The Gel’fand-Kapranov-Zelevinski HGS, the D-Brane Superpotential, and the Witten Mirror

The D-brane superpotential, a section of special holomorphic line bundles of the complex and Kähler product moduli space, and hence the correlation-function generator, is pivotal for at least three central reasons in the construction of Calabi-Yau compactifications: first, for allowing M-theory compactifications on Kovalev twisted connected sum ${{G}_{2}}$ -manifolds. Second, for giving rise to Yukawa couplings, and consequently, allows us to determine the vacuum of the low energy N=1 effective theory. And third, to allow for topological CY properties such as Jones-Witten and Ooguri–Vafa invariants. In this post, we will analyze the D-brane superpotential for pseudo-Fermat Calabi-Yau manifolds via mirror symmetry and the Gel’fand-Kapranov-Zelevinski hypergeometric system in the context of Saito’s differential equations of deformation theory of singularities and derive an Aganagic-Klemm-Vafa type mirror symmetry based on the Witten equation. First, let us set the stage. Dp-brane solutions preserving half supersymmetry have general form:

$\displaystyle ds_{{Dp}}^{2}={{\Omega }^{{-1}}}\left[ {d{{t}^{2}}-ds_{p}^{2}} \right]-\Omega dx_{{5-p}}^{2}$

$\displaystyle {{e}^{{2\phi }}}={{\Omega }^{{1-p}}}$

$\displaystyle F_{{01...pm}}^{A}={{\partial }_{m}}{{H}^{A}}$

$\displaystyle {{M}_{{Kah}}}^{{AB}}={{\text{I}}_{{\text{Re}{{\text{p}}_{G}}}}}^{{AB}}+2{{\Omega }^{{-1}}}{{H}^{A}}{{H}^{B}}$

where the Hamiltonian metaplectic action in the Heisenberg representation on the Dp+1 dimensional worldvolume gives us:

$H = {\dot \psi ^{2\pi ik}}{\Im _i} + V_t^{p + 1}\not K + \oint_{p + i} {\delta _k^{{\rm{susy}}}} \left| {_{{B_{{\rm{Bos}}}}}} \right.d\,\Omega {({\phi _{si}})^{p + 1}}{\not H_i} + \lambda {\not H^i}$

where:

$\left\{ {\begin{array}{*{20}{c}}{{\Im _i} = - \not \partial {\phi _{si}}{T_{Dp}}d\,\Omega {{({\phi _{si}})}^{2\pi ik}}}\\{\not K = - {{\not \partial }_i}{{\widetilde E}^i} + {{( - 1)}^{p + 1}}{T_{Dp}}{S^{{\rm{Fer}}}}}\\{{{\not H}_i} = \widetilde P{\alpha _i}\widetilde E_i^\alpha {{\not \partial }_i}{\phi _{si}} + \widetilde E{{\not F}_{ij}}}\\{H = \frac{1}{{2\pi ik}}\left[ {{{\widetilde P}^2} + {{\widetilde E}^i}{{\widetilde E}^j}{G_{ij}} + T_{Dp}^2{e^{ - 2{\phi _{si}}}}{\rm{det}}\left( {{G_{ij}} + {{\not F}_{ij}}} \right)} \right]}\end{array}} \right.$

with:

$S = * {\left( {{{\not R}_{\mu \nu }}{\varepsilon ^{{\rm{Fer}}}}} \right)_p}$

and:

$E_i^\alpha = \delta \int {d\not E_m^\alpha } {\not \partial _i}{\dot X^m}$

where the Ramond-Ramond gauge-coupling sector is given by the action:

$\displaystyle \mathcal{L}_{G}^{{Loc}}=\sum\limits_{{b=1}}^{{N-1}}{{\frac{1}{{2g_{b}^{2}}}}}{{\int{\text{d}}}^{2}}\theta {{W}^{\alpha }}{{W}_{\alpha }}{{\delta }^{2}}\left( {\left( {1-{{e}^{{ib\phi }}}} \right)z} \right)$

and generally, the action of a $Dp$ -brane is given by a Dirac-Born-Infeld part coupled to a Chern-Simons WZ part:

$\displaystyle S_{{DBI}}^{{cs}}=-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\Phi }}}\Xi -{{T}_{p}}\int_{{{\mathcal{W}}'}}{{C_{F}^{B}}}$

with:

$\displaystyle \Xi \doteq \sqrt{{{{{\det }}_{{\left[ {a,b} \right]}}}\left( {{{g}_{{ab}}}+{{B}_{b}}+2\pi {\alpha }'{{F}_{{ab}}}} \right)}}$

$\displaystyle C_{F}^{B}\doteq \text{TrP}\left[ {C\wedge {{e}^{{-B}}}} \right]\wedge {{e}^{{2\pi {\alpha }'F}}}$

where $P$ is the worldvolume pullback with $p$ -orientifold action:

$\displaystyle {{S}_{{Op}}}={{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}_{\parallel }}}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\phi }}}\left( \Pi \right)-\Psi$

with:

$\displaystyle \Pi \doteq \sqrt{{-\det P\left[ {{{g}_{{\mu \nu }}}} \right]}}-{{2}^{{2-4}}}$

and

$\displaystyle \Psi \doteq -{{T}_{p}}\int_{{{{\mathcal{W}}_{\parallel }}}}{{P\left[ {{{C}_{{p+1}}}} \right]}}$

where the pullback to the $Dp$ -worldvolume yields the 10-D SYM action:

$\displaystyle {{S}_{{YM}}}=\frac{1}{{4g_{{_{{YM}}}}^{2}}}\int{{{{d}^{{10}}}}}x\left[ {\text{Tr}\left( {{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}} \right)+2\text{iTr}\left( {\bar{\psi }\,{{\Gamma }^{\mu }}{{D}_{\mu }}\psi } \right)} \right]$

with string coupling:

$\displaystyle \frac{1}{{{{g}_{s}}}}={{e}^{{-\Phi }}}$

and the 10-D SUGRA dimensionally reduced Type-IIB action is:

$\displaystyle S_{{DBI}}^{{c{s}'}}=-\frac{{{{T}_{p}}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{4{{g}_{s}}}}\int_{{{\mathcal{W}}'}}{{{{d}^{{p+1}}}\left( {\hat{F}+\hat{X}} \right)-\frac{{{{T}_{p}}}}{{{{g}_{s}}}}{{V}_{\vartheta }}}}$

with:

$\displaystyle \hat{F}\equiv {{F}_{{ab}}}{{F}^{{ab}}}$

$\displaystyle \hat{X}\equiv \frac{2}{{{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\partial }_{a}}{{X}^{m}}{{\partial }^{a}}{{X}_{m}}$

$\displaystyle {{V}_{\vartheta }}\equiv V_{{p+1}}^{{WV}}+\vartheta \left( {{{F}^{4}}} \right)$

and in the string-frame, the type-IIB SUGRA action is given by:

$\displaystyle {{S}_{{IIB}}}={{S}_{{NS}}}+{{S}_{R}}+{{S}_{{CS}}}$

with:

$\displaystyle {{S}_{{NS}}}=\frac{1}{{2k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}{{e}^{{-2\Phi }}}\left[ {{{R}_{{10}}}+4\left( {{{\partial }^{\mu }}\Phi } \right)\left( {{{\partial }_{\mu }}\Phi } \right)-\frac{1}{2}{{{\left| {{{H}_{{\left( 3 \right)}}}} \right|}}^{2}}} \right]$

$\displaystyle {{S}_{R}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}\left[ {{{{\left| {{{F}_{1}}} \right|}}^{2}}+{{{\left| {{{{\tilde{F}}}_{{\left( 3 \right)}}}} \right|}}^{2}}+\frac{1}{2}{{{\left| {{{{\tilde{F}}}_{{\left( 5 \right)}}}} \right|}}^{2}}} \right]$

$\displaystyle {{S}_{{CS}}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{C}_{{\left( 4 \right)}}}\wedge {{H}_{{\left( 3 \right)}}}\wedge {{F}_{{\left( 3 \right)}}}}}$

where the Calabi-Yau superpotential is:

$\displaystyle W=\int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {S,U} \right){{e}^{{-{{a}_{i}}{{T}_{i}}}}}$

where:

$\displaystyle \int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}$

is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:

$\displaystyle S={{e}^{{-\phi }}}+i{{C}_{0}}$

Given the presence of $E3$ -brane instantons, ${{T}_{i}}$ are of Kähler moduli Type-IIB-orbifold class:

$\displaystyle {{T}_{i}}={{e}^{{-\phi }}}{{\tau }_{1}}+i{{\rho }_{i}}$

with ${{\tau }_{i}}$ being the volume of the divisor ${{D}_{i}}$ and ${{\rho }_{i}}$ the 4-form Ramond-Ramond axion field corresponding to:

$\displaystyle {{\tau }_{i}}=\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J}}=\frac{1}{2}{{k}_{{ijk}}}+{{\,}^{j}}{{t}^{k}}$

and:

$\displaystyle {{\rho }_{i}}=\int_{{{{D}_{i}}}}{{{{C}_{4}}}}$

where $J$ is the Kähler form:

$\displaystyle J=\sum\limits_{i}{{{{t}_{i}}}}{{\eta }_{i}}$

and:

$\displaystyle \left\{ {{{\eta }_{i}}} \right\}\in {{H}^{{1,1}}}\left( {Y,\mathbb{Z}} \right)$

an integral-form basis and ${{k}_{{ijk}}}$ the associated intersection coefficients. Hence, the Kähler potential is given by:

$\displaystyle K=-2\text{In}\left( {\tilde{\mathcal{V}}+\frac{\xi }{{2g_{s}^{{3/2}}}}} \right)-\text{In}\left( {S+\tilde{S}} \right)-\text{In}\left( {-i\int_{Y}{{\Omega \wedge \tilde{\Omega }}}} \right)$

with ${\tilde{\mathcal{V}}}$ the Calabi-Yau volume, and in the Einstein frame, is given by:

$\displaystyle \mathcal{V}=\frac{1}{{3!}}\int_{Y}{{J\wedge J\wedge J=}}\frac{1}{6}{{k}_{{ijk}}}{{t}^{i}}{{t}^{j}}{{t}^{k}}$

The $F$ -term is given by:

$\displaystyle {{V}_{F}}={{e}^{K}}\left( {\sum\limits_{{i={{T}_{i}};j={{S}_{j}}}}{{{{K}^{{ij}}}}}{{D}_{i}}W{{D}_{j}}\tilde{W}-3{{{\left| {{{W}_{i}}} \right|}}^{2}}} \right)$

with the Large Volume Scenario $D$ -term is given by:

$\displaystyle {{V}_{D}}=\sum\limits_{{i=1}}^{N}{{\frac{1}{{\operatorname{Re}\left( {{{f}_{i}}} \right)}}}}{{\left( {\sum\limits_{j}{{Q_{j}^{{\left( i \right)}}{{{\left| {{{\phi }_{j}}} \right|}}^{2}}-{{{\hat{\xi }}}_{i}}}}} \right)}^{2}}$

with:

$\displaystyle \operatorname{Re}\left( {{{f}_{i}}} \right)\doteq {{e}^{{-\phi }}}\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J-}}{{e}^{{-\phi }}}\int_{{{{D}_{i}}}}{{\text{c}{{\text{h}}_{2}}}}\left( {{{\mathcal{L}}_{i}}-B} \right)$

and the Fayet-Illopoulos terms being:

$\displaystyle {{\hat{\xi }}_{i}}=-\text{Im}\left( {\frac{1}{{\tilde{\mathcal{V}}}}\int_{Y}{{{{e}^{{-\left( {B+iJ} \right)}}}}}{{\Gamma }_{i}}} \right)$

where ${{\Gamma }_{i}}$ are the $D7$ -brane charge-vectors. Then we saw that the Chern-Simons orientifold action gets a Calabi-Yau curvature correction in the form of:

$\displaystyle {{S}_{{{{O}_{p}},CS}}}=-{{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}^{\prime }}}}{{P\left[ C \right]}}\wedge \Theta$

with

$\displaystyle \Theta \doteq \sqrt{{\frac{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'T{{\mathcal{W}}^{\prime }}} \right)}}{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'N{{\mathcal{W}}^{\prime }}} \right)}}}}$

due to the Gauss–Codazzi equations:

$\displaystyle {{S}_{{{{O}_{p}}}}}={{2}^{{p-4}}}-{{T}_{p}}{{\int_{{{\mathcal{W}}'}}{\text{d}}}^{{p+1}}}\xi {{e}^{{-\phi }}}\sqrt{{-\det \left( {P\left[ {{{g}_{{\mu \nu }}}} \right]} \right)}}-{{2}^{{2-4}}}-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{P\left[ {{{C}_{{p+1}}}} \right]}}$

and the Ramond-Ramond term being:

$\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{Tr}}}\left( {{{e}^{{F/2\pi }}}} \right)$

which yields the Type-IIB Calabi-Yau three-fold superpotential:

$\displaystyle {{V}_{W}}=\int\limits_{X}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {\left( {{{e}^{{-\phi }}}+i{{C}_{0}}} \right),U} \right){{e}^{{-a\left( {{{e}^{{-\phi }}}{{\tau }_{i}}+i{{\rho }_{i}}} \right)}}}$

and where the topologically mixed Yang-Mills action is given by:

$\displaystyle {{\mathcal{L}}_{{TYM}}}\equiv -\frac{1}{4}e{{\tilde{F}}_{{\mu \nu }}}^{M}{{\tilde{F}}^{{\mu \nu N}}}{{\hat{M}}_{{MN}}}+\kappa {{\mathcal{L}}_{{CS}}}$

with the corresponding Chern-Simons action:

$\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{ch}}}\left( {\tilde{F}} \right)\wedge \sqrt{{\frac{{\hat{A}\left( {{{R}_{T}}} \right)}}{{\hat{A}\left( {{{R}_{N}}} \right)}}}}$

Generically, type-II compactifications on Calabi–Yau manifolds give rise to N=1 low energy effective theories given suitable background fluxes and space-filling D-branes wrapping internal special Lagrangian sub-manifolds and holomorphic divisors determined by period integrals satisfying systems of Picard-Fuchs differential equations where the period integrals of the holomorphic three-form whose cycles have boundaries wrapped by D-branes determine the effective superpotential. The associated superpotential of D-branes wrapping internal cycles of Calabi–Yau manifold $X$ is given as such:

$\displaystyle {{\mathcal{W}}_{{Brane}}}=\int_{X}{{\Omega \wedge \text{Tr}}}\left[ {A\wedge \bar{\partial }A+\frac{1}{2}A\wedge A\wedge A} \right]$

In the Type-II case, it is a linear sum of the period integrals:

$\displaystyle {{\mathcal{W}}_{{Brane}}}\left( {\varphi ,\xi } \right)={{{\hat{N}}}_{a}}{{{\hat{\Pi }}}^{a}}\left( {\varphi ,\xi } \right)$

with:

$\displaystyle {{{\hat{\Pi }}}^{a}}\left( {\varphi ,\xi } \right)=\int_{{{{\gamma }^{a}}\left( \xi \right)}}{{\Omega \left( \varphi \right)}}$

The Ramond-Ramond and Neveu-Schwarz background fluxes ${{H}^{{RR}}}$ , ${{H}^{{NS}}}$ yield the flux-superpotential:

$\displaystyle {{\mathcal{W}}_{{flux}}}=\int_{X}{{\Omega \wedge H=\int_{X}{{\Omega \wedge \left( {{{H}^{{RR}}}+\tau {{H}^{{NS}}}} \right)}}}}$

Thus, putting the two superpotentials together gives us:

$\displaystyle \mathcal{W}\left( {\varphi ,\xi } \right){{\mathcal{W}}_{{brane}}}\left( {\varphi ,\xi } \right)+{{\mathcal{W}}_{{flux}}}\left( \varphi \right)=\sum{{{{N}_{\Sigma }}}}{{\Pi }_{\Sigma }}\left( {\varphi ,\xi } \right)$

where ${{N}_{\Sigma }}$ is the D-brane topological charge and ${{\Pi }_{\Sigma }}\left( {\varphi ,\xi } \right)$ is the period-integral over the CY 3-form:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\Pi }_{\Sigma }}\left( {\varphi ,\xi } \right)=\int_{{{{\Gamma }^{\alpha }}\left( \xi \right)}}{{\Omega \left( \varphi \right)}}} \\ {{{\Gamma }^{\alpha }}\left( \xi \right)\in {{H}_{3}}\left( {Y,S,Z} \right)} \end{array}} \right.$

and the domain-wall tension is given by:

$\displaystyle \mathcal{T}\left( {\varphi ,\xi } \right)=\mathcal{W}\left( {C_{{\left( {\varphi ,\xi } \right)}}^{+}} \right)-\mathcal{W}\left( {C_{{\left( {\varphi ,\xi } \right)}}^{-}} \right)$

Now, in the A-model, the D-brane superpotential is defined in terms of closed/open holomorphic correlation-functional coordinates determined by OPE coefficients corresponding to the worldsheet chiral ring:

$\displaystyle \mathcal{W}\left( {t,\hat{t}} \right)=\sum\limits_{{\vec{k},\vec{m}}}{{{{G}_{{\vec{k},\vec{m}}}}}}{{q}^{{\text{d}\vec{k}}}}{{\hat{q}}^{{\text{d}\vec{m}}}}=\sum\limits_{{\vec{k},\vec{m}}}{{\sum\limits_{d}{{{{n}_{{\vec{k},\vec{m}}}}}}}}\frac{{{{q}^{{\text{d}\vec{k}}}}{{{\hat{q}}}^{{\text{d}\vec{m}}}}}}{{{{k}^{2}}}}$

where we have:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {q={{e}^{{2\pi \text{i}t}}}} \\ {\hat{q}={{e}^{{2\pi \text{i}\hat{t}}}}} \\ {{{n}_{{\vec{k},\vec{m}}}}\doteq \text{Ooguri-Vafa invariant}} \end{array}} \right.$

and the following relations follow for the D-brane superpotential and the Ooguri–Vafa invariants:

$\displaystyle \frac{{{{W}^{{\left( \pm \right)}}}\left( {z\left( q \right)} \right)}}{{{{\omega }_{0}}\left( {z\left( q \right)} \right)}}=\frac{1}{{{{{\left( {2\pi i} \right)}}^{2}}}}\sum\limits_{{k\,\in \,\ \text{odd}}}{{\sum\limits_{{{{d}_{1}}\ge 0,{{d}_{2}}\,\in \ \ \text{odd}}}{{n_{{{{d}_{1}},{{d}_{2}}}}^{{\left( \pm \right)}}}}}}\frac{{q_{1}^{{k{{d}_{1}}}}q_{2}^{{k{{d}_{2}}/2}}}}{{{{k}^{2}}}}$

and:

$\displaystyle {{\omega }_{0}}\left( z \right)=\sum\limits_{n}{{c\left( n \right){{z}^{n}}}}\sum\limits_{n}{{\frac{{\prod\nolimits_{j}{{\Gamma \left( {\sum\nolimits_{a}{{l_{{0j}}^{{\left( a \right)}}{{n}_{a}}+1}}} \right)}}}}{{\prod\nolimits_{i}{{\Gamma \left( {\sum\nolimits_{a}{{l_{i}^{{\left( a \right)}}{{n}_{a}}+1}}} \right)}}}}}}{{z}^{n}}$

where the mirror-symmetry map is given by:

$\displaystyle {{t}_{a}}=\frac{{{{\partial }_{a}}{{\omega }_{0}}}}{{{{\omega }_{0}}}}$

We now define a mirror pair of hypersurfaces $\left( {X,{{X}^{*}}} \right)$ in toric ambient spaces $\left( {V,{{V}^{*}}} \right)$ with corresponding fans $\left( {\Sigma \left( \Delta \right),\Sigma \left( {{{\Delta }^{*}}} \right)} \right)$ , where the hypersurface defining polynomial is given as such:

$\displaystyle \mathcal{P}=\sum\limits_{{i=0}}^{{p-1}}{{{{a}_{i}}}}\prod\limits_{{k=1}}^{4}{{X_{k}^{{\upsilon _{i}^{*},k}}}}\dot{\equiv }\mathcal{P}=\sum\limits_{{i=0}}^{{p-1}}{{{{a}_{i}}}}\prod\limits_{{{{\upsilon }_{j}}\in \,\Delta }}{{x_{j}^{{\left\langle {{{\upsilon }_{j}},\upsilon _{i}^{*}} \right\rangle +1}}}}$

