A proposal for (0 , 2) mirror symmetry of toric varieties Wei Gu Based on Wei Gu , Eric Sharpe arXiv : 1707 . 05274
Contents ◮ Review of (2,2) mirror symmetry ◮ Proposal for (0,2) mirror symmetry ◮ Future directions
Mirror symmetry ◮ Worldsheet SUSY algebra is invariant under the outer automorphism given by the exchange of the generators Q − ↔ Q − , F V ↔ F A (1) ◮ Q ± and Q ± are SUSY generators, while F V and F A are the generators of vector R-symmetry and axial R-symmetry respectively. ◮ For CYs, mirror symmetry is a duality between interpretations of an SCFT. One implication: rotating the Hodge diamond, which for 3-folds acts as H 1 , 1 1 ⇔ H 2 , 1 2 . ◮ In this talk, we’re going to mainly focus on mirror symmetry for Fano spaces, in which sigma model on Fano space mirror to a LG model, and in particular after reviewing (2,2), we’ll discuss a proposal for (0,2).
A-twist and B-twist A-twist M ′ = M + F V . Q A = Q + + Q − , B-twist M ′ = M + F A . Q B = Q + + Q − , M is the lorentz generator. One can easily find that A-twist and B-twist are exchanged under mirror symmetry. In this talk, we mainly focus on twisted theory.
(2,2) Lagrangian for general toric variety We consider a GLSM with gauge group U (1) k , and N chiral superfields with gauge charge Q a i . We require N > k . The lagrangian can be written as �� � � � � 1 Φ i e − Q a d 4 θ i V a Φ i − d 2 θ W ( Φ ) + c . c L = Σ a Σ a + e 2 a a i � � � � d 2 � + θ − t a Σ a + c . c . a This is the classical action, the first two terms stand for the kinetic terms of the system, while W is the superpotential.
In the A-twist, the σ ’s are topological observables. (Morrison, Plesser ’94). Benini, Zaffaroni ’15, Cyril, Cremonesi, Park ’15 compute correlation functions for A-twisted GLSM in general by localization. The effective twisted superpotential � � � a � �� � � � � Q a Q a − t a + − 1 W eff = Σ a log i Σ a . i a i (Λ 0 ) + � i log µ t a ( µ ) = t classical i Q a Λ 0 . µ is the physical scale which a we set 1 in the previous slides while Λ 0 is the cutoff scale which for defining the theory at UV. The chiral ring relations are �� � Q a � i Q a = q a . i σ a a i
(2,2) Mirror The dual theory is (Hori, Vafa ’00) � � � � � � � 1 d 4 θ = − ( Y i + Y i ) log Y i + Y i − Σ a Σ a L e 2 a a i �� � � � � d 2 θ e − Y i + c . c . Q a i Y i − t a + Σ a + a i i The matter maps between two theories are Y i + Y i = Φ i e − Q a i V a Φ i
In the B-twisted mirror, the TFT observable is y , the lowest component of Y . One can easily see that x = e − y is also a topological observable and used in B-model. In B-model, all of the terms except superpotential are Q -exact, but in B-model superpotential will not receive quantum correction.
Mirror symmetry For GLSM, now, we want to build more detailed maps between two sides. ⇔ Q A , F V Q B , F A g ( e − y ) f ( σ ) ⇔ � W eff ⇔ W = functions . Correlation functions Correlation The last three maps are only well-defined at the vacuum of the theories, while the first map can even define without at vacuum. The maps imply that if we know the detailed map between the σ and e − y , we can obtain the mirror side’s superpotential and compare the correlation functions between two sides.
Ansatz for the map between observable of two sides The D-term equations are the following constraints � i y i − t a = 0 , Q a i then � � e − y i � Q a i = q a . i We assume the above are the chiral ring relations for B-model, because the chiral ring of A-model is � i ( � i σ a ) Q a i = q a , then a Q a the ansatz of operator mirror map is � Q a i σ a ⇔ e − y i a Sometimes people write the lowest y as superfield Y without causing any confusion. One comment: one can use above to re-derive the terms � i e − Y i should appear in the B-model LG superpotential.
A-model correlation function A-model correlation function. Melnikov, Plesser ’05, Cyril, Cremonesi, Park ’15, Nekrasov, Shatashvili ’14. � σ v ) Z 1 − loop O ( � ( � σ v ) �O ( σ ) � ∼ 0 . = H ( � σ v ) � σ v ∈ V � � ∂ σ a ∂ σ b � H ( � σ ) = det ab W eff . The V denotes the solutions of ∂ � W eff = 0 , a = 1 , · · · , k , for ∂σ a and � a � r i − 1 � � Z 1 − loop Q a ( � σ ) = i σ a . 0 i r i is the R-charge for the chiral superfield Φ i .
