By Y. He

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Consider each point t the algebraic torus T n := (C∗ )n ≃ N ⊗ZZ C∗ ≃ hom(M,C∗ ) ≃ spec(C[M]) as a group homomorphism t : M → C∗ and each point x ∈ Xσ as a monoid homomorphism x : Sσ → C. Then we see that there is a natural torus action on the toric variety by the algebraic torus T n as x → t · x such that (t · x)(u) := t(u)x(u) for u ∈ Sσ . For σ = {0}, this action is nothing other than the group multiplication in T n = Xσ={0} . 2 The Delzant Polytope and Moment Map How does the above tie in together with what we have discussed on symplectic quotients?

1 The Gauged Linear Sigma Model According to [17], let us begin with neither the Calabi-Yau sigma model nor the LG theory with superpotenetial, let us begin instead with a linear sigma model with gauge group U(1). , 1). We choose W to be of the form W = P · G(si ) where G is a homogeneous polynomial of degree 5. On the other hand, SD is the D-term of Fayet-Illiopoulos, of the form D = −e2 i Qi |Xi |2 − r = −e2 i |si |2 − 5|p|2 − r . The bosonic part of our potential then becomes U = |G(si)|2 + |p|2 i | ∂G 2 1 | + 2 + 2|σ|2 ∂si 2e i Q2i |Xi |2 , with σ a scalar field in the (twisted) chiral multiplet.

Nite groups. Moreover, the Cartan matrices will correspond to certain graphs constructable from the latter. 43 Chapter 4 Finite Graphs, Quivers, and Resolution of Singularities We have addressed algebraic singularities, symplectic quotients and orbifolds in relation to finite group representations. It is now time to embark on a journey which would ultimately give a unified outlook. To do so we must involve ourselves with yet another field of mathematics, namely the theory of graphs. 1 Some Rudiments on Graphs and Quivers As we shall be dealing extensively with algorithms on finite graphs in our later work on toric singularities, let us first begin with the fundamental concepts in graph theory.

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Algebraic Singularities, Finite Graphs and D-Brane Theories by Y. He

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