Perturbative BV-BFV formalism with homotopic renormalization a case study Minghao Wang and Gongwang Yan

2025-05-02 0 0 657.33KB 32 页 10玖币

侵权投诉

Perturbative BV-BFV formalism with homotopic

renormalization: a case study

Minghao Wang and Gongwang Yan

April 3, 2023

Abstract

We report a rigorous quantization of topological quantum mechanics on R>0and I= [0,1]

in perturbative BV-BFV formalism. Costello’s homotopic renormalization is extended, and

incorporated in our construction. Moreover, BV quantization of the same model studied in

previous work [WY22] is derived from the BV-BFV quantization, leading to a comparison

between diﬀerent interpretations of “state” in these two frameworks.

Contents

1 Introduction 1

1.1 Mainresults......................................... 2

1.2 Organizationofthepaper ................................. 4

2 Algebraic Preliminaries 7

2.1 Perturbative BV quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Perturbative BV-BFV quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Moyal product and Weyl quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 BV-BFV Description of TQM on R>012

3.1 Thecontentoffreetheory ................................. 12

3.2 The content of interactive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 BV-BFV Description of TQM on Interval 22

5 Return to the BV Description 26

1 Introduction

Among mathematical frameworks for quantum ﬁeld theory (QFT), the one developed by Costello

[Cos11] is a candidate to systematically study perturbative gauge theories. This framework uses

Batalin-Vilkovisky (BV) formalism [BV81] to describe gauge symmetry, and contains a procedure

arXiv:2210.03009v3 [math-ph] 31 Mar 2023

which we call “homotopic renormalization” to deal with UV problem. These two ingredients are

combined together in the language of homological algebra. A successful example quantized in this

formalism is BCOV(Bershadsk-Cecotti-Ooguri-Vafa) theory, see [CL12][CL15][CL20]. The corre-

sponding string ﬁeld theory at all topology can only be quantized by this method so far, which is a

degenerate BV theory coupled with large N gauge ﬁeld.

However, this framework needs modiﬁcation when the spacetime manifold has boundary. In order

to deﬁne the theory still within BV formalism, we need to impose boundary condition and work on a

“restricted bulk ﬁeld space” only (see e.g., [Rab21, WY22, GRW20, GW19, Zen21] for this approach).

In this setting, renormalization is less developed than the closed spacetime case, and we can ﬁnd

discussions for speciﬁc models, for examples, in [Alb16, Rab21].

Alternatively, we could generalize BV formalism itself to study QFT on manifold with boundary.

A potential generalization is proposed by Cattaneo, Mnev and Reshetikhin [CMR14, CMR18], called

“BV-BFV formalism”1. Its central notion, called “modiﬁed quantum master equation (mQME)”,

characterizes that the anomaly to quantize the theory in BV formalism is controlled by certain

boundary data. This formalism also contains formulation for (gauge) ﬁeld theories on manifold with

corners, hence may help with topics such as functorial QFT and bulk-boundary correspondence.

While BV-BFV formalism has been shown fruitful even only at classical level (see e.g., [CS16,

CS19, Sch15, RS21, MS22]), it has not incorporated a systematic renormalization for quantization.

For topological ﬁeld theories, a successful example in quantum level can be found in [CMR20]. For

ﬁeld theories which are not topological, we need to add counter terms to make the contributions of

Feynman graphs ﬁnite. Counter terms on manifolds with boundaries in homotopic renormalization

has been discussed in [Rab21]. To quantize ﬁeld theories which are not topological in BV-BFV

formalism, We hope to adapt Costello’s homotopic renormalization to BV-BFV formalism. As the

ﬁrst step, we would like to clarify the relation between BV-BFV formalism and the approach within

BV formalism mentioned above.

1.1 Main results

We use AKSZ type [ASZK97] topological quantum mechanics (TQM) as the toy model to study the

above questions.

Based on previous work [WY22], we obtain a rigorous BV-BFV description of TQM on R>0

and I= [0,1], with homotopic renormalization incorporated. The mQME’s are stated in Deﬁnition

3.2.1 and Deﬁnition 4.0.2. Their generic solutions are described in Theorem 3.2.1 and Theorem

4.0.2, respectively. Then we derive and sharpen the BV description in [WY22] from these BV-BFV

constructions (see Proposition 5.0.1 and Proposition 5.0.2). We use 1D BF theory to demonstrate

our result in Example 4.0.1, 5.0.1.

