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Perhaps the proper beginning of Whitham theory is Whitham's work [[#References|[a75]]], [[#References|[a74]]], which can be viewed as a crucible of various averaging ideas subsequently developed in e.g. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a27]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a42]]], [[#References|[a49]]], [[#References|[a50]]], [[#References|[a61]]], [[#References|[a73]]] to theories involving multi-phase averaging, Hamiltonian systems and weakly deformed soliton lattices. The term "Whitham equations" then became associated with the moduli dynamics of Riemann surfaces and this fits naturally into work on topological field theories, Frobenius manifolds, renormalization groups, coupling constants, and Seiberg–Witten theory (cf. also [[Seiberg–Witten equations|Seiberg–Witten equations]]), along with singularity theory, isomonodromy deformations, quantum cohomology and [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300801.png" />-theory]], Gromov–Witten invariants, Witten–Dijkgraaf–Verlinde–Verlinde equations, etc. (see the references below or the survey material in [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]]).
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Perhaps the proper beginning of Whitham theory is Whitham's work [[#References|[a75]]], [[#References|[a74]]], which can be viewed as a crucible of various averaging ideas subsequently developed in e.g. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a27]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a42]]], [[#References|[a49]]], [[#References|[a50]]], [[#References|[a61]]], [[#References|[a73]]] to theories involving multi-phase averaging, Hamiltonian systems and weakly deformed soliton lattices. The term "Whitham equations" then became associated with the moduli dynamics of Riemann surfaces and this fits naturally into work on topological field theories, Frobenius manifolds, renormalization groups, coupling constants, and Seiberg–Witten theory (cf. also [[Seiberg–Witten equations|Seiberg–Witten equations]]), along with singularity theory, isomonodromy deformations, quantum cohomology and [[K-theory|$K$-theory]], Gromov–Witten invariants, Witten–Dijkgraaf–Verlinde–Verlinde equations, etc. (see the references below or the survey material in [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]]).
  
 
==Averaging.==
 
==Averaging.==
 
One of the most important applications of averaging theory and the Whitham equation is to the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]
 
One of the most important applications of averaging theory and the Whitham equation is to the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300802.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} u _ { t } - 6 u u _ { x } + u _ { xxx } = 0. \end{equation}
  
 
Key early papers on averaging for this equation include [[#References|[a27]]] and [[#References|[a49]]]. The basic ideas of G. Whitham are discussed in [[#References|[a74]]], [[#References|[a75]]]. Other important papers include [[#References|[a1]]], [[#References|[a2]]], [[#References|[a25]]], [[#References|[a65]]].
 
Key early papers on averaging for this equation include [[#References|[a27]]] and [[#References|[a49]]]. The basic ideas of G. Whitham are discussed in [[#References|[a74]]], [[#References|[a75]]]. Other important papers include [[#References|[a1]]], [[#References|[a2]]], [[#References|[a25]]], [[#References|[a65]]].
  
The key idea is averaging out fast scales; one introduces two scales: the "fast" scale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300803.png" /> and "slow" scale (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300805.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300806.png" /> small. One obtains a class of ( "finite-gap" ) solutions of the form
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The key idea is averaging out fast scales; one introduces two scales: the "fast" scale $( x , t )$ and "slow" scale ($X = \epsilon x$, $T = \epsilon t$), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300806.png"/> small. One obtains a class of ( "finite-gap" ) solutions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300807.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} u ( x , t ) = U = f _ { g } ( \theta _ { 1 } , \ldots , \theta _ { g } ), \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300808.png" /> is a [[Meromorphic function|meromorphic function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w1300809.png" /> variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008010.png" />, where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008013.png" /> depend only on the slow variables.
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where $f _ { g }$ is a [[Meromorphic function|meromorphic function]] of $g$ variables and $\theta _ { i } = \kappa _ { i } + \omega _ { i } + \widehat { \theta } _ { i }$, where the parameters $U$, $\kappa_i$, $\omega _ { i }$ depend only on the slow variables.
  
One can then write down the evolution equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008014.png" />-phase wave trains in terms of differentials on an associated Riemann surface.
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One can then write down the evolution equations for $g$-phase wave trains in terms of differentials on an associated Riemann surface.
  
 
The Whitham equation for the Korteweg–de Vries equation is given by
 
The Whitham equation for the Korteweg–de Vries equation is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \frac { \partial d \omega _ { 1 } } { \partial T } = \frac { \partial d \omega _ { 3 } } { \partial X }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008017.png" /> are Abelian differentials on the Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008018.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008019.png" /> (cf. also [[Differential on a Riemann surface|Differential on a Riemann surface]]; [[Abelian differential|Abelian differential]]; [[Riemann surface|Riemann surface]]), where
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008016.png"/> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008017.png"/> are Abelian differentials on the Riemann surface of genus $g$ given by $y ^ { 2 } = R _ { g } ( \lambda )$ (cf. also [[Differential on a Riemann surface|Differential on a Riemann surface]]; [[Abelian differential|Abelian differential]]; [[Riemann surface|Riemann surface]]), where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008020.png" /></td> </tr></table>
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\begin{equation*} R _ { g } ( \lambda ) = \prod _ { i = 0 } ^ { 2 g } ( \lambda - \lambda _ { i } ) \end{equation*}
  
and the branch points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008021.png" /> are real and are assumed to satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008022.png" />.
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and the branch points $\lambda _ { i }$ are real and are assumed to satisfy $\lambda _ { 0 } &lt; \ldots &lt; \lambda _ { 2 g }$.
  
 
Explicitly,
 
Explicitly,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a4} d \omega _ { 1 } ( \lambda ) = \frac { \prod _ { i = 1 } ^ { g } ( \lambda - \alpha _ { i } ) } { \sqrt { R _ { g } ( \lambda ) } } d \lambda \sim \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008024.png" /></td> </tr></table>
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\begin{equation*} \sim \frac { d \lambda } { \sqrt { \lambda } } + ( \text { holomorphic } ) , \text { as } \lambda \rightarrow \infty , \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} d \omega _ { 3 } ( \lambda ) = \frac { \lambda ^ { g + 1 } - \frac { 1 } { 2 } \sigma _ { 1 } \lambda ^ { g } + \beta _ { 1 } \lambda ^ { g - 1 } + \ldots + \beta _ { g } } { \sqrt { R _ { g } ( \lambda ) } } d \lambda \sim  \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008026.png" /></td> </tr></table>
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\begin{equation*} \sim \sqrt { \lambda } d \lambda + \text { (holomorphic), as } \lambda \rightarrow \infty. \end{equation*}
  
Here, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008027.png" /> are determined by
+
Here, the coefficients $\{ \alpha _ { j } , \beta _ { j } \}$ are determined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008028.png" /></td> </tr></table>
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\begin{equation*} \oint _ { A _ { j } } d \omega _ { 1 } = \oint _ { A _ { j } } d \omega _ { 3 } = 0 , j = 1 , \dots , g , \end{equation*}
  
the vanishing of the contour integral along the canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008029.png" />-cycle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008030.png" />.
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the vanishing of the contour integral along the canonical $A _ { j }$-cycle, and $\sigma _ { 1 } = \sum _ { i = 0 } ^ { 2 g } \lambda _ { i }$.
  
