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### Young inequality

Let $a,b,p,q$ be positive real numbers. Assume further that $1/p+1/q=1.$ The following inequality is called the Young inequality.

$\displaystyle\frac{a^{p}}{p}+\frac{b^{q}}{q}\geq ab.$

Since both $a,b$ are nonzero, we can divide both side of the above inequality by $a^{p}$ and obtain the following equivalent form:

$\displaystyle \frac{1}{p}+\frac{b^{q}}{qa^{p}}\geq\frac{b}{a^{p-1}}.$

Let us define a new variable $t=b/a^{p-1}.$ Using the relation $1/p+1/q=1,$ we can see that $t^{q}=b^{q}/a^{p}.$ Thus the inequality can be rewritten as

$\displaystyle \frac{t^{q}}{q}-t+\frac{1}{p}\geq 0.$

Let us define a function $y=f(t)$ on $[0,\infty)$ by

$\displaystyle f(t)=\frac{t^{q}}{q}-t+\frac{1}{p}.$

If we can show that for any $t\geq 0,$ $f(t)\geq 0,$ then the Young inequality holds. This fact can be proved by differential calculus. Solving $f'(t)=0$, we can see that $t=1$ is the only critical point of the function $y=f(t).$ Moreover, $f'(t)<0$ if $0\leq t<1$ and $f'(t)>0$ if $t>1$. We can also show that $f''(t)=(q-1)t^{q-2}>0$ for any $t\geq 0.$ We find that $t=1$ is the global minimum of the function if you analyze the slop and the convexity of the function by $f',f''.$ The following picture is the graph of the function in the case when $p=1/3.$

Thus we conclude that for any $t\geq 0,$$f(t)\geq f(1)=0$。We coomplete the proof of the Young inequality

### Application in Lebesgue integral theory

Let $(X,\mu)$ be a measure space and $L^{p}(X,\mu)$ be the $p$ (complex-valued) integrable  functions  on $X.$ Define the $p$-norm of $f\in L^{p}(X,\mu)$ by

$\displaystyle\|f\|_{p}=\left(\int_{X}|f|^{p}d\mu\right)^{1/p}.$

Using the Young inequality, we can show that

Proposition:  (Holder inequality)Let $p,q>0$ and $1/p+1/q=1.$ If $f\in L^{p}(X,\mu),$ $g\in L^{q}(X,\mu),$ then

$\displaystyle \int_{X}|f(x)g(x)|d\mu\leq \|f\|_{p}\|g\|_{q}.$

Proof:Dividing the inequality by $\|f\|_{p},\|g\|_{q},$ we have

$\displaystyle\int_{X}F(x)G(x)d\mu\leq 1,$

where $F(x)=|f(x)|/\|f\|_{p},$ $G(x)=\|g(x)\|/\|g\|_{q}.$ We have $\|F\|_{p}=\|G\|_{q}=1$ and $F,G\geq 0.$ Using the Young inequality, we obtain

$\displaystyle F(x)G(x)\leq \frac{F(x)^{p}}{p}+\frac{G(x)^{q}}{q}$

Integrating both side of the inequality, we get

$\displaystyle\int_{X}F(x)G(x)d\mu\leq\frac{1}{p}+\frac{1}{q}=1.$

Cor: (Discrete Holder inequality) Assume that$x_{k},y_{k},$ $1\leq k\leq n,$ are complex numbers. Then

$\displaystyle\sum_{k=1}^{n}|x_{k}y_{k}|\leq(\sum_{k=1}^{n}|x_{k}|^{p})^{1/p}(\sum_{k=1}^{n}|y_{k}|^{q})^{1/q}.$

### Integral over infinite dimensional space.

It is well-known how to define the Riemann integral or the Lebesgue integral of a measurable function on the finite dimensional Euclidean space $\mathbb{R}^{n}.$ Given an infinite dimensional separable Hilbert space $H$, for example, space of square summable sequences, and a functional $f$ on $H,$ can we have the infinite dimensional analogy $\int_{H}f$?

