arXiv subjects

Jan Troost

Publications and source records attributed to Jan Troost.

At least 19 recordsLinked to original sources

arXiv2026

Equivariant Interpolations in Topological Holography

We revisit equivariant Gromov-Witten theories on P1 and on P1 x C2. One can introduce three equivariant parameters associated to rotations of the sphere as well as the two planes. A number of points in the parameter space have known holographic duals. These include the symmetric orbifold point dual to the AdS3 x S3 x C2 string theory at string scale radius of curvature, the grand canonical Hurwitz theory and the product of two Kontsevich models. Within this framework, we discuss interpolations in the equivariant parameters. Firstly, we move between the small and large equivariant parameter regimes in Gromov-Witten theory on P1. At large equivariant parameter, the model is dominated by the pure topological gravity theories at the two fixed points while at small equivariant parameter the theory is equivalent to the grand canonical Hurwitz theory. The deformation is a solvable analogue for the interpolation in the transposition coupling in the moduli space of the AdS3/CFT2 duality. Moreover, we propose that the full equivariant correspondence between the Gromov-Witten theory on P1 x C2 and the symmetric orbifold of the equivariant plane can be embedded in string theory. On the boundary side of that correspondence, we analyze the scaling limit from the equivariant to the ordinary cohomology ring for the Hilbert scheme of points on the plane in terms of Jack symmetric polynomials. We explicitly compute several structure constants of the equivariant cohomology ring and point out their intriguing positivity and integrality properties.

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arXiv2026

The Open/Closed Gromov-Witten/Hurwitz Correspondence and Localized World Sheets for Completed Cycles

We discuss the open/closed version of the Gromov-Witten/Hurwitz correspondence. The duality equates the relative Gromov-Witten invariants and the count of covers of the target space with prescribed holonomies at boundaries. We clarify the projective large N limit as well as the role of the completed versus the ordinary cycles associated to the bulk and the boundary vertex operators respectively. We provide an example check of both the correspondence and the fact that cycles dual to closed strings need to be completed. Moreover, we identify the connected world sheets that contribute to an equivariantly localized amplitude in the bulk that is solely due to a completion term. We also propose a picture for the completed cycle combinatorics that involves a localization diagram glued to a cut-and-join string interaction.

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arXiv2025

Inverse Monoid Topological Quantum Field Theories and Open-Closed Grand Canonical Symmetric Orbifolds

We present an open-closed topological quantum field theory for inverse monoids which generalizes the theory of principle fiber bundles with finite gauge group over Riemann surfaces with boundary. The theory is constructed using the isomorphism between the semisimple inverse monoid algebra and a matrix algebra which lies at the heart of monoid structure and representation theory. An example that we study in detail is the Ivanov-Kerov monoid of partial permutations. We review motivations from string theory for the resulting grand canonical theory of covers with boundaries.

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arXiv2025

Mixed Solutions to the Liouville Equation

We enlarge the set of explicit classical solutions to the Liouville equation with three singularities to the cases with mixed hyperbolic and elliptic monodromies. We analyze the large hyperbolic monodromy limit of the solutions and the farthest geodesics looping one hyperbolic singularity. These two-dimensional geometries describe a time-symmetric spatial slice of a solution to three-dimensional general relativity. The geodesics are reinterpreted as snapshots of horizons of evolving black holes. We study the spatial slice with three horizons of very heavy black holes in some detail. We use uniform saddle point integration to present the Liouville and heavy black hole geometries in terms of simpler special functions. These make a detailed analysis of mixed particle and black hole geometries possible.

