Snowbird Lecture Series on the Gauged Linear Sigma Model Lecture 1: Preliminaries on Orbifolds and GW Theory What is an orbifold? Roughly: An orbifold is a space locally homeomorphic to the quotient of (an open subset of) Euclidean space by the action of a finite group. Definition (Analogous to manifold): Define an orbifold chart as: Define compatibility condition for orbifolds charts... Then an orbifold is a space X equipped with an equivalence class of orbifold atlases. Main observation: The orbifold chart remembers more than just the topological quotient Ũ/G, but also the isotropy of the action of G on Ũ. Example: Let Z 3 C via multiplication by nth roots of unity. Then we have an orbifold X = [C/Z 3 ] The underlying topological space is X = C/Z 3 = C, and there s a single global chart φ = id : R 2 C. Note that X remembers that the action has isotropy group Z 3 at the origin and trivial isotropy elsewhere. Example: If M is a smooth manifold and G is a finite group acting smoothly on M, there is a global quotient orbifold X = [M/G]. (Uses the fact that any x M with isotropy group G x is contained in a G x -invariant chart.)
Example: Let C C n+1 by λ (z 0,..., z n ) = (λ c 0 z 0,..., λ c n z n ) for coprime integers (c 0,..., c n ). The quotient P(c 0,..., c n ) := C n+1 /C can be given the structure of an orbifold, called weighted projective space. Specifically: X = C n+1 /C (as a topological quotient), and the coordinate point p i has isotropy group Z ci, while all other points have trivial isotropy. (Note: This is not a global quotient, in general.) Example: BG = [ /G] Fact: For each x X, there is a well-defined isotropy group G x : choose any orbifold chart U = Ũ/G with x U and any y Ũ that maps to x, and set G x := {g G gy = y}. Orbifold vector bundles In general, all of the geometric constructions one might associate to a manifold extend to orbifolds. The philosophy is: to define an orbifold analogue of something, take the corresponding thing on the charts Ũ, and then insist that it be equivariant with respect to the actions of the local groups G. This is easiest to explain in the case of global quotients. Definition: An orbifold vector bundle on a global quotient X = [M/G] is a vector bundle π : E M equipped with a G-action taking the fiber over x M linearly to the fiber over gx. Definition: A section of an orbifold vector bundle over [M/G] is a G- equivariant section of π. Can extend this definition to arbitrary orbifolds; locally, it s the same as the above. Orbifold cohomology (take one) First guess: We can define the tangent bundle of an orbifold. (E.g. for a global quotient [M/G], it s the bundle T M on which G acts by the derivative of its action on M.) From here, can define k T X, then differential forms, then orbifold de Rham cohomology HdR (X ).
Problem: Satake proved that H dr(x ) = H (X; R). In other words: orbifold de Rham cohomology doesn t see isotropy! Example: H dr (BG) = H dr ( ) Solution: First define orbifold quantum cohomology, and then restrict to the degree-zero part to recover the definition of orbifold (Chen Ruan) cohomology. Review of GW theory and quantum cohomology Let X be a complex manifold. Definition: The moduli space of genus-zero stable maps is M 0,n (X, β) := {(C; q 1,..., q n ; f) C genus-zero curve, f : C X}. This is equipped with evaluation maps ev i : M 0,n (X, β) X ev i (C; q 1,..., q n ; f) = f(q i ). Definition: Let α 1,..., α n H (X). Then the associated (genus-zero) GW invariant is: α 1 α n X 0,n,β := ev 1(α 1 ) ev n(α n ). [M 0,n (X,β)] vir Definition: The quantum cohomology of X is the vector space H (X)[[q]] equipped with the product defined by: ( ) α β, γ := q β α β γ X 0,3,β, β where ( ) is the Poincaré pairing: (ω, ν) := X ω ν. Exercise: When we set q = 0, quantum cohomology recovers usual cohomology with its usual product (intersection/cup/wedge). Next time: Quantum cohomology of orbifolds.
