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...4, 15, 16, 17, 18] and reviewed in conference proceedings including [19, 20, 21]. (There is also a significant mathematics literature on Gromov-Witten invariants of stacks and gerbes; see for example =-=[22, 23, 24, 25]-=- for a few representative examples.) Briefly, a stack is a manifold “paired with automorphisms.” (See e.g. [26, 27, 28] for a more technical definition.) At the same level of brevity, a gerbe is a sta...

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...oped by W. Chen and Y. Ruan and, secondly the author did not find in the literature a complete and explicit description of weighted projective spaces as toric Deligne-Mumford stacks. Afterward, using =-=[CCLT06]-=-, we prove two propositions 1 (cf. 4.14 and 4.17) on the value of some orbifold Gromov-Witten invariants with 3 marked points and we show in Section 6.c that these propositions imply an isomorphism be...

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...e 0, ∼= P(2) age 2. X 1 2 If 1i is the fundamental class of Xi and p = c1(O(1)), then φ0 = 10, φ1 = p10, φ2 = p 2 10, φ3 = p 3 10, φ4 = p 4 10, φ5 = 1 1 2 is a basis for H• orb (X; C). Theorem 1.6 in =-=[23]-=- shows that the restriction to the locus τ = tp of the untwisted J-function Jun (τ, z) of X is5 J un ∑ tp/z (tp, z) = ze d:d≥0 2d∈Z ∏ b:0<b≤d 〈b〉=〈d〉 Q d e dt (p + bz) 4 ∏ b:0<b≤2d 〈b〉=0 (2p + bz) 1 〈...

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...ic geometry; an algebrogeometric version of the theory has been developed by Abramovich, Graber, and Vistoli [1, 2]. An overview of this material, in compatible notation, can be found in Section 2 of =-=[15]-=-. Givental’s formalism was introduced in [26] and applied to Gromov–Witten theory in [16, 17, 39, 40, 50]. Expository accounts can be found in [27] and in Section 2 of [13].WALL-CROSSINGS IN TORIC GR...

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...f IX. (37) For this we argue exactly as in Proposition 4.1, combining Theorem 3.4 with [22, Theorem 1.7]. Theorem 3.4 here tells us how to modify the small J-function of P(1, 1, 3) and Theorem 1.7 in =-=[22]-=- tells us how to compute the small J-function of P(1, 1, 3). Proposition 5.3 then follows from (37) and Iritani’s Reconstruction Theorem [37, cf. Example 4.15]. □ Step 4: IX Determines LX. We have: Co...

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... ) he2 1 − q1q2q3 ) (5)For g ∈ {1/3, 1/4, 1/2, 3/4}, denote by 1g the class in A0 (P(w)(exp(2igπ))) where P(w) (exp(2igπ)) is the twisted sector corresponding to exp(2igπ). From [BCS05] or [Man05] or =-=[CCLT06]-=-, we deduce that the orbifold Chow ring A⋆ orb (P(1, 3, 4, 4)) is generated by H := c1(OP(w)(1)), E1 := 11/4, E2 := 11/2, E3 := 13/4 and E4 := 11/3, with relations: he3 HE4, E1E1 − 3HE2, E1E2 − 3HE3, ...

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...e give an example of this phenomenon. The weighted projective space 1. Introduction P(w0, . . .,wn) = C n+1 −{0} / C ∗ , z·(z0, . . .,zn) = (z −w0 z0, . . .,z −wn zn) provides a simple test case (see =-=[3]-=-, [2], [12]) for the recently developed theories of orbifold cohomology and orbifold quantum cohomology. Direct geometrical calculations are difficult, but mirror symmetry suggests an alternative and ...

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...with restrictions on nonperturbative sectors (sigma models on gerbes) in both the physics (see [4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]) and especially the math communities (see for example =-=[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]-=- for a sample). Such theories have also been discussed implicitly in a number of other places in the physics literature (see for example [50, 51, 52, 53, 54, 55, 56]). 6 Conclusions and outlook Supers...