Topological Strings on Toric geometries in the presence of Lagrangian branes

Topological strings Antoniadis:1993ze ; Neitzke:2004ni ; Assi:2014exa correspond to the subsector of full superstrings spectrum that is invariant under a non-trivial sub algebra of the extended supersymmetry algebra. The observables of this subsector describe various topological properties of the spacetime on which the observables are defined. In physical 4d effective field theory the topological strings with the so-called $A-twist$ generate the the following F-term that depends on the vector multiplet moduli

where $\mathcal{W}^{2}$ is composed of the Weyl multiplet superfield $\mathcal{W}_{\alpha\beta}$ and $F_{g}$ is the genus $g$ topological string free energy. $F_{g}$ for $g\geq 2$ describe coefficients in the scattering amplitude of $2g-2$ graviphotons. The presence of D-branes imply consistent boundary conditions on the world sheet boundaries. In the topological sector the boundary conditions should preserve the BRST symmetry of the world sheet. From the target space (CY) perspective in the A-model the world sheet boundaries are mapped to a particular submanifold $L$ whose dimension is equal to the half that of CY Neitzke:2004ni and the restriction of the CY Kähler form $\omega$ to $L$ vanishes. In general, open strings may end on $L$ resulting in the wrapping of an A-model brane on L.
The problem of computing the unrefined topological string amplitudes on the toric CY in the presence of both external as well as internal branes was solved by the technique of the topological vertex Aganagic:2003db . On the other hand the refined topological string amplitude in the presence of internal branes turns out to be a subtle problem. Certain surface operators in 4d gauge theory can be realised by wrapping D4-branes on the Lagrangian 3-cycles of the CY 3-fold with the other two directions extending along transverse $\mathbb{R}^{2}\subset\mathbb{R}^{4}$. These two transverse directions correspond to equivariant parameters $\epsilon_{1},\epsilon_{2}$ or $t=e^{-i\epsilon_{1}},q=e^{i\epsilon_{2}}$. The topological branes can be put on the external non-compact legs or the internal compact legs of the toric diagram associated to the CY. Progress has been made in Kozcaz:2018ndf , where authors discuss refinement of holonomies for refined open topological string amplitudes. This was the case for topological branes on the external leg of the toric web diagram. In the presence of internal brane(s), there is a mismatch between the results as computed by using refinement of holonomy prescription and the geometric transition.
In this note we suggest that by utilising the equivalence of topological string amplitudes with equivariant indices Li:2004ef on framed moduli spaces, it may be possible to compute the refined open string amplitudes in the presence of internal branes. To this end, the authors Bruzzo:2010fk proposed a conjecture that equates the refined open string invariants of the special lagrangian branes in toric Calabi Yau 3-folds, with the Witten index of the supersymmetric quantum mechanical model describing the BPS states attached to the surface defect. The Witten index can be interpreted as the Euler characteristic of the moduli space described by the quantum mechanical model. In other words the conjecture Bruzzo:2010fk equates the generating function of refined open string invariants of the special lagrangian branes with the instanton partition function in the presence of surface defects. The conjecture was checked to be true to high orders in the asymptotic expansion. According to the geometric engineering argument the $M5$ branes wrapped on a submanifold of the CY give rise to effective five dimensional gauge theories with surface defect.
In the case of partially compactified toric web diagram the refined open topological string amplitude is equated to the generating function of $\chi_{y}$ genus on the same moduli space Li:2004ef ; Hollowood:2003cv ; Chuang:2013wpa . The Kähler parameter of this new compact direction corresponds to a massive adjoint hyper in the 5d gauge theory Hollowood:2003gr . We generalise the results of Bruzzo:2010fk and state the conjectures in the case of partial compactification of the resolved conifold, $O(-1)\oplus O(-1)\to\mathbb{P}^{1}$ and the partial compactification of the total space of the canonical bundle $\mathcal{O}(-2,-2)$ of $P^{1}\times P^{1}$. In the fully compactified case we give the expression for the generating function of elliptic genus of the defect moduli space and speculate about its relevance for the refined open topological string amplitude in the presence of internal branes.
In section (2) we restate the conjecture proved in Bruzzo:2010fk , about the equivalence of partition function on quiver moduli space with open Gromov-Witten invariants of certain non-compact Calabi-Yau 3-folds. In section (3) we compute the generating function of $\chi_{y}$ genus on the quiver moduli space. The expression we obtain is not a polynomial in the mass-deformation parameter $y$. In the next section (3.3) we use analytic continuation to write the $\chi_{y}$-genus as a polynomial in $y$. In section 3.4 we compute the generating function $\chi_{y}$-genus for quiver moduli space for rank $r=2$, which corresponds to the refined open topological string partition function on the partially compactified web of the Hirzebruch surface $F_{0}$. In section (4) we briefly discuss the identification of the Donaldson-Thomas partition function of CY3-fold and K-theory partition function of 5d supersymmetric gauge theory motivated by the geometric engineering argument. This identification was obtained for the unrefined case by making a change of variables. Using certain consistency conditions we propose a generalisation of this change of variables to the refined case. In the last section (5) we given an expression for the elliptic genus on the same moduli space and suggest that it may be related to the open topological string partition function on non-compact Calabi-Yau 3-fold whose corresponding web diagram is fully compactified.The appendices (A,B,C) contain the refined topological vertex amplitudes for the remaining preferred directions and the appendix (D) contain a summary of the virtual equivariant localisation and fixed point theorems.

