Y˜31111233−2+2rA+3r+3r¯

Y˜:=Y˜1Y˜22Y˜31111433−2+4rA+3r+3r¯

Table 93d N=2 SU(4) with Image 19.Table 9SU(4)SU(2)SU(4)SU(2)U(1)U(1)U(1)U(1)R

AImage 3Image 211100rA

QImage 21Image 21010r

Q˜Image 611Image 2001r¯

M0:=QQ˜11Image 2Image 2011r+r¯

M2:=QA2Q˜11Image 2Image 22112rA+r+r¯

T:=A21Image 10112002rA

B:=Q411110404r

B1:=AQ21Image 2Image 31120rA + 2r

B¯1:=AQ˜21Image 211102rA+2r¯

Ybare:=Y1Y2Y3U(1)2 charge: 2111−4−4−24−4rA−4r−2r¯

YAdressed:=YbareA1Image 211−3−4−24−3rA−4r−2r¯

YQ˜dressed:=YbareQ˜21111−4−404 − 4rA − 4r

Table 10First dual of SU(4) with Image 19.Table 10SU(4)SU(2)SU(4)SU(2)U(1)U(1)U(1)U(1)R

aImage 3Image 211100rA

qImage 21Image 61−1−101 − rA − r

q˜Image 611Image 2−10−11−rA−r¯

M011Image 2Image 2011r+r¯

M211Image 2Image 22112rA+r+r¯

B11Image 2Image 31120rA + 2r

B¯11Image 211102rA+2r¯

YAdressed1Image 211−3−4−24−3rA−4r−2r¯

qq˜11Image 6Image 2−2−1−12−2rA−r−r¯

qa2q˜11Image 6Image 20−1−12−r−r¯

aq21Image 2Image 31−1−202 − rA − 2r

aq˜21Image 211−10−22−rA−2r¯

T ∼ a21Image 10112002rA

YQ˜dressed∼q41111−4−404 − 4rA − 4r

Y˜bare:=Y˜1Y˜2Y˜3U(1)2 charge: 2111242−2+2rA+4r+2r¯

Y˜adressed:=Y˜barea1Image 211342−2+3rA+4r+2r¯

B∼Y˜q˜dressed:=Y˜bareq˜211110404r

Table 11Second dual of SU(4) with Image 19.Table 11SU(4)SU(2)SU(4)SU(2)U(1)U(1)U(1)U(1)R

aImage 3Image 211100rA

qImage 21Image 61010r

q˜Image 611Image 2−2−2−12−2rA−r−r¯

M011Image 2Image 2011r+r¯

M211Image 2Image 22112rA+r+r¯

qq˜11Image 6Image 2−2−1−12−2rA−r−r¯

qa2q˜11Image 6Image 20−1−12−r−r¯

T ∼ a21Image 10112002rA

B ∼ q411110404r

B1 ∼ aq21Image 2Image 31120rA + 2r

YAdressed∼aq˜21Image 211−3−4−24−3rA−4r−2r¯

Y˜bare:=Y˜1Y˜2Y˜3U(1)2 charge: 21110022r¯

B1¯∼Y˜adressed:=Y˜barea1Image 211102rA+2r¯

YQ˜dressed∼Y˜q˜dressed:=Y˜bareq˜21111−4−404 − 4rA − 4r

Table 12Third dual of SU(4) with Image 19.Table 12SU(4)SU(2)SU(4)SU(2)U(1)U(1)U(1)U(1)R

aImage 3Image 211100rA

qImage 21Image 21−1−101 − rA − r

q˜Image 611Image 2121−1+rA+2r+r¯

B11Image 2Image 31120rA + 2r

B¯11Image 211102rA+2r¯

YAdressed1Image 211−3−4−24−3rA−4r−2r¯

M0∼qq˜11Image 2Image 2011r+r¯

M2∼qa2q˜11Image 2Image 22112rA+r+r¯

T ∼ a21Image 10112002rA

YQ˜dressed∼q41111−4−404 − 4rA − 4r

aq21Image 2Image 31−1−202 − rA − 2r

aq˜21Image 211342−2+3rA+4r+2r¯

Y˜bare:=Y˜1Y˜2Y˜3U(1)2 charge: 2111−20−22−2rA−2r¯

Y˜adressed:=Y˜barea1Image 211−10−22−rA−2r¯

B∼Y˜q˜dressed:=Y˜bareq˜211110404r

Table 13SU(6) with Image 21.Table 13SU(6)SU(6)SU(4)U(1)U(1)U(1)U(1)R

AImage 2211100rA

QImage 2Image 21010r

Q˜Image 61Image 2001r¯

M0:=QQ˜1Image 2Image 2011r+r¯

M2:=QA2Q˜1Image 2Image 22112rA+r+r¯

T4:=A41114004rA

B0:=Q61110606r

B1:=AQ31Image 221130rA + 3r

B3:=A3Q31Image 2213303rA + 3r

B¯1:=AQ˜311Image 6103rA+3r¯

B¯3:=A3Q˜311Image 63033rA+3r¯

YSU(4)bareU(1)2 charge: −411−6−6−46−6rA−6r−4r¯

YSU(4),Q˜dressed:=YSU(4)bareQ˜4111−6−606 − 6rA − 6r

YSU(4),Q˜Adressed:=YSU(4)bareQ˜A11Image 2−5−6−36−5rA−6r−3r¯

YSU(4),Q˜A3dressed:=YSU(4)bareA2Q˜A11Image 2−3−6−36−3rA−6r−3r¯

Table 14Self-dual of SU(6) with Image 21.Table 14SU(6)SU(6)SU(4)U(1)U(1)U(1)U(1)R

aImage 2211100rA

qImage 2Image 61010r

q˜Image 61Image 6−2−2−12−2rA−2r−r¯

M01Image 2Image 2011r+r¯

M21Image 2Image 22112rA+r+r¯

T4 ∼ a41114004rA

B0 ∼ q61110606r

B1 ∼ aq31Image 221130rA + 3r

B3 ∼ a3q31Image 2213303rA + 3r

YSU(4),Q˜Adressed∼aq˜311Image 2−5−6−36−5rA−6r−3r¯

YSU(4),Q˜A3dressed∼A3Q˜311Image 2−3−6−36−3rA−6r−3r¯

Y˜SU(4)bareU(1)2 charge: −411224−2+2rA+2r+4r¯

YSU(4),Q˜dressed∼Y˜SU(4),q˜dressed:=Y˜SU(4)bareq˜4111−6−606 − 6rA − 6r

B¯1∼Y˜SU(4),q˜adressed:=Y˜SU(4)bareq˜a11Image 6103rA+3r¯

B¯3∼Y˜SU(4),q˜a3dressed:=Y˜SU(4)barea2q˜a11Image 63033rA+3r¯

Table 153d N=2 SU(6) gauge theory with Image 25.Table 15SU(6)SU(5)SU(3)U(1)U(1)U(1)U(1)R

AImage 2211100rA

QImage 2Image 21010r

Q˜Image 61Image 2001r¯

M0:=QQ˜1Image 2Image 2011r+r¯

M2:=QA2Q˜1Image 2Image 22112rA+r+r¯

T4:=A41114004rA

B1:=AQ31Image 261130rA + 3r

B3:=A3Q31Image 2613303rA + 3r

B¯1:=AQ˜3111103rA+3r¯

B¯3:=A3Q˜31113033rA+3r¯

YSU(4)bareU(1)2 charge: −411−6−5−34−6rA−5r−3r¯

YSU(4),Q˜Adressed:=YSU(4)bareQ˜A11Image 2−5−5−24−5rA−5r−2r¯

YSU(4),Q˜A3dressed:=YSU(4)bareA2Q˜A11Image 2−3−5−24−3rA−5r−2r¯

Table 16USp(4) magnetic dual of SU(6) with Image 27.Table 16USp(4)SU(5)SU(3)U(1)U(1)U(1)U(1)R

