arXiv subjects

Jan Manschot

Publications and source records attributed to Jan Manschot.

At least 37 records · Page 2Linked to original sources

arXiv2021

Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD

We consider the order parameter $u=\left<{\rm Tr}ϕ^2\right>$ as function of the running coupling constant $τ\in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $τ$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $Γ$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $Γ\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.

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arXiv2021

Elliptic Loci of SU(3) Vacua

The space of vacua of many four-dimensional, $\mathcal{N}=2$ supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group $SU(2)$, this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group $SU(3)$ parametrizes an intricate family of genus two curves. We analyze this family using the so-called Rosenhain form for these curves. We demonstrate that two natural one-dimensional subloci of the space of $SU(3)$ vacua, $\mathcal{E}_u$ and $\mathcal{E}_v$, each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus $\mathcal{E}_u$ contains the points where mutually local dyons become massless, and is a fundamental domain for a classical congruence subgroup. Moreover, the locus $\mathcal{E}_v$ contains the superconformal Argyres-Douglas points, and is a fundamental domain for a Fricke group.

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arXiv2021

Topological correlators of $SU(2)$, $\mathcal{N}=2^*$ SYM on four-manifolds

We consider topologically twisted $\mathcal{N}=2$, $SU(2)$ gauge theory with a massive adjoint hypermultiplet on a smooth, compact four-manifold $X$. A consistent formulation requires coupling the theory to a ${\rm Spin}^c$ structure, which is necessarily non-trivial if $X$ is non-spin. We derive explicit formulae for the topological correlation functions when $b_2^+\geq 1$. We demonstrate that, when the ${\rm Spin}^c$ structure is canonically determined by an almost complex structure and the mass is taken to zero, the path integral reproduces known results for the path integral of the $\mathcal{N}=4$ gauge theory with Vafa-Witten twist. On the other hand, we reproduce results from Donaldson-Witten theory after taking a suitable infinite mass limit. The topological correlators are functions of the UV coupling constant $τ_{\rm uv}$ and we confirm that they obey the expected $S$-duality transformation laws. The holomorphic part of the partition function is a generating function for the Euler numbers of the matter (or obstruction) bundle over the instanton moduli space. For $b_2^+=1$, we derive a non-holomorphic contribution to the path integral, such that the partition function and correlation functions are mock modular forms rather than modular forms. We comment on the generalization of this work to the large class of $\mathcal{N}=2$ theories of class $S$.

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arXiv2020

Indefinite theta series and generalized error functions

Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice $A_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.

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arXiv2020

S-duality and refined BPS indices

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $\mathbb{P}^2$, and make them explicit for $U(4)$.

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arXiv2020

Effective gravitational couplings of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories

The low energy effective couplings of a four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory to topological invariants of the background gravitational field are described by two functions $A$ and $B$. These two functions play an important role in the study of topologically twisted four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and in the computation of central charges of $\mathcal{N}=2$ superconformal theories. In this paper, we compute $A$ and $B$ from the partition function in the $Ω$-background for $\mathrm{SU}(2)$ gauge theories. Our results not only confirm the predicted expressions of the effective gravitational couplings, but also give the previously undetermined overall multiplicative factors. We also analyze $A$ and $B$ for the $\mathrm{SU}(N)$ super-Yang-Mills theory, and confirm all the previous predictions.

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arXiv2019

Mocking the u-plane integral

The u-plane integral is the contribution of the Coulomb branch to correlation functions of N=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU(2), for an arbitrary four-manifold with (b1,b2+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg-Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell-Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

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arXiv2018

Vafa-Witten theory and iterated integrals of modular forms

Vafa-Witten (VW) theory is a topologically twisted version of N=4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P2. This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU(3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU(3) partition function is mathematically an example of a mock modular form of depth two.

