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Abstract. Let X 1 and X 2 be smooth proper Deligne-Mumford stacks with projective coarse moduli spaces. We prove a formula for orbifold Gromov-Witten invariants of the product stack X 1 × X 2 in terms of Gromov-Witten invariants of the factors X 1 and X 2 . As an application, we deduce a decomposition result for Gromov-Witten theory of trivial gerbes.

Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r > 1 one defines the stack r L/X of r-th roots of L. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus 0 twisted stable maps to r L/X. Our main results are explicit constructions of moduli stacks of genus 0 twisted stable maps to r L/X starting from moduli stack of genus 0 stable maps to X. As a consequence, we prove an exact formula expressing genus 0 Gromov-Witten invariants of r L/X in terms of those of X.Date: November 1, 2018. 1 2 ELENA ANDREINI, YUNFENG JIANG, AND HSIAN-HUA TSENG References 36The Gromov-Witten theoretic version of the decomposition conjecture, which can be formulated for arbitrary G-gerbes more general than G-banded gerbes, states that Gromov-Witten theory of the G-gerbe is equivalent to certain twist of the Gromov-Witten theory of someétale cover of the base. A detailed discussion of the conjecture in full generality can be found in [38]. For G-banded gerbes this conjecture states that the Gromov-Witten theory of a G-banded gerbe over X is equivalent to (certain twists of) the Gromov-Witten theory of the disjoint union of |Conj(G)| copies 1 of X after a change of variables. Computations of Gromov-Witten invariants ofétale gerbes is thus intimated connected to the decomposition conjecture.The simplest examples of G-gerbes are trivial gerbes. The trivial G-gerbe over a Deligne-Mumford stack X is the product X × BG. In [7] the computation of Gromov-Witten invariants of X × BG is handled as a special case of a general product formula for orbifold Gromov-Witten invariants of product Deligne-Mumford stacks X × Y. As a consequence the decomposition conjecture is proven for trivial G-gerbes.An interesting class of non-trivial gerbes is provided by root gerbes associated to line bundles. This is the first of two papers in which we study Gromov-Witten theory of root gerbes of line bundles over smooth projective varieties, with the decomposition conjecture in mind. The present paper is devoted to study the genus 0 Gromov-Witten theory of root gerbes.Let X be a smooth complex projective variety and L → X a line bundle. Given an integer r > 0, let G := r L/X → X 1 Here Conj(G) is the set of conjugacy classes of G.

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Narcissistic personality disorder is characterized by self-absorption, grandiosity, exploitation of others and lack of empathy. People with that disorder may switch from an overt form, mainly with grandiosity, to a covert presentation, with fears, hypersensitivity and dependence from others. Empathy represents a key point in detecting people affected by narcissistic personality disorder because, even if it is described as reduced, it plays a fundamental role in exploitation and manipulation. A systematic search of Literature without any language or time restriction, was performed combining thesaurus and free-search indexing terms related to Narcissistic personality disorder and empathy and produced 531 results. Fifty-two papers that analyzed possible issues in the empathic attitude of people with narcissistic personality disorder were included in this narrative review. Empathy is the capability of understating and feeling others emotions. It is not a unitary construct and can be distinguished in cognitive and affective. It might be channeled into prosocial and antisocial behaviors. A crucial trait identified in narcissistic empathy is affective dissonance that is closely related to rivalry as part of the dark tetrad (narcissism, machiavellianism, psychopathy, and sadism). Subjects affected by narcissistic personality disorder show greater impairment in affective aspects while their cognitive part of empathy appears preserved. Saving at least the cognitive aspects of empathy may contribute to therapeutic improvement of affective aspects.