Questions tagged [supersymmetry]

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1answer
62 views

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 ...
6
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0answers
128 views

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 ...
11
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0answers
510 views

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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1answer
221 views

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 ...
7
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1answer
371 views

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 $\mathfrak{h}^*[2]/W$ (...
4
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1answer
498 views

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 ...
2
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0answers
99 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 ...
7
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1answer
132 views

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 $...
10
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1answer
182 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 ...
5
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0answers
215 views

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 ...
10
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1answer
385 views

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 ...
9
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4answers
784 views

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 ...
5
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0answers
198 views

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 ($m,n$)-...
15
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6answers
1k views

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 ...
2
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0answers
195 views

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 ...
3
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1answer
203 views

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 ...
3
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1answer
115 views

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 ...
1
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1answer
107 views

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, \...
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1answer
71 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 ...
7
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1answer
273 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 ...
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0answers
52 views

$\mathbb{Z}_2$ 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 ...
9
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2answers
276 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 ...
8
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1answer
744 views

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 $C(A)$, ...
2
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1answer
95 views

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:...
23
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1answer
629 views

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 "$\Pi$" ...
3
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1answer
188 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(...
5
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1answer
496 views

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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2answers
312 views

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
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0answers
519 views

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 ...
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0answers
111 views

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
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1answer
420 views

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 ...
3
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0answers
94 views

Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related: (a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...
1
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0answers
157 views

Classification of (almost) contact structures on $S^3$

Question: Is there a classification of almost contact or contact structures on $S^3$? What is it and references? The motivation of this question is as follows: (1) There is one paper showing that a ...
7
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1answer
739 views

wrapping M5-branes on a Riemann surface

AdS/CFT gives us a way to use geometry to study field theory! I am trying to wrap M5-branes on a Riemann surface $\Sigma_{g}$. In my problem, for a Riemann surface in 11d, the normal bundle is max $SO(...
11
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1answer
374 views

Is there a version of supersymmetry for homogeneous spaces?

The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...
1
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0answers
110 views

About the massless supermultiplets in $2+1$ dimensional supersymmetry [closed]

I thought of cross-linking here this question that I had asked on physicsstackexchange. It would be a great help if someone can answer that.
8
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1answer
475 views

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 trees of }...
5
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0answers
312 views

Is the SUSY Algebra isomorphic for all Kähler Manifolds?

For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the ...
5
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1answer
381 views

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 ...
23
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2answers
822 views

Is super-vector spaces a “universal central extension” of vector spaces?

Is there some sense in which the category $sVect$ of super-vector spaces is the "maximal non-trivial extension" of $Vect$ as a symmetric monoidal category? Is the $\mathbb Z/2$ that shows up in the ...
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2answers
2k views

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,...
9
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1answer
1k views

I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt

Question: I am talking about the proof given on pages 50-52 of Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (...
3
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1answer
1k views

Witten Index, letter partition function and superconformal representations.

Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly. I would like to know of expository references and explanations on the ...
0
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1answer
836 views

Clifford Algebra and Gamma matrices: is this relation generally true for any dimension?

I expect the following relation to be vanishing. But it seems not that obvious. $\Gamma_{ab}^{\lambda}t^at^b \Gamma_{\lambda c(d)}t^c=0$ where $t^a$ are even ghosts, "$ab$" are indices for matrix ...
2
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2answers
530 views

Change of coordinates introduced through dx

Hi, I have a superspace spanned by 4 commuting coordinates + 2 anti-commuting ones $\{x^\mu,\theta^\alpha\}$, I have to do the change of coordinates $dx^\mu\to dy^\mu= dx^\mu+d\theta^\alpha \eta_\...
6
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1answer
1k views

N=(2,2) supersymmetry in two-dimensional Euclidean space

Can anyone describe (or give a reference for) the 2d superspace formulation of N=(2,2) SUSY in Euclidean signature? I'm reading Hori's excellent introduction to QFT in the book 'Mirror symmetry', and ...
15
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2answers
636 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? (...
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2answers
1k views

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 ...
11
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1answer
412 views

3/4-Lie superalgebras: how much of a theory can one develop?

Let $\mathfrak{s} = \mathfrak{s}_0 \oplus \mathfrak{s}_1$ be a real Lie superalgebra. (The ground field does not matter much, but at least one formula will not work as written if the characteristic ...
53
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11answers
8k views

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 ...