Congruences on K–theoretic Gromov–Witten invariants

We study K–theoretic Gromov–Witten invariants of projective hypersurfaces using a virtual localization formula under finite group actions. In particular, it provides all K–theoretic Gromov–Witten invariants of the quintic threefold modulo 41 , up to genus 19 and degree 40 . As an illustration, we give an instance in genus one and degree one. Applying the same idea to a K–theoretic version of FJRW theory, we determine it modulo 205 for the quintic polynomial with minimal group and narrow insertions, in every genus.


Geometry & Topology msp 0 Introduction
One of the first achievements of Gromov-Witten (GW) theory is the celebrated formula of Candelas, de la Ossa, Green and Parkes [4] computing genus-0 invariants of the quintic threefold in terms of a hypergeometric series solution of a Picard-Fuchs equation.It was a first instance of mirror symmetry and was proved by Givental [14] and Lian, Liu and Yau [28].
The K-theoretic version of GW theory, which we refer to as KGW theory, was constructed in Lee [25], and it is only recently that mirror symmetry in this context was Jérémy Guéré developed by Givental in his series of preprints starting with [15].It relates the KGW generating series to a q-hypergeometric function solution of a finite-difference equation.
Both GW and KGW theories rely on the notion of a perfect obstruction theory (see Behrend and Fantechi [2]), producing two fundamental objects on the moduli space M g;n .X; ˇ/ of stable maps to a given nonsingular variety X, namely the virtual cycle OEM g;n .X; ˇ/ vir living in the Chow ring of the moduli space, and the virtual structure sheaf O vir M g;n .X ;ˇ/ living in its K-theory of coherent sheaves.
Given insertions y i 2 CH .X /, or Y i 2 K 0 .X /, and psi-classes i 2 CH p .aOEM g;n .X; ˇ/ vir / 2 Q and p ! .A ˝Ovir M g;n .X ;ˇ/ / 2 Z; where p is the projection map to a point along which we take pushforwards in Chow or in K-theory. 1 Each theory has an important feature: the virtual cycle is puredimensional, leading to a degree condition on the insertions for the GW invariant to be nonzero, and KGW invariants are all integers.Moreover, the two theories are related via a Hirzebruch-Riemann-Roch theorem (see Tonita [38]), saying that all KGW invariants of a nonsingular variety X can be reconstructed from the knowledge of all GW invariants of the DM stacks OEX=.Z=M Z/ for all M 2 N ; see Givental [16,Main Theorem].
Let T be a torus.When the variety X carries a nontrivial T -action, so does the moduli space of stable maps, and the virtual cycle and virtual structure sheaf are T -equivariant.One then benefits from the virtual localization formula of Graber and Pandharipande [17] to reduce the computation of invariants to the T -fixed locus, which greatly simplifies the calculation.Unfortunately, the automorphism group of a smooth projective hypersurface such as the quintic threefold is finite (except in the special cases of quadrics, elliptic curves and K3 surfaces), so that there is no nontrivial T -action.
Let G be a finite cyclic group.In this paper, we take advantage of the fact that the (virtual) localization formula holds with no change under finite group actions.Since projective hypersurfaces X admit such actions, we can apply it to the study of KGW theory of X.However, we still have two difficulties.First, the G-fixed moduli space is in general quite involved and we cannot guarantee that it is smooth, so even after applying the virtual localization formula, we may not be able to finish the computation.Second, the (virtual) localization formula gives an answer in a localized ring.For instance, the G-equivariant K-theory of a point is isomorphic to the representation ring R.G/, which in the case of a finite cyclic group of order M yields ZOEX =.1 X M /.Instead of providing an answer in R.G/, the (virtual) localization formula only gives us the image in the localized complexified ring R.G/ C;loc , where we invert a maximal ideal corresponding to a nonzero element in G.The issue with the localized ring is that the map R.G/ !R.G/ C;loc is in general not injective.In our example, we have R.G/ C;loc ' C and the map sends X to a given primitive M th root of unity, so that the nonzero polynomial 1 C X C C X M 1 2 R.G/ is sent to 0 2 C. Notice that when the group is a torus T , the localization map R.T / !R.T / C;loc is injective; that is why we have no such issue in the previous paragraph.
We overcome the first difficulty by means of an "equivariant quantum Lefschetz theorem" that we developed for GW theory in [19, Section 2] and that we adapt to KGW theory and to finite group actions in Section 1; see Theorem 1.6.It compares the G-equivariant virtual structure sheaf of a hypersurface X P N to that of the ambient space P N , and then we use the T -action on the ambient space to apply the virtual localization formula.However, Theorem 1.6 requires that for every G-fixed stable map from a curve C to X, all stable components of C are contracted to a point in X.This condition could fail if the automorphism group of the curve is too big, leading us to impose restrictions on the genus of the curve and on the degree of the stable map.
The second difficulty is more serious.Indeed, we know the G-equivariant KGW invariant is of the form a 0 C a 1 X C C a M 1 X M 1 for some integers a i , and our goal would be the "nonequivariant" limit a 0 C C a M 1 , but we only have access to the complex number , where is a primitive M th root of unity.Luckily, KGW are integers, so that when M is a prime number, we can sum all these complex numbers for primitive roots and obtain the KGW invariant modulo M.
As a conclusion, we seek automorphisms of X of prime order with isolated fixed points.For instance, the quintic threefold can be realized as the zero locus in P 4 of the loop polynomial x 4 0 x 1 C C x 4  4 x 0 and the action .x0 ; : : : ; x 4 / D .x 0 ; 4 x 1 ; 16 x 2 ; 64 x 3 ; 256 x 4 /; where WD e 2i =41 ; Jérémy Guéré yields an automorphism of X of prime order 41, whose fixed points are coordinate points.As a result of Corollary 2.9, we obtain2 all KGW invariants of the quintic threefold up to genus 19 and degree 40, modulo 41.
Remark 0.1 It happens that 41 is the biggest prime number p for which there exists an automorphism of order p for a smooth quintic hypersurface; see Oguiso and Yu [30].
In Proposition 2.13, we provide an instance of this calculation in genus one.Precisely, we compute where E denotes the Hodge bundle and q is a formal variable, so that the inverse of 1 qE _ is defined as the geometric series in q of general term E k q k (here E is a line bundle).
Interestingly, if we can find automorphisms of prime orders for infinitely many primes and if we can handle the respective localization formulas, then we are able to determine KGW invariants as integers instead of modulo a prime number.We apply this idea to elliptic curves.There are indeed p-torsion points for every prime number p, so that a translation by this point is an automorphism of order p.Furthermore, the localization formula is trivial since there are no fixed points.We deduce the vanishing of all KGW invariants of an elliptic curve, with homogeneous insertions.
Similarly to GW theory, Fan, Jarvis and Ruan [12; 11] developed a quantum singularity theory for Landau-Ginzburg orbifolds.It is known as the FJRW theory and an algebraic construction has been established by Polishchuk and Vaintrob in [33].Precisely, they construct a matrix factorization over the moduli space of .W; G/-spin curves, where W is a nondegenerate quasihomogeneous polynomial and G is an admissible group of symmetries.We refer to Guéré [18] for details.
In Section 3, we explain how to construct a K-theoretic version of FJRW theory and we then pursue the same goal as for KGW theory: compute invariants by applying the localization theorem under finite group actions.We focus on the quintic polynomial with group 5 for clarity of the exposition and we find all its K-theoretic FJRW invariants with narrow insertions in every genus and modulo 205; see Corollary 3.11.
Here, we do not have restriction bounds on the genus of the curve.
In [18], we compute genus-0 FJRW invariants of chain polynomials using a characteristic class 3 c t W K 0 .S/ !H .S/OEOEt, which we can define for a line bundle L over a smooth DM stack S as and then extend multiplicatively.Genus-0 FJRW invariants of a chain polynomial where t j WD t .a 1 / .a j 1 / and R L j are the derived pushforwards of the universal line bundles over the moduli space of .W; G/-spin curves.It is remarkable that such a limit exists and the author has wondered since then whether other limits could exist, for instance when t tends to some root of unity.Interestingly, we prove4 for the quintic loop polynomial with group 5 and narrow insertions that such a limit exists for all genus when t tends to a 41st root of unity 41 .It then converges to a Z=41Z-equivariant version of the FJRW virtual cycle, defined as follows.The two-periodic complex obtained from the Polishchuk-Vaintrob matrix factorization naturally decouples as a direct sum of 41 two-periodic complexes. 5Each one of them provides a (virtual) cycle a k , for 0 Ä k Ä 40, and we define c We easily find similar results for other loop polynomials. 3Polishchuk and Vaintrob's definition of the virtual cycle in FJRW theory involves a Koszul complex of vector bundles; see [33].The class c t then appears naturally from the definition of the Koszul complex, as it involves exterior powers of vector bundles.Note also that c 1 recovers the top Chern class of a vector bundle.