 
Then the averaged solution of the Korteweg–de Vries equation is given by
 
Then the averaged solution of the Korteweg–de Vries equation is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008031.png" /></td> </tr></table>
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\begin{equation*} \overline { u } ( x , t ) = \frac { 1 } { 2 } \sum _ { i = 0 } ^ { 2 g } \lambda _ { i } - \sum _ { j = 0 } ^ { g } \alpha _ { j }. \end{equation*}
  
Note that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008032.png" />, the equation reduces to the dispersionless Korteweg–de Vries equation (Hopf–Burgers equation) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008034.png" />, i.e.,
+
Note that when $g = 0$, the equation reduces to the dispersionless Korteweg–de Vries equation (Hopf–Burgers equation) with $\lambda _ { 0 } = 2 \overline { u }$ and $\lambda _ { 1 } = \ldots = \lambda _ { 2 g } = \alpha _ { 1 } = \ldots = \alpha _ { g } = 0$, i.e.,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008035.png" /></td> </tr></table>
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\begin{equation*} \frac { \partial \overline { u } } { \partial T } = \overline { u } \frac { \partial \overline { u } } { \partial X }. \end{equation*}
  
 
The Whitham equation for the discrete Toda lattice (cf. [[Toda lattices|Toda lattices]]) is treated in [[#References|[a4]]] where shock formation is analyzed. Shocks for the Korteweg–de Vries equation are analyzed in [[#References|[a34]]], [[#References|[a35]]].
 
The Whitham equation for the discrete Toda lattice (cf. [[Toda lattices|Toda lattices]]) is treated in [[#References|[a4]]] where shock formation is analyzed. Shocks for the Korteweg–de Vries equation are analyzed in [[#References|[a34]]], [[#References|[a35]]].
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==General Whitham theory.==
 
==General Whitham theory.==
More generally, for any compact [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008036.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008037.png" /> and point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008038.png" />, the Baker–Akhiezer function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008039.png" /> gives rise to a KP-hierarchy (cf. also [[KP-equation|KP-equation]]). In particular, following [[#References|[a11]]], [[#References|[a27]]], [[#References|[a42]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008040.png" /> can be written as
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More generally, for any compact [[Riemann surface|Riemann surface]] $\Sigma _ { g }$ of genus $g$ and point $Q \sim \infty$, the Baker–Akhiezer function $\psi$ gives rise to a KP-hierarchy (cf. also [[KP-equation|KP-equation]]). In particular, following [[#References|[a11]]], [[#References|[a27]]], [[#References|[a42]]], $\psi$ can be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a6} \psi ( P ) = \operatorname { exp } \left( \sum t _ { n } \Omega _ { n } \right) \phi \left( \sum t _ { n } \overset{\rightharpoonup}{ V } _ { n } , P \right) , \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008042.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008045.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008046.png" /> is periodic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008048.png" /> for slow times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008049.png" /> defined via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008051.png" /> is a point in the Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008052.png" /> (cf. also [[Jacobi variety|Jacobi variety]]). Assuming, for simplicity, that the periods are incommensurable, by ergodicity one finds
+
where $d \Omega _ { n } \sim d ( \lambda ^ { n } ) + \ldots$ near $\infty$ and $\int _ { A _ { i } } d \Omega _ { n } = 0$ with $\int _ { B _ { i } } d \Omega _ { n } = V _ { i n } \sim ( \overset{\rightharpoonup}{ V _ { n } } ) _ { i }$. Here, $\phi$ is periodic and $\Omega _ { n } = \Omega _ { n } ( T _ { m } )$ with $\overset{\rightharpoonup} { V } _ { n } = \overset{\rightharpoonup}  { V } _ { n } ( T _ { m } )$ for slow times $T _ { m }$ defined via $T _ { m } = \epsilon t _ { m }$ and $\overset{\rightharpoonup} { \theta } = \sum t _ { n } \overset{\rightharpoonup} { V } _ { n }$ is a point in the Jacobian $\operatorname { Jac } ( \Sigma _ { g } )$ (cf. also [[Jacobi variety|Jacobi variety]]). Assuming, for simplicity, that the periods are incommensurable, by ergodicity one finds
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008053.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { 2 L } \int _ { - L } ^ { L } \phi d t _ { i } = \langle \phi \rangle = \left( \frac { 1 } { 2 \pi } \right) ^ { 2 g } \int \ldots \int \phi d ^ { 2 g } \theta . \end{equation*}
  
With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008054.png" /> corresponding to the adjoint Baker–Akhiezer function, one can think now of multi-scale analysis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008056.png" /> plus averaging over the fast times (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008057.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008058.png" /> is canonically specified). This corresponds to looking at an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008059.png" /> expansion and setting the average first-order term to zero, leading to the Whitham equations
+
With $\psi ^ { * }$ corresponding to the adjoint Baker–Akhiezer function, one can think now of multi-scale analysis of $\psi \psi ^ { * } d \widetilde { \Omega }$ with $\partial _ { i } \rightarrow \partial _ { i } + \epsilon ( \partial / \partial T _ { i } )$ plus averaging over the fast times (here, $d \tilde { \Omega } = d \lambda + O ( \lambda ^ { - 2 } ) d \lambda$ near $\infty$ is canonically specified). This corresponds to looking at an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008059.png"/> expansion and setting the average first-order term to zero, leading to the Whitham equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\begin{equation} \tag{a7} \frac { \Omega _ { n } } { \partial T _ { m } } = \frac { \partial \Omega _ { m } } { \partial T _ { n } }. \end{equation}
  
