\newbox\leftpage \newdimen\fullhsize \newdimen\hstitle \newdimen\hsbody \tolerance=10000\hfuzz=10000pt\def\fontflag{cm} % tolerance=1000 \hfuzz=20pt \def\printertype{ps: }% default postscript \def\qms{\def\printertype{qms: }% qms lasergrafix \ifx\answ\bigans\else\voffset=-.4truein\hoffset=.125truein\fi} \def\bigans{b } %\message{ big or little (b/l)? }\read-1 to\answ \def\answ{b } % \ifx\answ\bigans\message{(This will come out unreduced.} \magnification=1200\baselineskip=14pt plus 2pt minus 1pt \hsbody=\hsize \hstitle=\hsize %take default values for unreduced format % \else\message{(This will be reduced.} \let\lr=L \magnification=1000\baselineskip=14pt plus 2pt minus 1pt \voffset=-.31truein\vsize=7truein\hoffset=-.59truein% apple lw \hstitle=8truein\hsbody=4.75truein\fullhsize=10truein\hsize=\hsbody % \output={\ifnum\pageno=0 %%% This is the HUTP version \shipout\vbox{\special{\printertype landscape}\makeheadline \hbox to \fullhsize{\hfill\pagebody\hfill}}\advancepageno \else \almostshipout{\leftline{\vbox{\pagebody\makefootline}}}\advancepageno \fi} \def\almostshipout#1{\if L\lr \count1=1 \message{[\the\count0.\the\count1]} \global\setbox\leftpage=#1 \global\let\lr=R \else \count1=2 \shipout\vbox{\special{\printertype landscape} \hbox to\fullhsize{\box\leftpage\hfil#1}} \global\let\lr=L\fi} \fi \input amssym.def \input amssym.tex \font\bigfont=cmr10 scaled\magstep3 \def\section#1#2{\vskip32pt plus4pt \goodbreak \noindent{\bf#1. #2} \xdef\currentsec{#1} \global\eqnum=0 \global\thmnum=0} \newcount\thmnum \global\thmnum=0 \def\prop#1#2{\global\advance\thmnum by 1 \xdef#1{Proposition \currentsec.\the\thmnum} \bigbreak\noindent{\bf Proposition \currentsec.\the\thmnum.} {\it#2} } \def\define#1#2{\global\advance\thmnum by 1 \xdef#1{Definition \currentsec.\the\thmnum} \bigbreak\noindent{\bf Definition \currentsec.\the\thmnum.} {\it#2} } \def\lemma#1#2{\global\advance\thmnum by 1 \xdef#1{Lemma \currentsec.\the\thmnum} \bigbreak\noindent{\bf Lemma \currentsec.\the\thmnum.} {\it#2}} \def\thm#1#2{\global\advance\thmnum by 1 \xdef#1{Theorem \currentsec.\the\thmnum} \bigbreak\noindent{\bf Theorem \currentsec.\the\thmnum.} {\it#2} } \def\cor#1#2{\global\advance\thmnum by 1 \xdef#1{Corollary \currentsec.\the\thmnum} \bigbreak\noindent{\bf Corollary \currentsec.\the\thmnum.} {\it#2} } \def\conj#1#2{\global\advance\thmnum by 1 \xdef#1{Conjecture \currentsec.\the\thmnum} \bigbreak\noindent{\bf Conjecture \currentsec.\the\thmnum.} {\it#2} } \def\proof#1{{\it Proof#1.}} \newcount\eqnum \global\eqnum=0 \def\num{\global\advance\eqnum by 1 \eqno({\rm\currentsec}.\the\eqnum)} \def\eqalignnum{\global\advance\eqnum by 1 ({\rm\currentsec}.\the\eqnum)} \def\ref#1{\num \xdef#1{(\currentsec.\the\eqnum)}} \def\eqalignref#1{\eqalignnum \xdef#1{(\currentsec.\the\eqnum)}} \def\title#1{\centerline{\bf\bigfont#1}} \newcount\subnum \def\Alph#1{\ifcase#1\or A\or B\or C\or D\or E\or F\or G\or H\fi} \def\subsec{\global\advance\subnum by 1 \vskip12pt plus4pt \goodbreak \noindent {\bf \currentsec.\Alph\subnum.} } \def\newsubsec{\global\subnum=1 \vskip6pt\noindent {\bf \currentsec.\Alph\subnum.} } \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi\space\number\day, \number\year} \def\intsec{I} \def\landginzsec{II} \def\smoothcohsec{III} \def\compactcohsec{IV} \def\sqrintsec{V} \def\bC{\Bbb{C}} \def\bR{\Bbb{R}} \def\bZ{\Bbb{Z}} \def\tr{\mathop{\rm Tr}\nolimits} \def\del{\partial} \def\str{\mathop{\rm Str}\nolimits} \def\ol{\overline} \def\re{{\rm Re}} \def\dbr{\ol\partial} \def\dolb#1#2{\bigwedge\nolimits^{#1,#2}(E_M)} \def\dol{\bigwedge(E_M)} \def\h{{\cal H}} \def\w{{\cal W}_z} \def\d0{{\cal D}_0} \def\harm{{\rm Harm}_V(E_M)} \def\lapl{\ol\square_V} \def\hil{\bigwedge\nolimits_2(E_M)} \def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} \def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} {\baselineskip=12pt \nopagenumbers \line{\hfill \bf HUTMP 94/B331} \line{\hfill \bf \today} \vfill \title{The smooth cohomology of N=2 supersymmetric} \vskip1cm \title{ Landau-Ginzburg field theories} \vskip1in \centerline{{\bf Slawomir Klimek}$^{*}$ \footnote{$^1$}{Supported in part by the National Science Foundation under grant DMS--9206936} and {\bf Andrzej Lesniewski}$^{**}$\footnote{$^2$} {Supported in part by the Department of Energy under grant DE--FG02--88ER25065}} \vskip12pt \centerline{ $^{*}$Department of Mathematics} \centerline{IUPUI} \centerline{Indianapolis, IN 46205, USA} \vskip12pt \centerline{ $^{**}$Lyman Laboratory of Physics} \centerline{Harvard University} \centerline{Cambridge, MA 02138, USA} \vskip1in\noindent {\bf Abstract.