B twisted LG correlation function For B-model, the LG correlation functions were obtained much earlier by Vafa , which are � O ( X v ) �O ( X ) � = H ( X v ) , V where X = e − Y , and H ( X v ) = det ( ∂ Θ A ∂ Θ B W ). The Θ A are the fundamental variables in B-model and their detailed meaning are following � V A Y i = i Θ A + t i . A Solving D-term constraints � i Y i − t a = 0, we have i Q a � � Q a i V A Q a i t i = t a . = 0 , i i i
Example: CP 4 The charge matrix is � � Q a i = 1 1 1 1 1 , then V A is i − 1 1 1 − 1 V A = . i − 1 1 1 − 1
CP 4 The A-model twisted superpotential and B-model superpotential are 5 � � e − Y i W eff = − t σ + 5 σ (log σ − 1) , W = i =1 Chiral ring relations are σ 5 = q , X 5 = q There are five vacua. One can also compute that 5 � i σ a = 5 σ 4 , H ( X ) = 5 X 4 Q a H ( σ ) i =1 The correlation functions are � σ 5 k +4 � = q k , � X 5 k +4 � = q k , k ≥ 0 . for
The correlation functions match We can prove the correlation functions match in general under the observable map O ( σ ) ⇔ O ( X ) �� � � Q a H ( σ ) i σ a ⇔ H ( X ) a i The first map is about the match of chiral ring relations, and written the second one in details: �� � �� � Q a i Q b � i Q a �� � det i σ a i a Q a i σ a ab i a � V A i V B i ( Q a = det i σ a ) AB i , a We have used the operator mirror map � a Q a i σ a ⇔ e − y i . The above has been proved in Gu, Sharpe ’17.
Basics of (0,2) superfields The bose (0 , 2) chiral superfield Φ, in some representation of the gauge group, satisfying D + Φ = 0 . Its θ expansion is √ + ( D 0 + D 1 ) φ. 2 θ + ψ + − i θ + θ Φ = φ + + = 0. Here D α is now the gauge-covariant derivatives at θ + = θ
Basics of (0,2) superfields Fermi multiplets: anticommuting, negative chirality spinor super field Λ − , in some representation of the gauge group, obeying √ D + Λ − = 2 E , where E is some superfield satisfying D + E = 0 . The θ expansion of the fermi multiplet is √ √ + ( D 0 + D 1 ) λ − − + E . 2 θ + G − i θ + θ Λ − = λ − − 2 θ E is a holomorphic function of chiral superfields Φ i , it has Θ expansion √ 2 θ + ∂ E + ( D 0 + D 1 ) E ( φ i ) . ψ + , i − i θ + θ E (Φ i ) = E ( φ i ) + ∂φ i
(0,2) mirror symmetry (0,2) GLSM We consider A / 2 twisted abelian GLSM which is a deformation of a (2,2) theory. L = L gauge + L ch + L F + L D ,θ + L J � � 1 + Υ a Υ a d θ + d θ = 2 e 2 a � � � � i 1 d 2 θ Φ i Φ i − d 2 θ Λ − , j Λ − , j − 2 2 i j �� � � d θ + � � t a 1 d θ + Λ − , j J j | + 2 Υ a | θ + =0 + c . c − √ 2 a j
The appearance of function J i is related to the construction of hypersurface in (0 , 2) model, and it has the constraint that � E i J i = 0 . i On Coulomb branch, we have k � σ a E a E I = I (Φ) , a =1 for some holomorphic functions E a I (Φ), and the matter multiplets Φ I , Λ − , I acquire masses k � σ a ∂ J E a M IJ = ∂ J E I | φ =0 = I | φ =0 . a =1
Generic point of coulomb branch � eff = t a − t a Q a α log (det M α ) . α � α k α = N . The vacuum of A / 2-model thus corresponds to eff = ∂ � W eff t a = 0 , ∂ Υ a then we have � q a = e − t a . (det M α ) Q a α = q a , where α McOrist, Melnikov ’08.
P 1 × P 1 The most general possible (0,2) deformation is � M = A σ + B � σ, M = C σ + D � σ, where A , B , C , D are two by two matrices and for N = (2 , 2) locus A = D = I and B = C = 0. The chiral ring relations are det( A σ + B � σ ) = q 1 , det( C σ + D � σ ) = q 2 We will return to this example later.
Any B/2 LG model has the form � � � � � d 2 θ K L = Y i , Y i , F i , F i + K Σ a , Σ a , Υ a , Υ a � � � k N � � d θ + − + W + c . c . a =1 i =1 The Kahler potentials of B/2 LG model are Q -exact and do not contribute the correlation function, so we do not write them out explicitly. A very general expression for mirror W was proposed by Adams et.al ’03 k N � � � � i Υ a 2 ( Q a i Y i − t a ) + β ij F i e − Y j , W = − Σ a F a + a a =1 i =1 i , j where β ij are some parameters and β ij = − δ ij is (2,2) theory.
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