Questions tagged [supersymmetry]
The supersymmetry tag has no usage guidance.
63
questions
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The supermoduli space of supertori with odd spin structure and metaplectic group actions
I'm trying to understand the description of the supermoduli space of supertori with odd spin structure as a quotient of the super complex upper half plane $\mathbb{H}^{1|1}$. Such a description ...
13
votes
0
answers
847
views
What is a BPS state and why is it the cohomology of a moduli space?
The notion of a BPS state has existed in physics for a long time: I do not understand it completely, but I get the impression they are the supersymmetric analogue of ground states, and then physicists ...
18
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3
answers
2k
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Mathematical motivation for supergeometry
Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) ...
1
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0
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55
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Existence of a minimal ideal with a specific property
Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...
19
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3
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Does the super Temperley-Lieb algebra have a Z-form?
Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...
19
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7
answers
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Supermanifolds — elementary introduction?
I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting.
Elementary in the sense that there is plenty of motivation for ...
2
votes
0
answers
59
views
Link between Extended Supergauge Group and Twistor Theory
In a 1974 paper on supergauge transformations in four dimensions, Wess and Zumino considered an extended supergauge group which contains the conformal group as a subgroup. An interesting thing about ...
4
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0
answers
100
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Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed
I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ...
9
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1
answer
603
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Matrix-tree theorem via supersymmetry (i.e. Grassman algebras)
The matrix-tree theorem states the number of spanning trees of a graph $G$ is equal to a modified determinant of the adjacency matrix or "graph Laplacian", $\Delta_G$:
$$\#\{ \text{spanning ...
17
votes
2
answers
831
views
What is the conceptual significance of supercommutativity?
A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? (...
3
votes
1
answer
185
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Supersymmetric SYK Model in 3D?
In a 2017 article More on supersymmetric and 2d analogs of the SYK model by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study ...
7
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2
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342
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Is there Z_n graded supersymmetry?
I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such ...
2
votes
1
answer
149
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Notation on supergeometry — parity
I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...
6
votes
1
answer
226
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Chain rule for the superderivative
A one dimensional complex supermanifold $X$ is locally described by an ordinary complex coordinate $z$ and an anticommuting coordinate $\theta$, $\theta^2 = 0$.
The superderivative is the square root ...
3
votes
1
answer
233
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Supersymmetry charge $Q$ as anti-linear and anti-unitary operator
We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...
6
votes
0
answers
209
views
References for superhomology
This question concerns topological string theory.
It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
4
votes
2
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244
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Spectral Flow Invariance for Calabi-Yau Sigma Models
I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.
If $X$ is a Calabi-Yau $d$-...
2
votes
1
answer
86
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Sufficient conditions for unitarity of a representation of a Lie Superalgebra
Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...
11
votes
0
answers
588
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The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
6
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0
answers
209
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Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory
In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
7
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1
answer
549
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Implications of gauge symmetry breaking on the spectral side of geometric Langlands?
Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (...
6
votes
1
answer
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Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
3
votes
1
answer
158
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Reference request: coordinate ring of $OSP(2p|n)$
In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.
Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are ...
6
votes
1
answer
1k
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Question on Witten’s paper “Supersymmetry and Morse theory”
EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)).
This well known article applies some tools developed by physicists (e.g. path integrals) to ...
10
votes
1
answer
230
views
Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?
For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...
8
votes
1
answer
150
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Are there Type III codes with small but nonzero "index"?
Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
2
votes
0
answers
141
views
Two definitions of super-Virasoro algebra
Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...
5
votes
0
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245
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Localization principle in integration over supermanifolds
This post is closely related to the post Localization principle in supersymmetry
and can be considered as a continuation of it, although independent.
In § 9.3 of the book "Mirror symmetry" (K. Hori ...
11
votes
1
answer
586
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supersymmetry and the de Rham complex
In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is
given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior ...
10
votes
4
answers
1k
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Geometric or conceptual way to understand supersymmetry algebra
Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ...
