arXiv subjects

Victor A. Rodriguez

Publications and source records attributed to Victor A. Rodriguez.

At least 19 recordsLinked to original sources

arXiv2026

The Runkel-Watts string

We introduce and solve a new family of two-dimensional string theories obtained by coupling Liouville CFT to the generalized Runkel-Watts CFT. By taking an appropriate limit of the complex Liouville string, we derive an all-genus formula for its string amplitudes, and formulate a duality with a matrix integral. We explain that it can be interpreted as 2d SU(2) Yang-Mills theory coupled to gravity. We show that it also directly relates to other minimal string constructions such as the A-series minimal string.

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arXiv2026

$c=1$ strings as a matrix integral

We study the perturbative $S$-matrix of the $c=1$ string and show that it admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve $\mathsf{x}(z) = 2\sqrt{2}\cos(z)$, $\mathsf{y}(z)=\sin(z)$. Combined with the famous duality to matrix quantum mechanics, this establishes a triality between three formulations of the theory: the worldsheet description, matrix quantum mechanics, and a matrix integral. Starting from the intersection number expressions for the complex Liouville string, we derive closed-form Feynman rule expressions for the $c=1$ amplitudes as intersection numbers on the moduli space of Riemann surfaces. The intersection theory naturally computes amplitudes corresponding to a discretized target space where momentum is conserved only modulo an integer. The physical $S$-matrix elements are recovered by restriction to the first `Brillouin zone' and analytic continuation to Lorentzian kinematics. We prove that these amplitudes satisfy perturbative spacetime unitarity directly from the intersection theory expressions, and show that they satisfy a Mirzakhani-type recursion relation. We show detailed agreement with the known matrix quantum mechanics results, providing strong evidence for the triality.

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arXiv2026

Asymptotic bootstrap for unitary matrix integrals at complex coupling

We apply an asymptotic bootstrap estimate method to the non-perturbative study of unitary matrix integrals. The method combines exact recursion relations with asymptotic control of large modes to achieve very high numerical precision without relying on positivity or semidefinite programming. We demonstrate its effectiveness in large-$N$ unitary matrix models by computing Wilson loop expectation values with sensitivity to exponentially small instanton effects and validating them against analytical instanton calculations. We further use the method to explore phase diagrams of unitary matrix models in complex 't Hooft coupling space, where positivity is absent, and observe that Stokes lines provide a useful proxy for additional phase boundaries. Our results show that asymptotic bootstrap estimates offer a practical and precise tool for probing the non-perturbative structure of unitary matrix integrals.

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arXiv2025

ADE Minimal Strings and Multi-Matrix Duals

We revisit ADE minimal string theories, focusing on the D- and E-series minimal models coupled to Liouville theory. Unlike the A-series, whose duals are solvable two-matrix models, these theories are conjectured to correspond to unsolvable four-matrix integrals. We compute sphere four-point and torus one-point amplitudes in the AMS, DMS, and EMS via direct numerical integration over moduli space, confirming/disproving some known results and providing new data where matrix-model predictions are unavailable. From amplitudes with conformal boundaries, we find evidence for multi-matrix structure in the D-series, including scaled ramp behavior in cylinder diagrams and deviations from the ZZ-instanton sector of two-matrix models. We also perform a preliminary positivity bootstrap to constrain critical points of the multi-matrix models relevant to the DMS string.

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arXiv2025

Goldilocks and the bootstrap

We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.

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arXiv2025

The complex Liouville string

We introduce the complex Liouville string, a solvable string theory defined by coupling two Liouville theories with complex conjugate central charges $c \in 13+i \mathbb{R}$ on the worldsheet. We compute its amplitudes from first principles and establish a duality with a double-scaled two-matrix integral. We also analyze general worldsheet boundaries and non-perturbative effects in the genus expansion. By expressing the complex Liouville string as a 2d dilaton gravity theory with a sine potential, we show that it admits both AdS$_2$ and dS$_2$ vacua.

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arXiv2025

The complex Liouville string: the matrix integral

We propose a duality between the complex Liouville string and a two-matrix integral. The complex Liouville string is defined by coupling two Liouville theories with complex central charges $c = 13 \pm i λ$ on the worldsheet. The matrix integral is characterized by its spectral curve which allows us to compute the perturbative string amplitudes recursively via topological recursion. This duality constitutes a controllable instance of holographic duality. The leverage on the theory is provided by the rich analytic structure of the string amplitudes that we discussed in arXiv:2409.18759 and allows us to perform numerous tests on the duality.

