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Brane Brick Models and Their Elliptic Genera 2016 Current Topics in String Theory : CFT Brane Brick Models and their Elliptic Genera Dongwook Ghim Seoul National University 2016. 12. 15 @ KIAS, Seoul 2016. 12. 15. KIAS This talk is based on … • 1506.03818 – Construction of theories SF, DG, SL, RKS, DY • 1510.01744 – Brane Brick SF, SL, RKS • 1602.01834 – (0,2) Triality SF, SL, RKS • 1609.01723 – Mirror perspective SF, SL, RKS, CV • 1609.07144 – Orbifold reduction SF, SL, RKS • 1612. to appear – Elliptic genus SF, DG, SL, RKS Collaborators: Sebastian Franco (CUNY), Sangmin Lee (SNU), Rak-Kyeong Seong (Uppsala), Daisuke Yokoyama (King’s college), Cumrun Vafa (Harvard) 2016. 12. 15. KIAS 2 Brane Brick Model 2d (0,2) gauge theory as a world-volume theory of D1-branes probing toric Calabi Yau 4-fold cones 2d (0,2) quiver gauge theory with U(N) gauge groups Quiver diagram N D1-branes toric CY 4-fold Brane configuration Toric diagram 2016. 12. 15. KIAS 3 Relatives and ancestors D(9-2n)-branes In type-IIB string theory, D(9-2n)-branes probing toric CY n-folds toric CYn dim, T-dual Symmetry n # of SUSY configuration 2 6d, N=(1,0) Necklace quiver - 3 4d, N=1 Brane tiling Seiberg duality 4 2d, N=(0,2) Brane brick GGP triality 2016. 12. 15. KIAS 4 2d (0,2) quiver gauge theories Gauge Theory SUSY multiplets of 2d (0,2) theories multiplets superfield component fields vector V (va, χ , χ ,D) − − chiral Φij (φ, +) Quiver diagrams of Fermi ⇤ij (λ ,G) − 2d (0,2) gauge theories • 2d (0,2) field-strength multiplet with U(N) gauge group + + + Y = χ i✓ (D iF01) i✓ ✓ @+χ − − − − − Φ • 2d (0,2) chiral ij • 2d (0,2) Fermi ⇤ij J-term E-term For SUSY, 2016. 12. 15. KIAS 5 Brane Brick Brane Configuration A graph-dual of periodic quiver gauge group = bulk of brick chiral fields = oriented surfaces graph dual Fermi fields = unorientable srfs or T-dual of brane configuration {arg(x2,3), arg(x4,5), arg(x6,7)}=T3 D1-branes toric CY 4-fold T-dual3 0 1 2 3 4 5 6 7 8 9 D4 NS5 NS5 branes wrap a holomorphic cycle 2016. 12. 15. KIAS 6 Why 3-torus? Toric Calabi-Ya u 4 Toric Calabi-Yau 4-fold has 4 isometries; 3 Each corresponds to U(1)R U(1) global symmetry ⇥ of 2d (0,2) gauge theory. Toric diagrams of toric CY4 cones toric CY 4-fold 2016. 12. 15. KIAS 7 So far… Graph dual Brane Brick Periodic Quiver Phase 3-torus Boundaries Toric diagram Quiver diagram mesonic moduli space 2016. 12. 