from qft to 2-d supersymetric tqt:mirror symmetry between …€¦ · from qft to 2-d supersymetric...

From QFT to 2-d Supersymetric TQT:Mirror symmetry betweenMath and Physic

PHILIPPE DURANDConservatoire National des Arts et Metiers

Mathematics and Statistics Department Laboratory M2N292 Rue Saint-Martin 75003 Paris

[email protected]

Abstract: Theoretical physics is taking an increasing part in the universe of mathematics. After calculus, vectorand tensorial analysis, topological theories make their entry into quantum field theories. More precisely, in thisdomain, topological theories are the most relevant. A fundamental theorem of the Atiyah has important repercus-sions in several branches of quantum physics in the geometric approach. We can cite the work of Alain Connes onnon-commutative geometry, but also all the developments due to Donaldson, E. Witten around Gauge theories, su-perstring and Mirror Symmetry. We present here an historical survey of some topological field theories, especiallyMirror Symmetry to understand the interpenetration between quantum physics and topology.

Key–Words: Superstring theory, Supersymmetry, Mirror Symmetry, K-theory

Received: February 19, 2020. Revised: March 17, 2020. Accepted: March 19, 2020. Published: March 20, 2020.

1 IntroductionThe introduction of supersymmetric theories into two-dimensional field theories has helped to strengthen thelinks between geometry and physics. The formalismof string theory takes root around two-dimensionalconformal fields theories, the special structure of thesource manifold(Riemann surface) and supersymme-try which extends Noether symmetry by the additionof superymetric variables made it possible to extendthe BRST formalism to make these theories topo-logical. The computation of correlation functions inwhich points are replaced by classes of cohomologies,lead to define spaces of instantons (moduli spaces)whose dimension calculated by the index theorem canbe made equal to zero by adding certain constraintsand lead to staging enumerative geometry. At thesame moment, Gromov and Witten [1] realize theimportance of the concept of ”curves” in symplecticgeometry, A little later, a very elaborate form of T-duality, the mirror symmetry, played a decisive role tocalculate some invariants of (Gromov Witten invari-ants) for the quintics treefold .The presentation willgive some of the stages of this intellectual develop-ment and the relevance of the coupling between geom-etry and physics. From the tools that will be exposedin this paper , our goal is to identify the baground ge-ometry and physics to understand the T-duality andmirror symmetry of models A and B. The past re-cents years have seen the gradual fusion of geometryand physics. This program initiated at the start of the

twentieth century by Einstein has not stopped grow-ing since. There are several key period. The first be-tween 1915 and 1930 sees the birth of Einstein’s gen-eral relativity and that of the works of Oscar Kaluzaand Felix Klein, the first attempt is to unify the fun-damental interactions known at this moment: Rela-tivity and electromagnetism. In addition returns toKlein the idea taken up by string theorists is then ofrolled dimensions. Then quantum mechanics comeson the scene making us forgetting geometry a bit infavor of functional analysis and probabilities. Geom-etry comes back on the scenein the fifties with thenotions of bundles and connections in differential ge-ometry. The Yang-Mills theory, an example of a non-Abelian gauge theory modeled on the bundle SU(2),comes into play. It prefigures the time of gauges theo-ries in four dimension. Finally a third period beginsat the start of the adaptation of supersymmetric mod-els to string theory in the nineties . A first model ofsuperstrings is born, Theory of superstrings of typeI . it continues until now. Geometry is present ev-erywhere; A string can be seen as a parameterized”complex” curve: a map of a Riemann surface in aLorentzian manifold (bosonic strings). A superstringevolves in a richer space, a special complex manifold:called Calabi-Yau. Thus we can modelise the classi-cal evolution of a string as the evolution of a point par-ticle in space time. One of the big differences lies inthe fact that an interval of time the source space of theclassical theory must be replaced by a richer topolog-

WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.4 Philippe Durand

E-ISSN: 2224-2880 39 Volume 19, 2020

ically space namely a riemann surface. In the sameway the target space is a particular Kahlerian manifold.That explains why the developments in algebraic ge-ometry mainly due to Alexander Grothendieck at thebeginning of the sixties, those of Atiyah about in-dex theorem and developments K-topological the-ory, those of Kodaira and Spencer on the theory of de-formations of complex manifold, those of Misha Gro-mov in the mid-eighties on the theory of holomorphiccurves will play a major role in understanding thisphysical theory.We will discuss a little bit about geometric model-ing of string theory and especially mirror symme-try. String theory developed considerably thanks tothe progress of complex geometry and algebraic ge-ometry in the fifties during the Cartan seminar andthen in the sixties thanks to Alexander Grothendieck.The beginning of the sixties sees the result of thedemonstration of the Index theorem by Atiyah [2]which reconciles analysis and topology. This willlater calculate the dimension of spaces, of modulusof curves. This theorem is in fact a smoother andmore flexible version of the Riemann-Roch theorem[4] obtained in algebraic geometry. With the proofof the index theorem, we see the development of theK-topological theory extending algebraic topology tofiber bundles and which will be of the greatest im-portance in open string and branes theory to dealwith the difficult question of topological defaults andhope to have an intrinsic and non-disturbing visionof the theory of everything. It should also be notedthat these analytical-topological developments will betaken up in the symplectic version of string theory.Faced with the rigidity of the notion of holomorphicfunctions, Misha Gromov will enrich the landscapeof complex and symplectic geometry by putting for-ward the notion of curves before that of functions.The genial idea comes back to being interested in thesections: holomorphic curves rather than in holomor-phic functions, thus the landscape of geometry willbe revisited through parametric curves (the parame-ters of evolution are found those of the classical stringevolving in a space-time; in parallel algebraic geom-etry is developed and the theory of deformation bornfrom Kodaira-Spencer’s [5] work in complex geome-try, derived functors enrich symplectic developments.Thus through derived functors algebraic geometerswill provide complementary tools study of stable bun-dles on algebraic curves and enrich the tools from theK -theory for the study of branes.