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arXiv2025

The complex Liouville string: the worldsheet

We introduce a new two-dimensional string theory defined by coupling two copies of Liouville CFT with complex central charge $c=13\pm i λ$ on the worldsheet. This string theory defines a novel, consistent and controllable model of two-dimensional quantum gravity. We use the exact solution of the worldsheet theory to derive stringent constraints on the analytic structure of the string amplitudes as a function of the vertex operator momenta. Together with other worldsheet constraints, this allows us to completely pin down the string amplitudes without explicitly computing the moduli space integrals. We focus on the case of the sphere four-point amplitude and torus one-point amplitude as worked examples. This is the first in a series of papers on the complex Liouville string: three subsequent papers will elucidate the holographic duality with a two-matrix integral, discuss worldsheet boundaries and non-perturbative effects, and connect the theory to de Sitter quantum gravity.

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arXiv2025

The complex Liouville string: worldsheet boundaries and non-perturbative effects

We investigate general observables of the complex Liouville string with worldsheet boundaries. We develop a universal formalism that reduces such observables to ordinary closed string amplitudes without boundaries, applicable to any worldsheet string theory, but particularly simple in the context of 2d or minimal string theories. We apply this formalism to the duality of the complex Liouville string with the matrix integral proposed in arXiv:2409.18759 and arXiv:2410.07345 and showcase the formalism by finding appropriate boundary conditions for various matrix model quantities of interest, such as the resolvent or the partition function. We also apply this formalism towards the computation of non-perturbative effects on the worldsheet mediated by ZZ-instantons. These are known to be plagued by extra subtleties which need input from string field theory to resolve. These computations probe and uncover the duality between the complex Liouville string and the matrix model at the non-perturbative level.

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arXiv2024

A Two-Dimensional String Cosmology

We study two-dimensional string theory on a time-dependent background, whose worldsheet description consists of Liouville theory at central charge c = 1 and Liouville theory at central charge c = 25, together with the conformal ghosts. We compute the tree-level three-point and four-point cosmological wavefunctions in string perturbation theory. The latter is evaluated numerically by decomposing the Liouville four-point correlation functions into Virasoro conformal blocks and three-point function coefficients and integrating over the moduli space of the four-punctured sphere string diagram. This computation numerically confirms a surprisingly simple conjectural result for the four-point wavefunction whose physical interpretation remains to be clarified.

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arXiv2024

The Virasoro Minimal String

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge $c\geq 25$ coupled to timelike Liouville CFT with central charge $26-c$. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter $c$ and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

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arXiv2023

Scattering From $(p,q)$-Strings in $\text{AdS}_5 \times \text{S}^5$

Motivated by understanding the scattering of gravitons and their superpartners from extended $(p, q)$-strings in type IIB string theory via AdS/CFT, we study an integrated two-point function of stress tensor multiplet operators in the presence of a half-BPS line defect in ${\cal N} = 4$ $SU(N)$ super-Yang-Mills theory. We determine this integrated correlator at the five lowest non-trivial orders in $1/\sqrt{N}$ at fixed Yang-Mills coupling and $θ$ angle. Our calculations are performed explicitly when the line defect is a Wilson line, in which case we find a finite number of perturbative contributions at each order in $1/\sqrt{N}$, as well as instanton contributions. Using $SL(2, \mathbb{Z})$ transformations, our results can also be applied to Wilson-'t Hooft line defects dual to extended $(p, q)$-strings in the bulk. We analyze features of these integrated correlators in the weak coupling expansion by comparing with open-closed amplitudes of type IIB string theory on $\text{AdS}_5\times \text{S}^5$, as well as in its flat space limit. We predict new higher-derivative interaction vertices on the D1-brane and, more generally, on $(p,q)$-strings.

