Showing papers on "Topological string theory published in 2019"

A Solvable Deformation of Quantum Mechanics

Abstract: The conventional Hamiltonian $H= p^2+ V_N(x)$, where the potential $V_N(x)$ is a polynomial of degree $N$, has been studied intensively since the birth of quantum mechanics. In some cases, its spectrum can be determined by combining the WKB method with resummation techniques. In this paper we point out that the deformed Hamiltonian $H=2 \cosh(p)+ V_N(x)$ is exactly solvable for any potential: a conjectural exact quantization condition, involving well-defined functions, can be written down in closed form, and determines the spectrum of bound states and resonances. In particular, no resummation techniques are needed. This Hamiltonian is obtained by quantizing the Seiberg-Witten curve of $\mathcal{N}=2$ Yang-Mills theory, and the exact quantization condition follows from the correspondence between spectral theory and topological strings, after taking a suitable four-dimensional limit. In this formulation, conventional quantum mechanics emerges in a scaling limit near the Argyres-Douglas superconformal point in moduli space. Although our deformed version of quantum mechanics is in many respects similar to the conventional version, it also displays new phenomena, like spontaneous parity symmetry breaking.

Elliptic Blowup Equations for 6d SCFTs. II: Exceptional Cases

Abstract: The building blocks of 6d $(1,0)$ SCFTs include certain rank one theories with gauge group $G=SU(3),SO(8),F_4,E_{6,7,8}$. In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d $\mathcal{N}=2$ superconformal $H_{G}$ theories. We also observe an intriguing relation between the $k$-string elliptic genus and the Schur indices of rank $k$ $H_{G}$ SCFTs, as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters.

Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases

Abstract: The building blocks of 6d (1, 0) SCFTs include certain rank one theories with gauge group G = SU(3), SO(8), F4, E6,7,8. In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d $$ \mathcal{N} $$ = 2 superconformal HG theories. We also observe an intriguing relation between the k-string elliptic genus and the Schur indices of rank k HG SCFTs, as a generalization of Del Zotto-Lockhart’s conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters.

Topological strings, strips and quivers

Abstract: We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Abstract: This work develops a new method to calculate non-perturbative corrections in one-dimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrodinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the trans-series structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential and the quantum mirror curves of local $${\mathbb {P}}^2$$ and local $${\mathbb {F}}_0$$ . In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors.

Topological string entanglement

Abstract: We investigate how topological entanglement of Chern-Simons theory is captured in a string theoretic realization. Our explorations are motivated by a desire to understand how quantum entanglement of low energy open string degrees of freedom is encoded in string theory (beyond the oft discussed classical gravity limit). Concretely, we realize the Chern-Simons theory as the worldvolume dynamics of topological D-branes in the topological A-model string theory on a Calabi-Yau target. Via the open/closed topological string duality one can map this theory onto a pure closed topological A-model string on a different target space, one which is related to the original Calabi-Yau geometry by a geometric/conifold transition. We demonstrate how to uplift the replica construction of Chern-Simons theory directly onto the closed string and show that it provides a meaningful definition of reduced density matrices in topological string theory. Furthermore, we argue that the replica construction commutes with the geometric transition, thereby providing an explicit closed string dual for computing reduced states, and Renyi and von Neumann entropies thereof. While most of our analysis is carried out for Chern-Simons on S3, the emergent picture is rather general. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry.

Twisted S-Duality

Abstract: S-duality is a nontrivial self-duality of type IIB string theory that exchanges strong and weak coupling. We give a mathematically rigorous description of how S-duality acts on a low-energy supersymmetry-protected sector of IIB string theory, using a conjectural description of such protected sectors in terms of topological string theory. We then give some applications which are of relevance to Geometric Langlands Theory and the representation theory of the Yangian.

Recent developments in topological string theory

Abstract: This review summarizes the recent developments in topological string theory from the author’s perspective, mostly focusing on aspects of research in which the author is involved. After a brief overview of the theory, we discuss two aspects of these developments. First, we discuss the computational progress in the topological string partition functions on a class of elliptic Calabi-Yau manifolds. We propose to use Jacobi forms as an ansatz for the partition function. For non-compact models, the techniques often provide complete solutions, while for compact models, though it is still not completely solvable, we compute to higher genus than previous works. Second, we explore a remarkable connection of refined topological strings on a class of non-compact toric Calabi-Yau threefolds with non-perturbative effects in quantum-mechanical systems. The connections provide rarely available exact quantization conditions for quantum systems and new insights on non-perturbative formulations of topological string theory.

Notes on the Hemisphere

Abstract: In these notes, we provide an introduction to the hemisphere partition function of 2d $(2,2)$ supersymmetric gauge theories, and discuss its relation to the "D-brane central charge" which were studied in superstring theory, in 2d supersymmetric quantum field theory, and in topological string theory. We also discuss relation to "macroscopic loop" in matrix models. They are mostly reviews of the work by the authors, but contains some new results such as the partition function for a rotated supersymmetry as well as the differential equations.

From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras

Abstract: We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum spectral curve whose generalized periods define a new set of moduli. We define homology cycles and differential forms of the quantum spectral curve, allowing to derive quantum analogs of the form-cycle duality and Riemann bilinear identities of classical geometry. A tau-function is introduced for this system in the form of a theta-series and in such a way that the variations of its coefficients with respect to moduli, isomonodromic or not, can be computed as quantum period integrals. This lays new grounds to relate our study to that of integrable hierarchies, isomonodromic deformation of meromorphic connections and non-perturbative topological string theory. In turn, we define amplitudes on the quantum spectral curve which have an interpretation in conformal field theory when the Lie algebra is assumed to be simply-laced: they coincide with correlation functions involving twisted chiral fields of an affine Lie algebra at level one. The singularities at the punctures are interpreted as primary fields of the associated Casimir W-algebra. The amplitudes are moreover related by W-constraints, so-called loop equations, allowing one to compute recursively a certain asymptotic expansion of the tau-function, namely the one corresponding both to the heavy-charge regime of conformal field theory and to the weak-coupling regime of topological string theory.

Topological string entanglement

Abstract: We investigate how topological entanglement of Chern-Simons theory is captured in a string theoretic realization. Our explorations are motivated by a desire to understand how quantum entanglement of low energy open string degrees of freedom is encoded in string theory (beyond the oft discussed classical gravity limit). Concretely, we realize the Chern-Simons theory as the worldvolume dynamics of topological D-branes in the topological A-model string theory on a Calabi-Yau target. Via the open/closed topological string duality one can map this theory onto a pure closed topological A-model string on a different target space, one which is related to the original Calabi-Yau geometry by a geometric/conifold transition. We demonstrate how to uplift the replica construction of Chern-Simons theory directly onto the closed string and show that it provides a meaningful definition of reduced density matrices in topological string theory. Furthermore, we argue that the replica construction commutes with the geometric transition, thereby providing an explicit closed string dual for computing reduced states, and Renyi and von Neumann entropies thereof. While most of our analysis is carried out for Chern-Simons on S^3, the emergent picture is rather general. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry.

Some aspects of topological string theory

Abstract: In the thesis we study topological aspects of string and M-theory. We derive a large N holomorphic string expansion for the Macdonald-deformed U(N) Yang-Mills theory on a closed Riemann surface. Macdonald deformation of two-dimensional Yang-Mills theory computes entropies of BPS black holes and it is also dual to refined topological string theory. In the classical limit, the expansion defines a new β-deformation for Hurwitz theory of branched covers wherein the refined partition function is a generating function for certain parameterized Euler characters. We also apply the large N expansion to observables corresponding to open surfaces and Wilson loops. We study AKSZ constructions for the A and B sigma-models of topological string theory within a double field theory formulation that incorporates backgrounds with geometric and non-geometric fluxes. AKSZ formulations provide natural geometric methods for constructing BV quantized sigma-models. After a section condition, we relate the Aand B-model to a three-dimensional Courant sigma-model, corresponding to a generalized complex structure, which reduces to the Aor B-models on the boundary. We introduce S-duality at the level of the three-dimensional sigma-model based on the generalized complex structure, which exchanges the related AKSZ field theories, and interpret it as topological S-duality of the Aand B-models. We also study AKSZ constructions for closed topological membranes on G2-manifolds. These membranes were originally introduced to be the worldvolume formulation for topological M-theory, which is intended to capture a topological sector of physical Mtheory. We propose two inequivalent AKSZ membrane theories, in each of which the two existing topological membranes appear as different gauge fixed versions, and their dimensional reductions give new AKSZ constructions for the topological A-model. We show that the two AKSZ membrane models originate through worldvolume dimensional reduction of a single AKSZ three-brane theory, which gives the higher Courant bracket of exceptional generalized geometry of M-theory as the underlying derived bracket. The thesis is based on three papers [1–3].

Special Geometry, Hessian Structures and Applications

Abstract: The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometries of four- and five-dimensional vector multiplets. We formulate ${\cal N}= 2$ theories in terms of Hessian structures and give various concrete applications of Hessian geometry, ranging from static BPS black holes in four and five space-time dimensions to topological string theory, emphasizing the role of the Hesse potential. We also discuss the r-map and c-map which relate the special geometries of vector multiplets to each other and to hypermultiplet geometries. By including time-like dimensional reductions, we obtain theories in Euclidean signature, where the scalar target spaces carry para-complex versions of special geometry.

Topological string theory and applications

Abstract: This thesis focuses on various applications of topological string theory based on different types of Calabi-Yau (CY) manifolds. The first type considered is the toric CY manifold, which is intimately related to spectral problems of difference operators. The particular example considered in the thesis closely resembles the Harper-Hofstadter model in condensed matter physics. We first study the non-perturbative sectors in this model, and then propose a new way to compute them using topological string theory. In the second part of the thesis, we consider partition functions on elliptically fibered CY manifolds. These exhibit interesting modular behavior. We show that for geometries which don't lead to non-abelian gauge symmetries, the topological string partition functions can be reconstructed based solely on genus zero Gromov-Witten invariants. Finally, we discuss ongoing work regarding the relation of the topological string partition functions on the so-called Higgsing trees in F-theory.

Black Hole Partition Function and Airy Equation

Abstract: We have used the Topological String Theory partition function in the scaling limit, to relate the Black Hole partition function and Topological String Theory partition function.