Self-Duality for Landau--Ginzburg models
Brian Callander, Elizabeth Gasparim, Rollo Jenkins, Lino Marcos Silva
aa r X i v : . [ m a t h . AG ] O c t SELF-DUALITY FOR LANDAU–GINZBURG MODELS
B. CALLANDER, E. GASPARIM, R. JENKINS, L. M. SILVA
Abstract.
P. Clarke describes mirror symmetry as a duality betweenLandau–Ginzburg models, so that the dual of an LG model is anotherLG model. We describe examples in which the underlying space is atotal space of a vector bundle on the projective line, and we show thatself-duality occurs in precisely two cases: the cotangent bundle and theresolved conifold. Introduction
For us a Landau–Ginzburg model (LG) is a variety X together with aregular function W : X → C called the superpotential. Clarke [1] showedthat one can state a generalised version of the Homological Mirror Symmetryconjecture of Kontsevich [4] as a duality between LG models. He also showedthat this correspondence generalises those of Batyrev–Borisov, Berglung–H¨ubsch, Givental, and Hori–Vafa.This paper is an exercise in understanding the details of this correspon-dence. We summarise the construction in [1], which, for a given LG model( X, W ), produces a dual ( X ∨ , W ∨ ). When ( X ∨ , W ∨ ) ∼ = ( X, W ), we call X self-dual. We then study the case when X is the total space of a vec-tor bundle on P and prove that self-duality occurs in only two cases: X = Tot( O ( − X = Tot( O ( − ⊕ O ( − The Character to Divisor Map
Let X be a toric variety of rank n with a torus embedding ι : T −→ X .The torus T = ( C ∗ ) n is an algebraic group, whose algebraic functions arecharacters, that is, group morphisms, χ : T −→ C ∗ . Let M denote the groupof characters of T , and N the group of one-parameter subgroups, naturallyidentified with the dual of M , Hom Z ( M, Z ). Let M R and N R denote thetensor products M ⊗ Z R and N ⊗ R Z , respectively.Since ι ( T ) is dense inside X , each character χ ∈ M can be thoughtof as a rational map, f χ : X C , which is nowhere zero on ι ( T ). Let R = { D , . . . , D r } denote the set of irreducible components of X \ ι ( T ).These are prime T -invariant Weil divisors and can be read off the momentpolytope for X . Since each D ∈ R is irreducible and X is normal, one cancompute the order of vanishing, ord D ( f χ ), of f χ along D . This defines amap, div ( X ) : M −→ Z R ; χ (ord D ( f χ ) , . . . , ord D r ( f χ )) . Choosing ordered generators for M and an ordering of R gives a matrix M div ( X ) ∈ Mat n × r ( Z ). For each D k ∈ R , let v k ∈ N be a generator for thecorresponding ray in the fan. By [2, Section 3.3], ord D k ( f χ ) = h χ, v k i . Thisimplies that, when the bases of N and M are dual, the rows of the matrix M div ( X ) are simply the generating vectors, v k .The cokernel of div ( X ) is the Chow group of X , written A n − ( X ).When X is a complete toric variety, the Chow group can be identified withthe second integral cohomology H ( X, Z ) and is torsion free. The followinglemma is from [1]. Lemma 2.1. [1, Cor. 4.5] If D , . . . , D c are T -invariant Cartier divisorsand X is the total space of the split bundle O Y ( − D ) ⊕ · · · ⊕ O Y ( − D c ) overa toric variety Y , then the character group of X decomposes as M X ∼ = M Y ⊕ Z σ ⊕ · · · ⊕ Z σ c , where σ j is a rational section of O Y ( D j ) whose divisor is D j , interpreted hereas a character of T . The T -invariant Weil divisors of X are the preimagesunder p of the T -invariant Weil divisors of Y as well as the total spaces X j of the c subbundles E ∨ j , where E ∨ j is the dual bundle to ker( π j : E → O ( D j )) .Furthermore, div X = (cid:18) div Y | D | · · · | D c id (cid:19) . with respect to the decomposition of M X above and Z R X = Z R Y ⊕ Z X ⊕ · · · ⊕ Z X c . The Infinitesimal Action on Monomials
Let E be a vector bundle on a K¨ahler manifold Y with a global section w ∈ H ( Y, E ). Assume that X = Tot( E ∨ ) is a toric variety. A superpotential W : X → C is a regular function on X . It can be determined by w as follows.In the category of coherent O Y -modules, there are isomorphisms H ( Y, E ) ∼ = Hom( O Y , E ) ∼ = Hom( E ∨ , O Y ) . Thus, w determines a morphism from E ∨ to O Y , or, equivalently, a regularfunction W on the total space of E ∨ . Since T acts freely on the embeddedtorus ι ( T ) ⊂ X , the zeroes of the function W must lie on the locus of T -invariant divisors. Thus, W ◦ ι : T → C ∗ is a homomorphism of algebraicgroups, which may be expressed as a finite linear sum of characters of T : ι ∗ W = s X i =1 a i ξ i , for scalars a i ∈ C and characters ξ i ∈ M . Set Ξ := { ξ , . . . , ξ s } .The scalars { a , . . . , a s } depend on the initial choice of embedding ι . Inturn, the map ι is determined by a point x ∈ X ; namely, the image of 1 ∈ T . ELF-DUALITY FOR LANDAU–GINZBURG MODELS 3
Write ι x for the map sending 1 to x . If x ′ = tx is another point in ι ( T ) forsome t ∈ T , we have ι ∗ x ′ W = s X i =1 a i ξ i ( t ) ξ i . Let ( C ∗ ) Ξ denote the space of all C ∗ -linear sums of monomials in Ξ – theseare regular functions on T . Now T acts on ( C ∗ ) Ξ as above; that is, if ι ∗ x W ∈ ( C ∗ ) Ξ and t ∈ T , then t · ι ∗ x W := ι ∗ t · x W . In order to eliminatethe dependence of ι ∗ W on the choice of embedding, we consider ι ∗ W as anelement of the quotient ( C ∗ ) Ξ /T . The kernel of the exponential map C n −→ T ; ( t , . . . , t n ) (e t , . . . , e t n ) is isomorphic to Z n , as is the lattice of one-parameter subgroups N . Let Z Ξ denote the kernel of the correspondingexponential map on C Ξ . The action of T on ( C ∗ ) Ξ gives a map f : T −→ ( C ∗ ) Ξ ; t t · ( ξ + · · · + ξ s ). Restricting the derivative f. : C r −→ C Ξ tothe kernel N of e ( − ) yields a map which we denote by mon : N −→ Z Ξ .Hence, the maps f , f., and mon define a morphism of the following shortexact sequences. 0 / / N / / mon (cid:15) (cid:15) C n e ( − ) / / f. (cid:15) (cid:15) T f (cid:15) (cid:15) / / / / Z Ξ / / C Ξ e ( − ) / / ( C ∗ ) Ξ / / N and an ordering of the monomials in Ξallows us to express the map mon as a matrix M mon ( X ) ∈ Mat n × s ( Z ) suchthat the k th row of this matrix is given by the n-tuple ( b , . . . , b n ) definedby the equation ξ k ( t , . . . , t n ) = t b · · · t b n n .4. Toric LG Models A toric Landau–Ginzburg model is a triple, ( X, W, K ), where X isa toric variety, W is a regular function on X and K ∈ A n − ( X ) ⊗ Z C / Z isan element of the Chow group (with C / Z coefficients). To such a model wehave associated linear maps div ( X ) and mon ( X ). Choosing an element L ∈ coker( mon ) ⊗ Z C / Z determines the linear data associated to ( X, W, K );namely, the pairs ( div , K ) and ( mon , L ). We now provide an inverse to thisconstruction.First we specify the conditions on R -linear data ( C, c ) for it to yield anappropriate toric variety. Let C : M → Z r be a linear map, and c ∈ Z r . Wesay that the R -linear data ( C, c ) is kopaseptic if(1) the polyhedral set P = { ξ ∈ M ; Cξ + c ≥ } associated to ( C, c )has non-empty interior; and
B. CALLANDER, E. GASPARIM, R. JENKINS, L. M. SILVA (2) there exists a surjection k : Z r → Z R X ( C,c ) sending standard genera-tors either to standard generators or to zero such that the followingdiagram commutes M Z r Z R X ( C,c ) C div X ( C,c ) k ,where R X ( C,c ) denotes the number of torus-invariant divisors of thetoric variety X ( C, c ).Condition 1 guarantees that the toric variety X ( C, c ) corresponding to thepolyhedral set of (
C, c ) is well-defined, and thus allows us to make sense ofcondition 2. Some of the inequalities Cξ + c ≥ k is almost uniquely determined, the only choice beingwhich redundant condition to drop.Now we need to determine precisely when a potential W (defined on atoric variety X ) is regular. Since it is regular if and only if all its monomialsare regular, and the mon matrix encodes all the information about thosemonomials, we can state our condition in terms of that matrix. Indeed, amonomial ξ is regular if and only if div ξ ≥
0, which implies the followinglemma.