2 Quantum field theory2.1 Field theory and symmetries

The concept of field is fundamental in physics. Afield ϕ is a function of a source space (world sheetfor string theory ) into a target space, M with a suf-ficient number of dimensions. So this given a ”pack-age” (Σ,M, ϕ) and a classical action: S where: Σ isthe source space, often a manifold: for The classicalmechanic of the point is the time axis (world line). Forthe conformal field theories like bosonic string theo-ries: Riemann surface ...Lagrangian density is a function on one or more fieldsand its first derivatives:

L = L(ϕ1, ϕ2, ..., ∂µϕ1, ∂µϕ2...) (1)

Classical action is the integral of the classical La-grangian density on space S =

∫Ldn+1x

Principle of least action: The minimization of the ac-tion: (δS = 0, leads to each field noted just ϕ to theEuler-Lagrange equation which gives the equations ofmotion of the particle

∂L∂ϕ− ∂µ(

∂L∂(∂µϕ)

) = 0 (2)

2.1.1 Noether Symmetries

Symmetry of the action: the role of symmetry inphysics is essential. We want, for example, such ac-tion invariant through a transformation like transla-tion, rotation ...: if ϕ→ ϕ+ δϕ alors S → S + δSNoether’s theorem say: through any symmetry, the ac-tion is the same: δS = 0 for example, in the case ofthe free particle, the symmetry by translation is equiv-alent to the conservation of momentum p = mx

2.2 From Field Theory to Quantum FieldTheory

A quantum field theory can be seen in through a clas-sical action quantified from the path integral. We canthen define the correlation functions dependent on se-lected observable. These functions depend, for exam-ple, the metric that is chosen on the target space.

2.2.1 Quantum fields, QFT

Uncertainty on the position or momentum in quantummechanic led to replace the classical solution (leastaction) by the partition function or the set of all possi-ble solutions: It is the path integral

Z =

∫Σ→M

e−S(ϕ)Dϕ (3)


E-ISSN: 2224-2880 40 Volume 19, 2020

Similarly, one can calculate correlation functions, orfunctions with n marked points.

< ϕ1(x1), ..., ϕn(xn) >=∫Σ→M

ϕ1(x1)...ϕn(xn)e−S(ϕ)Dϕ(4)

The simplest example which is the leitmotif ofthe theory of renormalization is the two-dimensionalmodel on lattice , is the Ising model: Here Σ is a lat-tice:

∧and M = +,−

| + | + | + | + |− − − −

| µ(t1) | − | − | + |− − − −

| + | + | − | + |− − − −

| + | + | + | µ(t2) |− − − −

| µ(t3) | + | + | + |

Figure 1: Ising model

The figure represents the Lattice and correlationsbetween some marked points.

Here the action is: S(µ) = − β2π

∑|t−s|=1 µ(t)µ(s)

Correlators are defined as mean values in somelattice points :

< µ(x1), µ(x2), ..., µ(xn) >=

ΣMap(Σ,M)µ(x1)...µ(xn)e−S(µ)DµΣMap(Σ,M)e

−S(µ)Dµ(5)

3 Quantum field theory for stringsThe quantum field theory adapted to string theory isthe conformal field theory [6]. this theory is very com-plex for the mathematician and it is not our purposeto describe it completely. Let us remember, however,that the evolution of a string can be described by a Rie-mann surface or complex curve. the adapted model isthe sigma model, and particularly, the supersymmet-ric sigma model. In this model, the source space is aRiemann surface and for the moment, the target spacesupposed to be a complex variety.

3.1 Moduli space of Riemann Surfaces

In the extended field theories used for string theory,in quantum field theory, the word line that leads tothe definition of Feymann diagrams is replaced by theworld sheet: a Riemann surface. Riemann’s theory ofsurfaces can nowadays be considered successful. Wecan see them from the angle of symplectic geometrywhere an important parameter is the area, or from theangle of complex geometry , the important parameteris then the complex structure J adopted. We will seefrom the angle of mirror symmetry, these parametersare in duality. A whole mathematical engineering canthen be carried out, leading to the definition of spacesof moduli of curves stabilized by the addition ofmarked points. These concepts have been well knownsince the work of Riemann and Roch. Conformalfield theory uses these results to define the conceptof sigma model, and the concepts of stable curvesand stable maps generalize the Feynman diagramsand n-point correlation functions of quantum fieldtheory.

3.2 The Mathematics of Gromov andGrothendieck

The global study of schemes or varieties is very dif-ficult. Grothendieck already started studying moduliof curves, more malleable, as that means study objectthrough its sections which is much easier. Gromovnoted that. There are very few holomorphic functionson a complex manifold or almost-complex like sym-plectic manifolds.The big idea of Gromov [7] is to consider the studyof a symplectic manifold through its sections that are:holomorphic curves: In mathematics the worldsheet isa curve (Riemann surface), the evolution of a bosonicstring that is a field of Σ → X is a parameterizedcurve. We can present the Gromov-Witten invari-ants by one of these two views Symplectic Geometry(Ruan, Tuan) [1], Algebraic Geometry (Pandaripande,Katz) [8].