It is clear that a GKZ hypergeometric differential system:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{{\hat{D}}}_{l}}\Pi \left( a \right)=0\ ,\ \left( {l\in L} \right)} \\ {{{{\hat{Z}}}_{i}}\Pi \left( a \right)=0\ ,\ \left( {i=0,1,...,p} \right)} \\ {\Pi \left( a \right)=\frac{1}{{{{{\left( {2\pi i} \right)}}^{4}}}}\int_{{\left| {{{X}_{k}}} \right|=1}}{{\frac{1}{P}\prod\limits_{{k=1}}^{4}{{\frac{{\text{d}{{X}_{k}}}}{{{{X}_{k}}}}}}}}} \end{array}} \right.$

with:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{{\hat{D}}}_{l}}=\prod\limits_{{{{l}_{i}}>0}}{{{{{\left( {\frac{\partial }{{\partial {{a}_{i}}}}} \right)}}^{{{{l}_{i}}}}}-\prod\limits_{{{{l}_{j}}<0}}{{{{{\left( {\frac{\partial }{{\partial {{a}_{j}}}}} \right)}}^{{-{{l}_{j}}}}}}}}}} \\ {{{{\hat{Z}}}_{j}}=\sum\limits_{{i=0}}^{p}{{\bar{\upsilon }_{{i,j}}^{*}{{\theta }_{{{{a}_{i}}}}}-{{\beta }_{j}}\quad ,\ \left( {j=0,1,...,n} \right)}}} \end{array}} \right.$

annihilates the integral periods, and the torus invariant Kähler coordinates of the system are given as such:

$\displaystyle {{z}_{a}}={{\left( {-1} \right)}^{{l_{0}^{a}}}}\prod\limits_{j}{{a_{j}^{{l_{j}^{a}}}}}$

generating the Mori cone. Now, since we have:

$\displaystyle {{\theta }_{a}}={{z}_{a}}{{\partial }_{{{{z}_{a}}}}}$

we get:

$\displaystyle {{{\mathrm B}}_{{\left\{ {{{l}^{a}}} \right\}}}}\left( {{{z}_{a}};{{\rho }_{a}}} \right)=\sum\limits_{{{{n}_{1}},...,{{n}_{\text{N}}}\in Z_{0}^{+}}}{{\frac{{\Gamma \left( {1-\sum\limits_{a}{{l_{0}^{a}\left( {{{n}_{a}}+{{\rho }_{a}}} \right)}}} \right)}}{{_{{{{\Pi }_{{i>0}}}}}\Gamma \left( {1+\sum\limits_{a}{{l_{i}^{a}\left( {{{n}_{a}}+{{\rho }_{a}}} \right)}}} \right)}}}}\times \prod\limits_{a}{{z_{a}^{{\left( {{{n}_{a}}+{{\rho }_{a}}} \right)}}}}$

Now let us consider Saito’s system of DE of deformations of a singularity. Take a sheaf $\Omega _{{\not{\wp }/T}}^{p}$ of germs of relative holomorphic $p$ forms for the natural projection $\pi :\not{\wp }={{\text{C}}^{3}}\times T\to T$ . One consider now sheaves on S with the following properties:

$\displaystyle {{\mathcal{H}}^{{\left( 0 \right)}}}={{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{3}/df\wedge d\left( {{{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{1}} \right)$

$\displaystyle {{\mathcal{H}}^{{\left( {-1} \right)}}}={{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{2}/\left( {d{{f}_{1}}\wedge {{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{1}+d\left( {{{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{1}} \right)} \right)$

We now take an image of a primitive form $\zeta$ in ${{H}^{0}}\left( {S,{{\mathcal{H}}^{{\left( {-1} \right)}}}} \right)$ :

$\displaystyle {{\mho }_{0}}\left( a \right)=\operatorname{Re}{{s}_{{\left\{ {{{a}_{1}}+{{f}_{1}}=0} \right\}}}}\left( {\frac{{dU\wedge dV\wedge dW}}{{{{a}_{1}}+{{f}_{1}}\left( {a,U,V,W} \right)}}} \right)$

Let us connect the Gel’fand-Leray form to the Saito period integral and the GKZ system. The defining singularity polynomial equation for singularity ${{U}^{2}}+{{V}^{2}}+{{W}^{{\mu +1}}}$ reflecting a monodromy action, is:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{f}_{\Sigma }}\left( {a,W} \right)+{{U}^{2}}+{{V}^{2}}={{a}_{1}}+{{f}_{1}}\left( {a,U,V,W} \right)} \\ {{{f}_{1}}\left( {a,U,V,W} \right)={{a}_{2}}W+...+{{a}_{\mu }}{{W}^{{\mu -1}}}+{{W}^{{\mu +1}}}+{{U}^{2}}+{{V}^{2}}} \end{array}} \right.$

for a deformation:

$\displaystyle {{W}^{{\mu +1}}}+{{U}^{2}}+{{V}^{2}}=0\subset {{\mathbb{C}}^{2}}$

where the parameters $\left( {{{a}_{1}},{{a}_{2}},...,{{a}_{\mu }}} \right)$ act as a coordinate system of:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {S\doteq \mathbb{C}\times T} \\ {T:={{\mathbb{C}}^{{\mu -1}}}} \end{array}} \right.$

There is then a natural map:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\varphi :\not{\wp }\to S} \\ {\left( {U,V,W,{{a}_{2}},...{{a}_{\mu }}} \right)\Rightarrow \left( {-{{f}_{1}}\left( {a,U,V,W} \right),{{a}_{2}},...{{a}_{\mu }}} \right)} \end{array}} \right.$

Now we focus on the sheaf $\displaystyle \Omega _{{\not{\wp }/T}}^{p}$ of germs of holomorphic p-forms for the projection $\displaystyle \pi :\not{\wp }={{\mathbb{C}}^{3}}\times T\to T$ of form:

$\displaystyle {{\mathcal{H}}^{{\left( 0 \right)}}}={{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{3}/d{{f}_{1}}\wedge d\left( {{{\varphi }_{*}}\Omega _{{\not{\wp }/T}}^{1}} \right)$

