SciPost Submission Page
Branes, quivers and wavefunctions
by Taro Kimura, Miłosz Panfil, Yuji Sugimoto, Piotr Sułkowski
 Published as SciPost Phys. 10, 051 (2021)
Submission summary
As Contributors:  Milosz Panfil 
Preprint link:  scipost_202012_00012v1 
Date accepted:  20210219 
Date submitted:  20201214 12:21 
Submitted by:  Panfil, Milosz 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider a large class of branes in toric strip geometries, both nonperiodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wavefunctions in various polarizations. We determine operations on quivers, as well as $SL(2,\mathbb{Z})$ transformations, which correspond to changing positions of these branes. Our results prove integrality of BPS multiplicities associated to this class of branes, reveal how they transform under changes of polarization, and imply all other properties of brane amplitudes that follow from the relation to quivers.
Published as SciPost Phys. 10, 051 (2021)
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021215 Invited Report
 Cite as: Anonymous, Report on arXiv:scipost_202012_00012v1, delivered 20210215, doi: 10.21468/SciPost.Report.2560
Report
The paper by Kimura, Panfil, Sugimoto and Sulkowski considers partition functions of branes on a class of toric varieties called strip geometries (and their periodic generalization), from three perspectives: that of the open topological string, of the Apolynomial, and of quiver representations, generalizing previous work on the geometries $\mathbb{C}^3$ and the conifold.
The computations of the paper are anchored in the topological vertex formalism, which permits the computation of the open topological string partition function on toric geometries in the presence of Lagrangian branes. From this vantage point, the authors derive the Apolynomials for such geometries, as well as, notably, the associated quivers. The former are difference operators which annihilate the partition function; their $q \rightarrow 1$ limit coincides with the mirror curve of the underlying toric geometry. Given the partition function, they are straightforward to derive. The latter give rise to socalled motivic generating series; the authors identify the quiver associated to a toric brane configuration by matching this series to the partition function.
Having studied the behavior of the partition function, the Apolynomial, and the associated quiver representation upon changing the position of the Lagrangian brane, the authors proceed to investigate how this operation can be encoded in terms of an integration of the partition function against an integration kernel. This is motivated by previous work in the literature, in which the brane partition function is interpreted as a wavefunction, and the change of brane position as a canonical transformation on the underlying system. Evaluating such integrals in the semiclassical limit (and permitting themselves a rescaling of the Kaehler parameters), the authors conclude that only a subgroup of the group of canonical transformations, generated by the operators $T$ and $S^2$, correspond to the operation of moving branes. Finally, the authors address the same question from the vantage point of the Apolynomial.
The paper concludes by working out numerous examples in detail.
This paper advances the investigation of an as yet somewhat mysterious relation, that between the open topological string partition function, its wave function interpretation, and the nature of the difference operator that annihilates it, on the computational front. It is clearly written and calculations are well presented. It contains novel results which should help in uncovering the underlying structures. I therefore recommend publication.
Anonymous Report 1 on 2021112 Invited Report
 Cite as: Anonymous, Report on arXiv:scipost_202012_00012v1, delivered 20210112, doi: 10.21468/SciPost.Report.2392
Report
The knotsquiver correspondence (P. Kucharski, M. Reineke, M. Stosic and P. Sulkowski, BPS states, knots and quivers, Phys. Rev. D 96, 121902 (2017)) states that several properties of the topological string on a toric threefold with a brane configuration can be encoded in a quiver, an oriented graph. Quiver representation theory is a very rich and important branch of algebra, with several applications in physics. In string theory this setup can be used to engineer a knot, for example with a brane configuration on the resolved conifold. Then the topological partition functions, which correspond to certain polynomials associated with the knot, can be recast as a quiver generating series.
The paper expands the scope of this correspondence in a different setup, namely strip geometries, a class of toric threefolds without compact 4cycles, with the insertion of a basic AganagicVafa lagrangian brane. In this case the brane partition functions behave as wavefunctions in different polarizations. The question addressed in the paper is how is this behaviour compatible with the quiver description. The authors determine operations on quivers which reproduce the effect of moving the branes or doing an $SL(2,Z)$ transformation, and discuss the consequences of these properties.
For a given brane in a toric strip geometry the open topological string partition function can be computed using the formalism of the topological vertex. These computations are reviewed in Section 2 of the paper (and the formalism slightly extended in Appendix A). The resulting wave function $\psi (x)$ is annihilated by an operator $\hat{A} (\hat{x} , \hat{y})$, the socalled quantum curve (where $\hat{x}$ acts by multiplication and $\hat{y}$ by $\hat{y} f(x) = f (q x)$). The name originates from the fact that the $q \longrightarrow 1$ limit of this operator defines the classical mirror curve.
One of the main results of the paper is that this brane wave function can be rewritten as a motivic generating series for a symmetric quiver. Such motivic series arise in the theory of DonaldsonThomas invariants; they are many variable series which contain information about the moduli spaces of quiver representations and which can be decomposed uniquely as a product of quantum dilogarithms whose exponents are DonaldsonThomas invariants. This connection between physics, representation theory and DonaldsonThomas theory is one of the most interesting aspects of the knotsquiver correspondence. The authors extend the same arguments also to the case where the strip geometry is periodic where two opposite edges of the toric diagram of the strip, in the first and last vertex, are identified.
The topological string partition function for branes, on toric manifolds, is conjectured to behave as a wave function. A change in polarization corresponds to changing the position of a single brane on a fixed toric manifold from, say the $i^{th}$ to the $j^{th}$ vertex in the strip. This transformation is implemented via an integral transform determined by an element of $SL(2;Z)$. The authors check explicitly that only a subset of $SL(2;Z)$, namely those transformations generated by $T$ and $S^2$ lead to wavefunctions which can be written in quiver form. Physically these are the transformations which correspond to changing the location of a brane. The $S$ transformation makes an appearance only in special situations, such as the conifold where the Apolynomial is symmetric. The analysis of the integral kernel which determines the transformations is done at the semiclassical level; the authors conjecture it should hold at the full quantum level too.
This behaviour is traced back to the action of $SL(2;Z)$ on the Apolynomial in Section 4. There the authors show by direct computations that the Apolynomial is preserved only by the $S^2$ and $T$ operations generically, and by $S$ in special cases (affine space and resolved conifold). The authors also discuss the effect of these operations on the physical setup, as moving a brane from one vertex to another or within the same vertex.
The behaviour of the brane partition function described above, is finally checked explicitly in a few examples in Section 5.
The paper is well written and clear, and contains enough details to follow the computations (although I haven't gone through all of them). The results are a nice confirmation of certain aspects of the conjectural knotquiver correspondence. They are important since they provide further evidence for the correspondence and help to sharpen it. I therefore recommend the paper for publication.