[blockquote]In physics, you don't have to go around making trouble for yourself - nature does it for you ~ Frank Wilczek ... Why I always say: 'lucky we have mathematicians'[/blockquote]

In my last four posts, I derived two propositions about quantum cohomology, with mirror-symmetry for Calabi-Yau manifolds being an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of \(X\) , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror \(Y\)

The post Orbifold Quantum Cohomology, Gromov–Witten Theory and the Quantum-Product appeared first on George Shiber.

]]>In physics, you don't have to go around making trouble for yourself - nature does it for you ~ Frank Wilczek ... Why I always say: 'lucky we have mathematicians'

In my last four posts, I derived two propositions about quantum cohomology, with mirror-symmetry for Calabi-Yau manifolds being an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror . With the Euler constant and the Riemann zeta function and , I defined the Witten -flat connection

with and hence, defines the quantum cohomological -integral structure. The mirror-symmetric image of a compact toric orbifold is then given by a Landau–Ginzburg model, which is a pair of a torus and a Laurent polynomial . The Landau–Ginzburg model then defines a -model -module which is decidable by an integral local system generated by Lefschetz thimbles of . By mirror symmetry, it follows that the quantum -module of a toric orbifold is isomorphic to the -model -module and derived the following two propositions

- Proposition one: Let be a weak Picard-Fano projective toric orbifold defined by the initial data satisfying , then, in light of mirror-symmetry, we get the -integral structure on the quantum -module, and it corresponds to the natural integral local system of the -model -module under the mirror isomorphism

with the quantum cohomology central charge of given by

- Proposition two: Under the same assumptions as proposition one, the quantum cohomology central charge of the structure sheaf is given by the Picard-integral over the real Lefschetz thimble

In this post, let me study, in an algebraic setting, orbifold quantum cohomology, in the context of the integral structure associated to the -group and the -class and derive a third proposition about the quantum product and how it is related to a power series . Let be the smooth Deligne–Mumford meta-stack over , with the inertia stack of over the fiber product of the diagonal morphisms . Now, points on are given by pairs for and with the pair-stabilizer. Let T be the Witten-index set of components of and the distinguished Euclidean-element corresponding to the trivial stabilizer: setting

one gets

with . Now pair a rational number to each connected component of and we get the degree shifting pair number of . Now define

as the eigenspace decomposition of relative to the stabilizer action, with acting on via

We hence can define:

and is independent of the choice of a point . I am now in a position to construct the orbifold cohomology group as the sum of the even degree cohomology of , :

with the degree of the orbifold cohomology a fractional number in general and factors in the right-hand side being the same as the cohomology group of qua topological space. Note now, there is an involution characterized by inducing an involution . So, let's then define the orbifold Poincaré pairing

with , the -components of , . The crucial properties of the orbifold Poincaré pairing is that it is symmetric, non-degenerate over and of degree , where . Since we can assume without loss of generality that the Poincaré-coarse moduli space of is projective, it follows that the genus zero Gromov– Witten invariants are integrals of the form

with

and a non-negative integer. Hence, is the virtual Yau-fundamental class of the Poincaré moduli stack of genus zero, -pointed meta-stable maps to of super-degree and is the symplectic evaluation map at the -th marked point, is the first Chern class of the line bundle over whose fiber at a meta-stable map is the cotangent space of the coarse curve at the -th marked point. Now Let and be bases of which are dual with respect to the orbifold Poincaré pairing, that is,

Then it follows that the orbifold quantum product is a family of commutative, associative products on parametrized by , which is defined by the formula

long-form,

with the element of the group ring corresponding to . Now, decomposing as

we finally get, by the Picard-divisor rank formula, the desired result of this post:

- proposition three: the quantum product can be viewed as a formal power series in and

vertically,

with

being the orbifold Poincaré 'term' and implies that the product defines an analytic family of commutative rings over , hence yielding the following deep (as we shall see) relation:

We shall get deeper next post.

