Recent MathOverflow Questions

Engineering Mathematics I [on hold]

Math Overflow Recent Questions - Sat, 11/24/2018 - 13:01

Given the vectors x1 = [2 1 2], x2 = [1 -1 –2], x3 = [1 1 1]. Test if these are linearly independent (li). If they are li, write x3 as a linear sum of x1 and x2. Is such a linear sum unique? Justify your answer.

Higher homotopical information in racks and quandles

Math Overflow Recent Questions - Sat, 11/24/2018 - 12:49

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.

Q1. a $\star$ a = a

Q2. (a $\star$ b) $\bar\star$ b = (a $\bar\star$ b) $\star$ b = a

Q3. (a $\star$ b) $\star$ c = (a $\star$ c) $\star$ (b $\star$ c)

When we drop out the first axiom we obtain a rack, by definition. Quandles generalize basic properties of the conjugation in a group (where $a \star b = b^{-1}ab$ and $a\ \bar\star \ b = bab^{−1}$), but they are also useful in knot theory.

Nevertheless, I'm interested mainly in the homotopy theory of these objects. In fact, following this article by Eisermann, we can define arrows

  • $a \xrightarrow{b}c$ for each triple $a,b,c \in Q$ with $a \star b = c$.

  • $a' \xleftarrow{b'}c'$ for each triple $a',b',c' \in Q$ with $a'\ \bar\star \ b' = c'$.

Then we have a notion of homotopy, built in the following way (see the article for details).

First define a combinatorial path between two elements $q,q'\in Q$ to be a sequence of arrows going in both ways, such that the first arrow is given by the action of an element of the quandle on $q$ and the last is given by the action of another one on $q'$.

Definition 1 Let $P(Q)$ be the category having as objects the elements $q\in Q$ and as morphisms from $q$ to $q'$ the set of combinatorial paths from $q$ to $q'$. Composition is given by juxtaposition: $$(a_0 \to \cdots \to a_m) \circ (a_m \to \cdots \to a_n) = (a_0 \to \cdots \to a_m \to \cdots \to a_n).$$

Then we can construct an homotopy as in the following definition.

Definition 2 Two combinatorial paths are homotopic if they can be transformed one into the other by a sequence of the following local moves and their inverses:

(H1) $a\xrightarrow{a}a$ is replaced by $a$, or $a\xleftarrow{a}a$ is replaced by $a$.

(H2) $a\xrightarrow{b}a \star b\xleftarrow{b}a$ is replaced by $a$, or $a\xleftarrow{b}a \ \bar\star \ b \xrightarrow{b}a$ is replaced by $a$.

(H3) $a\xrightarrow{b}a \star b\xrightarrow{c}(a \star b) \star c$ is replaced by $a\xrightarrow{c} a \star c \xrightarrow{b\star c} (a \star c) \star (b \star c) $

It seems to me that this data can be extended to a simplicial set, whose 0-simplices are elements of $Q$, 1-simplices are arrows between them and higher simplices witness these homotopical information.

My question is

Does $P(Q)$ embed in a simplicial set which keeps track of these information? Is it possible that this simplicial set is actually an $\infty$-category having $P(Q)$ as homotopy category? Is there an analogous construction for racks?

Is there a Giambelli identity with dual representations?

Math Overflow Recent Questions - Sat, 11/24/2018 - 11:33

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.

The Giambelli identity states that if $a_1 > \dots > a_r$ and $b_1 > \dots > b_r$ are natural numbers with $b_1 \leq n-1$ then the determinant in the representation ring of the matrix $M$ with entries $M_{ij} = V_{ (a_i |b_j)} $ is an irreducible representation of $GL_n$ with highest weight vector $$( a_1 + 1,\dots, a_r +r, r^{b_r}, (r-1)^{b_{r-1} -b_r-1}, \dots, 1^{ b_1 - b_{2}-1} , 0 ^{n-1-b_1}).$$

Technically, the Giambelli identity is an identity in the ring of symmetric functions in infinitely many variables. We deduce this using the fact that the representation ring of $GL_n$ is the quotient ring of symmetric functions in $n$ variables, where each Schur function is sent to an irreducible representation or zero depending on the number of parts.

Let $a_1 > \dots a_r, b_1> \dots > b_{2r}, c_1> \dots c_r$ be natural numbers with $b_1 \leq n-1$.

