### 数学代写|表示论代写Representation theory代考|MATH7333

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• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|表示论代写Representation theory代考|Perturbative Quantization of Gauge Theories

We remain very brief in this subsection. The interested reader is invited to study the references [Res10, Mne17] for a thorough introduction to the perturbative quantization of gauge theories from the mathematical viewpoint, or the short introductory paper [Pol05] (which is devoted exclusively to Chern-Simons theory).
The quantity of interest in this paper is the Chern-Simons partition function $Z_{M}$. Its naive definition would be
$$Z_{M}=\int_{F_{M}} e^{\frac{t S_{M}}{}}$$
but attempts at defining an appropriate measure on $F_{M}$ have failed (see [GJ87] for a discussion of measures on field spaces in quantum field theory). In perturbative quantization, one tries to define $Z_{M}$ by extrapolating the behavior of a finitedimensional oscillatory integral $Z=\int_{F} e^{\frac{i}{h} S}$ in the $h \rightarrow 0$ limit. It is well known that if $S$ has non-degenerate critical points, then the integral concentrates in a neighborhood of them, and one can derive a series in powers of $h$ describing the asymptotic behavior. One can mimic the definition of this power series in the infinite-dimensional case if the critical points of $S_{M}$ are non-degenerate. However, the critical points of functionals invariant under a symmetry, such as the ChernSimons functional, are never non-degenerate, so one needs an additional method from physics, called gauge fixing. There are different variants of this method, the most commonly used being the Faddeev-Popov (FP) ghosts [FP67] and the BRST formalism. ${ }^{3}$ The idea is to embed the space of fields $F_{M}$ in the degree 0 part of a graded vector space $\mathcal{F}{M}$, and define a new functional $\mathcal{S}{M}: \mathcal{F}{M} \rightarrow \mathbb{R}$ such that $\mathcal{S}{M}$ has non-degenerate critical points, and $\left.S_{M}\right|{F{M}}=S_{M}$.

Both the FP ghosts and the BRST formalism are subsumed in the BV formalism named after Batalin and Vilkovisky, who introduced it in [BV77, BV81, BV83]. We will briefly discuss the BV formalism and its adaptation to the case with boundary, the BV-BFV formalism, in the next section.

## 数学代写|表示论代写Representation theory代考|BV Formalism

In the BV formalism one embeds the space of states $F_{M}$ into an odd symplectic $\mathbb{Z}$ graded vector space $\left(F_{M}, \omega\right)$, where the odd symplectic form $\omega$ is required to have $\mathbb{Z}$-degree – 1. The $\mathbb{Z}$-degree is referred to as ghost number. One needs to find a BV action $\mathcal{S}{M}$ such that $\left.\mathcal{S}{M}\right|{F{M}}=S_{M}$, which satisfies the Classical Master Equation (CME) $(\mathcal{S}, \mathcal{S})=0$, where $(\cdot, \cdot)$ is the odd Poisson bracket associated with $\omega$. Notice that if $\mathcal{S}{M}$ has a Hamiltonian vector field $\mathcal{Q}{M}$, then $\mathcal{Q}_{M}^{2}=0$, from the CME. This leads to the following definition of BV theory [CMR 14 ]:

Definition 3.1 A $B V$ vector space is a quadruple $(\mathcal{F}, \omega, \mathcal{Q}, \mathcal{S})$ where $\mathcal{F}$ is a $\mathbb{Z}$ graded vector space, $\omega$ is a symplectic form of degree $-1$, $\mathcal{}$ field, and $\mathcal{S}$ is a function of degree 0 , such that $\mathcal{Q}^{2}=0$ and
$$\iota_{\mathcal{Q}} \omega=\delta \mathcal{S}$$
A $d$-dimensional (linear) $B V$ theory is an association of a $B V$ vector space $\left(\mathcal{F}{M}, \omega{M}, \mathcal{Q}{M}, \mathcal{S}{M}\right)$ to every $d$-dimensional manifold $M$.

## 数学代写|表示论代写Representation theory代考|Effective Action and Residual Fields

In certain cases, it is not possible to find directly a Lagrangian which satisfies the requirement that the action restricted to it has a unique critical point. A good example is the case of abelian BF theory. In $d$ dimensions, the BV space of fields is $\mathcal{F}{M}=\Omega^{\bullet}(M)[1] \oplus \Omega^{\bullet}(M)[d-2]$, and the BV action functional is $$\mathcal{S}{M}[\mathrm{~A}, \mathrm{~B}]=\int_{M} \mathrm{~B} \wedge \mathrm{d} \mathrm{A} .$$
The critical points are given by closed forms, and the gauge symmetries are given by shifting A, B by exact forms. Hence if the de Rham cohomology $H^{\bullet}(M)$ is non-

trivial, there are critical points which are inequivalent under gauge transformations, also known as zero modes.

The solution to this problem is to choose a $\mathrm{BV}$ space of residual fields $\mathcal{V}{M}$ and a splitting $F{M}=\mathcal{V}{M} \times \mathcal{Y}$, such that one can find a gauge-fixing Lagrangian $\mathcal{L}{M} \subset \mathcal{Y}$. Elements of $\mathcal{V}{M}$ are known as residual fields, zero modes, infrared fields, or slow fields, whereas the elements of $\mathcal{Y}$ are called fluctuations, fast fields, or ultraviolet fields. The partition function $\psi{M}$ gets replaced by an effective action $\psi_{M}(\mathbf{x})$, which is a function of the residual fields and formally defined via BV pushforward:
$$\psi_{M}(\mathbf{x})=\int_{\xi \in \mathcal{L}{M} \subset \mathcal{Y}} e^{\frac{1}{\hbar} \mathcal{S}[\mathrm{x}, \xi]}$$ In the case of abelian BF theory one can choose as residual fields representatives of the cohomology: $\mathcal{V}{M}=H^{\bullet}(M)[1] \oplus H^{\bullet}(M)[d-2]$. One way to do this is to pick a Riemannian metric and use the harmonic representatives, a possible choice of gauge-fixing Lagrangian is then given by $\mathrm{d}^{*}$-exact forms. In the finite-dimensional case, the QME for the BV action implies that the effective action is $\Delta \mathcal{V}_{M}$-closed, i.e. closed with respect to the BV Laplacian on residual fields, and changes by a $\Delta$-exact term under a deformation of the gauge-fixing Lagrangian.

## 数学代写|表示论代写Representation theory代考|Perturbative Quantization of Gauge Theories

FP 幽灵和 BRST 形式主义都包含在以 Batalin 和 Vilkovisky 命名的 BV 形式主义中，他们在 [BV77, BV81, BV83] 中介绍了它。我们将在下一节简要讨论 BV 形式主义及其对有边界情况的适应，即 BV-BFV 形式主义。

## 数学代写|表示论代写Representation theory代考|Effective Action and Residual Fields

ψ米(X)=∫X∈大号米⊂是和1⁇小号[X,X]在阿贝尔 BF 理论的情况下，可以选择上同调的剩余场代表：在米=H∙(米)[1]⊕H∙(米)[d−2]. 一种方法是选择黎曼度量并使用谐波代表，然后给出规范固定拉格朗日的可能选择d∗- 确切的形式。在有限维情况下，BV 动作的 QME 意味着有效动作是Δ在米-封闭，即关于剩余场上的 BV 拉普拉斯算子封闭，并且变化Δ- 量规固定拉格朗日变形下的精确项。

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