 
==Seiberg–Witten theory.==
 
==Seiberg–Witten theory.==
Given a low-energy effective action for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008061.png" /> susy gauge theory with partition function
+
Given a low-energy effective action for an $N = 2$ susy gauge theory with partition function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\begin{equation} \tag{a8} Z ( t , \phi ) = \int _ { \phi _ { 0 } } \mathcal{D} \phi \operatorname { exp } [ S ( t , \phi ) ], \end{equation}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008063.png" /> fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008064.png" /> coupling constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008065.png" /> gauge group in the background, it turns out (e.g. in matrix models) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008066.png" /> will often be a tau-function of KP–Toda type via Ward identities and Virasoro (origin of integrability). Recall that tau-functions are basic ingredients in integrable system theory (cf. also [[KP-equation|KP-equation]]; [[Toda lattices|Toda lattices]]) and e.g.
+
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008063.png"/> fields, $t \sim $ coupling constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008065.png"/> gauge group in the background, it turns out (e.g. in matrix models) that $Z ( t , \phi )$ will often be a tau-function of KP–Toda type via Ward identities and Virasoro (origin of integrability). Recall that tau-functions are basic ingredients in integrable system theory (cf. also [[KP-equation|KP-equation]]; [[Toda lattices|Toda lattices]]) and e.g.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} \psi = \frac { \operatorname { exp } \left( \sum t _ { n } \lambda ^ { n } \right) \tau ( t_{ j} - ( 1 / j \lambda ^ { j } ) ) } { \tau ( t _ { j } ) }. \end{equation}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008068.png" /> one has an effective (classical-type) dynamics in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008069.png" /> variables and averaging corresponds in some sense to suppressing fast oscillations (which suggests a renormalization procedure); alternatively, it is also in some sense related to a quantization procedure in the first [[WKBJ approximation|WKBJ approximation]], which produces slow dynamics on the action variables (Hamiltonians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008070.png" /> Casimirs from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008071.png" />; cf. also [[Casimir element|Casimir element]]; [[Kac–Moody algebra|Kac–Moody algebra]]), which is equivalent in many situations to dynamics on the moduli of the underlying spectral curves. Thus, the quantum arena shifts to the quantum moduli space and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008072.png" /> appear as renormalized coupling constants in one approach and as deformation parameters of moduli in another. The tau-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008073.png" /> goes to a quasi-classical tau-function whose logarithm (after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008074.png" /> adjustment) is called the pre-potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008075.png" /> and this serves as a generating function for correlators and as a vehicle for expressing further renormalization effects. Consider (cf. [[#References|[a31]]], [[#References|[a32]]], [[#References|[a33]]], [[#References|[a37]]], [[#References|[a38]]], [[#References|[a39]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a62]]]) the following example of Seiberg–Witten Toda curves for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008076.png" /> susy Yang–Mills with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008078.png" />, no masses and moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008079.png" /> quantum moduli space of inequivalent vacua:
+
For $Z \sim \tau$ one has an effective (classical-type) dynamics in the $t$ variables and averaging corresponds in some sense to suppressing fast oscillations (which suggests a renormalization procedure); alternatively, it is also in some sense related to a quantization procedure in the first [[WKBJ approximation|WKBJ approximation]], which produces slow dynamics on the action variables (Hamiltonians $\sim$ Casimirs from $\widetilde{ g } = \text { Lie } ( G )$; cf. also [[Casimir element|Casimir element]]; [[Kac–Moody algebra|Kac–Moody algebra]]), which is equivalent in many situations to dynamics on the moduli of the underlying spectral curves. Thus, the quantum arena shifts to the quantum moduli space and the $T _ { n }$ appear as renormalized coupling constants in one approach and as deformation parameters of moduli in another. The tau-function $\tau$ goes to a quasi-classical tau-function whose logarithm (after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008074.png"/> adjustment) is called the pre-potential $F$ and this serves as a generating function for correlators and as a vehicle for expressing further renormalization effects. Consider (cf. [[#References|[a31]]], [[#References|[a32]]], [[#References|[a33]]], [[#References|[a37]]], [[#References|[a38]]], [[#References|[a39]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a62]]]) the following example of Seiberg–Witten Toda curves for $\mathcal{N} = 2$ susy Yang–Mills with $G = \operatorname {SU} ( N )$, $N = N _ { c }$, no masses and moduli $u _ { k } \in \mathcal{M} =$ quantum moduli space of inequivalent vacua:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
+
\begin{equation} \tag{a10} y ^ { 2 } = P ^ { 2 } - 4 \Lambda ^ { 2 N }, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008081.png" /></td> </tr></table>
+
\begin{equation*} y = \Lambda ^ { N } \left( w - \frac { 1 } { w } \right) , P = \lambda ^ { N } - \sum _ { 2 } ^ { N } u _ { k } \lambda ^ { N - k } = \Lambda ^ { N } \left( w + \frac { 1 } { w } \right) . \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008082.png" /> is the quantum scale, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008083.png" /> is a local coordinate at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008084.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008085.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008086.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008088.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008089.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008090.png" />. One defines
+
Here, $\lambda$ is the quantum scale, $\xi $ is a local coordinate at $\infty_{\pm}$ with $\Lambda \xi \sim w ^ {\mp ( 1 / N ) }$ with $w \rightarrow \infty$ at $\infty _+$ and $w \rightarrow 0$ at $\infty _-$, and $g = N - 1$. One defines
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008091.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
+
\begin{equation} \tag{a11} d \hat { \Omega } _ { n } = P _ { + } ^ { n / N } \left( \frac { d w } { w } \right) \end{equation}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008092.png" /></td> </tr></table>
+
\begin{equation*} d \Omega _ { n } = d \hat { \Omega } _ { n } - \sum _ { 1 } g \left( \oint _ { A _ { j } } d \hat { \Omega} _ { n }  \right) d \omega _ { j } \end{equation*}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008093.png" /> for technical reasons and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008094.png" /> holomorphic differentials). The standard Whitham theory is now based on
+
($n &lt; 2 N$ for technical reasons and $d \omega_{j} \sim$ holomorphic differentials). The standard Whitham theory is now based on
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008095.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
+
\begin{equation} \tag{a12} d S = \sum _ { 1 } ^ { M } T _ { n } d \widehat { \Omega } _ { n } = \sum _ { 1 } ^ { M } T _ { n } d \Omega _ { n } + \sum _ { 1 } ^ { g } \alpha _ { j } d \omega _ { j }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008097.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008098.png" />. One has then Whitham equations
+
where $M &lt; 2 N$ and $T _ { 0 } = 0$ for $N _ { f } = 0$. One has then Whitham equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w13008099.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
+
\begin{equation} \tag{a13} \frac { \partial d \Omega _ { A } } { \partial T _ { B } } = \frac { \partial d \Omega _ { B } } { \partial T _ { A } } \end{equation}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080101.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080102.png" /> independent. The pre-potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080103.png" /> arises via
+
with $\partial d S / \partial \alpha_j = d \omega_j$ and $\partial d S / \partial T _ { n } = d \omega _ { n }$ for $( T _ { n } , \alpha _ { j } )$ independent. The pre-potential $F$ arises via
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
+
\begin{equation} \tag{a14} \frac { \partial F } { \partial \alpha _ { j } } = \oint _ { B _ { j } } d S \end{equation}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080106.png" /> involves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080107.png" /> and the Seiberg–Witten differential is
+
and $\partial _ { n } F = ( 1 / 2 \pi i n ) \operatorname { Res } _ { 0 } \xi ^ { - n } d S$, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080106.png"/> involves $\infty_{\pm}$ and the Seiberg–Witten differential is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
+
\begin{equation} \tag{a15} d S _ { S W } = d \widehat { \Omega } _ { 1 } = \lambda \left( \frac { d w } { w } \right) = \lambda \frac { d P } { y } = \lambda \frac { d y } { P }. \end{equation}
  
Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080109.png" /> one has the Seiberg–Witten situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080110.png" /> and one writes then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080111.png" />.
+
Thus, for $T _ { n } = \delta _ { n , 1 }$ one has the Seiberg–Witten situation $F ^ { \text{SW} } = \widetilde { F }$ and one writes then also $a _ { i } = \alpha _ { i }$.
  
 
==General framework.==
 
==General framework.==
The Whitham formulation of I. Krichever, developed in great detail with D.H. Phong (cf. [[#References|[a43]]], [[#References|[a44]]], [[#References|[a45]]], [[#References|[a46]]], [[#References|[a47]]]), involves a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080112.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080113.png" /> punctures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080114.png" />. One picks in an ad hoc manner two Abelian differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080116.png" /> having certain properties and sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080117.png" /> as a Seiberg–Witten-type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080118.png" /> with corresponding differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080119.png" />. For suitable choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080121.png" /> the formulation is adequate for Seiberg–Witten-type situations and topological field theories with Witten–Dijkgraaf–Verlinde–Verlinde equations will arise as well.
+
The Whitham formulation of I. Krichever, developed in great detail with D.H. Phong (cf. [[#References|[a43]]], [[#References|[a44]]], [[#References|[a45]]], [[#References|[a46]]], [[#References|[a47]]]), involves a Riemann surface $\Sigma _ { g }$ with $M$ punctures $P _ { \alpha }$. One picks in an ad hoc manner two Abelian differentials $d E$ and $d Q$ having certain properties and sets $d S = Q d E$ as a Seiberg–Witten-type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times $T _ { A }$ with corresponding differentials $d \Omega _ { A }$. For suitable choices of $d E$ and $d Q$ the formulation is adequate for Seiberg–Witten-type situations and topological field theories with Witten–Dijkgraaf–Verlinde–Verlinde equations will arise as well.
  
 
==Soft susy breaking.==
 
==Soft susy breaking.==
 
There is another role for Whitham times, via (cf. [[#References|[a26]]], [[#References|[a55]]])
 
