arXiv subjects

Eric D'Hoker

Publications and source records attributed to Eric D'Hoker.

At least 19 recordsLinked to original sources

arXiv2026

Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection

The flat DHS connection $\mathcal J_{\mathrm{DHS}}$ constructed in arXiv:2602.01461 is smooth on the configuration space of $n$ points on a fixed compact Riemann surface $Σ$ of arbitrary genus $h$, takes values in an infinite-dimensional Lie algebra $\hat{\mathfrak t}_{h,n}$ and is invariant under the modular group $\mathrm{Sp}(2h,\mathbb Z)$. This paper is the first in a series for a program whose goal is to extend the connection $\mathcal J_{\mathrm{DHS}}$ to a global flat connection on the Teichmüller space $\mathcal T_{h,n}$ valued in the Lie algebra of derivations of $\hat{\mathfrak t}_{h,n}$. Upon the choice of local coordinates adapted to the map $\mathcal T_{h,n}\to \mathcal T_h$, such a connection splits into three pieces: $\mathcal J_{\mathrm{DHS}}$, a piece $\mathcal L$ corresponding to holomorphic directions of $\mathcal T_h$ and a third piece corresponding to anti-holomorphic directions in $\mathcal T_h$. In this paper, we isolate the system of equations satisfied by $\mathcal L$ and obtain its solution locally and explicitly. The construction of a global extension of $\mathcal J_{\mathrm{DHS}}$ to $\mathcal T_{h,n}$ and the extension of the meromorphic connection of arXiv:1112.0864 to $\mathcal T_{h,n}$, are relegated to future publications in this series.

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arXiv2026

Exact solutions to complex Type IIB supergravity for complex superalgebra $F(4)$ and its real forms

We construct the general local solutions to complexified Type IIB supergravity which are invariant under the complexified Lie superalgebra $F(4)$. The geometry is a product of complexified maximally symmetric spaces $\mathcal M_{6 {\mathbb C}}$ and $\mathcal M_{2 {\mathbb C}}$ warped over a complexified surface $Σ_{\mathbb C}$. We classify the reality conditions that may be imposed consistently to obtain real form solutions within real forms of complex Type IIB supergravity. The latter comprise standard Type IIB, Type IIB$^\star$ and IIB$^\prime$, as well as theories with $3$, $5$, $7$ and $9$ time-like directions. Our classification of real solutions is consistent with and exhausts the real forms of $F(4)$, whose classification we confirm by elementary methods. The geometry of each real form solution is a product of real maximally symmetric spaces $\mathcal M_6$ and $\mathcal M_2$ warped over a Riemann surface $Σ$, with various signatures. The real solutions include, among others, known $AdS_6 \times S^2 \times Σ$ and $AdS_2 \times S^6 \times Σ$ solutions to standard Type IIB as well as new solutions of the form $dS_{1,5}\times S^2 \times Σ$ in Type IIB$^\star$. There are no real forms of the complex solutions with $\mathfrak{so} (7;{\mathbb R}) \oplus \mathfrak{so} (3;{\mathbb R})$ symmetry. We discuss the relevance of the complex solutions, and of analytic continuations from $dS_{1,5}\times S^2\timesΣ$ to $S^6\times S^2\timesΣ$ within complex Type IIB, in connection with holography for the polarized IKKT model.

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arXiv2026

Fay identities for polylogarithms on higher-genus Riemann surfaces

A recent construction of polylogarithms on Riemann surfaces of arbitrary genus in arXiv:2306.08644 is based on a flat connection assembled from single-valued non-holomorphic integration kernels that depend on two points on the Riemann surface. In this work, we construct and prove infinite families of bilinear relations among these integration kernels that are necessary for the closure of the space of higher-genus polylogarithms under integration over the points on the surface. Our bilinear relations generalize the Fay identities among the genus-one Kronecker-Eisenstein kernels to arbitrary genus. The multiple-valued meromorphic kernels in the flat connection of Enriquez are conjectured to obey higher-genus Fay identities of exactly the same form as their single-valued non-holomorphic counterparts. We initiate the applications of Fay identities to derive functional relations among higher-genus polylogarithms involving either single-valued or meromorphic integration kernels.