} We compute the smooth cohomology (both unrestricted and compactly supported) of the supercharge of an ultraviolet cutoff $N = 2$ supersymmetric Landau - Ginzburg field theory. \vfill\eject} \section\intsec{Introduction} \medskip\newsubsec The two-dimensional $N = 2$ supersymmetric Landau - Ginzburg models (a.k.a. as Wess - Zumino models) were introduced in the physics literature in the seventies. For early results, see [CG] and references therein. Constructive field theory aspects of these models were a subject of many investigations, see [JL] and [J] for references. The Landau - Ginzburg models provide useful examples to study complex physical and mathematical phenomena of supersymmetric quantum field theory which are much harder to control in the four dimensional world. Recent revival of interest in the Landau - Ginzburg models stems largely from the fact that they seem to play a role in various ``compactification'' scenarios of string theory (see e.g. [CV] and references therein). Supersymmetric quantum field theories provide non-trivial examples of infinite dimensional non-commutative geometries [C]. In particular, supersymmetric field theories with $N=2$ supersymmetries lead naturally to structures which can be regarded as examples of non-commutative K\"ahler geometry. For the Landau - Ginzburg models, the underlying infinite dimensional geometry is flat. What makes them non-trivial is the non-linear self-interaction term in the Hamiltonian. One of the fundamental difficulties in studying the mathematical structures associated with this model is of technical character: to show that the Hamiltonian is well defined on a dense domain, and that its heat kernel is trace class. This requires a detailed analysis of a suitably regularized form of the Hamiltonian. \subsec In this paper, we choose a particular regularization, namely the sharp ultraviolet cutoff $M$. This amounts to suppressing all the modes with $|p| > M$ in the Fourier expansion of the field operators. The regularized Hamiltonian has then the following form. Let $\delta_M$ be the following regularized delta function, $$ \delta_M(\sigma)={{\sin(M+1/2)\sigma}\over{\sin\sigma/2}}.\ref{\deltaref} $$ By $\gamma_\mu$ we denote the two-dimensional Dirac matrices, $$ \gamma_0=\left(\matrix{0&1\cr 1&0\cr}\right),\qquad \gamma_1=\left(\matrix{0&-1\cr 1&0\cr}\right),\ref{\diracmatrices} $$ and define the chiral projections, $$ \Lambda_{\pm}={1\over 2}\,(I\pm\gamma_0\gamma_1).\ref{\lambdas} $$ Finally, define the following operator $$ D_V(\sigma)=i\gamma_1\partial_\sigma+\Lambda_+P_M\nabla^2V(\phi(\sigma)) +\Lambda_-P_M\ol{\nabla^2V(\phi(\sigma))},\num $$ where the projection operator $P_M$ is defined by $(P_Mf)(\sigma)= \int_{S^1}\delta_M(\sigma-\tau)f(\tau)\, d\tau$. Then $$ \eqalign{ \ol\square_V=&\int_{S^1}\, (\ol\pi(\sigma)\pi(\sigma)+\partial_\sigma \ol\phi(\sigma)\partial_\sigma\phi(\sigma))\, d\sigma + \int_{S^1}\, \ol\psi(\sigma) D_V(\sigma)\psi(\sigma)\, d\sigma \cr &+\int_{S^1\times S^1}\, \ol{\nabla V(\phi(\sigma))}\,\delta_M(\sigma-\tau) \,\nabla V(\phi(\tau))\, d\sigma\, d\tau ,\cr}\ref{\reghamref} $$ where $\pi(\sigma)$ is the canonical momentum operator, and where $$ \psi(\sigma)=\left(\matrix{\psi_1(\sigma)\cr \psi_2(\sigma)}\right), \qquad \ol\psi(\sigma)=\psi^*(\sigma)\,\gamma_0= (\psi_2^*(\sigma)\quad\psi_1^*(\sigma)).\num $$ Throughout this paper, we work with the regularized theory only. All our results are, however, uniform in the cutoff $M$, and we believe that they hold true for a suitably defined limit $M\rightarrow\infty$. \subsec A particular feature of the above regularized Hamiltonian is that it is a square of a fermionic operator, the supercharge. Specifically, with $$ \eqalign{ \ol Q_V = {1\over{\sqrt{2}}}\int_{S^1}\bigl(&-i\psi_1(\sigma)(\pi(\sigma)- \partial_{\sigma}\phi(\sigma)) + \psi_2(\sigma)(\ol\pi(\sigma)+ \partial_{\sigma}\ol\phi(\sigma))\cr &+i\psi_1(\sigma)\nabla V(\phi(\sigma))+\psi_2(\sigma)\ol{\nabla V(\phi(\sigma))}\bigr) d\sigma + {\hbox {herm. conj.