Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 5, 2022

A Landau–Ginzburg mirror theorem via matrix factorizations

  • Weiqiang He , Alexander Polishchuk , Yefeng Shen EMAIL logo and Arkady Vaintrob

Abstract

For an invertible quasihomogeneous polynomial 𝒘 we prove an all-genus mirror theorem relating two cohomological field theories of Landau–Ginzburg type. On the B-side it is the Saito–Givental theory for a specific choice of a primitive form. On the A-side, it is the matrix factorization CohFT for the dual singularity 𝒘 T with the maximal diagonal symmetry group.

Award Identifier / Grant number: DMS-1700642

Funding source: Simons Foundation

Award Identifier / Grant number: 587119

Funding statement: Supported by Simons Collaboration grant 587119 (Yefeng Shen), NSF grant DMS-1700642 (Alexander Polishchuk), and Russian Academic Excellence Project 5-100 (Alexander Polishchuk).

A Reconstruction for polynomials of the chain type

In this section we present a proof of the reconstruction result Proposition 4.7 for a polynomial of the chain type, 𝒘 = i = 1 n - 1 x i a i x i + 1 + x n a n .

Consider a nonzero correlator (4.6)

X = θ n , , θ n n , θ n - 1 , , θ n - 1 n - 1 , , θ 1 , , θ 1 1 , α , β ,

with

α = i = 1 n θ i p i , β = i = 1 n θ i q i ( 𝒘 , G 𝒘 ) ,

as in (4.7) (except that here we are using lower-case letters p i and q i which should not lead to a confusion since the weights of the variables do not appear in the appendix). Note that from the description of the standard basis in Table 2, it follows that

(A.1) p i a i - 1 , q i a i - 1 , and p i + q i 2 a i - 2 for  i = 1 , , n .

Together with formula (2.38) for ρ j ( i ) and the fact that i 0 , this gives the following constraints between the integers K 1 , , K n defined by (4.9) and (4.10):

(A.2) p i + q i = a i ( i - K i + 1 ) + ( i + 1 - K i + 1 + 1 ) - i - 2 ,
(A.3) p n + q n = a n ( n - K n + 1 ) - n - 2 ,
(A.4) a i K i + K i + 1 ( a i - 1 ) ( i - 1 ) + i + 1 ,
(A.5) a n K n ( a n - 1 ) ( n - 1 ) - 1 ,
(A.6) K i + K i + 1 ( 1 - a i ) ( 1 + K i )

for all i = 1 , , n - 1 . These equations further imply the following additional relations.

Lemma A.1.

We have:

  1. If K i < 0 for some i n - 1 , then K i + K i + 1 0 .

  2. If K i < 0 and K i + K i + 1 = 0 , then ( K i , K i + 1 ) = ( - 1 , 1 ) . In this case i = i + 1 = 0 , p i + q i = 2 a i - 2 , and p i + 1 + q i + 1 = i + 2 - K i + 2 - 1 .

  3. - 1 K n n and, if K n = - 1 , then n = 0 , p n + q n = 2 a n - 2 .

This leads to a complete description of possible collections ( K 1 , , K n ) with K n 0 .

Lemma A.2.

If K n 0 , then the tuple ( K 1 , , K n ) is of one of the following kinds:

  1. a concatenation of some ( 0 ) s and ( - 1 , 1 ) s with one ( 1 ) , in any order as long as it does not end with ( - 1 , 1 ) ,

  2. a concatenation of some ( 0 ) s and ( - 1 , 1 ) s with one of ( 1 ) , ( - 1 , 2 ) , or ( - 2 , 3 ) ending with two nonnegative numbers.

Proof.

First, we observe that K n - 1 0 . Indeed, the assumption K n - 1 - 1 together with n K n contradicts to (A.4).