Science is the belief in the ignorance of experts ~ Richard Feynman, again ... as I say, 'lucky we have mathematicians'

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]]>[blockquote]"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances" ~ Hannah Arendt[/blockquote]

In this post, I will begin the first part of a study of the cohomological-Dubrovin connection in the context of mirror-symmetry for Calabi-Yau Gromov–Witten theory. In particular, I will derive two propositions about quantum cohomology. In my last three posts, I finally derived, after studying the Witten Equation as well as the Landau-Ginzburg/Calabi-Yau correspondence, the Witten super-Dubrovin compactification Chern-Simons formula

The post Gromov–Witten Theory, Quantum Cohomology and Mirror Symmetry appeared first on George Shiber.

]]>"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances" ~ Hannah Arendt

In this post, I will begin the first part of a study of the cohomological-Dubrovin connection in the context of mirror-symmetry for Calabi-Yau Gromov–Witten theory. In particular, I will derive two propositions about quantum cohomology. In my last three posts, I finally derived, after studying the Witten Equation as well as the Landau-Ginzburg/Calabi-Yau correspondence, the Witten super-Dubrovin compactification Chern-Simons formula

with

and

the super-Dubrovin relation. Realizing that mirror symmetry for Calabi–Yau manifolds is interpretable as an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror . Now, while the B-model of the variations of Hodge structures has integral local system , the A-model does not. So: what is the integral local system in the A-model mirrored from the B-model?! I showed here that analysis on compact toric orbifolds entail that the K-group of should give the integral local system in the A-model. So, the genus zero Gromov-Witten theory induces a family of super-commutative algebras

on the cohomology group metaplectically parametrized by

which is the Witten quantum cohomology whose D-module is defined by a flat connection on the bundle , with a parameter , and is the Dubrovin connection studied by me here, and is quantum-cohomologically given by

with , where denotes a point on the base and is the directional derivative. One can then extend this connection in the direction of the parameter yielding a flat -bundle over

Now a symplectic solution to the differential equation

has general Picard-form

where is the Hodge grading operator

with is the master solution

Let me separate the terms

which is asymptotic to in the large radius limit

Now, with the Chern roots of the tangent bundle , one defines a Picard-transcendental characteristic class via

with

key for the Landau-Ginzburg/Calabi-Yau correspondence, with the Euler constant and the Riemann zeta function. For , define a -flat connection

with and hence, define the quantum cohomological -integral structure. The mirror-symmetric image of a compact toric orbifold is then given by a Landau–Ginzburg model, which is a pair of a torus and a Laurent polynomial . The Landau–Ginzburg model then defines a -model -module which is decidable by an integral local system generated by Lefschetz thimbles of . By mirror symmetry, it follows that the quantum -module of a toric orbifold is isomorphic to the -model -module: hence the two propositions of this post can be stated

- Proposition one: Let be a weak Picard-Fano projective toric orbifold defined by the initial data satisfying , then, in light of mirror-symmetry, we get the -integral structure on the quantum -module, and it corresponds to the natural integral local system of the -model -module under the mirror isomorphism

with the quantum cohomology central charge of given by

- Proposition two: Under the same assumptions as proposition one, the quantum cohomology central charge of the structure sheaf is given by the Picard-integral over the real Lefschetz thimble

In the nest post, I will go further and study Orbifold quantum cohomology.

Physics is becoming too difficult for the physicists ~ David Hilbert

The post Gromov–Witten Theory, Quantum Cohomology and Mirror Symmetry appeared first on George Shiber.

]]>[blockquote]“But in my opinion, all things in nature occur mathematically” ~ René Descartes[/blockquote]

In my last two posts on the Witten Equation, I showed that in the context of Picard-Lefschetz Theory, it entails the non-forking, super-stability and hyper-categoricity of M-theory, making it, up to isomorphism, the only unification theory possible between Einstein's ToGR and Quantum Field Theory, as well as derived the Dubrovin connection and related it to Kähler-Witten integral and showed that it is a necessary condition for the action-principle of any such unified 'Theory-of-Everything'. Let's delve deeper.