Jérémy Guéré
As a conclusion, we mention a future line of research.Chiodo and Ruan [9] and Chiodo, Iritani and Ruan [8] studied the so-called genus-0 Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence, which provides a striking relation between GW theory of a projective hypersurface and FJRW theory of the defining polynomial.Following Chiodo and Ruan [10], there is a similar correspondence in higher genus as well.
Since we expect the LG/CY correspondence to hold in K-theory as well, it would be interesting to probe a K-theoretic version for the quintic threefold, up to genus 19, degree 40, and modulo 41.
Another question we may ask is: what information do we get on GW invariants of the quintic threefold up to genus 19?The quintic threefold X is special, its virtual cycle (with no markings) is 0-dimensional, so that a lot of its GW invariants vanish.In fact, they are all deduced from some rational numbers n g;d 2 Q for nonnegative integers g and d, corresponding to its GW invariants without markings.As a consequence, we expect some simplifications in the Hirzebruch-Riemann-Roch theorem of Tonita [38] and Givental [16], and to find formulas expressing KGW invariants of X in terms of the n g;d .Moreover, it is proven in Fan and Lee [13], Guo, Janda and Ruan [20] and Chang, Guo and Li [5]  As we are able to compute all KGW invariants modulo 41 up to genus 19 and degree 40, we expect a lot of relations among these 61 unknowns.Moreover, KGW is not restricted by a degree condition on insertions, so we can also insert K-classes from P 4 , yielding indeed infinitely many relations among these 61 unknowns.Of course, we do not know yet how many of these relations are nontrivial.It would also be enlightening to express KGW invariants in terms of BPS numbers, which are integers as well; see [22] for a formula in genus zero.
Notation In this paper, we work over the complex numbers.We denote by G 0 .X / the Grothendieck group of coherent sheaves on a DM stack X and by K 0 .X / the Grothendieck ring of vector bundles on X.If a linear algebraic group G acts on X, then we denote by G 0 .G; X / and K 0 .G; X / the Grothendieck groups of G-equivariant coherent sheaves and vector bundles.They are identified when X is smooth, by Thomason [36].When X is a point, then it equals the representation ring R.G/ of the group G.The G-fixed locus inside X is denoted by X G .For an element h 2 G, we denote by X h the h-fixed locus.If V and W are G-equivariant vector bundles over X, then we denote by the lambda-structure in K-theory.We extend multiplicatively the notation to any element V 2 K 0 .G; X /.When we forget the group action, we simply denote it by t .V / 2 K 0 .X /OEOEt.Let G be a diagonalizable group.The complexified representation ring R.G/ C WD R.G/ ˝C is identified with the coordinate ring O.G/ of G. Hence, for every h 2 G, there is a corresponding maximal ideal denote the localizations.Assume X is smooth and let ÃW X h X be the inclusion of the h-fixed locus.The localization theorem says see Thomason [37].Note that G 1 is the evaluation of the formula above at t D 1.In general, it is not defined in K 0 .G; X / and it is only partially defined in K 0 .G; X / loc .Precisely, for a vector bundle V , the term G 1 .V / is invertible if V has no G-fixed part.This is the case in equation (2).Equation ( 2) is in particular true for finite groups G, even though the localization map R.G/ C ! R.G/ loc is not injective in that case.Moreover, we can relax the smoothness condition on X.Indeed, if X is singular but carries a G-equivariant perfect obstruction theory OEE 1 !E 0 , then there is a G-equivariant virtual structure sheaf O vir;G X 2 G 0 .G; X /; see Lee [25].The obstruction theory pulls back to the G-fixed locus X G and we denote by N vir Ã 2 K 0 .G; X G / the K-theoretic class of the dual of its G-moving part.The G-fixed part gives a perfect obstruction theory on X G and yields a virtual structure sheaf O vir X G .Furthermore, we have the virtual localization formula See Qu [34,Theorem 3.3] for the proof in the case where the group is a torus T , but the same proof holds word for word when we replace T by any diagonalizable group G.In particular, it applies to the moduli space M.X / of stable maps to a smooth DM stack X .