Lemma 4.1. W is regular if and only if div ◦ mon T ≥ . We now combine the above remarks into one definition. Let A and B behomomorphisms of free abelian groups of finite rank such that the domains of A and B have the same rank, and let K and L be elements in coker( A ) ⊗ Z C / Z and coker( B ) ⊗ Z C / Z , respectively. A pair ( A, K ) and (
B, L ) is called C / Z -linear data . Such data is said to be kopaseptic if(1) ( A, ( ℑ K )) is kopaseptic; and(2) the entries of the matrix A ◦ B T are all non-negative.Here ℑ K denotes the imaginary part of K .Given kopaseptic C / Z -linear data ( A, K ), (
B, L ), we can define the cor-responding toric Landau–Ginzburg model (
X, W, K ) given by(1) the toric variety X := X ( A, ℑ K ) determined by A and ℑ K ;(2) the regular function W := W ( B, L ) determined by B and L .The element K specifies a choice of complexified K¨ahler class for our Landau–Ginzburg model. ELF-DUALITY FOR LANDAU–GINZBURG MODELS 5 Self-duality
Let (
X, W, K ) be a toric Landau–Ginzburg model with linear data ( div ( X ) , K ),( mon , L ). Then the dual ( X ∨ , W ∨ , K ∨ ) of ( X, W, K ) is the toric Landau–Ginzburg model corresponding to the linear data obtained exchanging ( div , K )and ( mon , L ). Lemma 5.1.
Let ( X, W, K ) and ( Y, W ′ , K ′ ) be toric Landau–Ginzburg mod-els. Then ( X × Y, W + W ′ , K + K ′ ) is a toric Landau–Ginzburg model and div ( X × Y ) = div ( X ) ⊕ div ( Y ) and mon ( X × Y ) = mon ( X ) ⊕ mon ( Y ) .Proof. This follows directly from the definitions, given that the torus actionon X × Y agrees with the original actions on X and Y . (cid:3) This immediately implies the following.
Corollary 5.2.
Suppose ( X, W, K ) is a toric Landau–Ginzburg model whichis dual to ( X ∨ , W ′ , K ′ ) . Then ( X × X ∨ , W + W ′ , K + K ′ ) is self-dual. The CY Condition.
There are several inequivalent definitions of aCalabi–Yau manifold. Some authors require that the manifold be a compactcomplex K¨ahler manifold with a Ricci flat metric, while others use a strongercondition that implies the former: a compact complex K¨ahler manifold withtrivial canonical bundle. When a K¨ahler manifold is non-compact, the triv-iality of the canonical bundle does not necessarily imply the existence of acomplete Ricci flat metric. In this case we make the following definition.
Definition 5.3.
A complex K¨ahler manifold is
Calabi–Yau if it has trivialcanonical bundle and admits a complete Ricci-flat metric. Such a metric iscalled a
Calabi–Yau metric .The dual of a Calabi–Yau variety is expected to also be Calabi–Yau.6.
Self-duality for Bundles on P We now describe such dualities for the case when our variety X is thetotal space of a vector bundle on P .6.1. Rank 1.
Let X = Tot( O P ( − k )). For k < E has no global sections,so assume k ≥
0. The chart U := { [ z : 1] ; z ∈ C } of P determines a chart of X on which points may be described as pairs ( z, u ), where u is the coordinatefor the fibre of E ∨ | U . The point x = (1 ,
1) determines the embedding ι x ,so that an element ( t , t ) ∈ T acts on X by ( t , t ) · ( z, u ) = ( t z, t u ).Having embedded the torus this way, Laurent polynomials in t and t canbe interpreted both as characters of the torus T and as rational functions on X . This gives a basis for the group of characters M = h t , t i . Let ν , ν bethe dual basis for the one-parameter subgroups N . The T -invariant divisorsof X are f = { t = 0 } , f ∞ = { t = ∞} and ℓ = { t = 0 } . The momentpolytope for X is given by connecting the vertices (0 , , , k +1 , k = 2. B. CALLANDER, E. GASPARIM, R. JENKINS, L. M. SILVA
Figure 1.