There is another role for Whitham times, via (cf. [[#References|[a26]]], [[#References|[a55]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080122.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>
+
\begin{equation} \tag{a16} \hat{T} _ { n } = T _ { n } T _ { 1 } ^ { - 1 } , \hat { u } _ { k } = T _ { 1 } ^ { k } u _ { k }, \end{equation}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080123.png" /> (note <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080124.png" /> in the Seiberg–Witten situation). Then one defines
+
and $\hat { a } _ { i } = \alpha _ { i } ( u _ { k } , T _ { 1 } , \hat{T} _ { n &gt; 1 } = 0 ) = T _ { 1 } a _ { i } ( u _ { k } , \Lambda = 1 ) = a _ { i } ( \hat { u } _ { k } , \Lambda = T _ { 1 } )$ (note $T _ { 1 } \sim \Lambda$ in the Seiberg–Witten situation). Then one defines
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080125.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>
+
\begin{equation} \tag{a17} s _ { 1 } = - i \operatorname { log } ( \lambda ) \end{equation}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080126.png" /> and these are promoted to spurion superfields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080128.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080129.png" /> superfield language (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080131.png" /> are Grassmann variables while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080133.png" /> are auxiliary fields). One has a family of non-susy theories and soft susy breaking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080134.png" /> is achieved by fixing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080135.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080136.png" /> and using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080138.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080139.png" />) as susy breaking parameters (actually, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080140.png" /> alone will suffice). In any event, one can develop formulas involving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080143.png" /> derivatives of the pre-potential and eventually parametrize soft susy breaking terms induced by all of the Casimirs.
+
and $s _ { n } = - i \hat{T} _ { n }$ and these are promoted to spurion superfields $\mathcal{S} _ { n } = s _ { n } + \theta ^ { 2 } F _ { n }$ and $V _ { n } = ( 1 / 2 ) D _ { n } \theta ^ { 2 } \overline { \theta } ^ { 2 }$ in $\mathcal{N} = 1$ superfield language ($\theta$ and $\overline{\theta}$ are Grassmann variables while $D _ { n }$ and $F _ { n }$ are auxiliary fields). One has a family of non-susy theories and soft susy breaking $\mathcal{N} = 2 \rightarrow \mathcal{N} = 0$ is achieved by fixing $s _ { n } = 0$ for $n &gt; 1$ and using $D _ { n }$, $F _ { n }$ ($n \geq 1$) as susy breaking parameters (actually, the $F _ { n }$ alone will suffice). In any event, one can develop formulas involving $\lambda$, $\tilde{T} _ { n }$ and $\alpha_j$ derivatives of the pre-potential and eventually parametrize soft susy breaking terms induced by all of the Casimirs.
  
 
==Isomonodromy.==
 
==Isomonodromy.==
Various isomonodromy problems can be treated by multi-scale analysis to produce results indicating that isomonodromy deformations in WKB approximation correspond to modulation of isospectral problems (with Whitham-type equations as modulation equations). One can generate a pre-potential, period integrals, etc. as in Seiberg–Witten theory (see e.g. [[#References|[a66]]], [[#References|[a67]]], [[#References|[a68]]], [[#References|[a69]]], [[#References|[a70]]]). There are also isomonodromy connections to the Knizhnik–Zamolodchikov–Bernard equations (cf. [[#References|[a51]]], [[#References|[a52]]], [[#References|[a53]]], [[#References|[a63]]]); these equations arise in various ways in conformal field theory, geometric quantization of flat bundles, etc. Here one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080144.png" /> as flat vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080145.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080146.png" /> and smooth connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080147.png" />. "Flat" means zero curvature and with an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080148.png" /> this has the form
+
Various isomonodromy problems can be treated by multi-scale analysis to produce results indicating that isomonodromy deformations in WKB approximation correspond to modulation of isospectral problems (with Whitham-type equations as modulation equations). One can generate a pre-potential, period integrals, etc. as in Seiberg–Witten theory (see e.g. [[#References|[a66]]], [[#References|[a67]]], [[#References|[a68]]], [[#References|[a69]]], [[#References|[a70]]]). There are also isomonodromy connections to the Knizhnik–Zamolodchikov–Bernard equations (cf. [[#References|[a51]]], [[#References|[a52]]], [[#References|[a53]]], [[#References|[a63]]]); these equations arise in various ways in conformal field theory, geometric quantization of flat bundles, etc. Here one takes $F B ( \Sigma _ { g } , G )$ as flat vector bundles over $\Sigma _ { g }$ with $G = \operatorname{GL} ( N ,\bf C )$ and smooth connections $\mathcal{A} \sim ( A , \overline { A } )$. "Flat" means zero curvature and with an arbitrary $\kappa$ this has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080149.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>
+
\begin{equation} \tag{a18} ( \kappa \partial + A ) \psi = 0 \end{equation}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080150.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080151.png" /> (Beltrami differentials), so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080152.png" /> and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080153.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080154.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080155.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080156.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080157.png" />. Then (a18) becomes
+
and $( \overline { \partial } + \overline { A } ) \psi = 0$. Let $\mu \in \Omega ^ { - 1,1 } ( \Sigma _ { g } )$ (Beltrami differentials), so $\mu = \mu ( z , \bar{z} ) \partial _ { \bar{z} } \otimes d \bar{z}$ and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080153.png"/>, where $\text{l} = 3 g - 3$ ($g &gt; 1$) and $\mu _ { a } ^ { 0 }$ is a basis in $T \mathcal{M} _ { g }$. Then (a18) becomes
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080158.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a19)</td></tr></table>
+
\begin{equation} \tag{a19} ( \kappa \partial + A ) \psi = 0 \end{equation}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080159.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080160.png" /> be a homotopically non-trivial cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080161.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080162.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080163.png" /> and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080164.png" /> (path-ordered exponential), which yields a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080165.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080166.png" />. The independence of monodromy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080167.png" /> to complex structure deformation corresponds to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080168.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080169.png" />. Compatibility with (a19) requires
+
and $( \overline { \partial } + \mu \partial + \overline { A } ) \psi = 0$. Let $\gamma$ be a homotopically non-trivial cycle in $\Sigma _ { g }$ such that $( z_0 , \overline{z}_0 ) \in \gamma$ with $\psi ( z _ { 0 } , \overline{z} _ { 0 } ) = I$ and write $\mathcal{Y} ( \gamma ) = \psi ( z _ { 0 } , \overline{z} _ { 0 } ) | _ { \gamma } = P \operatorname { exp } ( \oint _ { \gamma } \mathcal{A} )$ (path-ordered exponential), which yields a representation of $\Pi _ { 1 } ( \Sigma _ { g } , z _ { 0 } )$ in $\operatorname {GL} ( N , \mathbf{C} )$. The independence of monodromy $\mathcal{Y}$ to complex structure deformation corresponds to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080168.png"/> for $a = 1 , \dots , \text{l}$. Compatibility with (a19) requires
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a20)</td></tr></table>
+
\begin{equation} \tag{a20} \partial _ { a } A = 0 \text { and } \partial \overline { A } = ( 1 / \kappa ) A \mu _ { a } ^ { 0 }. \end{equation}
  
These equations are Hamiltonian when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080171.png" /> has a symplectic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080172.png" /> with Hamiltonians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080173.png" />. Consider the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080174.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080175.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080176.png" /> (using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080177.png" /> as local coordinates). A gauge fixing plus flatness corresponds to reduction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080178.png" /> and one can (via WZW theory) fix the gauge to get a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080179.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080180.png" /> and equations
+
These equations are Hamiltonian when $F B ( \sigma _ { g } , G )$ has a symplectic form $\omega ^ { 0 } = \int \Sigma _ { g } \langle \delta A , \delta \overline { A } \rangle$ with Hamiltonians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080173.png"/>. Consider the bundle $\mathcal{P}$ over $\mathcal{M} _ { g }$ with fibre $F B$ (using $( A , \overline { A } , t \sim t _ { a } )$ as local coordinates). A gauge fixing plus flatness corresponds to reduction from $F B \rightarrow \widetilde { F B }$ and one can (via WZW theory) fix the gauge to get a bundle $\tilde {\cal  P }$ with fibre $\widetilde { F B }$ and equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080181.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a21)</td></tr></table>
+
\begin{equation} \tag{a21} ( \kappa \partial + L ) \psi = 0 \end{equation}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080182.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080183.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080184.png" /> comes from the gauge transformation. Putting in the canonical form via local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080185.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080186.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080187.png" />, one can write
+
with $( \overline { \partial } + \mu \partial + \overline{L}) \psi = 0$ and $( \kappa \partial _ { a} + M _ { a  } ) \psi = 0$, where $M _ { a }$ comes from the gauge transformation. Putting in the canonical form via local coordinates $( v _ { i } , u _ { i } )$ in $\widetilde { F B }$, where $i = 1 , \dots , M = ( N ^ { 2 } - 1 ) ( g - 1 )$, one can write
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080188.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a22)</td></tr></table>
+
\begin{equation} \tag{a22} \omega ^ { 0 } = ( \delta v , \delta u ) \end{equation}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080189.png" />. Using the Poincaré–Cartan invariant form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080190.png" /> there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080191.png" /> vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080192.png" /> which annihilate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080193.png" />. With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080194.png" />-structure this gives
+
with $\omega = \omega ^ { 0 } - ( 1 / \kappa ) \sum \delta H _ { \alpha } \delta t _ { \alpha }$. Using the Poincaré–Cartan invariant form $\Theta = ( u , \delta v ) - ( 1 / \kappa ) \sum H _ { a } \delta t _ { a }$ there exist $3 g - 3$ vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080192.png"/> which annihilate $\Theta$. With $\{ .\}_0 \sim \omega ^ { 0 }$-structure this gives
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080195.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a23)</td></tr></table>
+
\begin{equation} \tag{a23} \kappa \partial _ { s } H _ { r } - \kappa \partial _ { r } H _ { s } + \{ H _ { s } , H _ { r } \} _ { 0 } = 0. \end{equation}
  
