55 research outputs found
Superpotential algebras and manifolds
In this paper we study a special class of Calabi-Yau algebras (in the sense
of Ginzburg): those arising as the fundamental group algebras of acyclic
manifolds. Motivated partly by the usefulness of `superpotential descriptions'
in motivic Donaldson-Thomas theory, we investigate the question of whether
these algebras admit superpotential presentations. We establish that the
fundamental group algebras of a wide class of acyclic manifolds, including all
hyperbolic manifolds, do not admit such descriptions, disproving Ginzburg's
conjecture regarding them. We also describe a class of manifolds that do admit
such descriptions, and discuss a little their motivic Donaldson-Thomas theory.
Finally, some links with topological field theory are described.Comment: 31 pages, 2 figures, final version. Thanks to M. Kontsevich, V.
Ginzburg, M, Van den Bergh and B. Keller for helpful comments and
corrections. I've added some examples e.g. Klein bottl
Consistency conditions for brane tilings
Given a brane tiling on a torus, we provide a new way to prove and generalise
the recent results of Szendroi, Mozgovoy and Reineke regarding the
Donaldson-Thomas theory of the moduli space of framed cyclic representations of
the associated algebra. Using only a natural cancellation-type consistency
condition, we show that the algebras are 3-Calabi-Yau, and calculate
Donaldson-Thomas type invariants of the moduli spaces. Two new ingredients to
our proofs are a grading of the algebra by the path category of the associated
quiver modulo relations, and a way of assigning winding numbers to pairs of
paths in the lift of the brane tiling to R^2. These ideas allow us to
generalise the above results to all consistent brane tilings on K(pi,1)
surfaces. We also prove a converse: no consistent brane tiling on a sphere
gives rise to a 3-Calabi-Yau algebra.Comment: 28 pages, 4 figures. Many clarifications thanks to referee. Final
versio
Quasi-Hamiltonian reduction via classical Chern-Simons theory
This paper puts the theory of quasi-Hamiltonian reduction in the framework of
shifted symplectic structures developed by Pantev, To\"{e}n, Vaqui\'{e} and
Vezzosi. We compute the symplectic structures on mapping stacks and show how
the AKSZ topological field theory defined by Calaque allows one to neatly
package the constructions used in quasi-Hamiltonian reduction. Finally, we
explain how a prequantization of character stacks can be obtained purely
locally.Comment: 33 page
Gopakumar-Vafa invariants via vanishing cycles
In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of
Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal
is a modification of a recent approach of Kiem-Li, which is itself based on
earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants
are equivalent to other curve-counting theories such as Gromov-Witten theory
and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces,
our invariants agree with PT invariants for irreducible one-cycles. We also
give a counter-example to the Kiem-Li conjectures, where our invariants match
the predicted answer. Finally, we give examples where our invariant matches the
expected answer in cases where the cycle is non-reduced, non-planar, or
non-primitive.Comment: 63 pages, many improvements of the exposition following referee
comments, final version to appear in Inventione
Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory
The ADHM construction establishes a one-to-one correspondence between framed
torsion free sheaves on the projective plane and stable framed representations
of a quiver with relations in the category of complex vector spaces. This paper
studies the geometry of moduli spaces of representations of the same quiver
with relations in the abelian category of coherent sheaves on a smooth complex
projective curve . In particular it is proven that this moduli space is
virtually smooth and related byrelative Beilinson spectral sequence to the
curve counting construction via stable pairs of Pandharipande and Thomas. This
yields a new conjectural construction for the local Donaldson-Thomas theory of
curves as well as a natural higher rank generalization.Comment: 61 pages AMS Latex; v2: minor corrections, reference added; v3: some
proofs corrected using the GIT construction of the moduli space due to A.
Schmitt; main results unchanged; final version to appear in J. Geom. Phy
Shifted Symplectic Structures
This is the first of a series of papers about \emph{quantization} in the
context of \emph{derived algebraic geometry}. In this first part, we introduce
the notion of \emph{-shifted symplectic structures}, a generalization of the
notion of symplectic structures on smooth varieties and schemes, meaningful in
the setting of derived Artin n-stacks. We prove that classifying stacks of
reductive groups, as well as the derived stack of perfect complexes, carry
canonical 2-shifted symplectic structures. Our main existence theorem states
that for any derived Artin stack equipped with an -shifted symplectic
structure, the derived mapping stack is equipped with a
canonical -shifted symplectic structure as soon a satisfies a
Calabi-Yau condition in dimension . These two results imply the existence of
many examples of derived moduli stacks equipped with -shifted symplectic
structures, such as the derived moduli of perfect complexes on Calabi-Yau
varieties, or the derived moduli stack of perfect complexes of local systems on
a compact and oriented topological manifold. We also show that Lagrangian
intersections carry canonical (-1)-shifted symplectic structures.Comment: 52 pages. To appear in Publ. Math. IHE
The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity
We have made the largest-volume measurement to date of the transition to
large-scale homogeneity in the distribution of galaxies. We use the WiggleZ
survey, a spectroscopic survey of over 200,000 blue galaxies in a cosmic volume
of ~1 (Gpc/h)^3. A new method of defining the 'homogeneity scale' is presented,
which is more robust than methods previously used in the literature, and which
can be easily compared between different surveys. Due to the large cosmic depth
of WiggleZ (up to z=1) we are able to make the first measurement of the
transition to homogeneity over a range of cosmic epochs. The mean number of
galaxies N(<r) in spheres of comoving radius r is proportional to r^3 within
1%, or equivalently the fractal dimension of the sample is within 1% of D_2=3,
at radii larger than 71 \pm 8 Mpc/h at z~0.2, 70 \pm 5 Mpc/h at z~0.4, 81 \pm 5
Mpc/h at z~0.6, and 75 \pm 4 Mpc/h at z~0.8. We demonstrate the robustness of
our results against selection function effects, using a LCDM N-body simulation
and a suite of inhomogeneous fractal distributions. The results are in
excellent agreement with both the LCDM N-body simulation and an analytical LCDM
prediction. We can exclude a fractal distribution with fractal dimension below
D_2=2.97 on scales from ~80 Mpc/h up to the largest scales probed by our
measurement, ~300 Mpc/h, at 99.99% confidence.Comment: 21 pages, 16 figures, accepted for publication in MNRA
Measurement of the branching ratio of pi^0 -> e^+e^- using K_L -> 3 pi^0 decays in flight
The branching ratio of the rare decay pi^0 -> e^+e^- has been measured in
E799-II, a rare kaon decay experiment using the KTeV detector at Fermilab. The
pi^0's were produced in fully-reconstructed K_L -> 3 pi^0 decays in flight. We
observed 275 candidate pi^0 -> e^+e^- events, with an expected background of
21.4 +- 6.2 events which includes the contribution from Dalitz decays. We
measured BR(pi^0 -> e^+e^-, x>0.95) = (6.09 +- 0.40 +- 0.24) times 10^{-8},
where the first error is statistical and the second systematic. This result is
the first significant observation of the excess rate for this decay above the
unitarity lower bound.Comment: New version shortened to PRL length limit. 5 pages, 4 figures.
Published in Phys. Rev. Let
Regulation of dynein-driven microtubule sliding by the axonemal protein kinase CK1 in Chlamydomonas flagella
CK1 puts the brakes on dynein activity when added to purified axonemes in vitro, presumably to regulate how flagella bend
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A Moment of Mindfulness: Computer-Mediated Mindfulness Practice Increases State Mindfulness
Three studies investigated the use of a 5-minute, computer-mediated mindfulness practice in increasing levels of state mindfulness. In Study 1, 54 high school students completed the computer-mediated mindfulness practice in a lab setting and Toronto Mindfulness Scale (TMS) scores were measured before and after the practice. In Study 2 (N = 90) and Study 3 (N = 61), the mindfulness practice was tested with an entirely online sample to test the delivery of the 5-minute mindfulness practice via the internet. In Study 2 and 3, we found a significant increase in TMS scores in the mindful condition, but not in the control condition. These findings highlight the impact of a brief, mindfulness practice for single-session, computer-mediated use to increase mindfulness as a state
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