A standard D-brane construction of $5d$ gauge theory with eight supercharges involves Bruzzo:2010fk $D6$-branes wrapped on the holomorphic curves in a non-compact $K3$ surface $S$. The type of quiver is defined by the intersection matrix of the configuration of $(-2)$-rational curves after suitable resolution. In the present context only $A_{r}$-type singularities are considered which are amenable to analysis with toric geometry techniques. So the type IIA vacuum we consider is given by $S\times S^{1}\times\mathbb{R}^{5}\equiv S\times C^{*}\times\mathbb{R}^{4}$. The low energy limit of $D6$-branes wrapped on these rational curves gives rise to $5d$ quiver gauge theory with eight supercharges. In the more general type IIA superstrings setup we can add D4-branes wrapping the special Lagrangian submanifolds, along with D2 branes ending on these D4-branes. D2-branes are wrapped on $(-2)$-rational curves. In other words D4-branes serve as defect operators and D2 branes correspond to the BPS states bound to the defect operators. The world volume of D4-brane is $L\times\mathbb{R}^{2}$, where $L\subset T\times C^{*}$ is the special Lagrangian submanifold and $\mathbb{R}^{2}\subset\mathbb{R}^{4}$. By construction a toric Calabi Yau 3-fold $X$ admits a symplectic $U(1)^{3}$ action and the resulting moment map $\rho:X\to\mathbb{R}^{3}$ to the so-called Delzant polytope. The collection of $U(1)^{3}$ preserving compact and non-compact rational holomorphic curves of $X$ define its toric skeleton. The toric skeleton is mapped by $\rho$ to a trivalent graph $\Delta$ in $\mathbb{R}^{3}$. The special lagrangian submanifold $L$ under consideration is topologically equivalent to $S^{1}\times\mathbb{R}^{2}$ and is mapped to a half real line which intersects a 1-face of the graph $\Delta$. The external lagrangian cycles intersect the non-compact components of the skeleton, whereas the internal lagrangian cycles intersect compact components. In the web diagrams we always show the branes wrapped on the lagrangian cycles by dashed lines.
The D4-branes are wrapped on the Lagrangian cycles of the CY3-fold, with two directions extending along one of the $\mathbb{R}^{2}s$ of the transverse $\mathbb{R}^{4}$. For the unrefined case the choice of $\mathbb{R}^{2}$ is immaterial. For the refined case the two $\mathbb{R}^{2}$s inside $\mathbb{R}^{4}$ are rotated by $q=e^{i\epsilon_{1}},t=e^{-i\epsilon_{2}}$ corresponding to a particular choice of complex structure. Depending on which $\mathbb{R}^{2}$ the D4-brane is extended along, it is called either a q-brane or a t-brane Kozcaz:2018ndf . In the presence of D4-branes the open topological string amplitudes are the generating functions of BPS degeneracies of D2 branes. These D2-branes are wrapped on smooth curves whose boundaries lie on D4-branes. The boundary conditions are necessary for the complete specification of the open string amplitudes and are given by gauge invariant combinations of holonomy operators.
The pure $SU(r),r\geq 2$ gauge theories in 5d can be engineered by certain non-compact CY 3-folds. To construct these 3-folds , the total space of $\mathcal{O}(-1)\oplus\mathcal{O}(-1)\to\mathbb{P}^{1}$ is orbifolded by the action $(z_{1},z_{2})\to(e^{\frac{2\pi i}{r}}z_{1},e^{-\frac{2\pi i}{r}}z_{2})$ on the fiber coordinates $(z_{1},z_{2})$. The resulting space is singular and can be resolved to a smooth CY 3-fold which contains $(r-1)$ geometrically ruled surfaces glued together. It is equivalent to the resolved $A_{r-1}$ fibration over $\mathbb{P}^{1}$. The compact part of the geometry consist of $r-1$ Hirzebruch surfaces glued together and the normal geometry of a base $\mathbb{P}^{1}$ in the $p-1$-th and $p$-th Hirzebruch surfaces is $\mathcal{O}(-r+2p-2)\oplus\mathcal{O}(r-2p)$. Formally allowing the value $r=1$ corresponds to the resolved conifold, the total space of $\mathcal{O}(-1)\oplus\mathcal{O}(-1)\to\mathbb{P}^{1}$.
Partially compactifying the web diagrams, which becomes non-planar, changes the CY3-fold to an elliptic CY3-fold which has the structure of the form $\mathbb{C}^{2}/\mathbb{Z}_{r}\times_{f}T^{2}$. M-theory compactification on this geometry engineers $5d$ $\mathcal{N}=1^{*}$ gauge theory with a single adjoint hypermultiplet Hollowood:2003cv ; Iqbal:2008ra .
The fields of quantum mechanical system, describing the BPS states bound to the surface operators, arise from low energy modes of $D2-D6$, $D2-D4$ and $D2-D2$ configuration of D-branes in type IIA strings. The field content is summarised as