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arXiv2025

The String Dual to Two-dimensional Yang-Mills Theory Revisited

We propose that chiral two-dimensional Yang-Mills theory on a Riemann surface is dual to a deformed stationary subsector of the Gromov-Witten theory of that Riemann surface. Firstly, we argue that the algebraic structure that underlies the large N limit of the chiral gauge theory is a partial permutation Frobenius algebra of observables which codes covering maps of all degrees simultaneously. Secondly, we exploit the Gromov-Witten/Hurwitz correspondence to interpret chiral Yang-Mills theory as a finite deformation of a Gromov-Witten theory by an area-dependent transposition interaction and an operator that codes a compactification of Hurwitz space. The proposed string dual manifestly includes an integral over the moduli space of Riemann surfaces as well as the identification of closed string states as completed cycles.

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arXiv2024

Orbifolded Elliptic Genera of Non-Compact Models

We revisit the flavored elliptic genus of the N=2 superconformal cigar model and generalize the analysis of the path integral result to the case of real central charge. It gives rise to a non-holomorphic modular covariant function generalizing completed mock modular forms. We also compute the genus for angular orbifolds of the cigar and Liouville theory and decompose it in terms of discrete and continuous contributions. The orbifolded elliptic genus at fractional level is a completed mock modular form with a shadow related to U$(1)$ modular invariants at rational radius squared. We take the limit of the orbifolded genera towards a weighted ground state index and carefully interpret the contributions. We stress that the orbifold cigar and Liouville theories have a maximal and a minimal radius, respectively.

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arXiv2024

Symmetric Group Gauge Theories and Simple Gauge/String Dualities

We study two-dimensional topological gauge theories with gauge group equal to the symmetric group $S_n$ and their string theory duals. The simplest such theory is the topological quantum field theory of principal $S_n$ fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of $n$ are coded in a partial permutation algebra. We propose a generalization of the partial permutation algebra to any symmetric orbifold topological quantum field theory and show that the latter theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov-Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

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arXiv2023

The Operator Rings of Topological Symmetric Orbifolds and their Large N Limit

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise.

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arXiv2023

More On Pure Gravity with a Negative Cosmological Constant

We identify an ambiguity in the Chern-Simons formulation of three-dimensional gravity with negative cosmological constant that originates in an outer automorphism of the Lie algebra sl(2,R). It has important consequences for the stability of the theory in a space-time with boundary. We revisit the classical equivalence of three-dimensional gravity with a boundary Liouville theory both on and off the mass shell. Moreover, we provide further details on the quantum equivalence, the gauge symmetry that renders the spectrum diagonal, as well as the relation between asymptotically AdS3 metrics and polar boundary conditions. We thus set the proposal that the Liouville conformal field theory serves as a definition of a unitary theory of pure gravity in three dimensions with negative cosmological constant on a more stable footing.

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arXiv2023

The Chiral Ring of a Symmetric Orbifold and its Large N Limit

We analyze the chiral operator ring of the symmetric orbifold conformal field theory on the complex two-plane. We compute the large N limit of the ring and exhibit its factorized leading order behaviour. We moreover calculate all structure constants at the subleading and sub-subleading order. These features are coded as properties of the symmetric group and we review the relevant mathematical theorems on the product of conjugacy classes in the center of the group algebra. We illustrate the efficiency of the formalism by iteratively computing broad classes of higher point extremal correlators. We point out generalizations of our simplest of models and argue that our combinatorial analysis is relevant to the organization of the large N perturbation theory of generic symmetric orbifolds.

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arXiv2023

Topological Two-Dimensional Gravity on Surfaces with Boundary

We solve two-dimensional gravity on surfaces with boundary in terms of contact interactions and surface degenerations. The known solution of the bulk theory in terms of a contact algebra is generalized to include boundaries and an enlarged set of boundary operators. The latter allow for a linearization of the Virasoro constraints in terms of an extended integrable KdV hierarchy.