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arXiv2017

Donaldson-Witten theory and indefinite theta functions

We consider partition functions with insertions of surface operators of topologically twisted N=2, SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter $a$, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of a Q-exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate $\bar a$ using Zwegers' indefinite theta functions. In this way, we reproduce Göttsche's expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.

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arXiv2017

The Coulomb Branch Formula for Quiver Moduli Spaces

In recent series of works, by translating properties of multi-centered supersymmetric black holes into the language of quiver representations, we proposed a formula that expresses the Hodge numbers of the moduli space of semi-stable representations of quivers with generic superpotential in terms of a set of invariants associated to `single-centered' or `pure-Higgs' states. The distinguishing feature of these invariants is that they are independent of the choice of stability condition. Furthermore they are uniquely determined by the $χ_y$-genus of the moduli space. Here, we provide a self-contained summary of the Coulomb branch formula, spelling out mathematical details but leaving out proofs and physical motivations.

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arXiv2017

D3-instantons, Mock Theta Series and Twistors

The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2,Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2,Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.

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arXiv2017

Multiple D3-instantons and mock modular forms II

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest.

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arXiv2016

Intersection cohomology of moduli spaces of sheaves on surfaces

We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.

math.AG↗

arXiv2016

Multiple D3-instantons and mock modular forms I

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the M5-brane elliptic genus. We extend this analysis to the two-instanton level, where wall-crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kahler potential which must transform as a modular form under S-duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4-D2-D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson-Thomas invariants in a particular chamber). Modular invariance of the contact potential then requires that, in case where the D3-brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector-valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson-Thomas invariants of pure two-dimensional sheaves.

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arXiv2016

Sheaves on P2 and generalized Appell functions

A closed expression is given for the generating function of (virtual) Poincaré polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and Chern classes. This generating function is known to equal the partition function of topologically twisted gauge theory with $\mathcal{N}=4$ supersymmetry and gauge group $U(r)$, which localizes on the Hermitian Yang-Mills solutions of the gauge field. To classify and study the novel generating functions, the notion of Appell functions with signature $(n_+,n_-)$ is introduced. For $n_-=1$, these novel functions reduce to the known class of Appell functions with multiple variables or higher level.

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arXiv2015

Identities for generalized Appell functions and the blow-up formula

In this paper, we prove identities for a class of generalized Appell functions which are based on the $\operatorname{A}_2$ root lattice. The identities are reminiscent of periodicity relations for the classical Appell function, and are proven using only analytic properties of the functions. Moreover they are a consequence of the blow-up formula for generating functions of invariants of moduli spaces of semi-stable sheaves of rank 3 on rational surfaces. Our proof confirms that in the latter context, different routes to compute the generating function (using the blow-up formula and wall-crossing) do arrive at identical $q$-series. The proof also gives a clear procedure for how to prove analogous identities for generalized Appell functions appearing in generating functions for sheaves with rank $r>3$.

math.NT↗

arXiv2015

The asymptotic profile of $χ_y$-genera of Hilbert schemes of points on K3 surfaces

The Hodge numbers of the Hilbert schemes of points on algebraic surfaces are given by Göttsche's formula, which expresses the generating functions of the Hodge numbers in terms of theta and eta functions. We specialize in this paper to generating functions of the $χ_y(\mathrm{K3}^{[n]})$ genera of Hilbert schemes of $n$ points on K3 surfaces. We determine asymptotic values of the coefficients of the $χ_y$-genus for $n\to \infty$ as well as their asymptotic profile.

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arXiv2015

Quantum hypermultiplet moduli spaces in N=2 string vacua: a review

The hypermultiplet moduli space M_H in type II string theories compactified on a Calabi-Yau threefold X is largely constrained by supersymmetry (which demands quaternion-Kählerity), S-duality (which requires an isometric action of SL(2, Z)) and regularity. Mathematically, M_H ought to encode all generalized Donaldson-Thomas invariants on X consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on M_H, or rather the exact complex contact structure on its twistor space.

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