Extending Hori-Vafa. Richard S. Garavuso. University of Alberta. Collaboration: L. Katzarkov, M. Kreuzer, A. Noll

Transcription

1 Extending Hori-Vafa Richard S. Garavuso University of Alberta Collaboration: L. Katzarkov, M. Kreuzer, A. Noll

2 Outline. Hori-Vafa (hep-th/000) review (a) Hypersurface in WCP m (b) Complete intersection in WCP m. Extension to supervarieties (a) Hypersurface in WCP (m n) (b) Complete intersection in WCP (m n) 3. Period relations 4. Geometric interpretation 5. Supermanifold Hodge numbers 6. Conclusion

3 Hypersurface in WCP m M = {G = 0} WCP m (Q,...,Q m )[s] c (M) 0, Q i > 0 (i =,...,m) (,) U() gauged linear sigma model Z L = d 4 θ Φ i e QiV Φ i e ΣΣ i= Z t d θ «Z «Σ + c.c. + d θ PG(Φ) + c.c. t = r iϑ 3

4 superfield U() charge comments bosonic chiral Φ i Q i D ± Φ i = 0 P s D ± P = 0 bosonic vector V real bosonic twisted Σ D + Σ = 0 D Σ = 0 Σ = D + D V (x 0, x, θ ±, θ ± ) superspace Φ i = φ i + `θ + ψ +i + θ ψ i + θ + θ F i + P = p + Σ = σ + i `θ + λ + θ λ + θ + θ D + D± = θ ± iθ± x ± 0 «x D± = θ ± + iθ± x 0 ± «x 4

5 Φ i = φ i + θ + ψ +i + θ ψ i + θ + θ F i + P = p + Σ = σ + i θ + λ + θ λ + θ + θ D + Nonlinear sigma model phase r >> 0, σ = 0, p = 0 Pm (φ,...,φ m ) i= M = {G = 0} Q i φ i = r U() Landau-Ginzburg mirror period Z m Y Π = dy i dy P e Y P δ Q i Y i sy P t exp i= " # e Y i e Y P i= i= Re Y i = Φ i e Q i V Φ i, Re Y P = Pe sv P Φ i Φ i e iα i Y i Y i + ieα i 5

6 Complete intersection in WCP m C = l\ β= {G β = 0} WCP m (Q,..., Q m )[s,..., s l ] c (C) 0, Q i > 0 (i =,..., m) (,) U() gauged linear sigma model L = Z d 4 θ Z t Φ i e Q i V Φ i e ΣΣ i= 0 d θ «Z lx Σ + c.c. d θ P β G β (Φ) + c.c. A β= Landau-Ginzburg mirror period Z 0 Y m ly Π = dy dy Pβ e Y P β A i= β= 0 lx Q i Y i s β Y Pβ ta i= exp 4 β= lx e Y i i= β= e Y P β 3 5 6

7 Hypersurface in WCP (m n) M = {G = 0} WCP m (Q,..., Q m q,..., q n )[s] c (M) 0, Q i > 0 (i =,..., m), qa > 0 (a =,..., n) (,) U() gauged linear sigma model Z L = d 4 θ Φ i e Q i V Φ i + Ξ a e qav Ξ a e ΣΣ i= a= Z t d θ «Z «Σ + c.c. + d θ PG(Φ,Ξ) + c.c. Super Landau-Ginzburg mirror period Z Y m ny Π = dy i dy P e Y P dx a dη a dγ a i= a= δ Q i Y i sy P q a X a t i= " exp a= # e Y i e Y P e Xa ( + η a γ a ) i= a= Re X a = Ξ a e q av Ξ a Ξ a Ξ a e iβ a X a X a + i e β a 7

8 Φ i = φ i + `θ + ψ +i + θ ψ i + θ + θ F i + P = p + Ξ a = ξ a + `θ + b +a + θ b a + θ + θ χ a + Σ = σ + i `θ + λ + θ λ + θ + θ D + Nonlinear sigma model phase r >> 0, σ = 0, p = 0 M = {G = 0} ( (φ,...,φm ξ,...,ξn) ) Q i φ i + q a ξ a = r i= U() a= 8

9 Complete intersection in WCP (m n) l\ C = Gβ = 0 β= WCP m (Q,..., Q m q,..., q n )[s,..., s l ] c (C) 0, Q i > 0 (i =,..., m), qa > 0 (a =,..., n) (,) U() gauged linear sigma model Z L = d 4 θ Φ i e Q i V Φ i + Ξ a e qav Ξ a e ΣΣ i= a= 0 Z t d θ «Z lx Σ + c.c. d θ P β G β (Φ,Ξ) + c.c. A i= β= β= Super Landau-Ginzburg mirror period Z 0 Y m ly ny Π = dy dy Pβ e Y P β A dx a dη a dγ a 0 Q i Y i i= exp 4 lx s β Y Pβ β= e Y i i= a= q a X a ta a= lx e Y P β β= n X a= 3 e Xa ( + η a γ a ) 5 9

10 Period relations Π WCP (m n) (Q,...,Q m q,...,q n )[s] = Π WCP m (Q,...,Q m )[s,q,...,q n ] Π WCP (m n) (Q,...,Q m q,...,q n )[s,...,s l ] = Π WCP (m n+l ) (Q,...,Q m q,...,q n,s,...,s l )[s l ] = Π WCP (m n+l) (Q,...,Q m q,...,q n,s,...,s l ) 0

11 Geometric interpretation Consider the GLSM associated with M = {G = 0} WCP m (Q,..., Q m q,..., q n )[s] c (M) 0, Q i > 0 (i =,..., m), qa > 0 (a =,..., n). We obtain the super Landau-Ginzburg mirror period Z Π = δ exp m Y i= dy i dy P e Y P Q i Y i sy P i= " ny a= dx a dη a dγ a q a X a t a= e Y i e Y P i= Integrating over Y P yields Z Π = exp m Y i= ny a= " dy i "e t s my i= dx a dη a dγ a e Y i e s t i= Q i e Y i s my i= # e Xa ( + η a γ a ). a= # e Xa ( + η a γ a ). a= ny a= Q i e Y i s # e X qa a s ny a= e X qa a s

12 Now, consider the change of variables where e Y i = my j= y M ji j, e Xa = ny b= η a = x a ˆη a, γ a = x a ˆγ a, x N ba b, s = M ji Q i = i= N ba q a (j =,..., m ; b =,..., n). a= This change of variables is one-to-one up to the action of Γ : y j ω yj y j, x b ω xb x b, ˆη a ω x a ˆη a, ˆγ a ω x aˆγ a, my j= ω M ji y j =, ny b= ω N ba x b =, my j= In terms of the new variables, we obtain ω yj n Y b= ω x b =. where Π = ( ) m+n det (M ji ) det (N ba ) e t/s Z Y m ny dy i dx a dˆη a dˆγ a exp fw = + i= my i= j= y M ji j a= + x a a= + e t/s my j= ˆη aˆγ a n Y b= y j n Y b= x N ba b. x b h i fw,

13 Super Landau-Ginzburg orbifold W/Γ W = + m i= n a= m j= y M ji j ( + x a + e t/s m j= ˆη aˆγ a ) n b= y j n b= x N ba b. x b A-side super CY condition z } { Q i q a = s i= a= z B-side } { fw quasihomogeneous of degree s t When the A-side super CY condition is satisfied, we obtain { W = 0} M = Γ/ J WCP (m+n n), where ej : y j e πi n y j /s y j, x b e πi n x b /s x b, ˆη b e πi nˆη b /sˆη b, ˆγ b e πi nˆγ b /sˆγ b. 3

14 Supermanifold Hodge numbers.(a) Schimmrigk 993 & 996 (b) Candelas, Derrick & Parkes 993. Sethi, Nucl. Phys. B430 (994) 3. (a) Proposed correspondence between N = Landau-Ginzburg orbifolds with integral ĉ and N = nonlinear sigma models. (b) Heuristic method for computing supermanifold Hodge numbers. 3. Garavuso, Kreuzer & Noll JHEP 0903 (009) 007. Investigated the geometrical interpretations associated with Sethi s paper. 4. Categorical approach. 4

15 Consider a super Calabi-Yau hypersurface M = {G = 0} WCP (6 ) (,,,,,,3 q, q )[6], G = φ 6 + φ6 + φ6 3 + φ3 4 + φ3 5 + φ3 6 + φ 7 + ξ ξ. Via Sethi s proposed correspondence, for appropriately chosen (q, q ), M is associated with the Landau-Ginzburg orbifold where G bos /J bos, G bos = φ 6 + φ6 + φ6 3 + φ3 4 + φ3 5 + φ3 6 + φ 7 J bos : φ i e πiq i /s φ i. Using the techniques of Vafa (989) and Intriligator & Vafa (990), the Hodge diamond of G bos /J bos is determined to be The Hodge diamond of M should agree with this result. 5

16 Using Sethi s heuristic method for computing supermanifold Hodge numbers, one obtains for (q, q ) {(,5),(3,3)} untwisted sector z } { third twisted sector z } { For (q, q ) = (,4), the only contribution to the Hodge diamond arises from the untwisted sector and we obtain Thus, for all possible choices of (q, q ), Sethi s heuristic method does not yield agreement with the Landau-Ginzburg orbifold Hodge diamond. 6