Jérémy Guéré
Here, we specify the genus, the degree and the number of markings as M g;n .X ; ˇ/ when needed.
The letters GW stand for Gromov-Witten and KGW for K-theoretic Gromov-Witten.

Equivariant quantum Lefschetz theorem
This section is a generalization of [19,Section 2] to K-theory and to more general group actions.The main result is an "equivariant quantum Lefschetz" theorem which is of first importance in the next section.

Virtual localization formula
Let G be a linear algebraic group and X be a smooth DM stack equipped with a G-action.The moduli space M.X / of stable maps to X carries a G-action, a Gequivariant perfect obstruction theory, and thus a G-equivariant virtual structure sheaf Denote by ÃW M.X / G ,! M.X / the embedding of the G-fixed locus.By definition, the virtual normal bundle N vir Ã 2 K 0 .G; M.X / G / is the moving part of the pullback of the perfect obstruction theory to the fixed locus. 6The virtual localization formula (3) states 1

.2 Enhancement of the group
Let G T be an embedding of linear algebraic groups and X ,! P be an embedding of smooth DM stacks equipped with a G-action.We assume that the G-fixed loci of X and of P are equal; for every G-fixed stable map to P, all stable components of the source curve are sent to P G ; P is equipped with a T -action extending the G-action; the normal bundle of X ,! P is the pullback of a T -equivariant vector bundle N over P; X is the zero locus of a G-invariant section of the vector bundle N ; and the vector bundle N is convex up to two markings, ie for every stable map f W C ! P, where C is a smooth genus-0 orbifold curve with at most two markings, we have Let us first consider the G-fixed loci of the moduli spaces of stable maps and observe the fibered diagram Note that their duals are parts of the perfect obstruction theories of M.X / and of M.P/, the remaining parts being the perfect obstruction theory of the moduli space of stable curves itself.
The term E WD R f N , pulled back to M.P/ G , has a fixed and a moving part, that we denote respectively by E fix and E mov .
Proposition 1.1 The fixed part E fix is a vector bundle over the fixed moduli space M.P/ G .
Proof Let f W C ! P be a stable map belonging to M.P/ G .We denote by W C ! C the coarse map.It is enough to prove that Take the normalization W C ! C of the curves at all their nodes.We have where the superscripts refer respectively to fixed/nonfixed components of C under the map f .In particular, nonfixed components are unstable curves, ie the projective Jérémy Guéré line with one or two special points.By the normalization exact sequence, we obtain an exact sequence Since the normal bundle has a nontrivial G-action once restricted to the fixed locus of X (or equivalently of P), we have Therefore, it remains to see the vanishing of H 1 for nonfixed unstable curves C nf j , for j 2 J .The curve C nf j is isomorphic to P 1 with either one or two markings, hence f N / D 0 by our assumption of convexity up to two markings.
Denote by O vir M.P/ G the virtual structure sheaf obtained by the G-fixed part of the perfect obstruction theory R f T P .Proposition 1.2 We have M.P/ G /: Furthermore, in the localized equivariant K-theoretic ring, we have Proof It follows from the standard proof using convexity; we recall here the main arguments.
The DM stack X is the zero locus of a G-invariant section of the vector bundle N over the ambient space P.This section induces a map s from the moduli space of stable maps to P to the direct image cone f N ; see [6, Definition 2:1].Since the moduli space M.P/ G is fixed by the action of G, it maps to the fixed part of the direct image cone, that is, the vector bundle E fix .Hence we have the fibered diagram where the bottom map is the embedding as the zero section.The fixed part of the distinguished triangle (4) gives a compatibility datum of perfect obstruction theories for the fixed moduli spaces.Functoriality of the virtual structure sheaf gives [34].Applying the projection formula via the map j on both sides and using the Koszul resolution gives the first result.The second part of the statement follows from the moving part of the distinguished triangle (4).
By the virtual localization formula, the G-equivariant virtual structure sheaf satisfies where equalities happen in G 0 .G; M.P// loc .
Remark 1.3 If it were defined, the right-hand side would equal G 1 .R f N / _ ˝Ovir;G M.P/ ; using the virtual localization formula, but it is not clear that the G-lambda class of R f N is defined in G 0 .G; M.P// loc .However, we say that G 1 .R f N / _ is defined after localization7 to mean that its pullback to the fixed locus is defined.Now, we aim to extend the right-hand side of the equality to the T action.The inclusion of groups G ,! T yields a morphism W G 0 .T; M.P// !G 0 .G; M.P//; under which we get .O vir;T M.P/ / D O vir;G M.P/ : Unfortunately, the map is only partially defined when we localize equivariant parameters: the denominators could be nonzero in the T -localization but vanish in the G-localization.It is easier to work out this issue on the fixed locus of the moduli space.

