Thank you. First of all, I wish to thank the organizers for the invitation.
It is my great honor to have the opportunity to speak at this wonderful conference.
I also wish to thank the organizers for organizing such a wonderful conference.
I have learned so much exciting developments from other talks.
Today, I'll talk about Groom-of-Witten theory,
FJRW theory, and MSP fields.
The last part will be based on joint work with Huai-Liang Zhang-Jing Li and Wei-Ping Li.
In this talk, L capital I is generally and L small i is Wei-Ping Li.
But today, like Groom-of-Witten theory,
I will only talk about Groom-of-Witten theory of the quintic three-fold.
So just to fix the notation that W denote this from our quintic polynomial in five variables,
the quintic labial three-fold is defined by the zero locus of this quintic polynomial in P4.
And actually, we will consider Groom-of-Witten theory of the quintic three-fold
or amodotopological string amplitudes.
So we can take any smooth degree five hypersurfacing P4,
and they will give us the same Groom-of-Witten invariance.
So the Kaler moduli is one dimensional.
Okay, so in this talk, we will let Mu5 denote the group of fifth roots of unity,
and Zeta will denote this particular fifth roots of unity.
So the Murrow family is a one-parameter family compact clavial three-fold
obtained by Kreper resolution of this hypersurface, not in P4,
but in P4 mod, the action of this G.
So this G is like A1 to A5 are fifth roots of unity,
their product is one, and mod the diagonal, which acts trivially, and this preserve this equation.
Okay, so that's the Murrow family, and so the complex moduli is one dimensional.
And this passai is the local parameter of the complex moduli of the Murrow family,
and the complex complex moduli of the Murrow family can be identified with the weighted projective line.
So it looks like this.
So this is a unique or before point.
So where the transition function is given explicitly by that.
Okay, so this maximally-unipotent monitoring point is called large complex structure limit
in Nick Sheridan's talk, and when passai is one, it's the conical point, passai is zero,
it's the orbital point. So that's a unique orbital point in the moduli space.
So over this moduli space, there is a harsh line bundle whose fiber over passai is just the space
of holomorphic three forms. So we consider this complex moduli, so we know that if we consider
the B-model topological string, we have this global FG generating function of G B-model topological
string amplitudes, which is a section of this bundle. And so if we expand this near the large
radius, if we expand it near passai is equal to infinity using super low flat coordinate,
we obtain the Gromov generating function FG of G, Gromov with an invariance of the quintic
three-fold. So this is just an overview, so I will give mathematical definition of Gromov
with an invariance, FJRW invariance, and finally MSP theory by the end of this talk. So if we
expand this FGB near the orbital point, passai is equal to zero, then we get so-called FJRW
invariance. So standing for Fan Jarvis Run and Witten, FJRW invariance of the FMR quintic
polynomial. Yeah, so this roughly counts solutions to the Witten's equation. And it is like from the
A-model point of view, it is not clear why these two theories shoot the FJRW of W and the Gromov
with the quintic should be related. But motivated by this mirror symmetry picture, they should be
related. And that is the Landau-Ginzburg-Clabial correspondence proposed in early 90s by Witten.
So now next I am going to, so this is the motivation from string theory. So next I am
going to give you a summary of mathematical results. Yeah, so we know that the genus zero
mirror formula was conjectured by, yeah, so genus zero mirror formula for the quintic
threefold was conjectured by candelas and collaborators in early 90s. And it was proved
about 20 years ago by Givental and Lian Liao. So the genus one mirror formula was conjectured by
VCOV and this formula was proved about 10 years ago by Alexei Singer. And his work relies on the
reduced Gromov, genus one Gromov-Witten theory developed by Kiehl-Zinger and Li-Zinger. So after,
these are the first proofs and there are like other later proofs which I won't list all of them
there. So for example there are like more recent proofs using quiescent map approach.
So now on the top, so this mirror symmetry for the FJRW theory, this is proof in genus zero by
Kyodo and Run in 2008. And genus one was proved last year by Schreig-Wun Rose and this approach
uses the MSP theory that we developed. So then if you combine the top and the bottom and they manage
to prove the no Ginsburg-Clabial correspondence in genus zero and genus one. And so now the thing
is it turns out that the Gromov-Witten invariance of the quintic can be reformulated as Landau-Ginsburg
theory of KP4, the total space of the canonical limon over P4 with a superpotential. And the
FJRW theory can also be viewed as a mathematical theory of Landau-Ginsburg model C5 mod mu 5. And
these two are actually the GIT quotient with different stability conditions. Yeah so actually
this correspondence already appear in Hori's talk. So in the rest of my talk I'm going to
explain first first review this and recall the definition of this generating function
and then explain like the mathematical theory of this
and the equivalence and also
so
so here W is
where x1 to x5 are the homogeneous coordinate on the base and P is the coordinate on the fiber.