6
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0
answers
213
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Physical effects in supersymmetric theories of the underlying supermanifold being split or non-split?
Given an $m$-dimensional $C^{\infty}$ manifold $M$ with an $n$-dimensional vector bundle $E$ over $M$, one can use the transition functions of the manifold and the bundle to construct an ($m,n$)-...
2
votes
0
answers
232
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What exactly is the role of the mysterious manifold underlying the definition of a superspace?
In the intro to chapter 12.3 of this book about the applications of coherent states, it says that
classical spaces for bosons are real or complex vector spaces or manifolds, whereas classical spaces ...
4
votes
1
answer
794
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Definition of orthosymplectic supergroups
I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard.
The first version of the ...
1
vote
1
answer
180
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Construct super Poisson brackets on the coordinate rings of Lie super groups
On line 7 of page 61 of the book a guide to quantum groups, a Poisson bracket is defined on $\mathbb{C}[GL_n]$ for every classical $r$-matrix as follows.
Let $V$ be a vector space with a basis $v_1, \...
2
votes
1
answer
88
views
Trace of the chiral matrix of a subspace
Let $(V,Q)$ be a pair consisting of a $\mathbb{C}$-vector space $V$ together with a nondegenerate bilinear form $Q$ and let $V_0\subseteq V$ be a linear subspace such that $Q\vert_{V_0}$ is ...
25
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2
answers
2k
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What does the Tannakian formalism reconstruct when fed the category of chain complexes?
I've recently realized that there is a gap in my understanding of the Tannakian formalism for reconstructing an (algebraic) group from its category of (finite-dimensional) representations. To warm up,...
8
votes
1
answer
395
views
How do you get the spectral curve from a Calabi-Yau?
In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann ...
2
votes
0
answers
72
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$\mathbb{Z}_2$ graded analog of row operations for supermatrices
I'm working on some research involving supermatrices, and I was wondering if there was a $\mathbb{Z}_2$ graded analog of row operations for supermatrices.
It seems to me that it makes sense to have ...
10
votes
2
answers
355
views
What is the "quaternionic" super Brauer group?
In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
3
votes
1
answer
238
views
projective representation of supergroup
In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group SO(...
29
votes
1
answer
964
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Is there a symmetric monoidal 2-category "SuperDuperVect"?
Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called &...
10
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1
answer
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Mathematics of Chiral Rings
Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.
We now construct $C(A)$, ...
3
votes
1
answer
124
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An intrinsic supergeometric description of the Green–Schwarz supersymmetric action
The Green-Schwarz action is a natural supersymmetric extension of the Polyakov action (with a $B$-field which I will omit in what follows since it is not relevant to the question).
For a morphism $X:...
62
votes
11
answers
10k
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Why is the exterior algebra so ubiquitous?
The exterior algebra of a vector space V seems to appear all over the place, such as in
the definition of the cross product and determinant,
the description of the Grassmannian as a variety,
the ...
5
votes
1
answer
620
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Localization principle in supersymmetry
In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd ...
1
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2
answers
349
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Witten index non-trivial in the context of Quantum Mechanics?
Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$.
I will now consider the one-dimensional case on a compact set:
So ...
3
votes
0
answers
615
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Orthosymplectic group, matrix representations
We have the orthosymplectic $osp(n,m|2k)$. The bosonic part is $so(n,m)\times sp(2k)$. The lie algebra generators are given in eg
http://cds.cern.ch/record/524737/files/0110257.pdf$
where the group ...
5
votes
1
answer
389
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About the quantum spectrum of a certain potential.
Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...
1
vote
0
answers
139
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Double vector bundle vs vector bundle over supermanifolds
Double vector bundle is roughly a vector bundle over (horizontal) vector bundle.
Vector bundle is a supermanifold in a nature way (non-nature on the other way around).
My question: is a double ...
3
votes
1
answer
570
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A good reference for learning about super-differentiation & super-integration?
I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis.
Unfortunately both books lack a clear ...