294 research outputs found
GLSM's for gerbes (and other toric stacks)
In this paper we will discuss gauged linear sigma model descriptions of toric
stacks. Toric stacks have a simple description in terms of (symplectic, GIT)
quotients of homogeneous coordinates, in exactly the same
form as toric varieties. We describe the physics of the gauged linear sigma
models that formally coincide with the mathematical description of toric
stacks, and check that physical predictions of those gauged linear sigma models
exactly match the corresponding stacks. We also check in examples that when a
given toric stack has multiple presentations in a form accessible as a gauged
linear sigma model, that the IR physics of those different presentations
matches, so that the IR physics is presentation-independent, making it
reasonable to associate CFT's to stacks, not just presentations of stacks. We
discuss mirror symmetry for stacks, using Morrison-Plesser-Hori-Vafa techniques
to compute mirrors explicitly, and also find a natural generalization of
Batyrev's mirror conjecture. In the process of studying mirror symmetry, we
find some new abstract CFT's, involving fields valued in roots of unity.Comment: 43 pages, LaTeX, 3 figures; v2: typos fixe
Hyperelliptic Szpiro inequality
We generalize the classical Szpiro inequality to the case of a semistable
family of hyperelliptic curves. We show that for a semistable symplectic
Lefschetz fibration of hyperelliptic curves of genus , the number of
non-separating vanishing cycles and the number of singular fibers satisfy
the inequality .Comment: LaTeX2e, 27 page
Density of monodromy actions on non-abelian cohomology
In this paper we study the monodromy action on the first Betti and de Rham
non-abelian cohomology arising from a family of smooth curves. We describe
sufficient conditions for the existence of a Zariski dense monodromy orbit. In
particular we show that for a Lefschetz pencil of sufficiently high degree the
monodromy action is dense.Comment: LaTeX2e, 48 pages, Version substantially revised for publication. A
gap in the proof of the density for Lefschetz pencils is fixed. The case of
hyperelliptic monodromy is also treated in detai
Schematic homotopy types and non-abelian Hodge theory
In this work we use Hodge theoretic methods to study homotopy types of
complex projective manifolds with arbitrary fundamental groups. The main tool
we use is the \textit{schematization functor} , introduced by the third author as a substitute for the
rationalization functor in homotopy theory in the case of non-simply connected
spaces. Our main result is the construction of a \textit{Hodge decomposition}
on . This Hodge decomposition is encoded in an
action of the discrete group on the object
and is shown to recover the usual Hodge
decomposition on cohomology, the Hodge filtration on the pro-algebraic
fundamental group as defined by C.Simpson, and in the simply connected case,
the Hodge decomposition on the complexified homotopy groups as defined by
J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity
property with respect to a weight filtration, generalizing the fact that the
higher homotopy groups of a simply connected projective manifold have natural
mixed Hodge structures. As a first application we construct a new family of
examples of homotopy types which are not realizable as complex projective
manifolds. Our second application is a formality theorem for the schematization
of a complex projective manifold. Finally, we present conditions on a complex
projective manifold under which the image of the Hurewitz morphism of
is a sub-Hodge structure.Comment: 57 pages. This new version has been globally reorganized and includes
additional results and applications. Minor correction
Decomposition and the Gross-Taylor string theory
It was recently argued by Nguyen-Tanizaki-Unsal that two-dimensional pure
Yang-Mills theory is equivalent to (decomposes into) a disjoint union of
(invertible) quantum field theories, known as universes. In this paper we
compare this decomposition to the Gross-Taylor expansion of two-dimensional
pure SU(N) Yang-Mills theory in the large N limit as the string field theory of
a sigma model. Specifically, we study the Gross-Taylor expansion of individual
Nguyen-Tanizaki-Unsal universes. These differ from the Gross-Taylor expansion
of the full Yang-Mills theory in two ways: a restriction to single instanton
degrees, and some additional contributions not present in the expansion of the
full Yang-Mills theory. We propose to interpret the restriction to single
instanton degree as implying a constraint, namely that the Gross-Taylor string
has a global (higher-form) symmetry with Noether current related to the
worldsheet instanton number. We compare two-dimensional pure Maxwell theory as
a prototype obeying such a constraint, and also discuss in that case an
analogue of the Witten effect arising under two-dimensional theta angle
rotation. We also propose a geometric interpretation of the additional terms,
in the special case of Yang-Mills theories on two-spheres. In addition, also
for the case of theories on two-spheres, we propose a reinterpretation of the
terms in the Gross-Taylor expansion of the Nguyen-Tanizaki-Unsal universes,
replacing sigma models on branched covers by counting disjoint unions of stacky
copies of the target Riemann surface, that makes the Nguyen-Tanizaki-Unsal
decomposition into invertible field theories more nearly manifest. As the
Gross-Taylor string is a sigma model coupled to worldsheet gravity, we also
briefly outline the tangentially-related topic of decomposition in
two-dimensional theories coupled to gravity.Comment: 95 pages, LaTe
Cluster decomposition, T-duality, and gerby CFT's
In this paper we study CFT's associated to gerbes. These theories suffer from
a lack of cluster decomposition, but this problem can be resolved: the CFT's
are the same as CFT's for disconnected targets. Such theories also lack cluster
decomposition, but in that form, the lack is manifestly not very problematic.
In particular, we shall see that this matching of CFT's, this duality between
noneffective gaugings and sigma models on disconnected targets, is a worldsheet
duality related to T-duality. We perform a wide variety of tests of this claim,
ranging from checking partition functions at arbitrary genus to D-branes to
mirror symmetry. We also discuss a number of applications of these results,
including predictions for quantum cohomology and Gromov-Witten theory and
additional physical understanding of the geometric Langlands program.Comment: 61 pages, LaTeX; v2,3: typos fixed; v4: writing improved in several
sections; v5: typos fixe
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