And then actually we will explain this theory of MSP fields which is meant to interpolate this
two mathematical theories. So to fill the the gap on a model side. Okay so this is an old story so
let so this is the modulized space of genus G degree d stable maps with no bar point to Q.
It has virtual dimension zero for any genus and degree. So this is a degree zero
a homology class pair so pair it with one which is a degree zero co-homology class one gets a
rational number. This is the mathematical definition of the genus G degree d
Gromov-Witten invariance except for these two cases where the modulized space are empty.
So that fg be the generating function of genus G Gromov-Witten invariance summing over all degrees.
So in genus zero and one there are some logarithm singularity so there are monodromics. When genus
is greater or equal to two these are power series of Q variable. Oh I should say this so
back in like 2006 so Molligan-Panderpanda like provide an algorithm of evaluating all
N G D based on the degeneration formula of Gromov-Witten invariance provided generally.
So if you want to know any N G D like with the help of computer you can get it in finite time
but the algorithm yeah it's kind of very slow and this finite time is long compared with our
lifetime. So okay so this so so so okay so okay so we are done with this part.
Okay so then next thing I'm going to explain this part so so this yeah so this is so-called the
stable map developed by Huai Liang, Chang and Jun Li. It can be viewed as a mathematical theory
of the Landau-Ginzburg model on Kp4 with this super potential and genus zero was developed by
Gufeng and Sharpe who's here. So let me describe the moduli so we consider
yeah so this so so moduli space so if you consider this so U is a stable map to P4,
C is its domain which is a genus G nodal curve and N is a number of mark points so
this mark points are distinct and they are smooth points. So if you only look at U C Z this is just
a point in the moduli space of genus GM pointed degree D stable map to P4 but this so so this is
stable map together with P field so so P field where this row is a section of the pullback of
O minus five twisted by the dualizing sheath. So because so so when genus is zero this P field
is always zero so this is compact when the genus is greater than zero then because of this P field
we can rescale this P field this moduli space is not compact. So any map from any morphism to the
projectives four-dimensional projective space is describing terms of five sections of pullback of
O1 which do not vanish at the same time so one can slightly reformulate this as follows. So this is
so we have seed so we have a curve genus GM pointed pre-stable curve so it's a nodal curve with
and distinct smooth mark points and then there's a line bundle which is a degree D line bundle on
this nodal curve and we have five section which are no so they do not vanish at the same same time
and row is the section of this bundle and then you can like so this is somehow this this nowhere
zero condition is like the stability condition of the target viewed as a GIT quotient and this
is the stability condition of the domain curve. So these two are the same moduli problem but if
you formulate in this way it's more linear and then this phi one two and this phi one two phi
five row these are like sections of line bundles or fields they can be viewed as a
quantum version of the sixth homogeneous coordinate of KP4. So this super potential
so KP4 is like given by C6 yeah C6 mod out by the C star action of weight one one one one one minus
five so this super potential descends to yeah so this this polynomial descend to a super potential
on KP4 and yeah it's easy calculus to compute the critical locus is exactly this so this says it's
in the zero locus P4 and it's exactly the quintic inside the zero zero section P4
and so we have this so this using this super potential we can define some co-section which
is a map from the obstruction sheath over the moduli space of yeah stable map with P field
denoted by curly P to the so actually at any point so if you are so if you have any point
in this moduli space so this so this is the obstruction space
so it if you have a section that means at any point you get a vector here a co-section is at
any point you get a linear functional and the degeneracy locus is given by exactly this so when
phi satisfy this equation and rho is zero so you see like this equation exactly looks like the
corresponding equation of the critical locus and this is so this exactly defined so the this
degeneracy locus is exactly the stable map of genus g degree d to the quintic threefold sorry
this should be a n and this is compact so so somehow you should view this co-section on some
quantum version of the super potential and the degeneracy locus of the co-section is like the
quantum version of the critical locus and so kimli so if you have in general you you you have some
moduli space which is the link mover stack with a perfect obstruction theory you can apply the
baron fan tacky and you get a virtual fundamental cycle so now kimli has this construction if you
have a smooth just a delingman first step with a perfect obstruction theory together with uh this
co-section then they get the so-called co-section localized virtual cycle so this this co-section
localized virtual cycle is supported actually in this sub-stack which is the degeneracy locus
of the co-section uh so you can view it as an element in child group or homology group