}},} \ref{\superchargeref} $$ we have $\ol\square_V = \ol Q_V^2$. This property indicates that $\ol Q_V$ is a Dirac type operator, and the natural problem is to study the properties of its kernel and in particular its index. Physically, the kernel of $\ol Q_V$ consists of the (zero energy) ground states of the Hamiltonian. It is easy to see that $\ol Q_V$ has the structure $\ol Q_V = \ol\partial_V+\ol\partial_V^*$, where $\ol\partial_V$ is a coboundary operator, and where $\ol\partial_V^*$ denotes its hermitian adjoint. In this paper we are concerned with the cohomology groups of the operator $\ol\partial_V$ which we believe are related the space of harmonic forms of $\ol\square_V$. We explicitly compute the smooth and compactly supported cohomology groups of this operator, and show that they depend only on the singularity structure of the superpotential. We establish a vanishing theorem for the smooth cohomology associated with the operator $\ol\partial_V$: only cohomology groups of definite parity are non-trivial. Furthermore, we show that the cohomology groups are independent of the ultraviolet cutoff $M$. We believe that similar results hold for the space of harmonic forms of $\ol\square_V$, and present partial arguments to support this hypothesis. \subsec The paper is organized as follows. In Section II we define the smooth cohomology complex which corresponds to the Landau - Ginzburg model and state our main results. These results are proved in Sections III and IV. In Section V we define the square integrable cohomology corresponding to the Landau - Ginzburg model, formulate a technical conjecture, and study its consequences. \section\landginzsec{The Landau - Ginzburg complex} \medskip\newsubsec In this section we define the equivariant cohomology of a perturbed $\dbr$ operator on the space $E_M=\bC^n\times\bC^{2nM}$, where $n\geq 1$ and $M\geq 0$ are integers. We will regard $E_M$ as a vector bundle over $\bC^n$ with fiber $\bC^{2nM}$. We represent a point $z\in E_M$ as $z=(z_{-M},\ldots ,z_{-1},z_0,z_1,\ldots ,z_M)$, where $z_p\in\bC^n$. In the language of quantum field theory, $z_0$ represents the zero modes while the $z_p$'s with $1\leq |p|\leq M$ represent the excited modes. The integer $M$ is the ultraviolet cutoff. The relevant complex is defined as follows. We let $\dolb pq$ denote the space of smooth $(p,q)$-forms on $E_M$ and let $\dbr :\dolb pq \rightarrow\dolb p{q+1}$ and $\partial :\dolb pq \rightarrow \dolb {p+1}q$ denote the usual Dolbeault coboundary operators. Let $S^1$ be the circle of circumference $1$. There is a natural $S^1$-action on $E_M$, $$ z_p\rightarrow e^{-2\pi ip\sigma}z_p,\qquad\sigma\in S^1, \qquad |p|\leq M,\ref{\circleaction} $$ which is generated by the following holomorphic vector field: $$ K(z)=-2\pi i\sum_{|p|\leq M}\, pz_p\nabla_p,\num $$ where $\nabla_p=\partial/\partial z_p$. $K$ acts on forms by interior multiplication; we denote the corresponding operator mapping $\dolb pq$ into $\dolb {p-1}q$ by $i(K)$. Clearly, $i(K)^2=0$ and $\dbr\, i(K)+i(K)\,\dbr=0$. As a consequence, the operator $$ \dbr_0=\dbr+i(K)\num $$ satisfies $\dbr_0^2=0$. It is an $S^1$-equivariant version of the Dolbeault operator. \subsec We will be concerned with the cohomology of a perturbation of $\dbr_0$ by a holomorphic $1$-form. Let $\dol =\bigoplus_{p,q} \dolb pq$ be the Grassmann algebra over $E_M$. We define the usual $\bZ_2$-grading on $\dol$, $$ \dol = \dol^0\oplus\dol^1, \ref{\grading} $$ where $$ \dol^0 =\bigoplus_{p+q\, even}\dolb pq ,\qquad \dol^1 =\bigoplus_{p+q\, odd}\dolb pq .\num $$ For $\sigma\in S^1$ we define the following field operators on $\dol$: $$ \eqalign{ &\ol b^*(\sigma)=\sum_{|p|\leq M}\,e^{2\pi ip\sigma}d\ol z_p\wedge ,\cr &b^*(\sigma)=\sum_{|p|\leq M}\, e^{2\pi ip\sigma}dz_p\wedge ,\cr &\ol b(\sigma)=\sum_{|p|\leq M}\, e^{-2\pi ip\sigma}i(\ol\nabla_p),\cr &b(\sigma)=\sum_{|p|\leq M}\, e^{-2\pi ip\sigma}i(\nabla_p),\cr &\phi(\sigma)=\sum_{|p|\leq M}\, e^{-2\pi ip\sigma}z_p,\cr &\ol\phi(\sigma)=\sum_{|p|\leq M}\, e^{2\pi ip\sigma}\ol z_p,\cr &K(\sigma)=-2\pi i\,\sum_{|p|\leq M}\,p\, e^{-2\pi ip\sigma}z_p= \partial_\sigma\phi(\sigma),\cr &\ol K(\sigma)=2\pi i\,\sum_{|p|\leq M}\,p\, e^{2\pi ip\sigma}\ol z_p= \partial_\sigma\ol\phi(\sigma),\cr &\ol\pi(\sigma)=-i\sum_{|p|\leq M}\, e^{-2\pi ip\sigma}\dbr_p,\cr &\pi(\sigma)=-i\sum_{|p|\leq M}\, e^{2\pi ip\sigma}\partial_p\cr }\ref{\fieldoperators} $$ (these are actually vectors of operators; we will put the indices whenever necessary). In terms of these operators, $$ \dbr_0=i\int_{S^1}\ol b^*(\sigma)\,\ol\pi(\sigma)\, d\sigma +\int_{S^1}b(\sigma)\,\partial_\sigma\phi(\sigma)\, d\sigma.\num $$ Let now $V:\bC^n\rightarrow\bC$ be a holomorphic polynomial (the superpotential). We consider the following function on $E_M$: $$ h(z)=\int_{S^1}V(\phi(\sigma))\, d\sigma= \int_{S^1}V(\,\sum_{|p|\leq M}\, e^{-2\pi ip\sigma}z_p)\, d\sigma .