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Rigorously Computed Enumerative Norms as PrescribedRigorously Computed Enumerative Norms as Prescribed
through Quantum Cohomological Connectivity over Gromov –through Quantum Cohomological Connectivity over Gromov –
Witten InvariantsWitten Invariants
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CITATION
Bhattacharjee, Deep (2022): Rigorously Computed Enumerative Norms as Prescribed through Quantum
Cohomological Connectivity over Gromov – Witten Invariants. TechRxiv. Preprint.
https://doi.org/10.36227/techrxiv.19524214.v1
DOI
10.36227/techrxiv.19524214.v1
1
DRAFT VERSION – 1 RG – No. 70, Sec – deep@aatwri / (Proceedings - 2) / (Project Content Upload – 2)
PREPRINT https://www.researchgate.net/profile/Deep-Bhattacharjee
Rigorously Computed Enumerative Norms as Prescribed through
Quantum Cohomological Connectivity over Gromov – Witten Invariants
Deep Bhattacharjee*
Project Director, Development of G – Fuel at AATWRI, Electro–Gravitational Space Propulsion
Abstract: Novikov ring which has been used as a coefficient over the closed
symplectic manifold that when encountered through an extension, Big one, from
the ordinary cohomology to the quantum cohomology, then the ‘fuzzy’ quantum
nature could be more precisely described by the ‘quantum cup product’ inducing
the variations from the Riemann sphere connecting two points as analogous to
the J – Holomorphic curves or Pseudoholomorphic curves between two points as
in symplectic manifolds. Poincare duality as interpreted over the curves makes
the associativity over two bubble–manifolds through Gromov connections which
later makes a non–local invariance over the Gromov – Witten Invariants.
Complex graded Novikov ring being associated over Poincare duality finds its
way through the Riemann surface over genus – 0 and marked points – K, through
the perturbed Cauchy – Riemann Equation. For the n – point Gromov – Witten
Invariants, = 3 is taken for small quantum cohomology and 4 for big
quantum cohomological models. This proves essential in establishing the duality
(Topological) between Heterotic SO(32), Heterotic E8×E8 with Type II-A
supersymmetric strings in M – Theory.
Methods: Algebraic Topology and symplectic geometry being a vast domain of
mathematical studies, when it’s a subset called Quantum Cohomology and
Gromov – Witten Invariants are considered then, it’s almost impossible and
impracticable to jump straight into the desired topic. This is not a feasible idea
because there are several difficult notions and theories along with conjectures
that fall under this topic which again contains many sub–theories therefore, one
who is reading this paper and trying to grasp or get a hold of this subject would
find themselves completely lost, even if he knows the desired topics of this
paper. Thus, if I haven’t taken all the sub–concepts and explained them before
the Gromov – Witten Invariants, then this would naturally indicate that this paper
is for experts and professionals who are going through this paper just to see the
old things again. Thus, the reader will find several topics of explanations at the
beginning which is not at all recommended for them to skip, else when only
those terminologies appear in the Gromov - Witten Invariant section and
Quantum Cohomology section, they will find an ocean of difficulties in
understanding the concepts.
Keywords: De Rahm Cohomology – Novikov Rings – J-Holomorphic Curves – Hilbert Polynomial – Poincare Duality –
Deligne-Mumford Stack
*Email: deep@aatwri.in
Domain –
Algebraic Geometry
Symplectic Topology
Sub Domain –
Quantum Cohomology
Gromov – Witten Invariants
Areas –
Mathematics
Mathematical Physics
Date –
March, 2022
Project –
Proceedings – 2 : deep@aatwri
2
Topics covered in course of explaining the Quantum Cohomology and Gromov – Witten
Invariants:
Pages
Deligne – Mumford Stack 3
De Rahm Cohomology 4
Novikov Ring 5
J – Holomorphic Curve 6
Poincar Duality 7
Topological String Theory 8
Gromov – Witten Invariants 9
Acknowledgement 10
Conflicts of Interest 10
References 10
Each topic contains several subtopics which are not included in the appendix as a single subtopic
appears under other topics also.
