Linear Sigma Models and (0 , 2) Mirror Symmetry Ilarion Melnikov University of Chicago String Phenomenology 2008, University of Pennsylvania Based on work in progress with Jock McOrist.
Summary • Mirror Symmetry is a powerful tool in the study of string vacua that preserve (2 , 2) SUSY on the world-sheet. • There are many interesting (0,2) Heterotic compactifications! • Is there Mirror Symmetry off the (2 , 2) locus? Is it useful? • We wish to go beyond exactly soluble examples and describe moduli space, Yukawas, singularities, etc. • We study half-twisted sectors of (0 , 2) deformed linear sigma models for Calabi-Yau hypersurfaces in toric varieties and find encouraging clues: – A/2 model is independent of “complex structure” moduli and solved by quantum restriction formula; – B/2 model is independent of K¨ ahler moduli and reduces to classical geometry.
A Class of (0 , 2) Linear Sigma Models • d = 2 gauge theory, gauge group G ≃ U(1) n − d × K with matter fields Φ i (2 , 2) = (Φ i , Γ i ) , i = 0 , . . . , n with charges Q a i under U(1) n − d , and n − d neutral multiplets Σ a . • Chirality constraints: D Φ i = 0 D Σ a = 0 , D Γ i = E i (Φ , Σ) . �� � d 2 z dθ + L + h.c. • Action: S = S kin + , � n − d � n 8 πi Υ a log q a + Γ 0 P (Φ 1 , ..., Φ n ) + Φ 0 1 Γ i J i (Φ 1 , ... Φ n ) . L = a =1 i =1 • (0,2) SUSY constraint: E 0 P + � i> 0 E i J i = 0 .
A Little Nomenclature n − d n � � 8 πi Υ a log q a + Γ 0 P (Φ 1 , ..., Φ n ) + Φ 0 1 Γ i J i (Φ 1 , ... Φ n ) . L = a =1 i =1 i so that for generic q a , E i at • The V–model : Drop (Φ 0 , Γ 0 ) . Can choose Q a low energies this is a (0,2) NLSM with target-space V —a projective toric variety of dimension d and left-moving bundle E → V , a deformation of T V . • M–model : Reinstate (Φ 0 , Γ 0 ) . When L preserves U(1) L × U(1) R symme- try and � Q a Q a 0 = − i , i> 0 M–model believed to flow to a (0,2) fixed point.
The (2,2) Locus • (2 , 2) SUSY ⇔ E i = Q i a Σ a Φ i for all i , and J i = ∂P ∂ Φ i for i > 0 . • (2,2) Superpotentials: � � � − � dθ + L → dθ + dθ dθ + dθ − Φ 0 P (Φ i ) W ( q a ) + • May choose q a so that at low energies M-model reduces to (2,2) NLSM with target-space Calabi-Yau hypersurface M ≃ { P = 0 } ⊂ V . • Parameters: q a → complexified K¨ ahler (toric); Φ 0 P = � r c r µ r , µ r = � i (Φ i ) p ri , c r → complex structure (polynomial) .
(2 , 2) SUSY and “Linear” Mirror Symmetry 2 = 0 • Topological essentials: Q T S = S top + {Q T ,V } . ; Q T Twist Observables Correlators computation Q + + Q − Q T σ a = 0 � σ a 1 · · · σ a d − 1 � M ( q a ) A instanton sum Q + + Q − Q T µ r = 0 � µ r 1 · · · µ r d − 1 � M ( c r ) classical geometry B • Quantum Restriction: − Q a 0 σ a � σ a 1 · · · σ a d − 1 � M ( q a ) = � σ a 1 · · · σ a d − 1 � V . 1 − Q a 0 σ a • V-model may be solved by toric methods. • Compare M -model to W -model, W is Batyrev mirror of M : � σ a 1 · · · σ a d − 1 � M ( q a ) ↔ � � µ i 1 · · · � µ i d − 1 � W ( � c i ) , � Q a q a = � c i is a global mirror map. i i
(0,2) Deformations and the Half-Twists • Take E i = � a,j Σ a A ai j Φ j . Gauge invariance = ⇒ A a = diag( A a (1) , . . . , A a ( J ) ) , A a ( α ) are n α × n α matrices of parameters. ∂ Φ i by parameters γ while satisfying E 0 P + � • Deform J i = ∂P i Φ i J i ( γ ) = 0 . • Low Energy: (0,2) NLSM with target M and bundle F ( A, γ ) → M . F (0 , 0) = T M . • Two distinct half-twists ( A/2,B/2 ). Both have Q T = Q + . • Observables: Q T σ a = 0 in A/2 , Q T µ r = 0 in B/2 . • Q 2 T = 0 = ⇒ � σ a 1 · · · σ a d − 1 � M = F ( q, A, c, γ ) , � µ r 1 · · · µ r d − 1 � M = G ( q, A, c, γ ) . Is there further decoupling?
Localization and Zero Modes • Fixed point theorem: half-twisted path integral localizes onto field configu- rations annihilated by Q T . • In either half-twist, this locus is given by gauge instantons: D a + f a = 0 φ 0 = 0 → M n — V -model instanton moduli space z φ i = 0 D ¯ P = 0 → a complicated subset of M n • Expand the action around a point in M n and examine the fermion zero modes. Use holomorphy to show − A a 0 σ a � σ a 1 · · · σ a d − 1 � M = � σ a 1 · · · σ a d − 1 � V quantum restriction 1 − A a 0 σ a � µ r 1 · · · µ r d − 1 � M = G ( c, A ? , γ ) classical geometry.
An Example of Quantum Restriction � � 0 0 1 1 1 1 • V -model: V is resolved P 4 1 , 1 , 2 , 2 , 2 . Q a i = . 1 1 0 0 0 − 2 � � � � � � Γ 1 Φ 1 Σ 2 + ǫ 1 Σ 1 ǫ 2 Σ 1 Parameters: q 1 , q 2 , D = . Γ 2 Φ 2 ǫ 3 Σ 1 Σ 2 � 4 � • M -model: a degree hypersurface in V , any J i . 0 • Quantum Restriction: 1 � = 2 [(1 − 2 8 q 1 ) 2 − 2 18 q 2 − 1 . � σ 4 1 q 2 + 2 ǫ 1 (1 − 2 8 q 1 ) − 4 ǫ 2 ǫ 3 ] • Parameter democracy! • Singularity signals opening up of the “Coulomb” branch—a massless σ direction in field space predicted by the one-loop effective potential L eff = Υ a J a (Σ; q, A ) obtained by integrating out massive (Φ i , Γ i ) matter.
Conclusions, Comments, and Further Directions • Studied (0,2) deformations of (2,2) linear sigma models: A/2 correlators in M -model obtained by quantum restriction from V -model. B/2 correlators are given by classical computations. • Correlators of M and W models will help us determine a “linear” (0,2) mirror map. • Counting parameters in linear model is straight-forward and passes a number of checks. • V -model comments: – Computations are made tractable by effective potential techniques. – Deformed quantum cohomology is easily determined. • Likely that B/2 correlators are A -independent. Is this true? • Is there a “linear” (0,2) mirror map? • What is the (0,2) version of special geometry? • Can our techniques be applied to more generic (0,2) theories?
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