\num $$ Clearly, $h$ is holomorphic on $E_M$. Its differential $dh$ is a holomorphic $1$-form on $E_M$ and acts on $\dol$ by exterior multiplication. We denote the corresponding operator mapping $\dolb pq$ into $\dolb {p+1}q$ by $\delta_V$. In terms of the field operators \fieldoperators, $$ \delta_V =\int_{S^1}b^*(\sigma)\,\nabla V(\phi(\sigma))\, d\sigma.\num $$ Clearly, $\delta_V^2=0$, and $\dbr\,\delta_V+\delta_V\,\dbr=0$. Furthermore, $$ \eqalign{ i(K)\,\delta_V+\delta_V\, i(K)&=\int_{S^1}\nabla V(\phi(\sigma))\, \partial_\sigma\phi(\sigma)\, d\sigma\cr &=\int_{S^1}\,{d\over{d\sigma}}\, V(\phi(\sigma))\, d\sigma = 0.\cr } $$ As a consequence, the operator $$ \dbr_V=\dbr +i(K)+\delta_V=\dbr_0+\delta_V\num $$ satisfies $\dbr_V^2=0$. Note also that $\dbr_V$ is odd with respect to the $\bZ_2$-grading defined in \grading. We thus have a complex $$ \ldots\quad\mapright{\dbr_V}\quad\dol^0\quad\mapright{\dbr_V}\quad \dol^1\quad\mapright{\dbr_V}\quad\dol^0\quad\mapright{\dbr_V}\quad \ldots\, .\ref{\maincomplex} $$ Let $H^{\phantom{\dbr}*}_{\dbr_V}(E_M)$ denote the cohomology of this complex. For future reference, we observe that $H^{\phantom{\dbr}*}_{\dbr_V}(E_M)$ arises as the total cohomology of the following double complex: $$ \matrix{&&&\vdots&&\vdots&&\cr &&&&&&&\cr &&&\mapup{\dbr}&&\mapup{\dbr}&&\cr &&&&&&&\cr &\ldots&\mapright{\Delta_V}&\bigwedge\nolimits^{2k+1}(E_M)^0& \mapright{\Delta_V}&\bigwedge\nolimits^{2k+1}(E_M)^1 &\mapright{\Delta_V}&\ldots\cr &&&&&&&\cr &&&\mapup{\dbr}&&\mapup{\dbr}&&\cr &&&&&&&\cr &\ldots&\mapright{\Delta_V}&\bigwedge\nolimits^{2k}(E_M)^0& \mapright{\Delta_V}&\bigwedge\nolimits^{2k}(E_M)^1 &\mapright{\Delta_V}&\ldots\cr &&&&&&&\cr &&&\mapup{\dbr}&&\mapup{\dbr}&&\cr &&&&&&&\cr &&&\vdots&&\vdots&&\cr }\ref{\firstdoublecomplex} $$ where $$ \bigwedge\nolimits^q(E_M)^0=\bigoplus_{p\, even}\,\dolb pq,\qquad \bigwedge\nolimits^q(E_M)^1=\bigoplus_{p\, odd}\,\dolb pq,\num $$ and where $$ \Delta_V=i(K)+\delta_V.\num $$ Finally, let $\Omega^k(\bC^n)$ denote the space of holomorphic $k$-forms on $\bC^n$ and let $K^*_V(\bC^n)$ be the cohomology of the following Koszul complex: $$ \ldots\quad\mapright{dV}\quad\Omega^k(\bC^n) \quad\mapright{dV}\quad\Omega^{k+1}(\bC^n) \quad\mapright{dV}\quad\ldots\, .\ref{\koszul} $$ \subsec Our first main result is contained in the following theorem. \thm\firstmainthm{We have the following isomorphisms: $$ \eqalign{ &H^{\phantom{\dbr}0}_{\dbr_V}(E_M)\simeq \bigoplus_{j}\, K^{2j}_V(\bC^n),\cr &H^{\phantom{\dbr}1}_{\dbr_V}(E_M)\simeq \bigoplus_{j}\, K^{2j+1}_V(\bC^n).\cr }\ref{\isomorphisms} $$} \noindent We prove this theorem in Section III. The complex \maincomplex\ is precisely the complex which was studied in [CGP] and [KL1] in connection with $N=2$ supersymmetric quantum mechanics (which corresponds to $M=0$ in our notation). The above theorem asserts that the topological content of the regularized $N=2$ supersymmetric Landau - Ginzburg theory is identical to that of the zero mode limit of the theory. Furthermore, it is well known (see e.g. [GH]) that if the critical set of $V$, $cr(V)$, is finite, then $$ K^k_V(\bC^n)\simeq\cases{0,&if $k0$;\cr \Omega(E_M)^0, &if $q=0,\,p$ even;\cr \Omega(E_M)^1, &if $q=0,\,p$ odd;\cr}\num $$ and the claim follows. $\square$ We let $s:\,\bC^n\rightarrow E_M$ denote the natural holomorphic embedding, $$ \bC^n\ni z_0\,\longrightarrow\, s(z_0)=(0,\,\ldots\, ,0, z_0,0,\,\ldots\, ,0)\in E_M,\ref{\embedding} $$ and consider the spaces $$ \tilde\Omega^k(E_M)=\{\omega\in\Omega^k(E_M):\quad s^\sharp \omega = 0\},\num $$ and the corresponding spaces $\tilde\Omega(E_M)^*$ defined in analogy with \gradedomegas. In the above expression, $s^\sharp\omega$ denotes the pullback of $\omega$ under $s$. We note that if $s^\sharp\omega=0$, then also $s^\sharp\,\Delta_V \omega=0$, and so the following complex is defined: $$ \ldots\quad\mapright{\Delta_V}\quad\tilde\Omega(E_M)^0\quad \mapright{\Delta_V}\quad\tilde\Omega(E_M)^1\quad\mapright{\Delta_V} \quad\tilde\Omega(E_M)^0\quad\mapright {\Delta_V}\quad\ldots\, ,\ref{\secondauxcomplex} $$ We let $\tilde H^{\phantom{\dbr}*}_{\Delta_V}(E_M)$ denote its cohomology groups. \lemma\secondlemma{The complex \secondauxcomplex\ has a trivial cohomology.} \smallskip\noindent {\it Proof.