Chen Ruan cohomology Lecture 2: FJRW Theory Last time: Defined quantum cohomology of manifolds, and saw that when q = 0 (constant maps), it recovers usual cohomology. Orbifold morphisms, on the other hand, are rich and interesting, even when they re degree-zero ( constant ) maps! I won t tell you the careful definition of orbifold morphisms (it s kind of hard...the atlas on the source orbifold may need to be refined), but I ll tell you that an upshot of the definition of orbifold morphisms is: Fact: An orbifold morphism f : X Y induces a map f : X Y and homomorphisms λ x : G x G f(x) for each x X. Exercise: One very special case of an orbifold morphism, when X = [M/G] and Y = [N/H], is to take a continuous map f 0 : M N and a homorphism f 1 : G H such that f 0 is equivariant with respect to f 1. Check that the fact is true in this case. Upshot of fact: When X is an orbifold, we can still define M 0,n (X, β) := {(C; q 1,..., q n ; f) C genus-zero (orbifold) curve, f : C X }, but now there are two pieces of local data around each marked point q i : The image f(q i ) X ; The homomorphism λ qi. (Actually, the isotropy groups of orbifold curves are cyclic with a canonical generator, so this is encoded in an element of the isotropy group of X at f(q i ).) As a result, the natural evaluation maps land not in X but in: Definition: The inertia stack of X is IX = {(x, g) x X, g G x }. (This can itself be given the structure of an orbifold...)
Example: X = [C/Z r ]. Then IX = C. We give this the structure of an orbifold by letting Z r act by sending (x, g G x ) (hx, hgh 1 G hx ). Because Z r is abelian, this sends (x, g) to (hx, g); in other words, it preserves each component. Thus, as an orbifold, IX = [C/Z r ] [ /Z r ] [ /Z r ]. Conclusion: The Gromov Witten invariants of X take insertions from H (IX ), so quantum cohomology is a product structure on H (IX )[[q]]. Definition: The orbifold (Chen Ruan) cohomology of X is H CR(X ) = H (IX ). The product structure is the q = 0 limit of the quantum product. Now that we ve developed the necessary background on orbifolds, we re ready to discuss: FJRW (Fan Jarvis Ruan Witten) Theory Analogously to the GW theory of a space X, this theory consists of: A state space H, analogous to H (X); A moduli space M, analogous to M g,n (X, β); A notion of correlators (integrals against a virtual cycle on M associated to any choice of elements in H), analogous to GW invariants. The input data for FJRW theory is a polynomial W C[x 1,..., x N ], analogous to the input data for GW theory being X. Ultimate goal (upcoming lectures): LG/CY correspondence, an equivalence (in some cases) between FJRW theory of W and GW theory of {W = 0}. First, we restrict to the certain class of polynomials for which FJRW theory is defined: Definition: A polynomial W C[x 1,..., x N ] is quasihomogeneous if there exist positive integers w 1,..., w N (weights) and d (the degree) such that W (λ c 1 x 1,..., λ c N x N ) = λ d W (x 1,..., x N ) for all λ C. We will always require our polynomials to satisfy two further conditions:
1. Nondegeneracy: W defines a nonsingular hypersurface in P(w 1,..., w N ); 2. Invertibility: The number of monomials of W is equal to the number of variables, and the exponent matrix is invertible. Running example: W (x 1,..., x 5 ) = x 5 1 + + x 5 5, which has w 1 = = w 5 = 1 d = 5. The state space of FJRW theory Let W be a quasihomogeneous polynomial. Then we define H W := H CR([C N /J], W + ; Q), where ) J := (e 2πi c 1 d,..., e 2πi c N d (C ) N (note that this is cooked up so that W (g x) = W ( x) for all g J), and W + = W 1 (ρ) for ρ 0. Recall that H CR (X ) = H (IX ), where IX = {(x, g G x )}, so we can more explicitly write: H W := g J H (C N g /J, W + ; Q) Example: W (x 1,..., x 5 ) = x 5 1 + + x 5 5. Then ( J = e 2πi 1 5,..., e 2πi 5) 1 = Z5 (C ) 5. Thus, I[C 5 /J] = [C 5 /Z 5 ] [ /Z 5 ] [ /Z 5 ], since the action of 1 J fixes all of C 5 but the action of any nontrivial g J fixes only the origin. It follows that H W = H (C 5, W + ) Z 5 g 1 Z 5 H ({0}, ) = H (C 5, W + ) Z5 Q 4. Definition: An element g J is called narrow if it fixes only 0 C N, meaning that its associated component of H W is a copy of Q. Elements that are not narrow are called broad. In the above example, g = 1 is broad and all other g are narrow.
