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Operator Formalism of Gauge Theory

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Instanton Counting, Quantum Geometry and Algebra

Part of the book series: Mathematical Physics Studies ((MPST))

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Abstract

We have discussed classical and quantum geometric aspects of \(\mathcal {N} = 2\) gauge theory in four dimensions in the relation to various fields of physics and mathematics. One of the key ingredients in such aspects is the non-perturbative symmetry of gauge theory incorporated by instantons, i.e., covariance of path integral partition function under the adding/removing-instanton operation. In general, symmetry of the system is rephrased as the invariance (covariance) under the corresponding transformation, which is described as a group action, and we would be able to discuss the algebraic structure from the infinitesimal version of the transformation (group action). The purpose of this Part III is to explore the algebraic structure associated with the non-perturbative symmetry of gauge theory, namely the symmetry algebra of the instanton creation and annihilation.

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Notes

  1. 1.

    In this Chapter, we will mostly use the K-theory (5d \(\mathcal {N} = 1\) theory on a circle) convention.

  2. 2.

    We also denote the partition function contribution associated with the instanton configuration \(\mathcal {X}\) by \(Z_{\mathcal {X}} = Z[\mathcal {X}]\).

  3. 3.

    This is similar to the Fourier transformation: In the presence of the factor \(\mathrm {e}^{\mathrm {i}p x}\), the multiplication of the variable p is equivalent to the derivative with the x-variable, \(p \leftrightarrow -\mathrm {i}\partial _x\).

  4. 4.

    The partition function for 6d theory is instead given by the trace over the Fock space, which gives rise to the character of the corresponding module. See Sect. 8.2.

  5. 5.

    A similar construction is available for 4d \(\mathcal {N} = 2\) theory [25], and also for 6d \(\mathcal {N} = (1,0)\) theory as discussed in Chap. 8.

  6. 6.

    For example, we take \(|q_1| \ll |q_2^{-1}| < 1\) and \(|\mathrm {e}^{\mathsf {a}}| \sim |\mathrm {e}^{m}| \sim 1\). Then, define the ordering \(x \succ x'\) if \(|x| > |x'|\).

  7. 7.

    Precisely speaking, this agreement is up to the constant factor, which is independent of the instanton configuration, and is interpreted as the perturbative contribution.

  8. 8.

    This infinite series is justified using the Jackson integral with the base x denoted by \(\displaystyle \oint _x dz_{q_2} \, S_{i,z}\). See [7] for a related discussion in the context of q-deformation of Dotsenko–Fateev integral.

  9. 9.

    From this relation, we may also incorporate another zero mode in the \(\mathsf {A}\)-operator (and also \(\mathsf {Y}\)-operator), \(\mathsf {A}_{i,x} \rightarrow x^{\kappa _i} \, \mathsf {A}_{i,x}\), which corresponds to the Chern–Simons term.

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  1. Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, Dijon, France

    Taro Kimura

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Kimura, T. (2021). Operator Formalism of Gauge Theory. In: Instanton Counting, Quantum Geometry and Algebra. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-030-76190-5_6

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