} We observe that $\tilde H^{\phantom{\dbr}*}_{\Delta_V}(E_M)$ arises as the total cohomology of the following double complex: $$ \matrix{&&&\vdots&&\vdots&&\cr &&&&&&&\cr &&&\mapup{i(K)}&&\mapup{i(K)}&&\cr &&&&&&&\cr &\ldots&\mapright{\delta_V}&\tilde\Omega^{k-1}(E_M)& \mapright{\delta_V}&\tilde\Omega^k(E_M) &\mapright{\delta_V}&\ldots\cr &&&&&&&\cr &&&\mapup{i(K)}&&\mapup{i(K)}&&\cr &&&&&&&\cr &\ldots&\mapright{\delta_V}&\tilde\Omega^k(E_M)& \mapright{\delta_V}&\tilde\Omega^{k+1}(E_M) &\mapright{\delta_V}&\ldots\cr &&&&&&&\cr &&&\mapup{i(K)}&&\mapup{i(K)}&&\cr &&&&&&&\cr &&&\vdots&&\vdots&&\cr }\ref{\seconddoublecomplex} $$ We claim that $\tilde H^{\phantom{\dbr}*}_{\Delta_V}(E_M)\simeq 0$. To prove this, consider a column in \seconddoublecomplex: $$ 0\longrightarrow\quad\tilde\Omega^{(2M+1)n}(E_M)\quad \mapright{\delta_V}\quad\ldots\quad \mapright{\delta_V}\quad\tilde\Omega^1(E_M)\quad\mapright{\delta_V} \quad\tilde\Omega^0(E_M)\quad\longrightarrow 0.\ref{\columnsequence} $$ We will construct a homotopy operator for \columnsequence. We write $z_p=(z_{p1},\,\ldots\, ,\, z_{pM})$, $z_{p\alpha}\in\bC$, $|p|\leq M$, and represent a form $\theta\in\tilde\Omega^k(E_M)$ as $$ \theta(z)=\sum_m\sum_{\alpha_1,\ldots ,\alpha_m}\, \omega_{\alpha_1\ldots\alpha_m}(z)\, dz_{0\alpha_1}\wedge\ldots \wedge dz_{0\alpha_m},\ref{\genericform} $$ where the forms $\omega_{\alpha_1\ldots\alpha_m}(z)$ do not involve factors of $dz_{0\alpha},\, 1\leq\alpha\leq n$. Let now $\omega(z) =f(z)dz_{p_1\alpha_{p_1}}\wedge\ldots\wedge dz_{p_k\alpha_{p_k}}, \, |p_1|\geq 1,\ldots ,|p_k|\geq 1$, be a homogeneous component in $\omega_{\alpha_1\ldots\alpha_m}(z)$. We set $$ (J\omega)(z)={i\over{2\pi}}\sum_{|p|\geq 1,\alpha}\,{1\over p} \int_0^1\, t^k{\partial\over{\partial z_{p\alpha}}}\, f(z_0, tz^\prime)\, dt\, dz_{p\alpha}\wedge dz_{p_1\alpha_{p_1}}\wedge\ldots \wedge dz_{p_k\alpha_{p_k}},\num $$ where $z^\prime=(z_{-M},\ldots ,z_{-1},z_1,\ldots ,z_M)\in\bC^{2nM}$. This defines an operator $J:\,\tilde\Omega^k(E_M)\,\rightarrow\, \tilde\Omega^{k+1}(E_M)$. An elementary (if slightly tedious) computation shows that $$ (i(K)\, J+J\, i(K))\,\omega(z)=\int_0^1\,{d\over{dt}}\,(t^k f(z_0,tz^\prime))\, dt\, dz_{p_1\alpha_{p_1}}\wedge\ldots \wedge dz_{p_k\alpha_{p_k}}.\ref{\homotopyrel} $$ We claim that $\int_0^1\,{d\over{dt}}\,(t^kf(z_0,tz^\prime))\, dt=f(z)$. Indeed, this is clear if $k>0$. If $k=0$, then $\omega(z)=f(z)$, and $f(z_0,0)=s^\sharp\omega(z)=0$ which implies our claim. Consequently, \homotopyrel\ equals $\omega(z)$, which means that $i(K)\, J+J\, i(K)={\rm Id}$. We have thus shown that $J$ is a homotopy operator for the complex \columnsequence. It is now easy to show that the total cohomology of \seconddoublecomplex\ is zero. Indeed, the first filtration associated with \seconddoublecomplex\ is bounded, and thus $^\prime E^{*,*}_r$ converges. Since, by the above argument, the rows of \seconddoublecomplex\ have zero cohomologies, it follows that $^\prime E^{*,*}_1\simeq 0$, and the lemma is proven. $\square$ \subsec We are now ready to prove \firstmainthm. \noindent {\it Proof of \firstmainthm.} Observe that $$ \Omega(E_M)^*/\tilde\Omega(E_M)^*\simeq\Omega(\bC^n)^*,\num $$ which yields the following short exact sequence of complexes $$ 0\quad\longrightarrow\quad\tilde\Omega(E_M)^*\quad\longrightarrow \quad\Omega(E_M)^*\quad\longrightarrow\quad\Omega(\bC^n)^* \quad\longrightarrow\quad 0.\num $$ >From the associated long sequence of cohomology groups $$ \eqalign{ \ldots\quad&\longrightarrow\quad\tilde H^{\phantom{\dbr}0}_{\Delta_V}(E_M) \quad\longrightarrow\quad H^{\phantom{\dbr}0}_{\Delta_V}(E_M) \quad\longrightarrow\quad \bigoplus_{j}\, K^{2j}_V(\bC^n)\cr \quad&\longrightarrow\quad\tilde H^{\phantom{\dbr}1}_{\Delta_V}(E_M) \quad\longrightarrow\quad H^{\phantom{\dbr}1}_{\Delta_V}(E_M) \quad\longrightarrow\quad \bigoplus_{j}\, K^{2j+1}_V(\bC^n) \quad\longrightarrow\quad\ldots\, ,\cr} $$ and \secondlemma\ it follows that $H^{\phantom{\dbr}*}_{\delta_V}(E_M) \simeq \bigoplus_{j}\, K^{2j+*}_V(\bC^n)$. But by \firstlemma\ $H^{\phantom{\dbr}*}_{\Delta_V}(E_M)\simeq H^{\phantom{\dbr}*}_{\dbr_V}(E_M)$ and the theorem is proven. $\square$ \section\compactcohsec{The compactly supported cohomology} \medskip\newsubsec The proof of \secondmainthm\ is based on the following lemma. \lemma\thirdlemma{Let $U\subset E_M$ be an open set such that $U\cap s(cr(V))=\emptyset$ ($cr(V)$ is not assumed to be finite). Then the cohomology of $\dbr_V$ restricted to $U$ is trivial.} \smallskip\noindent {\it Proof.