These equations define flat connections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080196.png" /> and are referred to as a Whitham hierarchy of isomonodromic deformations. For a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080197.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080198.png" /> they take the form
+
These equations define flat connections in $\tilde {\cal  P }$ and are referred to as a Whitham hierarchy of isomonodromic deformations. For a given $f ( u , v , t )$ on $\tilde {\cal  P }$ they take the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080199.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a24)</td></tr></table>
+
\begin{equation} \tag{a24} \frac { d f } { d t _ { s } } = \kappa \partial _ { s } f + \{ H _ { s } , f \} \end{equation}
  
and one can introduce a pre-potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080200.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080201.png" /> giving Hamilton–Jacobi equations (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]])
+
and one can introduce a pre-potential $F$ on $\tilde {\cal  P }$ giving Hamilton–Jacobi equations (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a25)</td></tr></table>
+
\begin{equation} \tag{a25} \kappa \partial _ { s } F + H _ { s } \left( \frac { \delta F } { \delta u } , u , t \right) = 0. \end{equation}
  
Thus, one has a derivation of deformation equations, properly referred to as a Whitham hierarchy, which involves no averaging or multi-scale analysis. One can also compare the Baker–Akhiezer function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080203.png" /> in the Whitham hierarchy of isomonodromic deformations with elements of a certain Hitchin hierarchy (cf. also [[Hitchin system|Hitchin system]]) using a WKB approximation with fast times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080204.png" /> and slow times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080205.png" />.
+
Thus, one has a derivation of deformation equations, properly referred to as a Whitham hierarchy, which involves no averaging or multi-scale analysis. One can also compare the Baker–Akhiezer function $\psi$ in the Whitham hierarchy of isomonodromic deformations with elements of a certain Hitchin hierarchy (cf. also [[Hitchin system|Hitchin system]]) using a WKB approximation with fast times $t _ { S } ^ { H }$ and slow times $T _ { S } \sim t _ { s }$.
  
 
==Contact terms.==
 
==Contact terms.==
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080206.png" /> susy gauge theory on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080207.png" />-manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080208.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080209.png" />-plane integral for, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080210.png" /> situations, which can be related to a Toda theory with fast and slow (Whitham) times (cf. [[#References|[a55]]], [[#References|[a56]]], [[#References|[a57]]], [[#References|[a58]]], [[#References|[a59]]], [[#References|[a71]]], [[#References|[a72]]]).
+
For $\mathcal{N} = 2$ susy gauge theory on a $4$-manifold with $b _ { 2 + }  = 1$ there is a $u$-plane integral for, say, $\operatorname{SU} ( N )$ situations, which can be related to a Toda theory with fast and slow (Whitham) times (cf. [[#References|[a55]]], [[#References|[a56]]], [[#References|[a57]]], [[#References|[a58]]], [[#References|[a59]]], [[#References|[a71]]], [[#References|[a72]]]).
  
 
==Witten–Dijkgraaf–Verlinde–Verlinde.==
 
==Witten–Dijkgraaf–Verlinde–Verlinde.==
There is a beautiful and elaborate theory of B. Dubrovin and others based on Frobenius manifolds (cf. [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]]). This approach is especially pleasing since there is a great deal of motivation and natural structure. There are many connections to mathematics and physics and this approach has led to extensive development in Frobenius manifolds, quantum cohomology and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080211.png" />-theory, singularity theory, Witten–Dijkgraaf–Verlinde–Verlinde, etc. (see e.g. [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a30]]], [[#References|[a54]]]). A simple Hurwitz-space Korteweg–de Vries–Landau–Ginsburg model is as follows.
+
There is a beautiful and elaborate theory of B. Dubrovin and others based on Frobenius manifolds (cf. [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]]). This approach is especially pleasing since there is a great deal of motivation and natural structure. There are many connections to mathematics and physics and this approach has led to extensive development in Frobenius manifolds, quantum cohomology and $K$-theory, singularity theory, Witten–Dijkgraaf–Verlinde–Verlinde, etc. (see e.g. [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a30]]], [[#References|[a54]]]). A simple Hurwitz-space Korteweg–de Vries–Landau–Ginsburg model is as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080212.png" /> be the moduli space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080213.png" /> gap Korteweg–de Vries solutions based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080214.png" /> with ramification based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080215.png" />. One defines Whitham times
+
Let ${\cal M} _ { g , n  + 1}$ be the moduli space of $g$ gap Korteweg–de Vries solutions based on $L = \partial ^ { n + 1 } - q _ { 1 } \partial ^ { n - 1 } - \ldots - q _ { n }$ with ramification based on $W = p ^ { n + 1 } - q _ { 1 } p ^ { n - 1 } - \ldots - q _ { n }$. One defines Whitham times
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080216.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a26)</td></tr></table>
+
\begin{equation} \tag{a26} T _ { i } = - \frac { n + 1 } { n + 1 - i } \operatorname { Res } _ { \infty } W ^ { 1 - [ i / ( n + 1 ) ] } d p, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080217.png" /></td> </tr></table>
+
\begin{equation*} T _ { n + \alpha } = \frac { 1 } { 2 \pi i } \oint _ { A _ { \alpha } } p d W , T _ { g + n + \alpha } = \oint _ { B _ { \alpha } } d p, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080218.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130080/w130080219.png" />. These are flat times for a certain metric and determine a Whitham hierarchy, a Frobenius manifold and a topological field theory of Landau–Ginsburg type satisfying the Witten–Dijkgraaf–Verlinde–Verlinde equations (associativity equations for related field correlators).
+
where $1 \leq i \leq n$ and $1 \leq \alpha \leq g$. These are flat times for a certain metric and determine a Whitham hierarchy, a Frobenius manifold and a topological field theory of Landau–Ginsburg type satisfying the Witten–Dijkgraaf–Verlinde–Verlinde equations (associativity equations for related field correlators).
  
 
====References====
 
====References====
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Revision as of 17:43, 1 July 2020

Perhaps the proper beginning of Whitham theory is Whitham's work [a75], [a74], which can be viewed as a crucible of various averaging ideas subsequently developed in e.g. [a3], [a4], [a12], [a13], [a14], [a27], [a28], [a29], [a42], [a49], [a50], [a61], [a73] to theories involving multi-phase averaging, Hamiltonian systems and weakly deformed soliton lattices. The term "Whitham equations" then became associated with the moduli dynamics of Riemann surfaces and this fits naturally into work on topological field theories, Frobenius manifolds, renormalization groups, coupling constants, and Seiberg–Witten theory (cf. also Seiberg–Witten equations), along with singularity theory, isomonodromy deformations, quantum cohomology and $K$-theory, Gromov–Witten invariants, Witten–Dijkgraaf–Verlinde–Verlinde equations, etc. (see the references below or the survey material in [a5], [a6], [a7], [a8], [a9], [a10]).

Averaging.