15. KIAS 8 Outline Introduction to Brane brick done! A new elliptic genus Geometric formula of new elliptic genus Summary 2016. 12. 15. KIAS 9 A new elliptic genus 10 Elliptic genus: review Definition (a ; q)=Tr ( 1)F qHL qHR afi I i RR − i Property Y • q- independence • coupling-independence [Gadde, Gukov 13] Localization for (0,2) GLSM [Benini, Eager, Hori, Tachikawa 13] 0 a χ Y = χ a Qi φi +i +h.c • gaugino zero-mode saturated by L − a i ! X X d2ru dχ0dχ0 d2z χ Qaφ d2z χ Qbφ Z I ⇠ c c a i i i b j j j 1-loop c * i j + Z Z Y Ya,b Z X Z X • JK-residue for charges of bosonic fields 2016. 12. 15. KIAS 11 Why “new”? Index with non-commuting operator insertion [Cecotti, Fendley, Intriligator, Vafa 92] (a ; q)=Tr ( 1)F qHL qHR afi I i RR − O i Y U(1) decouplinge in every theory of brane brick models Ex. a quiver diagram of Co x C model Fermionic zero-mode from decoupled free U(1) [Israel, Sarkis 15] Ex. gaugino zero-mode without Yukawa term [Kologlu 16] a Y = χ a Qi φi +i +h.c L − a i ! X X Insertion of zero-mode part of R-symmetry current operator 0 0 0 JR = χ χ +( ) − − − ··· 2016. 12. 15. KIAS 12 Property (a ; q)=Tr ( 1)F J qHL qHR afi I i RR − R i Y •q -independencee not guaranteed in general [Cecotti, Fendley, Intriligator, Vafa 92] [Israel, Sarkis 15] •coupling-independence 0 0 0 With the insertion of JR = χ χ +( ) , − − − ··· 2 1 1 d z 0 0 ⌫ [ ] ⌫ [Φ,⇤, ] ⌫free[ ] (e, g)= Φ ⇤ χ χ e− e2 Q int A −Q m A e− g2 Q A I D D DADA Im⌧ Z ✓Z ◆ e e e @ 0 0 S I 2 = Φ ⇤ χ χ ⌫int e− @(1/e ) D D DADA ⌃ Q e Z ✓Z ◆ 0 0 S 0 0 S = Φ ⇤ e χ χ ⌫ e− χ χ ⌫ e− D D DADAQ int − Q int Z ✓Z⌃ ◆ Z⌃ e 2016. 12. 15. KIAS 13 Property (a ; q)=Tr ( 1)F J qHL qHR afi I i RR − R i Y • coupling-independencee (cont’d) @ 0 0 S 0 0 S I 2 = Φ ⇤ χ χ ⌫int e− χ χ ⌫int e− @(1/e ) D D DADAQ ⌃ − Q ⌃ e Z ✓Z ◆ Z =0 e Note that χ = D iF01 =0 and ⌫ = χ (D + iF01). Q − 6 − − However, gaugino zero-mode from free U(1) vector is NOT saturated by them: dχ0dχ0 1=0. Z Thus, is protected under RG flow I e 2016. 12. 15. KIAS 14 Computation • Decoupled gaugino: zero-mode-saturated one-loop determinant (eta-function) dχ0dχ0 χ0χ0 =1 Z 0 0 S 2 χ χ χ χ e− free = (m + n⌧)= i⌘(q) D D − Z (m,n)=0 Y6 • Refinement with fugacity a x, y, z corresponding i 2 { } to 3 global symmetries for regularization 1 ⌘(q)2 (ai; q)= JK-Res (Q u , ⌘) Z1-loop(u, ai; q) u=u ⇤ I W ⇤ | i u M⇤ | | ⇤2Xsing e r • For G0 = U(1) /U(1)overall , we need to read JK-residue of the one-loop determinant factors from the interacting sectors (vector, chiral and fermi multiplets) 2016. 12. 15. KIAS 15 Geometric formula for new elliptic genus 16 Building block : C4 theory field U(1)x U(1)y U(1)z U(1)R X +1/2 1/2 1/21/2 Y 1/2+1− /2 −1/21/2 Z −1/2 1/2+1− /21/2 D +1− /2+1− /2+1/21/2 ⇤1 1000 ⇤2 0100 4 Quiver diagram of C theory ⇤3 0010 Global charges of matter fields in C4 theory Its new elliptic genus 3 ⌘(q) ✓1(q, x)✓1(q, y)✓1(q, z) C4 (x, y, z; q)= . I ✓1(q, x/yz)✓1(q, y/xz)✓1(q, z/xy)✓1(q, pxyz) e p p p … made of the following elements ⇢ r 2 ⌘(q) ✓1(q, x ) 2⇡⌘(q) ZΦ = i ⇢ ,Z⇤ = i ,ZV = dua . ✓1(q, x ) ⌘(q) i ⇢ ⇢ a=1 Y2R Y2R Y 1 ⌘(q)2 (ai; q)= JK-Res (Q u , ⌘) Z1-loop(u, ai; q) . u=u ⇤ I W ⇤ | i u M⇤ | | ⇤2Xsing e 2016. 12. 15. KIAS 17 Toric diagram Toric diagram (in Z4, but projected to Z3) encodes the geometric data of transverse CY4 geometry. Toric diagrams of toric CY4 cones toric CY 4-fold 2016. 12. 15. KIAS 18 Geometric formula from triangulation Triangulation of toric diagram dual cone for each {a} 1 (wap ) = ✏apaq ar as ✏ vj vk vl . i (3!)2 ijkl aq ar as 3 x TM three {a} (minimal tetrahedron) Sum of elliptic genera (C4-like pieces) corresponding to each building block, a minimal tetrahedron 4 ap a (w )i y{ }(t)= t , 3 3 a p i ⌘(q) e=1 ✓1 q, ze{ }(t) i=1 (t)= . Y M 4 ⇣ a ⌘ a y1y4 a y2y4 a y3y4 J Q { } a p=1 ✓1 q, yp (t) z1{ }(t)= ,z2{ }(t)= ,z3{ }(t)= . { X}2TM y y y y y y e r 2 3 r 3 1 r 1 2 Q ⇣ ⌘ X (x, y, z; q)= (t) . ↵ β γ I JM ti=x i y i z i (inspired by) [Martelli, Sparks, Yau 05] e e 2016. 12. 15. KIAS 19 4 Example : C /Z2 (0011) 4 2 2⇡i⌘(q) ✓1(q, z) Z1-loop = − 2 2 ✓1(q, x/yz) ✓1(q, y/xz) p ✓1(q, ux)p✓1(q, x/u)✓1(q, uy)✓1(q, y/u) 1 1 ⇥ ✓1(q, u z/xy) ✓1(q, u− z/xy)✓1(q, upxyz)✓1(q, u− pxyz) p p JK-residue for Qu = +1 4 Quiver diagram of C /Z2 theory ⌘(q)3✓ (q, z) = 1 I ✓1(q, x/yz)✓1(q, y/xz)⇥ e ✓ (q, x3yz)✓ (q, xy3z) ✓ (q, x3y/z)✓ (q, xy3/z) 1 p 1 p + 1 1 ✓ (q, xy)✓ (q, xyz) ✓ (q, xy)✓ (q, xy/z) 1p 1 p p1 1 p x y t = ,t= ,t= xy , t =1 1 yz 2 zx 3 4 r r 3 t1t2 t1t3 t2t3 ⌘(q) ✓1(q, )✓1(q, )✓1(q, ) = pt4 pt4 pt4 I ✓ (q, t )✓ (q, t )✓ (q, t )✓ (q, t4 ) 1 1 1 2 1 3 1 t1t2t3 3 t1t2 t1 t2 e ⌘(q) ✓1(q, )✓1(q, )✓1(q, ) + pt4 t3pt4 t3pt4 1 t3t4 ✓1(q, t1)✓1(q, t2)✓1(q, t3− )✓1(q, ) 4 t1t2 Toric diagram of C /Z2 2016. 12. 15. KIAS 20 Discussion Gauge theory from brane brick incorporates geometry. Geometric formula Gauge theory formula a 3 3 2 ⌘(q) e=1 ✓1 q, ze{ }(t) 1 ⌘(q) = = JK-Res Z1-loop(u, ai; q) IM 4 ⇣ a ⌘ I W u=u i a pQ=1 ✓1 q, yp{ }(t) u ⇤ { X}2TM ti=fi(x,y,z) | | ⇤ e X Q ⇣ ⌘ e 1.
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