The moment polytope of Tot( O ( − ℓ , f , and f ∞ Remark . The unique value of k for which X is Calabi–Yau is k = 2. Proposition 6.2.
The toric variety X = Tot( O P ( − k )) belongs to a self-dual Landau–Ginzburg model ( X, W, K ) if and only if k = 2 .Proof. With respect to the fixed basis above, the rows of the div -matrix aregiven by the vectors normal to the edges of the moment polytope, which are(1 , , − , k ). Hence M div ( X ) = − k . A global section w of E is represented by a polynomial of degree k (weassume k ≥ P with the subvariety of X cut out by t = 0gives a superpotential W = a t + a t t + · · · + a k t k t for some a , . . . , a k ∈ C .For X to belong to a self-dual toric Landau–Ginzburg model, there mustexist a choice of basis for N and an ordering of Ξ such that M div ( X ) = M mon ( X ). Clearly, Ξ must have cardinality three, so Ξ is a subset of threeof the monomials in { t , . . . , t k t } . With the dual basis for M , the mon -matrix for X is given by M mon ( X ) = a b c , where a, b, c are distinct integers in { , . . . , k } . If a choice of basis for N existssuch that M mon ( X ) = M div ( X ), then there are (non-zero) integers λ, µ ∈ Z such that λ ( a, b, c ) + µ (1 , ,
1) = (1 , − , a + b − c = 0.Likewise, there exist (non-zero) integers λ ′ , µ ′ ∈ Z such that λ ′ ( a, b, c ) + µ (1 , ,
1) = (0 , k, k − a + b − kc = 0. Together these twoequations give ( k − a − c ) = 0, which, since a and c are distinct, impliesthat k = 2.It remains to show that, for k = 2, an element K ∈ A n − ⊗ Z C / Z can bechosen so that ( div , Im ( K )) is kopaseptic. The Chow group in this case isisomorphic to Z by an isomorphism sending the generator (1 , , −
2) in thecodomain of M div ( X ) to 1 ∈ Z . The polyhedral set defined by choosing t > ∈ A n − has non-empty interior and produces inward normals that ELF-DUALITY FOR LANDAU–GINZBURG MODELS 7 determine the fan for X . On the other hand, for t ≤
0, the relation from thethird row of M div ( X ) is made redundant. It follows that lifting (1 , , −
2) to C / Z gives a K such that ( X, W, K ) is self-dual. (cid:3)
Rank Two Bundles.
Now we consider the rank 2 bundles on P whosetotal space is Calabi–Yau, so E = O ( − k ) ⊕ O ( k + 2) on Y = P . Let X = W k := Tot ( E ∨ ). Note that W k ≃ W − k − , so we can assume k ≥ − Proposition 6.3.
The toric variety X = W k belongs to a self-dual toricLandau–Ginzburg model ( X, W, K ) if and only if k = 0 , − .Proof. As in the example above, the chart U := { [ z : 1] ; z ∈ C } of P gives a chart on X on which points may be described as triples, ( z, u, v ),where u is the coordinate along a fibre of O ( k ) | U and v is the coordinatealong a fibre of O ( − k − | U . Let T = ( C ∗ ) be embedded in X so that( t , t , t ) ∈ T acts by the rule ( t , t , t ) · ( z, u, v ) = ( t z, t u, t v ). Againwe let Laurent polynomials in t i represent both the characters of T and therational functions on X . With this notation, the T -invariant divisors are f = { t = 0 } , f ∞ = { t = ∞} , l = { t = 0 } and l = { t = 0 } . Let[ ∞ ] denote the divisor of P which is the intersection of P with f ∞ in X .Applying Lemma 2.1 with c = 2, D = − k [ ∞ ], D = ( k + 2)[ ∞ ], σ = t and σ = t gives the matrix M div ( X ) = − − k k + 20 1 00 0 1 . The following three cases describe the global sections of E = O ( − k ) ⊕ O (2 + k ) on P . H ( P , E ) = H ( P , O (1) ⊕ O (1)) ∼ = C [ x ] ⊕ C [ x ] when k = − H ( P , O (2) ⊕ O ) ∼ = C [ x ] ⊕ C when k = 0, H ( P , O ( k + 2)) ∼ = C [ x ] k +2 when k ≥ k ≥ div and mon matrices decompose into the direct sumof the div and mon matrices for Tot( O ( − k − X belongs to a self-dual Landau–Ginzburg model for k = 0 but notfor k ≥ k = −
1. A generic section of E is a pair of linear polynomials ina single variable. This produces the superpotential W = a t + a t t + b t + b t t on X , where a , a , b , b ∈ C . Judiciously order the monomials in W so that Ξ = { t t , t , t , t t } . Let s , s , s denote one-parameter subgroupsdual to the characters t , t , t . Finally, choose the basis N = h s s , s , s s i .With respect to these choices, M mon ( W k ) = M div ( W k ). The Chow group isisomorphic to Z , which we identify with the subgroup { ( t, t, − t, − t ) ; t ∈ Z } of the codomain of M div ( X ). Again, if t <
0, then the relations fromthe third and forth rows of M div ( X ) are redundant, but t > B. CALLANDER, E. GASPARIM, R. JENKINS, L. M. SILVA polytope with inward normals which define the fan for X . Choosing a liftingof (1 , , − , −
1) yields the required Chow group element K . (cid:3) Higher Rank Bundles.