3.3 Symplectic geometry and pseudoholo-morphic curves

Recall that a symplectic manifold is a differentialmanifold equipped with a 2-closed form ω : (M,ω).The simplest example is R2n and ω =

∑dxi ∧ dyi,

We can also consider the space Cn or create new com-plex structures J which are all integrable. Thus R2n

can be endowed with a structure of simplectic man-ifold which is not always the case. A parameterized


E-ISSN: 2224-2880 41 Volume 19, 2020

curve is a map: ϕ : (Σ, j)→ (M,J), where j is an al-most complex structure on Σ, J be an almost complexstructure on the target. A parametrized curve is holo-morphe if its differential verifies equations of Cauchy-Riemann: J dϕ = dϕ j, (the differential is C lin-ear).

3.4 Riemann-Roch formula moduli space ofcurves

For now we place ourselves in the more generalcontext of algebraic geometry. We recall here theRiemann-Roch formula for a curve, is the case ofRiemann surfaces or Riemann surfaces with markedpoints which interests us here: Mg, Mg,n respec-tively. That need to define The moduli space of curvesof genus g, and the moduli space of curves of genus gwith n marked points:Riemann Roch formula for a curveThe Riemann-Roch formula gives:

dimCH0(TΣ)−dimCH

1(TΣ) =

∫Σch(TΣ)td(TΣ)

(6)Specifically, dimCH

0(TΣ), consider the infinitesimalautomorphisms, dimCH

1(TΣ) is the complex dimen-sion of moduli space of curves. For a Riemann surfaceof genus g yields: Dimension ofMg:

dimCH0(TΣ)− dimCH

1(TΣ) = 3− 3g (7)

dimCMg = 3g − 3 + dimCH0(TΣ) (8)

Riemann-Roch formula for a map, curves, stablemaps Make first application of the formula above:H0(TΣ) count the number of marked points need tostabilize the curve thus:

g ≥ 2 : dim(H0(TΣ)) = 0): dim(H1(TΣ)) =dim(Mg) = 3g − 3if g = 1 : H0(TΣ) = C: dim(H1(TΣ)) =dim(Mg) = 1if g = 0 : dim(H0(TΣ)) = 3: H1(TΣ) = Mg is apoint.We will therefore focus our attention on stable curves(and maps) . Let φ : Σ→ X be a holomorphic curveThe Riemann-Roch formula gives in this case:

dimCH0(φ∗TX)− dimCH

1(φ∗TX)

=

∫Σch(φ∗TX)td(Σ) = n(1− g) +

∫Σφ∗c1(TX)

(9)

3.5 Curves, stable maps, moduli spaces

In genus 0, an automorphism of P 1(C) is determinedby the images of three distinct points, that is to say au-tomorphisms which fix two points form a no discretesubgroup PGL(2,C),we will therefore infinitesimal automorphisms. Toget rid of infinitesimal automorphisms: stabilize thecurve, we must add ”marked points” doing the curveis stabilized.A stable curve is a curve on which we added enoughmarked points to kill the infinitesimal automorphisms.A stable map is a parametric curve (Σ, (x1, ..., xk), ϕ)for which Σ is stable. We can now give the defini-tion of the moduli space of stable maps with k markedpoints:Mg,n(X,β).We define the space:Mg,n(X,β) = (Σ, (x1, ..., xn), ϕ), ϕ : Σ →X,ϕ∗(Σ) = β/ ∼ where ∼ are quotient by thegroup of reparameterization.The space considered above is not compact in gen-eral, we denoteMg,n(X,β) its compactification, thisamounts to add singular curves .

3.6 (Virtual) dimension space of holomor-phic curves

From the two previous formula (Riemann-Roch forcurve and parametric curve), we can deduce the”virtual” dimension of moduli space of holomorphiccurves. For this we can use an exact sequences.Consider the long exact sequence in cohomologyassociated to the exact sequence:0→ TΣ → f∗TX → NΣ/X → 0For details see Pandharipande works [9]The Dimension ofMg(X) is given by combining thetwo forms of the Riemann-Roch previous:dimvirtMg,n(X,β) = (dimX)(1 − g) +∫f∗(Σ) c1(TX) + 3g − 3 + n

We could find directly this result in symplectic case(varieties that are treated are thus symplectic Kahler)relying on the index of a Fredholm operator for anelliptic complex adhoc.

The spaces of modules of curves are compactedto take account of singularities, and stabilized by quo-tient by a group of automorphism. They correspondto orbifolds for physicists . In the language of alge-braic geometry, the adapted structure is that of alge-braic stack. We can determine a (virtual) dimension tothese spaces given by Riemann Roch’s theorem whichis a rigid version of the Atiyah Singer index theo-rem. One application in mathematics is the countingof curves passing through a certain number of markedpoints. it is enumerative geometry. In MathematicalPhysics, the correlation functions resulting from this


E-ISSN: 2224-2880 42 Volume 19, 2020

theory leads to the invariants of Gromov Witten (The-ory of supersymmetric strings, A- Model of Witten),or the invariants of Donaldson [10] For the study offour manifolds.