aImage 3112002rA

qImage 2Image 611232012rA+32r

q˜Image 21Image 6−52−52−12−52rA−52r−r¯

M01Image 2Image 2011r+r¯

M21Image 2Image 22112rA+r+r¯

T4 ∼ a21114004rA

B1 ∼ q21Image 261130rA + 3r

B3 ∼ aq21Image 2613303rA + 3r

YSU(4),Q˜Adressed∼q˜211Image 2−5−5−24−5rA−5r−2r¯

YSU(4),Q˜A3dressed∼aq˜211Image 2−3−5−24−3rA−5r−2r¯

B¯1∼YUSp(2)111103rA+3r¯

B¯3∼YUSp(2),a:=YUSp(2)a1113033rA+3r¯

Table 173d N=2 USp(4) with Image 28.Table 17USp(4)SU(5)SU(3)U(1)U(1)U(1)U(1)R

AImage 311100rA

QImage 2Image 21010r

Q˜Image 21Image 2001r¯

Mm:=QAmQ˜(m=0,1)1Image 2Image 2m11mrA+r+r¯

Bm:=AmQ2 (m = 0,1)1Image 31m20mrA + 2r

B¯m:=AmQ˜2(m=0,1)11Image 6m02mrA+2r¯

T2:=A21112002rA

YUSp(2)111−2−5−36−2rA−5r−3r¯

YUSp(2),A:=YUSp(2)A111−1−5−36−2rA−5r−3r¯

Table 18SU(6) magnetic dual of USp(4) with Image 29.Table 18SU(6)SU(5)SU(3)U(1)U(1)U(1)U(1)R

aImage 2211120012rA

qImage 2Image 61−16230−16rA+23r

q˜Image 61Image 6−56−53−12−56rA−53r−r¯

Mm (m = 0,1)1Image 2Image 2m11mrA+r+r¯

Bm ∼ a1+2mq3 (m = 0,1)1Image 31m20mrA + 2r

YUSp(2)∼aq˜3111−2−5−36−2rA−5r−3r¯

YUSp(2),A∼a3q˜3111−1−5−36−rA−5r−3r¯

T2 ∼ a41112002rA

YSU(4)bareU(1)2 charge: −41113533−2+13rA+53r+3r¯

B¯0∼YSU(4),q˜adressed:=YSU(4)bareq˜a1110022r¯

B¯1∼YSU(4),q˜a3dressed:=YSU(4)bareq˜a3111102rA+2r¯

Table 193d N=2 SU(8) gauge theory with Image 37.Table 19SU(8)SU(2)SU(6)U(1)U(1)U(1)R

AImage 3Image 2110rA

Q˜Image 61Image 201r¯

B¯1:=AQ˜21Image 2Image 312rA+2r¯

B¯5:=A5Q˜21Image 2Image 3525rA+2r¯

T4:=A41Image 381404rA

YSU(6)bareU(1)2 charge: −611−12−64−12rA−6r¯

YSU(6),Q˜dressed:=YSU(6)bareQ˜6111−1204 − 12rA

YSU(6),Adressed:=YSU(6)bareA1Image 21−11−64−11rA−6r¯

YSU(6),A5dressed:=YSU(6)bareA51Image 21−7−64−7rA−6r¯

Table 20Self-dual of SU(8) with Image 37.Table 20SU(8)SU(2)SU(6)U(1)U(1)U(1)R

aImage 3Image 2110rA

q˜Image 61Image 6−3−11−3rA−r¯

B¯11Image 2Image 312rA+2r¯

B¯51Image 2Image 3525rA+2r¯

YSU(6),Adressed1Image 21−11−64−11rA−6r¯

YSU(6),A5dressed1Image 21−7−64−7rA−6r¯

T4 ∼ a41Image 381404rA

Y˜SU(6)bareU(1)2 charge: −61166−2+6rA+6r¯

YSU(6),Q˜dressed∼Y˜SU(6),q˜dressed:=Y˜SU(6)bareq˜6111−1204 − 12rA

Y˜SU(6),adressed:=Y˜SU(6)barea1Image 2176−2+7rA+6r¯

Y˜SU(6),a5dressed:=Y˜SU(6)barea51Image 21116−2+11rA+6r¯

Table 213d N=2 SU(2N) with Image 43.Table 21SU(2N)SU(4)SU(2)U(1)U(1)U(1)U(1)U(1)R

AImage 44111000rA

A¯Image 45110100r¯A

QImage 46Image 4610010r

Q˜Image 471Image 460001r¯

Mk:=Q˜(AA¯)kQ1Image 46Image 46kk11krA+kr¯A+r+r¯

Hm:=A¯(AA¯)mQ21Image 441mm + 120mrA+(m+1)r¯A+2r

H¯m:=A(A¯A)mQ˜2111m + 1m02(m+1)rA+mr¯A+2r¯

BN:=AN111N000NrA

BN−1:=AN−1Q21Image 441N − 1020(N − 1)rA + 2r

BN−2:=AN−2Q4111N − 2040(N − 2)rA + 4r

B¯N:=A¯N1110N00Nr¯A

B¯N−1:=A¯N−1Q˜21110N − 102(N−1)r¯A+2r¯

Tn:=(AA¯)n111nn00nrA+nr¯A

YSU(2N−2)bareU(1)2 charge: −2(N − 1)11−(2N − 2)−(2N − 2)−4−24−2(N−1)rA−2(N−1)r¯A−4r−2r¯

YSU(2N−2),mdressed:=YSU(2N−2)bareA(AA¯)m111m + 3 − 2Nm + 2 − 2N−4−24+(m+3−2N)rA+(m+2−2N)r¯A−4r−2r¯

YSU(2N−2),A¯dressed:=YSU(2N−2)bareA¯N−11112 − 2N1 − N−4−24+(2−2N)rA+(1−N)r¯A−4r−2r¯

YSU(2N−2),Q˜dressed:=YSU(2N−2)bareA¯N−2Q˜21112 − 2N−N−404+(2−2N)rA−Nr¯A−4r

Table 22Self-dual of SU(2N) with Image 48.Table 22SU(2N)SU(4)SU(2)U(1)U(1)U(1)U(1)U(1)R

aImage 44111000rA

a¯Image 45110100r¯A

qImage 46Image 4710010r

q˜Image 471Image 46−(N − 1)−(N − 1)−2−12−(N−1)rA−(N−1)r¯A−2r−r¯

Mk1Image 46Image 46kk11krA+kr¯A+r+r¯

Hm∼a¯(aa¯)mq21Image 441mm + 120mrA+(m+1)r¯A+2r

YSU(2N−2),mdressed∼a(a¯a)mq˜2111m + 3 − 2Nm + 2 − 2N−4−24+(m+3−2N)rA+(m+2−2N)r¯A−4r−2r¯

BN ∼ aN111N000NrA

BN−1 ∼ aN−1q21Image 441N − 1020(N − 1)rA + 2r

BN−2 ∼ aN−2q4111N − 2040(N − 2)rA + 4r

B¯N∼a¯N1110N00Nr¯A

YSU(2N−2),A¯dressed∼a¯N−1q˜21112 − 2N1 − N−4−24+(2−2N)rA+(1−N)r¯A−4r−2r¯

Tn∼(aa¯)n111nn00nrA+nr¯A

Y˜SU(2N−2)bareU(1)2 charge: −2(N − 1)1100022r¯

H¯m∼Y˜SU(2N−2),mdressed:=Y˜SU(2N−2)barea(aa¯)m111m + 1m02(m+1)rA+mr¯A+2r¯

B¯N−1∼Y˜SU(2N−2),a¯dressed:=Y˜SU(2N−2)barea¯N−11110N − 102(N−1)r¯A+2r¯

YSU(2N−2),Q˜dressed∼Y˜SU(2N−2),q˜dressed:=YSU(2N−2)barea¯N−2q˜21112 − 2N−N−404+(2−2N)rA−Nr¯A−4r

Table 233d N=2 Spin(7) with (Nv,Ns)=(3,3).Table 23Spin(7)SU(3)SU(3)U(1)U(1)U(1)R

Q7Image 2110rv

S81Image 201rs

MQQ:=QQ1Image 101202rv

MSS:=SS11Image 10022rs

PA1:=SQS1Image 2Image 612rv + 2rs

PA2:=SQ2S1Image 6Image 6222rv + 2rs

PS3:=SQ3S11Image 10323rv + 2rs

Y:=Y12Y22Y3111−3−64 − 3rv − 6rs

Z:=Y1Y22Y3111−6−64 − 6rv − 6rs

Table 24Self-dual of Spin(7) with (Nv,Ns)=(3,3).Table 24Spin(7)SU(3)SU(3)U(1)U(1)U(1)R

q7Image 2110rv

s81Image 6−32−11−32rv−rs

MSS11Image 10022rs

PA11Image 2Image 4912rv + 2rs

PA21Image 6Image 6222rv + 2rs

PS311Image 10323rv + 2rs

Y111−3−64 − 3rv − 6rs

Z111−6−64 − 6rv − 6rs

MQQ ∼ qq1Image 101202rv

ss11Image 50−3−22 − 3rv − 2rs

sqs1Image 2Image 2−2−22 − 2rv − rs

sq2s1Image 6Image 2−1−22 − rv − 2rs

sq3s11Image 500−22 − 2rs

Y˜11166−2 + 6rv + 6rs

Z˜11136−2 + 3rv + rs

Table 253d N=2 Spin(8) with (Nv,Ns,Nc)=(4,3,0).Table 25Spin(8)SU(4)SU(3)U(1)U(1)U(1)R