Hence, we can re-write the image of the primitive form as such:

$\displaystyle {{\mho }_{0}}\left( a \right)=\operatorname{Re}{{s}_{{\left\{ {{{a}_{1}}+{{f}_{1}}=0} \right\}}}}\left( {\frac{{dU\wedge dV\wedge dW}}{{a+{{f}_{1}}\left( {a,U,V,W} \right)}}} \right)$

Now the Saito’s system is characterized as a set of differential equations satisfied by the PF:

$\displaystyle {{P}_{{ij}}}{{\mho }_{0}}\left( a \right)=\left\{ {\frac{{{{\partial }^{2}}}}{{{{\partial }_{{{{a}_{i}}}}}{{\partial }_{{{{a}_{j}}}}}}}-{{\nabla }_{{\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}}}}\frac{\partial }{{{{\partial }_{{{{a}_{j}}}}}}}-\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}*\frac{\partial }{{{{\partial }_{{{{a}_{j}}}}}}}} \right)\frac{\partial }{{{{\partial }_{{{{a}_{1}}}}}}}} \right\}{{\mho }_{0}}\left( a \right)=0$

$\displaystyle Q\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right){{\mho }_{0}}\left( a \right)=\left\{ {w\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right)\frac{\partial }{{{{\partial }_{{{{a}_{1}}}}}}}-N\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right)+\frac{3}{2}\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right\}{{\mho }_{0}}\left( a \right)=0$

We can now prove the following:

With ${{\beta }_{0}}\left( a \right),...,{{\beta }_{\mu }}\left( a \right)$ the roots of ${{\psi }_{0}}\left( W \right):={{a}_{1}}+{{a}_{2}}W+...+{{a}_{\mu }}{{W}^{{\mu -1}}}+{{W}^{{\mu +1}}}$ satisfying ${{\beta }_{0}}\left( a \right)+...+{{\beta }_{\mu }}\left( a \right)=0$ , the space of the solutions to Saito’s system is generated by:

$\displaystyle 1,{{\beta }_{0}}\left( a \right)-{{\beta }_{1}}\left( a \right),...,{{\beta }_{{\mu -1}}}\left( a \right)-{{\beta }_{\mu }}\left( a \right)$

with a singularity at the discriminant locus:

$\displaystyle div{{\psi }_{0}}\left( a \right)=\prod\nolimits_{{1\le i,j\le \mu }}{{{{{\left( {{{\beta }_{i}}\left( a \right)-{{\beta }_{j}}\left( a \right)} \right)}}^{2}}=0}}$

and where the monodromy group about the discriminant is isomorphic to the order- $\mu$ symmetric group ${{S}_{\mu }}$ among the roots ${{\beta }_{i}}\left( a \right)$ s.

Note that now we have a defining equation:

$\displaystyle {{f}_{\Sigma }}\left( {a,W} \right)+{{U}^{2}}+{{V}^{2}}$

where we take the canonical torus action ${{\left( {{{\mathbb{C}}^{*}}} \right)}^{2}}$ on $\left( {{{a}_{1}},...,{{a}_{{\mu +2}}}} \right)$ :

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{a}_{i}}\mapsto \lambda {{a}_{i}}} \\ {{{a}_{i}}\mapsto {{\lambda }^{{i-1}}}{{a}_{i}}} \end{array}} \right.\quad \left( {\lambda \in {{\mathbb{C}}^{*}}} \right)$

The Saito system now naturally relates to a Gel’fand-Kapranov-Zelevinski system with canonical period integral forms:

$\displaystyle \int_{C}{{{{\mho }_{0}}\left( a \right)}}={{\int_{C}{{\operatorname{Re}s}}}_{{{{f}_{\Sigma }}\left( {a,W} \right)+{{U}^{2}}+{{V}^{2}}=0}}}\left( {\frac{{dW\wedge dU\wedge dV}}{{{{f}_{\Sigma }}\left( {a,W} \right)+{{U}^{2}}+{{V}^{2}}}}} \right)$

integration is over homology 2-cycle curves on:

$\displaystyle {{f}_{\Sigma }}\left( {a,W} \right)+{{U}^{2}}+{{V}^{2}}=0\subset {{\mathbb{C}}^{3}}$

and:

$\displaystyle {{f}_{\Sigma }}\left( {a,W} \right)={{a}_{1}}+{{a}_{2}}W+...+{{a}_{{\mu +2}}}{{W}^{{\mu +1}}}$

and up to homotopic rescaling, the period integral satisfies a Gel’fand-Kapranov-Zelevinski system. Two central properties follow. The period integral satisfies:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\square }_{{l\in L}}}\int_{C}{{{{\mho }_{0}}\left( a \right)}}=0} \\ {{{{{\not{Z}}'}}_{{i,(i=1,2)}}}\int_{C}{{{{\mho }_{0}}\left( a \right)}}=0} \end{array}} \right.$

with respect to the Saito lattice-operator, with:

$\displaystyle \left( {\begin{array}{*{20}{c}} {{{{{\not{Z}}'}}_{1}}} \\ {{{{{\not{Z}}'}}_{2}}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {{{\theta }_{1}}+{{\theta }_{2}}+...+{{\theta }_{{\mu +2}}}} \\ {{{\theta }_{2}}+2{{\theta }_{3}}+...+\left( {\mu +1} \right){{\theta }_{{\mu +2}}}+1} \end{array}} \right)$

and the system is rank- ${\mu +1}$ irreducible and the $\mu$ -independent solutions are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\beta }_{0}}\left( a \right)-{{\beta }_{1}}\left( a \right),...,{{\beta }_{{\mu -1}}}\left( a \right)-{{\beta }_{\mu }}\left( a \right)} \\ {{{\beta }_{i}}\left( a \right)\in Roots\left[\!\left[ {\psi \left( W \right)={{a}_{1}}+{{a}_{2}}W+...+{{a}_{{\mu +2}}}{{W}^{{\mu +1}}}=0} \right]\!\right]} \end{array}} \right.$

So the root $\beta \left( a \right)$ of $\psi \left( W \right)$ satisfies:

$\displaystyle \frac{{\partial \beta }}{{\partial {{a}_{i}}}}=-\frac{{{{\beta }^{{i-1}}}}}{{{\psi }'\left( \beta \right)}}$

$\displaystyle \frac{{{{\partial }^{2}}\beta }}{{\partial {{a}_{i}}\partial {{a}_{j}}}}=\frac{1}{{{\psi }'\left( \beta \right)}}\frac{d}{{dx}}{{\left( {\frac{{{{x}^{{i+j-2}}}}}{{{\psi }'\left( x \right)}}} \right)}_{{{{\mid }_{{x=\beta }}}}}}$

Hence, the independent solutions of the Gel’fand-Kapranov-Zelevinski system:

are given by:

$\displaystyle 1\ ,\ \log {{\beta }_{0}}\left( a \right)-\log {{\beta }_{1}}\left( a \right)\ ,\ ...\ ,\ \log {{\beta }_{{\mu -1}}}\left( a \right)-\log {{\beta }_{\mu }}\left( a \right)$

whose monodromy transformations properties are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\omega \left( {x;\frac{J}{{2\pi i}}} \right)=1+\sum\nolimits_{{k=1}}^{\mu }{{\omega \left( a \right){{J}_{k}}}}} \\ {2\pi i{{\omega }_{k}}\left( x \right)=\log {{\beta }_{{k-1}}}\left( a \right)-\log {{\beta }_{k}}\left( a \right)} \end{array}} \right.\quad ,\quad \left( {k=1,...,\mu } \right)$

Hence,

$\displaystyle \log {{\beta }_{k}}-\log {{\beta }_{{k-1}}}{{\ }_{{{{,}_{{\ \left( {k=1,...\mu } \right)}}}}}}$

get annihilated by the operators ${{\hat{\not{Z}}}_{1}},{{\hat{\not{Z}}}_{2}}$ . Moreover, we have the following relationship holding:

$\displaystyle \frac{{{{\partial }^{2}}}}{{{{\partial }_{{{{a}_{i}}}}}{{\partial }_{{{{a}_{j}}}}}}}\log \beta =\frac{{{{\partial }^{2}}}}{{{{\partial }_{{{{a}_{k}}}}}{{\partial }_{{{{a}_{l}}}}}}}\log \beta \quad ,\quad \text{if}\quad i+j=k+l$

given that the following holds:

$\displaystyle \frac{{{{\partial }^{2}}}}{{{{\partial }_{{{{a}_{i}}}}}{{\partial }_{{{{a}_{j}}}}}}}\log \beta =-\frac{1}{{{{\beta }^{2}}}}\frac{{{{\beta }^{{i+j-2}}}}}{{{{{\left( {{\psi }'\left( \beta \right)} \right)}}^{2}}}}+\frac{1}{\beta }\frac{1}{{{\psi }'\left( \beta \right)dx}}\left( {\frac{{{{x}^{{i+j-2}}}}}{{{\psi }'\left( x \right)}}} \right)\left| {_{{x=\beta }}} \right.$

By recursion, we can derive:

$\displaystyle {{\prod\limits_{i}{{\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right)}}}^{{l_{i}^{+}}}}\log \beta ={{\prod\limits_{i}{{\left( {\frac{\partial }{{{{\partial }_{{{{a}_{i}}}}}}}} \right)}}}^{{l_{i}^{-}}}}\log \beta$

$\displaystyle {{\square }_{l}}\log \beta =0$

Now since by definition we have:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {2\pi i{{\omega }_{k}}\left( x \right)\sim \log \beta =0} \\ {{{x}_{k}}=\frac{{{{a}_{k}}{{a}_{{k+2}}}}}{{a_{{k+1}}^{2}}}} \end{array}} \right.$

the ${{{\omega }_{k}}\left( x \right)}$ ‘s are solutions to the Gel’fand-Kapranov-Zelevinski hypergeometric system, with:

$\displaystyle \psi \left( W \right)={{a}_{1}}+{{a}_{2}}W+...+{{a}_{{\mu +2}}}{{W}^{{\mu +1}}}=\prod\limits_{k}{{\left( {{{\lambda }_{k}}W+1} \right)}}$

Now we interpret the Chern-Simons part of the D-brane action as naturally inducing the charge vectors $Q_{i}^{a}/\upsilon _{i}^{b}$ defining the Calabi-Yau condition, and consider the following operators:

$\displaystyle {{\mathcal{L}}_{{{{Q}^{a}}}}}={{\prod\limits_{{Q_{i}^{a}>0}}{{\left( {\frac{\partial }{{\partial {{a}_{i}}}}} \right)}}}^{{Q_{i}^{a}}}}-{{\prod\limits_{{Q_{i}^{a}<0}}{{\left( {\frac{\partial }{{\partial {{a}_{i}}}}} \right)}}}^{{-Q_{i}^{a}}}}$