This is a rigorously computed paper for highly advanced readers. Almost every concept, that is
found feasible for this work is explained here. References are there at the end for a more
detailed understanding of the subject. However, the arena and perspectives of the Quantum
Cohomology defined through the Gromov – Witten Invariants are not at all limited to this. The
resources are huge and mathematics which has never been a completed theory, at least as of now,
where I meant the advanced mathematics, thus it is reasonable to understand mathematical physics
as associated with strings are still ongoing and subject to research for further modifications,
alterations, and additions. The prerequisite knowledge needed to study this paper is given below,
1. Strong knowledge of supersymmetric string field theory.
2. Concrete knowledge of algebraic topology.
3. Expertise in Riemann, Symplectic and Enumerative Geometries.
4. Needless to say, the names, meanings of the operators, or symbols are used here.
5. Last and final, a passion to know more and a hunger to digest the mysteries, symmetries,
norms, axioms, invariants and topological models of the farthest fathom of mathematical
physics or in turn mathematics (depending on the subject specialization of the readers or
researchers are having).
3
I Deligne – Mumford Stack
If is a quasi–compact manifold then the stack it is going to represent could be such that the notion of a virtual group
can be satisfied via finite automorphisms for any parameter in the Deligne – Mumford () stack. Generally, a virtual
group can be seen as a morphism in an invertible category where the morphisms parameterized by and satisfied a
composition if a function could be taken representing the same as such,
()
In case of the algebraic geometry, when the algebraic stack that is the generalization of the algebraic spaces, formed the
structure of the moduli spaces, then the stack can be represented by two classes,
A map from the domain to the stack as .
A diagonal morphism of the stack to the stack product as ×.
Over a given degree of the projective space, certain algebraic varieties are effective over a certain class of fixed
dimensions 1 having a degree in the projective space 1. Thus satisfying the Chow variety as,
,,
When this chow variety is refined in algebraic geometry then this provides the spatial parameterization of particular
schemes being the subset of that projective space, thus the Chow variety is refined as the Hilbert scheme. If the locus of
any such stable curves over this Hilbert scheme could be over the irreducible stack parameter then the scheme
suffices the relation,
551
()
Where, as regards to the homogeneous components, the growth dimensions can suffice over the Hilbert Polynomial with
the form,
=611
Such that, the embedded isomorphisms over the Deligne – Mumford stack is (,). Considering the genus 0 the
cohomology group can be given, 1 over its isomorphisms over algebraic group (2, ) with Genus – 0 and
connecting points – 3, 0,3 can be defined with ample
3 as,
1,0,
0,
2
Genus – 0 satisfies the norms of the sheaf dualization where no globally defined sections are present if the
2 takes
the value of -4 with the degree of the canonical bundle -2 as such the above equation satisfies,
1,0,
0 0,
2= 0
The nodal curves of genus satisfying automorphisms over marked points where they are smooth, denoted by
,
suffices the degrees as,
33 +
4
II De Rahm Cohomology
In algebraic topology the smooth manifolds are expressed as a category of the cohomological class by De Rahm
cohomology. The set of exact forms are closed with the quantitative invariance setting up this particular cohomology.
Cochain complex satisfies the image of homomorphisms from one image of the kernel to the next where the sequences are
the abelian groups. Being used as an invariant of the topological space, they maintain the singular homology of a
parameter over a manifold as the Initial function is 0Mx which when transformed over the exterior derivative
yields the 1 –form to 2 – form to 3 – form and so on… with the form closed 2= 0 expressed as,
20Mx
1Mx
2Mx
3Mx
kMx
Where the De Rahm cohomology group of composed of disconnected components over smooth
and closed manifold as,
The isomorphism maps for the singular cohomology groups , makes the connected map for the
with the symplectic norm as,
, ,
Satisfying the isomorphism map
, where a ring structure is established
in the cohomology class with the relation between De Rahm and singular cohomology.
For the simplicial complexes, the degrees of connectivity over topological manifolds are expressed by the Betti
Numbers, wherefore the infinite group of the torsion coefficients could be generated when the
order -2 cyclic groups relate the cohomology group over connecting points as,
+
2 =
For the compactly oriented Riemann manifold denoted by , if we choose the harmonic form = 0 where is the
harmonic then the have an isomorphic map,
(),
Given the equivalence class, the satisfies other 2 apart from the harmonic gamma every member
of equivalence class stratifies the relation as,
= + +
1
Thus, for = 0 with = 0 where =, the 0 makes the Hodge decomposition sates of the De
Rahm cosmology groups
kMx
over the integral,
,=
With norms satisfied the inner product on kMx when = 2 which is the orthogonally to , that is and on .
5
III Novikov Ring
Morse Theory provides the differentiable functions computed over a topological space, a manifold to study its properties
in rigorous details, Picard – Lefschetz theory takes that holomorphic domain over the critical points for an extension of
the Morse Theory in a complex space. Through the monodromy action of a fundamental holonomy group with the
generator with vanishing cycle for = 2 is given by,
=+1+1+2/2,
In the absence of the Picard – Lefschetz, Morse takes the function where the mapping suffices an extension of all points
to the elevation with its inverse mapped back to passing over the saddle points being a contour line itself with the
mapping as,
1(,]
Here, in quantum cohomology, when is replaced by 1 then Nov is the Novikov Ring with then the
relation satisfied as,
lim
Being the Nov satisfies the homomorphic map : Nov, where , a path connected
Hurewich homomorphism over with suffices,
, ()()
Thus, the Morse Homology over satisfying the chain complex makes the first cohomology class over as 1,
comes from the equivalence over through,
,
Here, the group ring as stated by marks a map of group – G and ring – R can be mapped by having the notions of
an existent scalar, in a ring – R with the respective mappings,
+ ()
Thus the complete final mapping being satisfied with the above two mappings with coefficients – R and linear element
composition – G is,
()
=
= (1)
In the same first cohomology class over as 1,, when assigned over 1, then, taking the integral rank
(as in of the chain complex , transforming to in a non-trivial way, the same rank when taken through then,
,
The equality of the Morse Theory satisfies the Novikov – Betti number over with the assigning parameter of
1, as .
6
IV J – Holomorphic Curve
The special unitary group () comes under the gauge groups sufficing the Yang–Mills Theory associated with Lie
algebras. The non – abelian Lie groups which portray the relation between electromagnetic, strong nuclear, weak nuclear
forces are reasoned by this Yang–Mills notion to describe the behaviors of the elementary particle which further got into
smaller sets as,
(1) (2) 3(5) (10)(8)
The simple non – abelian symmetry is given by the Lagrangian where the field strength is denoted by and Trace – Tr
having the form,
=1
2Tr2=1
4
Using this Yang–Mill functional, the 3 – dimensional closed manifolds that comes under homology groups could be
associated with the Floer homology which is also used to study low – dimensional manifolds in symplectic topology and
algebraic geometry with its relation to Morse Theory for the infinite dimensional manifolds. A real valued function is
preserved on an infinite dimensional manifold when its associated with Floer homology where the quantum field theory of
the topological invariants is associated with the Schwarz – type, then the action when computed over the manifold as
a form of integration of the 2 – form and scalar (where Aux is the auxiliary) then it satisfies the basic notions of
Chern – Simon theory. The action is computed as,
=
The J – holomorphic curve or Pseudoholomorphic curve when satisfies the Cauchy – Riemann equation as a smooth map
satisfies,
The 3 – Tuple relation of the holomorphic curve ,, where the inhomogeneous term satisfied the perturbed Cauchy
– Riemann equation as,
,=
The original Cauchy – Riemann equation satisfies with the J – holomorphic curve mapped from the Riemann surface
gives suffice the result,
,1
2+= 0
This is immensely helpful in studying the Gromov – Witten Invariants[will be explained in the last section] with closed domain C the
element of Deligne – Mumford moduli space of curves[explained in section I], C with genus – g (0 taken in this paper) and
marked points – 3 proved useful in quantum theories regarding the path integral formulations and Supersymmetric
string theories with the implementation of the over J – holomorphic curves that makes the reducible path
integrals of finite dimensional stable maps arises in the Type – IIA Strings in M – Theory that incorporates the Type – IIB
strings which formed F – Theory. Another important element is the 11 – dimensional supersymmetric gravity ( SUGRA).
All theories comprising M – Theory with Topological (T) and Strong – Weak (S) duality are – Type I, Type IIA, Type
IIB, Heterotic SO (32), Heterotic E8×E8, 11-D SUGRA.
7
V Poincar Duality
For the cohomology groups of manifolds that are compact without boundary, the isomorphisms over – dimensions
with – cohomology groups, the canonically defined results over a fundamental class the affine element being in
the homology group over oriented makes the mapping as,
⁔
Thus, dimension being and cohomology group being , when this isomorphic mapping over manifold M is computed to
() homology group then the fundamental results of this duality can be expressed as,
,
,
Poincar Duality indeed concreted by the proof of the existence of isomorphisms over chain complexes where the image
of a homomorphism is mapped to the kernel of the next, thus giving a chain structure. Thus the homomorphism induced
connected modules 0,1,2,,8 as defined over in the form , the mapping satisfied through,
1 0
01
12
23
34
0
Therefore, the cellular homology and the cohomology could in principle be defined for the cell of
the CW – decomposition of the through two relations,
1
For the distinguished element of the fundamental class the Poincar complex acts over the homology groups satisfying the
duality between homology and cohomology classes over the chain complex for the group over – dimensions as
given by,
⁔ 0
Thus, a close relation between the Poincar duality and Thom isomorphism could be given by the n – ranked vector
bundle over a restricted for any fibre then the isomorphism is defined over the vector map 0 to 1 with the
associated ring suffices,
0;+0,0 \ 1;
Apart from this, several dualities exist in mathematics and mathematical physics that connects different subgroups within
groups where in case of a few specific dualities, one is dual to another and among them in the M – Theory, the T
(Topological) duality depicts the winding numbers and S (Strong – Weak) depicts the charges being one dual to the other.
The other forms are Hodge duality, Alexander duality, Lefschetz duality. Coming to the T and S duality in M – theory the
connections are as follows,
Type I and Heterotic SO (32) : S - duality
Type II-A and Type II-B : T - duality 8,1 ×1
Type II-B : T - duality and S - duality upon itself
Heterotic SO(32) and Heterotic E8×E8 : T – duality 8,1 ×1
8
VI Topological String Theory
The vibrations of the fundamental strings through the 3 – genus of the complex Khler that is Calabi – Yau manifold with
a vanishing Ricci curvature suffices the origin of the particles of nature. Earlier versions of this theory is Bosonic,
however the Polchinsky action incorporated the SUSY over the Bosons with fermions thus enabling a theory of
supersymmetric strings which was in 5 – categories, that got glued by Edward Witten via M – They by the addition of 11
– dimensional SUGRA. Topological and Strong – Weak duality interconnects the 5 distinct theories in a unified schemes.
Predominately the Type II-B constituting both open and closed strings being also Topological and Strong – Weak dual to
its own made up the F – Theory as proposed by Cumrum Vafa.
In supersymmetric gauge theories, the prepotentials and superpotentials over = 2 and = 1 constitutes the
topological A – Model and B – Model respectively in four and five dimensions. Their combinations relates the
dimensional reducibility in topological M – Theory, however, the A – Model and B – Model are related by mirror
symmetries over mirror manifolds that assert the S – Duality in between them through a dimensional extension of NS5 –
Brane. Holomorphic duality being a important aspects of B – Models comprises the low – energy effective actions in the
quantum background.
The 2 – cycles over a Brane warping defines a deformed conifold that when gets holomorphic to Chern – simons theory
makes arrangements over the Kodaira – Spencer gravity via B – Models. The nonvanishing nature of this model always
takes up complex parts when regarding the resolving of the conifold in B – fields. The quite different approach taken by
the A – Models are,
() Chern –Simons theory makes up the open strings.
Khler gravity describes the closed strings.
The A and B – Models of the topological string theories contain holomorphic quantities being in a supersymmetric origin
of particles are related with Gromov – Witten[described in next sections] invariants, mirror symmetry and Chern – Simons theory.
Ordinary worldsheet string backgrounds are not topological but when Witten applied the topological twist by mixing
rotations over two (1) symmetries as, Lorentz and R = (2,2) (1) × (1) . This instead
makes up an exact BRST quantization where the theory is no more dynamic with all the configurations localized over the
precise configurations as topological strings.