An intricate analysis Bruzzo:2010fk shows that the moduli space of the supersymmetric vacua of the quantum mechanical model is isomorphic to the data defining ADHM type quiver. In a certain region of the moduli space it is interpreted in terms of certain generalised vector bundles.
The D2-brane effective action is derived by the dimensional reduction of the field contents of quiver diagram (1). The quiver diagram describes a vector space of supersymmetric flat directions parametrised by the fields $(A_{1},A_{2},I,J,B_{2},f,g,\sigma_{1},\sigma_{2})$ as follows

for hermitian inner product vector spaces $V_{1},V_{2}$ and $W$. The vacuum equations of the D2-branes effective action as given below define a moduli space

The moduli space parametrize $\mbox{U}(V_{1})\times\mbox{U}(V_{2})$ gauge inequivalent solutions to (2). An important result proven in Bruzzo:2010fk shows that the moduli space defined by the last set of equations is isomorphic to a different moduli space for generic values of Fayet Illuopolous parameters.The later moduli space comprises of stable representations of the enhanced ADHM quiver, also described in terms of framed torsion free sheaves on the projective plane. The virtual smoothness in our context means that moduli space of stable representations of the ADHM quiver can be embedded in a smooth variety which is a hyper kähler quotient.

The enhanced ADHM quiver is given in figure (3) along with the $relations$

A triple of vector spaces $(V_{1},V_{2},W)$ is assigned to the vertices $(e_{1},e_{2},e_{\infty})$ as a representation. Similarly the linear maps $(A_{1},A_{2},I,J,B,f,g)$ represent the arrows $(\alpha_{1},\alpha_{2},\xi,\eta,\beta,\phi,\gamma)$. The resulting quiver representation is moded out by the relations (5). If we define $(r,n_{1},n_{2}):=(\mbox{dim}(W),\mbox{dim}(V_{1}),\mbox{dim}(V_{2}))$, then this triple of positive integers define the numerical type of the quiver. The framing of the enhanced quiver representation corresponds to the existence of an isomorphism $\mbox{h}:\mbox{W}\to\mathbb{C}^{r}$. Moreover a stable¹¹1more precisely $\theta$-semistable representation of type $(r,n_{1},n_{2})$ implies the existence of a set of parameters $\theta=(\theta_{1},\theta_{2},\theta_{\infty})$ satisfying the relation $n_{1}\theta_{1}+n_{2}\theta_{2}+r\theta_{\infty}$ such that

• •

  Any subrepresentaion of type $(0,m_{1},m_{2})$ satisfies $m_{1}\theta_{1}+m_{2}\theta_{2}\leq 0$.