The post The Witten Super-Dubrovin 'Compactification Chern-Simons' Formula appeared first on George Shiber.

]]>“But in my opinion, all things in nature occur mathematically” ~ René Descartes

In my last two posts on the Witten Equation, I showed that in the context of Picard-Lefschetz Theory, it entails the non-forking, super-stability and hyper-categoricity of M-theory, making it, up to isomorphism, the only unification theory possible between Einstein's ToGR and Quantum Field Theory, as well as derived the Dubrovin connection and related it to Kähler-Witten integral and showed that it is a necessary condition for the action-principle of any such unified 'Theory-of-Everything'. Let's delve deeper. Keep in mind the Kähler-Witten integral of

with the Calabi-Yau smooth projective variety corresponding to the Witten-potential. This is key because the -invariants of the differential operators in the Dubrovin meromorphic flat connection on

are metaplectically connected to the secondary invariants Chern-Simons type, which is the Atiyah-Patodi formula, that the Witten equation entails are -invariants of such operators for spaces that fibre over . It is key to homeomorphically derive, from the Dubrovin connection, the corresponding Witten-fibrated Atiyah-Patodi-Singer super-Dubrovin connection

that acts on sections of an infinite-dimensional Hermitian vector bundle over , with arclength on because, as we shall see in future posts, it is essential for the principal action of M-theoretic braneworld cosmology.

First, note we have

with

the Kähler-invariant associated to the first Chern class and the connection and is the phase where is the determinant of the holonomy of . Let be a four-fold Calabi-Yau manifold, and a Hermitian vector bundle with Hermitian connection over and

given by

on . Now, since and super-extends over , letting and refer to the characteristic forms corresponding to the -class of and Chern character of , we can derive

Let us get to the Witten formula now. Letting

be a Calabi-Yau submersion, and with , refer to the exterior differentiation, with the Hodge -operator, let

and

Then clearly is self-adjoint, elliptic, and and and now write

where is arclength on . Then we have

hence Witten's formula for ) is locally computable on . Put

and let

denote the heat kernel of . Then for the operator and we get the of determinant line bundle global anomaly relation

Now, with , we have and the super-Dubrovin relation

and holding. Therefore, we can deduce

for

finally arriving at the Witten super-Dubrovin compactification Chern-Simons formula

with

Let us see where the mathematics leads us next, for

"We are servants rather than masters in mathematics" ~ Charles Hermite

The post The Witten Super-Dubrovin 'Compactification Chern-Simons' Formula appeared first on George Shiber.

]]>[blockquote]Mathematics is the music of reason ~

James Joseph Sylvester

[/blockquote]

In my last post in this series on the Witten-equation, I showed that the genus-zero Landau-Ginzburg/Calabi-Yau correspondence amounts to the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small \(J\)-functions coincide, thus, the 'Picard-Lefschetz' Witten relation

The post The Witten Equation, Quantum Cohomology and the Dubrovin Connection appeared first on George Shiber.

]]>Mathematics is the music of reason ~

James Joseph Sylvester

In my last post (part 1) in this series on the Witten-equation, I showed that the genus-zero Landau-Ginzburg/Calabi-Yau correspondence amounts to the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small -functions coincide, thus, the 'Picard-Lefschetz' Witten relation, for are

and

Keep the quantum product

in mind throughout the Witten Equation post-series.