4 Gromov Witten invariant, A-Sideof mirror symmetry

Witten [11], to show that the introduction of super-symmetry, made it possible to localize incalculablepath integrals around the critical configurations.

that are instantons. In the case of the superstringtheory, the A-model shows that these instantonsare precisely the holomorphic curves satisfying theCauchy Riemann conditions. Suddenly the path inte-grals become computable in particular, the correlationfunctions can be seen as the invariants of GromovWitten. In the appropriate cases, those where theinstabilities have been killed, these invariants becomenumbers. is called Gromov-Witten invariant quantity:

< [ω1], ...[ωn] >β=

∫[Mg,n(X,β)]

ev∗([ω1])∧...∧ev∗([ωn])

(10)In this script we used an evaluation map:

evi :Mg,n(X,β)→ X : (Σ, x1, ..., xn, ϕ) 7→ ϕ(xi)(11)

and the dual map:

ev∗i : H∗(X)→ H∗(Mg,n(X,β)) : [ωi] 7→ ev∗i ([ωi])(12)

Kontsevich [12], early nineteen nineties gave anelementary model by considering the holomorphiccurves of P 1(C) in P 2(C), he thus obtained an iter-ative formula allowing to know by how many pointpasses a curve of any degree: Inspired by Feymanndiagram he showed that There Nd the number ofcurves of degree d through 3d− 1 points, given by:

Nd =∑

d=d1+d2

Nd1Nd2(d21d

22C

3d−43d1−2 − d

31d2C

3d−43d1−1)

(13)By applying this formula, we find: N2 = 1, N3 = 12,N4 = 620

4.1 Back physics!

We return now to our case for the supersymmetric Amodel of Witten, take X Calabi-Yau 3 included inP4(C)It is known that Calabi Yau Ricci-flat is that means:

∫Σ ϕ∗(c1(X)) = 0 In These conditions, the dimen-

sion ofMg,n(X,β) is given by:

dim(Mg,n(X,β)) = −3 + 3g + dimX(1− g) + n(14)

If g = 0 and dimX = 3 we get: dim(M0,n(X,β)) =nIn M0,n(X,β)) as deg([H]) = codim[H] = 1 ittakes n hyperplanes in< [H], ..., [H] >β to make thisnon-trivial correlation function.For n = 3, we have the three points correlation func-tion or ”pant”: n = 3, < [H], [H], [H] >β , basic”lego” of topological fields theory .

4.2 Example 2: Calculating the correlationfunction at three points: the problems,the mirror symmetry

The calculation of correlation functions by directmethods namely ’A’side of the mirror is not easy: ifyou expand you have:< [H], [H], [H] >=

∑β < [H], [H], [H] >β

e−∫β ω.

A conjecture of Clemens [13] said we cannot calcu-late this number due to a problem of multiple cover-ings, which contradicts the expected dimension of 0for space applications module with 0 runs scored.To survive we must work on the other side of the mir-ror. mirror symmetry The objects we work with inenumerative geometry are Kahler forms and param-eterize deformations of this form. Mirror symmetrysays that one can express the same physical settingdeformations of complex structure on a variety mirrorand identifying the correlation functions from the twomodels.

5 Mirror symetry for closed stringsThe mirror symmetry postulate that there are twoviewpoints to represent the physics of fermionicstrings. The relevance comes from the fact that someproblems not solvable from a certain point of view areon the other. Mirror symmetry is a very accomplishedform of T-duality: appeared in bosonic theory whichstates that the partition function remains unchanged

in the change R ↔ 1

R, where R denotes the radius

of compactification of extra dimension. In a bosonicfield theory, where the source space is the cylinderand target a torus, one can notice that the T-dualityexchange symplectic structure deformation (area) anddeformation of complex structure. In this ”survey”,we will introduce tools from physics and mathemat-ics for understanding some aspects of mirror symme-


E-ISSN: 2224-2880 43 Volume 19, 2020

try. Especially we construct the mirror map locallyas historically Morrison [14] has presented this theoryfor closed strings. In addition, we briefly explain theimplications in enumerative geometry.

5.1 Theory ϕ4 and complex geometry

The model of Ginzburg Landau plays a decisive roleon the B-side of the mirror. We will look at the admis-sible deformations ( marginal distortions ) preservingcertain symmetries. Starting from the Lagrangian inϕ4:LLG = ∂µφ∂µφ − V (T, φ) where V (T, φ) =14!λ(t)φ4 + 1

2!µ2(t)φ2 At the critical temperature Tc

”mass”, µ2(Tc) = 0, so the correlation length (in-verse mass) is infinite. At this temperature the field

φ0, solution of∂

∂φV (T, φ) is zero three times degen-

erated. A small perturbation V (Tc, φ) → V (Tc, φ) +δµ2(T )φ2, solves the singularity and the is ”symme-try breaking”. The challenge is to find ways to per-turb marginally a potential in order to preserve thesymmetry and defined by the fact a critical family ofsuperpotentials.

5.2 Marginal deformations, Calabi-Yau hy-persurfaces

The superpotential is an holomorphic functionW : CM → C is chosen as a potentialV (x) =

∑N1 |∂iW(X)|2 =

∑N1 ∂iW(X)∂iW(X)∗

if we consider only one field, one can consider thefunctionW(X) = 1

(n+1)!Xn+1

The bosonic part of the supersymmetric Lagrangianis written then:LLGN=2 = −∂+X

∗∂−X + ∂−X∗∂+X + V (X)

There is: V (X) = 0 ⇔ ∂iW(X0) = 0, so it is

relevant to define the Chiral ring RW =C[X]

∂W(X)where the ratio is proportional to the polynomialsof ∂iW(X) : P (X) = P i(X)∂iW(X) defor-mations respecting the Chiral ring are given by:Wdef (X) =W(X) +

∑P∈RW tPP (X)

If we choose W(X,Y, Z) =1

3(X3 + Y 3 + Z3),

the deformed potential is given byWdef (X,Y, Z) =W(X,Y, Z) + t0 + t1X + t2Y + t3Z + t4XY +t5Y Z + t6ZX + t7XY ZOnly the non vanishing term µ = t7 preserves thecritical situation, and does not break the Z3 symmetry(X,Y, Z)→ (exp(2kiπ