Q8vImage 2110rv

S8s1Image 201rs

MQQ:=QQ1Image 101202rv

MSS:=SS11Image 10022rs

PA2:=SQ2S1Image 3Image 6222rv + 2rs

PS4:=SQ4S11Image 10424rv + 2rs

Y:=Y12Y22Y3Y4111−4−64 − 4rv − 6rs

Z:=Y1Y22Y3Y4111−8−64 − 8rv − 6rs

Table 26Self-dual of Spin(8) with (Nv,Ns,Nc)=(4,3,0).Table 26Spin(8)SU(4)SU(3)U(1)U(1)U(1)R

q8vImage 2110rv

s8s1Image 6−2−11 − 2rv − rs

MSS11Image 10022rs

PA21Image 3Image 6222rv + 2rs

PS411Image 10424rv + 2rs

Y111−4−64 − 4rv − 6rs

Z111−8−64 − 8rv − 6rs

MQQ ∼ qq1Image 101202rv

ss11Image 50−4−22 − 4rv − 2rs

sq2s1Image 3Image 2−2−22 − 2rv − 2rs

sq4s11Image 500−22 − 2rs

Y˜11186−2 + 8rv + 6rs

Z˜11146−2 + 4rv + 6rs

Table 273d N=2 Spin(8) with (Nv,Ns,Nc)=(4,2,1).Table 27Spin(8)SU(4)SU(2)U(1)U(1)U(1)U(1)R

Q8vImage 21100rv

S8s1Image 2010rs

S′8c11001rc

MQQ:=QQ1Image 1012002rv

MSS:=SS11Image 100202rs

MS′S′:=S′S′1110022rc

P1:=SQS′1Image 2Image 2111rv + rs + rc

P2A:=SQ2S1Image 312202rv + 2rs

P3:=SQ3S′1Image 6Image 23113rv + rs + rc

P4S:=SQ4S11Image 104204rv + 2rs

P4C:=S′Q4S′1114024rv + 2rc

Y111−4−4−24 − 4rv − 4rs − 2rc

Z111−8−4−24 − 8rv − 4rs − 2rc

Table 28Self-dual of Spin(8) with (Nv,Ns,Nc)=(4,2,1).Table 28Spin(8)SU(4)SU(2)U(1)U(1)U(1)U(1)R