Jérémy Guéré
Let M.P/ T ,! M.P/ denote the T -fixed locus of the moduli space.In particular, we have the inclusion y ÃW M.P/ T ,! M.P/ G .We notice that the moduli space M.P/ G is stable under the T -action from M.P/ and that the map y Ã is T -equivariant.Moreover Furthermore, in K-theory on the space M.P/ T we have the equality (5) N vir z Ãıy Ã D y Ã N vir z Ã C N vir y Ã : Indeed, let F be the pullback of the perfect obstruction theory from M.P/ to M.P/ T .By definition, the virtual normal bundle N vir z Ãıy Ã is the T -moving part F mov , which decomposes as F mov D F mov fix C F mov mov , where the subscript denotes the G-fixed/moving part.By definition, the virtual normal bundle y Ã N vir z Ã is the G-moving part of F, ie F mov mov , since there is no G-moving T -fixed part in F. The virtual normal bundle N vir y Ã identifies with F mov fix .
Remark 1.4 The virtual normal bundle N vir z Ã is defined on M.P/ G and we have a well-defined equality We also have seen the G-decomposition E D E fix C E mov over M.P/ G with E fix being a T -equivariant vector bundle.Indeed, the vector bundle N over P is T -equivariant, thus so are E and E fix .As a consequence, the equality Then its pushforward under the inclusion y Ã equals In particular, we have Proof By the virtual localization above and equation ( 5), we have The last sentence follows from the following property of .Let Z be a DM stack with a T -action and take A 2 K 0 .T; Z/ loc , B 2 G 0 .T; Z/ loc , a 2 K 0 .G; Z/ loc and b 2 G 0 .G; Z/ loc .If .A/ D a and .B/ D b are well-defined equalities, then .A ˝B/ is well defined and equals the localized class a ˝b.
The pushforward maps z Ã ! and commute when the latter is well defined.Precisely, the map z Ã is T -equivariant and for any localized class C 2 G 0 .T; M.P/ G / loc such that .C / is well defined in G 0 .G; M.P/ G / loc , the localized class z Ã !.C / is well defined under and we have

Equivariant quantum Lefschetz formula
Summarizing our discussion, we obtain the following.Proof Using previous equalities, we get Following Remark 1.3, the meaning of "defined after localization" is precisely

Automorphisms of loop hypersurfaces
Let X be a smooth degree-d hypersurface in P N .K-theoretic Gromov-Witten (KGW) theory is invariant under smooth deformations, so that we can choose any degree-d homogeneous polynomial P to define X, as long as it satisfies the Jacobian criterion for smoothness.Here, we will focus on loop polynomials, ie we take j=d and consider on P N the Z=M Z-action .x0 ; : : : ; x N / D .x0 ; x 1 ; u 2 x 2 ; : : : where u 0 WD 0 and u j C1 WD 1 .d1/u j .We have .d1/u N Á 1 modulo M , so that the hypersurface X is Z=M Z-invariant.Explicitly, we have We write M WD M N and G WD Z=M Z.
Geometry & Topology, Volume 27 (2023) Proposition 2.2 The group G acts on P N , leaves the hypersurface X invariant, and for all nonzero g 2 G, the g-fixed locus in P N consists of all coordinate points.Furthermore, assuming the Calabi-Yau condition N C 1 D d and assuming d is a prime number, then we have M D M .It holds in particular for the quintic hypersurface in P 4 .
Proof By the construction of M, we see that every u j with 1 Ä j Ä N is coprime with M. It implies that every pairwise difference u i u j is coprime with M. Indeed, let 0 Ä i < j Ä N , then we have so that it is enough to prove that d 1 is coprime with M. If a nonzero integer p divides M and d 1, then it divides M , and from its expression in terms of powers of d 1, we get 1 Á 0 modulo p, so that p D 1.
Therefore, we have, for all 0 Ä i < j Ä N and all 0 < k < M , and hence the statement about the fixed locus.
For the second statement, let d D N C 1 be a prime number.Then we have M Á 0 modulo d.Thus, if we have ku j Á 0 modulo M , then we have ku j Á 0 modulo d.But u j Á j modulo d, so that k D 0. As a consequence, every u j is coprime with M , and M D M .

Virtual localization formula
Gromov-Witten (GW) theory of P N and its K-theoretic version is computed by the virtual localization formula under the natural action of the torus T D .C / N .Unfortunately, there are in general no nontrivial torus-actions preserving a smooth degree-d hypersurface X, leading to many difficulties in the computation of its GW and KGW invariants.Nevertheless, we have an action of the finite group G on X.