Now, Lemma A.1 implies that if K i < 0 for some i < n - 1 , then we have K i - 1 0 and K i + K i + 1 0 . Since, by Lemma 4.5, we have K 1 + + K n = 1 , removing all pairs ( K i , K i + 1 ) with K i < 0 , will leave us with a tuple of nonnegative integers ( K 1 ~ , , K s ~ ) (recall that K n 0 ) such that K 1 ~ + + K s ~ 1 . Therefore at most one of them can be nonzero. Also, since the removed pairs ( K i , K i + 1 ) satisfy 0 K i + K i + 1 1 , for all but at most one of these pairs we have K i + K i + 1 = 0 , and so ( K i , K i + 1 ) = ( - 1 , 1 ) by Lemma A.1. If K i + K i + 1 = 1 , then from (A.6) it follows that ( K i , K i + 1 ) must be ( - 1 , 2 ) or ( - 2 , 3 ) . ∎

The correlator X in (4.6) can be reconstructed with the correlators of the following much simpler form.

Proposition A.3.

We can reconstruct correlators in (4.6) from correlators of the form

(A.7) X = θ n , , θ n n copies , α , β .

Proof.

Starting with a correlator X in (4.6) not in (A.7), we can choose i to be the largest index with i < n and i 1 . More precisely, X is of the following form:

X = θ i , 𝜽 S , α , β , i < n ,

where 𝜽 S is a tuple consisting of θ j s with j = n or j i . Now it is sufficient to prove that using (4.5), X can be reconstructed from correlators with fewer insertions and correlators of the form

Z = θ j , θ S , α , β , j > i .

Here the set 𝜽 S in Z is the same as that in X, but α , β can be different form α and β in X.

Notice that X = θ S , θ i , θ n p n α ~ , β for some p n 0 . If p n 1 , then we apply (4.5) with γ = β , δ = θ i , ϵ = θ n , and ϕ = θ n p n - 1 α ~ . The correlators with δ ϕ and δ γ are of the form

θ S , θ n , θ i θ n p n - 1 α ~ , β , θ S , θ n , θ n p n - 1 α ~ , θ i β .

They are both of the form Z. The correlator with ϵ γ equals θ S , θ i , θ n p n - 1 α , θ n β . By induction we can reconstruct X from Z and the correlator Y = θ S , θ i , α Y , β Y , where p n Y = 0 .

Similarly, we move all x n - 1 from α Y to β Y , and so on, until we move all θ i + 1 from α to β. Thus we reconstruct X from correlators Z, and the correlator Y = , θ i , α Y , β Y , where p i + 1 Y = = p n Y = 0 .

After reducing to the basis listed in Table 2, Y satisfies p k Y + q k Y a k - 1 for k > i . By Lemma A.1, we have K n Y 0 in Y. In the following argument, we focus on the reconstruction of Y, and drop the superscript Y on K, 𝒑 and 𝒒 .

Case K n = 1 . In this case 𝑲 is a concatenation of ( 0 ) s and ( - 1 , 1 ) s, followed by K n = 1 .

If 𝑲 = ( , - 1 , 1 , 1 ) , then = ( , 0 , 0 , * ) and 𝒑 + 𝒒 = ( , 2 a n - 2 - 2 , * , * ) by equation (A.2). Then we have n - 2 > i , but p n - 2 + q n - 2 a n - 2 , contradicting our assumption p k + q k a k - 1 on Y. Similarly, we reach a contradiction if there is an j > i such that ( K j , K j + 1 ) = ( - 1 , 1 ) .

Therefore, 𝑲 = ( , 0 ¯ , 0 , , 0 , 1 ) and = ( , 1 ¯ , 0 , , 0 , * ) , where the underline marks the i-th spot and i = 1 by equation (A.4). Possibly, i = n - 1 . If i n - 1 , then by assumption we have ( K i + 1 , i + 1 ) = ( 0 , 0 ) so p i + q i = 2 a i - 2 by (A.2). If i = n - 1 , then ( K i + 1 , i + 1 ) = ( K n , n ) where n K n . Then (A.2) shows p i + q i 2 a i - 2 so by (A.1) we know that p i + q i = 2 a i - 2 . Thus 𝒑 + 𝒒 = ( , 2 a i - 2 ¯ , * , , * , * ) . Now we have three cases. In each case we compute 𝒑 + 𝒒 by first using (A.4) to compute and then using (A.2), (4.7), and Lemma A.1.