q8vImage 21100rv

s8s1Image 2−2−101 − 2rv − rs

s′8c11−20−11 − 2rv − rc

MSS11Image 100202rs

MS′S′1110022rc

P11Image 2Image 2111rv + rs + rc

P2A1Image 312202rv + 2rs

P31Image 6Image 23113rv + rs + rc

P4S11Image 104204rv + 2rs

P4C1114024rv + 2rc

Y111−4−4−24 − 4rv − 4rs − 2rc

Z111−8−4−24 − 8rv − 4rs − 2rc

MQQ ∼ qq1Image 1012002rv

Y˜111842−2 + 8rv + 4rs + 2rc

Z˜111442−2 + 4rv + 4rs + 2rc

3d Self-dualitiesKeitaNiinii@itp.unibe.chAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, SwitzerlandAlbert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5BernCH-3012SwitzerlandEditor: Clay CórdovaAbstractWe investigate self-dualities in three-dimensional N=2 supersymmetric gauge theories. The electric and magnetic theories share the same gauge group. The examples include SU(2N), SO(7) and SO(8) gauge theories with various matter contents. In some examples, the duality exchanges the role of the baryon and Coulomb branch operators. In other examples, the Coulomb branch operator becomes an elementary field on the dual side. These self-dualities in turn teach us a correct quantum structure of the Coulomb moduli space of vacua. Some dualities exhibit symmetry enhancement.1IntroductionDuality is a very powerful tool of studying the low-energy dynamics in strongly-coupled gauge theories. In supersymmetric gauge theories with four supercharges, this type of duality is called Seiberg duality [1,2]. In general, the electric and magnetic (dual) theories have the different gauge groups and flow to the same infrared physics. In four dimensional spacetime, “self-dualities” where the electric and magnetic theories share the same gauge group were also found [3,4] in addition to the usual Seiberg dualities. It is important to study the self-dualities because these examples sometimes exhibit symmetry enhancement in the far-infrared limit and because the symmetry enhancement is in turn related to the existence of many sets of dualities [5–9].In this paper, we investigate the possibility of the self-dualities in the 3d N=2 supersymmetric gauge theories with various gauge groups and various matter contents. In four dimensions, this was extensively studied in [3,4]. In three dimensions, this was recently investigated in[10,11] for the USp(2N) gauge theories with an anti-symmetric matter. The SU(2) self-duality was also studied in [12]. We will mimic their approach and find similar self-dualities. In 4d, there are chiral anomalies and a particular U(1) global symmetry is anomalous. The 4d duality and the (Higgs branch) operator matching do not respect this broken symmetry. In 3d, the U(1) symmetry is restored and hence the naive dimensional reduction of the 4d self-dualities does not hold. In order to construct the 3d dualities and obtain the correct operator matching, generally speaking, we have to slightly modify the electric or magnetic theory: The Aharony duality [13] considered the U(N) gauge group instead of SU(N), where we can forget about the baryon operators. The Giveon-Kutasov duality [14] considered the Chern-Simons duality and the Coulomb branch is removed. The 3d SU(N) SQCD duality [15] has the complicated U(F−N) (non-simple) dual description. In these examples, the set of the gauge invariant operators becomes more simplified or complicated compared to the 4d cases and evades the mismatch of the moduli operators.We will find that the 3d self-dualities similar to 4d ones are indeed applicable by slightly reducing the matter contents. In 3d, there is an additional branch in the moduli space of vacua, which is called a Coulomb branch. The Coulomb branch is a flat direction of the scalar fields from the vector superfields. In [16], we studied the low-energy dynamics of the 3d N=2 “chiral” SU(N) theories with F fundamental and F¯ anti-fundamental matters (with F≠F¯). We there found that the “chiral” theory can have the richer structure of the quantum Coulomb branch, compared to the “vector-like” theory whose Coulomb branch is one-dimensional. When the theory is “chiral”, the Coulomb branch is dressed by chiral superfields and becomes multi-dimensional. The richness of the (dressed) Coulomb branch can remedy seemingly-invalid duality since we can find the operator matching between the Higgs and Coulomb branch operators. In order to make the Coulomb branch rich, we have to slightly change the matter contents of the 4d self-dualities. The resulting pair of the 3d theories has gauge anomalies if we assume that those self-dualities live in 4d. We will also discuss the connection between the 3d and 4d self-dualities. We will propose self-dualities for SU(2N), Spin(7) and Spin(8) cases and these self-dualities in turn teach us the correct quantum structure of the Coulomb moduli space.The rest of this paper is organized as follows. In Section 2, we will review the self-duality in the 3d N=2 SU(2) gauge theory with six fundamental matters, which would be a simplest case of the self-duality. In Section 3, we consider the self-duality in the 3d N=2 SU(4) gauge theory. In Section 4, we will consider the SU(6) self-duality with a third-order anti-symmetric tensor. In Section 5, we will consider the SU(8) self-duality with two anti-symmetric tensors. In Section 6, we will consider the SU(2N) self-duality which generalizes the SU(4) self-duality. In Section 7, we will move on to the self-dualities of the 3d N=2 Spin(N) gauge theories with vector and spinor matters. In Section 8, we will summarize our findings and give possible future directions.2SU(2) self-dualityFor illustrating how the 3d self-dualities work, let us consider the 3d N=2 SU(2) gauge theory with six fundamental matters (six doublets). The low-energy dynamics of this theory was studied in [17,18] and the self-duality was discussed in [12] (see also [10,11]). The Higgs branch is described by a meson composite MQQ:=QQ while the Coulomb branch is parametrized by a single coordinate Y. When Y obtains a non-zero vacuum expectation value, the gauge group is broken as SU(2)→U(1). The matter content and their quantum numbers are summarized in Table 1. The theory exhibits a manifest global SU(6) symmetry.We can regard this theory as the 3d N=2 SU(2) gauge theory with four fundamentals and two anti-fundamentals although the explicit SU(6) flavor symmetry is invisible. The Coulomb branch is completely the same as the previous one while the Higgs branch operator is decomposed into the meson M, baryon B and anti-baryon B¯ operators in Table 2. In the following subsections, we will review the SU(2) self-dualities of Table 1 or Table 2.2.1USp(2)=SU(2) self-dualFirst, we consider the USp dual of the theory in Table 1. Since the SU(2) group is a member of symplectic groups, we can use the Aharony duality [13] and obtain the USp(2N˜) dual description. The dual gauge group is again SU(2) in this case. The dual theory includes the meson and the Coulomb branch operator as elementary fields. Therefore, all the moduli coordinates are introduced as elementary fields on the dual side. Table 3 shows the matter contents and their quantum numbers of the USp(2) dual theory. The dual theory has a tree-level superpotential(2.1)W=MQQqq+YY˜, which is consistent with all the symmetries in Table 3. One can rewrite the USp(2) dual in such a way that the global SU(4)×SU(2)×U(1)×U(1) symmetry is manifest (see Table 4). The superpotential is decomposed into(2.2)W=Mqq˜+Bq2+B¯q˜2+YY˜. In the following subsections, we will construct other self-dual descriptions where M is only introduce as an elementary field or where B,B¯ and Y are introduced as elementary fields.2.2SU(2) “chiral” self-dualNext, we consider the “chiral” self-dual description. By regarding the SU(2) theory with Image 7 as an SU(2) theory with four fundamentals and two anti-fundamentals (Table 2), we can apply the “chiral” Seiberg duality [16]. The dual gauge group again becomes SU(2). The dual theory contains four fundamentals, two anti-fundamentals and a meson singlet M. The dual theory has a tree-level superpotential(2.3)W=Mqq˜. In this self-dual description, only the eight components of MQQ, which are denoted by M, are introduced as elementary fields. The quantum numbers of the dual fields are summarized in Table 5. The matching of the moduli operators is as follows.(2.4)QQ˜∼M,Q2∼q2Q˜2∼YSU(2),Y∼q˜2 The role of the Coulomb branch and the anti-baryonic operator is exchanged.2.3SU(2) third self-dualWe can construct the third self-dual description where the (anti-)baryonic operators and the Coulomb branch are introduced as elementary fields. The third dual description includes a tree-level superpotential(2.5)W=Bq2+Yq˜2+B¯Y˜SU(2), which is consistent with all the symmetries in Table 6. The global SU(6) symmetry is invisible and only the SU(4)×SU(2)×U(1) subgroup is manifest. The global symmetry will be enhanced only in the infrared limit. The meson MQQ is decomposed into B,B¯ and M∼qq˜ in this self-dual description. The Coulomb branch Y˜SU(2) of the third self-dual is lifted by the superpotential. The superconformal indices of this self-duality were calculated in [16] and the indices are consistent with the symmetry enhancement.3SU(4) self-dualityIn this section, we consider the self-duality in the 3d N=2 SU(4) gauge theory with two anti-symmetric tensors and some (anti-)fundamental matters. We will find the two self-dual examples. The first one is the self-dual of the vector-like theory. The second one is “chiral”.3.1SU(4) with Image 8The first example is the 3d N=2 SU(4) gauge theory with two anti-symmetric tensors and three flavors in the (anti-)fundamental representation. Since the anti-symmetric representation of SU(4) is real, the theory can be regarded as a “vector-like” theory in a four-dimensional sense. The theory is also regarded as the Spin(6) gauge theory and its Coulomb branch can be studied in the same manner as [19,20]. The Coulomb branch is now two-dimensional. The first coordinate corresponds to the breaking SU(4)→SU(2)×U(1)1×U(1)2 and is denoted as Z. The second one corresponds to the breaking SU(4)→SU(2)×SU(2)×U(1) and is denoted as Y. Since the theory is vector-like, these two operators are gauge-invariant. The Higgs branch is described by the following composite operators.