Jérémy Guéré
In cohomology or in Chow theory, the action of the finite group G is useless with respect to the localization formula.Indeed, we have, for example, A G .pt/ D C. On the other hand, we have K 0 .G; pt/ D R.G/, the representation ring of the group G.Moreover, there exists in K-theory a (virtual) localization formula under finite group actions.
Unfortunately, the (virtual) localization formula does not give a result in K 0 .G; pt/, but in a localized ring where we invert equivariant parameters.For instance, in the case of an abelian group G D Z=M Z, the representation ring (taken with complex coefficients) is and the multiplicative set we use for localization is generated by f1 X; : : : ; 1 X M 1 g: As a consequence, the localized ring is isomorphic to C and the map R.G/ C ! R.G/ C;loc is not injective.Precisely, the map sends X to a primitive M th root of unity , so that for every prime divisor p of M, the polynomial In conclusion, the (virtual) localization formula successfully computes a G-equivariant K-class expressed using roots of unity, but we cannot extract the "nonequivariant" limit corresponding to the map Nevertheless, we find a way to extract some information.Indeed, K-theoretic invariants have another important feature: they are integers.Therefore, when the order of the group is a prime number p, the defect of injectivity of the map R.G/ !R.G/ C;loc amounts to the uncertainty 1 C X C C X p 1 ; which equals p in the nonequivariant limit X 7 ! 1.To conclude, we are left with the desired integer modulo p.Furthermore, if we have several finite actions of different prime orders, we can increase our knowledge about the result.
Let us go back to the degree-d hypersurface X P N .The action of G D Z=M Z on P N leaving X invariant induces a G-action on the moduli spaces of stable maps to P N and to X, so that their virtual structure sheaves are G-equivariant, namely O vir M g .P N ;ˇ/ 2 G 0 .G; M g .P N ; ˇ// and O vir M g .X ;ˇ/ 2 G 0 .G; M g .X; ˇ//: By the virtual localization formula, we then obtain where ÃW M g .X; ˇ/fix ,! M g .X; ˇ/ denotes the G-fixed locus and N vir Ã denotes the moving part of the perfect obstruction theory on the fixed locus.At last, we get The next step is to use Theorem 1.6 to relate this formula to a formula for P N , where an explicit localization formula is available via the torus action.

Fixed locus
We easily check all the conditions listed in Section 1.2 but the second: every stable component of a fixed stable map is contracted.
We are able to prove it under the following restrictions on genus of the source curve and degree of the map.
Proposition 2.4 Let G D Z=M Z act on a smooth projective variety X so that, for every nonzero element h 2 G, the h-fixed locus X h consists of isolated points in X.Let f W C ! X be a stable map corresponding to a G-fixed point in the moduli space M g;n .X; ˇ/.We assume (6) g < 1 2 .p1/ and ˇ< M; where p is the greatest prime divisor of M. Then every stable component of the curve C is mapped to one of the G-fixed points in X.
Proof First, we claim that if f W C ! X is a G-fixed stable map of positive degree, then the group G is a subgroup of the group Aut.C / of automorphisms of C .Indeed, let 2 G be a primitive element.Since the stable map is fixed, we can choose an automorphism 1 2 Aut.C / of the curve C such that f .x/D f . 1 .x//for all x 2 C: We then define k WD k 1 for any k 2 N. Since the degree of the map f is positive and all but a finite number of points in X are not fixed by any element of G, we can choose a point x 2 C such that the points k f .x/D f .k .x//Jérémy Guéré are all distinct for 0 Ä k < M .Since the automorphism M is an automorphism of the stable map f , it is of finite order.Thus we can consider the smallest integer K 2 N such that K D id, so that we have K f .x/D f .K .x//D f .x/; and the integer M divides the integer K.As a consequence, the map sending 0 Ä k < M to k K =M embeds the group G in Aut.C /.Secondly, let us assume C is a stable curve and is not contracted.Since G is a subgroup of Aut.C /, the prime number p divides the order of Aut.C /.By [29, Proposition 3.6], we get8 p Ä 2g C 1, which is a contradiction.Lastly, we consider the case where C is not a stable curve (and therefore the degree of the map is positive).Let f be a dual graph representing the stable map f , where we represent every stable component of C by a vertex and every unstable component by an edge.Furthermore, we add on the graph labels to keep track of the genus, number of markings and degree.It is clear that every automorphism k of the curve C induces an automorphism of the dual graph f .Moreover, for each stable component D of C whose corresponding vertex is fixed in C , the restriction f jD W D !X is fixed by the group G.
We aim to show that the set V >0 of vertices with positive degree is empty.Assume it is not.Then, if the group G acts on V >0 without fixed points, the total degree of the map is at least M, which is a contradiction.Therefore, there is at least one fixed point, ie there exists a stable component D of C such that the restriction f jD is G-fixed.As we have seen above, the stable component D is then contracted to a point, which contradicts the fact that its corresponding vertex is in V >0 .
Remark 2.5 Proposition 2.4 also holds if the condition g < 1 2 .p1/ is replaced by "for every stable curve of genus less than g, there is no automorphism of order equal to M ".

Equivariant and congruent formulas
Let us apply Theorem 1.6 to our situation.Theorem 2.6 Let g, n and ˇbe nonnegative integers.Let X be a degree-d loop hypersurface in P N and take a subgroup H Z=M Z of order q acting on X via the action (7) k .x0 ; : : : ; x N / D .x0 ; k x 1 ; k u 2 x 2 ; : : : ; k u N x N / for k 2 H Z=M ZI see Section 2.1.This action depends on the choice of a primitive q th root of unity .Moreover, we have the usual T WD .C / N -action on P N and we see that it extends the H -action via the embedding ' W H ,! T WD .C / N sending k to .k ; k u 2 ; : : : ; k u N /.
Assume the bounds g < 1 2 .p1/ and ˇ< q; where p is the greatest prime divisor of q.Let A WD N n iD1 ‰ a i i ˝ev .Y i / denote some insertions of Psi-classes and K-classes Y i 2 K 0 .T; P N / coming from the ambient space.Then the corresponding H -equivariant K-theoretic GW invariant equals _ is only defined after localization, so we first apply the virtual localization formula to the left-hand side, then we compute it in K 0 .T; M g;n / loc D K 0 .M g;n / ˝C.t 1 ; : : : ; t N / as rational fractions in the Tequivariant parameters, then we specialize them to .t 1 ; : : : ; t N / D .; u 2 ; : : : ; u N / using ' W H ,! T and obtain a well-defined K-class in K 0 .M g;n / ˝C.Eventually we take its Euler characteristic and land in R.H / C;loc ' C, where the last isomorphism depends on the primitive q th root of unity .