In this post, part 2, I will connect the Witten equation

with the Gromov–Witten invariants and the Kähler-Witten integral in a deep way via the Dubrovin connection (see below) in the context of quantum cohomology, with the Dubrovin connection and the Gromov–Witten invariants playing a central role on the corresponding throat-bulk smooth projective variety. Let me change notation from my previous post: in this post, let from part 1 refer , the Calabi-Yau smooth projective variety corresponding to the Witten-potential, and the even part of , and the metaplectic moduli space of -pointed genus- stable maps to X of degree . The following Kähler-Witten integral of is central to this post

with and being the evaluation map at the -th marked point and being the universal cotangent line classes. Now we can derive

Let me fix bases , for satisfying

- being the identity element of

- is a nef-basis for

- each is Kähler-homogeneous

- and are Poincaré-pairing dual

with the rank of . To get to our quantum cohomology analysis, let the Novikov ring and for , we write with and now we can get to quantum cohomology. Letting be the coordinates defined by the basis such that satisfies , we hence get the genus-zero Gromov-Witten potential

via

with the first sum is over the set of degrees of effective curves in

So the quantum product can thus be only defined in terms of the third partial derivatives of as

where is bilinear over therefore defining a formal family of algebras on parametrized by and that is the quantum super-cohomology of and has a Hodge–Tate type: for . Since is a Gromov-Witten scheme over , letting be a topological neighbourhood of the origin in , then the Euler vector field on is

Note now that the grading operator is definable via

with projection to the first factor. Hence, the Dubrovin connection is a meromorphic flat connection on

defined by

and the coordinate on . Now, by the Poincaré pairing, the Dubrovin connection equips with a Frobenius manifold with extended structure connection, and thus the genus-zero Gromov–Witten potential converges to an analytic function, allowing a definition of a Fredholm-Calabi-Yau measure for the Kähler–Witten integral, due to the quantum product

and allows the Gromov–Witten invariants

to be metaplectic invariants on the homotopy group-manifold of again due to, and since it is, equipped with the quantum product .

As we shall see, these two relations are key to the unificational uniqueness, up to isomorphism, of M-theory and Branewold cosmology.

The post The Witten Equation, Quantum Cohomology and the Dubrovin Connection appeared first on George Shiber.

]]>[blockquote]'Edward Witten is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton' ~ John Horgan[/blockquote]

This is a series of posts that hopefully will culminate in a book I am working on. The aim is to show that the Witten equation in the context of Picard-Lefschetz Theory, leads to the non-forking, super-stability and hyper-categoricity of M-theory, thus making it the 'only-game-in-town' indeed. First, I will analyse the Witten nonlinear elliptic system of PDEs associated with a quasi-homogeneous polynomial-super-potential by showing its depth via the Landau-Ginzburg/Calabi-Yau correspondence.

The post M-Theory, the Witten Equation and Picard-Lefschetz Theory appeared first on George Shiber.

]]>'Edward Wittenis often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind sinceNewton' ~ John Horgan

This is a series of posts that hopefully will culminate in a book I am working on. The aim is to show that the Witten equation in the context of Picard-Lefschetz Theory, leads to the non-forking, super-stability and hyper-categoricity of M-theory, thus making it the 'only-game-in-town' indeed. First, I will analyse the Witten nonlinear elliptic system of PDEs associated with a quasi-homogeneous polynomial-super-potential by showing its depth via the Landau-Ginzburg/Calabi-Yau correspondence. The Witten equation is deceiving in its simplicity:

However, as you will see at the end of this post, it gets tricky:

with a quasi-homogeneous 'super-potential' polynomial and a section of the corresponding orbifold line bundle on a Riemann surface . To appreciate the depth of the Witten equation, one must understand the Landau-Ginzburg/Calabi-Yau correspondence: a connection between the geometry of Calabi-Yau hyper-complete intersections in projective space and the Landau-Ginzburg model, where the polynomials defining the intersections are interpreted as singularities, and the Calabi-Yau side is the Gromov-Witten theory of the hyper-complete intersection. To give a taste of such relation(s), take a nondegenerate collection of quasihomogeneous polynomials

with weights , degree with the following relation

On the Calabi-Yau 'side', one analyses the intersection in weighted projective space cut out by the polynomials: the cohomology of this intersection is quasimorphic to the state-space from which insertions to Gromov-Witten invariants of are selected. Now, for any choice of