} Consider the double complex \firstdoublecomplex\ with $E_M$ replaced by $U$. The second filtration associated with \firstdoublecomplex\ is bounded, and thus $^{\prime\prime}E^{*,*}_r$ converges. We claim that $^{\prime\prime}E^{*,*}_1\simeq 0$. Indeed, let $L$ be the following operator: $$ L=W(z)^{-1}\,\int_{S^1}\,(b^*(\sigma)\,\ol{\partial_\sigma\phi(\sigma)} +b(\sigma)\,\ol{\nabla V(\phi(\sigma))})\, d\sigma,\num $$ where $$ W(z)=\int_{S^1}\,|\partial_\sigma\phi(\sigma)|^2\, d\sigma +\int_{S^1\times S^1}\,\ol{\nabla V(\phi(\sigma))}\,\delta_M(\sigma-\tau) \nabla V(\phi(\tau))\,d\sigma d\tau,\ref{\wdef} $$ where the regularized delta function $\delta_M$ is given by \deltaref . Since $W(z)\neq 0$, for $z\not\in s(cr(V))$, $L$ is well defined. It is easy to see that $L\,\Delta_V+\Delta_V L={\rm Id}$, i.e. $L$ is a homotopy operator for $\Delta_V$. This proves that $^{\prime\prime}E^{*,*}_1\simeq 0$. $\square$ \subsec We can now prove \secondmainthm. \noindent {\it Proof of \secondmainthm.} Since the coboundary operator $\dbr_V$ is local, the complex of compactly supported smooth forms is a subcomplex of the complex of smooth forms. Let $i$ denote the natural injection. We claim $i$ induces an isomorphism of cohomologies. Indeed: \noindent {\it (i) $i$ is a monomorphism.} To prove this, we consider a compactly supported form $\omega$ such that $\omega = \dbr_V\eta,\, \eta\in\bigwedge(E_M)$. Let $D_1$ and $D_2$ be open balls such that ${\rm supp}\,\omega\cup s(cr(V))\subset D_1\subset\ol D_1\subset D_2$, and let $U$ be the complement of $\ol D_1$. Then $\dbr_V\eta=0$ on $U$. As a consequence of \thirdlemma, there is a smooth form $\zeta$ on $U$ such that $\eta=\dbr_V\zeta$ on $U$. We choose a smooth function $\chi$ on $E_M$ such that $$ \chi=\cases{0,&on $\ol D_1$;\cr 1,&on $D_2^c$,\cr}\ref{\chiref} $$ and set $\eta^\prime=\eta-\dbr_V(\chi\zeta)$. Then $\eta^\prime$ is compactly supported, cohomologous to $\eta$, and $\omega=\dbr_V \eta^\prime$ on $E_M$. This shows that $\omega$ is trivial in compactly supported cohomology. \noindent {\it (ii) $i$ induces an epimorphism.} Let $\omega$ be smooth and let $\dbr_V\omega=0$. Let $D_1$ and $D_2$ be two open balls in $E_M$ such that $s(cr(V))\subset D_1\subset\ol D_1\subset D_2$, and let $U$ be the complement of $\ol D_1$. As a consequence of \thirdlemma, there is a smooth form $\psi$ on $U$ such that $\omega=\dbr_V\psi$ on $U$. We choose $\chi$ as in \chiref\ and set $\omega^\prime = \omega-\dbr_V(\chi\psi)$. The $\omega^\prime$ is compactly supported, closed, and cohomologous to $\omega$. As a consequence, the cohomology class of $\omega$ contains a compactly supported representative. $\square$ \section\sqrintsec{The $L^2$-cohomology} \medskip\newsubsec We now turn to the analytic part of this study, namely the square integrable cohomology of the operator $\dbr_V$ and its relation to the previously studied smooth cohomologies. The content of this section has a largely conjectural character, as the proofs of some crucial technical results are still missing. Let $\star :\dolb pq \rightarrow \dolb {(2M+1)n-q}{(2M+1)n-p}$ be the Hodge star operator, and let $(\omega,\eta)=\int_{E_M}\, \star\omega\wedge\eta$ be the usual inner product defined on $\bigwedge^{p,q}_{comp}(E_M)$. We let $\bigwedge^{p,q}_2(E_M)$ denote the completion of $\bigwedge^{p,q}_{comp}(E_M)$ in the norm induced by this inner product. By $\hil$ we denote the direct sum of the above Hilbert spaces, and by $\hil_0$ and $\hil_1$ we denote its even and odd subspaces, respectively. The operators \fieldoperators\ act on $\hil$. Note that the fermionic operators are bounded while the bosonic operators are unbounded operators, defined on the dense invariant domain $\d0=\bigwedge_{comp}(E_M)$ (say). Note also that $\ol b^*(\sigma), b^*(\sigma), \ol \phi(\sigma), \ol K(\sigma)$ and $\ol \pi(\sigma)$ are the respective adjoints of $\ol b(\sigma), b(\sigma), \phi(\sigma), K(\sigma)$, and $\pi(\sigma)$. We will use the following notation $$ \ol b^*_p=d\ol z_p\wedge,\quad\ol b_p = i(\ol\nabla_p),\quad b^*_p=dz_p\wedge,\quad b_p = i(\nabla_p).\num $$ \subsec Consider now the Dirac type operator $$ \ol Q_V=\dbr_V+\dbr_V^*,\ref{\diracbar} $$ defined on $\d0$. Clearly, $\ol Q_V$ is symmetric and odd with respect to the $\bZ_2$-grading on $\hil$. Its square, $\ol\square_V=\ol Q_V^2\geq 0$, is given by $$ \eqalign{ \ol\square_V=&\int_{S^1}\, \bigl(\ol\pi(\sigma)\pi(\sigma)+\partial_\sigma \ol\phi(\sigma)\partial_\sigma\phi(\sigma)\bigr)\, d\sigma + \int_{S^1}\, \bigl(b^*(\sigma) \partial_\sigma\ol b^*(\sigma)-\ol b(\sigma)\partial_\sigma b(\sigma)\bigr) \, d\sigma \cr &+\int_{S^1}\, \bigl(\ol b^*(\sigma)\ol{\nabla^2V(\phi(\sigma))}b(\sigma) + b^*(\sigma)\nabla^2V(\phi(\sigma))\ol b(\sigma)\bigr)\, d\sigma \cr &+\int_{S^1\times S^1}\, \ol{\nabla V(\phi(\sigma))}\,\delta_M(\sigma-\tau) \,\nabla