Remark 2.7
The localization map R.H / !R.H / C;loc corresponds to the map ZOEX =.1 X q / !C sending the variable X to .
Remark 2.8 In Theorem 2.6, it is important that for every nonzero element h 2 H , the h-fixed locus consists of coordinate points in P N .It is guaranteed by Proposition 2.2 and the fact that H Z=M Z. Corollary 2.9 We take the same notation and assumptions as in Theorem 2.6.We further assume that the order q of the group H is a prime number.For each 1 Ä k < q, denote by B k 2 C the result of Theorem 2.6 when D e 2ik =q .Then the K-theoretic GW invariant of X equals .A ˝Ovir M g;n .X ;ˇ/ / Á .B 1 C C B q 1 / 2 Z=qZ:

Jérémy Guéré
Proof The H -equivariant Euler characteristic H .A ˝Ovir;H M g;n .X ;ˇ/ / lies in the representation ring R.H / ' ZOEX =.1 X q /, so there exist integers a 0 ; : : : ; a q 1 such that Our goal would be to compute but Theorem 2.6 only gives us P q 1 lD0 a l l 2 C.However, since q is a prime number, we can apply Theorem 2.6 to every primitive q th root of unity k for 1 Ä k < q.Summing the various results, we obtain leading to the congruence.
Remark 2.10 Assume the order q of the group H is not a prime number and choose a nonzero element h 2 H .Even when h is not a primitive element, we can apply Theorem 2.6 to the subgroup hhi, but we then have the bounds g < 1 2 .p1/ and ˇ< ord.h/;where ord.h/ denotes the order of the element h, and p is its greatest prime divisor.In order to obtain the KGW invariant in H , we then need to sum all the results of Theorem 2.6 for all nonzero elements h 2 H . Therefore, we have to restrict to the bounds g < 1 2 .p1/ and ˇ< p; where p is the smallest prime divisor of q.
Example 2.11 For the quintic threefold of Example 2.3, the specialization of equivariant parameters corresponding to G ,! T is .t0 ; : : : ; t 4 / D .1; ; 3 ; 13 ; 51 /; where 205 D 1: Moreover, we have a subgroup H WD Z=41Z Z=205Z, so that by Corollary 2.9, we are able to compute all KGW invariants modulo 41 up to genus 19 and degree 40.Moreover, by Remark 2.10, we are able to compute all KGW invariants modulo 205 in genera 0 and 1 up to degree 4.
Remark 2.12 Another way to realize the quintic hypersurface in P 4 is Then the group is .Z=5Z/ 4 , but to ensure that the g-fixed locus consists of isolated points for every element g of the group, we need to consider the subgroup G D Z=5Z, acting as x D .x0 ; x 1 ; 2 x 2 ; 3 x 3 ; 4 x 4 /: Furthermore, we observe that the G-fixed locus is empty.We then deduce that all KGW invariants in genera 0 and 1 and up to degree 4 vanish modulo 5.

Example of the quintic threefold
We illustrate Theorem 2.6 and Corollary 2.9 by a computation of the genus-one degreeone unmarked KGW invariant in the case of the quintic hypersurface in P 4 , modulo 205.
Proposition 2.13 Let X P 4 be a smooth quintic hypersurface.We find that 1 q 4 q 6 .1 q 4 /.1 q 6 / 2 Z=205ZOEOEq: In order to prove Proposition 2.13, we first write the general graph sum formula coming from torus localization and we then specialize to .g;n; ˇ/ D .1;0; 1/.
Following the general scheme of Theorem 2.6, we compute the K-theoretic class ˝A ˝Ovir;T M g;n .P 4 ;ˇ/ 2 K 0 .M g;n / ˝C.t 0 ; : : : ; t 4 /: It is done via the standard virtual localization formula of [17], lifted to K-theory, as a sum over dual graphs.Indeed, the class 1 is multiplicative in K-theory, just as the Euler class in cohomology, so that the whole proof of [17,Section 4] holds.Therefore, we take the same notation as in [17], to which we refer, for instance, for the description of graphs, except that we take the convention t j D e j with respect to their T -weights.
Let be a graph in the localization formula of P 4 .We denote by M the associated moduli space of stable curves and by A the group of automorphisms coming from the graph and from degrees of the edges, so that the corresponding fixed locus in M g;n .P 4 ; ˇ/ is the quotient stack OEM =A ; see [17].The contribution of the graph to the localization formula is of the form OEM =A I Contr.flags/Contr.vertices/Contr.edges/ ; where we have and where we write here the contribution of an edge linking the coordinate points p j and p j 0 .These formulae follow exactly from [17, Section 4], replacing the Euler class with the lambda class in K-theory.
Remark 2.14 In the contribution of vertices, we can rewrite the sum in terms of the lambda-structure as u .E/, with u WD t 5 i.v/ =t Let us now specialize the formula to .g;n; ˇ/ D .1;0; 1/.The graph has only two vertices v 1 and v 2 , of respective genera 1 and 0, and one degree-one edge in between.Moreover, as the vertex v 2 has valence one, it corresponds to a free point (not marked, not a node) rather than to a stable component of the curve.We denote by 0 Ä i 1 ¤ i 2 Ä 4 the indices of the coordinate points p i 1 and p i 2 to which the vertices v 1 and v 2 are sent by the stable map.Note also that such a graph has no automorphisms and the moduli space M is isomorphic to M 1;1 .Furthermore, we recall that the Hodge bundle E over M 1;1 is identified with the cotangent line ‰ 1 at the marking.As a consequence, the virtual localization formula equals Once we specialize to .t0 ; : : : ; t 4 / D .1; ; 3 ; 13 ; 51 /, where is any primitive root of unity of order 41, we notice that denominators never vanish, but the numerator could vanish; precisely, with the cyclic convention on indices, ie t 5 WD t 0 .Moreover, we have so that the specialization of the localization formula gives