  1. 𝑲 = ( , 0 , 0 ¯ , , 1 ) , 𝒑 + 𝒒 = ( , a i - 1 , M ¯ , , * ) ,

  2. 𝑲 = ( - 1 , 1 , , - 1 , 1 , 0 ¯ , , 1 ) , 𝒑 + 𝒒 = ( M , 0 , , M , 0 , M ¯ , , * ) ,

  3. 𝑲 = ( , 0 , - 1 , 1 , , - 1 , 1 , 0 ¯ , , 1 ) , 𝒑 + 𝒒 = ( , a r , M , 0 , , M , 0 , M ¯ , , * ) .

Here M = 2 a - 2 with the appropriate subscripts. In each case, we claim that there is an α ^ ( 𝒘 , G 𝒘 ) that satisfying α = θ i + 1 a i + 1 α ^ . We find a factor of θ i + 1 a i + 1 in α for each case as follows:

  1. Here α has a factor of θ i - 1 θ i a i - 1 , which by (3.29) is proportional to θ i + 1 a i + 1 in ( 𝒘 , G 𝒘 ) .

  2. Repeatedly apply (3.29) starting with a 1 θ 1 a 1 - 1 = - θ 2 a 2 .

  3. Repeatedly apply (3.29) starting with a r + 1 θ r θ r + 1 a r + 1 - 1 = - θ r + 2 a r + 2 .

Now apply (4.5) to Y with γ = θ i , δ = β , ϵ = θ i + 1 a i + 1 , and ϕ = α ^ . Then γ ϵ = θ i θ i + 1 a i + 1 vanishes by Corollary 3.13 and the other two correlators have the form 𝜽 S , θ i + 1 a i + 1 , α , β . Writing θ i + 1 a i + 1 = θ i + 1 θ i + 1 a i + 1 - 1 , and performing reconstruction scheme similar to the one in the proof of Lemma 4.4, we will obtain the correlator of the required form.

Case K n = 0 . In this case, (A.5) implies that there are three possibilities: n = 0 ; n = 1 ; or n = 2 , a n = 2 . Using (A.3) we see that the cases n = a n = 2 and n = 1 , a n 3 contradict our assumption that p n + q n a n - 1 . So it only remains to consider the cases n = 0 and n = 1 , a n = 2 .

Let us first consider the case n = 1 , a n = 2 . By (A.3), we have p n + q n = 1 = a n - 1 . If p n - 1 + q n - 1 > 0 , we assume without loss of generality that p n = 1 and q n - 1 > 0 and write α = θ n α . Applying (4.5) to Y with γ = θ i , δ = β , ϵ = θ n and ϕ = α , we can obtain the required correlator, since θ n - 1 θ n = 0 . If p n - 1 + q n - 1 = 0 , equations (A.2) and (A.4) imply that K n - 1 = 1 and n - 1 = 0 . Let i < n - 1 be the largest subscript such that i 0 .

There are three cases:[4]

  1. 𝑲 = ( , 0 , 0 ¯ , , 1 , 0 ) , 𝒑 + 𝒒 = ( , a i - 1 , M ¯ , , 0 , 1 ) ,

  2. 𝑲 = ( - 1 , 1 , , - 1 , 1 , 0 ¯ , , 1 , 0 ) , 𝒑 + 𝒒 = ( M , 0 , , M , 0 , M ¯ , , 0 , 1 ) ,

  3. 𝑲 = ( , 0 , - 1 , 1 , , - 1 , 1 , 0 ¯ , , 1 , 0 ) , 𝒑 + 𝒒 = ( , a r , M , 0 , , M , 0 , M ¯ , , 0 , 1 ) .