3 )X, exp(2kiπ3 )Y, exp(2kiπ

3 )Z)we have defined a family of continuous deformationsallowed:Wdef (X,Y, Z, µ) =

1

3(X3 + Y 3 + Z3) + µXY Z of

the initial potential. The hypersurface of a complexprojective space obtained by canceling this is the Thesimplest example of Calabi-Yau. Is an elliptic curveor complex torus .Calabi-Yau variety is a KahlerienneRicci flat which is to say: the canonical bundle istrivial. There are , among other, K3 surfaces involvedin branes theory and the quintic threefold for closedstrings. The easiest way to realize the Calabi-Yau isto consider an hypersurface of a complex projectivespace. There is a strong constraint between the degreeof a hypersurface and the dimension of the ambientspace. Subsection Example of a Calabi-Yau Let uswrite the exact sequence associated to a hypersurfaceof degree d:0→ OPn+1(−d)→ OPn+1 → OX → 0With exact long sequence in cohomology, one cancalculate the cohomology groups associated.

The Result: Hn(X,OX)=CCd−1n−1 = C (condition

Calabi-Yau) Necessarily d = n +2

• n = 1 (d = 3): Elliptic Curves

• n = 2 (d = 4): K3 Surfaces

• n = 3 (d = 5): Quintic threefold

5.3 Deformations, mirror symmetryWe can focus on two types of deformations of Calabi-Yau: the deformations of symplectic structure andthose from its complex structure. The tangent spaceof complex structure is H1(X,TX) ' H2,1(X),can consider the deformation of complex structure(J deformation). the Kahler structure is governed byH1,1(X) = H1(X,Ω1

X) There are two field theories( textbf CFT) supersymmetric duality in satisfactory:h2,1(X) = h1,1(MX) and h1,1(X) = h2,1(MX):deform the complex structure of X is equivalent todeform the volume of his mirror.diamond Hodge plots the Hodge numbers of acomplex manifoldHodge Diamond for Calabi-Yau Quintic Threefold

h0,0 = 1

h1,0 = 0 h0,1 = 0

h2,0 = 0 h1,1 = 1 h0,2 = 0

h3,0 = 1 h2,1 = 101 h1,2 = 101 h0,3 = 1

h2,0 = 0 h1,1 = 1 h0,2 = 0

h1,0 = 0 h0,1 = 0

h0,0 = 1

5.4 ”B side” Origin of Physics

On the A Side, supersymmetric constraints lead towhat the action does depends only on the the Kahlerform; instantons are holomorphic curves. The calcu-lation of correlation functions is difficult because it


E-ISSN: 2224-2880 44 Volume 19, 2020

takes into account correction on the degree curves (in-variants of Gromov-Witten) On the side ”B”, BRSTFormalism explained in the other side of the mirrorapplies here: instantons are constants maps from theworld-sheet Σ on a point of target spaceX . The corre-lation functions are simpler to calculate: they requireno instanton correction.If X is a Calabi-Yau 3, the 3 points correlation func-tion is:〈WAWBWC〉 =

∫X ΩjklAj ∧Bk ∧ Cl ∧ Ω

A,B,C belong to H1(X,TX) and depend on thecomplex structure, Ω is (3.0) top-form holomorphic.The two numbers h1,1(X)=1 and h2,1(X) = 101,count the number of deformation structures respec-tively Kahler and complex. The principle of mirrorsymmetry, gives h1,1(MX)=101 andh2,1(MX) = 1. He said in addition that correlationfunctions calculated from both sides of the mirror areidentical . The Mirror map associated parameter ofdeformation of Kalher structure with parameter of de-formation of complex structure. If a problem is diffi-cult at the A side, we can try to solve it at the B side.In mathematics passing through the mirror applica-tion can solves so important old problem of Enu-merative geometry

5.5 The Quintic and its mirror

Recall that the homogeneous quintic in P4, isobtained by canceling the superpotential: W=15(X5

0 + ...+X54 )

A marginal deformation of this superpotential isalmost the variety expected the variety mirror quinticis associated with the crepant resolution of:

(X0, ..., X4) ∈ P4/15(X5

0 + ...+X54 )−µX0...X4 =

0/G

with G = (a0, ..., a4) ∈ Z/5/∑ai =

0/Z/5=(a,a,a,a,a) ' (Z/5)3

5.6 Program

location The overall construction of the mirrormap is difficult. Should be localize and built themap at the neighborhood of a point.

Plan of the studyX denote the Calabi-Yau, MX its mirror .

• We must first calculate the Yukawa cou-pling H1(MX,TMX)

• Identify by the mirror map.

• Deduce the Yukawa couplings of H1,1(X)

One can deduce predictions about the number of ratio-nal curves inX We will briefly describe in the follow-ing useful mathematics for the B-side the Quintic.

5.7 Mirror-map

The principle of mirror symmetry says that< H,H,H >=< θ, θ, θ >, if tH denotes ancurve parameter in the module of Kahler X we setH = d

dt = 2πiq ddq its tangent vector (q = exp(2πit)Local coordinates for this module.The problem is to produce an image q(x) in modulecomplex deformations was: q = q(x), d

dq →dxdq

ddx

Then we can write H = 2πiq ddq )↔ θ = 2πiq dxdqddx

< H,H,H >= (2πiq ddq )3 < ddx ,

ddx ,

ddx >=

(2πi qxdxdq )3 < x d

dx , xddx , x

ddx >

5.8 Mathematical Tools

The simplest examples of Calabi-Yau are ellipticcurves. They will guide us to understand the tech-niques identified below and set the mirror applica-tion. Monodromy: For a flat bundle,go around a sin-gular object centred in t = 0 at constant distance|t|, (t:deformation parameter of a smooth family) isnamely monodromy in mathematics. In physics wetalk about loop Wilson. Residue map: We can gener-alize the formula for residues of a complex variablesfunction around z = 0 by replacing function differ-ential forms and point by hypersurface. This will bevery useful for calculating periods.