Let us simply assume that $H$ is the space of square summable real sequences. It is the vector space of sequences $\{a_{n}\}$ so that $a_{n}$ are all real numbers and $\displaystyle\sum_{n=1}^{\infty}a_{n}^{2}$ is convergent for each $\{a_{n}\}.$ The inner product on $H$ is given by

$\displaystyle\langle\{a_{n}\},\{b_{n}\}\rangle=\sum_{n=1}^{\infty}a_{n}b_{n}.$

With respect to the inner product, $H$ is an infinite dimensional real Hilbert space. We can think of $H$ as an infinite dimensional analogy of the Euclidean space. Let $\{e_{n}\}$ be the standard basis for $H.$ Given a functional $f:H\to\mathbb{R},$ we define the finite dimensional approximation of $f$ by

$f_{n}(x_{1},\cdots,x_{n})=f(x_{1}e_{1}+\cdots+x_{n}e_{n})$

and consider the Lebesgue integral (we may assume that $f_{n}$ are integrable  on $\mathbb{R}^{n}$ for all $n$)

$\displaystyle I_{n}(f)=\frac{1}{\sqrt{\pi^{n}}}\int_{\mathbb{R}^{n}}f_{n}(x_{1},\cdots,x_{n})dx_{1}\cdots dx_{n}.$

If the $\lim_{n\to\infty}I_{n}(f)$ limit exists, we denote

$\displaystyle \lim_{n\to\infty}I_{n}(f)=\int_{H}f(x)[Dx]$

and call it the functional integral of $f$ over $H.$

For example, let $A:H\to H$ be a positive definite Fredholm operator such that $A-1$ is a trace class operator. Then the notion of the determinant of $A$ is defined and can be calculated by the the following finite dimensional approximations. Let $H_n$ be the finite dimensional subspace of $H$ spanned by $\{e_{1},\cdots,e_{n}\}.$ Denote $P_{n}:H\to H_{n}$ the orthogonal projections for all $n.$ Let $A_{n}=P_{n}AP_{n}.$ Then

$\displaystyle\lim_{n\to\infty}\det A_{n}=\det A.$

Let us consider a functional $f$ on $H$ defined by

$f(x)=\exp(-\langle Ax,x\rangle),$ $x\in H.$

Then we know that for all $(x_{1},\cdots,x_{n})\in\mathbb{R}^{n},$ we have

$\displaystyle f_{n}(x_{1},\cdots,x_{n})=\exp(-\sum_{i,j=1}^{n}a_{ij}x_{i}x_{j}).$

The integral $I_{n}(f)$ is given by

$\displaystyle I_{n}(f)=\frac{1}{\pi^{n/2}}\int_{\mathbb{R}^{n}}f_{n}(x_{1},\cdots,x_{n})dx_{1}\cdots dx_{n}=\frac{1}{\sqrt{\det A_{n}}}.$

Thus $\lim_{n\to\infty}I_{n}(f)$ exists and equals $1/\sqrt{\det A}.$ Therefore we obtain

$\displaystyle \int_{H}\exp(-\langle Ax,x\rangle)[Dx]=\frac{1}{\sqrt{\det A}}.$

Remark. Since each $A_{n}$ is self-adjoint, we can choose an orthogonal matrix $U$ on $\mathbb{R}^{n}$ so that $U^{*}AU$ is the diagonal matrix $\mbox{diag}(\lambda_{1},\cdots,\lambda_{n}).$ Sinc e the integral is invariant under orthogonal transformation, we may assume that $A_{n}$ is a diagonal matrix and

$\displaystyle\sum_{i,j=1}^{n}A_{ij}x_{i}x_{j}=\sum_{i=1}^{n}\lambda_{i}x_{i}^{2}.$

Since the integral of $\displaystyle\int_{-\infty}^{\infty}e^{-\lambda x^{2}}dx=\sqrt{\pi/\lambda},$ we see that