• •

  Any subrepresentaion of type $(r,m_{1},m_{2})$ satisfies $m_{1}\theta_{1}+m_{2}\theta_{2}+r\theta_{\infty}\leq 0$

The following result gives criterion for generic stability conditions:
Given a quiver representation $\mathcal{R}$ of type $(r,n_{1},n_{2})$ and theta parameters satisfying $\theta_{2}>0,\theta_{1}+n_{2}\theta_{2}<0$, then the following three statements are equivalent

• •

  1. $\mathcal{R}$ is $\theta$-semistable

• •

  2.$\mathcal{R}$ is $\theta$-stable

• •

  3. (a) $f:V_{2}\to V_{1}$ is injective and $g:V_{1}\to V_{2}$ is identically zero
  (b) The data $\mathcal{A}=(V_{1},W,A_{1},A_{2},I,J)$ satisfies the ADHM stability conditions.

Consider the vector spaces $V_{1},V_{2},W$ with positive definite dimensions $n_{1},n_{2},r$ and the direct sum of vector spaces

$X(r,n_{1},n_{2})$ admits a $\mbox{GL}(V_{1})\times\mbox{GL}(V_{2})$ action defined as

It is clear that $\mathcal{G}=\mbox{GL}(V_{1})\times\mbox{GL}(V_{2})$ preserves the subset $X_{0}(r,n_{1},n_{2})\in X(r,n_{1},n_{2})$ defined by (2). The space $X_{0}(r,n_{1},n_{2})$ parametrises the representations $\mathcal{R}=(V_{1},V_{2},W,A_{1},I,J,B_{2},f,g)$ with two framed representations $\mathcal{R}_{1},\mathcal{R}_{2}$ being equivalent if the corresponding points in $X_{0}(r,n_{1},n_{2})$ belong to the same $GL(V_{1})\times GL(V_{2})$ orbit. The space $X_{0}(r,n_{1},n_{2})$ can be projectivized to a scheme as follows

with the notation $A(X_{0}(r,n_{1},n_{2}))^{\mathcal{G},\chi^{n}}:=\{f\in A(X_{0}(r,n_{1},n_{2}))|f(g.x)=\chi(g)^{n}f(x)\forall g\in G\}$. There exists an open subscheme $\mathcal{N}_{\theta}^{s}(r,n_{1},n_{2})$ of $\mathcal{N}_{\theta}^{ss}(r,n_{1},n_{2})$ whose $\mathcal{G}$-orbits are $\chi$-stable²²2The character function $\chi:\mathcal{G}\to\mathbb{C}^{\times}$, where $\mathcal{G}$ acts on the space $X(r,n_{1},n_{2})$, furnishes a definition of $\chi$-stability. For this consider the existence of a polynomial $q(x)$ on $X(r,n_{1},n_{2})$ such that $q((g_{1},g_{2}).x)=\chi(g_{1},g_{2})^{n}q(x)$ for positive definite integer $n$. If $q(x_{0})\neq 0$, $x_{0}$ is called $\chi$-semistable. Moreover if $\Delta\subset\mathcal{G}$ acts trivially on $X(r,n_{1},n_{2})$ such that $dim(\mathcal{G}.x_{0})=dim(\mathcal{G}/\Delta)$ and the action of $\mathcal{G}$ on all such $x_{0}$ is closed, then then $x_{0}$ is called $\chi$-stable.. In the parameter space defined by the inequalities $\theta_{2}>0,\theta_{1}+n_{2}\theta_{2}<0$ the framed representations of the enhanced quiver that satisfy condition $(3)$ can simply be denoted by $\mathcal{N}(r,n_{1},n_{2})$ by dropping subscripts and superscripts.
The matter couplings in the quantum mechanical model corresponds to the three tautological bundles $\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{W}$ on the moduli space $\mathcal{N}(r,n_{1},n_{2})$. These bunldes are defined by $\mathcal{W}=\mathcal{O}^{\oplus r}_{\mathcal{N}(r,n_{1},n_{2})}$, $\mathcal{L}_{1}=\mbox{det}(\mathcal{V}_{1})$ and $\mathcal{L}_{2}=\mbox{det}(\mathcal{V}_{2})$. Moreover several copies of $\mathcal{L}_{1},\mathcal{L}_{2}$ can be tensored to give rise to the mixed line bundles $\mathcal{L}_{p_{1},p_{2}}=\mathcal{L}_{1}^{\otimes p_{1}}\otimes\mathcal{L}_{2}^{\otimes p_{2}}$.
The existence of the morphism $s:\mathcal{N}(r,n_{1},n_{2})\to\mathcal{M}(r,n_{1}-n_{2})$, where $\mathcal{M}(r,n_{1}-n_{2})$ is the moduli space of ADHM data of type $(r,n)$, plays a simplifying role in the application of the equivariant fixed point theorems. The tangent space at a point of $\mathcal{N}(r,n_{1},n_{2})$ is isomorphic to the difference $H^{1}(\mathcal{C}(\mathcal{R}))-H^{2}(\mathcal{C}(\mathcal{R}))$ for $\mathcal{R}=(A_{1},A_{2},I,J,B_{2},f)$ representing a point of $\mathcal{N}(r,n_{1},n_{2})$, where the complex $\mathcal{C}(\mathcal{R})$ is defined by