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arXiv2026

Meromorphic higher-genus integration kernels via convolution over homology cycles

Polylogarithms on arbitrary higher-genus Riemann surfaces can be constructed from meromorphic integration kernels with at most simple poles, whose definition was given by Enriquez via functional properties. In this work, homotopy-invariant convolution integrals over homology cycles are shown to provide a direct construction of Enriquez kernels solely from holomorphic Abelian differentials and the prime form. Our new representation is used to demonstrate the closure of the space of Enriquez kernels under convolution over homology cycles and under variations of the moduli. The results of this work further strengthen the remarkable parallels of Enriquez kernels with the non-holomorphic modular tensors recently developed in an alternative construction of higher-genus polylogarithms.

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arXiv2026

Degenerations of flat connections on Riemann surfaces

The integration kernels for polylogarithm functions on a compact Riemann surface of arbitrary genus $h$ are shown to close as the surface undergoes a non-separating degeneration to one of genus $h{-}1$. Explicit formulas are obtained for the non-separating degeneration of the multivariable Enriquez connection for genus $h$ with an arbitrary number of variables to the Enriquez connection for genus $h{-}1$ with two additional punctures whose Lie algebra generators are related to the original ones by the characteristic Bernoulli generating functions known from the degeneration at $h=1$. Analogous degeneration formulas are obtained for the single-valued DHS kernels at the leading order in the real degeneration parameter that is adapted to relating modular tensors at genus $h$ and $h{-}1$.

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arXiv2026

Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.

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arXiv2026

Single-valued flat connections in several variables on arbitrary Riemann surfaces

Polylogarithms on Riemann surfaces may be constructed efficiently in terms of flat connections that can enjoy various algebraic and analytic properties. In this paper, we present a single-valued and modular invariant connection ${\cal J}_\text{DHS}$ on the configuration space $\text{Cf}_n(Σ)$ of an arbitrary number $n$ of points on an arbitrary compact Riemann surface $Σ$ with or without punctures. The connection ${\cal J}_\text{DHS}$ generalizes an earlier construction for a single variable and is built out of the same integration kernels. We show that ${\cal J}_\text{DHS}$ is flat on $\text{Cf}_n(Σ)$. For the case without punctures, we relate it to the meromorphic multiple-valued Enriquez connection ${\cal K}_\text{E}$ in $n$ variables on the universal cover $\tilde Σ$ of $Σ$ by the composition of a gauge transformation and an automorphism of the Lie algebra in which ${\cal J}_\text{DHS}$ and ${\cal K}_\text{E}$ take values. In a companion paper, we shall establish the equivalence between the flatness of these connections and the corresponding interchange and Fay identities, for arbitrary compact Riemann surfaces.

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arXiv2026

Relating flat connections and polylogarithms on higher genus Riemann surfaces

In this work, we relate two recent constructions that generalize classical (genus-zero) polylogarithms to higher-genus Riemann surfaces. A flat connection valued in a freely generated Lie algebra on a punctured Riemann surface of arbitrary genus produces an infinite family of homotopy-invariant iterated integrals associated to all possible words in the alphabet of the Lie algebra generators. Each iterated integral associated to a word is a higher-genus polylogarithm. Different flat connections taking values in the same Lie algebra on a given Riemann surface may be related to one another by the composition of a gauge transformation and an automorphism of the Lie algebra, thus producing closely related families of polylogarithms. In this paper we provide two methods to explicitly construct this correspondence between the meromorphic multiple-valued connection introduced by Enriquez in e-Print 1112.0864 and the non-meromorphic single-valued and modular-invariant connection introduced by D'Hoker, Hidding and Schlotterer, in e-Print 2306.08644.