with rational coefficient so apply them so huang and xunli obtain this and and moreover they
actually prove that so on this moduli space we have this uh virtual fundamental cycle coming
from like moduli space of stable map and in turns out that these two virtual classes differ by a
sign so they prove this so this is the yeah mathematical precise mathematical statement
of saying that these two theories are equivalent so now we have we are done with this
okay so next i'm going to move to the top
um so let gamma so so before we uh introduce this uh we need to introduce some other moduli
spaces moduli of five spin curves so let gamma be so that gamma one to gamma i be a non-trivial
fifth roots of unity so gamma is cmi uh sorry zeta mi
yeah so maybe i should um
okay so this is the moduli of gene energy gamma market five skin curves
you have a twisted gene a gl pointed priest yeah actually in this case it's uh yeah we
we have a stable curve so it looks like this
so each of this point looks are stat key points
and if you forget about the orbital and it can also have a node and
orbital structure at the node and if you start if you forget about the orbital structure it is just
a stable genus gl pointed curve so that's c and z and and for l it it is same bundle on this
orbit curve together with some isomorphism row from the fifth power of l to the
logarithm log dualizing shift which is just a dualizing shift twisted by this orbit curves
so this gamma j tells us the monodromy of this of this line bundle around this orbital point
so if you forget about this line bundle l in the row it's almost a finite map to the uh
to the moduli space of gene a gl pointed stable curve so it's a compact complex orbital of this
dimension it's a smooth proper smooth the link month instead and there are some condition on
the monodromy such that this moduli space is not empty so this is yeah this is very nice it's just
a compact complex orbital okay so now the um to define this fjrw so this fjrw invariance has been
defined by uh fun jarvis run uh for a very general quasi homogeneous polynomial w and also um
huai liang zhong jin li and and waypingly also develop theory for this generality but uh so
today we will just focus on this quin for my quintic polynomial so this is just so this is
so if you consider czl and row this is just a five spin curve defined in the previous slides
and then together with five fields so now v1 to v5 are sections of this line bundle l
so if we let c star axon
so with weight minus one minus one minus one minus one minus one five so kp4
is
well c5 mu5 is c5 times c minus zero
yeah so they are like different git so there are git quotients of the same action with difference
stability conditions and well since it's an isomorphism it can also be viewed
as a nowhere vanishing section of this line bundle um so the superpotential
is the for my quintic polynomial it descends to c5 mod mu5 and similarly so this superpotential
defines the co-section from the obstruction sheave over this modulized space of five spin
curve with five additional fields v1 to v5 uh so this v1 to v5 can be viewed as the quantum
version of the coordinates x1 to x5 and this co-section can be viewed as the quantum version
of the superpotential w the for my quintic and um and the critical point well you you
differentiate this and critical point is actually this and if you consider the reduced critical
reduced degeneracy locus of this co-section this is exactly when all the fields are zero
so you get this uh r5 spin curve inside this modulized inside this and again we um
so using so why long chong jin li and way ping li apply the uh construction of co-section localized
virtual cycle due to kim kim and li and they get this so-called um so you see it so they they get
this so-called like witten's top chung class uh which is actually an element in the shell group
or homology of this five spin modulized five spin curves so this modulized space is very good
compact and smooth orbitals but this this is difficult and complicated and one can compute
the virtual dimension and find that this doesn't depend on the genus is given by this recall that
if gamma is this then this is the virtual dimension uh so now we are ready to define um
um define this
so when all the mi's are two then the virtual dimension is zero so let's so
so this gr invariant is you fix genus and you take l l is the number of insertion where
you make all the mi to be two so this virtual cycle is a degree zero cycle and you uh pair it
with one you get a rational number so these are f j r w invariance also i mean these these are
defined when this is the condition where the modulized space is not empty so this are these
are defined when l is greater or equal to zero and also satisfy this condition well there are
some exceptional case where genus is one and l is zero that's the unstable case we define this to be
zero so this is the generating function of f j r w invariance okay so now we so so this f j r w
yeah so this is a mathematical theory of Landau-Ginsberg model on c5
mu 5 with super potential w okay so so now we have explained
so now we have explained like the g gromov g w theory f j of the f j r w theory in the title
so in the rest of the time we i'm going to explain the theory of MSP fields but before
to motivate the construction by looking at some geometry so this is a master space so consider a
c star action on c6 plus p1 so where the c star action um yeah so let me let me change this way
to the