Let $M$ be the matrix whose entries are $M_{ij} = V_{(a_i|b_j)}$ if $1 \leq i \leq r$ and $M_{ij} = V_{( c_{2r-i} | n-1-b_j )}^\vee$ if $ r+1 \leq i \leq 2r$.

Is $\det M$ an irreducible representation? Is it the one with highest weight vector $$(a_1+1,\dots, a+r+r, r^{b_{2r} }, (r-1)^{b_{2r-1}-b_{2r} -1},\dots, (-r+1)^{ b_1-b_2 -1}, (-r)^{n-1-b_{1}}, -c_r-r, \dots, -c_1-1)?$$

If not, does some similar formula hold for the class of this irreducible representation in the representation ring?

The motivation is that this would help generalize my work in arXiv:1810.01303 on the CFKRS conjecture for integral moments, from the moments conjectures to the ratios conjecture.

It doesn't seem possible to deduce this from any nice identity purely in the ring of symmetric functions, so the same approach might not work here.

Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?

Math Overflow Recent Questions - Sat, 11/24/2018 - 07:34

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1^2,x_1,x_2,x_2^2,x_3,x_4,x_3x_4\}$$ or from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\mathbb Z^4$ with $|x_1|<X_1$, $|x_2|<X_2$, $|x_3|<X_3$ and $|x_4|<X_4$. We see that $x_1,x_2$ and $x_3,x_4$ variables do not mix.

It seems the set of $x_1,x_2$ variables in first set satisfy the generalized lower triangle bound on page $16$ http://www.cits.rub.de/imperia/md/content/may/paper/jochemszmay.pdf and the overall set of variables $x_1,x_2,x_3,x_4$ also satisfy the generalized lower triangle bound in both sets.

In here $\lambda_1=\lambda_2=\lambda_3=\lambda_4=2$ and $D=1$.

Assume we have the additional condition that for a given $(x_1,x_2)\in\mathbb Z^2$ with $|x_1|<X_1$ and $|x_2|<X_2$ we have that there is an unique $(x_3,x_4)\in\mathbb Z^2$ with $f(x_1,x_2,x_3,x_4)=0$, $|x_3|<X_3$ and $|x_4|<X_4$ vice versa for a given $(x_3,x_4)\in\mathbb Z^2$ with $|x_3|<X_3$ and $|x_4|<X_4$ we have that there is an unique $(x_1,x_2)\in\mathbb Z^2$ with $f(x_1,x_2,x_3,x_4)=0$, $|x_1|<X_1$ and $|x_2|<X_2$. $W$ is highest absolute value of coefficient of $f(x_1X_1,x_2X_2,x_3X_3,x_4X_4)$.

  1. In this situation can the bound of $X_1^{\lambda_1}X_2^{\lambda_2}X_3^{\lambda_3}X_4^{\lambda_4}\leq W^\frac1D$ be improved to $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3}X_4^{\lambda_4})\leq W^\frac1D$$ or may be at least $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3/2}X_4^{\lambda_4/2})\leq W^\frac2{3D}$$ ($\lambda_3/2$ and $\lambda_4/2$ is based on guess that variables are disjoint and have separate control and $x_3,x_4$ do not satisfy generalized triangle bound with $\lambda_3=\lambda_4=1$ and assuming $x_1,x_2$ variables were not present in given polynomial will give $W^{\frac2{3D}}$ bound)?

  2. If not what is the best we can do at least for the case $X_1=X_2=X_3=X_4$?

Cross-posted: https://crypto.stackexchange.com/questions/64296/is-there-a-variation-of-coppersmiths-method-that-applies-to-disjoint-variable-s

"Solution" of finite cluster algebras

Math Overflow Recent Questions - Sat, 11/24/2018 - 02:37

Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?

Deriving condition to get correct asymptotic bound

Math Overflow Recent Questions - Fri, 11/23/2018 - 15:09

Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem (Theorem 7 here) for the sum of discrete random variables, $$ P(X=t)=\frac{1}{\sqrt{2\pi n\theta(1-\theta)}}\exp\left(-\frac{(t-n\theta)^2}{2n\theta(1-\theta)} \right)+o(n^{-1/2}) $$ for all $n\geq 1$ and uniformly in the integers $t$.