(3.1)M0:=QQ˜,M2:=QA2Q˜,T:=A2B:=Q4,B1:=AQ2,B¯1:=AQ˜2 Table 7 summarizes the quantum numbers of the matter fields and the moduli coordinates.The dual description is again given by the 3d N=2 SU(4) gauge theory with two anti-symmetric tensors and three flavors in the (anti-)fundamental representation. The dual theory includes the gauge-singlet chiral superfields M0,M2,B1,B¯1,Y and Z as elementary fields. These are identified with the moduli coordinates on the electric side. The magnetic description includes a tree-level superpotential(3.2)W=M0qa2q˜+M2qq˜+B1aq2+B¯1aq˜2+YZ˜+ZY˜, where Z˜ and Y˜ are the magnetic Coulomb branches corresponding to the breaking SU(4)→SU(2)×U(1)1×U(1)2 and SU(4)→SU(2)×SU(2)×U(1), respectively. The quantum numbers of the magnetic matter contents are summarized in Table 8. The charge assignment of the magnetic matters is determined by the above superpotential. Notice that the global symmetries on the magnetic side are correctly reduced by including the monopole potential. The matching of the chiral rings between the electric and magnetic theories is manifest from Table 8. The non-trivial identification of the gauge invariant operator is T:=A2∼a2.Let us check the validity of this self-duality. First, the parity anomaly matching is satisfied. We can test the USp(4) branch which can be realized by giving the expectation value to one of the anti-symmetric matters. Both the electric and magnetic theories flow to the USp(4) theory with an anti-symmetric matter and six fundamentals. The USp(4) self-duality was recently studied in [10,11] and we can reproduce it.We can also connect this duality to the 4d self-duality of the 4d N=1 SU(4) theory [3] with Image 12 via dimensional reduction and a real mass deformation [15,21]. The 4d electric description is the 4d N=1 SU(4) gauge theory with two anti-symmetric matters and four (anti-)fundamental flavors. The magnetic description is the 4d N=1 SU(4) gauge theory with two anti-symmetric matters, four (anti-)fundamental flavors, four gauge singlets M04d,M24d,B4d,B¯4d and a tree-level superpotential(3.3)Wmag4d=M04dqa2q˜+M24dqq˜+B14daq2+B¯14daq˜2, where the (anti-)fundamental flavor indices run from one to four since the global non-abelian symmetry is now SU(2)×SU(4)×SU(4). By putting the 4d self-dual pair on a circle, we obtain the S1×R3 self-duality with monopole superpotentials(3.4)WeleS1×R3=ηZ(3.5)WmagS1×R3=M04dqa2q˜+M24dqq˜+B14daq2+B¯4d1aq˜2+η˜Z˜, where η,η˜ are the dynamical scales of the 4d gauge interactions. In order to get rid of the monopole effects and to flow into the 3d limit, we introduce a positive real mass to one fundamental matter and a negative real mass to one anti-fundamental matter. This can be achieved by weakly gauging the flavor symmetries and by flowing to the Coulomb branch of the gauged flavor symmetry. The electric theory flows to Table 7 and the monopole potential is turned off [15,21]. On the magnetic side, the real masses are introduced also for the gauge singlets since they are not flavor singlets. The massless components of the meson and (anti-)baryon fields are decomposed into(3.6)M04d=(0M03d00000Z),M24d=(0M23d00000Y)(3.7)Ba4d=(0Ba3d000000),B¯a4d=(0B¯a3d000000), where a is an SU(2) fundamental index. The magnetic theory correctly flows to Table 8. The mechanism of the dynamical generation of the monopole superpotential in (3.2) is unclear in this dimensional reduction process but it is consistent with all the symmetries and correctly lifts the magnetic Coulomb branches.As another consistency check, we can compare the superconformal indices [22–25] of the electric and magnetic theories. The both sides produce the same index(3.8)I=1+x2/3(1t3u3v4+9tu+3v2)+x(6t2v+6u2v)+x4/3(1t6u6v8+9tu+4v2t3u3v4+45t2u2+36tuv2+6v4)+x5/3(54t3uv+18t2v3+6(t2+u2)t3u3v3+54tu3v+18u2v3)+x2(1t9u9v12+9tu+4v2t6u6v8+21t4v2+165t3u3+243t2u2v2+9(5t2u2+4tuv2+v4)t3u3v4+81tuv4+21u4v2+10v6−22)+⋯,where v is a fugacity for the first U(1) symmetry of the anti-symmetric matters and t,u are the fugacities for the remaining U(1) symmetries. The r-charges are fixed to rA=r=r¯=13 for convenience. We computed the indices up to O(x3) and found an exact agreement. The low-lying terms are easily identified with the moduli operators defined in Table 7 and Table 8. In the second term, x2/3t3u3v4 corresponds to the Coulomb branch Z. In the fourth term, 4x4/3t3u3v2 is interpreted as the sum of ZT and Y.3.2SU(4) with Image 13Next, we consider the self-duality in the 3d N=2 SU(4) gauge theory with two anti-symmetric matters, four fundamental matters and two anti-fundamental matters. The theory has no tree-level superpotential. The similar theory was studied in four-dimensions [3], where the theory was vector-like (four flavors) due to the gauge anomaly constraint. Of course, we can regard this theory as the Spin(6) theory with two vectors, four spinors and two complex conjugate spinors. The “chiral-ness” of the theory will allow us to construct various self-dual description.We first investigate the structure of the Coulomb moduli space of this theory. The bare Coulomb branch leads to the gauge symmetry breaking [19](3.9)SU(4)→SU(2)×U(1)1×U(1)2(3.10)(3.11)(3.12) and we denote the corresponding operator as Ybare. Along this Coulomb branch, the massive components are integrated out and the mixed Chern-Simons term is induced between the two U(1) gauge groups(3.13)keffU(1)1U(2)=2. The mixed Chern-Simons term makes the bare Coulomb branch operator Ybare gauge non-invariant. The U(1)2 charge of Ybare is −2. In order to construct the gauge invariant moduli coordinates, we can use the massless components from the anti-fundamental and anti-symmetric matters, Image 17 or Image 18. The dressed (gauge-invariant) Coulomb branch operators become(3.14)YAdressed:=YbareA(3.15)YQ˜dressed:=YbareQ˜2, where the flavor indices of the anti-quark chiral superfields are totally anti-symmetrized while YAdressed has a flavor index of the anti-symmetric matter. The Higgs branch is described by the following composite operators.(3.16)M0:=QQ˜,M2:=QA2Q˜,T:=A2B:=Q4,B1:=AQ2,B¯1:=AQ˜2 Table 9 shows the quantum numbers of the matter contents and the moduli coordinates.3.2.1First self-dualityWe start with the first dual description where the meson fields M0,M2, the baryon fields B1,B¯1 and the dressed Coulomb branch YAdressed are introduced as elementary fields. The dual theory includes a tree-level superpotential(3.17)W=M0qa2q˜2+M2qq˜+B1aq2+B¯1aq˜2+YAdressedY˜adressed. The quantum numbers of the dual matter contents are summarized in Table 10. The charge assignment is completely fixed by requiring that a2 should be identified with T:=A2 and from the above superpotential.The analysis of the Coulomb branch is the same as the electric theory but a different interpretation is necessary. The bare Coulomb branch Y˜bare corresponds to the breaking SU(4)→SU(2)×U(1)1×U(1)2 and requires the “dressing” procedure. The dressed operators are defined by(3.18)Y˜adressed:=Y˜barea(3.19)Y˜q˜dressed:=Y˜bareq˜2. The Coulomb branch dressed by the anti-symmetric matter a is lifted and excluded from the low-energy spectrum by the tree-level superpotential. The other operator Y˜q˜dressed is identified with the baryon operator B:=Q4. This is a bit surprising since Y˜q˜dressed does not include the dual quark superfield at all. The dual baryon q4 is identified with the dressed Coulomb branch YQ˜dressed. The matching of the other operators is found in Table 10.We can test various consistency conditions. The first self-dual description satisfies the parity anomaly matching. This 3d self-duality is connected to the 4d self-duality [3] via dimensional reduction as in the previous subsection. In this case, we have to introduce the real masses to the SU(2) subgroup of the SU(4) (anti-quark) flavor symmetry. The electric theory flows to Table 9. On the magnetic side, the anti-baryon operator is decomposed into B¯1 and YAdressed and we recover the description in Table 10.3.2.2Second self-dualityLet us consider the second self-dual description. The dual description is given by the 3d N=2 SU(4) gauge theory with two anti-symmetric matters, four fundamental matters, two anti-fundamental matters and two meson singlets M0,M2. The dual theory includes the tree-level superpotential(3.20)W=M0qa2q˜+M2qq˜. Table 11 shows the quantum numbers of the matter contents and the moduli coordinates. The charge assignment is determined by the above superpotential and by requiring the identification(3.21)B:=Q4∼q4,T:=A2∼a2. The Coulomb branch of the second dual description is defined in the same manner.