3D拓扑扭曲索引的分解：TTCSM在S2×S1上的路径积分研究

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本文探讨了在N=2 Chern-Simons物质理论（Topologically Twisted N=2 Chern-Simons Matter Theory, TTCSM）的背景下，对于三维空间S2（球面）上的路径积分。研究焦点在于对厄密（非辅助）场配置进行分析，这是一种重要的理论框架，因为这种配置有助于揭示拓扑性质和量子效应。首先，作者计算了TTCSM在S2乘以S1的一半（半个圆环）上的积分结果，这个过程被称为3D拓扑扭曲索引的第一次“卷积”形式。这种方法允许作者将理论在特定几何结构上的行为与拓扑不变量联系起来，这对于理解理论的物理特性至关重要。接着，作者将研究扩展到TTCSM在S2乘以完整的S1的情况，这里涉及到一个“带刺孔”的结构，这种变化可能会导致不同的物理效应或对称性的破坏。在这种情况下，理论的性质可能更加复杂，但提供了对理论在不同拓扑背景下的行为的深入洞察。最后，与前两者不同的是，当TTCSM与S2乘以S1结合时，矢量辅助场D被约束为反厄米特型（anti-hermitian），这可能是理论的一个关键约束条件，它可能会影响路径积分的结果和理论的稳定性。本文的主要贡献是给出了3D拓扑扭曲索引的具体公式，并且通过对不同几何设置的分析，揭示了TTCSM的内在对称性和拓扑性质。这些结果不仅对于理论物理学本身具有重要意义，而且可能对材料科学、凝聚态物理以及量子计算等领域有所启发，尤其是在寻找新的拓扑相变和量子纠缠现象的研究中。关键词包括：超对称 gauge 理论、Chern-Simons 理论、超对称性与对偶性。文章发表于《Journal of High Energy Physics》（JHEP），并且遵循开放获取原则，由SCOAP3资助，强调了作者的知识产权和开放科学理念。引用此文献时，需注意论文的DOI号：10.1007/JHEP04(2017)115。

JHEP04(2017)115

In a while we will comment about (2.12) and its relation with the term used in [1], that

we have written in the l.h.s. of (1.5).

In subsections to come the non covariant form of the algebra (2.3) and the localising

term (2.12) will be used to compute one loop determinants. This will prove to be useful

at the technical level. Mainly because with the aforementioned form of the algebra and

localising term, the explicit time dependence of the zero locus background σ

mentioned

in the introduction, (1.4), will be implicit.

From the term (

•



λ)Q



λ in the expansion of (2.12), and with the help of (2.3) is

straightforward to get the bosonic part



vector

≡ D













. (2.14)

It should be stressed that all equalities in this subsection do take into consideration total

derivatives. Notice that with the reality conditions (2.7) this bosonic term is positive

deﬁnite.

As for the fermionic part, one gets



•



λ) = −i

†

←−

, (2.15)

with

θ,φ

1 + σ

θ,φ

, P

:= D

. (2.16)

From (2.15) we obtain the fermionic part of (2.12)



vector

= −(

†

)





←−

(

←−

+ i csc θ

←−

) −i

←−





. (2.17)

Notice that this expression is non covariant. The use of this non covariant form results

convenient when computing 1-loop determinants. We stress that D

should be understood

as covariant derivative that includes the magnetic ﬂux potential felt by the eﬀective charge

of the corresponding gaugino component. Speciﬁcally, the spin connection part of the

covariant derivative is included in the latter mentioned magnetic ﬂux contribution. See

appendix B.3 for further details in this issue.

In appendix A.1 we show how to derive the localising action coming from (2.12) but by

using the covariant form of the algebra (2.1). Is straightforward to check that the result for

the action in the latter case is the same (2.14)+(2.17) but after substitution of (2.4). This

coincidence, checks the consistency of using the algebra in terms of the redeﬁned potential

A (2.3) instead of the usual one in terms of A (2.1).

It is also possible to use the orthodox localising term, the one used in [1] for instance,

which in the conventions used in this paper is given by



T r



†

λ − 4σD



. (2.18)

– 9 –

JHEP04(2017)115

After expanding (2.18) with the use of the non covariant form of the algebra (2.3) one

gets



vector

µν





+ D

+ 2σi

− 2σi

− 2iσ{

†

, λ}

−2i



D − i



, (2.19)



vector

:= i

†

Dλ − i



†

−



, (2.20)

with ?F



νβ

and P

∓

1∓σ

. Trace over the gauge indices is assumed in (2.19)

and (2.20). It is straightforward to show that after replacing the expression for the redeﬁned

gauge potential

A in terms of A, (2.4), in (2.20), one obtains the covariant form of the

localising action. Namely, the localising term (2.18) worked out with the covariant form of

the algebra (2.1)



vector

µν

+ (D

σ)

+ D

− iσ{

†

, λ} − iσ{

†

, P

−

λ}

+2 i D



σv

− v

σD + iσD

σ − 

µνβ

σD





vector

:= i

†

Dλ + iσ{

†

, P

−

λ} − iD



†

−



. (2.21)

Again, this coincidence checks the consistency of using the algebra in terms of the redeﬁned

potential

A (2.3) instead of the covariant one (2.1).