5.9 Elliptic curves

The elliptic curve Eτ = C/(1, τ) is a Calabi-Yau ofcomplex dimension 1. The volume form is given byΩ = dzHodge Numbers: Hodge diamond is:

h00

h10 h01

h11

=1

1 11

The one parameter family of deformations of an ellip-tic curve is:X3 + Y 3 + Z3 − 3ψXY Z = 0If α and β are homology cycles , they depend then ψ,we can find τ to from ratio periods:

∫α Ω,

∫β Ω.

Solving a differential equation called Picard-Fuchs


E-ISSN: 2224-2880 45 Volume 19, 2020

5.10 Family of elliptic curves, monodromy

degeneration of a family of curves

X [d]π ⊃ Xt[d]t 6= t′Xt ' X ′tD2 3 tt = 0X0 : sing.

Theoremmonodromy : EitherXt where t variesalong a loop in π1(D2 − 0, t0) around 0, (Wilsonloop), all elements of this family are diffeomor-phic .This induces an automorphism on homology:ϕ? ∈ Aut(Hn(Xt0),Z)) Pratical Example Let’sillustrate by taking a one parameter family of ellipticcurve Ct = (Y 2Z = X3 + X2Z − tZ3 ⊂ CP2

which is expressedin affine coordinates: Ct : y2 = x3 + x2 − t: ellipticcurve defined by an algebraic equation.The parameter t of the elliptic curve is the signatureof the variation of complex structure, the geometricexpression is Eτ = C/(1, τ(t)) when t revolvesaround the origin τ(t)→ τ(t) + 1 with τ function of:t: τ(t) = lnt

2πiAt the automorphism group:(τ(t)

1

)→(

1 10 1

)(τ(t)

1

)=

(τ(t) + 1

1

)We find that the complex structure varies but the

symplectic structure is unchanged.We justify the choice of new coordinate q(t) =e2πiτ(t), because a passage to the limit gives: whent → 0, Imτ(t) → +∞, so q(t), where q(t) isan holomorphic function of t which also go to 0;so it is a local coordinate for this family of ellipticcurves. We spoke in physics: large complex structurelimit(WSCC) in ”bijection” with large volume limit(LVL) on the A side

5.11 Application to the quintic

Is the quintic∑4

i=0 x5i −ψx0x1x2x3x4 = 0. As with

elliptic curves, Morrison showed by standing nearx = ( 1

ψ )−5 = 0 we could find t (side strain Kahler) from of ψ (or x).

As with elliptic curves, t =

∫γ1

Ω∫γ0

Ω, we must calculate

the periods φi(x) =∫γi

Ω, i = 1, 2. γ0 is invariant un-der monodromy around x = 0 (large complex struc-ture limit (WSCC )), ψ →∞ ψ →∞ γ1 → γ0 + γ1.Both quantities depend on ψ (or x)We can compute locally the three forms (*) Ω using aversion ”differential form” of theorem of residue (andthe theorem of implicit function) and deduce by direct

calculation , the first period.The equation called Picard Fuchs calculates the otherperiod. Finally, we find:φ0(x) =

∑∞n=0

5n!(n!)5

xn

φ1(x)=φ0(x)log(x) + f(x),withf(x) = 5

∑∞n=0

5n!(n!)5

(∑5n

j=n+11j )xn

5.12 Calculation of Yukawa couplingsquoted B

Let Θ(i) = (x ddx)(i), the equation of Picard-Fuch

written:

Θ(4)y + 2.55x1+55x

Θ(3)y + 7.54x1+55x

Θ(2)y + 2.54x1+55x

Θ(1)y +2.55x1+55x

Θy = 0It is applied to Ω:If Y =

∫MX Ω ∧Ω(3),

∫MX Ω ∧Ω =

∫MX Ω ∧Ω′ =∫

MX Ω∧Ω” = 0, twice differentiating the last equal-ity we have:

∫MX Ω ∧ Ω(4) + 2

∫MX Ω′ ∧ Ω(3) = 0

We deduce the differential equation (x ddx)Y =

−55x1+55x

Y

The solution is Y =< x ddx , x

ddx , x

ddx >= c2

1+55xWe must normalize Y in agreement with (*) and thendivide Ω by Φ0(x) Thus:< x d

dx , xddx , x

ddx >= c2

(1+55x)φ0(x)2For identifica-

tion,< H,H,H >= (2πi qx

dxdq )3 < x d

dx , xddx , x

ddx >=

c2(2πi qxdxdq

)3

(1+55x)φ0(x)2

5.13 Conclusions: Application to enumera-tive geometry

The parameter t Kahler deformation, expressed as afunction of the ratio of the first two periods, we get:

q = e2iπ

φ1(x)φ0(x) Where:

q = c1(x − 770x2 + ...) and converselyx = q

c1+ 770( qc1 )2 + ...

We can now calculate everything according to thevariable q:

< H,H,H >= (2πi)3(−c2 − 575( c2c1 )q −19575( c2

c21)q2 + ...)

It remains to calculate the constants c1 and c2.We remember that it was not enumerate rationalcurves of degree nd is known for first degrees, whichcalculates c1 and c2.< H,H,H >= 5+

∑∞d=1 ndd

3 qd

1−qd = 5+2875q+...


