Witten Index, 4-D N = 1 Gauge Theories, and Riemann Surfaces

By George Shiber Thursday, May 26, 2016Saturday, August 17, 2019 In Mathematical Physics, Mathematics, Physics, Super String Theory, Supersymmetry, Witten Theory Gauge Theories, Riemann Surfaces

In this post, I will discuss the Witten Index in the context of 4-D N = 1 gauge theories on ${\Sigma _g} \times {T^2}$ , a Riemann surface $\Sigma$ of genus g times a torus ${T^2}$ where in 2-D, the WI is

and by Jeffrey-Kirwan path-integration, we obtain

with ${Z_{cl,1l}}$ the classical and 1-loop contribution derived from the index of a topological twist on ${\Sigma _g} \times {T^2}$ via the partition function

and crucially noting that the effective action

$\tilde Z = \hat A\exp \int_{{\Sigma _g}} {\left[ {\hat BF + c\wp {\eta ^\dagger } \wedge \eta } \right]}$

is topological on ${\Sigma _g}$ . Take 4-D N = 1 gauge theories with a non-anomalous $U{(1)_R}$ $R$ -symmetry, located on ${\Sigma _g} \times {T^2}$ with a ${\Sigma _g}$ twist. Given that ${T^2}$ has modulus $\tau$ , the Lagrangian is given by

$L = L{\,_{YM}} + {L_{mat}} + {L_{{W^p}}}$

with ${L_{YM}}$ the supersymmetric Yang-Mills Lagrangian, ${L_{mat}}$ the matter kinetic Lagrangian, and ${L_{{W^p}}}$ superpotential interactions. For Abelian gauge-group factors reasons, we must also include a Fayet-Iliopoulos term

${L_{FI}} = i\frac{\zeta }{{2\pi }}D$

where the parameters of the background are the flux

$\tilde n = \frac{1}{{2\pi }}\int_\Sigma {{F^{flav}}}$

and flavor flat connection are

$v = 2\pi \oint_{A{\rm{ - cycle}}} A - 2\pi \tau \oint_{B{\rm{ - cycle}}} A$

and $v$ lives on a copy of the spacetime ${T^2}$ . Now, define

$\left\{ {\begin{array}{*{20}{c}}{q = {e^{2\pi i\tau }}}\\{x = {e^{iu}}}\\{y = {e^{iv}}}\end{array}} \right.$

Then the semi-classical and one-loop contribution ${Z_{{\rm{cl}},1l}}$ consists of the following pieces:

the semi-classical action contribution is from the FI term

${Z_{FI}} = {e^{ - {\rm{vol}}\left( {{T^2}} \right)\,\zeta \hat m}}$

the one-loop determinant for chiral multiplets is

with elliptic functions

$\eta (q) = {q^{1/24}}\prod\nolimits_{n = 1}^\infty {\left( {1 - {q^n}} \right)}$

and

So, the one-loop determinant for the off-diagonal vector multiplets is

whereas the vector multiplets contribution along the Cartan generators is

$Z_{1{\rm{ - loop}}}^{{\rm{Gauge,Cartan}}} = \eta {\left( q \right)^{2r\left( {1 - g} \right)}}{\left( {idu} \right)^r}$

and the fermionic zero modes on ${\Sigma _g}$ is

${\left( {{{\det }_{\left[ {ab} \right]}}\frac{{{{\not \partial }^2}\log {Z_{1{\rm{loop}}}}}}{{\not \partial i{u_a}\not \partial {{\hat m}_b}}}} \right)^g}$

Thus, combining, the final Abelian formula is

and there are no boundary contributions since the integration domain $\hat {\rm M}$ is compact.

In the non-Abelian contexts, W-bosons contributions are factored by re-summing over $\hat m$ and excluding the roots of the associated Baez AEs for which the Vandermonde determinant is zero

Now take $U(1)$ a supersymmetric Yang-Mills-Chern-Simons theory at level $k$ which is equivalent to bosonic Chern-Simons at level $k$ at low energies. The semi-classical and one-loop contribution is

${Z_{{\rm{cl,}}1l}} = {x^{\hat t}}{\xi ^{\hat m}}{x^{k\,\hat m}}$

turning on a background for the topological symmetry. Hence, we can derive

$Z = \sum\limits_{\hat m\, \in \mathbb{Z}} {\int_{{\rm{JK}}} {\frac{{dx}}{{2\pi ix}}\,} } {k^g}{x^{k\,\hat m\, + \hat t}}{\xi ^{\hat m}}$

with the charges of the points at infinity being ${Q_0} = - k$ and ${Q_\infty } = k$