(3.22)Y˜adressed:=Y˜barea(3.23)Y˜q˜dressed:=Y˜bareq˜2 The operator matching is manifest from Table 11 and readsB1:=AQ2∼aq2,B¯1:=aq˜2∼Y˜adressedYAdressed∼aq˜2,YQ˜dressed∼Y˜q˜dressed.Let us study the validity of the second self-dual. The parity anomaly matching condition is satisfied. This 3d self-duality is also connected to the 4d self-duality [3] as in the case of the first self-dual. By giving the expectation value to one of the anti-symmetric matters, we can reproduce the USp(4) self-duality [10,11].3.2.3Third self-dualityFinally, we present the third self-dual description (Table 12), where the baryon operators B1,B¯1 and the Coulomb branch operator YAdressed are introduced as elementary fields. The dual theory includes a tree-level superpotential(3.24)W=B1aq2+B¯1Y˜adressed+YAdressedaq˜2. Notice that the anti-baryon B¯1 couples to the dual Coulomb branch operator Y˜adressed and the Coulomb branch field YAdressed couples to the anti-baryon aq˜2. Table 12 shows the quantum numbers of the dual matter fields and the moduli coordinates. The matching of the moduli fields is straightforward:M0:=QQ∼qq˜,M2:=QA2Q˜∼qa2q˜,T:=A2∼a2B:=Q4∼Y˜q˜dressed,YQ˜dressed∼q4Let us check several consistencies of the self-duality. We can find that the dual description leads to the same parity anomalies as the electric side. This third self-duality is also connected to the 4d self-duality [3] as in the case of the first and the second self-dualities. By giving the expectation value to one of the anti-symmetric matters, we can reproduce the USp(4) self-duality studied in [10,11].As a final consistency check, let us compute the superconformal indices [22–25] of these four theories (one electric and three duals). We found that all the theories lead to the same indices and the result is(3.25)I=1+x2/3(8tu+3v2)+x(2t4u2v3+12t2v+2u2v)+x4/3(1t4v4+t4+36t2u2+32tuv2+6v4)+x5/3(2(8tu+3v2)t4u2v3+96t3uv+36t2v3+16tu3v+6u2v3)+x2(3t8u4v6+8t5u+78t4v2+4t4v2+120t3u3+8ut3v4+190t2u2v2+18t2u2v2+72tuv4+3u4v2+10v6−24)+⋯,where v is the fugacity for the U(1) symmetry of the anti-symmetric matter and t,u are the fugacities for the U(1) symmetries of the (anti-)fundamental matters. We set the r-charges as rA=r=r¯=13 for simplicity. We observed the agreement of the indices up to O(x3). The low-lying terms are easily identified with the moduli operators in Table 9 and the symmetric products of them.4SU(6) self-dualityIn this section, we will consider the self-duality in the SU(6) gauge theory with a third-order (totally) anti-symmetric tensor. The four-dimensional version of this duality was studied in [3]. We first discuss the self-duality and then derive the dualities between SU(6) and USp(4) gauge theories which can be obtained via a complex mass deformation in the following subsections.4.1SU(6) with Image 20The electric theory is the 3d N=2 SU(6) gauge theory with a third-order anti-symmetric matter, six fundamental and four anti-fundamental matters. Notice that this theory is “chiral” and has a gauge anomaly in 4d while the 3d version is anomaly-free. The “chiral-ness” of the theory will make the Coulomb branch richer and hence the self-duality well works even in 3d. In 4d, the SU(6) gauge theory with a third-order anti-symmetric matter and six fundamental flavors was considered and the self-duality was proposed [3]. In this case, the theory is “vector-like” and the corresponding 3d theory has a simple Coulomb branch structure and hence we could not find the self-duality.As in the 4d case [3], the Higgs branch is described by the following eight composite operatorsM0:=QQ˜,M2:=QA2Q˜,T4:=A4,B0:=Q6B1:=AQ3,B3:=A3Q3,B¯1:=AQ˜3,B¯3:=A3Q˜3. Table 13 summarizes the quantum numbers of the matter content and these moduli coordinates. Let us consider the Coulomb branch. When the bare Coulomb branch operator YSU(4)bare obtains the vacuum expectation value, the gauge group and the matter fields are decomposed into [19,26,27](4.1)SU(6)→SU(4)×U(1)1×U(1)2(4.2)6→40,−1+11,2+1−1,2(4.3)6‾→4‾0,1+1−1,−2+11,−2(4.4)20→61,0+6−1,0+4‾0,−3+40,3. The components with the U(1)1 charge are massive and integrated out form the low-energy spectrum. Since the theory is “chiral”, the mixed Chern-Simons term between the two U(1) subgroups is generated and then the bare Coulomb branch has a U(1)2 charge. In order to construct the gauge invariant moduli operator, we can combine YSU(4)bare with 4‾0,1∈Q˜ and 40,3∈A. The dressed Coulomb branch operators becomes(4.5)YSU(4),Q˜dressed:=YSU(4)bareQ˜4(4.6)YSU(4),Q˜Adressed:=YSU(4)bareQ˜A(4.7)YSU(4),Q˜A3dressed:=YSU(4)bareA2Q˜A. Notice that YSU(4)bareA2Q˜A∼YSU(4)bare(4‾0,−340,3)4‾0,140,3 should be regarded as an independent operator since we cannot express this contribution in terms of YSU(4),Q˜Adressed and the Higgs branch operators. The dressed Coulomb branch can have the flavor indices and their quantum numbers are summarized in Table 13.The dual description is again the 3d N=2 SU(6) gauge theory with a third-order anti-symmetric matter, six fundamental and four anti-fundamental matters. In addition, the magnetic theory includes the mesonic fields M0,M2 as elementary fields with a tree-level superpotential(4.8)W=M0qa2q˜+M2qq˜. The quantum numbers of the dual fields are summarized in Table 14. The charge assignment is determined by requiring the matching of the baryon operators constructed from the (dual) quarks Q,q and from the superpotential above.The Coulomb branch Y˜SU(4)bare again corresponds to the breaking SU(6)→SU(4)×U(1)×U(1), which must be dressed by the dual matter fields. Y˜SU(4),q˜dressed:=Y˜SU(4)bareq˜4 is identified with YSU(4),Q˜dressed and the other Coulomb branch operators are mapped to the anti-baryon operators. The other operator matching is summarized in Table 14. The role of the anti-baryons and the Coulomb branch operators is exchanged.We can show several consistency checks of this self-duality. First, the parity anomaly matching condition is satisfied. In addition, this self-duality is connected to the 4d self-duality via dimensional reduction and via the real mass deformation [15,17,21] as in the previous examples. In 4d, the self-duality was constructed for the 4d N=1 SU(6) gauge theory with Image 23. By putting this self-dual pair on a small circle, we get the self-duality with monopole superpotentials. In order to get rid of the monopole superpotential, we introduce the real masses to the SU(2) subgroup of the second SU(6) symmetry. The resulting dual pair exactly becomes the 3d duality discussed here. The matching of the moduli operators are highly changed in this flow.4.2SU(6)↔USp(4) dualityFrom the SU(6) self-duality above, one can derive the dual description of the 3d N=2 SU(6) gauge theory with Image 24. This is achieved by introducing a complex mass to one (anti-)fundamental flavor. The electric theory flows to the SU(6) theory with Image 24. The analysis of the Higgs and Coulomb branch coordinates is very similar to the previous subsection. The results are summarized in Table 15. Notice that YSU(4),Q˜dressed is not available since the number of anti-fundamentals is less than four.On the magnetic side, the complex mass deformation corresponds to the Higgsing of the SU(6) gauge symmetry. The SU(6) magnetic gauge group is broken to USp(4). The magnetic side flows to the 3d N=2 USp(4) gauge theory with an anti-symmetric matter and eight fundamentals. The theory also includes the gauge singlet fields M0 and M2. The global SU(8) symmetry is explicitly broken by the tree-level superpotential(4.9)W=M0qaq˜+M2qq˜. The Higgs branch is described by five composite operators a2,q2,aq2,q˜2 and aq˜2. The flat directions qq˜ and qa2q˜ are lifted by the above superpotential. The matching with the electric moduli fields are indicated in Table 16. The Coulomb branch was studied, for example, in [10,11] (see also [28,29]) and now two-dimensional. When the Coulomb branch YUSp(2) gets an expectation value, the gauge group is broken USp(2)×U(1). The bare Coulomb branch is gauge invariant and described by YUSp(2) and YUSp(2),a:=YUSp(2)a. Notice that, under this breaking, the anti-symmetric matter is reduced to a massless singlet and YUSp(2)a should be regarded as an independent gauge invariant operator. These Coulomb branch operators are mapped to the anti-baryonic operators B¯1 and B¯3, respectively.4.3USp(4)↔SU(6) dualityBy using the duality between the SU(6) and USp(4) gauge theories, we can think of the USp(4) theory as an electric description. The electric theory is the 3d N=2 USp(4) gauge theory with an anti-symmetric matter and eight fundamentals without a superpotential. Now, the USp(4) theory has an explicit SU(8) symmetry. However, we will keep only the SU(5)×SU(3) symmetry manifest in order to compare this theory with the magnetic theory. The quantum numbers of the matter fields are summarized in Table 17. The analysis of the moduli coordinates is identical to the previous case and the results are listed in Table 17.The magnetic side becomes the SU(6) gauge theory with Image 24 in addition to the gauge singlets Mm(m=0,1). The tree-level superpotential becomes(4.10)W=M0qa2q˜+M1qq˜. The analysis of the moduli coordinates is identical to the previous case and the matching of the electric and magnetic moduli fields is indicated in Table 18. In this dual description, the global SU(8) symmetry is not manifest and the symmetry will be enhanced to SU(8) in the far-infrared limit.5SU(8) self-dualityThe next example is the 3d N=2 SU(8) gauge theory with two anti-symmetric tensors and six anti-fundamental matters. The similar self-duality was studied in 4d [3]. The Higgs branch is parametrized by three composite operators(5.1)B¯1:=AQ˜2,B¯5:=A5Q˜2,T4:=A4.The