q-deformed group characters

Question

In a paper by Yuji Tachikawa, I found a q-deformed "2d Yang-Mills paritition function for a cylinder". Here it is (adapted):

$$ Z(q, x_L, x_R) = \mu(q, x_L)^{-1/2} \langle x_L | \bigg\[ \sum_{R \in \mathrm{Irr}(G)} | R \rangle e^{- aC_2(R) } \langle R | \bigg\] |x_R \rangle \mu(q, x_R)^{-1/2}$$

Here's some stuff to help you interpret:

• $G$ is a compact lie group and the Irreducible representations should be indexed by the root lattice.
• conjugacy classes are indexed by elements of maximal torus $\vec{x} \in \mathbb{T}^n \subset G$
• $C_2(R)$ is the quadratic Casimir of the representation.
• In my notation, borrowed from quantum mechanics $\langle R|x \rangle = \chi_R(x), \langle x|R \rangle=\overline{\chi_R(x)}$.
• $\displaystyle \mu(q, X) = \exp \left[ \sum_{n=1}^\infty \frac{-2q^n}{1-q^n}\chi_{\mathrm{adj}}(x^n) \right]$
• The partition function depends on the area $a$ of the cylinder.

In fact, let's turn this into a statement about the Laplacian: The $q$-dependence is hidden:

$$ e^{- a \Delta} = \sum_{R \in \mathrm{Irr}(G)} | R \rangle e^{- aC_2(R) } \langle R | $$

Let's set the area to $0$. From the last line, we should get the identity matrix. However,

$$ \sum_{R \in \mathrm{Irr}(G)} \langle x_L | R \rangle \overline{ \langle x_R | R \rangle } = \mu(q, x_L) \delta(x_L = x_R)$$

This really looks like orthogonality of characters for compact groups, except the right side should be the identity.

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**What are these characters $\langle x | R \rangle$ ?**

Originally, I wanted to ask about an analogue for finite $G$, but I don't even have a point of reference.

Answer 1

They are proportional to the Schur polynomials. For representation $\lambda \in \mathrm{Irr}(G), \chi \in \mathbb{T}^n \subset G$:

$$ \langle \lambda | x \rangle = \mu(q,x) \chi^\lambda(x) $$

This weight is is independent of the representation so we can do $\displaystyle \sum_\lambda$ no problem!

Here the Schur polynomial is defined by $\displaystyle \chi^\lambda(a) = \frac{ \det a_i^{\lambda_j + k - j}}{\det a_i^{k-j}}$

The number $\mu(q,x)$ is called superconformal index denoted $\mathcal{I}_q^V(a)$ in Section 6 of Gauge Theories and Macdonald Polynomials by Gadde, Rastelli, Razamat & Yan.

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This superconformal index is the trace over representations over a certain superalgebra. It can also be considered a matrix integral over Haar measure. In the case of the cylinder

$$ \mu(a,b) = \Delta(a) \mathcal{I}^V(a) \delta(a, b^{-1})$$

at least, in the zero area limit.

In special cases, they find functions $f^\alpha(a)$ (in our case $=\langle \lambda|x \rangle$) orthogonal with respect to "propagator measure":

$$ \oint [da] \Delta(a)\mathcal{I}^V(a) f^\alpha(a) f^\beta (a^{-1}) = \delta^{\alpha \beta}$$

Then they can define a new metric and structure constants

\begin{eqnarray} \mathcal{I}(a,b,c) &=& \sum_{\alpha, \beta, \gamma} C_{\alpha\beta\gamma}f^\alpha(a)f^\beta(b)f^{\gamma}(c)\\ \mu^{\alpha\beta} &=& \oint [da]\oint [db] \mu(a,b) f^\alpha(a)f^\beta(b)\\ \end{eqnarray}

These generalize the orthogonality of characters relations (at least for compact Lie groups).

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This "superconformal index" for a surface with punctures has an expression in terms of these generalized group characters.

$$ \mathcal{I}_{g,s} (a_1, a_2, \dots, a_n) = \sum_\alpha (C_{\alpha\alpha\alpha})^{2g-2+s} \prod_{i=1}^s f^\alpha(a_i) $$