The discussion is similar to the case K n = 1 .

Now assume n = 0 , and let i be the largest subscript such that ( K i , i ) ( 0 , 0 ) . Since 0 p i + q i , equation (A.2) shows K i i . Then (A.4) shows that ( K i - 1 , K i ) cannot be ( - 2 , 3 ) , ( - 1 , 2 ) , or ( - 1 , 1 ) . Six cases remain, and the reconstruction can be completed using the strategy analogous to the K n = 1 case (or see [12, p. 47]). ∎

Furthermore, we have:

Lemma A.4.

We can reconstruct correlators in (A.7) from correlators of the form

(A.8) X = θ n , θ n , α , β .

Proof.

Now we focus on correlator X in (A.7). From (A.5), since n 2 , we find

K n a n - 2 a n .

Thus K n 0 and equality is possible only if a n = 2 .

If K n = 0 and a n = 2 , then (A.5) shows that n 2 .

If K n 0 , then K n = 1 by Lemma A.2. Then (A.5) shows n 2 a n a n - 1 , so N = 2 , or n = a n = 3 , or a n = 2 and n = 3 or 4. We will show that in each case where n > 2 , the correlator does not satisfy (4.7), a contradiction.

If n = a n = 3 , then p n + q n = 3 a n - 5 = 2 a n - 2 . Either ( K n - 1 , n - 1 ) is ( 0 , 0 ) or it is ( 1 , 0 ) ; in each case, p n - 1 + q n - 1 1 . Without loss of generality p n - 1 1 , so that α has a factor of θ n - 1 θ n a n - 1 , violating (4.7).

Similarly, if a n = 2 and n = 3 or 4, we can check all possibilities for 𝑲 and and show that 𝒑 + 𝒒 violates (4.7). ∎

Finally, we only need to show:

Lemma A.5.

We can reconstruct correlators in (A.8) from correlators of the form

(A.9) X = θ n , θ n , θ n - 1 θ n a n - 2 , θ J - 1 .

Proof.

Let X be a correlator in (A.8) We know = ( 0 , , 0 , 2 ) . By (A.2)–(A.6), if M = 2 a - 2 , we have three possibilities for 𝑲 :

  1. 𝑲 = ( 0 , , 0 , 0 , 1 ) , 𝒑 + 𝒒 = ( a 1 - 1 , , a n - 2 - 1 , a n - 1 , 2 a n - 4 ) ,

  2. 𝑲 = ( - 1 , 1 , , - 1 , 1 , 1 ) , 𝒑 + 𝒒 = ( M , 0 , , M , 0 , 2 a n - 4 ) ,

  3. 𝑲 = ( 0 , , 0 , 0 , - 1 , 1 , , - 1 , 1 , 1 ) , 𝒑 + 𝒒 = ( a 1 - 1 , , a r - 1 - 1 , a r , M , 0 , , M , 0 , 2 a n - 4 ) .

In all cases, if X 0 , we must have

(A.10) X = θ n , θ n , θ 1 p 1 θ n - 1 p n - 1 θ n a n - 2 , θ 1 q 1 θ n - 1 q n - 1 θ n a n - 2 ,

where p i + q i = a i - 1 for i n - 2 and p n - 1 + q n - 1 = a n - 1 .

In the first case, both p n - 1 and q n - 1 are at least 1. If p n = a n - 1 , then X = 0 by (3.28), since it has a factor of θ n - 1 θ n a n - 1 . This shows that p n = q n = a n - 2 and (A.10) follows.

In the second case, α = θ 1 a 1 - 1 θ 3 a 3 - 1 θ n - 2 a n - 2 - 1 θ n p n . Relations (3.29) show

α θ 2 a 2 - 1 θ 4 a 4 - 1 θ n - 1 a n - 1 θ n p n .