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arXiv2025

Worldsheet fermion correlators, modular tensors and higher genus integration kernels

The cyclic product of an arbitrary number of Szegö kernels for even spin structure $δ$ on a compact higher-genus Riemann surface $Σ$ may be decomposed via a descent procedure which systematically separates the dependence on the points $z_i \in Σ$ from the dependence on the spin structure $δ$. In this paper, we prove two different, but complementary, descent procedures to achieve this decomposition. In the first procedure, the dependence on the points $z_i \in Σ$ is expressed via the meromorphic multiple-valued Enriquez kernels of e-print 1112.0864 while the dependence on $δ$ resides in multiplets of functions that are independent of $z_i$, locally holomorphic in the moduli of $Σ$ and generally do not have simple modular transformation properties. The $δ$-dependent constants are expressed as multiple convolution integrals over homology cycles of $Σ$, thereby generalizing a similar representation of the individual Enriquez kernels. In the second procedure, which was proposed without proof in e-print 2308.05044, the dependence on $z_i$ is expressed in terms the single-valued, modular invariant, but non-meromorphic DHS kernels introduced in e-print 2306.08644 while the dependence on $δ$ resides in modular tensors that are independent of $z_i$ and are generally non-holomorphic in the moduli of $Σ$. Although the individual building blocks of these decompositions have markedly different properties, we show that the combinatorial structure of the two decompositions is virtually identical, thereby extending the striking correspondence observed earlier between the roles played by Enriquez and DHS kernels. Both decompositions are further generalized to the case of linear chain products of Szegö kernels.

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arXiv2025

Cyclic products of higher-genus Szegö kernels, modular tensors and polylogarithms

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szegö kernels. In this paper we show that cyclic products of any number of Szegö kernels on a Riemann surface of arbitrary genus may be decomposed into linear combinations of modular tensors on moduli space that carry all the dependence on the spin structure $δ$. The $δ$-independent coefficients in these combinations carry all the dependence on the marked points and are composed of the integration kernels of higher-genus polylogarithms. We determine the antiholomorphic moduli derivatives of the $δ$-dependent modular tensors.

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arXiv2025

Constructing polylogarithms on higher-genus Riemann surfaces

An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini.

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arXiv2024

Cascading from $\mathscr{N}=2$ Supersymmetric Yang-Mills Theory to Confinement and Chiral Symmetry Breaking in Adjoint QCD

We argue that adjoint QCD in 3+1 dimensions, with any $SU(N)$ gauge group and two Weyl fermion flavors (i.e. one adjoint Dirac fermion), confines and spontaneously breaks its chiral symmetries via the condensation of a fermion bilinear. We flow to this theory from pure $\mathscr{N}=2$ SUSY Yang-Mills theory with the same gauge group, by giving a SUSY-breaking mass $M$ to the scalars in the $\mathscr{N} = 2$ vector multiplet. This flow can be analyzed rigorously at small $M$, where it leads to a deconfined vacuum at the origin of the $\mathscr{N}=2$ Coulomb branch. The analysis can be extended to all $M$ using an Abelian dual description that arises from the $N$ multi-monopole points of the $\mathscr{N} = 2$ theory. At each such point, there are $N-1$ hypermultiplet Higgs fields $h_m^{i = 1, 2}$, which are $SU(2)_R$ doublets. We provide a detailed study of the phase diagram as a function of $M$, by analyzing the semi-classical phases of the dual using a combination of analytic and numerical techniques. The result is a cascade of first-order phase transitions, along which the Higgs fields $h_m^i$ successively turn on, and which interpolates between the Coulomb branch at small $M$, where all $h_m^i = 0$, and a maximal Higgs branch, where all $h_m^i \neq 0$, at sufficiently large $M$. We show that this maximal Higgs branch precisely matches the confining and chiral symmetry breaking phase of two-flavor adjoint QCD, including its broken and unbroken symmetries, its massless spectrum, and the expected large-$N$ scaling of various observables. The spontaneous breaking pattern $SU(2)_R \to U(1)_R$, consistent with the Vafa-Witten theorem, is ensured by an intricate alignment mechanism for the $h_m^i$ in the dual, and leads to a $\mathbb{C}\mathbb{P}^1$ sigma model of increasing radius along the cascade.