The moduli space of FJRW theory Let W be a quasihomogeneous polynomial. Then we define where: M W g,n = {(C; q 1,..., q n ; L; φ)}, 1. (C; q 1,..., q n ) is a genus-g, n-pointed stable orbifold curve this has a precise meaning, but in particular, it means that C has nontrivial isotropy only at special points; 2. L is an orbifold line bundle on C (with a condition on multiplicities to get a proper moduli space ); 3. φ is an isomorphism L d = ωlog, where ω log = ω C ([q 1 ] + + [q n ]). Example: W (x 1,..., x 5 ) = x 5 1 + + x 5 5. Then M W g,n = M 1/5 g,n.
Lecture 3: The LG/CY Correspondence (for the quintic, in genus zero) The correlators of FJRW theory Recall that, when W is an invertible quasihomogeneous polynomial of degree d, we define M W g,n = {(C; q 1,..., q n ; L; φ) φ : L d ωlog }. This moduli space breaks up into components depending on the multiplicities of L: Definition: Let C be an orbifold curve, let L be an orbifold line bundle, and let q C be a point with isotropy group G q = Z r. Then, locally near q, L is a bundle on C with an action of Z r such that the projection map is equivariant; in particular, Z r acts on the fiber of L over q. The multiplicity of L at q is defined as the number m q {0, 1, 2,..., r 1} such that the action of Z r on L is ) ζ (x, v) = (e 2πi 1 r x, e 2πi mq r v in local coordinates around q. In M W g,n, the stability condition implies that all of the isotropy groups have order dividing d, so for a tuple (m 1,..., m n ) {0, 1,..., d 1}, we set M W g,(m 1,...,m n ) = {mult qi (L) = m i i} M W g,n. Correlators (narrow case): Recall that H W = H (C N g /J, W + ; Q) = H (C N g, W + ; Q) J g J g broad Choose n narrow elements of H W, say α 1,..., α n with α i = c i e gi and g i = (e 2πim i c 1 c d,..., e 2πim Nd ) i. Then Facts: α 1... α n W g,n = c 1 c n [M g,(m1,...,mn)] vir 1. g narrow Q{e g }. 1. There exists a virtual cycle [M g,(m1,...,m n )] vir. (Hard, we ll say some words later) 2. All of this can be extended to the case where α i is broad. (Harder)
Interlude: The idea of the LG/CY correspondence Reasonable questions at this point: 1. Where do all of these definitions come from? 2. What does this have to do with the GW theory of {W = 0} P(w 1,..., w N )? The answers to both of these questions come from putting FJRW theory in the context of the LG/CY correspondence. To tell that story, we ll focus specifically on the case where W (x 1,..., x 5 ) = x 5 1 + + x 5 5. Big picture: There are two theories associated to this polynomial: the FJRW theory of W and the GW theory of Q := {W = 0} P 4. Each has a state space, a moduli space, and correlators, and the LG/CY correspondence says: 1. The state spaces are isomorphic: H (Q) = H W. 2. The generating functions of (genus-zero) correlators match after certain identifications. The state space correspondence Let C act on C 5 C by The polynomial λ(x 1,..., x 5, p) = (λx 1,..., λx 5, λ 5 p). W (x 1,..., x 5, p) = p(x 5 1 + + x 5 5) gives a well-defined map out of the quotient (C 5 C)/C. This quotient is not separated, but it admits two maximal separated subquotients (GIT quotients): (C 5 \ {0}) C C C 5 (C \ {0}) C. = O P 4( 5) = [C 5 /Z 5 ]. In both cases, look at the relative Chen Ruan cohomology, relative to a fiber of W : H (O P 4( 5), W + ) = H (P 4, P 4 \ Q) = H (Q), H CR([C 5 /Z 5 ], W + ) =: H W. Thus, the state spaces of FJRW theory and GW theory arise in completely analogous ways. In fact, this framework can be leveraged (with the help of some exact sequences) to prove that H (Q) = H W.