One of the most important applications of averaging theory and the Whitham equation is to the Korteweg–de Vries equation

\begin{equation} \tag{a1} u _ { t } - 6 u u _ { x } + u _ { xxx } = 0. \end{equation}

Key early papers on averaging for this equation include [a27] and [a49]. The basic ideas of G. Whitham are discussed in [a74], [a75]. Other important papers include [a1], [a2], [a25], [a65].

The key idea is averaging out fast scales; one introduces two scales: the "fast" scale $( x , t )$ and "slow" scale ($X = \epsilon x$, $T = \epsilon t$), small. One obtains a class of ( "finite-gap" ) solutions of the form

\begin{equation} \tag{a2} u ( x , t ) = U = f _ { g } ( \theta _ { 1 } , \ldots , \theta _ { g } ), \end{equation}

where $f _ { g }$ is a meromorphic function of $g$ variables and $\theta _ { i } = \kappa _ { i } + \omega _ { i } + \widehat { \theta } _ { i }$, where the parameters $U$, $\kappa_i$, $\omega _ { i }$ depend only on the slow variables.

One can then write down the evolution equations for $g$-phase wave trains in terms of differentials on an associated Riemann surface.

The Whitham equation for the Korteweg–de Vries equation is given by

\begin{equation} \tag{a3} \frac { \partial d \omega _ { 1 } } { \partial T } = \frac { \partial d \omega _ { 3 } } { \partial X }, \end{equation}

where and are Abelian differentials on the Riemann surface of genus $g$ given by $y ^ { 2 } = R _ { g } ( \lambda )$ (cf. also Differential on a Riemann surface; Abelian differential; Riemann surface), where

\begin{equation*} R _ { g } ( \lambda ) = \prod _ { i = 0 } ^ { 2 g } ( \lambda - \lambda _ { i } ) \end{equation*}

and the branch points $\lambda _ { i }$ are real and are assumed to satisfy $\lambda _ { 0 } < \ldots < \lambda _ { 2 g }$.

Explicitly,

\begin{equation} \tag{a4} d \omega _ { 1 } ( \lambda ) = \frac { \prod _ { i = 1 } ^ { g } ( \lambda - \alpha _ { i } ) } { \sqrt { R _ { g } ( \lambda ) } } d \lambda \sim \end{equation}

\begin{equation*} \sim \frac { d \lambda } { \sqrt { \lambda } } + ( \text { holomorphic } ) , \text { as } \lambda \rightarrow \infty , \end{equation*}

\begin{equation} \tag{a5} d \omega _ { 3 } ( \lambda ) = \frac { \lambda ^ { g + 1 } - \frac { 1 } { 2 } \sigma _ { 1 } \lambda ^ { g } + \beta _ { 1 } \lambda ^ { g - 1 } + \ldots + \beta _ { g } } { \sqrt { R _ { g } ( \lambda ) } } d \lambda \sim \end{equation}

\begin{equation*} \sim \sqrt { \lambda } d \lambda + \text { (holomorphic), as } \lambda \rightarrow \infty. \end{equation*}

Here, the coefficients $\{ \alpha _ { j } , \beta _ { j } \}$ are determined by

\begin{equation*} \oint _ { A _ { j } } d \omega _ { 1 } = \oint _ { A _ { j } } d \omega _ { 3 } = 0 , j = 1 , \dots , g , \end{equation*}

the vanishing of the contour integral along the canonical $A _ { j }$-cycle, and $\sigma _ { 1 } = \sum _ { i = 0 } ^ { 2 g } \lambda _ { i }$.

Then the averaged solution of the Korteweg–de Vries equation is given by

\begin{equation*} \overline { u } ( x , t ) = \frac { 1 } { 2 } \sum _ { i = 0 } ^ { 2 g } \lambda _ { i } - \sum _ { j = 0 } ^ { g } \alpha _ { j }. \end{equation*}

Note that when $g = 0$, the equation reduces to the dispersionless Korteweg–de Vries equation (Hopf–Burgers equation) with $\lambda _ { 0 } = 2 \overline { u }$ and $\lambda _ { 1 } = \ldots = \lambda _ { 2 g } = \alpha _ { 1 } = \ldots = \alpha _ { g } = 0$, i.e.,

\begin{equation*} \frac { \partial \overline { u } } { \partial T } = \overline { u } \frac { \partial \overline { u } } { \partial X }. \end{equation*}

The Whitham equation for the discrete Toda lattice (cf. Toda lattices) is treated in [a4] where shock formation is analyzed. Shocks for the Korteweg–de Vries equation are analyzed in [a34], [a35].

The discrete Ablowitz–Ladik equations are analyzed in [a60].

The Whitham equations are also important in the analysis of the non-linear Schrödinger equation (cf. also Benjamin–Feir instability) and non-linear optics, see for example [a36], [a48], [a64] and references therein.

General Whitham theory.

More generally, for any compact Riemann surface $\Sigma _ { g }$ of genus $g$ and point $Q \sim \infty$, the Baker–Akhiezer function $\psi$ gives rise to a KP-hierarchy (cf. also KP-equation). In particular, following [a11], [a27], [a42], $\psi$ can be written as

\begin{equation} \tag{a6} \psi ( P ) = \operatorname { exp } \left( \sum t _ { n } \Omega _ { n } \right) \phi \left( \sum t _ { n } \overset{\rightharpoonup}{ V } _ { n } , P \right) , \end{equation}

where $d \Omega _ { n } \sim d ( \lambda ^ { n } ) + \ldots$ near $\infty$ and $\int _ { A _ { i } } d \Omega _ { n } = 0$ with $\int _ { B _ { i } } d \Omega _ { n } = V _ { i n } \sim ( \overset{\rightharpoonup}{ V _ { n } } ) _ { i }$. Here, $\phi$ is periodic and $\Omega _ { n } = \Omega _ { n } ( T _ { m } )$ with $\overset{\rightharpoonup} { V } _ { n } = \overset{\rightharpoonup} { V } _ { n } ( T _ { m } )$ for slow times $T _ { m }$ defined via $T _ { m } = \epsilon t _ { m }$ and $\overset{\rightharpoonup} { \theta } = \sum t _ { n } \overset{\rightharpoonup} { V } _ { n }$ is a point in the Jacobian $\operatorname { Jac } ( \Sigma _ { g } )$ (cf. also Jacobi variety). Assuming, for simplicity, that the periods are incommensurable, by ergodicity one finds

\begin{equation*} \frac { 1 } { 2 L } \int _ { - L } ^ { L } \phi d t _ { i } = \langle \phi \rangle = \left( \frac { 1 } { 2 \pi } \right) ^ { 2 g } \int \ldots \int \phi d ^ { 2 g } \theta . \end{equation*}

With $\psi ^ { * }$ corresponding to the adjoint Baker–Akhiezer function, one can think now of multi-scale analysis of $\psi \psi ^ { * } d \widetilde { \Omega }$ with $\partial _ { i } \rightarrow \partial _ { i } + \epsilon ( \partial / \partial T _ { i } )$ plus averaging over the fast times (here, $d \tilde { \Omega } = d \lambda + O ( \lambda ^ { - 2 } ) d \lambda$ near $\infty$ is canonically specified). This corresponds to looking at an expansion and setting the average first-order term to zero, leading to the Whitham equations

\begin{equation} \tag{a7} \frac { \Omega _ { n } } { \partial T _ { m } } = \frac { \partial \Omega _ { m } } { \partial T _ { n } }. \end{equation}

Seiberg–Witten theory.