:
Finally, we must take the opposite of the sum of these expressions over all primitive roots of order 41.First, we notice that the term inside the sum is a polynomial in q of degree at most two, so that it is enough to evaluate it at q 2 f0; 1; 2g.Using Sagemath, we find 1 q 4 q 6 .1 q 4 /.1 q 6 / Á .38q 2  C 16q C 2/ 1 q 4 q 6 .1 q 4 /.1 q 6 / 2 Z=41ZOEOEq: Furthermore, using Remark 2.12, we obtain the result of Proposition 2.13.

Special case of elliptic curves
In this section, we use the ideas behind Corollary 2.9 to prove that KGW theory with homogeneous insertions of an elliptic curve is trivial.
Proposition 2.16 Let E be an elliptic curve.Then for every genus g, degree ˇ, number of markings n and insertions A WD N n iD1 ‰ a i i ˝Yi , with 2g 2 C n > 0 and Y i 2 K 0 .E/ homogeneous K-classes, the corresponding KGW invariant vanishes: .A ˝Ovir M g;n .E;ˇ/ / D 0: Proof Let M be the largest possible order of an automorphism of a stable curve of genus g.Let p be any prime number larger than M C 1 and ˇC 1. Define G WD Z=pZ and take a G-torsion point x 2 E. Then the group G acts on the elliptic curve E by translation y 7 !y Cx, and for every nonzero element h 2 G, the h-fixed locus is empty.By Remark 2.5 and Proposition 2.4, the G-fixed locus in the moduli space of stable maps M g;n .E; ˇ/ is empty.Therefore, by the localization formula, the G-equivariant KGW invariant vanishes, so that we get .A ˝Ovir M g;n .E;ˇ/ / Á 0 2 Z=pZ for the nonequivariant limit.Since it is true for infinitely many prime numbers p, we obtain the vanishing in Z.
Remark 2.17 Interestingly, KGW invariants are deduced from GW invariants via a Kawazaki-Riemann-Roch theorem; see [38; 16].It would be instructive to compare Proposition 2.16 with GW theory of elliptic curves, which is nontrivial and described in [31; 32].

Remark 2.18
The same proof holds for abelian varieties.However, when the dimension of the abelian variety is greater than 2 and the degree-class ˇis nonzero, there is a trivial quotient of the obstruction theory, so that both GW and KGW theories are trivial.However, for degree-0 invariants, GW theory is nontrivial, but KGW theory is.
Remark 2. 19 The main idea in the proof of Proposition 2.16 is to use congruence relations for infinitely many prime numbers.Indeed, if we were able to find, for a smooth DM stack X, automorphisms of prime orders for infinitely many primes and to compute the localization formulae, then we would be able to know all KGW invariants of X.Therefore, a necessary condition is that the automorphism group of X must be infinite.However, it is not sufficient.For instance, some K3 surfaces have infinitely many symmetries, but it was shown by [23] that the maximal order of a finite group acting faithfully on a K3 surface is 3840.

Remark 2.20
Here are a few remarks on finiteness of automorphism groups.For projective hypersurfaces (except quadrics, elliptic curves, and K3 surfaces), every automorphism is projective and the automorphism group is finite.All Batyrev Calabi-Yau (CY) 3-folds have finite automorphism groups [35].Every projective variety of general type has finite automorphism group.CY varieties with Picard numbers 1 or 2 have finite automorphism groups.It is expected that most CY varieties with Picard number more than 4 have infinitely many automorphisms.In particular, it would be interesting to know whether the Schoen CY 3-fold has automorphisms of prime order for infinitely many primes and to study its KGW theory; see [21].

K-theoretic FJRW theory
Similarly to KGW theory, we aim in this section to compute the K-theoretic FJRW invariants of a Landau-Ginzburg (LG) orbifold modulo prime numbers.For simplicity of the exposition, we focus in this paper on the quintic polynomial with minimal group of symmetries.However, it is straightforward to apply the same ideas to an LG orbifold .W; H /, where W is an invertible polynomial and H is an admissible group, as long as we only insert Aut.W /-invariant states in the correlator.We refer to [18] for details.