If p n = a n - 1 , we have a factor of α equal to θ n - 1 θ n a n - 1 , and α = 0 by (3.28). Otherwise, p n = q n = a n - 2 and (A.10) follows.

In the last case, α has a factor equal to θ r p r θ r + 1 a r + 1 - 1 θ n - 2 a n - 2 - 1 θ n p n . As before we use relations (3.29) to rewrite it as

θ r p r - 1 θ r + 2 a r + 2 - 1 θ n - 1 a n - 1 θ n p n .

As before, if X 0 , then (A.10) follows.

Finally, we apply formula (4.5) to X in equation (A.10) with

γ = θ n , ϵ = θ n - 1 θ n a n - 2 , ϕ = θ 1 p 1 θ n - 1 p n - 1 - 1 , δ = θ 1 q 1 θ n - 1 q n - 1 θ n a n - 2 .

Then ϵ γ and γ δ have a factor of θ n - 1 θ n a n - 1 , and hence both are 0 by (3.28). The remaining term containing δ ϕ is exactly 𝔉 n . ∎

Acknowledgements

Weiqiang He and Yefeng Shen would like to thank Jérémy Guéré, Changzheng Li, Si Li, Kyoji Saito, and Rachel Webb for helpful discussions on mirror symmetry of LG models. Yefeng Shen thanks Marc Krawitz who, in a conversation around 2011, brought up an idea of using matrix factorizations to study his mirror Frobenius algebras [15]. We also thank IHES (Alexander Polishchuk) and IMS, ShanghaiTech University (Yefeng Shen) for hospitality and support.

References

[1] V. I. Arnold, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monogr. Math. 82, Birkhäuser, Boston 1985. 10.1007/978-1-4612-5154-5Search in Google Scholar

[2] P. Berglund and M. N. Henningson, Landau–Ginzburg orbifolds, mirror symmetry and the elliptic genus, Nuclear Phys. B 433 (1995), no. 2, 311–332. 10.1016/0550-3213(94)00389-VSearch in Google Scholar

[3] P. Berglund and T. Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), no. 1–2, 377–391. 10.1016/0550-3213(93)90250-SSearch in Google Scholar

[4] H.-L. Chang, J. Li and W.-P. Li, Witten’s top Chern class via cosection localization, Invent. Math. 200 (2015), no. 3, 1015–1063. 10.1007/s00222-014-0549-5Search in Google Scholar

[5] A. Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math. 144 (2008), no. 6, 1461–1496. 10.1112/S0010437X08003709Search in Google Scholar

[6] B. Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups (Montecatini Terme 1993), Lecture Notes in Math. 1620, Springer, Berlin (1996), 120–348. 10.1007/BFb0094793Search in Google Scholar

[7] C. Faber, S. Shadrin and D. Zvonkine, Tautological relations and the r-spin Witten conjecture, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 4, 621–658. 10.24033/asens.2130Search in Google Scholar

[8] H. Fan, T. Jarvis and Y. Ruan, The Witten equation and its virtual fundamental cycle, preprint (2007), https://arxiv.org/abs/0712.4025. Search in Google Scholar

[9] H. Fan, T. Jarvis and Y. Ruan, The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2) 178 (2013), no. 1, 1–106. 10.4007/annals.2013.178.1.1Search in Google Scholar

[10] A. B. Givental, Semisimple Frobenius structures at higher genus, Int. Math. Res. Not. IMRN 2001 (2001), 23, 1265–1286. 10.1155/S1073792801000605Search in Google Scholar

[11] J. Guéré, A Landau–Ginzburg mirror theorem without concavity, Duke Math. J. 165 (2016), no. 13, 2461–2527. 10.1215/00127094-3477235Search in Google Scholar