Suppose $t=n\theta+\sqrt{2\theta(1-\theta)n\log m+O(1)}$. I am interested in the relationship between $n$ and $m$, where we can assume, $m>>n>>1$. For example, in my application, $m\approx 20000$ and $n\approx 200$ seems to work well. Intuitively, $m$ should grow much faster than $n$.

I'm interested in finding a theoretical relationship between $n$ and $m$ such that quantity $mP(X=t)=O(1)$. Now if the $o(n^{-1/2})$ remainder were not there, then I can reason the following,

\begin{align*} mP(X=t)&=O(mn^{-1/2}\exp(-O(\log m)))\\ &=O\left(\exp\left(\log m-\log n^{1/2}-O(\log m)\right)\right)\\ &= O\left(\exp\left(\log m^{r}-\log n^{1/2}\right)\right)\text{ for some constant $r>0$}\\ &= O\left(\frac{m^{r}}{n^{1/2}}\right) \end{align*}

This suggests that $m\leq n^\gamma$, where $\gamma = \frac{1}{2r}>0$ for $r>0$ can be a reasonable relationship.

How do I handle that remainder $o(n^{-1/2})$?

Representations of degenerate Clifford algebras

Math Overflow Recent Questions - Fri, 11/23/2018 - 07:43

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ideal generated by $v\otimes v + b(v,v) \mathbb{1}$ for all $v \in V$. $Cl(V,b)$ is a $\mathbb{Z}_2$-graded unital real associative algebra.

Every such $(V,b)$ is isomorphic to $\mathbb{R}^{r+s+t}$ with bilinear form defined by polarisation from the quadratic form $$ q(x) = \sum_{i=1}^{r + s + t} \varepsilon_i x_i^2, $$ where $$ \varepsilon_i = \begin{cases} 0 & i = 1,\dots,r\\ 1 & i = r+1,\dots,r+s\\ -1 & i = r+s+1, \dots, r+s+t .\end{cases} $$ Let $Cl(r,s,t)$ denote the Clifford algebra of $\mathbb{R}^{r+s+t}$ and the above bilinear form.

As $\mathbb{Z}_2$-graded unital real associative algebras, $$ Cl(r,s,t) \cong \Lambda \mathbb{R}^r \hat\otimes Cl(s,t), $$ where $\hat\otimes$ is the $\mathbb{Z}_2$-graded tensor product and where $Cl(s,t):= Cl(0,s,t)$ are the standard Clifford algebras associated to non-degenerate bilinear forms.

The representations of $Cl(s,t)$ are well-known: there are either one or two simple modules (up to isomorphism) depending on $s,t$ and every finite-dimensional module is a direct sum of simples.

I am interested in the representations of $Cl(r,s,t)$ for $r=1$, but more generally for $r>0$.

For $(s,t) = (1,0)$ and $(0,1)$, it is easy to work this out "by hand". The resulting category of representations is no longer semisimple, but it is not hard to show that any finite-dimensional module is a direct sum of indecomposable (but not simple) modules.

But before attempting to study the case of general $(s,t)$, I wonder whether there is some technology out there which can be brought to bear on this problem.

More concretely, I have a couple of

Questions

  1. Would a knowledge of the indecomposable modules of $\Lambda \mathbb{R}$ and $Cl(s,t)$ be sufficient to determine the indecomposable modules of their $\mathbb{Z}_2$-graded tensor product? If so, how?

  2. Is there a classification of indecomposable finite-dimensional modules of the exterior algebra $\Lambda \mathbb{R}^r$ for $r>1$? If so, where?

Method of Coppersmith optimal for multivariate?

Math Overflow Recent Questions - Thu, 11/22/2018 - 19:51

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials https://eprint.iacr.org/2012/108.pdf but seems to fall much short of general result for univariate case.

Is it believed that the Coppersmith method is best possible for multivariate case and what do we know about the multivariate case?

Hadamard theorem about embedding

Math Overflow Recent Questions - Thu, 11/22/2018 - 14:06

The following theorem is commonly attributed to Hadamard.

Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex set.

Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897) (for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).

Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]

I hope someone here knows this paper and can help me.