We recall the following definition.
Definition 6.4.
A vector bundle on a curve is polystable if it is isomorphicto a sum of stable bundles with the same slope.The following theorem of Hori is from [3, Theorem 32.8.8].
Theorem 6.5. (Hori) A holomorphic vector bundle admits a Calabi–Yaumetric if and only if it is polystable.
Theorem 6.6.
Let X be the total space of a vector bundle on P . Suppose,additionally, that such a bundle is Calabi–Yau. Then X is self-dual if andonly if X = O ( − or X = O ( − ⊕ O ( − .Proof. The previous sections deal with the rank one and two cases. TheGrothendieck splitting lemma states that a rank n bundle E on P splits asa sum of line bundles E ∼ = O ( a ) ⊕ · · · ⊕ O ( a n ). The total space of E hastrivial canonical bundle if and only if P a i = − E is a sum of two line bundles, O ( a ) ⊕ O ( b ), with a ≥ b , then the slopeof O ( a ) is greater than or equal to the slope of E . Induction on the rank r of E , for r ≥
2, shows that vector bundles on P of rank r ≥ P are the line bundles. Itfollows that a vector bundle on P is polystable if and only if it is of theform O ( a ) ⊕ · · · ⊕ O ( a )for some a . Therefore, vector bundles on P with rank greater than two donot satisfy the Calabi–Yau condition required for self-duality. (cid:3) Remark . We expect that the hypothesis that the bundle is Calabi–Yaucan be removed from this theorem.
Remark . The Calabi–Yau condition used in Theorem 6.6 is stronger thanthe commonly used definition that only requires triviality of the canonicalbundle. Using the latter, one can apply the algebraic argument from theproof of Proposition 6.2 to show that a Calabi–Yau vector bundle on P can also be a direct sum of O ( −
2) or O ( − ⊕ with O ⊕ k for some k ≥ Acknowledgements
We thank Patrick Clarke for patiently explaining details of his work, andTony Pantev for enlightening discussions.Elizabeth Gasparim was supported by Fapesp grant 2012/10179-5 andRollo Jenkins was supported by Faepex-PRP-Unicamp under grant numberPRP/FAEPEX 005/2014 and Fapesp grant 2013/17654-3.
ELF-DUALITY FOR LANDAU–GINZBURG MODELS 9
References [1] Clarke P.,
Duality for Toric Landau–Ginzburg Models , arXiv:0803.0447(2008)[2] Fulton W.,
Introduction to Toric Varieties , Annals of Math. Studies
PrincetonUniversity Press, Princeton, NJ (1993).[3] Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., ZaslowE.,
Mirror symmetry , Clay Math. Monographs American Mathematical Society andClay Mathematics Institute, Providence RI (2003).[4] Kontsevich M.,
Homological algebra of Mirror Symmetry , Proc. International Con-gress of Mathematicians (Zurich, 1994), Birkh¨auser, Basel (1995) 120–139.
Brian Callander
E-mail address : [email protected] Rollo Jenkins
E-mail address : [email protected] Elizabeth Gasparim
E-mail address : [email protected] Lino Marcos Silva
E-mail address : [email protected]@gmail.com