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arXiv2022

Long Strings and Quasinormal Winding Modes

We compute the path integral for a particle on the covering group of SL(2,R) using a decomposition of the Lie algebra into adjoint orbits. We thus intuitively derive the Hilbert space of the particle on the group including discrete and continuous representations. Next, we perform a Lorentzian hyperbolic orbifold of the partition function and relate it to the Euclidean BTZ partition function. We use the particle model to inform further discussion of the spectral content of the one loop vacuum amplitude for strings on BTZ black hole backgrounds. We argue that the poles in the loop integrand code contributions of long string modes that wind the black hole. We moreover identify saddle point contributions of quasinormal winding modes.

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arXiv2022

Path Integrals on sl(2,R) Orbits

We quantise orbits of the adjoint group action on elements of the sl(2,R) Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL(2,R) on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl(2,R) representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL(2,R) of PSL(2,R) as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantisation problem by indicating applications.

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arXiv2021

Twisted Strings in Three-dimensional Black Holes

We revisit the classical dynamics of fundamental strings in BTZ black holes with NSNS flux. We analyse probe strings in the black hole background and determine their classical energy using a Nambu-Goto action. Three-dimensional gravity has a non-local effect on the metric surrounding a massive object which can be measured by winding strings. We also study the behaviour of short and long strings in the Wess-Zumino-Witten orbifold and show that their energy matches the long string probe analysis. In the process, we fix subtleties in the Lorentzian choice of NSNS flux. In Euclidean signature, we analyse the one-loop fundamental string partition function on the BTZ black hole and interpret the off-shell degrees of freedom as those of a coset orbifold. Finally, we speculate on how to identify the quantum Lorentzian string spectrum from the Euclidean orbifold partition function.

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arXiv2021

The $AdS_3 \times S^1$ Chiral Ring

We study $AdS_3 \times S^1 \times Y$ supersymmetric string theory backgrounds with Neveu-Schwarz-Neveu-Schwarz flux that are dual to ${\cal N}=2$ superconformal theories on the boundary. We classify all worldsheet vertex operators that correspond to space-time chiral primaries. We compute space-time chiral ring structure constants for operators in the zero spectral flow sector using the operator product expansion in the worldsheet theory. We find that the structure constants take a universal form that depends only on the topological data of the ${\cal N}=2$ superconformal theory on $Y$.

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arXiv2021

String Scale Thermal Anti-de Sitter Spaces

We study finite temperature string scale $AdS_3$ backgrounds. One background is $AdS_3 \times S^1 \times T^2$ in which the anti-de Sitter space-time and the circle are at the radius $\sqrt{\alpha'}$. Using path integral techniques, we show that the bulk spectrum includes a continuum of states as well as Ramond-Ramond ground states that agree with those of the symmetric orbifold of the two-torus after second quantization. We also examine the one-loop free energy of the background $AdS_3 \times S^1$ at curvature radius $\sqrt{2 \alpha'/3}$. In the space-time NSNS sector, the string theory spontaneously breaks conformal symmetry as well as R-charge conjugation symmetry. We prove that the minimum in the boundary energy is reached for a singly wound string. In the RR sector, we classify the infinite set of ground states with fractional R-charges. Moreover, we remark on the behaviour of critical temperatures as the curvature scale becomes smaller than the string scale. In an appendix, we derive the Hawking-Page transition in string theory by integrating a world sheet one-point function.

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arXiv2020

Superstrings in Thermal Anti-de Sitter Space

We revisit the calculation of the thermal free energy for string theory in three-dimensional anti-de Sitter spacetime with Neveu-Schwarz-Neveu-Schwarz flux. The path integral calculation is exploited to confirm the off-shell Hilbert space and we find that the Casimir of the discrete representations of the isometry group takes values in a half-open interval. We extend the free energy calculation to the case of superstrings, calculate the boundary toroidal twisted partition function in the Ramond-Ramond sector, and prove lower bounds on the boundary conformal dimension from the bulk perspective. We classify Ramond-Ramond ground states and construct their second quantized partition function. The partition function exhibits intriguing modular properties.

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arXiv2020

The Topological Symmetric Orbifold

We analyse topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the quantum cohomology of the Hilbert scheme on the boundary.

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