E-ISSN: 2224-2880 46 Volume 19, 2020

We deduce c2 = −5(2πi)3

, c1 = −1 Finally, we canenumerate the number curve of a rational quintic ofP4 all degrees < H,H,H >= 5 +

∑∞d=1 nd

d3qd

1−qd =

5 + 2875 q1−q + 609250.23 q2

1−q2 + ...

6 Homological mirror symmetryIt was previously presented a mirror symmetry involv-ing the closed strings, supposed to translate gravita-tion. One may wonder if such a theory can be adaptedto open strings: the answer is yes. We have seen thatmirror symmetry was an elaborate form of T -duality.But how to translate the number of winding in the caseof open strings. The answer to the previous questionexists: you can wind an open string provided that itsends are fixed on a branes. Thus the mirror symme-try of the open strings is based on this duality ”openstrings-branes”.In axiomatic theory of homology, we often considerthe relative homology of a space by a subspace:H∗(A,B). In fact we can do the same in topologi-cal field theory; One can replace in theory of fields thesource space Σ by the pair (Σ, ∂Σ), and thus think, forexample of the evolution of an open string whose ex-tremities are constrained to remain fixed on a brane(s).This approach would describe all of the particles, notjust gravitation. in this theory, in fact, the particlesof the standard model U(1) × SU(2) × SU(3) areseen as open strings fixed on a brane: the universe inwhich we evolve. A D- Geometric Brane is a triple(L,E,∇E) where L is a sub-manifold of dimensionreal p: L included in (M,E), a vector bundle and∇E a connection on E. An open string of origin(L,E,∇E) and target (K,F,∇F ) is a map X fromI = [0, 1] ∈M with X(0) ∈ L, X(1) ∈ K . We alsocall ”open string story”, a map from of the open Rie-mann surface . Σ = I×R ∈M withX(0, t) ∈ L andX(1, t) ∈ K these fields are supposed to be holomor-phic curves on the A side: Symplectic side. But whya bundle E and a connection?: A elementary parti-cle is charged by a gauge field for the electron it is theMaxwell field of groupU(1), For a quark the charge isits color: Yang Mills field, so we associate with an el-ementary particle a bundle E and its associated con-nection. By analogy , at each end of an open string, isassociated the couple (E,∇E). In the story of a par-ticle, or of the ends of an open string, we are led toconsider the holonomy of the connection (linear map)from E|φ(t1) to E|φ(t2). By identifying the N branesbetween them, we can describe the interactions be-

tween open strings. At the origin of an open stringliving on the brane i, we specify the target brane k.we obtain an index called Chan-Paton. the introduc-tion of N branes therefore introduces a gauge theoryU(N). This theory is non commutative if n > 1. Onebrane is necessary to describe the electomagnetism, atleast 2 are necessary to describe interactions involvingthe gauge group SU(2). By playing with this, we canmodel the different theories of gauges. To find parti-cles in this modeling, it is necessary to consider inter-sections of branes: this is the case where the lengthof the rope joining two branes tends towards 0

6.1 Categorization of mirror symmetry andopen string theory

To define the mirror symmetry of open strings, Kont-sevich categorizes the complex-symplectic exchange.He invents homological mirror symmetry.The mainidea is to replace the objects of mirror symmetry,namely the symplectic manifolds by objects belong-ing to two categories ”mirrors of each other”: namelychain complexes. The Fukaya category contains thechain complexes on the symplectic side (side A of themirror), a subcategory of coherent sheaves, modelizethe complex geometry on side B. The mirror applica-tion will also be replaced by a functor between thesethe two categories. Is not the first example of fieldtheory to use category language: Chern-Simon theoryuses categorifications from knot-theory. Kontsevich’sstatement in 1994 says that:There is an equivalence of category between Thebounded derived category of coherent sheavesDb(X) on Calabi- Yau X and Fukaya category of itsmirror MX .

6.2 Symplectic geometry side A

Briefly, We will always consider in this context,holomorphic curves but now the Riemann’s surfaceswill be a surface with boundary We will describe thecurves in this context more precisely. A holomorphiccurve (associated with lagrangians L1, L2) in asymplectic variety (M,ω) is:u : R × [0, 1] → M with J an almost complexstructure such as:∂Ju = 0: Cauchy Riemann conditions, u(s, 0) ∈ L0,u(s, 1) ∈ L1

lims→+∞u(s, t) = p, lims→−∞u(s, t) = qWe can stay on the A side and do enumerativegeometry: Curve counting (holomorphic discs).


E-ISSN: 2224-2880 47 Volume 19, 2020

Here the option will be to categorize from Floer’shomology and to find an equivalent category in the Bside. On the symplectic side, in open string theory thegood theory is Floer’s homology. This homology is akind of of Morse homology adapted to the languageof symplectic geometry. It was invented by Floerto demonstrate Arnold’s conjecture in symplecticgeometry.Formally the Floertheory is an adaptationof that of Morse in infinite dimension. The startingmanifold is replaced by a path space. Instead ofcritical points, we will have to define the notion of”critical paths ...”Let P(L0, L1) be the space of the end paths in the twoLagrangian L0 and L1; We then define a functional ofthe space of (homotopies) of paths γ in R:(γ, [u])→

∫[0,1]2 u

∗ωthis quantity represents the symplectic area whichreplaces the height function of Morse theory.Let v a vector field, The differential of the previousexpression is dA.v =

∫[0,1] ω(γ, v)dt: critical points

are constant paths (immediate).The trajectories of the gradient areJ-holomorphic maps

6.2.1 Floer’s Complex

Let: CF (L0, L1) = Λ|L0∩L1|, free module generatedby the points of intersection of the two Lagrangian L0

and L1, we define a differential:∂p =

∑deg(q)=deg(p)−1(M(p, q, [u], J)/R)T

∫u∗ωq

As in Morse homology, we show that ∂ defines a com-plex:broken trajectories

∐r∈L0∩L1

(M(p, r, [u], J)/R) ×(M(r, q, [u], J)/R)compacts a space of dimension 1 so add a even num-ber of points at the edges: the square of the edge istherefore zeroNote: in the exact Lagrangian case, we can avoidT∫u∗ω and work modulo 2, exactly as in Morse, We

just defined the Floer complex! The physicist will beable to see that there is an homology managing thebrane intersection points ...