Sketch of Polishchuk-Vaintrob construction
Let W .x 1 ; : : : ; x 5 / be a quintic polynomial in five variables and let 5 act on C 5 by multiplication by a fifth root of unity.The moduli space used in FJRW theory of .W; 5 / is the moduli space S 1=5 g;n , which parametrizes .C; 1 ; : : : ; n ; L; /.Precisely, the curve C is an orbifold genus-g stable curve with isotropy group 5 at the markings 1 ; : : : ; n and at the nodes (and trivial everywhere else), L is a line bundle on C, and Let be the projection from the universal curve to S 1=5 g;n and L be the universal line bundle.In [33], Polishchuk and Vaintrob constructed resolutions R .L ˚5/ D OEA !B by vector bundles over S 1=5 g;n such that there exists some morphism ˛W Sym 4 A ! B _ corresponding to the differentiation of the polynomial W ; see [18] for details.Taking p W X !S 1=5 g;n to be the total space of the vector bundle A, then the morphism ˛is interpreted as a global section of p B _ over X, and the map ˇW A ! B coming from the resolution is interpreted as a global section of p B. As a consequence, we obtain a Koszul matrix factorization PV WD f˛; ˇg WD .ƒB _ ; ˛^ C Ã ˇ/ 2 D.X; ˛.ˇ// of potential ˛.ˇ/ over the space X, and the support of this matrix factorization is exactly the moduli space S where we need to consider rigidified moduli spaces; see [33] for details and notation.
In general, to any triangulated category C we associate a Grothendieck group K 0 .C/ by taking the free abelian group generated by the objects of the category and then modding out the relation OEA OEB C OEC D 0 for every distinguished triangle A ! B ! C .Furthermore, any functor f W C 1 !C 2 of triangulated dg categories induces a morphism of groups When the category is the derived category of coherent sheaves on a smooth DM stack, we recover the usual K-theory of the stack.In our situation, we find formula (11).for the FJRW virtual cycle of .W; 5 /, where t j WD t a j .This formula is only valid in genus 0 and we do not expect the left-hand side to converge in positive genus when t ! 1.However, by Theorem 3.8, we see that the formula converges for every genus when t ! .
In order to get congruences for the nonequivariant limit, we need to consider a subgroup of Aut.W / with prime order and whose fixed locus in X is compact.The only invertible polynomial for which it is possible is the loop polynomial, together with the subgroup 41 acting on X as .41 x 1 ; 37 41 x 2 ; 16 41 x 3 ; 18 41 x 4 ; 6 41 x 5 /; where 41 WD e 2i =41 : Remark 3.10 The prime decomposition of 205 is 5 41, so we could also hope for a congruence modulo 5.However, the subgroup 5 acts trivially on X.Indeed, it acts as .5 x 1 ; 5 x 2 ; 5 x 3 ; 5 x 4 ; 5 x 5 /; where 5 WD e 2i =5 ; which is rescaled by the automorphism group of the .W; 5 /-spin curve, so that the fixed locus is X.Nevertheless, from its definition using the quintic Fermat polynomial, we observe that the virtual structure sheaf decomposes into five identical summands, each one corresponding to the so-called 5-spin theory.It is then divisible by five in the K-theoretic ring with Z coefficients.As a consequence, all FJRW correlators of the quintic vanish modulo 5.However, since the cohomology is taken with Q-coefficients, we do not obtain congruence results on the virtual cycle.An idea would be to guess a formula for the K-class R as an integral linear combination of (natural) vector bundles over S 1=5 g;n ./. Since the virtual cycle is pure-dimensional and the right-hand side of the formula above is most likely not pure-dimensional when we take a generic R, only special integral coefficients in this linear combination would work.

Theorem 1 . 6 (
equivariant quantum Lefschetz) Let X ,! P be a G-equivariant embedding of smooth DM stacks satisfying assumptions listed at the beginning of this section.Then we havez j !O vir;G M.X / D T 1 .R f N / _ ˝Ovir;T M.P/ 2 G 0 .G; M.P// loc ; Jérémy Guéréwhere z j is the embedding of moduli spaces and is the specialization of T -equivariant parameters into G-equivariant parameters.Here, the T -equivariant lambda class T 1 .R f N / _ is defined after localization; see Remark 1.3.

Remark 3 . 1
has several components depending on the monodromies WD . 1 ; : : : ; n / 2 n 5 at the markings; we denote by S 1=5 g;n ./ the corresponding component.Assume all monodromies are nonzero; this is known as the narrow condition.Then the pairing ˛.ˇ/ is the zero function over X, and the matrix factorization PV becomes a two-periodic Geometry & Topology, Volume 27 (2023) complex, exact off the moduli space S 1=5 g;n ./. Therefore, we can define the pushforward along the projection map p in the category of matrix factorizations, yielding p .PV/ 2 D.S 1=5 g;n ./; 0/ ' D b .S 1=5 g;n .//; where on the right we have the derived category of coherent sheaves.If one allows trivial monodromies (ie one considers broad insertions), then the pairing ˛.ˇ/ does not vanish and we rather end with a functor ˆW D .A ; W / !D H .S rig g;n ./; 0/ ' D.OES rig g;n ./=H /; U 7 !p .ev .U / ˝PV/;

Ã 2 K
0 .S 1=5 g;n .// ˝C: Proof In the G-equivariant K-theory of the space X, the matrix factorization equalsPV D G 1 p B _ 2 K 0 .G; X /;and by the localization formula we getPV D Ã ! .G 1 .B _ A _ // 2 K 0 .G; X / locin the localized ring, where ÃW S 1=5 g;n ./ ,! X is the zero section.Taking the pushforward along the projection map p, we obtain the G-equivariant virtual structure sheafO _ A _ / 2 K 0 .G; S 1=5 g;n .// loc ' K 0 .S 1=5 g;n .// ˝C:If V is a vector bundle, we can express the -structure in terms of Adams operators via the formulap .V _ / D exp Â X lÄ 1 p l l ‰ l .V / Ã :Moreover, if the action of a group G on the vector bundle V is by rescaling fibers with 2 G, thenG 1 .V _ / D 1 .V _ / D exp
is the relative cotangent line at the i th marked point.It is more convenient to work with cyclic groups.Therefore, in the 2-loops polynomial case, we prefer to use G D 195 , where the G-action on X is .15x 1 ; 60 x 2 ; 240 x 3 ; 65 x 4 ; 260 x 5 /; where D e 2i =195 : Definition 3.7 Let l 2 Z.The Adams operation ‰ l in K-theory is defined on a line bundle L over a space S as ‰ l .L/ WD L ˝l ;and then extended as a ring homomorphism‰ l W K 0 .S/ !K 0 .S/: Theorem 3.8 Consider the two following situations: W is the loop polynomial , G WD 205 , a primitive 205 th root of unity, and .a 1 ; : : : ; a 5 / D .1;4; 16; 64; 256/.