This brings the study on TQM to a new stage, and we would like to present the result for TQM

on R>0(Theorem 3.2.1 and Proposition 5.0.1) here. Concretely, the TQM is of AKSZ type with

target being a ﬁnite dimensional graded symplectic vector space V. A Lagrangian decomposition

V=L⊕L0is chosen. We have:

1“BFV” for Batalin-Fradkin-Vilkovisky [BF83, FV75].

•Given a functional I0=π∗(J∂) + RR>0I∂with certain I∂∈ O(V)[[~]], J∂∈ O(L)[[~]], 2it

induces a consistent BV-BFV interactive theory with polarization V=L⊕L0and a BFV

operator Ωleft

L0(H∂,−) if and only if

I∂?~I∂= 0 and H∂=−eJ∂/~?~I∂?~e−J∂/~,

where ?~denotes the Moyal product on O(V)[[~]], and Ωleft

L0(H∂,−) denotes the Weyl quanti-

zation of H∂∈ O(V)[[~]] on O(L0)[[~]].

•The functional I0=π∗(J∂) + RR>0I∂above induces a consistent BV interactive theory with

“boundary condition L” if and only if3

I∂?~I∂= 0,and Ωright

L(eJ∂/~, I∂)=0,

where Ωright

L(−, I∂) denotes the Weyl quantization of I∂on O(L)[[~]].

We would like to stress the following aspects of the story, which should persist in more general

settings.

Homotopic renormalization in BV-BFV formalism

Perturbative BV-BFV formalism involves a BV structure, a splitting and a BFV operator which all

need to be properly regularized (or, renormalized) in order to rigorously quantize a generic theory.

For TQM, we only need to solve this problem for the former two structures.

A renormalized BV structure has been constructed in [Rab21] using homotopic renormalization,

endowing the “restricted bulk ﬁeld space” ELwith a diﬀerential BV algebra (O(EL),d, ∂Kt) for each

renormalization scale t > 0, see (3.6). As for the splitting, we propose a notion:

•The renormalized splitting (at scale t)is the map θt:L0→ E determined by

θt(l0) = 2(I(α)(l0⊗ −)⊗1) ¯

P(0, t)|C1

for l0∈L0(details see Deﬁnition 3.1.2).

This is deﬁned according to the renormalized BV structure associated to EL, as it depends on the

“propagator from scale 0 to scale t”¯

P(0, t). θtﬂows with tin a way compatible with the homotopic

renormalization group ﬂow (3.9, 3.15) of relevant BV structures:

•For ∀ε, Λ>0, the renormalized splittings θε, θΛmake this diagram commute:

O(E)[[~]] Iθε

−→ O(L0)⊗ O(EL)[[~]]

↓e~∂P(ε,Λ) ↓e−I(α)/~(1 ⊗e~∂P(ε,Λ) )eI(α)/~

O(E)[[~]]

IθΛ

−−→ O(L0)⊗ O(EL)[[~]]

where Iθε,IθΛ:O(E)→ O(L0)⊗ O(EL) denote the algebraic isomorphisms induced by θε, θΛ,

respectively (details see Theorem 3.1.1).

2O(V),O(L) are function rings on V, L, respectively. ~is the formal “quantum parameter”. π∗(J∂) is a boundary

term and RR>0I∂denotes the Ω•(R>0)-linear extension of I∂followed by integration over R>0.

3By imposing “boundary condition L” we deﬁne a “restricted bulk ﬁeld space” EL⊂ E in (3.4).

As a consistency check, the renormalized splitting interpolates between the ill-deﬁned “extension by

zero” in the original work [CMR18] and the “bulk to boundary propagator” known to physicists:

θt

“scale 0limit”

“extension by zero”

“scale 1limit”

“bulk to boundary propagator”

(the “scale ∞” here is not taking naive t→+∞in our setting, but corresponds to the “∆∞” in the

proof of [WY22, Proposition 2.3.1]).

Above should be the content of homotopic renormalization in BV-BFV formalism which applies

in general.

In this paper, structures purely on boundary (e.g., the BFV operator) do not need regularization.

If the dimension of spacetime is larger than one, the boundary ﬁeld space will be typically inﬁnite

dimensional, which suggests that homotopic renormalization should also involve the BFV operator.

We leave this consideration for later study.

BV-BFV formalism and the approach within BV formalism

From the BV-BFV description of TQM we read out (5.2), which is the QME written in [WY22,

Section 3]. Moreover, we characterize its generic solutions in Proposition 5.0.1 based on discussions

for mQME. This suggests that BV-BFV formalism could imply the approach within BV formalism.