Calabi – Yau consists of the 6 – dimensional complex or generalized Khler manifolds. These are the defining source of
the scattering amplitudes over the = (2,2) SUSY sigma models having holomorphic curves over 2 – real dimensions
making the correlation functional attachments to the cohomology rings as best described by the Gromov – Witten
Invariants. For this A – Model, the N – D2 Branes when stacked up then the Lagrangian submanifolds of the worldsheet
supersymmetry establishes over U(N) Chern – Simon Theory. The String – Brane interaction can be best described by,
3 = 0
B – Model however takes over warp of submanifolds that coexists with different Brane configurations where low –
dimensional Branes are reduced by the reducing dimensions suffices,
(1)
Holomorphic 0 – Submanifolds
(1)
Holomorphic 2 – Submanifolds
(3)
Holomorphic 4 – Submanifolds
(5)
Holomorphic 6 – Submanifolds
Regarding the dependability of the superpotentials of the Models, the supercoordinates of A – Models are integrals over
1 or + while the B – Model takes the conjugate forms of ± where A – Models are not dependent on superpotentials
but on twisted superpotentials in holomorphic ways while the reverse is true for B – Model.
9
VII Gromov – Witten Invariants
For a nondegenerate 2 – form, closed differentiable manifolds, equipped with the algebraic geometry and differential
topology, the symplectic geometry makes sense over this particular form as analogous to Riemannian geometry which
relates the angles, lengths of the concerned metric space, while symplectic connects the relations or measures the oriented
areas. Thus, over the symplectic norms when it's integrated over the region then the required area produced given by,
=
The non-trivial de Rham cohomology group[described in section 2] (2nd) relates the symplectic manifold M as the realtion 2.
The Riemann geometry when considered spheres between two marked points, the symplectic geometry considers two
curves or geodesics as analogous to that spheres called the Pseudoholomorphic / J – Holomorphic curves[described in section IV]
with the surface area of a minimal magnitude.
Khler manifold is an important tool, more precisely the generalized one, or the complex one with vanishing Ricci
curvatures as the Calabi – Yau Manifold, which the symplectic geometry doesn’t takes into account, but Mikhail Gromov
pointed out that there are an abundance of complex structures permitted within this geometry where the holomorphic
transition maps are the requirements to satisfy. Those Pseudoholomorphic curves, that are subjected to a class of
invariants called the Gromov – Witten (GW) Invariants which war further deduced by Floer where the homology of a
symplectomorphism makes a nondegenerate form as in Floer Homology.
When the cup roduct of the cohomology theory that too being quantum when preserved over a space, then the
enumerative counting results in the formation of a Pseudoholomorphic curve over a symplectic topology. The strong
theory that have been trying to make the impossible, almost difficult in making a scale invariant approach between
General Relativity and Quantum Theory where out of 10 dimensions the Calabi – Yau contains 6 – compactified
dimensions over the regime of the symplected manifold which makes the finite dimensional moduly spaces associated
with J – Holomorphic cuves incorporates the genus with the invariants as GW. The Deligne – Mumford[discussed in section I]
moduli spaces with the curve as over genus and marked points through the computation of a 4-tuple relation
,,,where , are non-negative integers associated with 2 closed with 2D homology class in X, the relations that
can be given by the GW invariants as,
,,
The rational numbers of GW invariants could be calculated from cycles where marked points are mapped. If the
Deligne – Mumford curve have a subset of domain called , then its homology class could be
,,
, with homology class
=1 , then the rational number suffice over the connected
homology of two topology spaces making the Gromov Connections making each bubble attaches via the GW invariants.
If we think such 2 – manifolds as one then, naming the manifolds as on with their cross product acan be defined – Let the
one manifold be 1 and the other be 2, the relations are,
1 × 2
,
,,
=1 =,
,×
Thus if 1 in denoted as x and 2 as Y×Z, or if 1 in denoted as Y×Z and 2 as x, the associative property of
GW Invariant is expressed as,
x×Y×Z=x×Y×Z
10
Acknowledgements: Author is grateful to Mr. Ashis Kumar Behara and Dr. Ushashi Bhattachya for the valuable
suggestions and insights in course of the preparation of this paper. Without their valuable assistance and appreciations the
paper could not be produced by sole me.
Conflicts of Interest: I, the sole author of this paper have no competing or other conflicting interests as concerned over
this.
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