Smooth functions on sphere

Math Overflow Recent Questions - Thu, 11/22/2018 - 09:37

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

Probability density from standard domain - I

Math Overflow Recent Questions - Thu, 11/22/2018 - 02:36

Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?

When an isomorphism on cohomology with $Z$-coefficient is trivial will imply it is trivial with $Z_2$-coefficient also?

Math Overflow Recent Questions - Thu, 11/22/2018 - 01:38

Suppose $G$ is a group acting freely on a topological space $X$. Take $g$ is an element of $G$. Let $g^*_\mathbb Z$ and $g^*_{\mathbb Z _2}$ are the corresponding induce map on cohomology with $\mathbb Z$ -coefficient and $\mathbb Z _2$-coefficient respectively, i.e., $g^*_\mathbb Z:H^*(X;\mathbb Z ) \to H^*(X;\mathbb Z ) $ and $g^*_{\mathbb Z _2}: H^*(X;\mathbb Z _2 ) \to H^*(X;\mathbb Z _2) $. Suppose $g^*_\mathbb Z$ is the trivial map. My question is: If $g^*_\mathbb Z$ is the trivial map then does it follow that $g^*_{\mathbb Z _2}$ is also trivial? If not under which condition $g^*_\mathbb Z$ is trivial will imply $g^*_{\mathbb Z _2}$ is also trivial?

"Almost-ideals" in the (simple) Lie algebra of an algebraic group?

Math Overflow Recent Questions - Thu, 11/22/2018 - 01:06

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple.

Is it necessarily the case that there is no subspace $\mathfrak{v}\subset \mathfrak{g}$ with $0<\dim(\mathfrak{v})<\dim(\mathfrak{g})$ such that $\mathfrak{v}$ is invariant under $\mathrm{Ad}_g$ for every $g\in G(K)$?

Note: it is clear that there is no $\mathfrak{v}\subset \mathfrak{g}$ with $0<\dim(\mathfrak{v})<\dim(\mathfrak{g})$ such that $\mathfrak{v}$ is invariant under $\mathrm{Ad}$ for every $g\in G(\overline{K})$. It is also clear that the answer to the question above is "yes" when the number of elements of $K$ is larger than a constant depending only on $n$: since $\mathfrak{v}$ is not an ideal, there is a $v\in\mathfrak{v}$ such that all $g\in G(\overline{K})$ such that $\mathrm{Ad}_g(v)\in \mathfrak{v}$ lie in a proper subvariety of $G$.

Note 2: A friend has just proposed over the breakfast table that there are linear algebraic groups with no non-trivial rational points over $K$. That would obviously imply an answer of "no" to my question. I am not convinced that such a thing is really possible, at least not when we are talking about the group $G(K)$, $G$ simple (as opposed to more exotic groups of Lie type).

Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

Math Overflow Recent Questions - Thu, 11/22/2018 - 01:00

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:

Specifically, how do we define this ideal $\frak{a}$ that allows us to get this group $\Gamma=\Gamma(\frak{a})$? I have tried following up the reference to Geometric construction of cohomology for arithmetic groups by Millson and Raghunathan, but unfortunately despite my best efforts I have really struggled to follow the latter paper and extract anything useful, since I am not only new to hyperbolic geometry, but also completely unfamiliar with number theory.

If someone could concretely describe how to define such an ideal and the group $\Gamma$, I would really appreciate it!

Smooth dependence of convex functions on Monge-Ampère measure

Math Overflow Recent Questions - Wed, 11/21/2018 - 23:46

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as long as $f$ does not tend to $+\infty$ too fast at the boundary, for every $t>0$ there is a unique convex function $u_t\in C^0(\overline{\Omega})\cap C^\infty(\Omega)$ satisfying $$ \det D^2 u_t=t\,f,\quad u_t|_{\partial\Omega}=\phi. $$

Question. Does $u_t$ depend smoothly on $t$?

Why did Voevodsky abandon his work on "singletons"?

Math Overflow Recent Questions - Wed, 11/21/2018 - 21:47

In an interview (I link the Google translation), Voevodsky talks about how, in the late 2000s, he worked on the problem of "restoring the history of populations according to their modern genetic composition". Some of his unpublished papers on this topic are now available online. For example, a paper titled "Singletons" is available on the IAS website. Why did Voevodsky abandon the subject of this rather fleshed-out paper so suddenly?

is the functor of finite order elements in grp representable

Math Overflow Recent Questions - Wed, 11/21/2018 - 21:20

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable. I have the sense that it shouldn't be but I've so far failed to prove it in any way.