yeah so the c6 part is the same as the action on the platform and the action p1 is given by this
so the master space is obtained by a certain git quotient of this it is obtained by you take away
some unstable locus z and modeled by the c star action so the result is a six-dimensional toric
orbital which is not compact and this is the picture so uh so yeah actually because i mean
this triangle should really be a five simplex so if you can check if you let u1 to be zero
then u2 must be non-zero and this toric divisor is kp4 and if you let u2 to be equal to zero
u1 must be non-zero then this divisor is c5 mu5 so this master space contains both spaces that we
want and also when you let p when p is zero you get a copy of p5 and then also when all the xi's
are zero then u1 must be non-zero and looking at p and u2 you get this uh weighted projective
line and this point is the unique orbital point in this six-dimensional toric orbital
so this super potential actually descends to the defines a super potential on the master space
and it also restrict to this super potential on kp4 and c5 mod mu5
and also if you look at the critical locus yeah and the critical locus so this equation
is the equation of oh so the sixth coordinate of this p5 are xi's and p so this equation is
exactly the equation of the quintic cone so the cone over the quintic threefold in p4 so these
two divisor intersect at p4 and then union this weighted projective line so the whole thing is
smooth and non-compact and the red part the critical point is uh singular but compact
so the also on the master space so there is a c star action on the master space which is like
going into this direction just by rescaling u1 so to avoid confusion i call this after
we quotient by the previous c star action let's call this c star t and let's call the
coordinate little t so there's a t star action c star action master space by rescaling u1
and the fixed locus is exactly kp4 c5 mod mu5 and the point and the critical locus is
the quintic the point and the yeah stacky point
yeah so so basically you see that you on this master space there are eight homogeneous
coordinates so the so the definition of like modular of msp fields will be more complicated
but the idea is that to interpolate these two theory we should consider landau ginsberg model
on the master space with this super potential so um yeah so this is uh so this so so so it
consists of like yeah because there are it's so the master space is the
uh so now we need like two line bundles and eight fields
so first uh c is uh so the gene c is the genus of the nodal curve c and z is like you have um
l orbital points on c and
l and n are line bundles sorry these are line bundles on the domain curve c that's a typo
so now this vr field so phi has uh so so the degree of l tensor n is d0 and degree of n
is d infinity so these are so this line bundles are if you think in terms of maps to
the master space this line bundles correspond to the line bound of the divisor u1 is equal to zero
and u2 is equal to zero um so gamma tells us the monodromy of l
oh so these are just the fields so yeah so this so now the point is we need eight fields
which are quantum version of the eight coordinates on the master space and this nowhere vanishing
exactly correspond to the yeah the stability condition of the git quotient on the previous
slide so yeah so it's complicated but idea is like this is um comes from like quantum version
of this Landau-Ginzmer model on the master space and also this super potential determines a
co-section from the obstruction sheaf over this moduli space to the structure sheaf of the moduli
space similar to the previous cases and let's denote the degeneracy locus of this co-section
w minus so this co-section can be viewed as the quantum version of the super potential
and the degeneracy locus is the quantum version of the critical locus like the red part
so moreover so recall that there's this um there's this t star action x on this master space whose
fixed locus is this so this action is given by scaling u1 so the quantum version of this action
should act on this moduli space by rescaling the section nu1 so there is an action on this moduli
space and everything is t quiverant oh so i should say so when when we reformulate it in terms of this
so if you consider the moduli space of p field so so the torus action on p4
uh also acts on the moduli space of stable map of p fields but but the problem is it doesn't
this torus action does not preserve the co-section as you can see because it does not preserve this
super potential so one cannot use localization to compute it on this side
so but now on the master space we can use this t action um
so yeah so now in this setting it's much more difficult to prove that the degeneracy locus
okay so this is w yeah this is this is w minus is close and proper and of finite time so it's
much more difficult to prove that this is compact but the degeneracy locus is indeed compact
so one can so we can we can apply kim and lee's construction of co-section localized virtual cycle
to yes to obtain this cycle which is supported in the degeneracy locus
so this is just another notation for this and the virtual dimension can be computed
so now uh so for the baron fantaki virtual cycle
so we can use this we can use this virtual cycle to define invariance in the msp
theory of msp fields and because of this t action we can also use define consider
t equivariant theory of msp fields and this t equivariant theory
invariance can actually be computed by torus localization