Given a low-energy effective action for an $N = 2$ susy gauge theory with partition function

\begin{equation} \tag{a8} Z ( t , \phi ) = \int _ { \phi _ { 0 } } \mathcal{D} \phi \operatorname { exp } [ S ( t , \phi ) ], \end{equation}

with fields, $t \sim $ coupling constants and gauge group in the background, it turns out (e.g. in matrix models) that $Z ( t , \phi )$ will often be a tau-function of KP–Toda type via Ward identities and Virasoro (origin of integrability). Recall that tau-functions are basic ingredients in integrable system theory (cf. also KP-equation; Toda lattices) and e.g.

\begin{equation} \tag{a9} \psi = \frac { \operatorname { exp } \left( \sum t _ { n } \lambda ^ { n } \right) \tau ( t_{ j} - ( 1 / j \lambda ^ { j } ) ) } { \tau ( t _ { j } ) }. \end{equation}

For $Z \sim \tau$ one has an effective (classical-type) dynamics in the $t$ variables and averaging corresponds in some sense to suppressing fast oscillations (which suggests a renormalization procedure); alternatively, it is also in some sense related to a quantization procedure in the first WKBJ approximation, which produces slow dynamics on the action variables (Hamiltonians $\sim$ Casimirs from $\widetilde{ g } = \text { Lie } ( G )$; cf. also Casimir element; Kac–Moody algebra), which is equivalent in many situations to dynamics on the moduli of the underlying spectral curves. Thus, the quantum arena shifts to the quantum moduli space and the $T _ { n }$ appear as renormalized coupling constants in one approach and as deformation parameters of moduli in another. The tau-function $\tau$ goes to a quasi-classical tau-function whose logarithm (after adjustment) is called the pre-potential $F$ and this serves as a generating function for correlators and as a vehicle for expressing further renormalization effects. Consider (cf. [a31], [a32], [a33], [a37], [a38], [a39], [a40], [a41], [a62]) the following example of Seiberg–Witten Toda curves for $\mathcal{N} = 2$ susy Yang–Mills with $G = \operatorname {SU} ( N )$, $N = N _ { c }$, no masses and moduli $u _ { k } \in \mathcal{M} =$ quantum moduli space of inequivalent vacua:

\begin{equation} \tag{a10} y ^ { 2 } = P ^ { 2 } - 4 \Lambda ^ { 2 N }, \end{equation}

\begin{equation*} y = \Lambda ^ { N } \left( w - \frac { 1 } { w } \right) , P = \lambda ^ { N } - \sum _ { 2 } ^ { N } u _ { k } \lambda ^ { N - k } = \Lambda ^ { N } \left( w + \frac { 1 } { w } \right) . \end{equation*}

Here, $\lambda$ is the quantum scale, $\xi $ is a local coordinate at $\infty_{\pm}$ with $\Lambda \xi \sim w ^ {\mp ( 1 / N ) }$ with $w \rightarrow \infty$ at $\infty _+$ and $w \rightarrow 0$ at $\infty _-$, and $g = N - 1$. One defines

\begin{equation} \tag{a11} d \hat { \Omega } _ { n } = P _ { + } ^ { n / N } \left( \frac { d w } { w } \right) \end{equation}

and

\begin{equation*} d \Omega _ { n } = d \hat { \Omega } _ { n } - \sum _ { 1 } g \left( \oint _ { A _ { j } } d \hat { \Omega} _ { n } \right) d \omega _ { j } \end{equation*}

($n < 2 N$ for technical reasons and $d \omega_{j} \sim$ holomorphic differentials). The standard Whitham theory is now based on

\begin{equation} \tag{a12} d S = \sum _ { 1 } ^ { M } T _ { n } d \widehat { \Omega } _ { n } = \sum _ { 1 } ^ { M } T _ { n } d \Omega _ { n } + \sum _ { 1 } ^ { g } \alpha _ { j } d \omega _ { j }, \end{equation}

where $M < 2 N$ and $T _ { 0 } = 0$ for $N _ { f } = 0$. One has then Whitham equations

\begin{equation} \tag{a13} \frac { \partial d \Omega _ { A } } { \partial T _ { B } } = \frac { \partial d \Omega _ { B } } { \partial T _ { A } } \end{equation}

with $\partial d S / \partial \alpha_j = d \omega_j$ and $\partial d S / \partial T _ { n } = d \omega _ { n }$ for $( T _ { n } , \alpha _ { j } )$ independent. The pre-potential $F$ arises via

\begin{equation} \tag{a14} \frac { \partial F } { \partial \alpha _ { j } } = \oint _ { B _ { j } } d S \end{equation}

and $\partial _ { n } F = ( 1 / 2 \pi i n ) \operatorname { Res } _ { 0 } \xi ^ { - n } d S$, where involves $\infty_{\pm}$ and the Seiberg–Witten differential is

\begin{equation} \tag{a15} d S _ { S W } = d \widehat { \Omega } _ { 1 } = \lambda \left( \frac { d w } { w } \right) = \lambda \frac { d P } { y } = \lambda \frac { d y } { P }. \end{equation}

Thus, for $T _ { n } = \delta _ { n , 1 }$ one has the Seiberg–Witten situation $F ^ { \text{SW} } = \widetilde { F }$ and one writes then also $a _ { i } = \alpha _ { i }$.

General framework.

The Whitham formulation of I. Krichever, developed in great detail with D.H. Phong (cf. [a43], [a44], [a45], [a46], [a47]), involves a Riemann surface $\Sigma _ { g }$ with $M$ punctures $P _ { \alpha }$. One picks in an ad hoc manner two Abelian differentials $d E$ and $d Q$ having certain properties and sets $d S = Q d E$ as a Seiberg–Witten-type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times $T _ { A }$ with corresponding differentials $d \Omega _ { A }$. For suitable choices of $d E$ and $d Q$ the formulation is adequate for Seiberg–Witten-type situations and topological field theories with Witten–Dijkgraaf–Verlinde–Verlinde equations will arise as well.

Soft susy breaking.

There is another role for Whitham times, via (cf. [a26], [a55])

\begin{equation} \tag{a16} \hat{T} _ { n } = T _ { n } T _ { 1 } ^ { - 1 } , \hat { u } _ { k } = T _ { 1 } ^ { k } u _ { k }, \end{equation}

and $\hat { a } _ { i } = \alpha _ { i } ( u _ { k } , T _ { 1 } , \hat{T} _ { n > 1 } = 0 ) = T _ { 1 } a _ { i } ( u _ { k } , \Lambda = 1 ) = a _ { i } ( \hat { u } _ { k } , \Lambda = T _ { 1 } )$ (note $T _ { 1 } \sim \Lambda$ in the Seiberg–Witten situation). Then one defines

\begin{equation} \tag{a17} s _ { 1 } = - i \operatorname { log } ( \lambda ) \end{equation}

and $s _ { n } = - i \hat{T} _ { n }$ and these are promoted to spurion superfields $\mathcal{S} _ { n } = s _ { n } + \theta ^ { 2 } F _ { n }$ and $V _ { n } = ( 1 / 2 ) D _ { n } \theta ^ { 2 } \overline { \theta } ^ { 2 }$ in $\mathcal{N} = 1$ superfield language ($\theta$ and $\overline{\theta}$ are Grassmann variables while $D _ { n }$ and $F _ { n }$ are auxiliary fields). One has a family of non-susy theories and soft susy breaking $\mathcal{N} = 2 \rightarrow \mathcal{N} = 0$ is achieved by fixing $s _ { n } = 0$ for $n > 1$ and using $D _ { n }$, $F _ { n }$ ($n \geq 1$) as susy breaking parameters (actually, the $F _ { n }$ alone will suffice). In any event, one can develop formulas involving $\lambda$, $\tilde{T} _ { n }$ and $\alpha_j$ derivatives of the pre-potential and eventually parametrize soft susy breaking terms induced by all of the Casimirs.

Isomonodromy.