The moduli spaces We d like to repeat the above argument on the moduli level. Just as, above, we started with a big but badly-behaved quotient (C 5 C)/C containing both GW and FJRW theory, we ll start with a big but badly-behaved moduli space: Let X = {(C; q 1,..., q n ; L; x 1,..., x 5 ; p) x i Γ(L), p Γ(L 5 ω log )}, where β = deg(l). This is a non-separated, non-compact Artin stack, but inside it we can find two separated Deligne Mumford substacks: X GW = { x nowhere zero} = {(C; q 1,..., q n ; f : C P 4 ; p) p Γ(f O( 5) ω log )}, X FJRW = {p nowhere zero} = {(C; q 1,..., q n ; L; x 1,..., x 5, ϕ) x i Γ(L), ϕ : L 5 = ωlog }. (Note that these are still not compact.) In both cases, look at the locus Z GW/FJRW where (x 1,..., x 5, p) Γ(L 5 (L 5 ω log )) lands in Crit(W ) C 5 C. Specifically, so Crit(W ) = Q O P 4( 5), Crit(W ) = [0/Z 5 ] [C 5 /Z 5 ], Z GW = {f maps to Q, p = 0} = M g,n (Q, β), Z FJRW = {x 1 = = x 5 = 0} = M W g,n. Thus, the moduli spaces of FJRW and GW theory also arise in completely analogous ways, and this framework can be upgraded to give a uniform construction of the virtual cycles in the two theories (the cosection construction, due to Kiem Li and first applied in this setting by Chang Li Li). Unlike the state space correspondence, this does not immediately give an equivalence between correlators. However, Chiodo Ruan proved the following theorem: Theorem [Chiodo Ruan]: The genus-zero FJRW theory of W can be encoded in a generating function J FJRW (t) taking values in H W [[z 1, z], and the genuszero GW theory of Q can be encoded in a generating function J GW (q) taking values in H (Q)[[z 1, z]. After choosing a specific isomorphism H W = H (Q), these two generating functions are related by change of variables (in q and t), identifying q = t 5, and analytic continuation. Next lecture: Generalize this story to complete intersections in projective space, and to the GLSM in general. Last lecture: Some ideas of the proof of LG/CY.
Lecture 4: The Hybrid Model and the GLSM Recall: Last time, we stated an equivalence between the genus-zero FJRW theory of W = x 5 1 + + x 5 5 and the genus-zero GW theory of Q = {W = 0} P 4. Question: What is the FJRW-type theory corresponding to the GW theory of a complete intersection in projective space? (This theory, once we construct it, will be called the hybrid model. ) The state space of the hybrid model Generalize the variation of GIT perspective from last lecture. Namely, fix W 1,..., W r C[x 1,..., x N ] quasihomogeneous of the same weights w 1,..., w N and the same degree d, defining a nonsingular complete intersection Z = {W 1 = = W r = 0} P(w 1,..., w N ). Let C act on C N C r by λ(x 1,..., x N, p 1,..., p r ) = (λ w 1 x 1,..., λ w N x N, λ d p 1,..., λ d p r ). Then the polynomial W (x 1,..., x N, p 1,..., p r ) = p 1 W 1 ( x) + + p r W r ( x) gives a well-defined map out of the quotient (C N C r )/C. This quotient is not separated, but it admits two maximal separated subquotients: X + = (CN \ {0}) C r X C = CN (C r \ {0}) C r N = O P(w1,...,w N )( d) = O P(d,...,d) ( w i ). j=1 In both cases, look at the relative Chen Ruan cohomology, relative to a fiber of W : r O P(w1,...,w N )( d), W + H CR H CR j=1 = H CR (P(w 1,..., w N ), P(w 1,..., w N ) \ Z) = HCR(Z), ( N ) O P(d,...,d) ( w i ), W + =: H W. i=1 This gives us our definition of the state space in the hybrid model. i=1