$\displaystyle\int_{\mathbb{R}^{n}}\exp\left(\lambda_{1}x_{1}^{2}+\cdots+\lambda_{n}x_{n}^{2}\right)dx_{1}\cdots dx_{n}=\prod_{j=1}^{n}\sqrt{\frac{\pi}{\lambda_{i}}}.$

Since $\det A_{n}=\lambda_{1}\cdots\lambda_{n},$ we obtain that $I_{n}(f)=1/\sqrt{\det A_{n}}.$

### Hopf Algebra

A hopf algebra $A$ is a graded algebra $A=\bigoplus_{n\geq 0}A_{n}$ over a commutative ring $R$ so that there exists an identity $1\in A_{0}$ such that the map $R\to A_{0}$ defined by $r\mapsto r\cdot 1$ is an isomorphism and there exists a homomorphism of graded algebra $\Delta:A\to A\otimes A$ so that

$\Delta (\alpha)=\alpha\otimes 1+1\otimes \alpha+\sum_{0

for $\alpha\in A_{n}$, $n>0$ and $\alpha_{j}',\alpha_{j}''\in A_{j}$.

Let $A=R[\alpha]$ be the polynomial ring over $R$. Then $\Delta(\alpha)=\alpha\otimes 1+1\otimes \alpha$. Assume that $\alpha$ is odd dimensional. Then $(\alpha \otimes 1)(1\otimes \alpha)=\alpha\otimes\alpha$ and $(1\otimes\alpha)(\alpha\otimes 1)=-\alpha\otimes\alpha$. This would implies that $\Delta(\alpha)^{2}=\alpha^{2}\otimes 1+1\otimes\alpha^{2}=\Delta(\alpha^{2})$.

Let $\Lambda_{R}[\alpha]$ be the exterior algebra over $R$. Then $\alpha^{2}=0$. It is easy to see that $\Delta(\alpha)^{2}=0$.

An element $\alpha$ in a Hopf algebra $A$ is said to be primitive if $\Delta\alpha=\alpha\otimes 1+1\otimes\alpha$.

Exercise: Let $A,B$ be hopf algebras over $R$. On $A\otimes_{R} B$, we define $\Delta(a\otimes b)=\Delta(a)\otimes\Delta(b)$. Show that $A\otimes_{R}B$ is again a hopf algebra.

### Picard-Fuchs Equation

Let ${M}$ be a smooth projective algebraic variety of dimension ${n}$. Given a holomorphic ${n}$-form ${\omega}$ and ${\gamma_{1},\cdots,\gamma_{r}}$ be a basis for ${H_{n}(M)}$. (We assume that the dimension of ${H_{n}(M)}$ is ${r}$.) The periods of ${\omega}$ associated with the basis ${\{\gamma_{1},\cdots,\gamma_{r}\}}$ are the integrals:

$\displaystyle \int_{\gamma_{j}}\omega,\quad 1\leq j\leq r. \ \ \ \ \ (1)$

Let us consider a family of ${n}$-dimensional projective algebraic varieties ${\bar{\pi}:\bar{X}\rightarrow \bar{C}}$, where ${\bar{C}}$ is a compact Riemann surface. Assume that ${C}$ is an open subset of ${C}$ so that the induced family ${\pi:X\rightarrow C}$ has smooth fibers. Let ${\omega}$ be an ${n}$-form on a fiber ${X_{0}}$ of ${\pi}$ and ${\{\gamma_{j}:1\leq j\leq r\}}$ be a basis for ${H_{n}(X_{0})}$. Assume that ${\omega}$ can be extended to a family of ${n}$-forms ${\{\omega(z)\}}$ with ${\omega(z)\in X_{z}}$ for each ${z}$ and ${\{\gamma_{j}\}}$ can be extended to a family of basis ${\{\gamma_{j}(z)\}}$ for ${H_{n}(X_{z})}$, where ${z}$ is a local coordinates on ${C}$. Let ${v(z)}$ be the vector whose components are periods of ${\omega(z)}$ associated with the basis ${\{\gamma_{j}(z)\}}$:

$\displaystyle v(z)=\left[ \begin{array}{c} \int_{\gamma_{1}(z)}\omega(z) \\ \int_{\gamma_{2}(z)}\omega(z)\\ \vdots\\ \int_{\gamma_{r}(z)}\omega(z) \\ \end{array} \right]\in\mathbb{C}^{r}. \ \ \ \ \ (2)$

Define

$\displaystyle v_{j}(z)=\frac{d^{j}}{dz^{j}}v(z) \ \ \ \ \ (3)$

and denote

$\displaystyle d_{j}(z)=\mbox{span}\{v_{1}(z),\cdots,v_{j}(z)\}. \ \ \ \ \ (4)$

Since ${\mbox{span}\{v_{1}(z),\cdots,v_{j}(z)\}}$ is a vector subspace of ${\mathbb{C}^{r}}$ for each ${z}$, ${d_{j}(z)\leq r}$ for all ${z}$ and for all ${j}$. Then there exists ${s}$ so that

$\displaystyle v_{s}(z)\in \mbox{span}\{v_{1}(z),\cdots,v_{s-1}(z)\}. \ \ \ \ \ (5)$

This shows that for each ${z}$, there exists ${c_{j}(z)\in\mathbb{C}^{r}}$ so that

$\displaystyle v_{s}(z)=-\sum_{j=1}^{s-1}c_{j}(z)v_{j}(z). \ \ \ \ \ (6)$

In other words, we obtain the following Picard-Fuchs equation:

$\displaystyle \frac{d^{s}}{dz^{s}}v(z)+\sum_{j=1}^{s-1}c_{j}(z)\frac{d^{j}}{dz^{j}}v(z)=0. \ \ \ \ \ (7)$

Multiplying (7) by ${z^{s}}$, we obtain another equation:

$\displaystyle D^{s}v(z)+\sum_{j=1}^{s-1}b_{j}(z)D^{j}v(z)=0, \ \ \ \ \ (8)$

where ${D}$ is the differential operator ${zd/dz}$. Equation (8) is called the logarithmic form of the Picard-Fuchs equation.

### Natural Representation of Fundamental groups on Cohomology Groups coming from Fibration

Let ${X}$ be a space. A self homotopy equivalence is a map ${f:X\rightarrow X}$ so that there is another map ${g:X\rightarrow X}$ with the property that ${f\circ g}$ and ${g\circ f}$ are both homotopic to the identity map on ${X}$. Let ${[f]}$ be the equivalent class of a self homotopy equivalence ${f:X\rightarrow X}$. The set of all homotopy equivalent classes forms a group. Let me denote this group by ${G(X)}$. Since a homotopy equivalence ${f:X\rightarrow X}$ induces isomorphisms of ${f^{*}:H^{n}(X)\rightarrow H^{n}(X)}$ cohomology of groups ${H^{n}(X)}$ for each ${n}$ with coefficient in any group ${A}$, we obtain representations:

$\displaystyle G(X)\rightarrow \mbox{Aut}(H^{n}(X)) \ \ \ \ \ (1)$

defined by ${[f]\mapsto f^{*}}$. Similarly, we have a representation of ${G(X)}$ in ${H_{n}(X)}$ defined by ${[f]\mapsto f_{*}}$.

A fibration is a continuous map ${p:E\rightarrow B}$ so that it has the homotopy lifting propery. Let ${p:E\rightarrow B}$ be a fibration. Then all fibers ${E_{b}}$ are homotopy equivalent. Every path ${\alpha:[0,1]\rightarrow B}$ defines a homotopy class ${\alpha_{*}}$ of homotopy equivalences ${E_{\alpha(0)}\rightarrow E_{\alpha(1)}}$ which depends only on the homotopy class of ${\alpha}$ rel to endpoints. Therefore every element in ${\pi_{1}(B,b_{0})}$ defines a homotopy class of self homotopy equivalence of ${E_{b_{0}}}$. Thus we have a group homomorphism:

$\displaystyle \pi_{1}(B,b_{0})\rightarrow G(B) \ \ \ \ \ (2)$

defined by ${\alpha\mapsto \alpha_{*}}$. Thus we obtain a representation of ${\pi_{1}(B,b_{0})}$ on ${H^{n}(B)}$:

$\displaystyle \pi_{1}(B,b_{0})\rightarrow H^{n}(B). \ \ \ \ \ (3)$

Similarly, we have a representation of ${\pi_{1}(B,b_{0})}$ on ${H_{n}(B)}$.

### A supplement note for Leray spectral sequence in Griffiths book: Local system

Let ${\pi:E\rightarrow B}$ be a differentiable fiber bundle whose fiber ${F}$ is a compact differentiable manifold. Choose an open set ${U}$ in ${B}$ diffeomorphic to ${\mathbb{R}^{k}}$ to obtain a diffeomorphism ${\pi^{-1}(U)\cong U\times F}$. The cohomology of ${\pi^{-1}(U)}$ can be computed by the K\”{u}nneth formula which states that

$\displaystyle H^{k}(X\times Y,\mathbb{Q})=\bigoplus_{i+j=k}H^{i}(X,\mathbb{Q})\otimes H^{j}(Y,\mathbb{Q}), \ \ \ \ \ (1)$

for any spaces ${X}$ and ${Y}$. Then we have an isomorphism ${H^{q}(\pi^{-1}(U),\mathbb{Q})\cong H^{q}(F,\mathbb{Q})}$. This isomorphism suggests two definitions. The first suggestion gives us the definition of the higher direct image sheaf. It is very natural to consider the sheaf associated with the presheaf ${U\mapsto H^{q}(\pi^{-1}(U),\mathbb{Q})}$. In general, given a continuous map ${f:X\rightarrow Y}$, the ${q}$-th direct image sheaf ${R^{q}f_{*}\mathcal{F}}$ over ${Y}$ is the sheaf associated with the presheaf

$\displaystyle U\mapsto H^{q}(f^{-1}(U),\mathcal{F}). \ \ \ \ \ (2)$

The second suggestion is the notion of locally constant sheaves (or a local system).

Let ${A}$ be an abelian group. On a space ${X}$, we define a presheaf ${\mathcal{A}}$ by ${U\mapsto \mathcal{A}(U)=A}$. The sheaf ${\mathcal{A}^{+}}$ associated with the presheaf ${\mathcal{A}}$ is called a constant sheaf. A locally constant sheaf ${\mathcal{F}}$ over a space ${X}$ is a sheaf over ${X}$ with the property that there exists an open cover ${\{U_{i}\}}$ such that the restriction ${ \mathcal{F}|_{U_{i}}}$ are constant sheaves for all ${i}$.

By ${H^{q}(\pi^{-1}(U),\mathbb{Q})\cong H^{q}(F,\mathbb{Q})}$, it is very natural to think that the ${q}$-th direct image sheaf ${R^{q}f_{*}\mathbb{Q}}$ is isomorphic to the constant sheaf given by ${H^{q}(F,\mathbb{Q})}$ but this is not always true. It does depend on the fundamental group ${\pi_{1}(B,x_{0})}$ and is a locally constant sheaf. Since ${\pi:E\rightarrow B}$ is a fibration, by the homotopy lifting properties of fibrations, an element ${\gamma\in \pi_{1}(B,x_{0})}$ gives a homotopy equivalence ${\gamma_{*}}$ on the fiber ${E_{x_{0}}}$ and thus determines an isomorphism on the cohomology group ${H^{n}(E_{x_{0}},M)}$ (and also the homology group) for any ${n}$ and for any abelian group ${M}$. From here, we obtain a representation of ${\pi_{1}(B,x_{0})}$ on ${H^{n}(E_{x_{0}},\mathbb{Q})}$. Read