Various isomonodromy problems can be treated by multi-scale analysis to produce results indicating that isomonodromy deformations in WKB approximation correspond to modulation of isospectral problems (with Whitham-type equations as modulation equations). One can generate a pre-potential, period integrals, etc. as in Seiberg–Witten theory (see e.g. [a66], [a67], [a68], [a69], [a70]). There are also isomonodromy connections to the Knizhnik–Zamolodchikov–Bernard equations (cf. [a51], [a52], [a53], [a63]); these equations arise in various ways in conformal field theory, geometric quantization of flat bundles, etc. Here one takes $F B ( \Sigma _ { g } , G )$ as flat vector bundles over $\Sigma _ { g }$ with $G = \operatorname{GL} ( N ,\bf C )$ and smooth connections $\mathcal{A} \sim ( A , \overline { A } )$. "Flat" means zero curvature and with an arbitrary $\kappa$ this has the form

\begin{equation} \tag{a18} ( \kappa \partial + A ) \psi = 0 \end{equation}

and $( \overline { \partial } + \overline { A } ) \psi = 0$. Let $\mu \in \Omega ^ { - 1,1 } ( \Sigma _ { g } )$ (Beltrami differentials), so $\mu = \mu ( z , \bar{z} ) \partial _ { \bar{z} } \otimes d \bar{z}$ and set , where $\text{l} = 3 g - 3$ ($g > 1$) and $\mu _ { a } ^ { 0 }$ is a basis in $T \mathcal{M} _ { g }$. Then (a18) becomes

\begin{equation} \tag{a19} ( \kappa \partial + A ) \psi = 0 \end{equation}

and $( \overline { \partial } + \mu \partial + \overline { A } ) \psi = 0$. Let $\gamma$ be a homotopically non-trivial cycle in $\Sigma _ { g }$ such that $( z_0 , \overline{z}_0 ) \in \gamma$ with $\psi ( z _ { 0 } , \overline{z} _ { 0 } ) = I$ and write $\mathcal{Y} ( \gamma ) = \psi ( z _ { 0 } , \overline{z} _ { 0 } ) | _ { \gamma } = P \operatorname { exp } ( \oint _ { \gamma } \mathcal{A} )$ (path-ordered exponential), which yields a representation of $\Pi _ { 1 } ( \Sigma _ { g } , z _ { 0 } )$ in $\operatorname {GL} ( N , \mathbf{C} )$. The independence of monodromy $\mathcal{Y}$ to complex structure deformation corresponds to for $a = 1 , \dots , \text{l}$. Compatibility with (a19) requires

\begin{equation} \tag{a20} \partial _ { a } A = 0 \text { and } \partial \overline { A } = ( 1 / \kappa ) A \mu _ { a } ^ { 0 }. \end{equation}

These equations are Hamiltonian when $F B ( \sigma _ { g } , G )$ has a symplectic form $\omega ^ { 0 } = \int \Sigma _ { g } \langle \delta A , \delta \overline { A } \rangle$ with Hamiltonians . Consider the bundle $\mathcal{P}$ over $\mathcal{M} _ { g }$ with fibre $F B$ (using $( A , \overline { A } , t \sim t _ { a } )$ as local coordinates). A gauge fixing plus flatness corresponds to reduction from $F B \rightarrow \widetilde { F B }$ and one can (via WZW theory) fix the gauge to get a bundle $\tilde {\cal P }$ with fibre $\widetilde { F B }$ and equations

\begin{equation} \tag{a21} ( \kappa \partial + L ) \psi = 0 \end{equation}

with $( \overline { \partial } + \mu \partial + \overline{L}) \psi = 0$ and $( \kappa \partial _ { a} + M _ { a } ) \psi = 0$, where $M _ { a }$ comes from the gauge transformation. Putting in the canonical form via local coordinates $( v _ { i } , u _ { i } )$ in $\widetilde { F B }$, where $i = 1 , \dots , M = ( N ^ { 2 } - 1 ) ( g - 1 )$, one can write

\begin{equation} \tag{a22} \omega ^ { 0 } = ( \delta v , \delta u ) \end{equation}

with $\omega = \omega ^ { 0 } - ( 1 / \kappa ) \sum \delta H _ { \alpha } \delta t _ { \alpha }$. Using the Poincaré–Cartan invariant form $\Theta = ( u , \delta v ) - ( 1 / \kappa ) \sum H _ { a } \delta t _ { a }$ there exist $3 g - 3$ vector fields which annihilate $\Theta$. With $\{ .\}_0 \sim \omega ^ { 0 }$-structure this gives

\begin{equation} \tag{a23} \kappa \partial _ { s } H _ { r } - \kappa \partial _ { r } H _ { s } + \{ H _ { s } , H _ { r } \} _ { 0 } = 0. \end{equation}

These equations define flat connections in $\tilde {\cal P }$ and are referred to as a Whitham hierarchy of isomonodromic deformations. For a given $f ( u , v , t )$ on $\tilde {\cal P }$ they take the form

\begin{equation} \tag{a24} \frac { d f } { d t _ { s } } = \kappa \partial _ { s } f + \{ H _ { s } , f \} \end{equation}

and one can introduce a pre-potential $F$ on $\tilde {\cal P }$ giving Hamilton–Jacobi equations (cf. Hamilton–Jacobi theory)

\begin{equation} \tag{a25} \kappa \partial _ { s } F + H _ { s } \left( \frac { \delta F } { \delta u } , u , t \right) = 0. \end{equation}

Thus, one has a derivation of deformation equations, properly referred to as a Whitham hierarchy, which involves no averaging or multi-scale analysis. One can also compare the Baker–Akhiezer function $\psi$ in the Whitham hierarchy of isomonodromic deformations with elements of a certain Hitchin hierarchy (cf. also Hitchin system) using a WKB approximation with fast times $t _ { S } ^ { H }$ and slow times $T _ { S } \sim t _ { s }$.

Contact terms.

For $\mathcal{N} = 2$ susy gauge theory on a $4$-manifold with $b _ { 2 + } = 1$ there is a $u$-plane integral for, say, $\operatorname{SU} ( N )$ situations, which can be related to a Toda theory with fast and slow (Whitham) times (cf. [a55], [a56], [a57], [a58], [a59], [a71], [a72]).

Witten–Dijkgraaf–Verlinde–Verlinde.

There is a beautiful and elaborate theory of B. Dubrovin and others based on Frobenius manifolds (cf. [a15], [a16], [a17], [a18], [a19], [a20], [a21], [a22], [a23], [a24]). This approach is especially pleasing since there is a great deal of motivation and natural structure. There are many connections to mathematics and physics and this approach has led to extensive development in Frobenius manifolds, quantum cohomology and $K$-theory, singularity theory, Witten–Dijkgraaf–Verlinde–Verlinde, etc. (see e.g. [a15], [a16], [a17], [a18], [a19], [a20], [a21], [a30], [a54]). A simple Hurwitz-space Korteweg–de Vries–Landau–Ginsburg model is as follows.

Let ${\cal M} _ { g , n + 1}$ be the moduli space of $g$ gap Korteweg–de Vries solutions based on $L = \partial ^ { n + 1 } - q _ { 1 } \partial ^ { n - 1 } - \ldots - q _ { n }$ with ramification based on $W = p ^ { n + 1 } - q _ { 1 } p ^ { n - 1 } - \ldots - q _ { n }$. One defines Whitham times

\begin{equation} \tag{a26} T _ { i } = - \frac { n + 1 } { n + 1 - i } \operatorname { Res } _ { \infty } W ^ { 1 - [ i / ( n + 1 ) ] } d p, \end{equation}

\begin{equation*} T _ { n + \alpha } = \frac { 1 } { 2 \pi i } \oint _ { A _ { \alpha } } p d W , T _ { g + n + \alpha } = \oint _ { B _ { \alpha } } d p, \end{equation*}

where $1 \leq i \leq n$ and $1 \leq \alpha \leq g$. These are flat times for a certain metric and determine a Whitham hierarchy, a Frobenius manifold and a topological field theory of Landau–Ginsburg type satisfying the Witten–Dijkgraaf–Verlinde–Verlinde equations (associativity equations for related field correlators).

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How to Cite This Entry:
Whitham equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitham_equations&oldid=24014
This article was adapted from an original article by A. BlochR. Carroll (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article