and any

we have a Gromov-Witten invariant

which is the hyper-intersection number on the moduli space of stable maps to

Hence, the genus-zero invariants are encoded by a -function

with

central for Calabi-Yau n-foldings, and with

holding, and runs over a basis for . On the Landau-Ginzburg side, the polynomials are interpretable as the equations for singularities in . Hence, since the state space and its members can be used as the insertions to hybrid invariants

and are metaplectic hyper-intersection numbers on a moduli space parameterizing stable maps to projective space together with a collection of line bundles on the source curve whose tensor powers satisfy equations determined by the polynomials , we get the -functional genus-encoding relation

and this time,

is key for to solving the equations for singularities. The genus-zero Landau-Ginzburg/Calabi-Yau correspondence is the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small -function coincide. So one gets, along the way to show the depth of the Witten equation, a derivation of the Picard-Fuchs equation

and

where . Note now by explicit change of variables

for -valued functions and , and thus, the -function matches , hence

For the Landau-Ginzburg side, we have

for the cubic

The key fact about -functions is the fact that the family lives on the Lagrangian cone on which the -function is a slice, as will be shown. Hence, we get

with

key to Calabi-Yau n-folding, hence getting the 'Picard-Lefschetz' Witten relation, for ,

In the next post, I will make contact with quantum cohomology.

The post M-Theory, the Witten Equation and Picard-Lefschetz Theory appeared first on George Shiber.

]]>[blockquote]Ex nihilo nihil fit ~ Parmenides[/blockquote]

In this post, in order to solve the Wheeler-deWitt problem-of-time discussed by me, and since string/M-theoretic braneworld cosmology offers the only solution in a unified C*-Heisenberg-algebraic sense, I must first analyse a field theory on a D-brane sigma worldvolume and study the Dirac-Born-Infeld action

The post The Dirac-Born-Infeld Action, D-Branes and Spacetime Symmetries appeared first on George Shiber.

]]>Ex nihilo nihil fit ~ Parmenides

In this post, in order to solve the Wheeler-deWitt problem-of-time I discussed in my last post, and since string/M-theoretic braneworld cosmology offers the only solution in a unified C*-Heisenberg-algebraic sense, I must first analyse a field theory on a D-brane sigma world-volume and study the Dirac-Born-Infeld (DBI) action. Let me start by giving a proof that the DBI action has a symmetry under the transformation

of full

on a generalized Riemannian structure

First, I must prove the following relation

with the Riemannian metric on

and the metric on

Now, since is the pullback of the embedding map defined by the field , we have

Thus, the determinants on the l. h. s. are those of the matrices, while the determinant on the r.h.s. is that of the matrix distinguished by the index . Now, the r. h. s. of

is the Lagrangian of the Dirac-Born-Infeld action and can be proven by combining the following relations of various determinants

1) Let be the symmetric part of given by

and

and the inverse matrix of , then we have

2) From

and the definition of , we have

3) Now, by using the explicit expression for , it follows that

Using these relations in 1), 2), 3), one can prove the representation of the DBI action given in

which is

The integral of the scalar density agrees with the DBI action

when evaluated on the leaf of at , hence, the DBI action is invariant under the world-volume diffeomorphism on the D-brane.

Now I am in a position to prove that the DBI action is not only invariant under the world-volume diffeomorphism but also under the full target space diffeomorphism and the B-field gauge transformation. Towards this end,

can be re-written as an integral over the target space as

where

is a Dirac delta function interpreted as a distribution along directions. The infinitesimal transformation of the full diffeomorphism and the B-field gauge transformation are parametrized by . So the transformation of the integrand is gotten from that of

and that of

And the glorious result is

with

being central to full diffeomorphism. This is the expected result since is a section of

On the other hand, the delta function transforms as

Now, combining, we get

as desired, hence, the transformation of the integrand in the DBI action

is a total derivative and the DBI action is invariant under full target space diffeomorphisms and B-field gauge transformations and the invariance within the static gauge is what will allow us to further analyse the Wheeler-deWitt problem-of-time.

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]]>