V(\phi(\tau))\, d\sigma\, d\tau ,\cr}\ref{\laplacebar} $$ where $\delta_M$ is given by \deltaref. The Friedrichs extension of $\ol\square_V$ defines a positive self-adjoint operator which we denote by the same symbol. The above Laplace operator is precisely the Hamiltonian of the regularized $N=2$ supersymmetric Landau - Ginzburg quantum field theory. We define $X$ to be the generator of the circle action \circleaction\ on $\hil$. Explicitly, $$ X=\int_{S^1}\,\bigl(\partial_\sigma\ol\phi(\sigma)\ol\pi(\sigma) -\pi(\sigma)\partial_\sigma\phi(\sigma)\bigr)\, d\sigma +\int_{S^1}\,\bigl(b^*(\sigma)i\partial_\sigma b(\sigma) -i\partial_\sigma\ol b(\sigma)\ol b^*(\sigma)\bigr)\, d\sigma, \ref{\momentum} $$ or, in terms of the Fourier modes, $$ X=\sum_{|p|\leq M}\, p(\ol b_p^*b_p+b^*_pb_p) + \sum_{|p|\leq M}\, p(z_p\nabla_p+\ol z_p\ol\nabla_p).\num $$ In the physics language, $X$ is the momentum operator. To make contact with the usual field theoretical expressions for the Hamiltonian \laplacebar\ and the momentum operator \momentum, we introduce the Dirac field operators, $$ \psi_1(\sigma)={1\over{\sqrt 2}}\,(\ol b(\sigma)-ib(\sigma)^*),\qquad \psi_2(\sigma)={1\over{\sqrt 2}}\,(b(\sigma)+i\ol b(\sigma)^*). \ref{\diracfields} $$ Then $\ol\square_V$ is given by \reghamref , and $$ X=\int_{S^1}\,(\partial_\sigma\ol\phi(\sigma)\ol\pi(\sigma) -\pi(\sigma)\partial_\sigma\phi(\sigma))\, d\sigma +\int_{S^1}\,\ol\psi(\sigma)i\gamma_0\partial_\sigma\psi(\sigma) \, d\sigma.\num $$ \subsec Below we formulate a conjecture concerning the analytic properties of $\ol\square_V$ which will be relevant for our purposes. The particular properties whose validity we conjecture to hold are motivated by the results obtained in [KL1], [AO], and [BI] for the case of supersymmetric quantum mechanics, $M=0$. To formulate the conjecture, we need to introduce a technical assumption on the superpotential $V$. We say that $V$ is elliptic if, for each multiindex $\alpha$, there exist positive constants $\epsilon_\alpha$ and $C_\alpha$ such that for all $z$, $$ |\partial^{\alpha}V(z)|\geq \epsilon_\alpha|z|^{d-|\alpha|}-C_\alpha , \ref{\ellref} $$ where $d$ denotes the algebraic degree of $V$. In other words, we exclude superpotentials $V$ which have flat directions. \conj\techconj{Let $V$ be an elliptic superpotential. Then: \item{(i)} {for all $t\geq 0$, $$ \tr(\exp\{-\, t\,\ol\square_V\})<0.\ref{\traceclass} $$} \item{(ii)} {Every eigenvector $\omega$ of $\lapl$ is smooth. Furthermore, there exist constants $a>0$ and $C$ such that $$ |\omega(z)|\leq C\exp\{-a|z|\}.\ref{\expdecay} $$}} \smallskip\noindent \subsec We now consider the square integrable cohomology $H^{\phantom{\dbr}*}_{\dbr_V,2}(E_M)$, defined as the cohomology of the complex \maincomplex\ with $\dolb pq$ replaced by the corresponding space $\bigwedge^{p,q}_{2}(E_M)$ of square integrable forms, and with the operator $\dbr_V$ defined as the closure of the corresponding operator with domain $\d0$. Let $\harm=\{\omega\in\hil:\,\,\lapl\,\omega=0\}$ denote the space of harmonic forms of $\lapl$, and let $\harm^*$ denote the corresponding even and odd subspaces. The following corollary to \techconj\ is a version of Hodge's theorem. \cor\hodgethm{Let $V$ be elliptic and assume that \techconj\ is true. Then: \item{(i)} {$\dim\,\harm <\infty$;} \item{(ii)} {we have the decomposition $$ \hil^*=\harm^*\oplus\dbr_V\bigl(\dbr^*_VG_V\hil^*\bigr) \oplus\dbr_V^*\bigl(\dbr_VG_V\hil^*\bigr),\num $$ where $G_V$ is a self-adjoint compact operator;} \item{(iii)} {there is a canonical isomorphism $$ H^{\phantom{\dbr}*}_{\dbr_V,2}(E_M)\simeq\harm^*.\num $$}} \smallskip\noindent {\it Proof.} The proof follows the standard arguments (see e.g. [GH]). Part (i) follows immediately from part (i) of \techconj . To prove part (ii), we set $$ G_V=\cases{0, &on $\harm^*$;\cr \lapl^{-1}, &on the orthogonal complement of $\harm^*$.\cr} $$ Part (iii) is a consequence of part (ii). $\square$ \subsec The next corollary to \techconj\ is a vanishing theorem for $\ol Q_V$ (a special case of the vanishing theorem was first conjectured in [JL]). It states that the kernel of $\ol Q_V$ consists of forms which are either purely bosonic or purely fermionic, depending on the superpotential. \smallskip \cor\vanthm{Assume that \techconj\ is true. Then there is an isomorphism, $$ \harm^*\simeq H^{\phantom{\dbr}*}_{\dbr_V}(E_M).\num $$ In particular, the kernel of $\ol Q_V$ consists of elements of definite parity.} \smallskip\noindent {\it Proof.