Computing with a subvector subspace equipped with a prescribed basis over finite fileds

Math Overflow Recent Questions - Wed, 11/21/2018 - 21:05

Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]_V$, the vector representing $u$ with respect to $B_V$.

How do we effectively compute $[u]_U$ when $\mathbb{k}$ is a finite field, say the one with two elements?

Is this pullback of non-degenerate form invertible?

Math Overflow Recent Questions - Wed, 11/21/2018 - 20:34

For a fiber bundle $M\longrightarrow N$ where $\dim N=n$, a non-degenerate 1-form $\theta$ on $M$ generates the differential ideal $\mathcal{I}$, and the Lagrangian $\mathcal{L}$ is an $n$-form on $M$. All these data form a non-degenerate variational problem.

In Griffiths' book Exterior Differential Systems and the Calculus of Variations page 84, he writes that

For non-degenerate variational problems the rank of $\theta$ is everywhere the maximum possible value $n$.

Then the question comes: Suppose there's another non-degenerate variational problem whose fiber bundle $W\longrightarrow N$ fibers on the same base $N$. Let $\varphi$ be the non-degenerate 1-form on $W$, then it's rank is restricted to be $n$.

If there is a smooth map $f:M\longrightarrow W$, we can induce a pullback $f^*:\varphi\longrightarrow\theta$. We know 1-form is a section of cotangent space (which is a vector space). A constant rank linear transformation between two vector space is invertible. Does the same rank $n$ of $\varphi$ and $\theta$ mean that the $f^*$ is invertible?

Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

Math Overflow Recent Questions - Wed, 11/21/2018 - 20:31

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-Singer index theorem. I cannot find any clue how one can derive that formula from APS index theorem.

In the following, I will briefly explain equation 2.19 in that paper, and will write the Atiyah-Patodi-Singer index theorem. Please help me figure out how one can derive that equation from APS index theorem.

Let $Y$ be a closed three dimensional manifold. Let $G$ be a compact simple gauge group, whose Lie algebra is denoted by $\mathfrak{g}$. Let $E$ be a $G$-bundle over $Y$, with connection $1$-form $A\in\Omega^{1}(Y,\mathfrak{g})$. The Chern-Simons action is given by

$$I[A]=\frac{1}{4\pi}\int_{Y}\mathrm{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right).$$

Let $D_{A}$ be the covariant derivative. Then, one is interested in the operator

$$L=\bigg( \begin{matrix} \ast D_{A}&-D_{A}\ast\\D_{A}\ast&0 \end{matrix} \bigg).$$

To be specific, one is interested in $L_{-}$, the restriction of $L$ on odd forms, i.e.

$$L_{-}=L|_{\Omega^{1}(Y)\oplus\Omega^{3}(Y)}.$$

One defines its eta-invariant

$$\eta_{L_{-}}(A)=\lim_{s\rightarrow 0}\sum_{j}\frac{\mathrm{sign}(v_{j})}{|v_{j}|^{s}}$$

where $v_{j}$ are non-zero eigenvalues of the operator $L_{-}$.

Similarly, one defines the eta-invariant for the trivial gauge $A=0$, denoted by $\eta_{L_{-}}(0)$. With this trivial gauge, one is interested in the operator

$$L=\bigg( \begin{matrix} \ast d&-d\ast\\d\ast&0 \end{matrix} \bigg),$$

restricted on odd forms.

Equation 2.16 in Quantum Field Theory and Jones Polynomial:

$$\frac{1}{4}\left(\eta_{L_{-}}(A)-\eta_{L_{-}}(0)\right)=\frac{c_{2}(G)}{2\pi}I[A]$$

Witten claimed that this is a result from Atiyah-Patodi-Singer index theorem.

The original statement of APS index theorem is from Spectral Asymmetry and Riemannian Geometry I, which is very hard to read for physics students. In the following, I copy the statement of APS index theorem from Aspects of Boundary Problems in Analysis and Geometry, which is easier to read.