[12] W. He, S. Li, Y. Shen and R. Webb, Landau–Ginzburg mirror symmetry conjecture, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 8, 2915–2978. 10.4171/JEMS/1155Search in Google Scholar

[13] T. J. Jarvis, T. Kimura and A. Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, Compos. Math. 126 (2001), no. 2, 157–212. 10.1023/A:1017528003622Search in Google Scholar

[14] M. Kontsevich and Y. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. 10.1007/BF02101490Search in Google Scholar

[15] M. Krawitz, FJRW rings and Landau–Ginzburg mirror symmetry, preprint (2009), https://arxiv.org/abs/0906.0796. Search in Google Scholar

[16] M. Krawitz, FJRW rings and Landau–Ginzburg mirror symmetry, Ph.D. thesis, University of Michigan, 2010. Search in Google Scholar

[17] M. Krawitz and Y. Shen, Landau–Ginzburg/Calabi–Yau correspondence of all genera for elliptic orbifold 1 , preprint (2011), https://arxiv.org/abs/1106.6270. Search in Google Scholar

[18] C. Li, S. Li and K. Saito, Primitive forms via polyvector fields, preprint (2013), https://arxiv.org/abs/1311.1659. Search in Google Scholar

[19] M. Kreuzer, The mirror map for invertible LG models, Phys. Lett. B 328 (1994), no. 3–4, 312–318. 10.1016/0370-2693(94)91485-0Search in Google Scholar

[20] M. Kreuzer and H. Skarke, On the classification of quasihomogeneous functions, Comm. Math. Phys. 150 (1992), no. 1, 137–147. 10.1007/BF02096569Search in Google Scholar

[21] C. Li, S. Li, K. Saito and Y. Shen, Mirror symmetry for exceptional unimodular singularities, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 1189–1229. 10.4171/JEMS/691Search in Google Scholar

[22] Y. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloq. Publ. 47, American Mathematical Society, Providence 1999. 10.1090/coll/047Search in Google Scholar

[23] T. Milanov, Analyticity of the total ancestor potential in singularity theory, Adv. Math. 255 (2014), 217–241. 10.1016/j.aim.2014.01.009Search in Google Scholar

[24] T. Milanov and Y. Shen, Global mirror symmetry for invertible simple elliptic singularities, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 1, 271–330.10.5802/aif.3012Search in Google Scholar

[25] A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics, Contemp. Math. 276, American Mathematical Society, Providence (2001), 229–249. 10.1090/conm/276/04523Search in Google Scholar

[26] A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations, Duke Math. J. 161 (2012), no. 10, 1863–1926. 10.1215/00127094-1645540Search in Google Scholar

[27] A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. reine angew. Math. 714 (2016), 1–122. 10.1515/crelle-2014-0024Search in Google Scholar

[28] K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 775–792. Search in Google Scholar

[29] K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 1231–1264. 10.2977/prims/1195182028Search in Google Scholar

[30] K. Saito, The higher residue pairings K F ( k ) for a family of hypersurface singular points, Singularities. Part 2 (Arcata 1981), Proc. Sympos. Pure Math. 40, American Mathematical Society, Providence (1983), 441–463. 10.1090/pspum/040.2/713270Search in Google Scholar

[31] S. Shadrin, BCOV theory via Givental group action on cohomological fields theories, Mosc. Math. J. 9 (2009), no. 2, 411–429. 10.17323/1609-4514-2009-9-2-411-429Search in Google Scholar

[32] C. Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), no. 3, 525–588. 10.1007/s00222-011-0352-5Search in Google Scholar

[33] E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook 1991), Publish or Perish, Houston (1993), 235–269. Search in Google Scholar

Received: 2020-11-26
Revised: 2022-07-20
Published Online: 2022-10-05
Published in Print: 2023-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 1.3.2024 from https://www.degruyter.com/document/doi/10.1515/crelle-2022-0057/html
Scroll to top button