bare Coulomb branch YSU(6)bare [19,26,27] corresponds to the gauge symmetry breaking(5.2)SU(8)→SU(6)×U(1)1×U(1)2(5.3)(5.4) The components charged under the U(1)1 subgroup are all massive and integrated out, which results in the mixed Chern-Simons term keffU(1)1,U(1)2=6. Therefore, the bare Coulomb branch YSU(6)bare is not gauge invariant and has the U(1)2 charge. In order to construct the gauge invariant coordinates, we can use Image 32 and Image 33. The dressed Coulomb branch operators are defined as(5.5)(5.6)YSU(6),Adressed:=YSU(6)bare10,6∼YSU(6)bareA(5.7) whose flavor indices are manifested in Table 19. Notice that YSU(6),Adressed×T4 has ten independent components while Image 36 has twelve components. The difference can be identified with an independent operator YSU(6),A5dressed.Let us move on to the magnetic description (Table 20) which is again the 3d N=2 SU(8) gauge theory with two anti-symmetric tensors and four anti-fundamental matters. The elementary gauge singlet fields on the magnetic side are B¯1,B¯5,YSU(6),Adressed and YSU(6),A5dressed. The theory includes a tree-level superpotential(5.8)W=B¯1a5q˜2+B¯5aq˜2+YSU(6),AdressedY˜SU(6),a5dressed+YSU(6),A5dressedY˜SU(6),adressed which lifts almost all the moduli fields on the magnetic side except for T4∼a4. We can easily check that the self-duality satisfies the parity anomaly matching condition. As in Section 3, this self-duality can be related to the 4d self-duality [3] of the SU(8) gauge theory with Image 39 via dimensional reduction and the real mass deformation of the SU(2) subgroup in the global SU(8) symmetry. The massless components B¯178 and B¯578 of the 4d singlet chiral superfields are identified with the two electric Coulomb branch operators, YSU(6),Adressed and YSU(6),A5dressed. Although the mechanism of the dynamical generation of the monopole potential is unclear, the above superpotential is consistent with all the symmetries.As a consistency check of the self-duality, we can derive the self-duality of the 3d N=2 USp(8) theory with one anti-symmetric matter and six fundamentals [10,11]. This flow can be achieved by introducing the vev to one of the anti-symmetric matters as follows.(5.9)〈A2〉=v(iσ20000iσ20000iσ20000iσ2) The electric SU(8) theory is Higgsed to the USp(8) theory with Image 40. On the magnetic side, the vev for A2 is mapped to the vev of a2 with the same form(5.10)〈a2〉=v′(iσ20000iσ20000iσ20000iσ2). The magnetic side becomes the USp(8) theory with Image 40. The gauge singlet chiral superfields are decomposed into the singlets introduced in [10,11] and the same superpotential is reproduced.6SU(2N) self-dualityIn this section, we will generalize the self-duality of the SU(4) gauge theory with Image 41 which was discussed in Section 3. The electric theory is the 3d N=2 SU(2N) gauge theory with an anti-symmetric flavor, four fundamental and two anti-fundamental matters. The matter contents and their quantum numbers are summarized in Table 21. The similar 4d theory and its self-dual were studied in [3]. The Higgs branch is the same as the 4d case with a small modification and described byMk:=Q˜(AA¯)kQ,Hm:=A¯(AA¯)mQ2,H¯m:=A(A¯A)mQ˜2BN:=AN,BN−1:=AN−1Q2,BN−2:=AN−2Q4B¯N:=A¯N,B¯N−1:=A¯N−1Q˜2,B¯N−1:=A¯N−1Q˜2,Tn:=(AA¯)n, where k=0,⋯,N−1, m=0,⋯,N−2 and n=1,⋯,N−1. Let us consider the Coulomb branch which was absent in 4d. The bare Coulomb branch operator YSU(2N−2)bare [19,26,27] leads to the gauge symmetry breaking where the Coulomb branch corresponds to the U(1)1 subgroup. The components with a non-zero U(1)1 charge are massive and integrated out from the low-energy spectrum. Since the theory is “chiral”, the mixed Chern-Simons term is generated between the U(1)1 and U(1)2 subgroups. The mixed Chern-Simons term makes the bare Coulomb branch operator charged under the U(1)2 gauge group. In order to construct the gauge invariant moduli coordinate from the bare Coulomb branch, we can use various massless components of the matter fields and define the Higgs-Coulomb composite operators. The dressed Coulomb branch operators are defined in Table 21. They have no flavor index and are only charged under the global U(1) symmetries.The magnetic description is again given by the 3d N=2 SU(2N) gauge theory with an anti-symmetric flavor, four fundamental and two anti-fundamental matters. By following the analysis [3] of the 4d N=1 SU(2N) self-duality, the magnetic theory here includes only the meson operators Mk as elementary fields. The magnetic theory contains a tree-level superpotential(6.1)W=∑k=0N−1Mkq˜(aa¯)N−1−kq, which is consistent with all the symmetries in Table 22. The analysis of the magnetic theory is the same as the electric one and summarized in Table 22.The matching of the moduli coordinates is described in Table 22. The role of the Coulomb and Higgs branch (constructed from the anti-quarks) operators is exchanged under the duality. The parity anomaly matching is satisfied on both sides. As a further consistency check, when A and A¯ obtain the vacuum expectation values, the both theories flow to the SU(2)N gauge theory, where each gauge group contains six fundamentals. As known in [12], the SU(2) theory with six fundamentals has the self-dual description. For N=2, the magnetic self-dual corresponds to the second self-dual description in Section 3.2.2. We can derive this self-duality from the 4d self-duality as in Section 3 by putting the 4d self-dual pair on a circle and by introducing the real masses to two anti-fundamental matters.7Spin(N) self-dualityIn this section, we study the self-dualities in the 3d N=2 Spin(N) gauge theory with vector matters and spinor matters. The corresponding 4d self-dualities were investigated in [3,4]. By following the same spirit as [3,4], we will try to construct the Spin(N) self-dualities. We could find the 3d self-dualities only for Spin(7) and Spin(8) cases. The self-dual examples in 3d contain one spinor matter fewer than the 4d cases. In 4d, the self-dualities were constructed also for Spin(N≥9) cases. However, in 3d, those one-spinor-less theories show s-confinement [20] and the self-duality seems not available. The analysis of the Spin(N≥9) self-dualities would be left as a future direction and we here focus on the Spin(7) and Spin(8) cases.7.1Spin(7) self-dualityLet us first consider the self-duality in the 3d N=2 Spin(7) gauge theory with three vectors and three spinors. The Coulomb branch of the Spin(7) theory was studied in [30] (see also [21,31]). The Coulomb branch is now two-dimensional and parametrized by the two operators Y,Z. When the operator Y obtains a non-zero vacuum expectation value, the gauge group is broken as so(7)→so(5)×u(1). Along this branch, the spinor matters are all massive and integrated out from the low-energy dynamics while the vector matters reduce to the massless SO(5) vectors which can make the vacuum of the low-energy SO(5) theory stable. As a result, the flat direction labeled by Y can become a quantum flat direction. Along the Z branch, on the other hand, the gauge group is broken as so(7)→so(3)×su(2)×u(1). The massless components of the vector matter make only the SO(3) dynamics stable while the massless components of the spinor matters can make both the SO(3)×SU(2) dynamics stable. Therefore, the Spin(7) theory with spinor matters can have this flat direction.The Higgs branch is the same as the 4d case. There are five composite operatorsMQQ:=QQ,MSS:=SSPA1:=SAS,PA2:=SQ2S,PS3:=SQ3S, where the spinor flavor indices are anti-symmetrized in PA1,PA2 and symmetrized in MSS,PS3. The quantum numbers of the matter contents and the moduli coordinates are summarized in Table 23.We consider the dual description of the Spin(7) with (Nv,Ns)=(3,3). The magnetic theory is again the 3d N=2 Spin(7) theory with three vectors and three spinors. In addition, the dual theory includes all the moduli coordinates of the electric theory as elementary fields except for MQQ. The tree-level superpotential takes(7.1)W=MSSsq3s+PA1sq2s+PA2sqs+PS3ss+YZ˜+ZY˜, which is consistent with all the symmetries in Table 24. The above superpotential lifts almost all the flat directions of the magnetic theory and the gauge invariant operator coincide with the electric side. Only the non-trivial identification is the vector meson MQQ:=QQ∼qq. The magnetic Coulomb branch is also two-dimensional and parametrized by Y˜,Z˜. These are excluded from the chiral ring by the F-flatness conditions of Z and Y respectively.Several consistency checks of the Spin(7) self-duality can be performed. First, the parity anomaly matching is satisfied. By giving the vev to one vector, the electric theory flows to the 3d N=2 SU(4) gauge theory with Image 51. The dual gauge group is also broken to SU(4). The magnetic superpotential is decomposed into (3.2) and corresponds to the self-dual description studied in Section 3. We can also check that these two descriptions exhibit the same superconformal indices [22–25]. The index is computed as(7.2)ISpin(7)=1+6x(t2+u2)+9tu2x3/4+x(1t6u6+21t4+45t2u2+21u4)+x5/4(60t3u2+54tu4)+x3/2(6t6u4+56t6+6t4u6+180t4u2+225t2u4+56u6)+x7/4(9t5u4+225t5u2+1t3u6+435t3u4+189tu6)+x2(1t12u12+126t8+525t6u2+21t6u2+1125t4u4+45t4u4+795t2u6+21t2u6+126u8−18)+⋯,where t is a fugacity for the vector U(1) symmetry and u is a fugacity for the spinor U(1) symmetry. The r-charges are set to rv=rs=14 for simplicity. The low-lying operators correspond to the moduli coordinates listed in Table 23. The Coulomb branch Z corresponds to