Cutting and gluing with running couplings in 
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 QCD

Aspman, Johannes; Furrer, Elias; Manschot, Jan

doi:10.1103/physrevd.105.025021

Cited by 13 publications

(65 citation statements)

References 89 publications

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“…We identify several loci in the space of masses where u transforms as a modular form for both τ UV and τ . This extends our recent work [9] on theories with N f ≤ 3 to N f = 4. In [9], we determined fundamental domains for the running coupling τ for the asymptotically free theories by analyzing in detail the order parameter u as function of the effective coupling τ ∈ H. We have demonstrated that for generic masses this function has branch points, with the consequence that the fundamental domain for τ is in general not of the form Γ\H for a congruence subgroup Γ ⊂ SL(2, Z).…”

supporting

confidence: 87%

“…Let us recall a method for finding explicit expressions for the Coulomb branch parameter u. This was recently discussed in detail in [9]. The J -invariant of the SW curve (2.4) is a rational function J (u, m, τ 0 ) in u, the masses m = (m 1 , m 2 , m 3 , m 4 ) and e i (τ 0 ).…”

Section: Order Parameters and Bimodular Formsmentioning

confidence: 99%

“…is the modular lambda function (a Hauptmodul for Γ(2)) with ϑ i the Jacobi theta functions (A.5), and j is the modular j-function (A.12) . The reason for this is that in certain mass configurations, the sextic equation constructed from the rational function J (u, m, τ 0 ) defines a field extension of C(SL(2, Z)) with intermediate field 2)), such that the sextic equation factors over C(Γ( 2)) into products of lower degree polynomials [9]. 8…”

Section: Order Parameters and Bimodular Formsmentioning

confidence: 99%

“…Actually, da du is not invariant under the monodromy around ∞, but multiplied by −1. Instead of da du , we may consider da du 2 , which is monodromy invariant [9,32]. In the pure (N f = 0) SU(2) case, it is a modular form of weight 2 for Γ 0 (4).…”

Section: Case Amentioning

confidence: 99%

“…The analysis of the A, B, C and D theories may suggest that the order parameter u m for a generic mass m transforms with weight (0, 2) under G τ × G τ 0 for some subgroup G ⊆ SL(2, Z). This is however not true in general, as for generic masses there are branch points and associated branch cuts, which spoil the modularity [9]. The discussion in [9] for N f ≤ 3 suggests that for a fixed τ or fixed τ 0 , there is a natural choice of fundamental domain F(m) ⊆ H for u m , such that u m : F(m) → B 4 is one-to-one.…”

Section: Generic Massmentioning

confidence: 99%

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Four flavours, triality and bimodular forms

Aspman,

Furrer,

Manschot

2021

Preprint

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We consider N = 2 supersymmetric SU(2) gauge theory with N f = 4 massive hypermultiplets. The duality group of this theory contains transformations acting on the UV-coupling τ UV as well as on the running coupling τ . We establish that subgroups of the duality group act separately on τ UV and τ , while a larger group acts simultaneously on τ UV and τ . For special choices of the masses, we find that the duality groups can be identified with congruence subgroups of SL(2, Z). We demonstrate that in such cases, the order parameters are instances of bimodular forms with arguments τ and τ UV . Since the UV duality group of the theory contains the triality group of outer automorphisms of the flavour symmetry SO(8), the duality action gives rise to an orbit of mass configurations. Consequently, the corresponding order parameters combine to vector-valued bimodular forms with SL(2, Z) acting simultaneously on the two couplings.

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supporting

confidence: 87%

Section: Order Parameters and Bimodular Formsmentioning

confidence: 99%

Section: Order Parameters and Bimodular Formsmentioning

confidence: 99%

Section: Case Amentioning

confidence: 99%

Section: Generic Massmentioning

confidence: 99%

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Four flavours, triality and bimodular forms

Aspman,

Furrer,

Manschot

2021

Preprint

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Partition functions of non-Lagrangian theories from the holomorphic anomaly

Fucito¹,

Grassi²,

Morales³

et al. 2023

J. High Energ. Phys.

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The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

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The u-plane integral, mock modularity and enumerative geometry

et al. 2022

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We revisit the low-energy effective U(1) action of topologically twisted $${\mathcal {N}}=2$$ N = 2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of $$\mathcal Q$$ Q -exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces $$X=\Sigma _g \times \mathbb {CP}^1$$ X = Σ g × CP 1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface $$\Sigma _g$$ Σ g , via an equivalent topological A-model on $$\mathbb {CP}^1$$ CP 1 , we will be able to express the generating function of genus zero Gromov–Witten invariants of the moduli space of flat rank one connections over $$\Sigma _g$$ Σ g in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

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Cutting and gluing with running couplings in N=2 QCD

Cited by 13 publications

References 89 publications

Four flavours, triality and bimodular forms

Four flavours, triality and bimodular forms

Partition functions of non-Lagrangian theories from the holomorphic anomaly

The u-plane integral, mock modularity and enumerative geometry

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