6.2.2 Categorization of Floer’s homology

We can define a product structure on the Floercomplexes:CF ∗(L0, L1)⊗ CF ∗(L1, L2)→ CF ∗(L0, L2)The module space will be M(p, q, r, [u], J)) ofdimension deg(r)− (deg(p) + deg(r)The product:

p.q =∑

r,deg(r)=deg(p)+deg(q)(M(p, q, r, [u], J)/R)T∫u∗ωr

we can reiterate:CF ∗(L0, L1)⊗CF ∗(L1, L2)...⊗CF ∗(Ln−1, Ln)→CF ∗(L0, Ln) We can then define a ∞-categorywhose objects are Lagrangian, and morphisms arethe Floer complexes, . We categorized the symplecticside..

7 Complex geometry on B-side

On this side we have already seen that dominates com-plex geometry. his categorization will lead to definethe category of coherent sheaves. We cannot givemore details here. As in the case of closed strings,it will be a question of defining an mirror map allow-ing to go from side B to side A. The first examplethat validates this theory was given by Zaslow [15], inthe context of the Lagrangian associated with ellipticcurves (complex torus).

8 ConclusionUnfortunately, in this paper we cannot mention allthe studies undertaken so far. It seems that the non-commutativity that we discover in boundary field the-ories is interesting. it evokes the symmetry groupsof the particles of the standard model. Another ap-proach proposed by Alain Connes is possible, it isnon-commutative geometry. The theory of the in-dex can also be considered in this context of non-commutativity . In his theory, Alain Connes does notneed to speak about supersymmetry. It uses a triplet(A, H, D), A being the deformation of the algebra ofcontinuous functions :semi-direct product of algebraof C∞ functions on a manifold by matrice spaces ac-counting for the internal symmetries of the particles ofthe standard model. It might be interesting to under-stand the links between the elaborate gauge theoriesproposed by Witten using supersymmetry and that ofAlain Connes using non-commutative algebras ...

References:

[1] D.McDuff, D.Salamon, J-holomorphic Curvesand Quantum Cohomology, Univ. Lecture series6, A.M.S., 1994.

[2] M. F. Atiyah and I. M. Singer The index of el-liptic Operators on compact manifolds, Journal:Bull. Amer. Math. Soc. 69 (1963), 422-433


E-ISSN: 2224-2880 48 Volume 19, 2020

[3]

[4] P.Griffiths, J.Harris, Principle of algebraic ge-ometry, Wiley-interscience, New York, 1978.

[5] K. Kodaira Complex Manifolds and Deforma-tions of Complex Structures, Springer, ClassicsIn Mathemtics, 2005.

[6] K. Gawedzki, Lectures on Conformal field the-ory, in Quantum Fields and String: A Course formathematicians, Vol 2 , A.M.S

[7] Gromov, M. . ”Pseudo holomorphic curves insymplectic manifolds”. Inventiones Mathemati-cae. 82 (2), 1985: 307–347.

[8] D.A. Cox, S. Katz,Mirror symetrie and Al-gebraic geometry, Mathematical Surveys andMonographs V68, A.M.S 1999.

[9] K.Hori, S. Katz, A. Klemm, R. Pandharipande,R. Thomas, C. Vafa, R. Vakil, E. Zaslow ,MirrorSymmetry Clay Mathematics Monographs, Vol 1A.M.S, 2003.

[10] S.Donaldson, P. Kronheimer, The geometry offour manifolds, Oxford University Press, 1990.

[11] E. Witten,”Topological quantum field theory”,Comm. Math. Phys., Volume 117, Number 3(1988), 353-386 D.A. Cox, S. Katz,Mirror syme-trie and Algebraic geometry, Mathematical Sur-veys and Monographs V68, A.M.S 1999.

[12] M. Kontsevich, Y.Manin, Gromov-WittenClasss, Quantum Cohomology, and enumerativeGeometry, Mirror symetry II, A.M.S/IP Studiesin advanced Math., Vol 1 A.M.S, 1997, pp.607-653.

[13] H. Clemens, A Scrapbook of Complex CurveTheory, G.S.M Vol 55, A.M.S Studies in ad-vanced Math., Vol 1 A.M.S, 1997, pp. 607-653.

[14] H. Morrison, Mirror symmetry and rationalcurves on quintic threefolds: a guide for mathe-maticians,J. Amer. Math. Soc. 6 (1993) 223–247

[15] A. Polishchuk, E. Zaslow, CategoricalMirror Symmetry: The Elliptic Curve,Adv.Theor.Math.Phys.2:443-470,1998

[16] D.McDuff, D.Salamon, Introduction to symplec-tic geometry Oxford University Press, 1995.

[17] C. Voisin Symetrie miroir, Panoramas etsyntheses Numero 2, 1996 S.M.F


E-ISSN: 2224-2880 49 Volume 19, 2020

from qft to 2-d supersymetric tqt:mirror symmetry between …€¦ · from qft to 2-d supersymetric...

Documents