The moduli space of the hybrid model Let X = {(C; q 1,..., q n ; L; x 1,..., x N ; p 1,..., p r ) x i Γ(L w i ), p j Γ(L d ω log )}. (Note that this is almost the same as maps into (C N C r )/C, except that we have to make a choice of where to put extra ω log s. That choice is referred to in the physics literature as the R-charge of the theory.) This is a non-separated, non-compact Artin stack, but the loci where the map lands in either of the two GIT quotients inside (C N C r )/C are separated Deligne Mumford substacks: X GW = { x nowhere zero} = {(C; q 1,..., q n ; f : C P( w); p 1,..., p N ) p j Γ(f O( d) ω log )}, X hyb = { p nowhere zero} = {(C; q 1,..., q n ; L; x 1,..., x N ; f : C P r 1 x i Γ(L w i ), f O(1) = L d ω log }. In both cases, look at the locus Z GW/hyb where N (x 1..., x N, p 1,..., p r ) Γ L w i i=1 r (L d ω log ) j=1 lands in Crit(W ) C N C r. Specifically: r Crit(W ) = Z O P(w1,...,w N )( d) and so j=1 Crit(W ) = P(d,..., d) N O P(d,...,d) ( w i ), i=1 Z GW = {f maps to Z, p = 0} = M g,n (Z, β), Z hyb = { x = 0} = {(C; q 1,..., q n ; L; f : C P r 1 L d = f O( 1) ω log }. This last line is the definition of the hybrid moduli space. (Note the name hybrid makes sense: it s a mix of the GW theory of P r 1 and the FJRW theory of a degree-d polynomial.)
The correlators of the hybrid model Again, the fact that these moduli spaces arise as the critical locus of a polynomial can be used to give them natural virtual cycles via the cosection construction. They also both have natural evaluation maps to the corresponding GIT quotient ev i : Z GW/hyb X +/, using the fact that ω log is trivial on the divisor of a marked point. Thus, we can pull back elements of H (X +/ ) and integrate them against the virtual cycle to define correlators. Subtlety: We defined the state space to be H CR (X +/, W + ), not H CR (X +/ ). On the GW side, this means that while our state space is we actually only allow insertions from H CR(X +, W + ) = H CR(Z), H CR(X + ) = H (P( w)); in other words, only ambient insertions. On the hybrid side, we can decompose HCR(X, W + ) = ( N ) H O P N 1( w i ), W + H (P N 1 ), g broad i=1 g narrow where g J is narrow if its fixed locus is the zero section P( d) X. In particular, the narrow part of the state space is naturally a subspace of H CR (P( d)) = H CR (X ), so we can define correlators for narrow insertions. Fact: On the GW side, these correlators recover the usual (ambient) GW invariants of Z. (Not obvious, because the virtual cycle is defined differently than usual. Proved for the quintic by Chang Li.) The LG/CY correspondence states the following (under restrictive assumptions): Theorem [C]: Suppose that Z is a Calabi Yau threefold and w 1 = = w N = 1. Then the genus-zero hybrid generating function J hyb (u) matches the genus-zero GW generating function J GW (q) after an isomorphism on the state spaces, an identification and change of variables in q and t, and analytic continuation. There is a version of the correspondence for (almost) any Calabi Yau Z in weighted projective space, assuming w i d for all i; this was proved in C Ross based on work of Lee Priddis Shoemaker. However, one needs more technical machinery to state it carefully, so we ll omit it here.
The general GLSM All of this can be generalized much further, to the gauged linear sigma model, developed mathematically by Fan Jarvis Ruan. Input: 1. A GIT quotient X = [V θ G], where V is a complex vector space and G GL(V ); 2. A polynomial function W : X C; 3. An R-charge, which is an action of C on V. The rough idea, then, is that the moduli space parameterizes Landau Ginzburg maps from curves into the critical locus of W, which look like maps except that they have additional ω log s dictated by the R-charge. In order for the resulting moduli space to be compact, it is sometimes important to weaken the notion of maps to quasimaps, an idea first introduced by Ciocan-Fontanine and Kim. These also play a key role in the proof of the LG/CY correspondence, so we ll discuss them next lecture.