} Since $\ol\square_V$ is elliptic, the harmonic forms of $\ol\square_V$ are smooth, and so we have a homomorphism $i:\harm^*\rightarrow H^{\phantom{\dbr}*}_{\dbr_V}(E_M)$. We assert that $i$ is an isomorphism. \noindent {\it (i) $i$ is a monomorphism.} Let $\omega$ be harmonic, and let $[\omega]$ be its image in $H^{\phantom{\dbr}*}_{\dbr_V}(E_M)$. Assume that $[\omega] = 0$, i.e. $\omega = \ol\partial_V\eta$. We claim that this implies $\omega = 0$. For the proof, we need the following result [H]. \smallskip \lemma\dolbeaultlemma{There exists an operator $J:\bigwedge^{p,q}(\bC^D) \rightarrow\bigwedge^{p,q-1}(\bC^D)$, $q\geq 1$, such that $J$ maps polynomially bounded forms into polynomially bounded forms and $$ J\ol\partial + \ol\partial J = I. \ref{\homotopyeqref} $$} We verify easily the identity $(\ol\partial+\Delta_V)(I+\Delta_V J) =(I+\Delta_V J)\ol\partial$, which implies that $$ \ol\partial (I+\Delta_V J)^{-1}=(I+\Delta_V J)^{-1} (\ol\partial+\Delta_V), \num $$ where the inverse is defined by a formal power series (note that this formal power series terminates, so no convergence questions arise). This, in turn, implies that $$ \ol\partial_V=\ol\partial_VJ(I+\Delta_V J)^{-1}\ol\partial_V.\num $$ Applying this identity to $\eta$ yields $$ \omega=\ol\partial_VJ(I+\Delta_V J)^{-1}\omega, $$ and so $\omega = \ol\partial_V\eta^{\prime}$, with $\eta^{\prime} =J(I+\Delta_V J)^{-1}\omega$. Since $\omega$ is bounded, and $J$ maps polynomially bounded forms into polynomially bounded forms, this implies that $\eta^{\prime}$ is polynomially bounded, and our claim follows. \noindent {\it (ii) $i$ is an epimorphism.} By \secondmainthm , every smooth cohomology class $[\omega]$ has a compactly supported representative $\omega_0$. In particular, $\omega_0$ is square integrable, and so by \hodgethm\ it is cohomologous to a harmonic form. $\square$ The above corollary can be also rephrased as the following index theorem. \smallskip \cor\indthm{Assume that \techconj\ is true. Then the index of the Dirac operator $\ol Q_V$ is given by $$ {\rm Ind}(\ol Q_V) = (-1)^n\#cr(V).\num $$} \subsec We end this section by describing the $N=2$ supersymmetry structure of the Landau-Ginzburg theory and the underlying K\"ahler geometry. We define a second coboundary operator, $$ \partial_V=i\int_{S^1}\, b^*(\sigma)\pi(\sigma)\, d\sigma - \int_{S^1}\,\ol b(\sigma)\partial_\sigma\ol\phi(\sigma)\, d\sigma + \int_{S^1}\,\ol b^*(\sigma)\ol{\nabla V(\phi(\sigma))}\, d\sigma.\num $$ and the corresponding Dirac type operator, $$ Q_V=\partial_V+\partial_V^*.\ref{\dirac} $$ We verify that the square of $Q_V$ is equal to the square of $\ol Q_V$, and that $\ol Q_V$ and $Q_V$ anticommute. Let us record these facts in the form of the following algebra. As operators on $\d0$, $$ \{\ol Q_V,\ol Q_V\}=2\,\ol\square_V,\qquad \{Q_V,Q_V\}=2\,\ol\square_V,\qquad \{Q_V,\ol Q_V\}=0.\ref{\firstalgebra} $$ where $\{.\, .\}$ denotes the anticommutator. In fact, \firstalgebra\ is a consequence of an algebra satisfied by the coboundary operators and their adjoints: $$ \eqalign{ &\{\dbr_V,\dbr_V\}=0,\quad \{\dbr_V,\partial_V\}=-iX,\quad \{\dbr_V,\dbr^*_V\}=\ol\square_V,\quad \{\dbr_V,\partial^*_V\}=0,\cr &\{\partial_V,\partial_V\}=0,\quad \{\partial_V,\partial^*_V\}=\ol\square_V,\quad \{\partial_V,\dbr_V^*\}=0,\quad \{\partial^*_V,\partial^*_V\}=0,\cr &\{\partial^*_V,\dbr^*_V\}=iX,\quad \{\dbr^*_V,\dbr^*_V\}=0.\cr}\ref{\relations} $$ These relations are reminiscent of the algebraic relations arising in ordinary K\"ahler geometry. We also note that the Hilbert space $\hil$ carries a representation of $sl(2)$. Namely, we define the operators $$ \eqalign{ &h=\int_{S^1}\, (\ol b(\sigma)\ol b^*(\sigma)-b^*(\sigma)b(\sigma))\, d\sigma=i\int_{S^1}\bigl(\ol\psi_1(\sigma)\ol\psi_2(\sigma)- \psi_1(\sigma)\psi_2(\sigma)\bigr)\, d\sigma,\cr &L=\int_{S^1}\,b^*(\sigma)\ol b(\sigma)^*\, d\sigma =\int_{S^1} \ol\psi_1(\sigma)\psi_1(\sigma)\, d\sigma,\cr &\Lambda =\int_{S^1}\,\ol b(\sigma)b(\sigma)\, d\sigma =\int_{S^1} \ol\psi_2(\sigma)\psi_2(\sigma)\, d\sigma,\cr}\ref{\sltwogen} $$ and verify that they satisfy the following set of relations: $$ [h,\, L]=-2L,\qquad [h,\,\Lambda]=2\Lambda,\qquad [\Lambda,\, L]=h.\ref{\sltworel} $$ \vfill\eject \centerline{\bf References} \baselineskip=12pt \frenchspacing \bigskip \item{[AO]} Arai, A., and Ogurisu, O.: Meromorphic $N=2$ Wess - Zumino supersymmetric quantum mechanics, {\it J. Math. Phys.}, {\bf 32}, 2427--2434 (1991). \item{[BI]} Borisov, and N. V. Ilinski, K. 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