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首页2d (0,2)对偶与4-单纯形:构建三维量子场论接口
本文探讨了在三维超弦理论(3d N=2 XYZ模型和3d N=2 SQED)中的一个重要概念——(0,2)对偶性。作者提出了一种将简单的2d N=(0,2)对偶接口与最基础的四维几何结构——四面体(4-单形)相联系的观点。这个接口作为桥梁,使得3d理论与2d理论之间的对称性转换得以实现。 研究者进一步扩展了这种对偶性的框架,将它应用到更广泛的具有边界的四维三角化空间上,特别是那些具有2d N=(0,2)接口的四维三角形。他们特别关注了与四维三角剖分局部改变相关的接口的红外对偶,这些改变由Pachner移动(3,3)、(2,4)和(4,2)规则控制。通过计算超对称半指数,作者验证了这些对偶性,这是理论物理中用来检验对称性的一种强大工具。 文章的核心部分还讨论了如何通过不同的场论选择,构建能够独立捕捉四面体几何和Pachner移动的二维理论。这与传统意义上的接口有所不同,这些独立的2d理论展示了阿贝尔N=2 Gadde-Gukov-Putrov的(0,2)对偶行为的全新视角。这种理论的发展不仅深化了我们对2d对偶与四维几何之间关系的理解,也为探索更高维度的量子场论和拓扑学提供了新的实证手段。 这篇论文提供了一套新的、更为具体的工具和技术,用于研究最近几年在2d对偶与四维几何之间日益丰富的互动,有助于推动理论物理学中高维对称性和拓扑性质的深入探究。通过这些研究成果,研究者希望能够揭示更深层次的物理规律和宇宙结构的奥秘。
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JHEP08(2019)132
3
0
2
1
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3
0
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1
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2
4
3
0
2
1
4
3
0
2
1
4
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4
3
4
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0
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1
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1
3
)
(2,3)
Figure 2. The (2,3) Pachner move, interpreted as relating two different sequences of flips.
(Thus ( ∂∆
5
)
−
omits vertices 0, 2, and 4; while (∂∆
5
)
+
omits vertices 1, 3, and 5.) Note
that (∂∆
5
)
±
are triangulations of the same 4d polyhedron, homeomorphic to a standard
4-disc D
4
, which plays a role analogous to the bipyramid above.
Next, we claim that the boundary of both (∂∆
5
)
+
and (∂∆
5
)
−
can be split symmet-
rically into two 3-dimensional octahedra:
∂
(∂∆
5
)
±
= octahedron
−
∪ octahedron
+
; or topologically, S
3
D
3
∪
S
2
D
3
.
2
1
3
4
0
5
2
1
3
4
0
5
'
'
octahedron
−
octahedron
+
[
(2.8)
Combinatorially, the boundary of (∂∆
5
)
−
consists a collection of tetrahedra obtained by
omitting a vertex from any of the three pentachora [12345], [01345], [01235] that appear
in (2.7). We find:
∂[12345] = [2345] ∪ [1345] ∪ [1245] ∪ [1235] ∪ [1234]
∂[01345] = [1345] ∪ [0345] ∪ [0145] ∪ [0135] ∪ [0134]
∂[01235] = [1235] ∪ [0235] ∪ [0135] ∪ [0125] ∪ [0123]
(2.9)
Of the 15 tetrahedra in this list, three of them (in bold) appear twice. The tetrahedra
that appear twice are glued together pairwise, and do not contribute to the total boundary
–8–
JHEP08(2019)132
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[12345]
[01345]
[01235]
[02345]
[01245]
[01234]
left
right
front
back
top
bottom
)