with the differentials defined as

Note that $H^{1}(\mathcal{C}(\mathcal{R}))$ parametrises the infinitesimal deformations of $\mathcal{R}$ and $H^{2}(\mathcal{C}(\mathcal{R}))$ the obstructions to the deformations. Moreover the conditions $H^{0}(\mathcal{C}(\mathcal{R}))=H^{3}(\mathcal{C}(\mathcal{R}))=0$ imply the stability of a framed representation. The equivariant virtual Euler characteristic of this determinant line bundle is arranged into a partition function of the quantum mechanical model.

The application of virtual equivariant localization requires the determination of fixed points of the torus action on the moduli space of the stable representations of the nested ADHM quivers. The moduli space, $\mathcal{N}(r,n+d,d)$, for which we compute the Euler characteristic, the $\chi_{y}$ genus and the elliptic genus is described by the stable representations of an enhanced ADHM quiver of type $(n+d,d,r)$. Note that the parameter $r$ is the rank of the 5d gauge group and $(n+d,d)$ are related to the number of $D2,D2^{\prime}$ branes. The BPS counting function is the Witten index of the supersymmetric quantum mechanics. The supersymmetric ground states are in one to one correspondence with cohomology classes in $\oplus_{i}H^{0,i}(\mathcal{N}(r,n_{1},n_{2}))$. Note that in the limit of decoupling the surface operator $\mathcal{N}(r,n+d,d)$ collapses to $\mathcal{M}(r,n)$ where $\mathcal{M}(r,n)$ is the suitably compactified and non-singular moduli space of instantons on $\mathbb{C}^{2}$. The later moduli space is isomorphic to the moduli space of rank $N$ torsion free sheaves with second Chern class $k$ on $\mathbb{P}^{2}$. So there exists a morphism $q:\mathcal{N}(r,n+d,d)\to\mathcal{M}(r,n)$. This morphism is equivariant with respect to the torus action. This makes it possible to classify the torus fixed loci in $\mathcal{N}(r,n+d,d)$.
In the following $(\mu,\nu)$ is a pair of nested Young diagrams satisfying the properties

• •

  $\nu\subseteq\mu$

• •

  if $(i,j)\in\mu\setminus\nu,then\quad(i+1,j)\notin\mu$

Similarly if we denote an ordered sequence of Young diagrams by $\bf{\underline{\mu}}=\{\mu^{1},...,\mu^{r}\}$ and $\bf{\underline{\nu}}=\{\nu^{1},...,\nu^{r}\}$ then it is a nested sequence if it is pairwise $(\mu^{a},\nu^{a})$ nested. The numerical type of this sequence is given by $(|\underline{\mu|},|\underline{\nu}|)$. For nested sequence the defining algebraic inequalities are described as

where $c^{a},e^{a}$ denote the number of columns of $\mu^{a}$ and $\nu^{a}$ respectively and $a=1,...,r$ and $i\geq 0$. The tangent space to the moduli space at a fixed point $(\mu^{a},\nu^{a})$ of the torus $\bf{T}=\mathbb{C}^{\times}\times\mathbb{C}^{\times}\times(\mathbb{C}^{\times})^{r}$ is regarded as an element of the representation ring of the torus action and is given by the expression

where $R_{1},...,R_{r},Q_{1},Q_{2}$ denote the one dimensional representations of $\bf{T}$ with their characters represented by $\rho_{1},...,\rho_{r},q_{1},q_{2}$ respectively.
According to this formula the fixed point locus is a finite set of points in one to one correspondence with pairs of nested sequences $(\underline{\mu},\underline{\nu})=(\mu^{a},\nu^{a})_{1\leq a\leq r}$ of length $r$. The type of Young diagrams is $(|\underline{\mu}|,|\underline{\nu}|)=(n+d,n)$.