xt6u6 while Y appears as x7/4t3u6. The higher order terms are fermion contributions and the symmetric products of the moduli operators.7.2Spin(8) self-duality7.2.1(Nv,Ns,Ns′)=(4,3,0)Let us move on to the self-duality in the 3d N=2 Spin(8) gauge theory with four vectors and three spinors without a tree-level superpotential. The Coulomb branch of the Spin(8) theory was studied in [20]. The Higgs branch of the Spin(8) theory is identical to the 4d Higgs branch [3] with reduction of Ns from four to three. We need to introduce the four Higgs branch operatorsMQQ:=QQ,MSS:=SSPA2:=SQ2S,PS4:=SQ4S, where the spinor flavor indices of MSS,PS4 are symmetrized and the indices of PA2 are anti-symmetric. Table 25 summarizes the quantum numbers of these operators.The Coulomb branch is now two-dimensional and parametrized by Y and Z [20]. The first Coulomb branch Y corresponds to the breaking so(8)→so(6)×u(1) and the second Coulomb branch corresponds to so(8)→so(4)×su(2)×u(1). Along the Y branch, all the components of the spinor matters are massive while the vector matters reduce to the massless SO(6) vectors which make the vacuum of the low-energy SO(6) theory stable. Along the Z branch, the massless components of the vector and spinor matters can make the vacuum of the low-energy SO(4)×SU(2) theory stable. The quantum numbers of the Coulomb branch coordinates are shown in Table 25.The magnetic theory is again the 3d N=2 Spin(8) gauge theory with four vectors and three spinors. The difference is that the magnetic description includes the five gauge singlet chiral superfields, MSS,PA2,PS4,Y and Z with the tree-level superpotential(7.3)W=MSSsq4s+PA2sq2s+PS4ss+ZY˜+YZ˜, which lifts up various flat directions of the magnetic theory. The quantum numbers of the magnetic fields and the moduli coordinates are summarized in Table 26. The charge assignment is determined by the above superpotential.The operator matching is clear from Table 25 and Table 26. Almost all the moduli fields are introduced as elementary fields on the magnetic side. The meson constructed from the vector matters is identified with(7.4)MQQ:=QQ∼qq. The Coulomb branch operators Y˜,Z˜ of the magnetic theory are all lifted by the above superpotential.Let us check the validity of this self-duality. First, the parity anomaly matching is satisfied. By applying the self-duality twice, we recover the electric theory. By giving the vev to a single vector field, we can flow to the Spin(7) self-duality which was discussed in the previous subsection. We can also test the superconformal indices of these two Spin(8) theories. The two theories give us the same index as follows(7.5)ISpin(8)(4,3,0)=1+x(1t8u6+10t2+6u2)+x(1t16u12+6t8u4+10t6u6+55t4+78t2u2+21u4)+x3/2(1t24u18+6t16u10+10t14u12+21t8u2+78t6u4+220t6+56t4u6+516t4u2+318t2u4+56u6)+x2(1t32u24+6t24u16+10t22u18+21t16u8+78t14u10+56t12u12+715t8+56t8+2370t6u2+318t6u2+2436t4u4+516t4u4+938t2u6+230t2u6+126u8−25)+⋯,where t and u are the fugacities for the vector and spinor U(1) symmetries, respectively. We set the r-charges to rv=rs=14 for simplicity. The second term corresponds to Z, MQQ and MSS. The third term consists of the symmetric products of these three fields and PA2. PS4 appears as t4u2x3/2 while Y corresponds to x3/2t4u6.7.2.2(Nv,Ns,Ns′)=(4,2,1)Finally, we study the self-duality of the 3d N=2 Spin(8) gauge theory with four vectors, two spinors and a single conjugate spinor. The Higgs branch is described by the following composite operators.MQQ:=QQ,MSS:=SS,MS′S′:=S′S′P1:=SQS′,P2A:=SQ2S,P3:=SQ3S′P4S:=SQ4S,P4C:=S′Q4S′ The quantum numbers of these operators are summarized in Table 27. The Coulomb branch is the same as the previous case and two-dimensional. The Y coordinate corresponds to the breaking so(8)→so(6)×u(1) while the Z coordinate is related to the breaking so(8)→so(4)×su(2)×u(1). These flat directions are made stable and supersymmetric by massless components of the vector and spinor matters. The quantum numbers of the corresponding operators are also summarized in Table 27.Let us consider the self-dual description. The dual theory is again the 3d N=2 Spin(8) gauge theory with four vectors, two spinors and a single conjugate spinor. In addition, the dual theory includes the elementary gauge singlet fields, MSS,MS′S′,P1,P2A,P3,P4S,P4C,Y and Z with the tree-level superpotential(7.6)W=MSSsq4s+MS′S′s′q4s′+P1sq3s′+P2Asq2s+P3sqs′+P4Sss+P4Cs′s′+YZ˜+ZY˜. Therefore, on the dual side, all the moduli coordinates are introduced as elementary fields except for the meson constructed from the vector matters. The non-trivial matching of the chiral ring generators is MQQ:=QQ∼qq. The charge assignment of the dual fields is determined by the superpotential and the meson matching. The quantum numbers of the dual moduli fields are summarized in Table 28.We can check several consistencies of this self-duality. First, the parity anomaly matching is satisfied. By applying the self-duality twice, we can go back to the electric description. By giving a vacuum expectation value to a single vector, the electric and magnetic theories flow to the Spin(7) duality which was studied in the previous subsections. Finally, we can test the superconformal indices of this self-duality. The indices of the electric and magnetic theories become(7.7)ISpin(8)(4,2,1)=1+x(1t8u4v2+10t2+3u2+v2)+8tuvx3/4+x(1t16u8v4+55t4+36t2u2+10t2v2+10t2+3u2+v2t8u4v2+6u4+3u2v2+v4)+x5/4(8t7u3v+88t3uv+24tu3v+8tuv3)+x3/2(1t24u12v6+220t6+228t4u2+56t4v2+78t2u4+72t2u2v2+10t2v4+10t2+3u2+v2t16u8v4+56t4+36t2u2+10t2v2+6u4+3u2v2+v4t8u4v2+10u6+6u4v2+3u2v4+v6)+x7/4(8t15u7v3+520t5uv+312t3u3v+88t3uv3+8(11t2+3u2+v2)t7u3v+48tu5v+24tu3v3+8tuv5)+⋯,where t,u and v are the fugacities corresponding to the three U(1) global symmetries. We set the r-charges to rv=rs=rc=14 for simplicity. The Coulomb branch Z is represented as x12t8u4v2 while Y appears as x32t4u4v2. The other terms are identified with the Higgs branch operators and the symmetric products of the gauge invariant operators in Table 27.8Summary and discussionIn this paper, we constructed the self-dualities in 3d N=2 supersymmetric gauge theories with various gauge groups and various matter contents. The self-dual examples include the SU(2), SU(4), SU(6), SU(8), SU(2N), Spin(7) and Spin(8) gauge theories with the particular matter contents. The 3d self-dualities are related to the 4d self-dualities via dimensional reduction and a real mass deformation. In 3d, there is an additional moduli space called the Coulomb branch and the matching of the moduli coordinates becomes more complicated, compared to the 4d case. The (dressed) Coulomb branch is mapped to the (anti-)baryon operator, to the gauge singlet chiral superfields or to the magnetic Coulomb branch. For some self-dual examples, we computed the superconformal indices of the self-dual pair and observed the beautiful agreement of the indices. For the SU(6) case, we also derived the duality between the USp(4) and SU(6) gauge theories, which exhibits the symmetry enhancement.Since we are interested in the 3d self-duality, the matter contents are very restricted. In (Seiberg-like) dualities, almost all the meson operators are introduced as elementary fields on the magnetic side and these meson (gauge-singlet) fields must couple with the magnetic composite operators via tree-level superpotentials. This is mandatory because the magnetic gauge-invariant composites should be excluded from the moduli space. Those superpotentials are available only for a very particular choice of the gauge group and matter contents. In the SU(N) gauge theory with an anti-symmetric matter, for instance, a 4d non-self duality is not known while the 4d self-duality is known for a particular matter content. In this paper, we constructed the similar 3d self-duality for the particular matter content. For the Spin(N) theories with generic vector and spinor matters, the Seiberg duality is constructed in 4d and described by the SU(N˜) gauge theory [32–37]. It is important to construct the corresponding 3d (non-self) duality for the Spin(N) groups.It is worth further checking the validity of our self-dualities. For example, one can compute the partition function on both sides of the self-dualities. This would be an independent check of our self-dualities. It would be important to study the connection between the 3d and 4d self-dualities by using the 4d superconformal indices and the 3d partition function, which are related to each other via dimensional reduction. Although we found the relation between the 3d and 4d self-dualities, the mechanism of the dynamical generation of the monopole potential like (3.2) was unclear. The monopole potential is usually generated by the Affleck-Harvey-Witten superpotential [38] from the low-energy monopole. The interaction between the monopole and a singlet field is regarded as the interaction between two monopoles by dualizing one monopole into a scalar singlet. However, in our example, the singlet comes from the one component of the Higgs branch operators. Currently, we don't know how to generate such interactions from the monopole argument. We will leave this problem for a future work. For the Spin(N) self-dualities, we could not find the relation between the 3d and 4d Spin(N) self-dualities. This would be also left as a future problem. Although we constructed the various self-dualities in this paper, we did not exhaust all the possibilities of the 3d self-dualities. 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