Given the natural assumptions: $k > 0$ , $\eta < 0$ and $x = 0$

we get

$Z = \left\{ {\begin{array}{*{20}{c}}{ - {k^g}{\xi ^{ - \hat t}}/k\quad {\rm{, if }}\hat t = 0\,\quad \bmod k}\\{0\,\,\,\,\,\,\,{\rm{,}}\,\,\,{\rm{otherwise}}}\end{array}} \right.$

where the known and central ${k^g}$ is the number of ground states of $U{(1)_k}$ Chern-Simons on ${\Sigma _g}$ .

To Witten all roads lead: take $U{(1)_k}$ with $N$ chiral multiplets of charges ${Q_i}$ and $R$ -charge 1 so one does not face the parity anomalies in the $R$ -symmetry. In the Witten-index context, the semi-classical and one-loop contribution is

${Z_{{\rm{cl,}}1l}} = {\xi ^{\hat m}}{x^{k\,\hat m\, + \hat t}}{\prod\limits_{i = 1}^N {\left( {\frac{{{x^{{Q_i}/2}}{y_i}^{1/2}}}{{1 - {x^{{Q_i}}}{y_i}}}} \right)} ^{{Q_i}\,\hat m\, + \,{{\hat n}_i}}}$

with $\left( {{y_i},{{\hat n}_i}} \right)$ controlling the flavor-symmetries-background for up to one combination that could be reabsorbed into $\left( {x,\hat m} \right)$ , and $\left( {\xi ,\hat t} \right)$ controls the background for the topological symmetry, and, in order not to face a gauge-parity anomaly, we insist that the following equivalence holds:

$k + \frac{1}{2}\sum\nolimits_i {Q_i^2} \in \mathbb{Z} \cong k + \frac{1}{2}\sum\nolimits_i {{Q_i}} \in \mathbb{Z}$

To compute the Witten index, set ${\hat n_i} = \hat t = 0$ , and then the poles are at

$\left\{ {\begin{array}{*{20}{c}}{{{\hat n}_i} = \hat t = 0}\\{x = 0}\\\infty \end{array}} \right.$

hence the sum over $\hat m$ generates the following

with ${x_{(\alpha )}}$ being the roots of the BaezAEs:

${e^{iB}} = \xi {x^k}{\prod\limits_{i = 1}^N {\left( {\frac{{{x^{{Q_i}/2}}y_i^{1/2}}}{{1 - {x^{{Q_i}}}{y_i}}}} \right)} ^{{Q_i}}} = 1$

hence the index ${Z_{{T^3}}}$ is the Witten index of the theory

On the flip-side,

becomes, up to sign-rescaling, the number of solutions to the BaezAEs.

Now, to compute the number of solutions, divide the chiral multiplets into two groups:

$\left\{ {\begin{array}{*{20}{c}}{{I_ + }\quad :\quad {Q_i} > 0}\\{{I_ - }\quad :\quad {Q_i} < 0}\end{array}} \right.$

Thus our WI- equation is

WI-E

Defining the positive numbers

$\left\{ {\begin{array}{*{20}{c}}{{n_ + } = \frac{1}{2}\sum\limits_{i \in \,{I_ + }} {Q_i^2} }\\{{n_ - } = \frac{1}{2}\sum\limits_{i \in \,{I_ - }} {Q_i^2} }\end{array}} \right.$

we can conclude that the number of solutions to WI-E is

${I_{{W_{itt}}}} = \left\{ {\begin{array}{*{20}{c}}{\left| k \right| + {n_ + } + {n_ - }\quad {\rm{, }}\;{\rm{if }}\left| k \right| \ge \left| {{n_ + } - {n_ - }} \right|}\\{\max \left( {2{n_ + },2{n_ - }} \right)\quad {\rm{,}}\;\;{\rm{if}}\;\;\left| k \right| \le \left| {{n_ + } - {n_ - }} \right|}\end{array}} \right.$

yielding the total Witten index

with

$\left\{ {\begin{array}{*{20}{c}}{{s_i} \equiv \Theta \left( {{n_i}F\left( {{\sigma _{{Q_i}}}} \right)} \right)}\\{\Theta (x) = \left\{ {\begin{array}{*{20}{c}}{1{\rm{ : }}x > 0}\\{0{\rm{ : }}x < 0}\end{array}} \right.}\end{array}} \right.$