Besides, if (5.2) is satisﬁed, the mQME in Deﬁnition 3.2.1 can be reinterpreted as a condition

that the map I(0,t)deﬁned in (5.10) is a cochain map:

I(0,t):O(L)[[~]],−1

~Ωright

L(−, H∂)→(O(EL)[[~]],d + ~∂Kt+{It|EL,−}t),

see (5.11) and its following discussion. This translation connects the “wave function” interpreta-

tion of “state” inherited in BV-BFV formalism and the “structure map” interpretation of “state”

from factorization algebra perspective in [CG16, CG21]. It may inspire further comparative studies

between BV-BFV formalism and other frameworks.

Conﬁguration space techniques

The space R>0[n] introduced in Deﬁnition 3.1.1 simpliﬁes our analysis of crucial properties of TQM,

including the UV ﬁniteness (Proposition 3.2.1) and the BV anomaly (Lemma 3.2.1). Such a conﬁg-

uration space technique originates from [Kon94, AS94, GJ94], and reﬂects the power of geometric

considerations. For QFT’s in 2D, another geometric renormalization method of regularized integral

introduced in [LZ21] (see also [GL21]) may applies. Maybe this method can be incorporated in

BV-BFV formalism as well.

1.2 Organization of the paper

The paper is organized as follows.

In Section 2 we brieﬂy introduce perturbative BV formalism and perturbative BV-BFV formalism,

and ﬁx notations on structures such as Moyal product and Weyl quantization for later use.

Section 3 is devoted to TQM on R>0. In Section 3.1, we formulate homotopic renormalization in

BV-BFV formalism for free theory. Rigorous mQME for interactive theory is stated in Section 3.2,

followed by a description of its generic solutions. For TQM on interval, parallel results are obtained

in Section 4.

In Section 5, we extract the BV description of TQM in [WY22] from the BV-BFV description in

Section 3 and Section 4. Then, mQME is reinterpreted from the perspective of factorization algebra

developed in [CG16, CG21].

Acknowledgments We would like to thank Si Li, Nicolai Reshetikhin, Kai Xu, Eugene Rabinovich,

Philsang Yoo, Brian Williams, Owen Gwilliam, Keyou Zeng for illuminating discussion. We espe-

cially thank Si Li for invaluable conversation and guidance on this work. This work was supported

by National Key Research and Development Program of China (NO. 2020YFA0713000). Part of

this work was done in Spring 2022 while M. W. was visiting Center of Mathematical Sciences and

Applications at Harvard and Perimeter Institute. He thanks for their hospitality and provision of

excellent working enviroment. Research at Perimeter Institute is supported in part by the Govern-

ment of Canada through the Department of Innovation, Science and Economic Development Canada

and by the Province of Ontario through the Ministry of Colleges and Universities.

Convention

•Let Vbe a Z-graded k-vector space. We use Vmto denote its degree mcomponent. Given

homogeneous element a∈Vm, we let |a|=mbe its degree.

−V[n] denotes the degree shifting of Vsuch that V[n]m=Vn+m.

−V∗denotes its linear dual such that V∗

m= Homk(V−m, k). Our base ﬁeld kwill mainly be

−Symm(V) and ∧m(V) denote the m-th power graded symmetric product and graded skew-

symmetric product respectively. We also denote

Sym(V) := M

m>0

Symm(V),d

Sym(V) := Y

m>0

Symm(V).

The latter is a graded symmetric algebra with the former being its subalgebra. We will

omit the multiplication mark for this product in expressions (unless confusion occurs).

−We call O(V) := d

Sym(V∗) the function ring on V.

−V[[~]], V ((~)) denote formal power series and Laurent series respectively in a variable ~

valued in V.

•We use the Einstein summation convention throughout this work.

•We use (±)Kos to represent the sign factors determined by Koszul sign rule. We always assume

this rule in dealing with graded objects.

−Example: let jbe a homogeneous linear map on V, then j∗denotes the induced linear

map on V∗: for ∀f∈V∗, a ∈Vbeing homogeneous,

j∗f(a) := (±)Kosf(j(a)) with (±)Kos = (−1)|j||f|here.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PerturbativeBV-BFVformalismwithhomotopicrenormalization:acasestudyMinghaoWangandGongwangYanApril3,2023AbstractWereportarigorousquantizationoftopologicalquantummechanicsonR>0andI=[0;1]inperturbativeBV-BFVformalism.Costello'shomotopicrenormalizationisextended,andincorporatedinourconstruction.Moreover,...

展开>> 收起<<

Perturbative BV-BFV formalism with homotopic renormalization a case study Minghao Wang and Gongwang Yan.pdf

共32页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Perturbative BV-BFV formalism with homotopic renormalization a case study Minghao Wang and Gongwang Yan

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: