A classical model for derived critical loci

Joyce, Dominic

doi:10.4310/jdg/1442364653

Cited by 76 publications

(152 citation statements)

References 33 publications

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“…The two handlebodies yield complex Lagrangians

L_{0}

L_{1} \subset X_{i r r}^{τ} false(normalΣ false)

, which can be equipped with spin structures. The intersection

L_{0} \cap L_{1}

is an oriented d‐critical locus in the sense of Joyce . Applying the work of Bussi , we obtain from here a perverse sheaf of vanishing cycles,

P_{L_{0}, L_{1}}^{•}

, on

L_{0} \cap L_{1}

.…”

Section: Introductionmentioning

confidence: 89%

“…Proposition is a consequence of the following general lemma, which appeals to theory of d‐critical loci introduced by Joyce in . A (complex‐analytic) d‐critical locus is a complex‐analytic space

X

along with a section

s

of a certain sheaf

S_{X}^{0}

which locally parametrizes different ways of writing

X

as the critical locus of a holomorphic function; see [, Definition 2.5]. Lemma Consider two complex spin Lagrangians

L_{0}, L_{1}

in a complex symplectic manifold

M

.…”

Section: Computational Toolsmentioning

confidence: 99%

“…We know that locally, the Lagrangian

L_{1}

is the graph of

d f

, for some holomorphic function

f : U \to C

, where

U

is a neighborhood of

x

L_{0}

. This turns

X

into a

d

‐critical locus in the sense of [, Definition 2.5]. In our case, the sheaf

S_{X}^{0}

is computed in [, Example 2.16] in the algebraic setting, but the same calculation applies in the complex analytic setting: The sections

s

S_{X}^{0}

are elements of the module

false(z^{n + 1} false) / false(z^{2 n} false)

over

O / (z^{n})

s = a_{n + 1} z^{n + 1} + a_{n + 2} z^{n + 2} + \dots + a_{2 n - 1} z^{2 n - 1} + false(z^{2 n} false) .

Such a section

s

specifies the structure of

X

as a

d

‐critical locus (provided that

a_{n + 1} \neq 0

).…”

Section: Computational Toolsmentioning

confidence: 99%

“…In our case, the sheaf

S_{X}^{0}

is computed in [, Example 2.16] in the algebraic setting, but the same calculation applies in the complex analytic setting: The sections

s

S_{X}^{0}

are elements of the module

false(z^{n + 1} false) / false(z^{2 n} false)

over

O / (z^{n})

s = a_{n + 1} z^{n + 1} + a_{n + 2} z^{n + 2} + \dots + a_{2 n - 1} z^{2 n - 1} + false(z^{2 n} false) .

Such a section

s

specifies the structure of

X

as a

d

‐critical locus (provided that

a_{n + 1} \neq 0

). Once this structure is determined, by Proposition 2.22 in , we find that, if

m

denotes the complex dimension of

L_{0}

, then there are local holomorphic coordinates

z_{1}, \dots, z_{m}

near

x \in L_{0}

in which we can write

f false(z_{1}, \dots, z_{m} false) = s false(z_{1} false) + z_{2}^{2} + \dots + z_{m}^{2} .

The perverse sheaf

P_{L_{0}, L_{1}}^{•} |_{X}

…”

Section: Computational Toolsmentioning

confidence: 99%

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A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three‐manifolds, similar to the three‐manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf‐theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two‐bridge and torus knots, the SL(2,C) invariant is determined by the l‐degree of the trueÂ‐polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.

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“…The two handlebodies yield complex Lagrangians

L_{0}

L_{1} \subset X_{i r r}^{τ} false(normalΣ false)

, which can be equipped with spin structures. The intersection

L_{0} \cap L_{1}

is an oriented d‐critical locus in the sense of Joyce . Applying the work of Bussi , we obtain from here a perverse sheaf of vanishing cycles,

P_{L_{0}, L_{1}}^{•}

, on

L_{0} \cap L_{1}

.…”

Section: Introductionmentioning

confidence: 89%

X

along with a section

s

of a certain sheaf

S_{X}^{0}

which locally parametrizes different ways of writing

X

as the critical locus of a holomorphic function; see [, Definition 2.5]. Lemma Consider two complex spin Lagrangians

L_{0}, L_{1}

in a complex symplectic manifold

M

.…”

Section: Computational Toolsmentioning

confidence: 99%

“…We know that locally, the Lagrangian

L_{1}

is the graph of

d f

, for some holomorphic function

f : U \to C

, where

U

is a neighborhood of

x

L_{0}

. This turns

X

into a

d

‐critical locus in the sense of [, Definition 2.5]. In our case, the sheaf

S_{X}^{0}

is computed in [, Example 2.16] in the algebraic setting, but the same calculation applies in the complex analytic setting: The sections

s

S_{X}^{0}

are elements of the module

false(z^{n + 1} false) / false(z^{2 n} false)

over

O / (z^{n})

s = a_{n + 1} z^{n + 1} + a_{n + 2} z^{n + 2} + \dots + a_{2 n - 1} z^{2 n - 1} + false(z^{2 n} false) .

Such a section

s

specifies the structure of

X

as a

d

‐critical locus (provided that

a_{n + 1} \neq 0

).…”

Section: Computational Toolsmentioning

confidence: 99%

“…In our case, the sheaf

S_{X}^{0}

is computed in [, Example 2.16] in the algebraic setting, but the same calculation applies in the complex analytic setting: The sections

s

S_{X}^{0}

are elements of the module

false(z^{n + 1} false) / false(z^{2 n} false)

over

O / (z^{n})

s = a_{n + 1} z^{n + 1} + a_{n + 2} z^{n + 2} + \dots + a_{2 n - 1} z^{2 n - 1} + false(z^{2 n} false) .

Such a section

s

specifies the structure of

X

as a

d

‐critical locus (provided that

a_{n + 1} \neq 0

). Once this structure is determined, by Proposition 2.22 in , we find that, if

m

denotes the complex dimension of

L_{0}

, then there are local holomorphic coordinates

z_{1}, \dots, z_{m}

near

x \in L_{0}

in which we can write

f false(z_{1}, \dots, z_{m} false) = s false(z_{1} false) + z_{2}^{2} + \dots + z_{m}^{2} .

The perverse sheaf

P_{L_{0}, L_{1}}^{•} |_{X}

…”

Section: Computational Toolsmentioning

confidence: 99%

See 2 more Smart Citations

A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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“…In Section 4, we outline some physical realizations of derived schemes implicit in the mathematics literature . Specifically, we discuss how some properties of two‐dimensional Landau–Ginzburg models are encapsulated by derived schemes, as derived critical loci and derived zero loci, and how renormalization group flow again realizes equivalences.…”

Section: Introductionmentioning

confidence: 99%

Categorical Equivalence and the Renormalization Group

Sharpe

2019

Fortschritte der Physik

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In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and ‘decomposition,’ in which two‐dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D‐branes, antibranes, and tachyons. In the third part of this article, we review how Landau–Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs.

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Introduction

Toda

2024

Categorical Donaldson-Thomas Theory for Local Surfaces

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A classical model for derived critical loci

Cited by 76 publications

References 33 publications

A sheaf‐theoretic SL(2,C) Floer homology for knots

A sheaf‐theoretic SL(2,C) Floer homology for knots

Categorical Equivalence and the Renormalization Group

Introduction

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