Hironori Mori (Osaka Univ.)
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1 Abelian 3d mirror symmetry on RP 2 S 1 Hironori Mori (Osaka Univ.) Particle Theory saka arxiv: , collaborated with Akinori Tanaka (RIKEN), Takeshi Morita (Osaka Univ.) 2015/06/09, String Theory in Greater RIKEN
2 3d mirror symmetry SQE XYZ model L SQE = L SYM + L (q=+1) (q= 1) Q + L Q L XYZ = L X + L Y + L Z + XY Z + (c.c.) Gauge coupling Yukawa coupling IR: same fixed point (superconformal) I SQE = I XYZ Superconformal index is invariant along RG flow
3 Plan Superconformal index Mirror symmetry on RP 2 S 1 Summary & Outlook
4 Superconformal index #(BPS) having quantum numbers commuting with {Q, Q } Refinement of Witten index =Tr H ( 1) ˆFxĤ+ĵ
5 Superconformal index #(BPS) having quantum numbers commuting with {Q, Q } I(x, )=Tr H ( 1) ˆFxĤ+ĵ 3 ˆf Ex) = +8x (fermions if -) state 8 bosons Ĥ + ĵ 3 ˆf 2 4
6 Superconformal index #(BPS) having quantum numbers commuting with {Q, Q } I(x, )=Tr H ( 1) ˆFxĤ+ĵ 3 ˆf Localization = e S[ ] t V Boundary conditions
7 Superconformal index Boundary conditions on Matter multiplet ( :vector with N f ) M N i 1 M i 1 N Z with M2 = N 2 1-loop =1 (depending on M, N) N T M =1 Vector multiplet (U(1) gauge) V (P) A A A,y + A,y + i 1 i 1 + A + A A,y A,y + i 1 + i 1
8 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) A A A,y + A,y A + A A,y A,y + i 1 i 1 + i 1 + i 1 +
9 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity A A A,y + A,y + i 1 i 1 + A + A A,y A,y + i 1 + i 1 ( Z2-holonomy ) ( Wilson line ) e i H A flat = ±1 (mod 2 ) Locus: A = + { =±1} holonomy 0 2 d 2
10 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity A A A,y + A,y + i 1 i 1 + A + A A,y A,y + i 1 + i 1 ( Z2-holonomy ) ( Wilson line ) e i H A flat = ±1 (mod 2 ) Locus: A = + I = { =±1} holonomy 0 2 d 2 Z 1-loop
11 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1
12 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 ( monopole ) ( Wilson line ) B 2Z ± (mod 2 ) Locus: A = +
13 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 ( monopole ) ( Wilson line ) B 2Z ± (mod 2 ) Locus: A = + B 2Z ±=0,
14 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 ( monopole ) ( Wilson line ) B 2Z ± (mod 2 ) Locus: A = + I = B 2Z ± =0, Z 1-loop
15 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 B 2Z ± =0, Z 1-loop
16 Superconformal index Boundary conditions on Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 B 2Z ± =0, Z 1-loop
17 Superconformal index Boundary conditions on Matter multiplet ( :vector with N f ) M N i 1 M i 1 N Z with M2 = N 2 1-loop =1 (depending on M, N) N T M =1 Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 B 2Z ± =0, Z 1-loop
18 Plan Superconformal index Mirror symmetry on RP 2 S 1 Summary & Outlook
19 Mirror symmetry on RP 2 S 1 Boundary conditions on Matter multiplet ( :vector with N f ) M N i 1 M i 1 N Z with M2 = N 2 1-loop =1 (depending on M, N) N T M =1 Vector multiplet (U(1) gauge) V (P) Parity Charge conjugation + Parity A A A,y + A,y + i 1 2 i 1 d + 2 Z 1-loop { =±1} holonomy 0 A + A A,y A,y + i 1 + i 1 B 2Z ± =0, Z 1-loop
20 Mirror symmetry on RP 2 S 1 V (P) SQE XYZ model To decide M, N: correspondence of moduli e + X e Y Q Q Z
21 Mirror symmetry on RP 2 S 1 V (P) SQE XYZ model To decide M, N: correspondence of moduli e + e X Y e e + Y X Q Q Q Q Z Z M = M =
22 Mirror symmetry on RP 2 S 1 V (P) SQE XYZ model To decide M, N: correspondence of moduli e + e e e + X Y single matter onons 2 Q Q Q Q Z Z M = M =
23 Mirror symmetry on RP 2 S 1 V (P) SQE XYZ model I SQE = I XYZ I SQE = I XYZ =3 q 1 8 (q 2 ; q 2 ) (q; q 2 ) a 1 4 a 1 2 q a 1 4 a 1 2 q 3 2 s (a, q; q 2 ) (a 1 q, q 2 ; q 2 ) s (a, q 3 ; q 2 ) (a 1 q 3,q 2 ; q 2 ) 2 1(a 1 q 2,a 1 q; q; q 2, aq 2 s ) 2 1(a 1 q 2,a 1 q 3 ; q 3 ; q 2, aq 2 s ) q ã 4 ã q 2 s (ã 1 2 q 1+ s ; q) (ã 1 2 q s ; q) (ã; q 2 ) (ã 1 q; q 2 ) 2 1 : basic hypergeometric series New!! (z; q) = k=0 (1 zq k ) n 0 ( ; q) n (q; q) n z n = ( z; q) (z; q) q-binomial theorem
24 Mirror symmetry on RP 2 S 1 SQE XYZ model To decide M, N: correspondence of moduli e + e + X X e e Y Y Q Q QQ Z Z single matter onons 2 M = M =
25 Mirror symmetry on RP 2 S 1 SQE XYZ model I SQE = II XYZ SQE = I XYZ 1 2 q 1 8 (q; q 2 ) (q 2 ; q 2 ) (a 1 2 q; q) (a 1 2 ; q) + ( a 1 2 q; q) ( a 1 2 ; q) 1 1(a 1 2 ; a 1 2 q; q, q 1 2 a 1 2 w) 1 1( a 1 2 ; a 1 2 q; q, q 1 2 a 1 2 w) 1 1 : bilateral basic hypergeometric series q 1 8 New!! (ã 1 2 w 1 q 1 2, ã 1 2 wq 1 2, ã 1 q; q 2 ) (ã 1 2 wq 1 2, ã 1 2 w 1 q 1 2, ã; q 2 ) (z; q) = (1 zq k ) k=0 1 1(a; b; q, z) = (q, b/a, az, q/az; q) (b, q/a, z, b/az; q) Ramanujan s sum + Product-to-sum identity of theta functions
26 Plan Superconformal index Mirror symmetry on RP 2 S 1 Summary & Outlook
27 Summary & Outlook (1)Classify parity conditions and compute superconformal indices on RP 2 S 1 V (P) { =±1} holonomy 0 2 d 2 Z 1-loop B 2Z ± =0, Z 1-loop (2)Give exact proof of SQE= XYZ model as new mathematical identities on RP 2 S 1 V (P) q-binomial theorem Ramanujan s sum
28 Summary & Outlook N f 1. Extend to flavors/non-abelian group 2. Insertion of Wilson/Votex loops 3. Find "Holomophic blocks 4. Apply to 3d-3d correspondence 5. From brane construction in string theory
arxiv: v1 [hep-th] 29 Oct 2015
![arxiv: v1 [hep-th] 29 Oct 2015](/thumbs/87/95754305.jpg "arxiv: v1 [hep-th] 29 Oct 2015")
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