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<latexit sha1_base64="E+ql3SgwjIInsFzh7/ZNPWlwHRk=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMVWwtpKJvtpl262Q27E6WE/gwvHhTx6q/x5r9x2+agrQ8GHu/NMDMvSgU36HnfTmlldW19o7xZ2dre2d2r7h+0jco0ZS2qhNKdiBgmuGQt5ChYJ9WMJJFgD9Hoeuo/PDJtuJL3OE5ZmJCB5DGnBK0UdO/4YIhEa/XUq9a8ujeDu0z8gtSgQLNX/er2Fc0SJpEKYkzgeymGOdHIqWCTSjczLCV0RAYssFSShJkwn508cU+s0ndjpW1JdGfq74mcJMaMk8h2JgSHZtGbiv95QYbxZZhzmWbIJJ0vijPhonKn/7t9rhlFMbaEUM3trS4dEk0o2pQqNgR/8eVl0j6r+17dvz2vNa6KOMpwBMdwCj5cQANuoAktoKDgGV7hzUHnxXl3PuatJaeYOYQ/cD5/AJEbkW0=</latexit>
<latexit sha1_base64="E+ql3SgwjIInsFzh7/ZNPWlwHRk=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMVWwtpKJvtpl262Q27E6WE/gwvHhTx6q/x5r9x2+agrQ8GHu/NMDMvSgU36HnfTmlldW19o7xZ2dre2d2r7h+0jco0ZS2qhNKdiBgmuGQt5ChYJ9WMJJFgD9Hoeuo/PDJtuJL3OE5ZmJCB5DGnBK0UdO/4YIhEa/XUq9a8ujeDu0z8gtSgQLNX/er2Fc0SJpEKYkzgeymGOdHIqWCTSjczLCV0RAYssFSShJkwn508cU+s0ndjpW1JdGfq74mcJMaMk8h2JgSHZtGbiv95QYbxZZhzmWbIJJ0vijPhonKn/7t9rhlFMbaEUM3trS4dEk0o2pQqNgR/8eVl0j6r+17dvz2vNa6KOMpwBMdwCj5cQANuoAktoKDgGV7hzUHnxXl3PuatJaeYOYQ/cD5/AJEbkW0=</latexit>
(3,3)
Figure 3. The (3,3) Pachner move, interpreted as relation between sequences of (2,3) and (3,2)
moves.
∂
(∂∆
5
)
−
; put differently, they are internal 3-faces in the triangulation of (∂∆
5
)
−
. The
actual boundary ∂
(∂∆
5
)
−
consists of the remaining nine tetrahedra,
∂
(∂∆
5
)
−
=
[2345] ∪ [1245] ∪ [0345] ∪ [0145]
← oct
−
∪
[1234] ∪ [0134] ∪ [0235] ∪ [0125] ∪ [0123]
← oct
+
(2.10)
We split the boundary as indicated into two octahedra oct
±
. The first octahedron, trian-
gulated into four tetrahedra, is shown on the far left of figure 3. The second octahedron,
triangulated into five tetrahedra, is shown on the far right of figure 3.
Successive cobordisms through the three pentachora [12345], [01345], [01235] may now
be interpreted as a sequence of (2,3) and (3,2) Pachner moves that take us from oct
−
to oct
+
. This sequence of moves is shown in the top part of figure 3. Explicitly, cobor-
dism through [12345] is a (2,3) move; cobordism through [01345] is a (3,2) move; and
cobordism through [01235] is another (2, 3) move. Note that the three internal tetrahedra
[1345], [1235], [0135] that appeared twice in (2.9) all play a role in figure 3. Namely, they
are the tetrahedra that are both ‘created’ and subsequently ‘annihilated’ by moves in the
top sequence.
Similarly, we may compute the triangulated boundary of (∂∆
5
)
+
by first considering
∂[02345] = [2345] ∪ [0345] ∪ [0245] ∪ [0235] ∪ [0234]
∂[01245] = [1245] ∪ [0245] ∪ [0145] ∪ [0125] ∪ [0124]
∂[01234] = [1234] ∪ [0234] ∪ [0134] ∪ [0124] ∪ [0123]
(2.11)
After removing the internal (repeated) tetrahedra, we find
∂
(∂∆
5
)
+
=
[2345] ∪ [1245] ∪ [0345] ∪ [0145]
← oct
−
∪
[1234] ∪ [0134] ∪ [0235] ∪ [0125] ∪ [0123]
← oct
+
= ∂
(∂∆
5
)
−
.
(2.12)
– 9 –
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