um so for the usual um virtual fundamental class by baron fantaki greeber pander panda and baron
they have virtual localization of the virtual fundamental cycle and cheng and kim and lee
prove the virtual look uh torus yeah so there are two so there are like torus localization of the
co-section localized virtual cycle so it looks like the following so your uh so your fixed locus
of the modular space yeah can be the components are yeah labeled by label graph so this omega gamma
w gamma are components and then the co-section localized virtual class are basically like the
push forward so this is the inclusion of the component and the co-section when you restrict
to this w gamma you get the restricted co-section and this virtual bundle is the so this virtual
classes you you have this torus action on the perfect obstruction theory the fixed part give
defines this co-section localized virtual cycle on this fixed locus and the moving part defines the
um virtual normal bundle this is the Euler class and
and so because like we have this picture we know that the fixed locus of the master space is just
kp for a point and c5 mu 5 so it's not surprising that like if you look at so each if we fix a graph
and this is up to some finite cover it's just a product of moduli space of these three types
so to summarize the so this is the virtual cycle in the in the theory of stable map
with p fields so this is the um moduli cycle in the f j r w theory so in the middle points
essentially just give you the theory gromov witten theory of a point which is known
so the t quiver and msp theory after this torus localization can be reduced to gromov witten
theory of the quintic gromov witten theory of point which is known by yeah so so and then the
gromov f j r w theory of the former quintic okay so then now we want to get some relation
between gromov witten theory of the quintic and f j r w invariance which is at least some weak
version of this landau ginsberg labial correspondence so let's consider so let's integrate don't insert
any class and get some correlator uh so when the virtual dimension is greater than zero this is a
zero and you can one can compute this zero one can compute this zero by virtual localization and
should be expressed in terms of um the gromov witten invariance of the quintic and f j r w invariance
of the former quintic so
so first let's actually further restrict to the case where there's no mark points so the
virtual dimension is just so first if we consider the case where d zero is zero then the relation
actually is like almost purely in the f j r w invariance and when um so yeah so these are like
so maybe let me give you a simplified version of this theorem so this is just some counting
dimension so to summarize so if you have um so so on the f j r w side so if you so we know we
already know like the genus zero okay so before our theory of msp field so kyodoran already proved
the genus zero mirror conjecture for the f j r w theory so if you know so for a given positive
genus if you know the generating function f g prime for all g prime less than g and this is the
initial values that you need to know so in particular because of so in particular like
this should completely determine f one from f zero so for um the quintic so suppose that we
know the f j r w invariance up to genus g and also the gromov also the gromov witten invariance of
the quintic threefold up to genus less than g then the initial condition one needs to know is
this this actually comes from yeah so we have a relation we when the virtual dimension is
yeah so we still need those dimension but in any case like the number of unknown is actually
linear so so after this like last year using localization on this moduli space of m m sp
fields they proved the mural con genus one mural conjecture for the f j r w theory so so finally
yeah okay so um so finally i just want to compare with some like prediction from physics
so
so we know that um about yeah almost
oh sorry
so this is this is more than 10 years ago so they use so so they have mural conjecture so they
have some algorithm to compute the like f g b the b model topological string up to
genus lesso equal to 51 and this this uses the early work of
yama gucci and yao okay so based on my understanding this goes roughly as following so you have
so so they do this so by so genus zero and one is like self so holomorphic anomaly equation
when you integrate it like you have three g minus two uh holomorphic ambiguity
and then you minus one so this is the n g zero at the
large complex structure limit and so then this this this can be computed and then you have
so this is more mysterious
yeah so so last week i heard clams computation at this point so we still don't know how to
make sense for the a model at this point but the advantage is like in the middle we just
get a theory of a point which is known so then at the orbital point
so this gives some extra condition which is like this
so this is roughly like this so in any case this is consistent with like the initial condition
is like linear in linear in the genus so then finally they have this cast nouveau bound which
is some vanishing for embedded curve or bps invariance if we assume like a goba kumawafa
conjecture relating gromov with any variance and the bps invariance and there's this parabola
so yeah so the 51 is yeah about here so this is okay so um yes so yeah i hope that using this
approach we can hopefully yeah proof like using this approach to prove the genus to
mural conjecture thank you