arXiv subjects

Babak Haghighat

Publications and source records attributed to Babak Haghighat.

At least 37 records · Page 2Linked to original sources

arXiv2021

E-string Quantum Curve

In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.

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arXiv2021

Elliptic Quantum Curves of 6d SO(N) theories

We discuss supersymmetric defects in 6d $\mathcal{N}=(1,0)$ SCFTs with $\mathrm{SO}(N_c)$ gauge group and $N_c-8$ fundamental flavors. The codimension 2 and 4 defects are engineered by coupling the 6d gauge fields to charged free fields in four and two dimensions, respectively. We find that the partition function in the presence of the codimension 2 defect on $\mathbb{R}^4\times \mathbb{T}^2$ in the Nekrasov-Shatashvili limit satisfies an elliptic difference equation which quantizes the Seiberg-Witten curve of the 6d theory. The expectation value of the codimension 4 defect appearing in the difference equation is an even (under reflection) degree $N_c$ section over the elliptic curve when $N_c$ is even, and an odd section when $N_c$ is odd. We also find that RG-flows of the defects and the associated difference equations in the 6d $\mathrm{SO}(2N+1)$ gauge theories triggered by Higgs VEVs of KK-momentum states provide quantum Seiberg-Witten curves for $\mathbb{Z}_2$ twisted compactifications of the 6d $\mathrm{SO}(2N)$ gauge theories.

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arXiv2021

Elliptic Blowup Equations for 6d SCFTs. IV: Matters

Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity- and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an $ε_1,ε_2$ expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of $-2$ curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$ SCFTs with various matter representations.

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arXiv2021

Fibre-base duality of 5d KK theories

We study circle compactifications of 6d superconformal field theories giving rise to 5d rank 1 and rank 2 Kaluza-Klein theories. We realise the resulting theories as M-theory compactifications on local Calabi-Yau 3-folds and match the prepotentials from geometry and field theory. One novelty in our approach is that we include explicit dependence on bare gauge couplings and mass parameters in the description which in turn leads to an accurate parametrisation of the prepotential including all parameters of the field theory. We find that the resulting geometries admit "fibre-base" duality which relates their six-dimensional origin with the purely five-dimensional quantum field theory interpretation. The fibre-base duality is realised simply by swapping base and fibre curves of compact surfaces in the local Calabi-Yau which can be viewed as the total space of the anti-canonical bundle over such surfaces. Our results show that such swappings precisely occur for surfaces with a zero self-intersection of the base curve and result in an exchange of the 6d and 5d pictures.

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arXiv2020

Elliptic Quantum Curves of Class $\mathcal{S}_k$

Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $\mathcal{N}=(1,0)$ SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a $\mathbb{C}^2/\mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $\mathcal{N}=2$ theories are known as class $\mathcal{S}_k$. We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.

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arXiv2020

4d N=1 from 6d D-type N=(1,0)

Compactifications of 6d N=(1,0) SCFTs give rise to new 4d N=1 SCFTs and shed light on interesting dualities between such theories. In this paper we continue exploring this line of research by extending the class of compactified 6d theories to the D-type case. The simplest such 6d theory arises from D5 branes probing D-type singularities. Equivalently, this theory can be obtained from an F-theory compactification using "-2"-curves intersecting according to a D-type quiver. Our approach is two-fold. We start by compactifying the 6d SCFT on a Riemann surface and compute the central charges of the resulting 4d theory by integrating the 6d anomaly polynomial over the Riemann surface. As a second step, in order to find candidate 4d UV Lagrangians, there is an intermediate 5d theory that serves to construct 4d domain walls. These can be used as building blocks to obtain torus compactifications. In contrast to the A-type case, the vanishing of anomalies in the 4d theory turns out to be very restrictive and constraints the choices of gauge nodes and matter content severely. As a consequence, in this paper one has to resort to non-maximal boundary conditions for the 4d domain walls. However, the comparison to the 6d theory compactified on the Riemann surface becomes less tractable.

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arXiv2020

Riemann-Theta Boltzmann Machine

A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques.

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arXiv2019

Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.

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arXiv2019

M5 branes and Theta Functions

We propose quantum states for Little String Theories (LSTs) arising from M5 branes probing A- and D-type singularities. This extends Witten's picture of M5 brane partition functions as theta functions to this more general setup. Compactifying the world-volume of the five-branes on a two-torus, we find that the corresponding theta functions are sections of line bundles over complex 4-tori. This formalism allows us to derive Seiberg-Witten curves for the resulting four-dimensional theories. Along the way, we prove a duality for LSTs observed by Iqbal, Hohenegger and Rey.

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arXiv2018

Blowup Equations for 6d SCFTs. I

We propose novel functional equations for the BPS partition functions of 6d (1,0) SCFTs, which can be regarded as an elliptic version of Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic Calabi-Yau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.

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arXiv2018

ADE String Chains and Mirror Symmetry

6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the $A$ case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the $D$ and $E$ cases as well.

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arXiv2018

D-type fiber-base duality

M5 branes probing D-type singularities give rise to 6d (1,0) SCFTs with $SO \times SO$ flavor symmetry known as D-type conformal matter theories. Gauging the diagonal $SO$-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits.

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arXiv2018

Mirror Symmetry and Modularity

This is a review article on mirror symmetry and aspects of it related to the theory of modular forms. We describe this topic along its historical development and connect to some more recent results toward the end. The article is for publication in a special issue of ICCM Notices.

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arXiv2017

6d String Chains

We consider bound states of strings which arise in 6d (1,0) SCFTs that are realized in F-theory in terms of linear chains of spheres with negative self-intersections 1,2, and 4. These include the strings associated to N small E8 instantons, as well as the ones associated to M5 branes probing A and D type singularities in M-theory or D5 branes probing ADE singularities in Type IIB string theory. We find that these bound states of strings admit (0,4) supersymmetric quiver descriptions and show how one can compute their elliptic genera.

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arXiv2016

M-strings in thermodynamic limit: Seiberg-Witten geometry

We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.

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arXiv2016

From strings in 6d to strings in 5d

We show how recent progress in computing elliptic genera of strings in six dimensions can be used to obtain expressions for elliptic genera of strings in five-dimensional field theories which have a six-dimensional parent. We further connect our results to recent mathematical results about sheaf counting on ruled surfaces.

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arXiv2015

F-Theory, Spinning Black Holes and Multi-string Branches

We study 5d supersymmetric black holes which descend from strings of generic $\mathcal{N}=(1,0)$ supergravity in 6d. These strings have an F-theory realization in 6d as D3 branes wrapping smooth genus $g$ curves in the base of elliptic 3-folds. They enjoy $(0,4)$ worldsheet supersymmetry with an extra $SU(2)_L$ current algebra at level $g$ realized on the left-movers. When the smooth curves degenerate they lead to multi-string branches and we find that the microscopic worldsheet theory flows in the IR to disconnected 2d CFTs having different central charges. The single string sector is the one with maximal central charge, which when wrapped on a circle, leads to a 5d spinning BPS black hole whose horizon volume agrees with the leading entropy prediction from the Cardy formula. However, we find new phenomena where this branch meets other branches of the CFT. These include multi-string configurations which have no bound states in 6 dimensions but are bound through KK momenta when wrapping a circle, as well as loci where the curves degenerate to spheres. These loci lead to black hole configurations which can have total angular momentum relative to a Taub-Nut center satisfying $J^2 > M^3$ and whose number of states, though exponentially large, grows much slower than those of the large spinning black hole.

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arXiv2015

Calabi-Yau modular forms in limit: Elliptic Fibrations

We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

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