$\displaystyle {{{\tilde{\not{Z}}}}_{b}}=\sum\limits_{{i=0}}^{n}{{\upsilon _{i}^{b}}}{{a}_{i}}\frac{\partial }{{\partial {{a}_{i}}}}-{{{\hat{\beta }}}_{b}}$

characterizing the Gel’fand-Kapranov-Zelevinski hypergeometric ${\hat{\beta }}$ -system:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\mathcal{L}}_{{{{Q}^{a}}}}}\tilde{\pi }\left( a \right)=0} \\ {{{{\tilde{\not{Z}}}}_{b}}\tilde{\pi }\left( a \right)=0} \end{array}} \right.$

Now, for the Calabi-Yau manifolds that arise as hypersurfaces in toric varieties, the above Gel’fand-Kapranov-Zelevinski equations are analytically related by a Witten deformation to the above Picard-Fuchs equations. Under the change of variables:

$\displaystyle {{z}_{a}}=\prod\limits_{i}{{{{{\left( {{{a}_{i}}} \right)}}^{{Q_{i}^{a}}}}}}$

and:

$\displaystyle \tilde{\pi }\left( a \right)=\frac{1}{{{{a}_{0}}}}\pi \left( z \right)$

the corresponding Gel’fand-Kapranov-Zelevinski system obtained allows us to derive the open-string instanton effects exactly as in the bulk Gel’fand-Kapranov-Zelevinski for the closed-string case. Take the following operators:

$\displaystyle {{\mathcal{L}}_{{{{q}^{\alpha }}}}}={{\prod\limits_{{q_{i}^{\alpha }>0}}{{\left( {\frac{\partial }{{\partial {{a}_{i}}}}} \right)}}}^{{q_{i}^{\alpha }}}}-{{\prod\limits_{{q_{i}^{\alpha }<0}}{{\left( {\frac{\partial }{{\partial {{a}_{i}}}}} \right)}}}^{{-q_{i}^{\alpha }}}}$

$\displaystyle {{{\tilde{\not{Z}}}}_{{b\beta }}}=\sum\limits_{{i=0}}^{n}{{\upsilon _{i}^{\beta }}}{{a}_{i}}\frac{\partial }{{\partial {{a}_{i}}}}-{{{\hat{\beta }}}_{\beta }}$

which is the Gel’fand-Kapranov-Zelevinski HGS that naturally corresponds to the boundary toric variety, and under the c.o.v.:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{z}_{a}}=\prod\limits_{i}{{{{{\left( {{{a}_{i}}} \right)}}^{{Q_{i}^{a}}}}}}} \\ {{{s}_{\alpha }}=\prod\limits_{i}{{{{{\left( {{{a}_{i}}} \right)}}^{{q_{i}^{\alpha }}}}}}} \end{array}} \right.$

yields an open/closed bulk/boundary duality supporting Witten mirror flops and flips in the dual character-torus lattice of the Calabi-Yau:

$\displaystyle {{\mathcal{L}}_{{boundary}}}{{/}^{{MS}}}{{\mathcal{L}}_{{bulk}}}$

For the Aganagic-Klemm-Vafa system, the solution corresponding to the D-brane superpotential are given as so:

$\displaystyle {{\partial }_{{\hat{\omega }}}}W\left( {\hat{\omega }} \right)=\sum\limits_{{\vec{k},m}}{{a\left( {\vec{k},m} \right){{z}^{{\vec{k}}}}}}{{s}^{m}}$

where $W$ is the Witten superpotential satisfying the Witten equation:

$\bar{\partial }{{u}_{i}}+\frac{{\bar{\partial }\bar{W}}}{{\partial {{u}_{i}}}}=0$

whose Picard-Lefschetz virtual fundamental cycles:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{{\left[ {{\bar{\mathcal{W}}}'_{{g,k}}^{{rig}}\left( {{{k}_{{{{j}_{1}}}}},...,{{k}_{{{{j}_{k}}}}}} \right)} \right]}}^{{vir}}}} \\ {\deg \quad 2\left( {\left( {{{{\hat{c}}}_{W}}-3} \right)\left( {g-1} \right)+k-\sum\limits_{i}{{{{\iota }_{{{{\gamma }_{i}}}}}}}} \right)-\sum\limits_{i}{{{{N}_{{{{\gamma }_{i}}}}}}}} \end{array}} \right.$

naturally induce Witten mirror moduli flops and flips bulk/boundary duality associated to the Aganagic-Klemm-Vafa disk instantons defined by the Gel’fand-Kapranov-Zelevinski system.

It can be shown that all flips and flops in the dual character-torus lattice of the Calabi-Yau moduli space corresponding to an M-theory compactification arise naturally in this way.

The Gel’fand-Kapranov-Zelevinski HGS, the D-Brane Superpotential, and the Witten Mirror

George Shiber

The D3-Brane, Self-Duality, the Super-Hamiltonian Action and the M5-Brane

M5-Branes, Taub-NUT Spaces, BPS States, and the Wess–Zumino–Witten–Novikov Model

George Shiber

The D3-Brane, Self-Duality, the Super-Hamiltonian Action and the M5-Brane

M5-Branes, Taub-NUT Spaces, BPS States, and the Wess–Zumino–Witten–Novikov Model

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