As one could expect a priori, the diﬀerence between the localising term (2.12) and (2.18)

is the Q



- exact term





2Dλ +



(4σD)



, (2.22)

which, naively, in virtue of the localisation argument, means both localising terms are set

to compute the same observables. However one must not forget that the for the reality

conditions (2.7), the bosonic part of the localising term to be used here, (2.12), is positive

deﬁnite, meanwhile the bosonic part of (2.18) is not.

Next, the same analysis will be performed for the matter sector of the theory. Let us

deﬁne the following localising term



Matter

:= Q





iγ

†

+ F

†



+ i

†

λφ



. (2.23)

In appendix A.2, it will be seen how this localising term diﬀers from the one used in [1],

by using the covariant form of the algebra (2.1) and (2.8). The diﬀerence being a Q



exact

term, as shall be seen below and thenceforth both localising terms are set to compute the

same observables.

The expansion of (2.23) under the redeﬁned form of the Q



algebra, (2.3) and (2.10),

is given by the sum of the following bosonic and fermionic terms



Matter

= (

φ)

†

φ +

†



σ + iD − 

µν

(qV

µν

+ W

µν

)



†

F +



i 

µν

†



, (2.24)



Matter

= i

†

ψ − i

†

λ φ − i

†

−

ψ − i



†



. (2.25)

– 10 –

JHEP04(2017)115

The saddle point expansion of these Lagrangians about the Q



localising locus will

coincide with the one of [1]. Notice the presence of the term

†

σφ. This term provides

the ﬂux dependence that in [1] comes from the complex value of D. Speciﬁcally the Q



zero

locus condition will ﬁx D

σ to the density −F

. To show the diﬀerence of the localising

term (2.23) with the orthodox one, the one used in [1], let us write down the latter one in

our conventions. The orthodox localising term for matter in the conventions of this paper

is given by the super derivative





−

†

ψ − 2i

†

σφ



, (2.26)

where the



algebra has not been deﬁned yet. We postpone the writing of



until

next subsection, equation (2.55), in order not to make too clumsy the presentation in this

subsection.

After expanding (2.26) with the redeﬁned form of the Q



algebra (2.3) (2.10) one

obtains



Matter

= (

φ)

†

φ +

†



σ + iD − 

µν

(qV

µν

+ W

µν

)



†

F + 2

†

φ +



i 

µν

†



, (2.27)



Matter

:= i

†

ψ−2 i

†

σ ψ−i

†

λ φ−i

†

ψ−i



†



. (2.28)

With the use of the redeﬁnition of

A in terms of A (2.4), this expansion presented before

coincides with the expansion in terms of the original form of the algebra (2.1) and (2.8)

which is given by



Matter

:= D

†

φ +

†



iD + σ

− 

µν

(qV

µν

+ W

µν

)



†

F + D



†

σφ + i 

µν

†



, (2.29)



Matter

:= i

†

ψ − i

†

σ ψ − i

†

λ φ − i

†

ψ − iD



†



, (2.30)

Again, this coincidence checks the consistency of using the redeﬁned form of the alge-

bra (2.10). Let us take a moment aside to comment about the importance of total deriva-

tives when dealing with BPS conﬁgurations of the kind (1.4). Notice that should one

discard the total derivative term D



†

σφ



in (2.29) the quadratic expansion about the

zero loci of the form (1.2)+(1.4) would not depend on the ﬂuxes m. Indeed, discarding such

a term would be a mistake, because its integration over S

× I

is non trivial. This is the

reason why using the form (2.27) instead of (2.29) is advisable. However notice that (2.29)

has an explicit dependence on σ that will give rise to an explicit dependence on t in the

quadratic expansion. Such a dependence can be absorbed by adding up a Q



exact term.

The ﬁnal and homogeneous in time quadratic operator expansion would be coming from

the new Q



- exact localising term (2.23) whose bosonic part has been written in (2.24).

The diﬀerence between the super-derivative localising term (2.26) and the one to be

used to compute one loop determinants (2.23), is the Q



exact term





2 i

†

σ ψ



, (2.31)

– 11 –

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