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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 thatx_{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}.

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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<i<n}\alpha_{i}'\otimes \alpha_{n-i}''

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.

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)}.

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})}.

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.

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