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arXiv2023

Approaching Argyres-Douglas theories

The Seiberg-Witten solution to four-dimensional $\mathcal{N}=2$ super-Yang-Mills theory with gauge group $\text{SU}(N)$ and without hypermultiplets is used to investigate the neighborhood of the maximal Argyres-Douglas points of type $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$. A convergent series expansion for the Seiberg-Witten periods near the Argyres-Douglas points is obtained by analytic continuation of the series expansion around the $\mathbb{Z}_{2N}$ symmetric point derived in arXiv:2208.11502. Along with direct integration of the Picard-Fuchs equations for the periods, the expansion is used to determine the location of the walls of marginal stability for $\text{SU}(3)$. The intrinsic periods and Kähler potential of the $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$ superconformal fixed point are computed by letting the strong coupling scale tend to infinity. We conjecture that the resulting intrinsic Kähler potential is positive definite and convex, with a unique minimum at the Argyres-Douglas point, provided only intrinsic Coulomb branch operators with unitary scaling dimensions $Δ>1$ acquire a vacuum expectation value, and provide both analytical and numerical evidence in support of this conjecture. In all the low rank examples considered here, it is found that turning on moduli dual to $Δ\leq 1$ operators spoils the positivity and convexity of the intrinsic Kähler potential.

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arXiv2023

Cyclic products of Szegö kernels and spin structure sums I: hyper-elliptic formulation

The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its $SL(2,\mathbb C)$ covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann $\vartheta$-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.

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arXiv2022

Lectures on modular forms and strings

The goal of these lectures is to present an informal but precise introduction to a body of concepts and methods of interest in number theory and string theory revolving around modular forms and their generalizations. Modular invariance lies at the heart of conformal field theory, string perturbation theory, Montonen-Olive duality, Seiberg-Witten theory, and S-duality in Type IIB superstring theory. Automorphic forms with respect to higher arithmetic groups as well as mock modular forms enter in toroidal string compactifications and the counting of black hole microstates. After introducing the basic mathematical concepts including elliptic functions, modular forms, Maass forms, modular forms for congruence subgroups, vector-valued modular forms, and modular graph forms, we describe a small subset of the countless applications to problems in Mathematics and Physics, including those mentioned above.

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arXiv2022

Exploring the Strong-Coupling Region of $SU(N)$ Seiberg-Witten Theory

We consider the Seiberg-Witten solution of pure $\mathcal{N} =2$ gauge theory in four dimensions, with gauge group $SU(N)$. A simple exact series expansion for the dependence of the $2 (N-1)$ Seiberg-Witten periods $a_I(u), a_{DI}(u)$ on the $N-1$ Coulomb-branch moduli $u_n$ is obtained around the $\mathbb{Z}_{2N}$-symmetric point of the Coulomb branch, where all $u_n$ vanish. This generalizes earlier results for $N=2$ in terms of hypergeometric functions, and for $N=3$ in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential $K = \frac{1}{2π} \sum_I \text{Im}(\bar a_I a_{DI})$, which is single valued on the Coulomb branch. Evidence is presented that $K$ is a convex function, with a unique minimum at the $\mathbb{Z}_{2N}$-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

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arXiv2022

Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the alpha' expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)_R-preserving amplitudes such as for five gravitons, and for U(1)_R-violating amplitudes such as for one dilaton and four gravitons. At each order in alpha', the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D^2 R^5 and D^4 R^5 are found to match those of D^4 R^4 and D^6 R^4, respectively, as required by non-linear supersymmetry. To the next order, a D^6 R^5 effective interaction arises, which is independent of the supersymmetric completion of D^8 R^4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D^6 R^5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)_R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

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arXiv2022

Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures

The contribution from even spin structures to the genus-two amplitude for five massless external NS states in Type II and Heterotic superstrings is evaluated from first principles in the RNS formulation. Using chiral splitting with the help of loop momenta this problem reduces to the evaluation of the corresponding chiral amplitude, which is carried out using the same techniques that were used for the genus-two amplitude with four external NS states. The results agree with the parity-even NS components of a construction using chiral splitting and pure spinors given in earlier companion papers arXiv:2006.05270 and arXiv:2008.08687.

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