Suppose that ${\rho:\pi_{1}(B,x_{0})\rightarrow \mbox{Aut}(V)}$ is a representation on a vector space ${V}$ and ${\widetilde{B}}$ is the universal covering space for ${B}$. The associated vector bundle ${V_{\rho}=\widetilde{B}\times_{\pi_{1}}V}$ gives a locally constant sheaf ${\mathcal{V}_{\rho}}$ on ${B}$ whose sections over an open set ${U}$ of ${B}$ are those which lift to constant sections of ${\widetilde{B}\times V}$. We call ${\mathcal{V}_{\rho}}$ the sheaf associated with the representation ${\rho:\pi_{1}(B,x_{0})\rightarrow \mbox{Aut}(V)}$. The direct image sheaf ${R^{q}f_{*}\mathbb{Q}}$ is the locally constant sheaf associated with the representation of ${\pi_{1}(B,x_{0})}$ on ${H^{q}(F_{x_{0}},\mathbb{Q})}$.

### Witten’s conjecture: Kontsevich’s Theorem

1. Witten’s Conjecture: Kontevich’s Theorem

Let ${\mathcal{M}_{g,n}}$ be the moduli space of smooth curves of genus ${g}$ with ${n}$-marked points and ${\bar{\mathcal{M}}_{g,n}}$ is its Deligne-Mumford compactification. A point in ${\bar{\mathcal{M}}_{g,n}}$ is of the form ${(C,x_{1},\cdots,x_{n})}$, where ${C}$ is a stable curve of genus ${g}$ and ${x_{i}}$ are smooth points on ${C}$. Let ${\mathbb{L}_{i}}$ be the line bundle over ${\bar{\mathcal{M}}_{g,n}}$ whose fiber over a point ${(C,x_{1},\cdots,x_{n})}$ is the cotangent line ${T_{x_{i}}^{*}C}$. Denote the Chern class of the line bundle ${\mathbb{L}_{i}}$ by ${\psi_{i}=c_{1}(\mathbb{L}_{i})}$. Introduce a sequence of variables ${\{\tau_{i}:i\geq 0\}}$. Define the intersection index:

$\displaystyle \langle \tau_{d_{1}}\cdots \tau_{d_{n}}\rangle = \int_{\bar{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots \psi_{n}^{d_{n}}, \ \ \ \ \ (1)$

where ${d_{1},\cdots,d_{n}}$ are nonnegative integers. If ${d_{1}+\cdots+d_{n}\neq 3g-3+n}$, we set ${\langle \tau_{d_{1}}\cdots \tau_{d_{n}}\rangle=0}$. Then we obtain a linear functional:

$\displaystyle \langle\cdot\rangle:\mathbb{Q}[\tau_{0},\tau_{1},\cdots]\rightarrow\mathbb{Q}. \ \ \ \ \ (2)$

Let ${\{t_{j}:j\geq 0\}}$ be another sequence of variables. Define a formal power series ${F}$ in the variable ${\{t_{i}\}}$ by

$\displaystyle F(t_{0},t_{1},\cdots)=\langle\exp\left(\sum_{i=0}^{\infty}t_{i}\tau_{i}\right)\rangle. \ \ \ \ \ (3)$

Witten states that the series ${F}$ coincides with the partition function in the standard matrix model theory and obeys the K.D.V hierarchy. The first equation is the classical KdV equation:

$\displaystyle \frac{\partial U}{\partial t_{1}}=U\frac{\partial U}{\partial t_{0}}+\frac{1}{12}\frac{\partial^{3}U}{\partial t_{0}^{3}}, \ \ \ \ \ (4)$

where ${U=\partial^{2}F/\partial t_{0}^{2}}$. In 1991, M. Kontevich proved the following theorem: The series ${\exp(F)}$ in variables ${T_{2i+1}=t_{i}/(2i+1)!!}$ is a ${\tau}$-function for the KdV hierarchy. It follows from this theorem that the Witten’s conjecture is true.