Lecture 5: Wall-Crossing and the Proof of LG/CY Quasimaps As in last lecture, let W 1,..., W r C[x 1,..., x N ] be quasihomgeneous polynomials of the same weights w 1,..., w N and the same degree d, defining a nonsingular complete intersection Z = {W 1 = = W r = 0} P(w 1,..., w N ). Recall that to define the moduli spaces in GW theory and the hybrid model, we started with X = {(C; q 1,..., q n ; L; x 1,..., x N ; p 1,..., p r ) x i Γ(L w i ), p j Γ(L d ω log )}. and imposed that either x or p had no common zeroes. Now, let s weaken that, allowing them to have common zeroes in a controlled way. Let ɛ Q +, and define X GW,ɛ X to be the locus where: 1. x = 0 at only finitely many, nonspecial points q, and for all such q, ord q ( x) 1/ɛ; 2. The bundle L ɛ ω log is ample ( ) deg(l)>1/ɛ on rational tails deg(l)>0 on rational bridges The idea, here, is that once we impose that ord q ( x) 1/ɛ, then in order to obtain a compact moduli space we must specify what happens as x approach a common zero of order > 1/ɛ. In this case, C sprouts a rational tail. But in order to obtain a separated moduli space, we must allow only those rational tails that are needed; this is why we impose deg(l) > 1/ɛ on rational tails. Rational bridges arise when a zero of x approaches a special point, and again, separatedness requires that we allow only those rational bridges that are needed. Now, restrict further to the locus Z GW,ɛ XGW,ɛ where ( x, p) lands in Crit(W ) = Z, i.e., where W j ( x) = 0 Γ(L d ) j = 1,..., r, p = 0. This recovers the moduli space of stable quasimaps to Z, which was introduced and extensively studied by Ciocan-Fontanine and Kim.
All of this can be carried out on the hybrid side, also: are defined by: Z hyb,ɛ Xhyb,ɛ X 1. For all q C, we have ord q ( p) 1/ɛ; 2. The bundle (L d ω log ) ɛ ω log is ample. Again, taking ɛ 0 recovers our old hybrid moduli space. Why quasimaps? 1. The moduli space for the asymptotic stability condition ɛ = 0+ is known to be compact in some cases where the ɛ 0 moduli space is not; for example, in the analogue of the hybrid model for hypersurfaces of different degree. 2. Quasimaps are related to mirror symmetry: there s a generating function J ɛ (q) of genus-zero quasimap invariants for all ɛ, and when ɛ = 0+, it s Givental s I-function. 3. They provide a path toward proving the LG/CY correspondence. The proof of LG/CY in genus zero For the sake of exposition, we ll restrict again to the quintic so N = 5 and r = 1. W = x 5 1 + + x 5 5, Step 1: Prove wall-crossing formulas expressing that J GW, differs from J GW,0+ by an explicit change of variables (Ciocan-Fontanine Kim) and similarly for J FJRW, and J FJRW,0+ (Ross Ruan). Step 2: Calculate the ɛ = 0+ generating functions explicitly: J GW,ɛ (q) =: I GW (q) = z β 0 b=1 β b=1 q β 5β J FJRW,ɛ (t) =: I FJRW (t) =. (5H + bz), (H + bz)5 Verify that these two functions assemble bases of solutions to the same differential equation under the identification q = t 5, and hence I hyb (t) differs from the analytic continuation of I GW (q) to the t-coordinate patch by an isomorphism of the state space that changes the basis of solutions.
Towards higher-genus LG/CY The above outline suggests a path toward the LG/CY correspondence in all genus: prove wall-crossing theorems relating the generating functions of genusg GW/hybrid invariants as ɛ varies, then try to relate the ɛ = 0+ generating functions on the two sides to one another. The first step of this program has been carried out: Theorem (Ciocan-Fontanine Kim, C Janda Ruan, Zhou): For any complete intersection Z P(w 1,..., w N ) of hypersurfaces of the same degrees, we have q β [Z GW/hyb,ɛ ] vir β = β 0,β 1,...,β k qβ0 ( k k! b β c q β i ev n+i i=1 ( µ ɛ βi ( ψ n+i ) ) [Z GW/hyb, ] vir The second step, though, has only been carried out thus far in genus 0 and genus 1 (Ross Guo); Dusty will speak on this next week. )