Derived categories of quadric fibrations and intersections of quadrics

GLSMs for partial flag manifolds

“…We will also describe a tentative proposal for understanding the mathematical relationship between non-birational phases: we propose that they should be understood in terms of Kuznetsov's homological projective duality [6,7,8]. In this paper we will only begin to outline the relevance of Kuznetsov's work -a much more thorough description, and further application to abelian GLSM's, will appear in [9].…”

“…In the past, it was thought that geometric phases of gauged linear sigma models must be related by birational transformations, but recently in [5][section 12.2] and [4], counterexamples have appeared. We interpret the different phases as being related by Kuznetsov's homological projective duality [6,7,8], an idea that will be explored much further in [9]. We use that proposal that gauged linear sigma models realize Kuznetsov's duality to make a proposal for the physical interpretation of Calabi-Yau complete intersections in G(2, N) for N even, which has been mysterious previously.…”

Some recent advances in gauged linear sigma models

“…Instead of being related by a birational transformation, these phases are instead related by a newer notion, namely Kuznetsov's homological projective duality [21,22,23]. Describing homological projective duality in detail is beyond the scope of these lectures, but we can observe briefly that not only the novel examples discussed above but also more traditional GLSM phases [17,18,19,20] naturally fit into that framework, and so many now believe that all GLSM phases are related by homological projective duality.…”

“…(Strictly speaking, because of the Z 2 orbifold along the fibers, they form a module over the sheaf of even parts of the Clifford algebra.) Mathematically, such matrix factorizations define what is known as a noncommutative resolution of the branched double cover [16,22]. Noncommutative resolutions, in the pertinent sense, are defined by their sheaves, and our claim is ultimately just an unraveling of definitions.…”

“…[2,3,4,5,6,7,8,9,10] and references therein) to new developments in Calabi-Yau compactifications, including realizations of non-complete-intersection target spaces [11,12,13,14,15], non-birational phases [11,12,15,16,17,18,19,20] (leading to modern conjectures that all phases of GLSM's are related by a new notion known as "homological projective duality" [17,18,19,20,21,22,23]), nonperturbative realizations of geometry [11,12,15,16,24,25,26], realizations of noncommutative resolutions [12,16,25,26], and localization techniques [27,28]. Those localization techniques have been applied to deduce new methods for computing Gromov-Witten invariants [24,29], computations of elliptic genera directly in GLSM's [30,31,…”