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arXiv2023

The Torus One-Point Diagram in Two-Dimensional String Cosmology

We calculate numerically the torus one-point string diagram in the two-dimensional string cosmology background by decomposing the one-point functions in $c=1$ and $c=25$ Liouville CFT into torus one-point Virasoro conformal blocks and integrating over the fundamental domain of the torus moduli space. We find a remarkably simple result as a function of the outgoing closed string energy. This torus one-point diagram is expected to contribute to the one-point cosmological wavefunction at order $g_s$, and to the four-point cosmological wavefunction at order $g_s^2$ through the disconnected product of the torus one-point diagram and the sphere three-point diagram.

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arXiv2022

Bootstrapping the Ising Model on the Lattice

We study the statistical Ising model of spins on the infinite lattice using a bootstrap method that combines spin-flip identities with positivity conditions, including reflection positivity and Griffiths inequalities, to derive rigorous two-sided bounds on spin correlators through semi-definite programming. For the 2D Ising model on the square lattice, the bootstrap bounds based on correlators supported in a 13-site diamond-shaped region determine the nearest-spin correlator to within a small window, which for a wide range of coupling and magnetic field is narrower than the precision attainable with Monte Carlo methods. We also report preliminary results of the bootstrap bounds for the 3D Ising model on the cubic lattice.

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arXiv2022

D-instanton Effects in Type IIB String Theory

We study D-instanton contributions to supergraviton scattering amplitudes in the ten-dimensional type IIB superstring theory beyond the leading non-perturbative order. In particular, we determine the one-D-instanton contribution to maximal R-symmetry violating (MRV) amplitudes with arbitrary momenta at the first subleading order in string coupling, as well as the effects of a D-/anti-D-instanton pair at leading nontrivial order in the momentum expansion. These results confirm a number of predictions of S-duality, and unveil some previously unknown pieces of type IIB string amplitudes. Our computation is based on the Neveu-Schwarz-Ramond formalism with picture changing operators and vertical integration. The naive on-shell prescription for D-instanton mediated amplitudes, based on integration over the moduli space of worldsheet geometries as well as the moduli space of D-instanton boundary conditions, suffers from potential open string divergences and regularization ambiguities that are in principle resolved in the framework of open+closed superstring field theory. In this paper, the "on-shell ambiguities" of one-D-instanton MRV amplitudes are resolved by arguments involving supersymmetry and soft limits, part of which is verified by a string field theoretic computation in a highly nontrivial manner.

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arXiv2022

The S-Matrix of 2D Type 0B String Theory Part 1: Perturbation Theory Revisited

We study the perturbative S-matrix of closed strings in the two-dimensional type 0B string theory from the worldsheet perspective, by directly integrating correlation functions of ${\cal N}=1$ Liouville theory. The latter is computed numerically using recurrence relations for super-Virasoro conformal blocks. We show that the tree level 3- and 4-point amplitudes are in agreement with the proposed dual matrix quantum mechanics. The non-perturbative aspects of the duality will be analyzed in a companion paper.

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arXiv2022

The S-Matrix of 2D Type 0B String Theory Part 2: D-Instanton Effects

We study the effect of D-instantons on closed string scattering amplitudes in the two-dimensional type 0B string theory from the worldsheet perspective. We find that the contribution from a pair of D-instanton and anti-D-instanton to the closed string reflection amplitude, with a suitable contour prescription for the integration over the D-instanton moduli space, agrees with the corresponding leading non-perturbative corrections in the proposed dual matrix quantum mechanics. This analysis is further extended to thermal observables defined at finite temperature. The infrared divergence in charged processes is understood through the measure factor for charged D-instantons, and can be treated with spacetime dimensional regularization.

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arXiv2019

ZZ Instantons and the Non-Perturbative Dual of c = 1 String Theory

We study the effect of ZZ instantons in c = 1 string theory, and demonstrate that they give rise to non-perturbative corrections to scattering amplitudes that do not saturate unitarity within the closed string sector. Beyond the leading non-perturbative order, logarithmic divergences are canceled between worldsheet diagrams of different topologies, due to the Fischler-Susskind-Polchinski mechanism. We propose that the closed string vacuum in c = 1 string theory is non-perturbatively dual to a state of the matrix quantum mechanics in which all scattering states up to a given energy with no incoming flux from the "other side" of the potential are occupied by free fermions. Under such a proposal, we find detailed agreement of non-perturbative corrections to closed string amplitudes in the worldsheet description and in the dual matrix model.

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