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...rem of W-Constraints for Simple Singularities Si-Qi Liu, Di Yang, Youjin Zhang Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China May 14, 2013 Abstract In a recent paper =-=[3]-=-, Bakalov and Milanov proved that the total descendant potential of a simple singularity satisfies theW-constraints, which come from theW-algebra of the lattice vertex algebra associated to the root l...

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...m 1. The notation t that appears here has nothing to do with the deformation parameters that we introduced before. In order to avoid confusion, from now on we put am := a tm, a ∈ h∆, m ∈ Z. Following =-=[2]-=-, we define bosonic fields Xt(α, λ) = ∂λ f̂α(t, λ), (16) and propagators Pα,β(t, λ; µ − λ) = ∂λ∂µ lim ǫ→0 ∫ t−λ 1 t−(ui(t)+ǫ)1 I(0)α (t′, µ − λ) • I(0)β (t′, 0) (17) where α, β ∈ ∆, for each λ ∈ D∗ we...

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...ions can be expressed in terms of period integrals and phase forms, which suggests that they should be compared to the correlation functions of the twisted Vertex algebra representation introduced in =-=[2]-=-. 1.1. Preliminary notation. Let f ∈ OC2l+1,0 be the germ of a holomorphic function with an isolated critical point at 0. We fix a miniversal deformation F(t, x), t ∈ B and a primitive form ω in the s...

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...|σb| if χ|σb| is odd. Notice that |σb| = lcm(a1, a2, a3), the least common multiple of a1, a2, a3. Remark 28. The notation SF is motivated from the notion of a Seifert form in singularity theory (cf. =-=[3, 4]-=-). We do not claim that (65) is a Seifert form, although it would be interesting to investigate whether definition (65) can be interpreted as a linking number between α and β. 38 TODOR MILANOV, YEFENG...

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...nd Kac-Schwarz [18]. For algebraic background on W -algebras, we refer to Fateev-Lukyanov [8], Feigin-Frenkel [9, 10] and Frenkel-Kac-Radul-Wang [12]. More recently, 1 2 JIAN ZHOU Bakalov and Milanov =-=[2, 3]-=- constructed W-constraints for Frobenius manifolds associated with simple singularities, and they conjectured their constraints uniquely determine the partition function up to a factor. Recently, Liu,...

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...owed a in a series of publications [6, 8, 9, 25] that the total descendant potential of an A, D or E type singularity satisfies the Kac–Wakimoto hierarchy [17]. Recently Bakalov and Milanov showed in =-=[2]-=- that this potential is also a highest weight vector for the corresponding W -algebra. For type A Fukuma, Kawai and Nakayama [7] showed that these W constraints can be obtained completely from the str...

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...owed a in a series of publications [6, 8, 9, 25] that the total descendant potential of an A, D or E type singularity satisfies the Kac–Wakimoto hierarchy [17]. Recently Bakalov and Milanov showed in =-=[2]-=- that this potential is also a highest weight vector for the corresponding W -algebra. For type A Fukuma, Kawai and Nakayama [7] showed that these W constraints can be obtained completely from the str...

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...e celebrated string equation [1] L−1Z(t) = 0, (1.10) where L−1 := ∑ p≥r+1,r∤p tp ∂ ∂tp−r + 1 2 r−1∑ k=1 tktr−k − ∂ ∂t1 . (1.11) It can also be uniquely determined by using only the WAr−1 -constraints =-=[1, 3, 16]-=-. According to Witten’s r-spin conjecture [19], which was proved by Faber, Shadrin and Zvonkine in [11] and in a more general setting by Fan, Jarvis and Ruan in [12], logZ is also the generating funct...

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...g Frobenuis mainfolds are nonsemisimple. 1. Introduction Since the pioneering discovery by Zamolodchikov in [8], W-algebras have been an active field of theoretical and mathematical physics, see e.g. =-=[20, 21, 32, 33, 43, 51, 47, 48, 49, 50]-=- and references therein. It is well known that classical realizations of W-algebras appear naturally as Poisson brackets of integrable hierarchies of Lax type. For example, the Virasoro algebra W2 is ...