Let the closed three dimensional manifold $Y$ be the boundary of a compact, oriented, four dimensional Riemannian manifold $M$, i,e, $\partial M=Y$. Let $S(M)$ be a spin bundle over $M$, then one has the splitting $S(M)=S^{+}(M)\oplus S^{-}(M)$ into chiral halves. Let $E$ be a Hermitian vector bundles over $M$. The twisted Dirac operator on $M$ is defined as

$$\mathcal{D}=\bigg( \begin{matrix} 0&D^{+}\\D^{-}&0 \end{matrix} \bigg),$$

with

$$D^{+}:\Gamma(M,S^{+}(M)\otimes E)\rightarrow\Gamma(M,S^{-}(M)\otimes E)$$

$$D^{-}:\Gamma(M,S^{-}(M)\otimes E)\rightarrow\Gamma(M,S^{+}(M)\otimes E)$$

where $\Gamma(M,S^{\pm}(M)\otimes E)$ is the set of sections of the bundle $S^{\pm}(M)\otimes E$.

In addition, one assumes the following conditions:

  1. $M$ has a collar neighborhood $N=[0,1)_{s}\times Y$ near $Y$, where the metric is a product

    $$g=ds^{2}+h$$

    with $h$ a metric on $Y$.

    $\hspace{3cm}$

  2. Denote the space of square-integrable spinors on $Y$ by $L^{2}(Y,S(Y))$. Near the boundary $Y$, the Dirac operator $\mathcal{D}$ is of product type on the collar of the following form:

    $$\mathcal{D}=\Gamma^{s}(\partial_{s}+D_{Y})$$

    where $\Gamma^{s}:S^{\pm}(N)\otimes E|_{N}\rightarrow S^{\mp}(N)\otimes E|_{N}$ is unitary mapping

    $$L^{2}(Y,S(Y)\otimes E|_{Y})\rightarrow L^{2}(Y,S(Y)\otimes E|_{Y})$$

    of spinors on $Y$, and

    $$D_{Y}:\Gamma(Y,S(Y)\otimes E|_{Y})\rightarrow\Gamma(Y,S(Y)\otimes E|_{Y})$$

    is the self-adjoint Dirac operator on $Y$.

Denote the APS eta-invariant for $D_{Y}$ by

$$\eta_{D_{Y}}(s)=\sum_{\lambda\in\mathrm{spec}(D_{Y})\backslash\left\{0\right\}}m_{\lambda}\frac{\mathrm{sign}(\lambda)}{|\lambda|^{s}}$$

where $m_{\lambda}$ is the multiplicity of the eigenvalue $\lambda$.

Let $\widehat{M}$ be the non-compact elongation of $M$ defined as follows:

$$\widehat{M}=(-\infty,0]_{s}\times Y\cup_{\partial M}M$$

$\hspace{4cm}$

One denotes the extension of $D_{M}$ on $\widehat{M}$ by $\widehat{D}$.

Atiyah-Padoti-Singer:

$$\mathrm{ind}(\widehat{D})=\int_{M}\hat{A}(TM)\mathrm{ch}(E)-\frac{1}{2}\left(\eta(D_{Y})+\dim\ker D_{Y}\right)$$

$$=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}\left(F\wedge F\right)+\frac{\dim E}{192\pi^{2}}\int_{M}\mathrm{Tr}\left(R\wedge R\right)-\frac{1}{2}\left(\eta(D_{Y})+\dim\ker D_{Y}\right)$$

Please tell me how I can derive Witten's formula (2.16) from the above APS index formula. The quadratic Casimir $c_{2}(G)$ is suppposed to come from replacing the trace in the adjoint representation by trace in the fundamental representation in the Chern-Simons action. However, from the APS index formula, I cannot see anything in the adjoint representation.

I've seen some "physical" derivations of the formula (2,16) from Gauge Dependence of the Eta-Function in Chern-Simons Field Theory and the Vilkovisky-DeWitt Correction, and Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, but they made me even more confused.

What is even worse, I found a more genetic formula generalizing Witten's formula (2.16) from Lectures on Quantization of Gauge Systems (equation 60 iin page 53) and Computer Calculation of Witten's 3-Manifold Invariant (equation 1.31 in page 86).

Also, in this physics paper Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups (equation 4.2 in page 33), the APS index theorem looks very different from the original one, with a factor of the quadratic Casimir.

Please tell me where this second order Casimir is coming from.

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